Preface

This book is a self-contained collection of scholarly papers targeting the audience of practic‐ ing researchers, academics, postgraduate students and other scientists whose work relates to waveguide researches and waveguide technologies. This book intends to provide the read‐ ers with a comprehensive overview of the current state of the art in waveguide technologies. It is divided into three sections with a total of 19 chapters. These chapters are written by several authors: researchers, scientists and experts in specific research fields related to radio‐ frequency (RF)/microwave, photonic and optical engineering. The editor would like to take this opportunity to thank all the authors for their valuable contributions. In fact, each chap‐ ter provides an introduction on the specific waveguide technologies as well as a detailed explanation of the methodology on how to solve the raised issues, which include both aca‐ demic and industry aspects.

The first section contains five chapters related to RF and microwave waveguide technolo‐ gies. The second section consists of ten chapters that focus on photonic and optical wave‐ guide technologies. Recently, the rapid development of waveguides has had a significant impact on the current industrial, electrical/electronic, clinical, and communication fields. The products of waveguide contributions are high-speed circuits, real-time and wireless control systems, high-quality information and communication (IC) components, low-attenu‐ ation transmission lines, nanoantennas, micro/nanoelectronic devices, and so on. In addi‐ tion, waveguides are also used as biological and chemical sensory devices.

The third section includes four chapters on waveguide analytical solutions. Analytical anal‐ ysis has played a very important role in the academic microwave engineering and associat‐ ed industries. Engineers in the microwave field will be able to understand the operating background of waveguide devices through analytical analysis because this is the fundamen‐ tal knowledge of waveguide designs before entering into further applications. Based on the analytical models, the time spent on waveguide (antennas, communication components, sensors, etc.) design and measurement setup can be shortened.

> **Kok Yeow You, Editor** Communication Engineering Department Faculty of Electrical Engineering Universiti Teknologi Malaysia Malaysia

**Section 1**

**RF and Microwave Waveguides**

**RF and Microwave Waveguides**

**Chapter 1**

**Provisional chapter**

**Broadband Slotted Waveguide Array Antenna**

**Broadband Slotted Waveguide Array Antenna**

DOI: 10.5772/intechopen.78308

This chapter describes the design and development of broadband slotted waveguide array (SWA) antenna. Conventional SWA antenna offers a few percentages of bandwidth, which can be enhanced using proposed novel differential feeding technique which electrically divides large resonating SWA into wideband subarrays by creating virtual shorts. This chapter discusses concepts to achieve broadband nature of SWA antennas, design, development, and characterization of edge fed slotted waveguide array antenna, coupling slot fed SWA antenna, and high efficiency broadband slotted waveguide array. The developed SWA antennas are characterized and their measured results are presented. The developed prototype of proposed SWA antenna demonstrates measured return loss better than −17 dB over 7.6% bandwidth and achieves 90.2% antenna efficiency. This chapter also briefs about planar broadband SWA antenna and

**Keywords:** slotted waveguide array, broadband antenna, array antenna, high efficiency

Slotted waveguide array (SWA) antenna technology has been utilized by many spaceborne missions such as Radarsat-1, SIR-X, ERS-1/2, and Sentinel-1, because SWA technology has several advantages like high efficiency, good mechanical strength, high power handling capacity, and manufacturing ease. However, the main drawback of this technology is narrow impedance bandwidth, which limits its applications to support high resolution SAR systems for civil and military applications. Moreover, the traveling wave type SWA provides wide

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Yogesh Tyagi and Pratik Mevada

Yogesh Tyagi and Pratik Mevada

http://dx.doi.org/10.5772/intechopen.78308

**Abstract**

**1. Introduction**

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

its prototype development and characterization.

antenna, virtual short, coupling slot

bandwidth, but its efficiency is very low.

#### **Broadband Slotted Waveguide Array Antenna Broadband Slotted Waveguide Array Antenna**

DOI: 10.5772/intechopen.78308

Yogesh Tyagi and Pratik Mevada Yogesh Tyagi and Pratik Mevada

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.78308

**Abstract**

This chapter describes the design and development of broadband slotted waveguide array (SWA) antenna. Conventional SWA antenna offers a few percentages of bandwidth, which can be enhanced using proposed novel differential feeding technique which electrically divides large resonating SWA into wideband subarrays by creating virtual shorts. This chapter discusses concepts to achieve broadband nature of SWA antennas, design, development, and characterization of edge fed slotted waveguide array antenna, coupling slot fed SWA antenna, and high efficiency broadband slotted waveguide array. The developed SWA antennas are characterized and their measured results are presented. The developed prototype of proposed SWA antenna demonstrates measured return loss better than −17 dB over 7.6% bandwidth and achieves 90.2% antenna efficiency. This chapter also briefs about planar broadband SWA antenna and its prototype development and characterization.

**Keywords:** slotted waveguide array, broadband antenna, array antenna, high efficiency antenna, virtual short, coupling slot

#### **1. Introduction**

Slotted waveguide array (SWA) antenna technology has been utilized by many spaceborne missions such as Radarsat-1, SIR-X, ERS-1/2, and Sentinel-1, because SWA technology has several advantages like high efficiency, good mechanical strength, high power handling capacity, and manufacturing ease. However, the main drawback of this technology is narrow impedance bandwidth, which limits its applications to support high resolution SAR systems for civil and military applications. Moreover, the traveling wave type SWA provides wide bandwidth, but its efficiency is very low.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Various bandwidth improvement techniques have been explored and reported in the literature, which includes reducing waveguide wall thickness, reducing waveguide cross section, widening slot width, and modifying slot shapes, e.g., dumbbell shape and elliptical shape [1–4]. When SWA is targeted for space applications, the reduction in wall thickness may not be suitable as it has to survive severe vibration and thermal loads. When SWA is targeted for high power and low attenuation systems, the reduction in waveguide cross section shows inferior performance. Moreover, slot widening and slot shaping affect the cross-polarization performance. SWA with modified slot shapes also needs high manufacturing accuracy. Although these techniques provide bandwidth improvement, other antenna parameters are compromised due to the constraints of respective techniques.

Sub-arraying technique is widely used and considered as most effective to broaden the bandwidth of conventional SWA. In this technique, SWA is divided into subarrays [3], so that better control over variation of normalize impedance (*z*) or admittance (*y*) with frequency is achieved. Generally, coupling slots are used to feed these subarrays. Being the resonating structure, coupling slot restricts the bandwidth performance. Therefore, a novel differential feedingave technique has been presented to remove the constraints introduced by coupling slots. WR-90 (22.86 mm × 10.16 mm in cross section) waveguide has been selected and the design of broadband SWA has been carried out at 9.65 GHz center frequency. Design and demonstration of SWA with conventional feeding techniques, i.e., edge feeding and coupling slot feeding have also been discussed and compared with proposed broadband SWA design.

In this chapter, Section 2 discusses the conventional bandwidth enhancement approach and various feeding mechanisms of subarrays are followed in Section 3. Mathematical proof of the presence of virtual short created due to the proposed differential feeding technique has been discussed in Section 4. Design and simulation of 10 elements linear SWAs is outlined in Section 5. Design of linear array is extended to planar array and it is presented in Section 6. The measured results of developed prototypes are provided in Section 7 with brief discussion.

### **2. Subarray: conventional technique for bandwidth enhancement**

Many techniques have been discussed in the literature to enhance the bandwidth of traditional slotted waveguide array antenna. As it has been discussed in the previous section, the use of thin wall waveguides, reduced cross sectional waveguide, wide slots, and modified slot shapes are the reported techniques to improve the bandwidth. However, these techniques are capable to improve few percent of bandwidth. Hence, they are not suitable to achieve expected wideband performance.

branch of WR-90 waveguide. From **Figure 1(a)**, it can be observed that reactive part is very high and crosses zero at 9.65 GHz. The frequency at which zero crossing occurs is the resonance frequency of slot. When the number of slots reaches up to 6, the variation of reactive part of input admittance tends to zero over the frequency band of interest, which confirms the realizability of wideband SWA. When the number of slots is further increased, the reactive part of admittance starts to destabilize near zero and eventually provides strong resonant behavior. This phenomenon is also validated by plotting the impedance variation for different number of slots on smith chart, as shown in **Figure 1(b)**. It can be observed that varying the number of slots results in variation in impedance loop. By adjusting this impedance loop near normalized impedance

**Figure 1.** (a) Variation of reactive component of input admittance for different number of longitudinal slots and (b)

Broadband Slotted Waveguide Array Antenna http://dx.doi.org/10.5772/intechopen.78308 5

variations of different number of longitudinal slots using smith chart.

of 1 by proper selection of number of slots, the antenna bandwidth can be improved.

Fundamentally, the bandwidth enhancement can be achieved by reducing the reactive effects of the input admittance over the frequency band of interest. By proper selection of number of slot elements in the edge fed linear SWA antenna for the required bandwidth, nearly zero reactive part of input admittance can be obtained for the desired frequency band, which subsequently results in wide bandwidth performance. **Figure 1(a)** shows the case study of variation of reactive part of input admittance with frequency for different number of slots in a single

Various bandwidth improvement techniques have been explored and reported in the literature, which includes reducing waveguide wall thickness, reducing waveguide cross section, widening slot width, and modifying slot shapes, e.g., dumbbell shape and elliptical shape [1–4]. When SWA is targeted for space applications, the reduction in wall thickness may not be suitable as it has to survive severe vibration and thermal loads. When SWA is targeted for high power and low attenuation systems, the reduction in waveguide cross section shows inferior performance. Moreover, slot widening and slot shaping affect the cross-polarization performance. SWA with modified slot shapes also needs high manufacturing accuracy. Although these techniques provide bandwidth improvement, other antenna parameters are

Sub-arraying technique is widely used and considered as most effective to broaden the bandwidth of conventional SWA. In this technique, SWA is divided into subarrays [3], so that better control over variation of normalize impedance (*z*) or admittance (*y*) with frequency is achieved. Generally, coupling slots are used to feed these subarrays. Being the resonating structure, coupling slot restricts the bandwidth performance. Therefore, a novel differential feedingave technique has been presented to remove the constraints introduced by coupling slots. WR-90 (22.86 mm × 10.16 mm in cross section) waveguide has been selected and the design of broadband SWA has been carried out at 9.65 GHz center frequency. Design and demonstration of SWA with conventional feeding techniques, i.e., edge feeding and coupling slot feeding have also been discussed and compared with proposed broadband SWA design.

In this chapter, Section 2 discusses the conventional bandwidth enhancement approach and various feeding mechanisms of subarrays are followed in Section 3. Mathematical proof of the presence of virtual short created due to the proposed differential feeding technique has been discussed in Section 4. Design and simulation of 10 elements linear SWAs is outlined in Section 5. Design of linear array is extended to planar array and it is presented in Section 6. The measured results of developed prototypes are provided in Section 7 with brief discussion.

Many techniques have been discussed in the literature to enhance the bandwidth of traditional slotted waveguide array antenna. As it has been discussed in the previous section, the use of thin wall waveguides, reduced cross sectional waveguide, wide slots, and modified slot shapes are the reported techniques to improve the bandwidth. However, these techniques are capable to improve few percent of bandwidth. Hence, they are not suitable to achieve

Fundamentally, the bandwidth enhancement can be achieved by reducing the reactive effects of the input admittance over the frequency band of interest. By proper selection of number of slot elements in the edge fed linear SWA antenna for the required bandwidth, nearly zero reactive part of input admittance can be obtained for the desired frequency band, which subsequently results in wide bandwidth performance. **Figure 1(a)** shows the case study of variation of reactive part of input admittance with frequency for different number of slots in a single

**2. Subarray: conventional technique for bandwidth enhancement**

expected wideband performance.

compromised due to the constraints of respective techniques.

4 Emerging Waveguide Technology

**Figure 1.** (a) Variation of reactive component of input admittance for different number of longitudinal slots and (b) variations of different number of longitudinal slots using smith chart.

branch of WR-90 waveguide. From **Figure 1(a)**, it can be observed that reactive part is very high and crosses zero at 9.65 GHz. The frequency at which zero crossing occurs is the resonance frequency of slot. When the number of slots reaches up to 6, the variation of reactive part of input admittance tends to zero over the frequency band of interest, which confirms the realizability of wideband SWA. When the number of slots is further increased, the reactive part of admittance starts to destabilize near zero and eventually provides strong resonant behavior. This phenomenon is also validated by plotting the impedance variation for different number of slots on smith chart, as shown in **Figure 1(b)**. It can be observed that varying the number of slots results in variation in impedance loop. By adjusting this impedance loop near normalized impedance of 1 by proper selection of number of slots, the antenna bandwidth can be improved.

The optimum number of radiating slots is selected by properly adjusting the size of impedance loop in smith chart. However, the bandwidth enhancement is not achieved by only size adjustment. Once the size of the impedance loop is adjusted which indicates the optimum number of radiating slots are selected, the impedance loop is moved to normalized impedance of 1 by using the impedance matching section. The impedance matching section maintains the variation of reflection coefficient below desired value, resulting in wide bandwidth performance. The discussed concept has been verified using circuit simulation in Advanced Design Studio (ADS). In this simulation, slotted waveguide array antenna having three longitudinal slots has been modeled using three parallel *RLC* network and inbuilt waveguide transmission line component. The circuit equivalent of three elements resonant SWAs is shown in **Figure 2(a)**. Here, slots are modeled as parallel *R*, *L* and *C* and they are connected by transmission line section having *Z0* and *β* equivalent to WR-90 waveguide at 9.6 GHz. The values of *L* and *C* have been selected such that LC network resonates at center frequency of 9.6 GHz. The value of *R* has been selected to provide normalized admittance of 1 at the input of the array. The circuit schematic in ADS is shown in **Figure 2(c)**. Here, separate extra impedance matching section has been added to adjust the location of impedance loop in smith chart. The return loss plot and impedance on smith chart with/without impedance matching section are presented in **Figure 3**. It can be observed that, without matching section at the input, the circuit shows highly resonating behavior. After adding matching section, the impedance loop has been shifter around the normalized impedance of 1, which results in wider impedance bandwidth.

In the case of slotted waveguide array antenna, the concept of impedance overloading of slots can be used as impedance matching technique to shift the impedance loop in smith chart, in place of adding extra impedance matching section. To validate the concept, five elements linear slotted waveguide array antenna has been designed as shown in **Figure 4(a)**. From **Figure 4(b)**, it can be observed that the selection of five radiating slots for the array provides optimum bandwidth.

Broadband Slotted Waveguide Array Antenna http://dx.doi.org/10.5772/intechopen.78308 7

**Figure 4.** (a) Five elements linear slotted waveguide array antenna, (b) return loss performance, and (c) smith chart.

**Figure 3.** Simulation results of circuit layout.

**Figure 2.** (a) Equivalent circuit representation of three elements resonant SWAs, (b) admittance modeling as parallel RLC, and (c) circuit layout of SWA in ADS.

**Figure 3.** Simulation results of circuit layout.

The optimum number of radiating slots is selected by properly adjusting the size of impedance loop in smith chart. However, the bandwidth enhancement is not achieved by only size adjustment. Once the size of the impedance loop is adjusted which indicates the optimum number of radiating slots are selected, the impedance loop is moved to normalized impedance of 1 by using the impedance matching section. The impedance matching section maintains the variation of reflection coefficient below desired value, resulting in wide bandwidth performance. The discussed concept has been verified using circuit simulation in Advanced Design Studio (ADS). In this simulation, slotted waveguide array antenna having three longitudinal slots has been modeled using three parallel *RLC* network and inbuilt waveguide transmission line component. The circuit equivalent of three elements resonant SWAs is shown in **Figure 2(a)**. Here, slots are modeled as parallel *R*, *L* and *C* and they are connected by transmission line

have been selected such that LC network resonates at center frequency of 9.6 GHz. The value of *R* has been selected to provide normalized admittance of 1 at the input of the array. The circuit schematic in ADS is shown in **Figure 2(c)**. Here, separate extra impedance matching section has been added to adjust the location of impedance loop in smith chart. The return loss plot and impedance on smith chart with/without impedance matching section are presented in **Figure 3**. It can be observed that, without matching section at the input, the circuit shows highly resonating behavior. After adding matching section, the impedance loop has been shifter around the normalized impedance of 1, which results in wider impedance bandwidth.

**Figure 2.** (a) Equivalent circuit representation of three elements resonant SWAs, (b) admittance modeling as parallel

and *β* equivalent to WR-90 waveguide at 9.6 GHz. The values of *L* and *C*

section having *Z0*

6 Emerging Waveguide Technology

RLC, and (c) circuit layout of SWA in ADS.

In the case of slotted waveguide array antenna, the concept of impedance overloading of slots can be used as impedance matching technique to shift the impedance loop in smith chart, in place of adding extra impedance matching section. To validate the concept, five elements linear slotted waveguide array antenna has been designed as shown in **Figure 4(a)**. From **Figure 4(b)**, it can be observed that the selection of five radiating slots for the array provides optimum bandwidth.

**Figure 4.** (a) Five elements linear slotted waveguide array antenna, (b) return loss performance, and (c) smith chart.


slots) which are added in the path of high power signal. These factors are the major causes to limit the use of coupling slot feeding technique for large wideband SWAs. The design and simulation of 10 elements linear SWAs excited with coupling slot has been carried out and

Broadband Slotted Waveguide Array Antenna http://dx.doi.org/10.5772/intechopen.78308 9

**Figure 6.** (a) Geometry of 10 elements slotted waveguide array antenna excited using differential feeding technique and

**Figure 5.** Geometry of 10 elements slotted waveguide array antenna fed coupling slots.

discussed in subsequent sections.

(b) presence of virtual short.

**Table 1.** Slot parameters.

Mutual coupling between the elements may be neglected due to considerably small array size. Therefore, Stevenson's formula can be used to compute slot lengths and offsets. Stevenson's formula provides the slot parameters for designed frequency only. Hence, the design of SWA with the computed slot parameters shows highly resonating nature, as shown in **Figure 4(b)**. It can be observed that the achieved return loss bandwidth is 2%. As it has been discussed earlier, the bandwidth of slotted waveguide array antenna can be improved using impedance overloading. To implement impedance over loading, slot length and offsets are tuned so that resultant admittance value slightly deviates from normalized value 1. For small SWA, this process can be carried out manually. **Table 1** lists slot admittance and their physical parameters for impedance overloading and non-overloading cases. **Figure 4(b)** shows the return loss performance of slotted waveguide array antenna incorporated with impedance overloading concept. It can be observed that the achieved return loss bandwidth is 15%, which is noteworthy improvement. Moreover, more detail about impedance overloading is explained in [4].

Furthermore, the linear array of five slots does not provide required gain, beamwidth, and desired gain ripples, if the pattern is shaped. Therefore, most effective approach to achieve overall performance of slotted waveguide array antenna is subarray approach. This technique exploits the advantages provided by dividing the large narrowband array into small wideband array ("Subarray"). The concept of subarray and its related theoretical studies have been published in the literature. But, the physical process of bandwidth broadening using subarray is not explained in detail.

### **3. Feeding mechanisms for subarray**

The feeding mechanism of subarrays of the slotted waveguide array antenna plays a crucial role in limiting its bandwidth performance. Conventionally, the power to each subarray branch is delivered using feeding waveguide connected with subarrays by means of coupling slots. **Figure 5** shows the geometry of the 10-element slotted waveguide array antenna excited using coupling slots. Being the resonating element, coupling slots have their own resonant behavior and hence, the variation of reactive part of input admittance with frequency. Overall antenna resulting input admittance behavior is governed by the combined effect of radiating slots and coupling slots. Therefore, inherent wide bandwidth behavior of radiation slots, achieved using sub-array technique is degraded. Furthermore, coupling slots are used in feeding section of SWA which has to handle larger power than radiation branch of waveguide. Multipaction margin is also affected as narrow gap structures (i.e., coupling slots) which are added in the path of high power signal. These factors are the major causes to limit the use of coupling slot feeding technique for large wideband SWAs. The design and simulation of 10 elements linear SWAs excited with coupling slot has been carried out and discussed in subsequent sections.

**Figure 5.** Geometry of 10 elements slotted waveguide array antenna fed coupling slots.

Mutual coupling between the elements may be neglected due to considerably small array size. Therefore, Stevenson's formula can be used to compute slot lengths and offsets. Stevenson's formula provides the slot parameters for designed frequency only. Hence, the design of SWA with the computed slot parameters shows highly resonating nature, as shown in **Figure 4(b)**. It can be observed that the achieved return loss bandwidth is 2%. As it has been discussed earlier, the bandwidth of slotted waveguide array antenna can be improved using impedance overloading. To implement impedance over loading, slot length and offsets are tuned so that resultant admittance value slightly deviates from normalized value 1. For small SWA, this process can be carried out manually. **Table 1** lists slot admittance and their physical parameters for impedance overloading and non-overloading cases. **Figure 4(b)** shows the return loss performance of slotted waveguide array antenna incorporated with impedance overloading concept. It can be observed that the achieved return loss bandwidth is 15%, which is noteworthy improvement.

**Without admittance overloading With admittance overloading**

Furthermore, the linear array of five slots does not provide required gain, beamwidth, and desired gain ripples, if the pattern is shaped. Therefore, most effective approach to achieve overall performance of slotted waveguide array antenna is subarray approach. This technique exploits the advantages provided by dividing the large narrowband array into small wideband array ("Subarray"). The concept of subarray and its related theoretical studies have been published in the literature. But, the physical process of bandwidth broadening using subarray

The feeding mechanism of subarrays of the slotted waveguide array antenna plays a crucial role in limiting its bandwidth performance. Conventionally, the power to each subarray branch is delivered using feeding waveguide connected with subarrays by means of coupling slots. **Figure 5** shows the geometry of the 10-element slotted waveguide array antenna excited using coupling slots. Being the resonating element, coupling slots have their own resonant behavior and hence, the variation of reactive part of input admittance with frequency. Overall antenna resulting input admittance behavior is governed by the combined effect of radiating slots and coupling slots. Therefore, inherent wide bandwidth behavior of radiation slots, achieved using sub-array technique is degraded. Furthermore, coupling slots are used in feeding section of SWA which has to handle larger power than radiation branch of waveguide. Multipaction margin is also affected as narrow gap structures (i.e., coupling

Moreover, more detail about impedance overloading is explained in [4].

Slot admittance 0.1945 − j\*0.016 0.2291 − j\*0.016

Slot length 15.37 15.40 Slot offset 3.28 3.60

is not explained in detail.

**Table 1.** Slot parameters.

8 Emerging Waveguide Technology

**3. Feeding mechanisms for subarray**

**Figure 6.** (a) Geometry of 10 elements slotted waveguide array antenna excited using differential feeding technique and (b) presence of virtual short.

Bandwidth and power handling performance of SWA can be preserved by eliminating resonating components from waveguide feed network design. This chapter describes the differential feeding technique which does not add any resonating structure in waveguide feeder network and maintains the effective bandwidth of subarray. **Figure 6(a)** shows the geometry of the proposed technique. Here, the edges of SWA are fed with 180° out of phase signal, which creates virtual short in the center of SWA. Presence of virtual short bifurcates the linear array and hence, effectively preserves the bandwidth achieved by subarraying. The presence of virtual short can be substantiated by observing current distribution on differentially excited waveguide, as shown in **Figure 6(b)**. The mathematical proof to validate the theory of virtual short is also detailed in Section 4. Moreover, two edges can be fed using E-plane T-junction which has inherent characteristics to provide 180° out of phase signals, irrespective of frequency of operation [5]. Therefore, the compensation of 180° phase over broadband of frequency can be achieved. Furthermore, the feeding network does not contain any resonating structures like coupling slots, to limit power handling capacity. The design and simulation results of 10 elements linear slotted waveguide array antenna incorporating the differential feeding technique is discussed in Section 5.

For TE10 mode,

phase). For *Ey*

at *z* = *zl*

Hence,

Now, for *Hx*

at *z* = *zl*

/2.

*Ex*

*Ez*

*Hy*

*Ey*

*Hx*

*Hz*

(*Ey*

(−ve *z* directed wave)

(*Ey*

(*Eytotal*)*<sup>z</sup>*<sup>=</sup>

field component,

(*Hx*

(*Eytotal*)*<sup>z</sup>*<sup>=</sup>

(*A*10)*<sup>z</sup>* <sup>=</sup> <sup>0</sup> <sup>=</sup> (*A*10)*<sup>z</sup>* <sup>=</sup> *zl* <sup>=</sup> <sup>1</sup>

+ ) *z*=0 , (*Hx* + )*z*=*zl* <sup>+</sup> = 0

Broadband Slotted Waveguide Array Antenna http://dx.doi.org/10.5772/intechopen.78308 11

<sup>+</sup> = 0

<sup>+</sup> = 0

Here, excitation coefficient *A10* for both inputs are *(A10)z* = 0 = 1 and *(A10)z* = *zl* = −1 (180° out of

= −\_\_*<sup>π</sup> <sup>a</sup>* sin( \_\_\_ *x <sup>a</sup>* ) *e* <sup>−</sup>*jβ<sup>z</sup> z*

= \_\_*<sup>π</sup> <sup>a</sup>* sin( \_\_\_ *x <sup>a</sup>* ) *e* <sup>−</sup>*jβz*(*zl* −*z*)

*z* \_\_*l* 2 = (*Ey* + )*<sup>z</sup>*=<sup>0</sup> + (*Ey* + )*<sup>z</sup>*=*zl*

> *z* \_\_*l* 2 = 0

<sup>=</sup> *<sup>β</sup><sup>z</sup> <sup>π</sup>* \_\_\_\_\_ *<sup>a</sup>* sin(

\_\_\_ *x <sup>a</sup>* ) *e* <sup>−</sup>*jβz*( *z* \_\_*l* 2)

*β*\_\_\_\_*<sup>z</sup>* \_\_ *π <sup>a</sup>* sin( \_\_\_ *x <sup>a</sup>* ) *e* <sup>−</sup>*jβ<sup>z</sup> z*

<sup>+</sup> = −\_\_\_ *A*<sup>10</sup> *<sup>ε</sup>* \_\_ *π <sup>a</sup>* sin( \_\_\_ *x <sup>a</sup>* ) *e* <sup>−</sup>*jβ<sup>z</sup> z*

<sup>+</sup> = *A*<sup>10</sup>

<sup>+</sup> = −*j* \_\_\_\_ *A*<sup>10</sup> ( \_\_ *π a* ) 2 cos( \_\_\_ *x <sup>a</sup>* ) *e* <sup>−</sup>*jβ<sup>z</sup> z*

field component, at *z* = 0 (+ve *z* directed wave)

+ )*<sup>z</sup>*=<sup>0</sup>

+ )*<sup>z</sup>*=*zl*

#### **4. Theory of virtual short**

Circuit theory has been applied to establish the virtual short theory in SWA. A branch in a circuit can be considered as short when the potential difference between two nodes of a branch is zero and the current flow is high in that branch. The presence of virtual short can be proved using similar concept in waveguide. In waveguide, *E*-field and *H*-field are used to derive the short circuit condition. The cross section of waveguide along length direction is shown in **Figure 7**. Here, operating frequency of 9.65 GHz has been selected. As WR-90 waveguide having cross sectional dimension of (*Wa* = 22.86 mm) × (*Wb* = 10.16 mm) is used, only TE10 mode is supported at these frequency. As it can be seen in **Figure 7**, *Z* = 0 end is excited with 1 < 0° and *Z = Zl* is excited with 1 < 180°. Therefore, *Ey* component of the signals is canceled and *Hx* components of it are added at the center of waveguide, which satisfy the short circuit condition. Here, there is no conductor at the center of waveguide. However, the short circuit condition is fulfilled, which substantiate the presence of virtual short at the center of waveguide.

**Figure 7.** Cross section of WR-90 waveguide along length direction.

For TE10 mode,

Bandwidth and power handling performance of SWA can be preserved by eliminating resonating components from waveguide feed network design. This chapter describes the differential feeding technique which does not add any resonating structure in waveguide feeder network and maintains the effective bandwidth of subarray. **Figure 6(a)** shows the geometry of the proposed technique. Here, the edges of SWA are fed with 180° out of phase signal, which creates virtual short in the center of SWA. Presence of virtual short bifurcates the linear array and hence, effectively preserves the bandwidth achieved by subarraying. The presence of virtual short can be substantiated by observing current distribution on differentially excited waveguide, as shown in **Figure 6(b)**. The mathematical proof to validate the theory of virtual short is also detailed in Section 4. Moreover, two edges can be fed using E-plane T-junction which has inherent characteristics to provide 180° out of phase signals, irrespective of frequency of operation [5]. Therefore, the compensation of 180° phase over broadband of frequency can be achieved. Furthermore, the feeding network does not contain any resonating structures like coupling slots, to limit power handling capacity. The design and simulation results of 10 elements linear slotted waveguide array antenna incorporating

Circuit theory has been applied to establish the virtual short theory in SWA. A branch in a circuit can be considered as short when the potential difference between two nodes of a branch is zero and the current flow is high in that branch. The presence of virtual short can be proved using similar concept in waveguide. In waveguide, *E*-field and *H*-field are used to derive the short circuit condition. The cross section of waveguide along length direction is shown in **Figure 7**. Here, operating frequency of 9.65 GHz has been selected. As WR-90 waveguide having cross sectional dimension of (*Wa* = 22.86 mm) × (*Wb* = 10.16 mm) is used, only TE10 mode is supported at these frequency. As it can be seen in **Figure 7**, *Z* = 0 end is excited with 1 < 0° and

ponents of it are added at the center of waveguide, which satisfy the short circuit condition. Here, there is no conductor at the center of waveguide. However, the short circuit condition is

fulfilled, which substantiate the presence of virtual short at the center of waveguide.

component of the signals is canceled and *Hx*

com-

the differential feeding technique is discussed in Section 5.

is excited with 1 < 180°. Therefore, *Ey*

**Figure 7.** Cross section of WR-90 waveguide along length direction.

**4. Theory of virtual short**

10 Emerging Waveguide Technology

*Z = Zl*

$$\begin{aligned} E\_x^\* &= 0 \\\\ E\_y^\* &= -\frac{A\_{10}}{\varepsilon} \frac{\pi}{d} \sin\left(\frac{\pi x}{d}\right) e^{-\beta \varepsilon z} \\\\ E\_z^\* &= 0 \\\\ H\_x^\* &= A\_{10} \frac{\beta\_z}{\alpha \rho \mu \epsilon} \frac{\pi}{d} \sin\left(\frac{\pi x}{d}\right) e^{-\beta \varepsilon z} \\\\ H\_y^\* &= 0 \\\\ H\_z^\* &= -j \frac{A\_{10}}{\rho \mu \mu \epsilon} \left(\frac{\pi}{d}\right)^2 \cos\left(\frac{\pi x}{d}\right) e^{-\beta \varepsilon z} \end{aligned}$$

Here, excitation coefficient *A10* for both inputs are *(A10)z* = 0 = 1 and *(A10)z* = *zl* = −1 (180° out of phase). For *Ey* field component, at *z* = 0 (+ve *z* directed wave)

*z*

$$\left(E\_y^{\ast}\right)\_{z\ast 0} = -\frac{\pi}{\varepsilon d} \sin\left(\frac{\pi \chi}{d}\right) e^{-|\beta|z}$$

at *z* = *zl* (−ve *z* directed wave)

$$\left(E\_y^{\star}\right)\_{z=z\_{\parallel}} = \frac{\pi}{\epsilon a} \sin\left(\frac{\pi \chi}{a}\right) e^{-\beta(z\_{\parallel}z\_{\parallel})z}$$

Hence,

$$\begin{aligned} \left(E\_{\text{total}}\right)\_{z \simeq\_{\frac{z\_i}{2}}^{z\_i}} &= \left(E\_y^+\right)\_{z \simeq 0} + \left(E\_y^+\right)\_{z \simeq z\_i} \\\\ \left(E\_{\text{total}}\right)\_{z \simeq \frac{z\_i}{2}} &= 0 \end{aligned}$$

Now, for *Hx* field component,

$$\left(A\_{\rm 10}\right)\_z = 0 \,\,=\left(A\_{\rm 10}\right)\_z = z\_{\rm 1} = 1$$

$$\left(H\_{\chi}^{\ast}\right)\_{z=0'}\left(H\_{\chi}^{\ast}\right)\_{z=\overline{z}\_{\gamma}} = \frac{\beta\_{\circ}\pi}{\overline{\alpha\rho\epsilon\pi t}}\sin\left(\frac{\pi\chi}{a}\right)e^{-\beta\left(\frac{\overline{z}}{2}\right)t}$$

at *z* = *zl* /2. Hence,

$$\left(H\_{\text{xtotal}}\right)\_{z \simeq \frac{z\_i}{2}} = \text{ } \left(H\_{\text{x}}^{+}\right)\_{z \simeq 0}$$

Now, for *Hz* field component,

$$\left(A\_{\rm 10}\right)\_z = 0 = 1 \text{ and } \left(A\_{\rm 10}\right)\_z = z\_l = -1$$

Hence,

$$\left(H\_{ztotal}\right)\_{z=\frac{z}{2}} = \mathbf{0}$$

Therefore,

$$\begin{aligned} \left(E\_{\text{total}}\right)\_{z \simeq \frac{z}{2}} &= \text{0} \\\\ \left\{H\_{\text{total}}\right\}\_{z \simeq \frac{z}{2}} &= \text{2} \left\{H\_{\text{x}}^{\ast}\right\}\_{z \simeq 0} \\\\ \left\{H\_{\text{z} \text{total}}\right\}\_{z \simeq \frac{z}{2}} &= \text{0} \end{aligned}$$

This fulfills the condition of short circuit node.

#### **5. Design and simulation of linear slotted waveguide array antenna**

Elliot's design technique has been applied to design SWA [6–8]. Elliot's design technique synthesizes slot lengths and offsets and incorporate mutual coupling between slots. This technique uses admittance variation of isolated shunt slot as a function of slot lengths and resonant length and the offset from center. Ansys HFSS 2014 has been used to compute the admittance variation and the real and imaginary part of admittance is plotted in **Figure 8(b)** and **(c)**. These plots are fed as input to Elliot's design technique and slot lengths and offsets are synthesized such that normalized input admittance (y) obtains value '1' at center frequency 9.65 GHz. Ten-element conventional edge fed SWA using synthesized slot lengths and offsets has been designed and shown in **Figure 9**.

are combined in parallel. Therefore, slot lengths and offsets are to be adjusted so that normalized input impedance of each subarray becomes value '2'. Consequently, parallel combina-

**Figure 8.** (a) Dimensional details of isolated shunt slot, variations of (b) real part of slot admittance, and (c) imaginary

Broadband Slotted Waveguide Array Antenna http://dx.doi.org/10.5772/intechopen.78308 13

Subarrays can be excited by proper waveguide plumbing incorporated with coupling slots and waveguide tees. Such feeding network effectively governs the impedance bandwidth of SWA. In Section 3, the conventional coupling slot-based excitation technique and proposed differential feeding technique have been broadly detailed. To compare these techniques, 10-elements linear SWAs have been designed and optimized to achieve maximum bandwidth

tion of five element array results normalized admittance of 1 at input.

**Figure 9.** Geometry of conventional SWA with 15 mm slot length and 3.5 mm slot offset.

part of slot admittance functions of slot length and slot offset.

3D simulation of conventional edge fed SWA has been carried out and the results are presented in Section 7. Simulation shows 2.8% 17-dB impedance bandwidth which does not cater to current need of bandwidth for communication and remote sensing applications. As it has been discussed in previous sections, 10 elements array can be split into two five element arrays, which will considerably increase the return loss bandwidth. Here, five element arrays

Hence,

12 Emerging Waveguide Technology

Now, for *Hz*

Hence,

Therefore,

(*Hxtotal*)

field component,

(*Hztotal*)

(*Eytotal*)*<sup>z</sup>*<sup>=</sup>

(*Hxtotal*)

(*Hztotal*)

This fulfills the condition of short circuit node.

offsets has been designed and shown in **Figure 9**.

(*A*10)*<sup>z</sup>* <sup>=</sup> <sup>0</sup> <sup>=</sup> <sup>1</sup> and (*A*10)*<sup>z</sup>* <sup>=</sup> *zl* <sup>=</sup> −1

*z*= *z* \_\_*l* 2

= 2 (*Hx* + ) *z*=0

*z*= *z* \_\_*l* 2 = 0

> *z* \_\_*l* 2 = 0

= 2 (*Hx* + ) *z*=0

*z*= *z* \_\_*l* 2 = 0

Elliot's design technique has been applied to design SWA [6–8]. Elliot's design technique synthesizes slot lengths and offsets and incorporate mutual coupling between slots. This technique uses admittance variation of isolated shunt slot as a function of slot lengths and resonant length and the offset from center. Ansys HFSS 2014 has been used to compute the admittance variation and the real and imaginary part of admittance is plotted in **Figure 8(b)** and **(c)**. These plots are fed as input to Elliot's design technique and slot lengths and offsets are synthesized such that normalized input admittance (y) obtains value '1' at center frequency 9.65 GHz. Ten-element conventional edge fed SWA using synthesized slot lengths and

3D simulation of conventional edge fed SWA has been carried out and the results are presented in Section 7. Simulation shows 2.8% 17-dB impedance bandwidth which does not cater to current need of bandwidth for communication and remote sensing applications. As it has been discussed in previous sections, 10 elements array can be split into two five element arrays, which will considerably increase the return loss bandwidth. Here, five element arrays

*z*= *z* \_\_*l* 2

**5. Design and simulation of linear slotted waveguide array antenna**

**Figure 8.** (a) Dimensional details of isolated shunt slot, variations of (b) real part of slot admittance, and (c) imaginary part of slot admittance functions of slot length and slot offset.

**Figure 9.** Geometry of conventional SWA with 15 mm slot length and 3.5 mm slot offset.

are combined in parallel. Therefore, slot lengths and offsets are to be adjusted so that normalized input impedance of each subarray becomes value '2'. Consequently, parallel combination of five element array results normalized admittance of 1 at input.

Subarrays can be excited by proper waveguide plumbing incorporated with coupling slots and waveguide tees. Such feeding network effectively governs the impedance bandwidth of SWA. In Section 3, the conventional coupling slot-based excitation technique and proposed differential feeding technique have been broadly detailed. To compare these techniques, 10-elements linear SWAs have been designed and optimized to achieve maximum bandwidth

with frequency for various designed SWAs is shown in **Figure 10**. As it can be observed from **Figure 10**, ~90% efficiency is achieved over the band of interest for the proposed broadband SWA. Furthermore, SWA fed using coupling slots shows lowest efficiency, which can be attributed to high insertion loss added by coupling slots. **Figure 11** shows the plot of variation of realized gain with frequency for the presented SWAs. As it can be observed, the behaviors of realized gain and antenna efficiency with frequency are same. Moreover, gain flatness of proposer SWA has significant improvement as compared to conventional edge fed SWA and

Broadband Slotted Waveguide Array Antenna http://dx.doi.org/10.5772/intechopen.78308 15

**6. Design and simulation of planar slotted waveguide array antenna**

Section 7 shows the plots of simulated RL and far field patterns.

**7. Prototype fabrication and measurement**

In SWA, resonant slots are cut on the broadwall of the waveguide. Therefore, when any radiating structure working on the same band of interest is placed near these resonant slots, strong coupling between the resonant slots on waveguide and radiating structure occurs and reduces the impedance bandwidth of the SWA. This effect has been examined for the proposed broadband SWA by designing 2 × 10 planar broadband SWA using proposed differential feeding technique. **Figure 12** shows the design model of planar SWA. Here, two linear broadband SWAs are placed in proximity and combined using 1:2 waveguide feeder network employed with E- and H-plane tees. As it has been discussed, coupling between two linear SWAs should degrade the impedance bandwidth of planar SWA. However, proper optimization of feeder network can cancel the coupled energy, subsequently preserving the bandwidth of planar broadband SWA. About 7.8% 17-dB return loss bandwidth has been achieved in simulation.

The presented antenna designs are fabricated and characterized for experimental validation of proposed technique. The realized prototypes of conventional SWA, SWA excited with coupling slots, and SWA with differential feeding technique are shown in **Figure 13**. Agilent E8363B microwave network analyzer with SOLT WR-09 calibration kit has been used to measure the return loss performance of these prototypes. **Figure 14** shows the comparison

**Figure 12.** Geometry of 2 × 10 planar broadband SWA antenna with differential feeding mechanism.

coupling slot fed SWA.

**Figure 10.** Comparison of total antenna efficiency between edge fed, coupling slot fed, and differential fed SWA.

for both type of feeding techniques. The simulated and their measured performances are compared and discussed in Section 7. When SWA is excited with coupling slots, 4.6% 17-dB RL bandwidth has been achieved, which is considerably less as compared to the expected bandwidth from subarray technique. The reduction in bandwidth can be ascribed to the addition of admittance variation of resonating coupling slots to the admittance variation of subarrays.

The chapter discusses the differential feeding technique which does not require resonating elements and hence, maintains the impedance bandwidth of subarray. After proper design and optimization of 10-elements linear SWAs with proposed feeding technique, 7.5% 17-dB RL bandwidth has been achieved, which is significantly large as compared to the RL bandwidth achieved when subarrays are fed using coupling slots. However, the achieved return loss bandwidth is not same as the RL bandwidth provided by a single subarray because of frequency sensitive nature of waveguide bends and E plane T junction.

Antenna efficiency is one of the important factors and it is the function of the reflection coefficient while considering other parameters constant. Therefore, if the variation of reflection coefficient with frequency is maintained below certain desired level (here, it is −17 dB), the antenna efficiency remains the same over the band of interest. Total antenna efficiency variation

**Figure 11.** Comparison of realized antenna gain between edge fed, coupling slot fed, and differential fed SWA.

with frequency for various designed SWAs is shown in **Figure 10**. As it can be observed from **Figure 10**, ~90% efficiency is achieved over the band of interest for the proposed broadband SWA. Furthermore, SWA fed using coupling slots shows lowest efficiency, which can be attributed to high insertion loss added by coupling slots. **Figure 11** shows the plot of variation of realized gain with frequency for the presented SWAs. As it can be observed, the behaviors of realized gain and antenna efficiency with frequency are same. Moreover, gain flatness of proposer SWA has significant improvement as compared to conventional edge fed SWA and coupling slot fed SWA.

## **6. Design and simulation of planar slotted waveguide array antenna**

In SWA, resonant slots are cut on the broadwall of the waveguide. Therefore, when any radiating structure working on the same band of interest is placed near these resonant slots, strong coupling between the resonant slots on waveguide and radiating structure occurs and reduces the impedance bandwidth of the SWA. This effect has been examined for the proposed broadband SWA by designing 2 × 10 planar broadband SWA using proposed differential feeding technique. **Figure 12** shows the design model of planar SWA. Here, two linear broadband SWAs are placed in proximity and combined using 1:2 waveguide feeder network employed with E- and H-plane tees. As it has been discussed, coupling between two linear SWAs should degrade the impedance bandwidth of planar SWA. However, proper optimization of feeder network can cancel the coupled energy, subsequently preserving the bandwidth of planar broadband SWA. About 7.8% 17-dB return loss bandwidth has been achieved in simulation. Section 7 shows the plots of simulated RL and far field patterns.

**Figure 12.** Geometry of 2 × 10 planar broadband SWA antenna with differential feeding mechanism.

### **7. Prototype fabrication and measurement**

**Figure 10.** Comparison of total antenna efficiency between edge fed, coupling slot fed, and differential fed SWA.

14 Emerging Waveguide Technology

frequency sensitive nature of waveguide bends and E plane T junction.

for both type of feeding techniques. The simulated and their measured performances are compared and discussed in Section 7. When SWA is excited with coupling slots, 4.6% 17-dB RL bandwidth has been achieved, which is considerably less as compared to the expected bandwidth from subarray technique. The reduction in bandwidth can be ascribed to the addition of admittance variation of resonating coupling slots to the admittance variation of subarrays. The chapter discusses the differential feeding technique which does not require resonating elements and hence, maintains the impedance bandwidth of subarray. After proper design and optimization of 10-elements linear SWAs with proposed feeding technique, 7.5% 17-dB RL bandwidth has been achieved, which is significantly large as compared to the RL bandwidth achieved when subarrays are fed using coupling slots. However, the achieved return loss bandwidth is not same as the RL bandwidth provided by a single subarray because of

Antenna efficiency is one of the important factors and it is the function of the reflection coefficient while considering other parameters constant. Therefore, if the variation of reflection coefficient with frequency is maintained below certain desired level (here, it is −17 dB), the antenna efficiency remains the same over the band of interest. Total antenna efficiency variation

**Figure 11.** Comparison of realized antenna gain between edge fed, coupling slot fed, and differential fed SWA.

The presented antenna designs are fabricated and characterized for experimental validation of proposed technique. The realized prototypes of conventional SWA, SWA excited with coupling slots, and SWA with differential feeding technique are shown in **Figure 13**. Agilent E8363B microwave network analyzer with SOLT WR-09 calibration kit has been used to measure the return loss performance of these prototypes. **Figure 14** shows the comparison

**Figure 13.** Fabricated structures (a) conventional slotted waveguide array, (b) slotted waveguide array antenna using coupling slots, and (c) SWA using differential feeding mechanism.

The radiation pattern and gain measurement of each fabricated prototype has also been carried out at in-house anechoic chamber. **Figure 16** shows proposed SWA antenna mounted on device under test (DUT) unit at anechoic chamber for far field measurement. Measured radiation patterns at 9.65 GHz for each prototype are shown in **Figure 17**. To obtain the gain of the array under test, the received power level difference between array under test and a standard gain horn has been measured and the results are presented in **Table 3** with their simulated performance. Good correlation between simulated and measurement of far-field parameters is obtained. Moreover, **Figure 18** shows the variation of measured gain with frequency of

**Length (mm) Height (mm) Width (mm)**

Broadband Slotted Waveguide Array Antenna http://dx.doi.org/10.5772/intechopen.78308 17

**Figure 15.** Simulated return loss performance of SWA with physical short and virtual short.

Conventional slotted waveguide array 36 14 50 Slotted waveguide array antenna using coupling slots 36 14 9.5 SWA using differential feeding mechanism 35 17 50

**Table 2.** Dimensions of fabricated SWA structures.

**Figure 16.** SWA using differential feeding mechanism mounted on DUT of anechoic chamber.

**Figure 14.** Simulated and measured return loss performance of (a) conventional slotted waveguide array, (b) slotted waveguide array antenna using coupling slots, and (c) SWA using differential feeding mechanism.

between measured and simulated return loss performance and **Table 3** lists the simulated and measured percentage bandwidth of each prototype. From **Figure 14**, it can be said that the simulated and measured results are in close agreement. Moreover, **Table 2** also compares the size of the fabricated SWA structure. The fabricated structures include the baseplates and interface plates which are not actual part of the antenna. They are connected with structure to perform S-parameter and far field characterization.

To validate the theory of virtual short in SWA, analysis has been carried out by placing physical short in the location of virtual short and its return loss performance is shown in **Figure 15**. No deviation from return loss performance of SWA with virtual short can be observed.


**Table 2.** Dimensions of fabricated SWA structures.

between measured and simulated return loss performance and **Table 3** lists the simulated and measured percentage bandwidth of each prototype. From **Figure 14**, it can be said that the simulated and measured results are in close agreement. Moreover, **Table 2** also compares the size of the fabricated SWA structure. The fabricated structures include the baseplates and interface plates which are not actual part of the antenna. They are connected with structure to

**Figure 14.** Simulated and measured return loss performance of (a) conventional slotted waveguide array, (b) slotted

waveguide array antenna using coupling slots, and (c) SWA using differential feeding mechanism.

**Figure 13.** Fabricated structures (a) conventional slotted waveguide array, (b) slotted waveguide array antenna using

To validate the theory of virtual short in SWA, analysis has been carried out by placing physical short in the location of virtual short and its return loss performance is shown in **Figure 15**. No deviation from return loss performance of SWA with virtual short can be observed.

perform S-parameter and far field characterization.

coupling slots, and (c) SWA using differential feeding mechanism.

16 Emerging Waveguide Technology

**Figure 15.** Simulated return loss performance of SWA with physical short and virtual short.

The radiation pattern and gain measurement of each fabricated prototype has also been carried out at in-house anechoic chamber. **Figure 16** shows proposed SWA antenna mounted on device under test (DUT) unit at anechoic chamber for far field measurement. Measured radiation patterns at 9.65 GHz for each prototype are shown in **Figure 17**. To obtain the gain of the array under test, the received power level difference between array under test and a standard gain horn has been measured and the results are presented in **Table 3** with their simulated performance. Good correlation between simulated and measurement of far-field parameters is obtained. Moreover, **Figure 18** shows the variation of measured gain with frequency of

**Figure 16.** SWA using differential feeding mechanism mounted on DUT of anechoic chamber.

A prototype of 2 × 10 planar SWA antenna has also been realized and characterized to verify the simulation data. The developed prototype model is shown in **Figure 19** and its measured RL performance and far field pattern with their simulation data is plotted in **Figures 20** and **21**.

Broadband Slotted Waveguide Array Antenna http://dx.doi.org/10.5772/intechopen.78308 19

**Figure 18.** Variation of measured gain with frequency for SWA with differential feeding mechanism.

**Figure 19.** Fabricated structures of 2 × 10 planar SWA using differential feeding mechanism.

**Figure 20.** Simulated and measured return loss performance of 2 × 10 planar SWA using differential feeding mechanism.

**Figure 17.** Simulated and measured radiation pattern of (a) conventional slotted waveguide array, (b) slotted waveguide array antenna using coupling slots, and (c) SWA using differential feeding mechanism.


**Table 3.** Measured bandwidth.

SWA with differential feeding mechanism. 0.88 dB gain variation over 800 MHz frequency range has been achieved.

Moreover, another most important mechanical aspects are antenna mass and its fabrication. Although the proposed SWA has high mass as compared to conventional SWAs, it can be reduced by the optimization of waveguide feeder network while incorporating mass as important factor in cost function and by using CFR- (Carbon Fiber Reinforced Plastic) based waveguides. In addition, the proposed SWA is complex to fabricate as compared to conventional SWAs. But, the recent evolution in mechanical fabrication processes has expanded the scope of fabricating complex waveguide structure within desired mass.

A prototype of 2 × 10 planar SWA antenna has also been realized and characterized to verify the simulation data. The developed prototype model is shown in **Figure 19** and its measured RL performance and far field pattern with their simulation data is plotted in **Figures 20** and **21**.

**Figure 18.** Variation of measured gain with frequency for SWA with differential feeding mechanism.

**Figure 19.** Fabricated structures of 2 × 10 planar SWA using differential feeding mechanism.

SWA with differential feeding mechanism. 0.88 dB gain variation over 800 MHz frequency

**Figure 17.** Simulated and measured radiation pattern of (a) conventional slotted waveguide array, (b) slotted waveguide

**(%)**

Edge fed SWA 2.8 2.7 16.60 16.35 SWA fed by coupling slot 4.6 4.8 16.61 16.21

**Measured BW** 

7.5 7.6 16.62 16.32

**Simulated gain** 

**Measured gain** 

**(dB)**

**(dB)**

array antenna using coupling slots, and (c) SWA using differential feeding mechanism.

**(%)**

**9.65 GHz Simulated BW** 

Moreover, another most important mechanical aspects are antenna mass and its fabrication. Although the proposed SWA has high mass as compared to conventional SWAs, it can be reduced by the optimization of waveguide feeder network while incorporating mass as important factor in cost function and by using CFR- (Carbon Fiber Reinforced Plastic) based waveguides. In addition, the proposed SWA is complex to fabricate as compared to conventional SWAs. But, the recent evolution in mechanical fabrication processes has expanded the

scope of fabricating complex waveguide structure within desired mass.

range has been achieved.

SWA fed by differential feeding

18 Emerging Waveguide Technology

**Table 3.** Measured bandwidth.

mechanism

**Figure 20.** Simulated and measured return loss performance of 2 × 10 planar SWA using differential feeding mechanism.

engineers of Microwave Sensors Antenna Divison (MSAD) and Antenna Measurment

Broadband Slotted Waveguide Array Antenna http://dx.doi.org/10.5772/intechopen.78308 21

Microwave Sensors Antenna Division (MSAD), Space Applications Centre (SAC), Indian

[1] Wang W, Zhong S-S, Zhang Y-M, Liang X-L. A broadband slotted ridge waveguide antenna array. IEEE Transactions on Antennas and Propagation. August 2006;**54**(8).

[2] Sekretarov SS, Vavriv DM.A wideband slotted waveguide antenna array for Sar systems. Progress In Electromagnetics Research M. 2010;**11**:165-176. DOI: 10.2528/PIERM10010606

[3] Zhao HC, Xu RR, Wu W. Broadband waveguide slot array for SAR. Electronics Letters.

[4] Silver S. Microwave Antenna Theory and Design. 1st ed. The United States of America:

[5] David M. Pozar, Microwave Engineering. 3rd ed. Singapore: John Wiley & Sons, Inc.;

[6] Elliott RS, Kurtz LA. The Design of Small Slot Arrays. IEEE Transactions on Antennas

[7] Elliott RS. An improved design procedure for small arrays of shunt slots. IEEE Transactions on Antennas and Propagation. January 1983;**AP-31**(1). DOI: 10.1109/TAP.

[8] Kim DY, Elliott RS. A design procedure for slot arrays fed by single-ridge waveguide. IEEE Transactions on Antennas and Propagation. November 1988;**36**(11). DOI:

and Propagation. March 1978;**AP-26**(2). DOI: 10.1109/TAP.1978.1141814

Facility (AMF) for their help.

Yogesh Tyagi and Pratik Mevada\*

DOI: 10.1109/TAP.2006.879216

\*Address all correspondence to: pratik@sac.isro.gov.in

Space Research Organization (ISRO), Ahmedabad, India

January 2011;**47**(2). DOI: 10.1049/el.2010.3009

MacGraw-Hill Book Company, Inc.; 1949

**Author details**

**References**

2005

1983.1143002

10.1109/8.9701

**Figure 21.** Simulated and measured radiation pattern of 2 × 10 planar SWA using differential feeding mechanism.

From **Figure 20**, it can be concluded that the planar array preserves the RL bandwidth of single linear array and the measured and simulated performances are in close match.

#### **8. Conclusion**

In this chapter, high efficiency broadband 10-element linear SWA integrated with differential feeding technique has been discussed in detail and also compared with conventional SWAs. The differential feeding technique has been proposed to eliminate the drawbacks of conventional SWAs excited with edge feeding and/or coupling slots. Mathematical justification of virtual short transpired due to differential feeding has also been presented and validated.

Ten-element linear SWAs employed with edge feeding, coupling slot feeding, and differential feeding have designed and simulated. The simulated results are also compared to validate the proposed advantages of differential feeding. About 7.5% return loss bandwidth has been achieved using differential feeding. Moreover, antenna efficiency and gain flatness performances are superior in the case of differential feeding. Presented designs of SWA are fabricated and characterized. In addition, 2 × 10 planar broadband SWA has also been designed, developed, and characterized. About 7.8% return loss bandwidth has been achieved in simulation. The measured performance of planar broadband SWA is also compared with simulated performance and good agreement has been achieved between them.

Furthermore, the proposed differential feeding techniques have shown many advantages like broad bandwidth, high power handling capacity, and the requirement of low fabrication tolerances. Therefore, this technique is the best suitable for SWA designs at millimeter wave and sub-millimeter wave.

### **Acknowledgements**

The authors thank and appreciate the support and encouragement provided by Shri Tapan Misra, Director, SAC during the course of this work. The authors also wish to thank all engineers of Microwave Sensors Antenna Divison (MSAD) and Antenna Measurment Facility (AMF) for their help.

### **Author details**

Yogesh Tyagi and Pratik Mevada\*

\*Address all correspondence to: pratik@sac.isro.gov.in

Microwave Sensors Antenna Division (MSAD), Space Applications Centre (SAC), Indian Space Research Organization (ISRO), Ahmedabad, India

### **References**

From **Figure 20**, it can be concluded that the planar array preserves the RL bandwidth of single

**Figure 21.** Simulated and measured radiation pattern of 2 × 10 planar SWA using differential feeding mechanism.

In this chapter, high efficiency broadband 10-element linear SWA integrated with differential feeding technique has been discussed in detail and also compared with conventional SWAs. The differential feeding technique has been proposed to eliminate the drawbacks of conventional SWAs excited with edge feeding and/or coupling slots. Mathematical justification of virtual short transpired due to differential feeding has also been presented and validated.

Ten-element linear SWAs employed with edge feeding, coupling slot feeding, and differential feeding have designed and simulated. The simulated results are also compared to validate the proposed advantages of differential feeding. About 7.5% return loss bandwidth has been achieved using differential feeding. Moreover, antenna efficiency and gain flatness performances are superior in the case of differential feeding. Presented designs of SWA are fabricated and characterized. In addition, 2 × 10 planar broadband SWA has also been designed, developed, and characterized. About 7.8% return loss bandwidth has been achieved in simulation. The measured performance of planar broadband SWA is also compared with simu-

Furthermore, the proposed differential feeding techniques have shown many advantages like broad bandwidth, high power handling capacity, and the requirement of low fabrication tolerances. Therefore, this technique is the best suitable for SWA designs at millimeter wave

The authors thank and appreciate the support and encouragement provided by Shri Tapan Misra, Director, SAC during the course of this work. The authors also wish to thank all

linear array and the measured and simulated performances are in close match.

lated performance and good agreement has been achieved between them.

**8. Conclusion**

20 Emerging Waveguide Technology

and sub-millimeter wave.

**Acknowledgements**


**Chapter 2**

Provisional chapter

**Photonic Crystal Waveguides**

Photonic Crystal Waveguides

Dmitry Usanov and Alexander Skripal

Dmitry Usanov and Alexander Skripal

http://dx.doi.org/10.5772/intechopen.76797

of microwave photonic crystals.

matched loads, low-dimensional photonic crystal

Abstract

1. Introduction

Additional information is available at the end of the chapter

The original results of theoretical and experimental studies and the properties of microwave one-dimensional waveguide photonic crystals have been generalized. Methods for describing the electrodynamic characteristics of photonic crystals and their relationship with the parameters of periodic structures filling the waveguides have been presented. The results of an investigation on the characteristics of microwave waveguide photonic crystals made in the form of dielectric matrices with air inclusions have been presented. The model of effective dielectric permittivity has been proposed for describing the characteristics of the investigated photonic crystals containing layers with a large number of air inclusions. New types of microwave low-dimensional waveguide photonic crystals containing periodically alternating elements that are sources of higher type waves have been described. The possibility of effective control of the amplitude-frequency characteristics of microwave photonic crystals by means of electric and magnetic fields has been analyzed. Examples of new applications of waveguide photonic crystals in the microwave range have been given: the measuring parameters of the materials and semiconductor nanostructures that play the role of the microwave photonic crystals' periodicity defect; the resonators of near-field microwave microscopes; small-sized matched loads for centimeter and millimeter wavelength ranges on the basis

DOI: 10.5772/intechopen.76797

Keywords: microwave photonic crystals, forbidden bands, defect mode, electrically controlled characteristics, measurement of micro- and nanostructures, microwave

The idea of the possibility of a continuous spectrum of electron energies decay into a set of alternating allowed and forbidden bands in the direction of electron motion and wave propagation in the presence of spatial periodicity of the deformation field was for the first

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Additional information is available at the end of the chapter

#### **Chapter 2** Provisional chapter

#### **Photonic Crystal Waveguides** Photonic Crystal Waveguides

Dmitry Usanov and Alexander Skripal Dmitry Usanov and Alexander Skripal

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.76797

#### Abstract

The original results of theoretical and experimental studies and the properties of microwave one-dimensional waveguide photonic crystals have been generalized. Methods for describing the electrodynamic characteristics of photonic crystals and their relationship with the parameters of periodic structures filling the waveguides have been presented. The results of an investigation on the characteristics of microwave waveguide photonic crystals made in the form of dielectric matrices with air inclusions have been presented. The model of effective dielectric permittivity has been proposed for describing the characteristics of the investigated photonic crystals containing layers with a large number of air inclusions. New types of microwave low-dimensional waveguide photonic crystals containing periodically alternating elements that are sources of higher type waves have been described. The possibility of effective control of the amplitude-frequency characteristics of microwave photonic crystals by means of electric and magnetic fields has been analyzed. Examples of new applications of waveguide photonic crystals in the microwave range have been given: the measuring parameters of the materials and semiconductor nanostructures that play the role of the microwave photonic crystals' periodicity defect; the resonators of near-field microwave microscopes; small-sized matched loads for centimeter and millimeter wavelength ranges on the basis of microwave photonic crystals.

DOI: 10.5772/intechopen.76797

Keywords: microwave photonic crystals, forbidden bands, defect mode, electrically controlled characteristics, measurement of micro- and nanostructures, microwave matched loads, low-dimensional photonic crystal

#### 1. Introduction

The idea of the possibility of a continuous spectrum of electron energies decay into a set of alternating allowed and forbidden bands in the direction of electron motion and wave propagation in the presence of spatial periodicity of the deformation field was for the first

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

time proposed by Keldysh [1]. Periodic semiconductor structures with predetermined parameters of layers were called semiconductor superlattices. The wide interest to the problem of their creation appeared after the article was published in 1970 by Esaki and Tsu, who proposed to make such structures by changing the doping or composition of the layers [2]. The periods in such structures had values from 5 to 20 nm. The number of layers reached several hundred.

vicinity of 6 GHz was observed in a solution with a chemical self-oscillating Briggs-Rauscher reaction characterized by the presence of periodically located regions with differ-

2. One-dimensional microwave photonic crystals based on rectangular

The results of theoretical analysis on the characteristics of one-dimensional microwave photonic crystals made on the basis of a rectangular waveguide and their experimental investigation are given, for example, in [14]. The one dimensionality of the crystal means that the dielectric structure that fills the waveguide has a periodicity of dielectric permittivity in one direction (along the wave propagation direction, along the Z axis). The authors of [14] studied structures in the form of alternating layers with high dielectric permittivity h and low l (hlpairs) and layers of the h1lh<sup>2</sup> type (h<sup>1</sup> and h<sup>2</sup> are dielectrics with different permittivities). In the theoretical description, the scattering matrix method was used. The arrangement of the dielec-

εð Þ¼ x; y; z þ d εð Þ x; y; z , 0 ≤ x ≤ a, 0 ≤ y ≤ b, z < ∞, (1)

<sup>10</sup> , (3)

2

, <sup>k</sup><sup>2</sup> <sup>¼</sup> ð Þ <sup>π</sup>=<sup>a</sup>

2 .

<sup>1</sup>,<sup>2</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>μ0ε1,<sup>2</sup> <sup>þ</sup> <sup>k</sup>

(2)

Photonic Crystal Waveguides

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http://dx.doi.org/10.5772/intechopen.76797

ε<sup>1</sup> ¼ ε0εr<sup>1</sup> for � w ≤ z ≤ 0 <sup>ε</sup><sup>2</sup> <sup>¼</sup> <sup>ε</sup>0εr<sup>2</sup> for <sup>0</sup> <sup>≤</sup> <sup>z</sup> <sup>≤</sup> <sup>v</sup> :

2.1. Microwave photonic crystals: structures with forbidden bands

tric layers in the waveguide can be characterized by the following relationships:

Here εr<sup>1</sup> and εr<sup>2</sup> are the relative permittivities of the two materials and ε<sup>0</sup> is the dielectric

When continuity conditions are used on surfaces z ¼ 0, z ¼ v and z ¼ �w for tangential field

sin <sup>β</sup>1<sup>w</sup> sin <sup>β</sup>2<sup>v</sup> , <sup>β</sup><sup>2</sup>

The authors of [14] proposed to obtain reflection and transmission coefficients using the generalized scattering matrix, which is a result of the application of continuity conditions on media boundaries. The results of measurements and calculation of the transmission coefficient (S21), characterized by the presence of a forbidden band for microwave photonic crystals consisting of 10 and 20 separate two-layer elements, are presented in [14]. The authors of [14] refer some of the differences between experiment and theory [14] to the imperfection of the

arccos LTEz

εð Þ¼ x; y; z þ d

components, the equation for finding the wave number γ is obtained:

1þβ<sup>2</sup> 2 2β1β<sup>2</sup>

<sup>γ</sup> <sup>¼</sup> <sup>1</sup> v þ w

constant of the vacuum, d ¼ v þ w.

<sup>10</sup> <sup>¼</sup> cos <sup>β</sup>1<sup>w</sup> cos <sup>β</sup>2<sup>v</sup> � <sup>β</sup><sup>2</sup>

where

waveguide walls.

LTEz

ent permittivities [13].

waveguides

The author [3] gave the definition of photonic crystals as materials whose crystal lattice has a periodicity of the permittivity leading to the appearance of the "forbidden" frequencies range, called the photonic band gap. Yablonovich [4] and John [5] proposed to create structures with a photonic band gap, which can be considered as an optical analog of the band gap in semiconductors. In this case, the forbidden band is the frequency range in which the existence of light in the inner part of the crystal is forbidden. The type of defect or disturbance of periodicity in this instance can be different. Such structures have to be created artificially in contrast to natural crystals. In this case, the size of the basic unit element of a photonic crystal should be comparable with the light wavelength. The manufacturing of such structures involves the use of electron-beam and X-ray lithography [6].

As the advantage of such photonic crystals, the author [3] notes the possibility of an exact description of their properties that coincide with the experiment, in contrast to superlattices.

Structures with spatial periodicity of elements were also used in the microwave range to reduce the phase velocity of the wave in comparison with the speed of light in special waveguides, called delay-line structures [7]. The authors of [7] called them "a kind of artificial crystals, the cells of which have large sizes."

Delay-line structures are used in various types of vacuum microwave electronics devices. The specificity of the delay-line systems is the choice of their basic elements from metals and the need to take into account in the design the possibility of passing the electron beam interacting with the field. Examples of periodic structures used in delay-line systems are "meander," "counterpins," systems with alternating diaphragms, and so on.

In the microwave range, the photonic crystal can be realized both on waveguides with dielectric filling [8, 9] and on flat transmission lines with periodically changing stripe structure [10]. There are examples of the creation of photonic crystals in the optical, infrared, ultraviolet, microwave ranges. Creating a photonic crystal for the microwave range is the simplest. It should be noted that in the theoretical description of the properties of such structures, unlike, for example, from superlattices, it is not necessary to take into account the properties of transition layers, quantum size effects, the specificity of technological processes. This opens the possibility to more accurately examine the properties of photonic crystals associated with periodicity and, in particular, to use the results of a theoretical description to measure the parameters of their layers as a result of solving the corresponding inverse problem.

Materials with the properties of photonic crystals are also known in nature. They include, for example, noble opal [11], spicules of natural biomineral crystals, the basal spicules of glass sea sponges [12]. The time-varying forbidden band for the frequency region in the vicinity of 6 GHz was observed in a solution with a chemical self-oscillating Briggs-Rauscher reaction characterized by the presence of periodically located regions with different permittivities [13].

### 2. One-dimensional microwave photonic crystals based on rectangular waveguides

#### 2.1. Microwave photonic crystals: structures with forbidden bands

The results of theoretical analysis on the characteristics of one-dimensional microwave photonic crystals made on the basis of a rectangular waveguide and their experimental investigation are given, for example, in [14]. The one dimensionality of the crystal means that the dielectric structure that fills the waveguide has a periodicity of dielectric permittivity in one direction (along the wave propagation direction, along the Z axis). The authors of [14] studied structures in the form of alternating layers with high dielectric permittivity h and low l (hlpairs) and layers of the h1lh<sup>2</sup> type (h<sup>1</sup> and h<sup>2</sup> are dielectrics with different permittivities). In the theoretical description, the scattering matrix method was used. The arrangement of the dielectric layers in the waveguide can be characterized by the following relationships:

$$
\varepsilon(\mathbf{x}, y, z + d) = \varepsilon(\mathbf{x}, y, z), \quad 0 \le \mathbf{x} \le a, \quad 0 \le y \le b, z < \infty,\tag{1}
$$

$$\varepsilon(\mathbf{x}, y, z + d) = \begin{cases} \varepsilon\_1 = \varepsilon\_0 \varepsilon\_{r1} \text{ for } -\varpi \le z \le 0 \\\varepsilon\_2 = \varepsilon\_0 \varepsilon\_{r2} \text{ for } 0 \le z \le v \end{cases} \tag{2}$$

Here εr<sup>1</sup> and εr<sup>2</sup> are the relative permittivities of the two materials and ε<sup>0</sup> is the dielectric constant of the vacuum, d ¼ v þ w.

When continuity conditions are used on surfaces z ¼ 0, z ¼ v and z ¼ �w for tangential field components, the equation for finding the wave number γ is obtained:

$$\gamma = \frac{1}{v+w} \arccos\left(L\_{T\mathbb{E}\_{\mathbb{I}0}^{v}}\right) \tag{3}$$

where

time proposed by Keldysh [1]. Periodic semiconductor structures with predetermined parameters of layers were called semiconductor superlattices. The wide interest to the problem of their creation appeared after the article was published in 1970 by Esaki and Tsu, who proposed to make such structures by changing the doping or composition of the layers [2]. The periods in such structures had values from 5 to 20 nm. The number of layers reached

The author [3] gave the definition of photonic crystals as materials whose crystal lattice has a periodicity of the permittivity leading to the appearance of the "forbidden" frequencies range, called the photonic band gap. Yablonovich [4] and John [5] proposed to create structures with a photonic band gap, which can be considered as an optical analog of the band gap in semiconductors. In this case, the forbidden band is the frequency range in which the existence of light in the inner part of the crystal is forbidden. The type of defect or disturbance of periodicity in this instance can be different. Such structures have to be created artificially in contrast to natural crystals. In this case, the size of the basic unit element of a photonic crystal should be comparable with the light wavelength. The manufacturing of such structures involves the use

As the advantage of such photonic crystals, the author [3] notes the possibility of an exact description of their properties that coincide with the experiment, in contrast to superlattices. Structures with spatial periodicity of elements were also used in the microwave range to reduce the phase velocity of the wave in comparison with the speed of light in special waveguides, called delay-line structures [7]. The authors of [7] called them "a kind of artificial

Delay-line structures are used in various types of vacuum microwave electronics devices. The specificity of the delay-line systems is the choice of their basic elements from metals and the need to take into account in the design the possibility of passing the electron beam interacting with the field. Examples of periodic structures used in delay-line systems are "meander,"

In the microwave range, the photonic crystal can be realized both on waveguides with dielectric filling [8, 9] and on flat transmission lines with periodically changing stripe structure [10]. There are examples of the creation of photonic crystals in the optical, infrared, ultraviolet, microwave ranges. Creating a photonic crystal for the microwave range is the simplest. It should be noted that in the theoretical description of the properties of such structures, unlike, for example, from superlattices, it is not necessary to take into account the properties of transition layers, quantum size effects, the specificity of technological processes. This opens the possibility to more accurately examine the properties of photonic crystals associated with periodicity and, in particular, to use the results of a theoretical description to measure the parameters of their layers as a result of solving the corresponding

Materials with the properties of photonic crystals are also known in nature. They include, for example, noble opal [11], spicules of natural biomineral crystals, the basal spicules of glass sea sponges [12]. The time-varying forbidden band for the frequency region in the

several hundred.

24 Emerging Waveguide Technology

inverse problem.

of electron-beam and X-ray lithography [6].

crystals, the cells of which have large sizes."

"counterpins," systems with alternating diaphragms, and so on.

 $L\_{T\Xi\_{10}^{\varepsilon}} = \cos\left(\beta\_1 w\right)\cos\left(\beta\_2 v\right) - \frac{\beta\_1^2 + \beta\_2^2}{2\beta\_1\beta\_2}\sin\left(\beta\_1 w\right)\sin\left(\beta\_2 v\right),$  $\beta\_{1,2}^2 = \omega^2\mu\_0\varepsilon\_{1,2} + k^2, k^2 = \left(\pi/a\right)^2.$ 

The authors of [14] proposed to obtain reflection and transmission coefficients using the generalized scattering matrix, which is a result of the application of continuity conditions on media boundaries. The results of measurements and calculation of the transmission coefficient (S21), characterized by the presence of a forbidden band for microwave photonic crystals consisting of 10 and 20 separate two-layer elements, are presented in [14]. The authors of [14] refer some of the differences between experiment and theory [14] to the imperfection of the waveguide walls.

To calculate the reflection S<sup>11</sup> and the transmission S21, coefficients of the electromagnetic wave at its normal incidence on an N layer structure, a matrix of wave transfer between regions with different values of wave propagation constants, can be used, similar to [15–17]:

$$\mathbf{T}(z\_{j,i+1}) = \begin{pmatrix} \frac{\mathcal{V}\_{j+1} + \mathcal{V}\_{j}}{2\mathcal{V}\_{j+1}} e^{(\mathcal{V}\_{j+1} - \mathcal{V}\_{j})z\_{j,i+1}} & \frac{\mathcal{V}\_{j+1} - \mathcal{V}\_{j}}{2\mathcal{V}\_{j+1}} e^{(\mathcal{V}\_{j+1} + \mathcal{V}\_{j})z\_{j,i+1}} \\ \frac{\mathcal{V}\_{j+1} - \mathcal{V}\_{j}}{2\mathcal{V}\_{j+1}} e^{-(\mathcal{V}\_{j+1} + \mathcal{V}\_{j})z\_{j,i+1}} & \frac{\mathcal{V}\_{j+1} + \mathcal{V}\_{j}}{2\mathcal{V}\_{j+1}} e^{-(\mathcal{V}\_{j+1} - \mathcal{V}\_{j})z\_{j,i+1}} \end{pmatrix},\tag{4}$$

which connects the coefficients Aj, Bj and Ajþ1, Bjþ1, which determine the amplitudes of the incident and reflected waves on both sides of the boundary zj,jþ1, by the relation:

$$
\mathbf{T}\begin{pmatrix} A\_{j+1} \\ B\_{j+1} \end{pmatrix} = \mathbf{T}(z\_{j,j+1}) \cdot \begin{pmatrix} A\_j \\ B\_j \end{pmatrix}.\tag{5}
$$

The results of the theoretical and experimental investigation of the resonant features that appear in the allowed and forbidden bands of microwave photonic crystal in the case of creating periodicity disturbance are given in [18]. As a waveguide photonic crystal, a waveguide section with a structure that is periodically alternating layers of two types of dielectrics with different values of thickness and permittivity was used. The dimensions and materials of the layers were chosen in such a way that in the frequency range 8–12 GHz, two allowed and one forbidden band were observed for the propagation of electromagnetic waves. The parameters of the first and the last layers of the photonic crystal were the same. The results of calculating the power transmission coefficient |S12|2 of an electromagnetic wave using the abovementioned relationships for the 11-layer photonic crystal without disturbances in the

Photonic Crystal Waveguides

27

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From the results of the calculation presented in Figure 3, it follows that an increase of layers in number causes the decrease of the width of the first band gap completely located within the 3 cm wavelength range and the increase in the width of both the left and right allowed bands in this wavelength range. With a number of layers larger than 27, these changes are less than 10 MHz. This behavior of the characteristics of an electromagnetic propagation through a photonic crystal is due to the following circumstance. The allowed band has a "dissected" frequency response and consists of a set of resonances, the number of which is determined by the number of identical elements that form the photonic crystal. Therefore, the increase of the number of photon crystal layers causes the increase of the number of resonances determining the width of the allowed band and, consequently, its width increases. At the same time, the

Figure 2. The results of calculating the square of the modulus of the transmission coefficient of an electromagnetic wave

through the 11-layer structure without disturbance (curve 1) and with the disturbed central layer (curve 2).

case of H<sup>10</sup> wave propagation are shown in Figure 2 (curve 1).

Figure 1. Structure consisting of N layers.

The coefficients ANþ<sup>1</sup> and B0, which determine the amplitudes of the wave transmitted through the multilayer structure (Figure 1) and the wave reflected from it, are connected to the coefficient A0, determining the amplitude of the incident wave, by the following relation:

$$
\begin{pmatrix} A\_{N+1} \\ 0 \end{pmatrix} = \mathbf{T}\_N \cdot \begin{pmatrix} A\_0 \\ B\_0 \end{pmatrix}' \tag{6}
$$

where

$$\mathbf{T}\_{N} = \begin{pmatrix} \mathbf{T}\_{N}[1,1] & \mathbf{T}\_{N}[1,2] \\ \mathbf{T}\_{N}[2,1] & \mathbf{T}\_{N}[2,2] \end{pmatrix} = \prod\_{j=N}^{0} \mathbf{T}\_{j,(j+1)} = \mathbf{T}(\mathbf{z}\_{N,N+1}) \cdot \mathbf{T}(\mathbf{z}\_{N-1,N}) \underbrace{\mathbf{T}(\mathbf{z}\_{1})}\_{\ldots} \cdot \mathbf{T}(\mathbf{z}\_{0,1}) \tag{7}$$

a transmission matrix of an N layer structure.

The reflection <sup>S</sup><sup>11</sup> <sup>¼</sup> <sup>B</sup><sup>0</sup> <sup>А</sup><sup>0</sup> and transmission <sup>S</sup><sup>21</sup> <sup>¼</sup> ANþ<sup>1</sup> <sup>А</sup><sup>0</sup> coefficients of an electromagnetic wave interacting with a layered structure are determined by the following relationships:

$$\mathbf{S}\_{11} = -\frac{\mathbf{T}\_N[2,1]}{\mathbf{T}\_N[2,2]},\\\mathbf{S}\_{21} = \frac{\mathbf{T}\_N[1,1] \cdot \mathbf{T}\_N[2,2] - \mathbf{T}\_N[1,2] \cdot \mathbf{T}\_N[2,1]}{\mathbf{T}\_N[2,2]}.\tag{8}$$

When calculating S<sup>11</sup> and S21, we used the matrixes of wave transfer between regions with different values of wave propagation constants <sup>γ</sup><sup>j</sup> and <sup>γ</sup><sup>j</sup>þ<sup>1</sup>.

The following expressions are used to calculate the reflection |S11|2 | and transmission |S12|2 coefficients of the electromagnetic wave expressed in terms of power (in dB unit):

$$\left|\left|\mathbf{S}\_{11}\right|^2 = 10\log\left|\mathbf{S}\_{11}\right|^2, \left|\mathbf{S}\_{21}\right|^2 = 10\log\left|\mathbf{S}\_{21}\right|^2\tag{9}$$

Figure 1. Structure consisting of N layers.

To calculate the reflection S<sup>11</sup> and the transmission S21, coefficients of the electromagnetic wave at its normal incidence on an N layer structure, a matrix of wave transfer between regions with

> <sup>γ</sup><sup>j</sup>þ<sup>1</sup> � <sup>γ</sup><sup>j</sup> <sup>2</sup>γ<sup>j</sup>þ<sup>1</sup>

<sup>γ</sup><sup>j</sup>þ<sup>1</sup> <sup>þ</sup> <sup>γ</sup><sup>j</sup> <sup>2</sup>γ<sup>j</sup>þ<sup>1</sup>

> Bj � �

e

e <sup>γ</sup>jþ1þ<sup>γ</sup> ð Þ<sup>j</sup> zj,jþ<sup>1</sup>

1

: (5)

, (6)

Tj,ð Þ <sup>j</sup>þ<sup>1</sup> ¼ Tð Þ� zN,Nþ<sup>1</sup> Tð Þ zN�1,N … Tð Þ� z1, <sup>2</sup> Tð Þ z0,<sup>1</sup> (7)

<sup>А</sup><sup>0</sup> coefficients of an electromagnetic wave

<sup>T</sup>N½ � <sup>2</sup>; <sup>2</sup> : (8)

<sup>2</sup> (9)

CCCA, (4)

� <sup>γ</sup>jþ1�<sup>γ</sup> ð Þ<sup>j</sup> zj,jþ<sup>1</sup>

e <sup>γ</sup>jþ1�<sup>γ</sup> ð Þ<sup>j</sup> zj,jþ<sup>1</sup>

� <sup>γ</sup>jþ1þ<sup>γ</sup> ð Þ<sup>j</sup> zj,jþ<sup>1</sup>

which connects the coefficients Aj, Bj and Ajþ1, Bjþ1, which determine the amplitudes of the

¼ T zj,jþ<sup>1</sup>

The coefficients ANþ<sup>1</sup> and B0, which determine the amplitudes of the wave transmitted through the multilayer structure (Figure 1) and the wave reflected from it, are connected to the coefficient A0, determining the amplitude of the incident wave, by the following

<sup>¼</sup> <sup>T</sup><sup>N</sup> � <sup>A</sup><sup>0</sup>

<sup>T</sup>N½ � <sup>2</sup>; <sup>2</sup> , S<sup>21</sup> <sup>¼</sup> <sup>T</sup>N½ �� <sup>1</sup>; <sup>1</sup> TN½ �� <sup>2</sup>; <sup>2</sup> <sup>T</sup>N½ �� <sup>1</sup>; <sup>2</sup> <sup>T</sup>N½ � <sup>2</sup>; <sup>1</sup>

When calculating S<sup>11</sup> and S21, we used the matrixes of wave transfer between regions with

The following expressions are used to calculate the reflection |S11|2 | and transmission |S12|2

2 , Sj j <sup>21</sup>

<sup>2</sup> <sup>¼</sup> 10 log j j <sup>S</sup><sup>21</sup>

B0 � �

� � � Aj

different values of wave propagation constants, can be used, similar to [15–17]:

e

incident and reflected waves on both sides of the boundary zj,jþ1, by the relation:

ANþ<sup>1</sup> 0 � �

<sup>¼</sup> <sup>Y</sup> 0

<sup>А</sup><sup>0</sup> and transmission <sup>S</sup><sup>21</sup> <sup>¼</sup> ANþ<sup>1</sup>

interacting with a layered structure are determined by the following relationships:

coefficients of the electromagnetic wave expressed in terms of power (in dB unit):

<sup>2</sup> <sup>¼</sup> 10 log j j <sup>S</sup><sup>11</sup>

j¼N

Ajþ<sup>1</sup> Bjþ<sup>1</sup> � �

<sup>γ</sup><sup>j</sup>þ<sup>1</sup> <sup>þ</sup> <sup>γ</sup><sup>j</sup> <sup>2</sup>γ<sup>j</sup>þ<sup>1</sup>

0

BBB@

<sup>γ</sup><sup>j</sup>þ<sup>1</sup> � <sup>γ</sup><sup>j</sup> <sup>2</sup>γ<sup>j</sup>þ<sup>1</sup>

T zj,jþ<sup>1</sup> � � <sup>¼</sup>

26 Emerging Waveguide Technology

<sup>T</sup><sup>N</sup> <sup>¼</sup> <sup>T</sup>N½ � <sup>1</sup>; <sup>1</sup> <sup>T</sup>N½ � <sup>1</sup>; <sup>2</sup> TN½ � 2; 1 TN½ � 2; 2 � �

The reflection <sup>S</sup><sup>11</sup> <sup>¼</sup> <sup>B</sup><sup>0</sup>

a transmission matrix of an N layer structure.

<sup>S</sup><sup>11</sup> ¼ � <sup>T</sup>N½ � <sup>2</sup>; <sup>1</sup>

different values of wave propagation constants <sup>γ</sup><sup>j</sup> and <sup>γ</sup><sup>j</sup>þ<sup>1</sup>.

j j S<sup>11</sup>

relation:

where

The results of the theoretical and experimental investigation of the resonant features that appear in the allowed and forbidden bands of microwave photonic crystal in the case of creating periodicity disturbance are given in [18]. As a waveguide photonic crystal, a waveguide section with a structure that is periodically alternating layers of two types of dielectrics with different values of thickness and permittivity was used. The dimensions and materials of the layers were chosen in such a way that in the frequency range 8–12 GHz, two allowed and one forbidden band were observed for the propagation of electromagnetic waves. The parameters of the first and the last layers of the photonic crystal were the same. The results of calculating the power transmission coefficient |S12|2 of an electromagnetic wave using the abovementioned relationships for the 11-layer photonic crystal without disturbances in the case of H<sup>10</sup> wave propagation are shown in Figure 2 (curve 1).

From the results of the calculation presented in Figure 3, it follows that an increase of layers in number causes the decrease of the width of the first band gap completely located within the 3 cm wavelength range and the increase in the width of both the left and right allowed bands in this wavelength range. With a number of layers larger than 27, these changes are less than 10 MHz. This behavior of the characteristics of an electromagnetic propagation through a photonic crystal is due to the following circumstance. The allowed band has a "dissected" frequency response and consists of a set of resonances, the number of which is determined by the number of identical elements that form the photonic crystal. Therefore, the increase of the number of photon crystal layers causes the increase of the number of resonances determining the width of the allowed band and, consequently, its width increases. At the same time, the

Figure 2. The results of calculating the square of the modulus of the transmission coefficient of an electromagnetic wave through the 11-layer structure without disturbance (curve 1) and with the disturbed central layer (curve 2).

crystal were measured with the Agilent PNA-L Network Analyzer N5230A. A comparison of the

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In [19], the results of an investigation on the characteristics of a waveguide microwave photonic crystal made in the form of dielectric matrices with air inclusions are presented. Ceramics (Al2O3) with a large number of air inclusions and polystyrene were used as materials of dielectric layers. A waveguide photonic crystal consisting of 11 layers was studied in the frequency range 8–12 GHz (Figure 4a). Odd layers were made of Al2O3 ceramics (ε<sup>r</sup> ¼ 9:6), even layers were made of polystyrene (ε<sup>r</sup> ¼ 1:05). Thickness of odd lengths was dAl2O3 ¼ 1:0mm, even dfoam ¼ 13:0 mm. The layers completely filled the cross-section of the waveguide. In ceramic layers, a large number of air inclusions were created in the form of square-through holes, which form a periodic structure

On the basis of numerical modeling using the finite element method in the CAD ANSYS HFSS, the influence of the volume fraction of air inclusions on the amplitude-frequency characteristics of the transmission coefficient of a photonic crystal was studied. The volume fraction of air inclusions was controlled by changing the size of the holes ahole in the ceramics plates with a fixed amount of 36 in each of the plates. The simulation was carried out for three sizes of holes, equal to 0.75, 1.2 and 1.65 mm, which corresponds to a volume fraction of air inclusions equal to 8.5, 23 and 43%, respectively. The disturbance of the periodicity of the photonic structure was created by changing the thickness of the central (sixth) polystyrene layer. The thickness of

As follows from the results of numerical calculation of the amplitude-frequency characteristics of the transmission coefficient of a photonic crystal by the finite-element method in the absence of the periodicity disturbance in the photonic structure, the increase in the volume fraction of air inclusions in the ceramic layers leads to the shift of the forbidden band of the photonic crystal toward shorter wavelengths and to the decrease of its depth. The results of the calculation of the photonic crystal amplitude-frequency characteristics by the finite element method in the presence of the disturbance of the photonic structure periodicity in the form of a central

The presence of the disturbance of the photonic structure periodicity in the form of the change of the thickness of the central (sixth) polystyrene layer led to the appearance of a transmission

Figure 4. Model of the photonic crystal: 1—layers of ceramics with square through holes, 2—polystyrene layers, 3 disturbance in the form of a polystyrene layer with changed thickness (a). The ceramics layer with square through holes (b).

the sixth modified (disturbed) polystyrene layer was 2.25, 2.75, 3.49 and 6.0 mm.

polystyrene layer of different thickness are shown in Figure 5 (solid curves).

results of calculation and experiment shows their good quantitative coincidence.

in the plane of the layer (Figure 4b).

Figure 3. Dependences of the width of the forbidden and left allowed bands on the number of layers of a photonic crystal without disturbance: □, ●—theoretical and experimental values of the width of the left-allowed band, ◊, ▲—theoretical and experimental values of the band gap width.

width of the forbidden band decreases. The Q-value of the resonances that form the allowed band, the so-called "allowed levels," as follows from the results of calculations and experiments, is maximum near the lower and upper boundaries of the allowed band.

Figure 2 (curve 2) shows the calculations results of the frequency dependences of the square of the modulus of the electromagnetic wave transmission coefficient S<sup>21</sup> in the presence of a disturbance in the photonic crystal in the form of, for example, a central layer of smaller thickness d6. In this case, a resonance feature with a transmission coefficient close to unit appears in the band gap. This feature is an impurity mode resonance whose location can be controlled by varying the dielectric layer thickness. The disturbance in a photonic crystal, as shown in [5], also leads to a change in the number of resonances formed in the allowed band and their positions on the frequency axis. When disturbance in the form of a central layer of a smaller thickness is created in a photonic crystal, the number of resonances in the allowed band decreases by one in comparison with the photonic crystal without disturbance, and the width of the forbidden band containing the impurity mode resonance increases substantially (Figure 2, curve 2). This effect is especially evident for a small number of photonic crystal periods. It should be noted that the width of the forbidden band containing an impurity mode resonance with a number of layers tending to infinity turns out to be approximately equal to the width of the forbidden band of a photonic crystal without disturbances.

The one-dimensional waveguide photonic crystals consisting of 7 and 11 layers completely filling a cross-section of rectangular waveguide of a 3-cm wavelength were used in the experiments. The odd layers were made of Al2O3 ceramics (ε<sup>r</sup> ¼ 9:6), and the even ones were made of Teflon (ε<sup>r</sup> ¼ 2:1). The length of odd segments is 1 mm and the length of even segments is 44 mm. The disturbance was created by changing the length of the central layer, which led to the appearance of an impurity mode resonance in the forbidden band of a photonic crystal. The length of the central disturbed (Teflon) layer was chosen equal to 14 mm. The frequency dependences of the reflection and transmission coefficients of microwave radiation interacting with a photonic crystal were measured with the Agilent PNA-L Network Analyzer N5230A. A comparison of the results of calculation and experiment shows their good quantitative coincidence.

In [19], the results of an investigation on the characteristics of a waveguide microwave photonic crystal made in the form of dielectric matrices with air inclusions are presented. Ceramics (Al2O3) with a large number of air inclusions and polystyrene were used as materials of dielectric layers. A waveguide photonic crystal consisting of 11 layers was studied in the frequency range 8–12 GHz (Figure 4a). Odd layers were made of Al2O3 ceramics (ε<sup>r</sup> ¼ 9:6), even layers were made of polystyrene (ε<sup>r</sup> ¼ 1:05). Thickness of odd lengths was dAl2O3 ¼ 1:0mm, even dfoam ¼ 13:0 mm. The layers completely filled the cross-section of the waveguide. In ceramic layers, a large number of air inclusions were created in the form of square-through holes, which form a periodic structure in the plane of the layer (Figure 4b).

On the basis of numerical modeling using the finite element method in the CAD ANSYS HFSS, the influence of the volume fraction of air inclusions on the amplitude-frequency characteristics of the transmission coefficient of a photonic crystal was studied. The volume fraction of air inclusions was controlled by changing the size of the holes ahole in the ceramics plates with a fixed amount of 36 in each of the plates. The simulation was carried out for three sizes of holes, equal to 0.75, 1.2 and 1.65 mm, which corresponds to a volume fraction of air inclusions equal to 8.5, 23 and 43%, respectively. The disturbance of the periodicity of the photonic structure was created by changing the thickness of the central (sixth) polystyrene layer. The thickness of the sixth modified (disturbed) polystyrene layer was 2.25, 2.75, 3.49 and 6.0 mm.

width of the forbidden band decreases. The Q-value of the resonances that form the allowed band, the so-called "allowed levels," as follows from the results of calculations and experi-

Figure 3. Dependences of the width of the forbidden and left allowed bands on the number of layers of a photonic crystal without disturbance: □, ●—theoretical and experimental values of the width of the left-allowed band, ◊, ▲—theoretical

Figure 2 (curve 2) shows the calculations results of the frequency dependences of the square of the modulus of the electromagnetic wave transmission coefficient S<sup>21</sup> in the presence of a disturbance in the photonic crystal in the form of, for example, a central layer of smaller thickness d6. In this case, a resonance feature with a transmission coefficient close to unit appears in the band gap. This feature is an impurity mode resonance whose location can be controlled by varying the dielectric layer thickness. The disturbance in a photonic crystal, as shown in [5], also leads to a change in the number of resonances formed in the allowed band and their positions on the frequency axis. When disturbance in the form of a central layer of a smaller thickness is created in a photonic crystal, the number of resonances in the allowed band decreases by one in comparison with the photonic crystal without disturbance, and the width of the forbidden band containing the impurity mode resonance increases substantially (Figure 2, curve 2). This effect is especially evident for a small number of photonic crystal periods. It should be noted that the width of the forbidden band containing an impurity mode resonance with a number of layers tending to infinity turns out to be approximately equal to

The one-dimensional waveguide photonic crystals consisting of 7 and 11 layers completely filling a cross-section of rectangular waveguide of a 3-cm wavelength were used in the experiments. The odd layers were made of Al2O3 ceramics (ε<sup>r</sup> ¼ 9:6), and the even ones were made of Teflon (ε<sup>r</sup> ¼ 2:1). The length of odd segments is 1 mm and the length of even segments is 44 mm. The disturbance was created by changing the length of the central layer, which led to the appearance of an impurity mode resonance in the forbidden band of a photonic crystal. The length of the central disturbed (Teflon) layer was chosen equal to 14 mm. The frequency dependences of the reflection and transmission coefficients of microwave radiation interacting with a photonic

ments, is maximum near the lower and upper boundaries of the allowed band.

and experimental values of the band gap width.

28 Emerging Waveguide Technology

the width of the forbidden band of a photonic crystal without disturbances.

As follows from the results of numerical calculation of the amplitude-frequency characteristics of the transmission coefficient of a photonic crystal by the finite-element method in the absence of the periodicity disturbance in the photonic structure, the increase in the volume fraction of air inclusions in the ceramic layers leads to the shift of the forbidden band of the photonic crystal toward shorter wavelengths and to the decrease of its depth. The results of the calculation of the photonic crystal amplitude-frequency characteristics by the finite element method in the presence of the disturbance of the photonic structure periodicity in the form of a central polystyrene layer of different thickness are shown in Figure 5 (solid curves).

The presence of the disturbance of the photonic structure periodicity in the form of the change of the thickness of the central (sixth) polystyrene layer led to the appearance of a transmission

Figure 4. Model of the photonic crystal: 1—layers of ceramics with square through holes, 2—polystyrene layers, 3 disturbance in the form of a polystyrene layer with changed thickness (a). The ceramics layer with square through holes (b).

peak in the forbidden band of the photonic crystal, the position of which is determined by the size of this disturbance.

photonic crystal (curves) calculated at the values of the effective permittivity εef found from the

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The dependence of the effective permittivity of the layers on the volume fraction of air inclusions, calculated from the solution of the inverse problem, coincides with the dependence described by Maxwell-Garnett ratio [20] with the error of 7.47%, the Bruggeman ratio [21] with

A distinctive feature of microwave photonic crystals is the ability to provide various functions necessary for the operation of microwave circuits [23–26] with a relatively small number of elements that make up a photonic crystal. A small number of elements forming microwave photonic crystals is associated with the need for compactness of the devices created on their basis. It is of scientific and practical interest to create multielement microwave photonic crystals characterized by small dimensions. To solve this kind of problem, it is possible to use as a photonic crystal the structure that excites higher type waves whose wavelengths are substantially shorter than the wavelength in the waveguide of the main type, as suggested by the authors in [27]. Therefore, the dimensions of devices at higher type waves become significantly smaller than similar devices on the main type of the wave. In this regard, they can be called low dimensional. The results of studies of waveguide photonic crystals in the form of alternating dielectric layers (even elements) of a photonic crystal and thin metal plates (odd elements) partially overlapping the waveguide cross-section are presented in [27]. Between the plates and the wide walls of the waveguide, there were gaps. Each of the plates created gap of the same width along the entire length of the plate. The gaps between the odd metal plates and the waveguide were created at one of the wide walls of the waveguide and the gaps between the even metal plates and the waveguide at the opposite wide wall of the waveguide. In this case, the gap is a source of higher type waves, forming in its vicinity the so-called near field. The structure of the investigated low-dimensional waveguide microwave photonic crystal is shown in Figure 7. The following materials were used as the dielectric material: polystyrene (ε<sup>r</sup> ¼ 1:02) and teflon (ε<sup>r</sup> ¼ 2:1). The dielectric layers completely filled the cross section (23 � 10 mm) of the waveguide. The metal plates 50 μm thick were made of aluminum. The

width of the gap S did not exceed one-tenth of the size of the waveguide narrow wall.

Figure 6. Amplitude-frequency characteristics of the transmission coefficients of a photonic crystal calculated using the values of the effective permittivity εef, found from the inverse problem solution, of ceramics layers with different fraction of air inclusions with the disturbed sixth layer of 6.0 mm, εef, rel. units: 1–9.6 (x<sup>1</sup> ¼ 0%), 2–8.44 (x<sup>1</sup> ¼ 8:5%), 3–6.78

the error of 4.50% and the Lichteneker ratio [22] with the error of 24.02%.

inverse problem solution.

(x<sup>1</sup> ¼ 23:0%), 4–4.82 (x<sup>1</sup> ¼ 43:0%).

Measurement of the amplitude-frequency characteristic of the transmittance of the investigated photonic crystal in the 3 cm wavelength range was carried out using the Agilent PNA-L Network Analyzer N5230A. The results of experimental studies of the amplitude-frequency characteristics of the transmittance of a photonic crystal without disturbance and for different thicknesses of the disturbed polystyrene layer d6foam are shown in Figure 5 (discrete curves). Comparison of the calculation and the experiment results indicates their good agreement.

The layers of investigated photonic crystals containing a large number of air inclusions can be considered as composite materials, which are dielectric matrices based on ceramics with fillers in the form of air inclusions. It is known that the dielectric properties of composite materials can be characterized by the value of the effective permittivity εef determined by the dielectric permittivities of the matrix ε<sup>1</sup> and filler ε<sup>2</sup> and their volume fractions. The results of the investigation on the possibility of describing the amplitude-frequency characteristics of the transmission coefficient of the investigated photonic crystal, using the model of the "effective" medium [20–22], are given in [19]. The photonic crystal consists of alternating homogeneous layers with effective dielectric permittivity and polystyrene. Its amplitude-frequency characteristic was calculated using a wave transfer matrix between regions with different values of the propagation constant of the electromagnetic wave, determined by the effective permittivity of the ceramic layers with air inclusions and the dielectric permittivity of the polystyrene.

To determine the effective permittivity εef, it is necessary to solve the inverse problem [16]. According to the frequency dependences of the transmittance of a photonic crystal consisting of periodically alternating polystyrene and ceramic layers with air inclusions, the inverse problem was solved using the least squares method.

Figure 6 shows the frequency dependences of the power transmission coefficient of the photonic crystal with a different fraction of air inclusion in ceramic layers with the disturbed sixth polystyrene layer of thickness d6foam ¼ 6:0 mm (circles) calculated using the finite element method in the CAD ANSYS HFSS and the frequency dependences of the transmission coefficients of a

Figure 5. Calculated (continuous curves) and experimental (discrete curves) amplitude-frequency characteristics of the transmittance of a photonic crystal for various volume fractions of air inclusions with disturbed sixth polystyrene layer of 6.0 mm. The volume fraction of inclusions: 1–0, 2–8.5, 3–23.0 and 4–43.0%.

photonic crystal (curves) calculated at the values of the effective permittivity εef found from the inverse problem solution.

peak in the forbidden band of the photonic crystal, the position of which is determined by the

Measurement of the amplitude-frequency characteristic of the transmittance of the investigated photonic crystal in the 3 cm wavelength range was carried out using the Agilent PNA-L Network Analyzer N5230A. The results of experimental studies of the amplitude-frequency characteristics of the transmittance of a photonic crystal without disturbance and for different thicknesses of the disturbed polystyrene layer d6foam are shown in Figure 5 (discrete curves). Comparison of the calculation and the experiment results indicates their good agreement.

The layers of investigated photonic crystals containing a large number of air inclusions can be considered as composite materials, which are dielectric matrices based on ceramics with fillers in the form of air inclusions. It is known that the dielectric properties of composite materials can be characterized by the value of the effective permittivity εef determined by the dielectric permittivities of the matrix ε<sup>1</sup> and filler ε<sup>2</sup> and their volume fractions. The results of the investigation on the possibility of describing the amplitude-frequency characteristics of the transmission coefficient of the investigated photonic crystal, using the model of the "effective" medium [20–22], are given in [19]. The photonic crystal consists of alternating homogeneous layers with effective dielectric permittivity and polystyrene. Its amplitude-frequency characteristic was calculated using a wave transfer matrix between regions with different values of the propagation constant of the electromagnetic wave, determined by the effective permittivity of the ceramic layers with air inclusions and the dielectric permittivity of the polystyrene.

To determine the effective permittivity εef, it is necessary to solve the inverse problem [16]. According to the frequency dependences of the transmittance of a photonic crystal consisting of periodically alternating polystyrene and ceramic layers with air inclusions, the inverse

Figure 6 shows the frequency dependences of the power transmission coefficient of the photonic crystal with a different fraction of air inclusion in ceramic layers with the disturbed sixth polystyrene layer of thickness d6foam ¼ 6:0 mm (circles) calculated using the finite element method in the CAD ANSYS HFSS and the frequency dependences of the transmission coefficients of a

Figure 5. Calculated (continuous curves) and experimental (discrete curves) amplitude-frequency characteristics of the transmittance of a photonic crystal for various volume fractions of air inclusions with disturbed sixth polystyrene layer of

problem was solved using the least squares method.

6.0 mm. The volume fraction of inclusions: 1–0, 2–8.5, 3–23.0 and 4–43.0%.

size of this disturbance.

30 Emerging Waveguide Technology

The dependence of the effective permittivity of the layers on the volume fraction of air inclusions, calculated from the solution of the inverse problem, coincides with the dependence described by Maxwell-Garnett ratio [20] with the error of 7.47%, the Bruggeman ratio [21] with the error of 4.50% and the Lichteneker ratio [22] with the error of 24.02%.

A distinctive feature of microwave photonic crystals is the ability to provide various functions necessary for the operation of microwave circuits [23–26] with a relatively small number of elements that make up a photonic crystal. A small number of elements forming microwave photonic crystals is associated with the need for compactness of the devices created on their basis. It is of scientific and practical interest to create multielement microwave photonic crystals characterized by small dimensions. To solve this kind of problem, it is possible to use as a photonic crystal the structure that excites higher type waves whose wavelengths are substantially shorter than the wavelength in the waveguide of the main type, as suggested by the authors in [27]. Therefore, the dimensions of devices at higher type waves become significantly smaller than similar devices on the main type of the wave. In this regard, they can be called low dimensional. The results of studies of waveguide photonic crystals in the form of alternating dielectric layers (even elements) of a photonic crystal and thin metal plates (odd elements) partially overlapping the waveguide cross-section are presented in [27]. Between the plates and the wide walls of the waveguide, there were gaps. Each of the plates created gap of the same width along the entire length of the plate. The gaps between the odd metal plates and the waveguide were created at one of the wide walls of the waveguide and the gaps between the even metal plates and the waveguide at the opposite wide wall of the waveguide. In this case, the gap is a source of higher type waves, forming in its vicinity the so-called near field. The structure of the investigated low-dimensional waveguide microwave photonic crystal is shown in Figure 7. The following materials were used as the dielectric material: polystyrene (ε<sup>r</sup> ¼ 1:02) and teflon (ε<sup>r</sup> ¼ 2:1). The dielectric layers completely filled the cross section (23 � 10 mm) of the waveguide. The metal plates 50 μm thick were made of aluminum. The width of the gap S did not exceed one-tenth of the size of the waveguide narrow wall.

Figure 6. Amplitude-frequency characteristics of the transmission coefficients of a photonic crystal calculated using the values of the effective permittivity εef, found from the inverse problem solution, of ceramics layers with different fraction of air inclusions with the disturbed sixth layer of 6.0 mm, εef, rel. units: 1–9.6 (x<sup>1</sup> ¼ 0%), 2–8.44 (x<sup>1</sup> ¼ 8:5%), 3–6.78 (x<sup>1</sup> ¼ 23:0%), 4–4.82 (x<sup>1</sup> ¼ 43:0%).

Based on numerical simulation, using the finite element method in the CAD ANSYS HFSS, the influence of the thickness of dielectric layers, the gap width and the number of layers of the photonic crystal structure on the amplitude-frequency characteristics of the transmission and reflection coefficients of a photonic crystal was investigated.

was described in [28]. A violation of the periodicity of a photonic crystal can be a different size of the central dielectric layer or a modified size of the capacitive gap of the diaphragm adjacent to the disturbed layer. In [28], the results of studies of the low-dimensional photonic crystals described above (the inset to Figure 8) are presented. The diaphragms of 50 μm thick were made of aluminum. The thickness of each dielectric layer in the photonic crystal without "disturbances" was 3 mm. Thus, the total longitudinal dimension of the crystal without breaking the periodicity was ~15 mm. Figure 8 shows the experimental and calculated using the ANSYS HFSS CAD the transmission coefficient S<sup>21</sup> of an eleven-layer photonic crystal without disturbance (curves 3, 4) and with disturbance (curves 1, 2) as a central dielectric (teflon, ε<sup>r</sup> ¼ 2:1) layer

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The measurements were carried out using the Agilent PNA-L Network Analyzer N5230A. Comparison of the calculations and measurements results indicates their qualitative agreement. The existing difference may be due to ignoring the attenuation in the waveguide walls,

As follows from the results shown in Figure 8, the introduction of the disturbance in the form of a changed size of the central dielectric layer led to the appearance of a defect mode resonance and a significant change in the width and depth of the "forbidden" band. At the same time, the position of the defect mode on the frequency scale essentially depends not only on the thickness of the "disturbed" dielectric layer but also on the value of the capacitive gap of

Periodic structures based on resonators as retarding systems for vacuum microwave devices and microwave filters have been described as early as in the 1960s [28, 29]. They were intended

Figure 8. Calculated (dashed and doted curves) and experimental (solid curves) amplitude-frequency characteristics of transmittance S<sup>12</sup> of 11-layer low-dimensional photonic crystal: curves 1, 2—with "disturbed" sixth central teflon layer of 1 mm; curves 3 and 4—without the periodicity disturbance. On the inset: a photonic crystal model: 5—rectangular

waveguide segment, 6—dielectric layers, 7—capacitive diaphragms.

with thickness reduced to 1 mm, with a fixed value of the gap S = 1 mm.

which is essential for higher types of waves.

the diaphragms.

The finite-element method was used to calculate the amplitude-frequency characteristic of the transmission S<sup>12</sup> and the reflection S<sup>11</sup> coefficients of a nine-layer photonic crystal consisting of five consistently alternating metal plates with gaps and four dielectric layers for different thicknesses of the dielectric layers h. It follows from the calculation results that the amplitudefrequency characteristic of the transmittance S<sup>12</sup> of the structure under study has a "band" character. The amplitude-frequency characteristic of such a photonic crystal consists of specific alternating "allowed" and "forbidden" bands. The frequency position of the resonance peaks of the coefficient S<sup>11</sup> corresponds to the position of the peaks of the coefficient S12. The decrease in the thickness h of the dielectric layers of the structure, as well as the increase in the width of the gap S, led to a shift in the amplitude-frequency characteristic of the photonic crystal toward shorter wavelengths and the increase in the width of the "allowed" band. At the same time, the width and depth of the "forbidden" band decreased. It was found that with the number of metal plates in the photonic crystal structure equal to m, the number of resonances on its amplitude-frequency characteristic is m1.

Comparison of the results of calculations and experimental studies of a photonic crystal created in accordance with the model described above indicates their good qualitative agreement.

The use of metal plates with gaps in the structure of a photonic crystal has made it possible to substantially reduce its longitudinal dimension to 12.25 mm, which is approximately five times smaller than the longitudinal dimension of a photonic crystal created on elements made of alternating layers of dielectrics with different permittivity.

The disturbance of periodicity in a low-dimensional microwave photonic crystal, as well as in the ordinary one, should lead to the appearance of a defect (impurity) mode resonance. The theoretical definition of the conditions for its appearance and their experimental realization

Figure 7. Model of a photonic crystal: 1—rectangular waveguide segment, 2—dielectric layers, 3—thin metal plates, S—gap width, h—dielectric layer thickness.

was described in [28]. A violation of the periodicity of a photonic crystal can be a different size of the central dielectric layer or a modified size of the capacitive gap of the diaphragm adjacent to the disturbed layer. In [28], the results of studies of the low-dimensional photonic crystals described above (the inset to Figure 8) are presented. The diaphragms of 50 μm thick were made of aluminum. The thickness of each dielectric layer in the photonic crystal without "disturbances" was 3 mm. Thus, the total longitudinal dimension of the crystal without breaking the periodicity was ~15 mm. Figure 8 shows the experimental and calculated using the ANSYS HFSS CAD the transmission coefficient S<sup>21</sup> of an eleven-layer photonic crystal without disturbance (curves 3, 4) and with disturbance (curves 1, 2) as a central dielectric (teflon, ε<sup>r</sup> ¼ 2:1) layer with thickness reduced to 1 mm, with a fixed value of the gap S = 1 mm.

Based on numerical simulation, using the finite element method in the CAD ANSYS HFSS, the influence of the thickness of dielectric layers, the gap width and the number of layers of the photonic crystal structure on the amplitude-frequency characteristics of the transmission and

The finite-element method was used to calculate the amplitude-frequency characteristic of the transmission S<sup>12</sup> and the reflection S<sup>11</sup> coefficients of a nine-layer photonic crystal consisting of five consistently alternating metal plates with gaps and four dielectric layers for different thicknesses of the dielectric layers h. It follows from the calculation results that the amplitudefrequency characteristic of the transmittance S<sup>12</sup> of the structure under study has a "band" character. The amplitude-frequency characteristic of such a photonic crystal consists of specific alternating "allowed" and "forbidden" bands. The frequency position of the resonance peaks of the coefficient S<sup>11</sup> corresponds to the position of the peaks of the coefficient S12. The decrease in the thickness h of the dielectric layers of the structure, as well as the increase in the width of the gap S, led to a shift in the amplitude-frequency characteristic of the photonic crystal toward shorter wavelengths and the increase in the width of the "allowed" band. At the same time, the width and depth of the "forbidden" band decreased. It was found that with the number of metal plates in the photonic crystal structure equal to m, the number of resonances on its

Comparison of the results of calculations and experimental studies of a photonic crystal created in accordance with the model described above indicates their good qualitative agreement.

The use of metal plates with gaps in the structure of a photonic crystal has made it possible to substantially reduce its longitudinal dimension to 12.25 mm, which is approximately five times smaller than the longitudinal dimension of a photonic crystal created on elements made

The disturbance of periodicity in a low-dimensional microwave photonic crystal, as well as in the ordinary one, should lead to the appearance of a defect (impurity) mode resonance. The theoretical definition of the conditions for its appearance and their experimental realization

Figure 7. Model of a photonic crystal: 1—rectangular waveguide segment, 2—dielectric layers, 3—thin metal plates, S—gap

reflection coefficients of a photonic crystal was investigated.

32 Emerging Waveguide Technology

amplitude-frequency characteristic is m1.

width, h—dielectric layer thickness.

of alternating layers of dielectrics with different permittivity.

The measurements were carried out using the Agilent PNA-L Network Analyzer N5230A. Comparison of the calculations and measurements results indicates their qualitative agreement. The existing difference may be due to ignoring the attenuation in the waveguide walls, which is essential for higher types of waves.

As follows from the results shown in Figure 8, the introduction of the disturbance in the form of a changed size of the central dielectric layer led to the appearance of a defect mode resonance and a significant change in the width and depth of the "forbidden" band. At the same time, the position of the defect mode on the frequency scale essentially depends not only on the thickness of the "disturbed" dielectric layer but also on the value of the capacitive gap of the diaphragms.

Periodic structures based on resonators as retarding systems for vacuum microwave devices and microwave filters have been described as early as in the 1960s [28, 29]. They were intended

Figure 8. Calculated (dashed and doted curves) and experimental (solid curves) amplitude-frequency characteristics of transmittance S<sup>12</sup> of 11-layer low-dimensional photonic crystal: curves 1, 2—with "disturbed" sixth central teflon layer of 1 mm; curves 3 and 4—without the periodicity disturbance. On the inset: a photonic crystal model: 5—rectangular waveguide segment, 6—dielectric layers, 7—capacitive diaphragms.

to be used as delay-line structures in these devices, providing the optimal interaction of the electron beam with the electromagnetic wave. The investigated structure consists of periodically located metal resonant diaphragms at a distance l from each other, deposited to a dielectric substrate (Figure 9) [30].

On the basis of numerical simulation using the finite element method in the CAD ANSYS HFSS, the influence of substrates with different dielectric permittivity on the microwave reflection and transmission coefficients of the structure was investigated. The amplitudefrequency characteristics of a photonic crystal made up of metal diaphragms deposited on a dielectric substrate with a through aperture for different widths of the aperture and the amplitude-frequency characteristics of a photonic crystal made up of diaphragms on dielectric substrates with apertures filled with a material with the permittivity ε<sup>2</sup> were analyzed. The increase in the width of the aperture, for a fixed permittivity of the substrate, causes the increase in the width and depth of the band gap. Moreover, the low-frequency edge of the band remains stationary around 9 GHz, and the extension occurs due to the shift from the high-frequency edge of the forbidden band to the high-frequency region. The same tendency is observed when the permittivity of insulator within the aperture increases.

The photonic crystal based on diaphragms without dielectric substrates consisted of 6 aluminum diaphragms of 10 μm fixed between two layers of 2 mm polystyrene placed in a rectangular waveguide. The width and height of the apertures of the diaphragms of the photonic crystal were chosen equal to 14 and 1 mm, respectively. This provided the appearance on the frequency dependences of the transmission and reflection coefficients (the dashed curves in Figure 10a and b, respectively) in the frequency range 8–12 GHz of one allowed and one forbidden bands. The same figures show the results of measurements of the amplitudefrequency characteristics of a photonic crystal with a disturbance of periodicity (solid curves) in the form of a modified distance L between the central diaphragms.

A comparison of the experimental dependences presented in Figure 10 with the results of the calculation of the amplitude-frequency characteristics of the photonic crystal based on diaphragms without dielectric substrates in the case of the disturbance presence and the disturbance absence with parameters corresponding to the experimental sample described above

Figure 10. Experimental amplitude-frequency characteristics of the transmission coefficient (a) and reflection (b) of a photonic crystal based on diaphragms without dielectric substrates in the case of the disturbance absence (curve 1) and the disturbance presence (curve 2). Distance between diaphragms l ¼ 27 mm, length of slots a<sup>1</sup> ¼ 14 mm, size of distur-

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The ability to control the amplitude-frequency characteristics of microwave photonic crystals opens the prospect of expanding their field of application. This possibility was considered, in particular, in [31]. The authors of [31] took the design of the microwave photonic crystal used in [32] as a filter. Polystyrene, ceramics Al2O3 (thickness of 1 mm) and polycrystalline yttriumiron garnet (YIG) (thickness of 1 mm) were used as the material forming the photonic crystal layer. The control of the amplitude-frequency characteristics of the investigated structures by magnetic field is provided by the increase in the real part of the ferrite magnetic susceptibility under increasing magnetic field. This leads to the increase in the microwave field concentration in the ferrite and to the increase in the phase shift of the wave as it passes through the ferrite

In [33], the possibility of creating the waveguide photonic crystal with a tunable frequency position of a transparency window associated with the periodicity disturbance in the photonic crystal and with attenuation in this window controlled by p-i-n diodes was shown. The 11 layer microwave photonic crystal designed for operation in a 3-cm wavelength range was created, consisting of 11 alternating layers of ceramics Al2O3 (ε<sup>r</sup> ¼ 9:6, thickness of 1 mm) and polystyrene (ε<sup>r</sup> ¼ 1:1, thickness of 1 mm). The disturbance of periodicity was provided by using polystyrene plate as the sixth layer. To provide control of the transmittance in the transparency band the p-i-n-diode array was used, which was placed into the waveguide in

plate and, as a consequence, to the shift of the amplitude-frequency characteristic.

indicates good qualitative and quantitative agreement.

bance in the central layer L ¼ 20 mm.

conjunction with the photonic crystal (Figure 11).

2.2. Control of the characteristics of microwave photonic crystals

Figure 9. The scheme of the microwave photonic crystal, where l is the distance between the diaphragms.

Figure 10. Experimental amplitude-frequency characteristics of the transmission coefficient (a) and reflection (b) of a photonic crystal based on diaphragms without dielectric substrates in the case of the disturbance absence (curve 1) and the disturbance presence (curve 2). Distance between diaphragms l ¼ 27 mm, length of slots a<sup>1</sup> ¼ 14 mm, size of disturbance in the central layer L ¼ 20 mm.

A comparison of the experimental dependences presented in Figure 10 with the results of the calculation of the amplitude-frequency characteristics of the photonic crystal based on diaphragms without dielectric substrates in the case of the disturbance presence and the disturbance absence with parameters corresponding to the experimental sample described above indicates good qualitative and quantitative agreement.

#### 2.2. Control of the characteristics of microwave photonic crystals

to be used as delay-line structures in these devices, providing the optimal interaction of the electron beam with the electromagnetic wave. The investigated structure consists of periodically located metal resonant diaphragms at a distance l from each other, deposited to a

On the basis of numerical simulation using the finite element method in the CAD ANSYS HFSS, the influence of substrates with different dielectric permittivity on the microwave reflection and transmission coefficients of the structure was investigated. The amplitudefrequency characteristics of a photonic crystal made up of metal diaphragms deposited on a dielectric substrate with a through aperture for different widths of the aperture and the amplitude-frequency characteristics of a photonic crystal made up of diaphragms on dielectric substrates with apertures filled with a material with the permittivity ε<sup>2</sup> were analyzed. The increase in the width of the aperture, for a fixed permittivity of the substrate, causes the increase in the width and depth of the band gap. Moreover, the low-frequency edge of the band remains stationary around 9 GHz, and the extension occurs due to the shift from the high-frequency edge of the forbidden band to the high-frequency region. The same tendency is

The photonic crystal based on diaphragms without dielectric substrates consisted of 6 aluminum diaphragms of 10 μm fixed between two layers of 2 mm polystyrene placed in a rectangular waveguide. The width and height of the apertures of the diaphragms of the photonic crystal were chosen equal to 14 and 1 mm, respectively. This provided the appearance on the frequency dependences of the transmission and reflection coefficients (the dashed curves in Figure 10a and b, respectively) in the frequency range 8–12 GHz of one allowed and one forbidden bands. The same figures show the results of measurements of the amplitudefrequency characteristics of a photonic crystal with a disturbance of periodicity (solid curves)

observed when the permittivity of insulator within the aperture increases.

in the form of a modified distance L between the central diaphragms.

Figure 9. The scheme of the microwave photonic crystal, where l is the distance between the diaphragms.

dielectric substrate (Figure 9) [30].

34 Emerging Waveguide Technology

The ability to control the amplitude-frequency characteristics of microwave photonic crystals opens the prospect of expanding their field of application. This possibility was considered, in particular, in [31]. The authors of [31] took the design of the microwave photonic crystal used in [32] as a filter. Polystyrene, ceramics Al2O3 (thickness of 1 mm) and polycrystalline yttriumiron garnet (YIG) (thickness of 1 mm) were used as the material forming the photonic crystal layer. The control of the amplitude-frequency characteristics of the investigated structures by magnetic field is provided by the increase in the real part of the ferrite magnetic susceptibility under increasing magnetic field. This leads to the increase in the microwave field concentration in the ferrite and to the increase in the phase shift of the wave as it passes through the ferrite plate and, as a consequence, to the shift of the amplitude-frequency characteristic.

In [33], the possibility of creating the waveguide photonic crystal with a tunable frequency position of a transparency window associated with the periodicity disturbance in the photonic crystal and with attenuation in this window controlled by p-i-n diodes was shown. The 11 layer microwave photonic crystal designed for operation in a 3-cm wavelength range was created, consisting of 11 alternating layers of ceramics Al2O3 (ε<sup>r</sup> ¼ 9:6, thickness of 1 mm) and polystyrene (ε<sup>r</sup> ¼ 1:1, thickness of 1 mm). The disturbance of periodicity was provided by using polystyrene plate as the sixth layer. To provide control of the transmittance in the transparency band the p-i-n-diode array was used, which was placed into the waveguide in conjunction with the photonic crystal (Figure 11).

Figure 11. The arrangement of the photonic crystal and the p-i-n-diode array.

The control voltage regulated in the range 0–700 mV was applied to the p-i-n-diodes matrix. The matrix of p-i-n-diodes, in the absence of bias voltage, introduces a weak perturbation into the photonic crystal, and its characteristics remain practically unchanged. With increasing bias voltage, this perturbation increases due to the enrichment of the i-region by the charge carriers, and specificity for the photonic crystal resonant transmission decreases. The experimental frequency dependences |S12|2 , arg S<sup>12</sup> and |S11|2 , arg S<sup>11</sup> are shown in Figure 12 for different bias voltage values on the p-i-n-diode with the thickness of the disturbed layer d<sup>6</sup> ¼ 5 mm. As it follows from the results in this figure, the use of the microwave photonic crystal makes it possible to create the microwave switch with electrically adjustable characteristics from �1.5 to �25 dB when the bias voltage on p-i-n diodes varies from 0 to 700 mV.

It is known that the ring-type structures demonstrate properties of a photonic crystal, such as the presence of forbidden and allowed bands. The band nature of the spectrum in such structures is provided due to multiple reflections from the inhomogeneity in the structure. Such devices in microstrip design are given in [34, 35]. The characteristics of such structures in waveguide design are given in [36]. The element of the type "metal pin with a gap" is used in them as inhomogeneity. This inhomogeneity provides the appearance of a resonant feature in the forbidden band of the system under investigation, called the defect mode resonance or "transparency window." Accordingly, the blocking peak may appear in the allowed band (passband). Figure 13 shows the design of a microwave element based on diaphragm, the system of coupled frame elements with "pin with a gap" type inhomogeneity located 20 mm to the right of the diaphragm and the semiconductor n-i-p-i-n-structure located in the gap between the pin and the frame element.

As it follows from the calculation results, the change in the conductivity of the control element from 10�<sup>3</sup> to 105 S/m leads to a change in the transmission coefficient at 9.44 GHz, corresponding to the blocking peak, in the range �36.79 to �1.01 dB. The system whose construction was described above was investigated experimentally. The bias voltage applied to the n-i-p-i-n structure in the 0–9 V range resulted in the change of the transmission coefficient from �25 to �1.5 dB at 9.644 GHz.

2.3. Application of microwave photonic crystals

for different values of the voltage on the p-i-n-diode, d<sup>6</sup> ¼ 5:5 mm.

Different directions of practical use of the unique characteristics of microwave photonic crystals were considered in [25]. One of the common names of microwave photonic crystals is structures with a forbidden band. At the same time, it is known that, in addition to the forbidden band (an analog of the band gap in semiconductors), microwave photonic crystals demonstrated the frequency band in which the wave propagates, practically without attenuation. The authors of [37] proposed to use this property to create waveguide broadband

Figure 12. Experimental dependences of the square of the modulus (a, b) and phase (b, d) of the transmission (a, b) and the reflection (c, d) coefficients of electromagnetic radiation in the region of the transparency window of a photonic crystal

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Figure 12. Experimental dependences of the square of the modulus (a, b) and phase (b, d) of the transmission (a, b) and the reflection (c, d) coefficients of electromagnetic radiation in the region of the transparency window of a photonic crystal for different values of the voltage on the p-i-n-diode, d<sup>6</sup> ¼ 5:5 mm.

#### 2.3. Application of microwave photonic crystals

The control voltage regulated in the range 0–700 mV was applied to the p-i-n-diodes matrix. The matrix of p-i-n-diodes, in the absence of bias voltage, introduces a weak perturbation into the photonic crystal, and its characteristics remain practically unchanged. With increasing bias voltage, this perturbation increases due to the enrichment of the i-region by the charge carriers, and specificity for the photonic crystal resonant transmission decreases. The experimental

bias voltage values on the p-i-n-diode with the thickness of the disturbed layer d<sup>6</sup> ¼ 5 mm. As it follows from the results in this figure, the use of the microwave photonic crystal makes it possible to create the microwave switch with electrically adjustable characteristics from �1.5 to

It is known that the ring-type structures demonstrate properties of a photonic crystal, such as the presence of forbidden and allowed bands. The band nature of the spectrum in such structures is provided due to multiple reflections from the inhomogeneity in the structure. Such devices in microstrip design are given in [34, 35]. The characteristics of such structures in waveguide design are given in [36]. The element of the type "metal pin with a gap" is used in them as inhomogeneity. This inhomogeneity provides the appearance of a resonant feature in the forbidden band of the system under investigation, called the defect mode resonance or "transparency window." Accordingly, the blocking peak may appear in the allowed band (passband). Figure 13 shows the design of a microwave element based on diaphragm, the system of coupled frame elements with "pin with a gap" type inhomogeneity located 20 mm to the right of the diaphragm and the semiconductor n-i-p-i-n-structure located in the gap

As it follows from the calculation results, the change in the conductivity of the control element from 10�<sup>3</sup> to 105 S/m leads to a change in the transmission coefficient at 9.44 GHz, corresponding to the blocking peak, in the range �36.79 to �1.01 dB. The system whose construction was described above was investigated experimentally. The bias voltage applied to the n-i-p-i-n structure in the 0–9 V range resulted in the change of the transmission coefficient from �25 to �1.5 dB

, arg S<sup>11</sup> are shown in Figure 12 for different

, arg S<sup>12</sup> and |S11|2

�25 dB when the bias voltage on p-i-n diodes varies from 0 to 700 mV.

Figure 11. The arrangement of the photonic crystal and the p-i-n-diode array.

frequency dependences |S12|2

36 Emerging Waveguide Technology

between the pin and the frame element.

at 9.644 GHz.

Different directions of practical use of the unique characteristics of microwave photonic crystals were considered in [25]. One of the common names of microwave photonic crystals is structures with a forbidden band. At the same time, it is known that, in addition to the forbidden band (an analog of the band gap in semiconductors), microwave photonic crystals demonstrated the frequency band in which the wave propagates, practically without attenuation. The authors of [37] proposed to use this property to create waveguide broadband

Figure 13. Model of the microwave element based on the diaphragm and the system of coupled frame elements containing inhomogeneity in the form of "pin with a gap" designs: 1—waveguide, 2—diaphragm, 3—aperture, 4—frame element, 5–7—inhomogeneities of "pin with a gap" type.

which the metal-dielectric structure was placed in a segment of a microwave photonic crystal that disturbed its periodicity. In the "forbidden" band of the photonic crystal, the transparency "window," that is sensitive to the sought parameters of the metal-dielectric structure, is seen.

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Figure 14. Experimental frequency dependence of VSWR in the range 25.95–37.50 GHz.

One of the most successful and promising methods for diagnosing materials and structures that allow measurements with high spatial resolution is near-field microwave microscopy [42–44]. A key element of the near-field microwave microscope is a probe with an aperture size much smaller than the wavelength of the microwave radiation. In near-field microwave microscopes, the field of a non-propagating wave type is used as a probing source [16, 42, 43]. This field that is formed if the central conductor of the coaxial line goes beyond the outer conductor [15]. The authors [13] called the microwave resonator connected to the probe as the main element of the near-field microwave microscope, which provides to a greater extent its

The increase in the sensitivity of the resonator to the perturbation introduced into it through the probe causes the increase in the sensitivity and resolution of the microwave microscope as

The design of the resonator in conjunction with the probe element for a near-field microwave microscope is described in [45]. In [46], the results of studying the possibility of using the onedimensional photonic crystal in which a tunable resonator, serving as a load, allows to control the resonance features in the reflection spectrum of a near-field microwave microscope probe were presented. As such a load, the authors of [46] chose the cylindrical resonator with the coupling element extending beyond the cavity, the end of which is used as a probe of a nearfield microscope to monitor the parameters of the dielectric plate with different values of the permittivity and the thickness of nanometer metal layers deposited on dielectric plates.

The general view of the probe of a near-field microwave microscope on the basis of a cylindrical microwave resonator with the coupling frame element and the one-dimensional photonic

The probe based on the cylindrical resonator with a coupling frame element was connected to a segment of the waveguide photonic crystal 9 with the disturbance of the periodicity. The

high sensitivity and resolution.

crystal is shown in Figure 15.

a whole.

matched loads. The problem of constructing such loads remains one of the topical problems in microwave radioelectronics at the present time. Matched microwave loads are widely used both independently and as elements of complex functional devices: directional couplers, summators, power meters, measuring bridges, microwave filters, and so on [38, 39].

The results of studies of frequency dependences of the reflection coefficients of electromagnetic radiation from one-dimensional photonic crystals containing, along with periodically changing dielectric filling, nanometer metal layers are presented in [9, 10]. Computer simulations were carried to determine the possibility of using one-dimensional photonic crystals for creating matched loads. When calculating the reflection S<sup>11</sup> and the propagation S<sup>21</sup> coefficients of an electromagnetic wave, the matrix of wave transfer between regions with different values of the propagation constant of the electromagnetic wave was used, as was described in detail in [10, 16, 17]. During computer simulation in [2], the possibility of creating waveguide matched loads in an 8- and 3-cm wavelength bands was demonstrated.

The authors of [40] showed, as a result of solving the optimization problem, including the choice of the surface resistance of a nanometer metal film, that it was possible to create the matched load of the described above type with the voltage standing wave ratio (VSWR) not more than 1.1 in the frequency bands 8.15–12.05, 12.05–17.44 and 17.44–25.95 GHz. The longitudinal dimensions of the loads were less than 14.5, 10.0, 9.0 mm, respectively. For the frequency range 25.95–37.50 GHz (Figure 14), the VSWR of the created load was less than 1.15 with a longitudinal dimension of 6.5 mm.

It is known that reflection and transmission spectra of electromagnetic radiation can be used to measure the thickness and electrical conductivity of a semiconductor in metal-semiconductor structures [15–17]. The sensitivity of these methods depends significantly on how much these spectra are changed under changing the values of the semiconductor parameters. The determination of these parameters from the results of measurements of spectral dependences is called the solution of the inverse problem. The authors of [41] proposed to use for measurements of the parameters of nanometer metal layers on insulating substrates waveguide photonic structures in

Figure 14. Experimental frequency dependence of VSWR in the range 25.95–37.50 GHz.

matched loads. The problem of constructing such loads remains one of the topical problems in microwave radioelectronics at the present time. Matched microwave loads are widely used both independently and as elements of complex functional devices: directional couplers,

Figure 13. Model of the microwave element based on the diaphragm and the system of coupled frame elements containing inhomogeneity in the form of "pin with a gap" designs: 1—waveguide, 2—diaphragm, 3—aperture, 4—frame

The results of studies of frequency dependences of the reflection coefficients of electromagnetic radiation from one-dimensional photonic crystals containing, along with periodically changing dielectric filling, nanometer metal layers are presented in [9, 10]. Computer simulations were carried to determine the possibility of using one-dimensional photonic crystals for creating matched loads. When calculating the reflection S<sup>11</sup> and the propagation S<sup>21</sup> coefficients of an electromagnetic wave, the matrix of wave transfer between regions with different values of the propagation constant of the electromagnetic wave was used, as was described in detail in [10, 16, 17]. During computer simulation in [2], the possibility of creating waveguide matched

The authors of [40] showed, as a result of solving the optimization problem, including the choice of the surface resistance of a nanometer metal film, that it was possible to create the matched load of the described above type with the voltage standing wave ratio (VSWR) not more than 1.1 in the frequency bands 8.15–12.05, 12.05–17.44 and 17.44–25.95 GHz. The longitudinal dimensions of the loads were less than 14.5, 10.0, 9.0 mm, respectively. For the frequency range 25.95–37.50 GHz (Figure 14), the VSWR of the created load was less than 1.15

It is known that reflection and transmission spectra of electromagnetic radiation can be used to measure the thickness and electrical conductivity of a semiconductor in metal-semiconductor structures [15–17]. The sensitivity of these methods depends significantly on how much these spectra are changed under changing the values of the semiconductor parameters. The determination of these parameters from the results of measurements of spectral dependences is called the solution of the inverse problem. The authors of [41] proposed to use for measurements of the parameters of nanometer metal layers on insulating substrates waveguide photonic structures in

summators, power meters, measuring bridges, microwave filters, and so on [38, 39].

loads in an 8- and 3-cm wavelength bands was demonstrated.

with a longitudinal dimension of 6.5 mm.

element, 5–7—inhomogeneities of "pin with a gap" type.

38 Emerging Waveguide Technology

which the metal-dielectric structure was placed in a segment of a microwave photonic crystal that disturbed its periodicity. In the "forbidden" band of the photonic crystal, the transparency "window," that is sensitive to the sought parameters of the metal-dielectric structure, is seen.

One of the most successful and promising methods for diagnosing materials and structures that allow measurements with high spatial resolution is near-field microwave microscopy [42–44]. A key element of the near-field microwave microscope is a probe with an aperture size much smaller than the wavelength of the microwave radiation. In near-field microwave microscopes, the field of a non-propagating wave type is used as a probing source [16, 42, 43]. This field that is formed if the central conductor of the coaxial line goes beyond the outer conductor [15]. The authors [13] called the microwave resonator connected to the probe as the main element of the near-field microwave microscope, which provides to a greater extent its high sensitivity and resolution.

The increase in the sensitivity of the resonator to the perturbation introduced into it through the probe causes the increase in the sensitivity and resolution of the microwave microscope as a whole.

The design of the resonator in conjunction with the probe element for a near-field microwave microscope is described in [45]. In [46], the results of studying the possibility of using the onedimensional photonic crystal in which a tunable resonator, serving as a load, allows to control the resonance features in the reflection spectrum of a near-field microwave microscope probe were presented. As such a load, the authors of [46] chose the cylindrical resonator with the coupling element extending beyond the cavity, the end of which is used as a probe of a nearfield microscope to monitor the parameters of the dielectric plate with different values of the permittivity and the thickness of nanometer metal layers deposited on dielectric plates.

The general view of the probe of a near-field microwave microscope on the basis of a cylindrical microwave resonator with the coupling frame element and the one-dimensional photonic crystal is shown in Figure 15.

The probe based on the cylindrical resonator with a coupling frame element was connected to a segment of the waveguide photonic crystal 9 with the disturbance of the periodicity. The

Figure 15. Probe of a near-field microwave microscope based on a cylindrical microwave cavity 1 with a communication coupler 6 and a one-dimensional photonic crystal 9. Insert A is the frame coupling element. Insert B is the cylindrical microwave resonator with the coupling frame element and the measured sample 10.

The inset of Figure 17 shows the dependences of the microwave reflection coefficient measured at various fixed frequencies (10.77908, 10.77895 and 10.77850 GHz) in the vicinity of the reflection coefficient minimum on the thickness of the nanometer metal layer on ceramics

Figure 16. Frequency dependences of the microwave reflection coefficient in the vicinity of the resonance frequency at a fixed gap equal to 18 μm between the probe and samples with different dielectric permittivity. Curve 1 corresponds to the absence of the measured sample (ε<sup>r</sup> ¼ 1), 2—teflon (ε<sup>r</sup> ¼ 2:0), 3—paper-based laminate (ε<sup>r</sup> ¼ 2:5), 4—textolite; (ε<sup>r</sup> ¼ 3:4),

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In [47], it is proposed to implement a method for simultaneous measurement of the substrate thickness of a semiconductor structure, the thickness and conductivity of a heavily doped epitaxial layer, the mobility of free charge carriers in this layer using a one-dimensional microwave photonic crystal. The one-dimensional waveguide photonic crystal consisted of 11 layers, forming a structure of periodically alternating elements, each of which included two

Figure 17. Frequency dependences of the microwave reflection coefficient in the vicinity of the resonance frequency at a fixed gap equal to 18 μm between the probe and the samples under study with different thicknesses of the nanometer metal layer Cr: (1) without metallization, (2) d ¼ 3 nm, (3) d ¼ 7 nm, (4) d ¼ 9 nm, (5) d ¼ 13 nm, (6) d ¼ 30 nm, and (7) d ¼ 180 nm.

Al2O<sup>3</sup> plates placed at a fixed distance near the tip of the probe.

5—ceramics Al2O<sup>3</sup> (ε<sup>r</sup> ¼ 9:6), 6—silicon (ε<sup>r</sup> ¼ 11:7).

one-dimensional waveguide photonic crystal consisting of 11 layers was used in the frequency range 8–12 GHz. The odd layers were made of ceramics Al2O3 (ε<sup>r</sup> ¼ 9:6), even layers were made of teflon (ε<sup>r</sup> ¼ 2:1). The length of odd segments was 1 mm, of even segments varied in the range from 7 to 14 mm. The disturbance of periodicity was created by changing the length of the sixth central layer. The length of the disturbed sixth (teflon) layer varied in the range from 3 to 4 mm.

The tip of the probe approaching to the sample caused abrupt change in the input impedance of the probe and the reflection coefficient of the microwave wave from the probe. The magnitude of its change depends on the parameters of the test sample, such as electrical conductivity, permittivity, thickness.

Figure 16 shows the results of measurements of the frequency dependence of the microwave reflection coefficient in the vicinity of the resonant frequency at a fixed gap (18 μm) between the probe and the samples under study with different dielectric permittivity. Figure 16 shows the dependences of the microwave reflection coefficient measured at various fixed frequencies in the vicinity of the reflection coefficient minimum on the dielectric permittivity of the samples placed at a fixed distance near the tip of the probe.

The investigated resonance system can also be used to measure samples in the form of dielectric plates with nanometer metal layers of different thicknesses. Figure 17 shows the results of measurements of the frequency dependences of the microwave reflection coefficient in the vicinity of the resonance frequency at a fixed gap (18 μm) between the probe and the samples under study with different thicknesses of the nanometer metal layer (Cr) in the metalinsulator structure. The thickness of the deposited nanometer metal layers was measured using the Agilent Technologies AFM5600 atomic force microscope.

Figure 16. Frequency dependences of the microwave reflection coefficient in the vicinity of the resonance frequency at a fixed gap equal to 18 μm between the probe and samples with different dielectric permittivity. Curve 1 corresponds to the absence of the measured sample (ε<sup>r</sup> ¼ 1), 2—teflon (ε<sup>r</sup> ¼ 2:0), 3—paper-based laminate (ε<sup>r</sup> ¼ 2:5), 4—textolite; (ε<sup>r</sup> ¼ 3:4), 5—ceramics Al2O<sup>3</sup> (ε<sup>r</sup> ¼ 9:6), 6—silicon (ε<sup>r</sup> ¼ 11:7).

The inset of Figure 17 shows the dependences of the microwave reflection coefficient measured at various fixed frequencies (10.77908, 10.77895 and 10.77850 GHz) in the vicinity of the reflection coefficient minimum on the thickness of the nanometer metal layer on ceramics Al2O<sup>3</sup> plates placed at a fixed distance near the tip of the probe.

one-dimensional waveguide photonic crystal consisting of 11 layers was used in the frequency range 8–12 GHz. The odd layers were made of ceramics Al2O3 (ε<sup>r</sup> ¼ 9:6), even layers were made of teflon (ε<sup>r</sup> ¼ 2:1). The length of odd segments was 1 mm, of even segments varied in the range from 7 to 14 mm. The disturbance of periodicity was created by changing the length of the sixth central layer. The length of the disturbed sixth (teflon) layer varied in the range

Figure 15. Probe of a near-field microwave microscope based on a cylindrical microwave cavity 1 with a communication coupler 6 and a one-dimensional photonic crystal 9. Insert A is the frame coupling element. Insert B is the cylindrical

The tip of the probe approaching to the sample caused abrupt change in the input impedance of the probe and the reflection coefficient of the microwave wave from the probe. The magnitude of its change depends on the parameters of the test sample, such as electrical conductivity,

Figure 16 shows the results of measurements of the frequency dependence of the microwave reflection coefficient in the vicinity of the resonant frequency at a fixed gap (18 μm) between the probe and the samples under study with different dielectric permittivity. Figure 16 shows the dependences of the microwave reflection coefficient measured at various fixed frequencies in the vicinity of the reflection coefficient minimum on the dielectric permittivity of the

The investigated resonance system can also be used to measure samples in the form of dielectric plates with nanometer metal layers of different thicknesses. Figure 17 shows the results of measurements of the frequency dependences of the microwave reflection coefficient in the vicinity of the resonance frequency at a fixed gap (18 μm) between the probe and the samples under study with different thicknesses of the nanometer metal layer (Cr) in the metalinsulator structure. The thickness of the deposited nanometer metal layers was measured

samples placed at a fixed distance near the tip of the probe.

microwave resonator with the coupling frame element and the measured sample 10.

using the Agilent Technologies AFM5600 atomic force microscope.

from 3 to 4 mm.

permittivity, thickness.

40 Emerging Waveguide Technology

In [47], it is proposed to implement a method for simultaneous measurement of the substrate thickness of a semiconductor structure, the thickness and conductivity of a heavily doped epitaxial layer, the mobility of free charge carriers in this layer using a one-dimensional microwave photonic crystal. The one-dimensional waveguide photonic crystal consisted of 11 layers, forming a structure of periodically alternating elements, each of which included two

Figure 17. Frequency dependences of the microwave reflection coefficient in the vicinity of the resonance frequency at a fixed gap equal to 18 μm between the probe and the samples under study with different thicknesses of the nanometer metal layer Cr: (1) without metallization, (2) d ¼ 3 nm, (3) d ¼ 7 nm, (4) d ¼ 9 nm, (5) d ¼ 13 nm, (6) d ¼ 30 nm, and (7) d ¼ 180 nm.

layers, was considered. The measured structure, placed on the boundary of the disturbed central teflon layer and the next ceramics Al2O<sup>3</sup> layer, was oriented in two ways relative to the direction of the electromagnetic wave propagation. The location of the sample inside the disturbed layer and its orientation relative to the disturbed layer in the photonic crystal are shown in Figure 18 (configuration 1 and 2).

The investigated samples were epitaxial arsenide-gallium structures of thickness ts ¼ t þ tsub consisting of an epitaxial layer with a thickness t and electrical conductivity σ and a semiinsulating substrate with thickness tsub. The sought values of the investigated sample parameters are determined by the numerical method as a result of solving the system of equations:

$$\frac{\partial S(t\_{\rm sub}, t, \sigma)}{\partial t} = 0; \frac{\partial S(t\_{\rm sub}, t, \sigma)}{\partial t\_{\rm sub}} = 0; \frac{\partial S(t\_{\rm sub}, t, \sigma)}{\partial \sigma} = 0 \tag{10}$$

global minimum in the coordinate space ð Þ tsub; t; σ; S tð Þ ; σ , and the contour maps (Figure 19) are characterized by the presence of closed trajectories near the minimum. This fact determines possibility to obtain the thickness and electrical conductivity of the semiconductor layer from

Figure 20. Arrangement of the photonic crystal and epitaxial semiconductor structure in the waveguide. 1—ceramics Al2O3 layer, 2—teflon layer, 3—disturbed teflon layer, 4—tested gallium arsenide structure, which includes: 5—high-

resistance substrate, 6—heavily doped semiconductor layer. N and S are poles of the electromagnet.

Figure 19. The form of the residual function in space and the contour maps in the planes of the sought parameters a), b) — tð Þ sub; σ for the gallium arsenide sample with an epitaxial layer of thickness t ¼ 13:14 μm and electrical conductivity

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the solution of the system of equations (Eq. (9)).

<sup>σ</sup> <sup>¼</sup> <sup>71</sup>:73 Om�<sup>1</sup> <sup>m</sup>�<sup>1</sup> grown on a high-resistivity substrate of thickness <sup>t</sup>sub <sup>¼</sup> <sup>480</sup>:<sup>3</sup> <sup>μ</sup>m.

The residual function S tð Þ sub; t; σ defined by expression.

$$S(t\_{\rm sub}, t, \sigma) = \sum\_{i=1}^{K} \left( \begin{aligned} & \left( \left| S\_{211}(\omega\_{i}, \ t\_{\rm sub}, t, \sigma) \right|^{2} - \left| S\_{211\,\rm exp} \right|^{2} \right)^{2} + \left( \left| S\_{111}(\omega\_{i}, \ t\_{\rm sub}, t, \sigma) \right|^{2} - \left| S\_{111\,\rm exp} \right|^{2} \right)^{2} + \\ & + \left( \left| S\_{212}(\omega\_{i}, \ t\_{\rm sub}, t, \sigma) \right|^{2} - \left| S\_{212\,\rm exp} \right|^{2} \right)^{2} + \left( \left| S\_{112}(\omega\_{i}, t\_{\rm sub}, \ t, \sigma) \right|^{2} - \left| S\_{112\,\rm exp} \right|^{2} \right)^{2} \right) \end{aligned} \tag{11}$$

for the case when the thickness of the heavily doped GaAs layer was 13.14 μm, its electrical conductivity <sup>σ</sup> <sup>¼</sup> <sup>71</sup>:73 Om�<sup>1</sup> <sup>m</sup>�<sup>1</sup> , and the thickness of the substrate tsub ¼ 480:3 μm, has a

Figure 18. Arrangement of the semiconductor structure relative to the disturbed layer in the waveguide microwave 8 photonic crystal: (1) heavily doped semiconductor layer, (2) high-resistance substrate, (3) disturbed central layer, (4) and (5) periodically alternating layers with different values of permittivity.

layers, was considered. The measured structure, placed on the boundary of the disturbed central teflon layer and the next ceramics Al2O<sup>3</sup> layer, was oriented in two ways relative to the direction of the electromagnetic wave propagation. The location of the sample inside the disturbed layer and its orientation relative to the disturbed layer in the photonic crystal are

The investigated samples were epitaxial arsenide-gallium structures of thickness ts ¼ t þ tsub consisting of an epitaxial layer with a thickness t and electrical conductivity σ and a semiinsulating substrate with thickness tsub. The sought values of the investigated sample parameters are determined by the numerical method as a result of solving the system of equations:

> ∂S tð Þ sub; t; σ ∂tsub

> > � � �

> > > � � �

for the case when the thickness of the heavily doped GaAs layer was 13.14 μm, its electrical

Figure 18. Arrangement of the semiconductor structure relative to the disturbed layer in the waveguide microwave 8 photonic crystal: (1) heavily doped semiconductor layer, (2) high-resistance substrate, (3) disturbed central layer, (4) and

¼ 0;

∂S tð Þ sub; t; σ

<sup>þ</sup> <sup>S</sup><sup>111</sup> <sup>ω</sup><sup>i</sup> j j ð Þ ; <sup>t</sup>sub; <sup>t</sup>; <sup>σ</sup> <sup>2</sup> � <sup>S</sup><sup>11</sup>i1 exp

, and the thickness of the substrate tsub ¼ 480:3 μm, has a

<sup>þ</sup> <sup>S</sup><sup>112</sup> <sup>ω</sup>i; <sup>t</sup> j j ð Þ sub; <sup>t</sup>; <sup>σ</sup> <sup>2</sup> � <sup>S</sup><sup>11</sup>i2 exp

<sup>2</sup> � �<sup>2</sup>

<sup>2</sup> � �<sup>2</sup>

<sup>∂</sup><sup>σ</sup> <sup>¼</sup> <sup>0</sup> (10)

� � �

> � � �

þ

1

CCCA, (11)

� � �

> � � �

shown in Figure 18 (configuration 1 and 2).

S tð Þ¼ sub; <sup>t</sup>; <sup>σ</sup> <sup>X</sup><sup>K</sup>

42 Emerging Waveguide Technology

i¼1

0

BBB@

conductivity <sup>σ</sup> <sup>¼</sup> <sup>71</sup>:73 Om�<sup>1</sup> <sup>m</sup>�<sup>1</sup>

∂S tð Þ sub; t; σ

The residual function S tð Þ sub; t; σ defined by expression.

(5) periodically alternating layers with different values of permittivity.

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>0</sup>;

<sup>S</sup><sup>211</sup> <sup>ω</sup><sup>i</sup> j j ð Þ ; <sup>t</sup>sub; <sup>t</sup>; <sup>σ</sup> <sup>2</sup> � <sup>S</sup><sup>21</sup>i1 exp

<sup>þ</sup> <sup>S</sup><sup>212</sup> <sup>ω</sup>i; <sup>t</sup> j j ð Þ sub; <sup>t</sup>; <sup>σ</sup> <sup>2</sup> � <sup>S</sup><sup>21</sup>i2 exp

<sup>2</sup> � �<sup>2</sup>

<sup>2</sup> � �<sup>2</sup>

� � �

> � � �

Figure 19. The form of the residual function in space and the contour maps in the planes of the sought parameters a), b) — tð Þ sub; σ for the gallium arsenide sample with an epitaxial layer of thickness t ¼ 13:14 μm and electrical conductivity <sup>σ</sup> <sup>¼</sup> <sup>71</sup>:73 Om�<sup>1</sup> <sup>m</sup>�<sup>1</sup> grown on a high-resistivity substrate of thickness <sup>t</sup>sub <sup>¼</sup> <sup>480</sup>:<sup>3</sup> <sup>μ</sup>m.

global minimum in the coordinate space ð Þ tsub; t; σ; S tð Þ ; σ , and the contour maps (Figure 19) are characterized by the presence of closed trajectories near the minimum. This fact determines possibility to obtain the thickness and electrical conductivity of the semiconductor layer from the solution of the system of equations (Eq. (9)).

Figure 20. Arrangement of the photonic crystal and epitaxial semiconductor structure in the waveguide. 1—ceramics Al2O3 layer, 2—teflon layer, 3—disturbed teflon layer, 4—tested gallium arsenide structure, which includes: 5—highresistance substrate, 6—heavily doped semiconductor layer. N and S are poles of the electromagnet.

To determine the mobility of free charge carriers of semiconductor heavily doped layers, the investigated epitaxial semiconductor structure was placed in the E plane at the center of the cross section of a rectangular waveguide after a waveguide photonic crystal. The magnetic induction vector of the magnetic field B ! was directed perpendicular to the narrow walls of the waveguide (Figure 20).

Acknowledgements

Author details

References

Dmitry Usanov and Alexander Skripal \*

the Solid State. 1962;8:2265-2267

1997;386:143-146. DOI: 10.1038/386143a0

Saratov State University, Russia

10.1109/22.910561

\*Address all correspondence to: skripala\_v@info.sgu.ru

The research was carried out with the financial support of the Ministry of Education and

Photonic Crystal Waveguides

45

http://dx.doi.org/10.5772/intechopen.76797

[1] Keldysh LV. The influence of ultrasound on the electronic spectrum of a crystal. Physics of

[2] Esaki L, Tsu R. Superlattice and negative differential conductivity in semiconductors. IBM Journal of Research and Development. 1970;14:61-65. DOI: 10.1147/rd.141.0061

[3] Joannopulos J, Villeneuve PR, Fan S. Photonic crystals: Putting a new twist on light. Nature.

[4] Yablonovitch E. Inhibited spontaneous emission in solid-state physics and electronics. Physical Review Letters. 1987;58:2059-2063. DOI: 10.1103/PhysRevLett.58.2059

[5] John S. Strong localisation of fotons in certain disordered dielectric superlattices. Physical

[8] Tae-Yeoul, Chang K. Uniplanar one-dimensional photonic-bandgap structures and resonators. IEEE Transactions on Microwave Theory and Techniques. 2001;49:549-553. DOI:

[9] Usanov D, Skripal Al, Abramov A, Bogolyubov A, Skvortsov V, Merdanov M. Measurement of the metal nanometer layer parameters on dielectric substrates using photonic crystals based on the waveguide structures with controlled irregularity in the microwave band. In: Proceedings of 37th European Microwave Conference; 8–12 October 2007;

Review Letters. 1987;58:2486-2491. DOI: 10.1103/PhysRevLett.58.2486

Munich, Germany. pp. 198-201. DOI: 10.1109/EUMC.2007.4405160

[6] Zaitsev DF. Nanophotonics and Its Application. Moscow: Acteon; 2012. 445p

[7] Silin RA, Sazonov VP. Delay-Line Systems. Moscow: Soviet Radio; 1966. 628p

Science of the Russian Federation (state task №8.7628.2017/BCH).

To find the mobility of free charge carriers by frequency dependences S21ð Þ ω and S11ð Þ ω , the least squares method was used, in which the mobility value μ corresponding to the minimum value of the sum of the squares of the differences between the calculated <sup>S</sup><sup>21</sup> <sup>ω</sup>; <sup>μ</sup> <sup>2</sup> and <sup>S</sup><sup>11</sup> <sup>ω</sup>; <sup>μ</sup> <sup>2</sup> and the experimental <sup>S</sup>21exp 2 and S11exp <sup>2</sup> values of the squares of the modules of the transmission and reflection coefficients measured under the influence of the magnetic field and without it. The sought mobility values for two measured structures with thicknesses of heavily doped semiconductor layers of t ¼ 2:17 and 13:14 μm were determined numerically and amounted to 0.591 m2 /(V\*s) and 0.72 m2 /(V\*s), respectively.

### 3. Conclusions

This chapter presents the results of theoretical and experimental studies of one-dimensional microwave photonic crystals based on rectangular waveguides characterized by the presence of forbidden and allowed bands for the propagation of electromagnetic waves. Methods for describing the electrodynamic characteristics of photonic crystals and their relationship with the parameters of periodic structures filling the waveguides have been presented. The change in the width of the allowed and forbidden bands of a photonic crystal has been described when a defect mode occurs because of the creation of the disturbance of the periodicity of the photonic crystal structure. The change in the amplitude-frequency characteristics of microwave photonic crystals has been considered. The types of disturbances and defect modes have been described. The results of the investigation on the characteristics of waveguide microwave photonic crystals created in the form of dielectric matrices with air inclusions have been presented. It has been shown that the layers of the investigated photonic crystals containing a large number of air inclusions can be considered as composite materials. New types of lowdimensional microwave waveguide photonic crystals containing periodically alternating elements that are the source of higher type waves have been described. The possibility of effective control of the amplitude-frequency characteristics of microwave photonic crystals by electric and magnetic fields has been analyzed. The examples of applications of waveguide microwave photonic crystals such as methods for measuring the parameters of materials and semiconductor nanostructures that play the role of periodicity disturbance of the microwave photonic crystals; resonators of near-field microwave microscopes; small-sized broadband matched loads of centimeter and millimeter wavelengths ranges based on microwave photonic crystals have been given.

### Acknowledgements

To determine the mobility of free charge carriers of semiconductor heavily doped layers, the investigated epitaxial semiconductor structure was placed in the E plane at the center of the cross section of a rectangular waveguide after a waveguide photonic crystal. The magnetic

To find the mobility of free charge carriers by frequency dependences S21ð Þ ω and S11ð Þ ω , the least squares method was used, in which the mobility value μ corresponding to the minimum value of the sum of the squares of the differences between the calculated <sup>S</sup><sup>21</sup> <sup>ω</sup>; <sup>μ</sup>

> and S11exp

the transmission and reflection coefficients measured under the influence of the magnetic field and without it. The sought mobility values for two measured structures with thicknesses of heavily doped semiconductor layers of t ¼ 2:17 and 13:14 μm were determined numerically

This chapter presents the results of theoretical and experimental studies of one-dimensional microwave photonic crystals based on rectangular waveguides characterized by the presence of forbidden and allowed bands for the propagation of electromagnetic waves. Methods for describing the electrodynamic characteristics of photonic crystals and their relationship with the parameters of periodic structures filling the waveguides have been presented. The change in the width of the allowed and forbidden bands of a photonic crystal has been described when a defect mode occurs because of the creation of the disturbance of the periodicity of the photonic crystal structure. The change in the amplitude-frequency characteristics of microwave photonic crystals has been considered. The types of disturbances and defect modes have been described. The results of the investigation on the characteristics of waveguide microwave photonic crystals created in the form of dielectric matrices with air inclusions have been presented. It has been shown that the layers of the investigated photonic crystals containing a large number of air inclusions can be considered as composite materials. New types of lowdimensional microwave waveguide photonic crystals containing periodically alternating elements that are the source of higher type waves have been described. The possibility of effective control of the amplitude-frequency characteristics of microwave photonic crystals by electric and magnetic fields has been analyzed. The examples of applications of waveguide microwave photonic crystals such as methods for measuring the parameters of materials and semiconductor nanostructures that play the role of periodicity disturbance of the microwave photonic crystals; resonators of near-field microwave microscopes; small-sized broadband matched loads of centimeter and millimeter wavelengths ranges based on microwave photonic crystals

/(V\*s), respectively.

was directed perpendicular to the narrow walls of the

<sup>2</sup> values of the squares of the modules of

 <sup>2</sup> and

!

 2

 

/(V\*s) and 0.72 m2

induction vector of the magnetic field B

<sup>2</sup> and the experimental <sup>S</sup>21exp

waveguide (Figure 20).

44 Emerging Waveguide Technology

3. Conclusions

have been given.

and amounted to 0.591 m2

<sup>S</sup><sup>11</sup> <sup>ω</sup>; <sup>μ</sup>

The research was carried out with the financial support of the Ministry of Education and Science of the Russian Federation (state task №8.7628.2017/BCH).

### Author details

Dmitry Usanov and Alexander Skripal \*

\*Address all correspondence to: skripala\_v@info.sgu.ru

Saratov State University, Russia

### References


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[18] Usanov DA, Nikitov SA, Skripal' AV, Ponomarev DV. Resonance features of the allowed and forbidden bands of the microwave photonic crystal with periodicity defects. Journal of Communications Technology and Electronics. 2013;58:1035-1040. DOI: 10.1134/S1064

[19] Usanov DA, Skripal AV, Merdanov MK, Gorlitskii VO. Dielectric matrices with air cavities as a waveguide photonic crystal. Technical Physics. 2016;61:221-226. DOI: 10.1134/

[20] Maxwell-Garnett JC. Colours in metal glasses and in metallic films. Philosophical Trans-

[21] Bruggeman DAG. Dielectric constant and conductivity of mixtures of isotropic materials.

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**Chapter 3**

**Provisional chapter**

**High-Gain Amplifier Module Integrating a Waveguide**

**Waveguide into the Module Case for Millimeter Wave** 

**High-Gain Amplifier Module Integrating a** 

DOI: 10.5772/intechopen.76622

**into the Module Case for Millimeter Wave Applications**

A high-gain amplifier module with integrated waveguide (WG) has been presented for millimeter wave applications. In order to improve the isolation between the amplification stages in the multi-stage amplifier module, an isolated WG is integrated into the module case. It is possible to effectively suppress the oscillation occurring in the high gain stage. Microstrip line (MSL)-to-WG transitions are designed and fabricated on a 5 mil thick RT5880 substrate for interconnection of the isolated WG, input/output WG and amplifier PCB in a cascaded two-stage high gain amplifier module. The transition loss of −0.44 dB is obtained at 40 GHz and return-loss (S11) bandwidth below −10 dB is from 34.1 to 50 GHz. The fabricated high-gain amplifier module shows a high gain over 39.7 dB from 38 to

**Keywords:** waveguide, isolation, microstrip-to-waveguide transition, amplifier module

Recently, millimeter-wave (mm-wave) frequency bands have attracted attention as various applications such as radar sensors [1–3] as well as wireless communication. In particular, due to very widely available bandwidth, frequency bands for a variety of wireless communication applications such as point-to-point wireless communications, radio-on-fiber (RoF) links [4],

One of the key issues for the commercialization of mm-wave systems is reproducible and inexpensive packaging technology in addition to active integrated circuit (IC) technology.

5G cellular wireless networks [5], etc. are shifting to the mm-wave frequency band.

41 GHz. At 38.7 GHz, its maximum gain of 44.25 dB is achieved.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Young Chul Lee

Young Chul Lee

**Abstract**

**1. Introduction**

**Applications**

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.76622


#### **High-Gain Amplifier Module Integrating a Waveguide into the Module Case for Millimeter Wave Applications High-Gain Amplifier Module Integrating a Waveguide into the Module Case for Millimeter Wave Applications**

DOI: 10.5772/intechopen.76622

#### Young Chul Lee Young Chul Lee

[37] Usanov DA, Skripal AV, Abramov AV, Bogolyubov AS, Skvortsov VS, Merdanov MK. Broadband waveguide matched loads based on photonic crystals with nanometer metal

[38] Helszajn J. Passive and Active Microwave Circuits. New York, Chichester, Brisbane, Toronto:

[39] Lee KA, Guo Y, Stimson PA, Potter KA, Jung-Chih C, Rutledge DB. Thin-film powerdensity meter for millimeter wavelengths. IEEE Transactions on Antennas and Propaga-

[40] Usanov DA, Meshchanov VP, Skripal' AV, Popova NF, Ponomarev DV, Merdanov MK. Centimeter- and millimeter-wavelength matched loads based on microwave photonic

[41] Usanov DA, Skripal AV, Abramov AV, Bogolyubov AS, Skvortsov VS, Merdanov MK. Using waveguide photonic structures to measure the parameters of nanometer metal layers on insulating substrates. Electronics. 2007;6:25-32. In: Proceedings of the Universities [42] Anlage SM, Steinhauer DE, Feenstra BJ, Vlahacos CP, Wellstood FC. Near-field microwave microscopy of materials properties. Microwave Superconductivity. 2001;375:239-

[43] Usanov DA. Nearfield Scanning Microwave Microscopy and Its Applications. Saratov:

[44] Usanov DA, Gorbatov SS. Near Field Effects in Electrodynamic Systems with Inhomogeneities and their Application in Microwave Technology. Saratov: Saratov University Pub-

[45] Usanov DA, Skripal' AV, Frolov AP, Nikitov SA. Microwave near-field microscope based on a photonic crystal with a cavity and a controlled coupling element as a probe. Journal of Communications Technology and Electronics. 2013;12:1130-1136. DOI: 10.1134/S1064

[46] Usanov DА, Nikitov SA, Skripal AV, Orlov VE, Frolov AP. The patent of the Russian Federation for utility model 144 869 U1 IPC The device for measuring the permittivity of plates and thicknesses of nanometer conductive films. G01N 22/00 B82B 1/00 Application:

[47] Usanov DA, Nikitov SA, Skripal' AV, Ponomarev DV, Latysheva EV. Multiparametric measurements of epitaxial semiconductor structures with the use of one-dimensional microwave photonic crystals. Journal of Communications Technology and Electronics.

crystals. Technical Physics. 2017;62:243-247. DOI: 10.1134/S106378421702027X

layers. Radioelectronics. 2009;1:73-80. In: Proceedings of the Universities

John Wiley & Sons; 1978. 284p

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tion. 1991;39:425-428. DOI: 10.1109/8.76347

269. DOI: 10.1007/978-94-010-0450-3\_10

lishing House; 2011. 392p

226913120176

Saratov University Publishing House; 2010. 100p

2013125178/07 dated 05/30/2013; Pub. 10/09/2014; Bul. 25

2016;61:42-49. DOI: 10.1134/S1064226916010125

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.76622

#### **Abstract**

A high-gain amplifier module with integrated waveguide (WG) has been presented for millimeter wave applications. In order to improve the isolation between the amplification stages in the multi-stage amplifier module, an isolated WG is integrated into the module case. It is possible to effectively suppress the oscillation occurring in the high gain stage. Microstrip line (MSL)-to-WG transitions are designed and fabricated on a 5 mil thick RT5880 substrate for interconnection of the isolated WG, input/output WG and amplifier PCB in a cascaded two-stage high gain amplifier module. The transition loss of −0.44 dB is obtained at 40 GHz and return-loss (S11) bandwidth below −10 dB is from 34.1 to 50 GHz. The fabricated high-gain amplifier module shows a high gain over 39.7 dB from 38 to 41 GHz. At 38.7 GHz, its maximum gain of 44.25 dB is achieved.

**Keywords:** waveguide, isolation, microstrip-to-waveguide transition, amplifier module

### **1. Introduction**

Recently, millimeter-wave (mm-wave) frequency bands have attracted attention as various applications such as radar sensors [1–3] as well as wireless communication. In particular, due to very widely available bandwidth, frequency bands for a variety of wireless communication applications such as point-to-point wireless communications, radio-on-fiber (RoF) links [4], 5G cellular wireless networks [5], etc. are shifting to the mm-wave frequency band.

One of the key issues for the commercialization of mm-wave systems is reproducible and inexpensive packaging technology in addition to active integrated circuit (IC) technology.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Typically, active IC chips are assembled into metal or dielectric substrate carriers using wirebonding or flip-chip [6] interconnect and eventually encapsulated in plastic packages or metal housings. Due to the integration of various materials and structures in a compact, limited packaging space, unwanted substrate modes [6], cavity resonance [7], feedback, or crosstalk [8, 9] occur within the packaging module. In the previous papers [6–9], this phenomenon was well analyzed and the causes were identified and design rules or various methods for suppressing them were presented. For example, the resistivity [6] of the flip chip carrier, the resonance condition of the cavity [7], the chip mounting design rule [8], and the resistance coating of the lid [10] were investigated. Several modules [1, 2, 6, 11] have been successfully developed to reflect these attempts. However, in the case of a high-gain amplification block requiring a gain of 30 dB or more, the stability problem is caused by the feedback effect [8, 9] of the reflected signal due to structural discontinuities in the packaging. That is, the radiated signals are reflected by the surrounding structures, enter the input stage of the amplification block, and are amplified, so that the entire module oscillates. Therefore, to eliminate the oscillation of the amplification block, small and medium gain amplifier modules [2, 12, 13] are connected in series using an external WG until the required gain is satisfied. An attenuator or filter is inserted between the amplifier modules to adjust the gain or remove unwanted waves [14]. However, these methods lead to bulky and expensive mm-wave radio system due to expensive additional components.

In this work, a 40 dB high-gain amplifier module integrating the isolation WG has been demonstrated for 40 GHz radio system applications. Because of the isolation WG as well as input and output WG into the metal case of the amplifier module, a low-loss and wideband MSLto-WG transition is designed on the 5-mil thick RT5880 substrate to interconnect the amplifier IC mounted PCB with integrated WG. The simulated and tested results of the transition have been presented. The high-gain amplifier module was fabricated and its measured performance is analyzed.

**3. Design and measurement of the low-loss and wideband MSL-to-WG** 

**Figure 1.** The metal case integrating an isolation WG for high-gain amplifier module applications.

**Figure 2** presents a configuration of the MSL probe transition, transition module for measurement, and opening in the WG side to insert the transition. For this MSL-to-WG transition, a simple electric (E)-plane transition [17, 18] is used because of easy and simple design. In this transition, a TE10-mode energy in the WG couples to quasi-TEM-mode one in the MSL. The MSL transition consists of an E-plane probe, impedance transformer, and 50-Ω MSL. They are printed on a 5 mil thick substrate with a permittivity of 2.2 [19]. The size of the E-probe is 383 ×

optimization. Its size is 295 μm × 1000 μm. The 363 μm wide MSL is designed for the 50-Ω impedance and its length between two transitions is 18.0 mm considering the cavity size in the metal case as shown in **Figure 2(b)**. A back-to-back structure is required for measurement of the fabricated MSL-to-WG transition. Two sets of the back-to-back structured transition are required to apply to the metal case as shown in **Figure 2(b)**. The E-probe of the transition is

. The simple high-impedance matching line with is designed for easy design and

High-Gain Amplifier Module Integrating a Waveguide into the Module Case for Millimeter…

http://dx.doi.org/10.5772/intechopen.76622

51

**transition**

1465 μm2

### **2. Metal case integrating an isolation WG for high-gain amplifier module**

In general, a proactive approach to suppressing this feedback effect is to effectively isolate the two adjacent amplification stages. **Figure 1** shows the metal case inserting a 15.7 mm long isolation WG between two cavities for high-gain amplifier module applications [15]. This method allows high-gain amplification without oscillation because of good isolation between two enclosed cavities in the metal case. In the each cavity, the PCB mounting an amplifier IC is assembled. Because signals radiated due to discontinuities in the metal case or PCB assembly are confined within each cavity, the amplifier IC assembled in another cavity is protected. In this work, the MSL-to-WG transition in the isolation WG as well as an input and output WG port should be designed. The WG is based on WR22 WG, whose size is 2.84 × 5.68 mm2 . Two commercial amplifier ICs [16] are used for high-gain amplification block with the gain of 40 dB. The signal line on the PCB is the 50 Ω microstrip line (MSL). Considering the inserted isolation WG and port WGs, four MSL-to-WG transitions are needed. The main key design issue is the low-loss and wide-band transition.

High-Gain Amplifier Module Integrating a Waveguide into the Module Case for Millimeter… http://dx.doi.org/10.5772/intechopen.76622 51

Typically, active IC chips are assembled into metal or dielectric substrate carriers using wirebonding or flip-chip [6] interconnect and eventually encapsulated in plastic packages or metal housings. Due to the integration of various materials and structures in a compact, limited packaging space, unwanted substrate modes [6], cavity resonance [7], feedback, or crosstalk [8, 9] occur within the packaging module. In the previous papers [6–9], this phenomenon was well analyzed and the causes were identified and design rules or various methods for suppressing them were presented. For example, the resistivity [6] of the flip chip carrier, the resonance condition of the cavity [7], the chip mounting design rule [8], and the resistance coating of the lid [10] were investigated. Several modules [1, 2, 6, 11] have been successfully developed to reflect these attempts. However, in the case of a high-gain amplification block requiring a gain of 30 dB or more, the stability problem is caused by the feedback effect [8, 9] of the reflected signal due to structural discontinuities in the packaging. That is, the radiated signals are reflected by the surrounding structures, enter the input stage of the amplification block, and are amplified, so that the entire module oscillates. Therefore, to eliminate the oscillation of the amplification block, small and medium gain amplifier modules [2, 12, 13] are connected in series using an external WG until the required gain is satisfied. An attenuator or filter is inserted between the amplifier modules to adjust the gain or remove unwanted waves [14]. However, these methods lead to bulky and expensive mm-wave radio system due to expensive additional components. In this work, a 40 dB high-gain amplifier module integrating the isolation WG has been demonstrated for 40 GHz radio system applications. Because of the isolation WG as well as input and output WG into the metal case of the amplifier module, a low-loss and wideband MSLto-WG transition is designed on the 5-mil thick RT5880 substrate to interconnect the amplifier IC mounted PCB with integrated WG. The simulated and tested results of the transition have been presented. The high-gain amplifier module was fabricated and its measured per-

**2. Metal case integrating an isolation WG for high-gain amplifier** 

In general, a proactive approach to suppressing this feedback effect is to effectively isolate the two adjacent amplification stages. **Figure 1** shows the metal case inserting a 15.7 mm long isolation WG between two cavities for high-gain amplifier module applications [15]. This method allows high-gain amplification without oscillation because of good isolation between two enclosed cavities in the metal case. In the each cavity, the PCB mounting an amplifier IC is assembled. Because signals radiated due to discontinuities in the metal case or PCB assembly are confined within each cavity, the amplifier IC assembled in another cavity is protected. In this work, the MSL-to-WG transition in the isolation WG as well as an input and output WG port should be designed. The WG is based on WR22 WG, whose size is 2.84 × 5.68 mm2

Two commercial amplifier ICs [16] are used for high-gain amplification block with the gain of 40 dB. The signal line on the PCB is the 50 Ω microstrip line (MSL). Considering the inserted isolation WG and port WGs, four MSL-to-WG transitions are needed. The main key design

.

formance is analyzed.

50 Emerging Waveguide Technology

issue is the low-loss and wide-band transition.

**module**

**Figure 1.** The metal case integrating an isolation WG for high-gain amplifier module applications.

### **3. Design and measurement of the low-loss and wideband MSL-to-WG transition**

**Figure 2** presents a configuration of the MSL probe transition, transition module for measurement, and opening in the WG side to insert the transition. For this MSL-to-WG transition, a simple electric (E)-plane transition [17, 18] is used because of easy and simple design. In this transition, a TE10-mode energy in the WG couples to quasi-TEM-mode one in the MSL. The MSL transition consists of an E-plane probe, impedance transformer, and 50-Ω MSL. They are printed on a 5 mil thick substrate with a permittivity of 2.2 [19]. The size of the E-probe is 383 × 1465 μm2 . The simple high-impedance matching line with is designed for easy design and optimization. Its size is 295 μm × 1000 μm. The 363 μm wide MSL is designed for the 50-Ω impedance and its length between two transitions is 18.0 mm considering the cavity size in the metal case as shown in **Figure 2(b)**. A back-to-back structure is required for measurement of the fabricated MSL-to-WG transition. Two sets of the back-to-back structured transition are required to apply to the metal case as shown in **Figure 2(b)**. The E-probe of the transition is

inserted into the side opening of the WG. Of course, its position and size should be optimized

High-Gain Amplifier Module Integrating a Waveguide into the Module Case for Millimeter…

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53

Using electromagnetic (EM) analysis software [20], transitions were designed and analyzed. In **Figure 3**, the design model and results are presented. The transitions are designed to be assembled in the metal case. Since the WG must be integrated in the metal case, it is divided into two parts, the body and the lid. In **Figure 3(a)** and **(b)**, the metal case integrating transitions and the designed results are presented, respectively. In these designed results, an input

**Figure 5.** Measured results of the fabricated two-set transition (TR) in the back-to-back structure [an inset: an adapter

and the final dimensions are shown in **Figure 2(c)**.

**Figure 4.** Fabricated transitions and assembled in the metal case.

connection, AD: adapters].

**Figure 2.** The configuration of the MSL probe transition (a), transition module consisting of four MSL-to-WG transitions (b), and opening in the WG side into which the transition is inserted.

**Figure 3.** The metal case integrating transitions (a) and the designed results of the transition (b).

inserted into the side opening of the WG. Of course, its position and size should be optimized and the final dimensions are shown in **Figure 2(c)**.

Using electromagnetic (EM) analysis software [20], transitions were designed and analyzed. In **Figure 3**, the design model and results are presented. The transitions are designed to be assembled in the metal case. Since the WG must be integrated in the metal case, it is divided into two parts, the body and the lid. In **Figure 3(a)** and **(b)**, the metal case integrating transitions and the designed results are presented, respectively. In these designed results, an input

**Figure 4.** Fabricated transitions and assembled in the metal case.

**Figure 3.** The metal case integrating transitions (a) and the designed results of the transition (b).

**Figure 2.** The configuration of the MSL probe transition (a), transition module consisting of four MSL-to-WG transitions

(b), and opening in the WG side into which the transition is inserted.

52 Emerging Waveguide Technology

**Figure 5.** Measured results of the fabricated two-set transition (TR) in the back-to-back structure [an inset: an adapter connection, AD: adapters].

return loss (S11) less than −10 dB and insertion loss (S21) lower than −0.52 dB are obtained from 34.09 GHz up to 50 GHz.

the amplifier IC and the fabricated high-gain amplifier module. Landing patterns to mount a SMT-type amplifier IC [16] are designed on the RT5880 substrate by referring to application note [21] as shown in **Figure 6(a)**. In this substrate, MSL-to-WG transitions are also included to connect with WG. The designed PCB were fabricated and assembled with several components. Two PCBs were assembled in the metal case. The assembled high-gain amplifier module and the lid closed one are shown in **Figure 6(b)**. Its overall size is 79 × 42 × 32 mm3

High-Gain Amplifier Module Integrating a Waveguide into the Module Case for Millimeter…

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The insertion and return losses of the fabricated high-gain amplifier module were measured at DC bias conditions (Vd = 5 V and Id = 1000 mA) and the measured results are presented in **Figure 7**. For comparison purposes, the characteristics plotted based on the data in a datasheet of the single amplifier IC [16] are also shown. For the high gain amplifier module, the gain more than 39.7 dB was measured at 38–41 GHz. At 38.7 GHz, the maximum gain of

Considering a single amplifier IC with the gain of 20 dB, a gain of 40 dB for the high-gain amplifier module connecting two amplifier ICs in series means that the transition loss is negligible. Therefore, these results demonstrate that the isolation WG provides good isolation between two amplifier ICs and suppress effectively feedback effects in the high-gain amplification block. Compared to the return losses (|S11| data and |S22| data) of the single amplifier IC, the return loss (|S22| meas) at output connection part of the high-gain amplifier module is

**Figure 7.** Measured results of the fabricated high-gain amplifier module, compared to amplifier IC (AMMP-6441) ones

from its data sheet [M: measurement and amp. IC: data sheet of an amplifier IC].

44.25 dB is obtained [15].

.

55

The designed transitions were realized on the RT5880 substrate in commercial PCB foundry and two sets of the fabricated transitions were assembled on the metal case for the high-gain amplifier module as shown in **Figure 4**.

The measured loss characteristics of the adapters and fabricated transitions are presented as shown in **Figure 5**. Since the input and output port of the fabricated metal case is the WR22 WG, a 2.4 mm-male cable-to-WR22 WG adapters were used for connection with a vector network analyzer (VNA, Agilent N5250A) as shown in an inset of **Figure 5**. Losses of the adapters and the assembled two-set transitions were tested using the standard open-short-load (SOL) calibration from 30 to 50 GHz. The measured insertion and return loss of the adapters (AD) are from −0.26 to −0.33 dB and less than −10 dB, respectively, from 30 to 50 GHz. For the assembled two-set transitions (TR), the insertion losses of −2.9 and − 2.5 dB are observed at 38 and 41 GHz, respectively. This measured insertion loss of the transition includes several loss components came from the adapters, two 18 mm long MSLs with the loss of −0.0239 dB/mm, and the MSL-to-WG transitions. By considering these loss components, the loss per a single transition is −0.32 and −0.44 dB at 38 and 41 GHz, respectively. Its measured return loss of the transition is below −10 dB from 34.1 to 50 GHz. The operational bandwidth (BW) of the transition for a return loss of −10 dB is 15.9 GHz.

### **4. Fabrication and measurement of the high-gain amplifier module**

The high-gain amplifier module was designed and fabricated for the purpose of demonstration of the isolation WG integrated in its metal case. **Figure 6** presents PCB layout to mount

**Figure 6.** PCB layout to mount the amplifier IC (a) and the fabricated high-gain amplifier module (b).

the amplifier IC and the fabricated high-gain amplifier module. Landing patterns to mount a SMT-type amplifier IC [16] are designed on the RT5880 substrate by referring to application note [21] as shown in **Figure 6(a)**. In this substrate, MSL-to-WG transitions are also included to connect with WG. The designed PCB were fabricated and assembled with several components. Two PCBs were assembled in the metal case. The assembled high-gain amplifier module and the lid closed one are shown in **Figure 6(b)**. Its overall size is 79 × 42 × 32 mm3 .

return loss (S11) less than −10 dB and insertion loss (S21) lower than −0.52 dB are obtained

The designed transitions were realized on the RT5880 substrate in commercial PCB foundry and two sets of the fabricated transitions were assembled on the metal case for the high-gain

The measured loss characteristics of the adapters and fabricated transitions are presented as shown in **Figure 5**. Since the input and output port of the fabricated metal case is the WR22 WG, a 2.4 mm-male cable-to-WR22 WG adapters were used for connection with a vector network analyzer (VNA, Agilent N5250A) as shown in an inset of **Figure 5**. Losses of the adapters and the assembled two-set transitions were tested using the standard open-short-load (SOL) calibration from 30 to 50 GHz. The measured insertion and return loss of the adapters (AD) are from −0.26 to −0.33 dB and less than −10 dB, respectively, from 30 to 50 GHz. For the assembled two-set transitions (TR), the insertion losses of −2.9 and − 2.5 dB are observed at 38 and 41 GHz, respectively. This measured insertion loss of the transition includes several loss components came from the adapters, two 18 mm long MSLs with the loss of −0.0239 dB/mm, and the MSL-to-WG transitions. By considering these loss components, the loss per a single transition is −0.32 and −0.44 dB at 38 and 41 GHz, respectively. Its measured return loss of the transition is below −10 dB from 34.1 to 50 GHz. The operational bandwidth (BW) of the transition for a return loss of −10 dB is 15.9 GHz.

**4. Fabrication and measurement of the high-gain amplifier module**

**Figure 6.** PCB layout to mount the amplifier IC (a) and the fabricated high-gain amplifier module (b).

The high-gain amplifier module was designed and fabricated for the purpose of demonstration of the isolation WG integrated in its metal case. **Figure 6** presents PCB layout to mount

from 34.09 GHz up to 50 GHz.

54 Emerging Waveguide Technology

amplifier module as shown in **Figure 4**.

The insertion and return losses of the fabricated high-gain amplifier module were measured at DC bias conditions (Vd = 5 V and Id = 1000 mA) and the measured results are presented in **Figure 7**. For comparison purposes, the characteristics plotted based on the data in a datasheet of the single amplifier IC [16] are also shown. For the high gain amplifier module, the gain more than 39.7 dB was measured at 38–41 GHz. At 38.7 GHz, the maximum gain of 44.25 dB is obtained [15].

Considering a single amplifier IC with the gain of 20 dB, a gain of 40 dB for the high-gain amplifier module connecting two amplifier ICs in series means that the transition loss is negligible. Therefore, these results demonstrate that the isolation WG provides good isolation between two amplifier ICs and suppress effectively feedback effects in the high-gain amplification block. Compared to the return losses (|S11| data and |S22| data) of the single amplifier IC, the return loss (|S22| meas) at output connection part of the high-gain amplifier module is

**Figure 7.** Measured results of the fabricated high-gain amplifier module, compared to amplifier IC (AMMP-6441) ones from its data sheet [M: measurement and amp. IC: data sheet of an amplifier IC].

noticeably improved, but the return loss (|S11| meas) at its input connection part is degraded. In general, the return loss of an assembled amplifier module follows the return loss characteristic of the transition. Therefore, the improvement of |S22| meas is due to the characteristics of the transition. However, the deterioration of |S11| meas at the input stage is caused by alignment and PCB fabrication process problems during assembly of the WG and PCB transitions.

[4] Yaakob S, Samsuri NM, Mohamad R, Farid NE, Azmi IM, Hassan SMM, Khushairi N, Rahim SAEA, Rahim AIA, Rasmi A, Zamzuri AK, Idrus SM, Fan S. Live HD video transmission using 40GHz radio over fibre downlink system. In: IEEE 3rd International Conference on Photonics (ICP). 2012. pp. 246-249. DOI: 10.1109/ICP.2012.6379854

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57

[5] Al-Falahy N, Omar YA. Technologies for 5G networks: Challenges and opportunities. IT

[6] Tessmann A, Riessle M, Kudszus S, Massler H. A flip-chip packaged coplanar 94 GHz amplifier module with efficient suppression of parasitic substrate effects. IEEE Microwave and Wireless Components Letters. 2004;**14**:145-147. DOI: 10.1109/LMWC.2004.827115 [7] Dhar J, Arora RK, Dasgupta A, Rana SS. Enclosure effect on microwave power amplifier.

[8] Krems T, Tessmann A, Haydl WH, Schmelz C, Heide P. Avoiding cross talk and feedback effects in packaging coplanar millimeter-wave circuits. In: IEEE MTT-S International Microwave Symposium. Vol. 2. 1998. pp. 1091-1094. DOI: 10.1109/MWSYM.1998.705183

[9] Beilenhoff K, Heinrich W. Excitation of the parasitic parallel-plate line mode at coplanar discontinuities. In: IEEE MTT-S International Microwave Symposium. Vol. 3. 1997.

[10] Yook J-G, Katehi LPB, Simons RN, Shalkhauser KA. Experimental and theoretical study of parasitic leakage/resonance in a K/Ka-band MMIC package. IEEE Transactions on

[11] Lee YC, Chang W-I, Park CS. Monolithic LTCC SiP transmitter for 60GHz wireless communication terminals. In: IEEE MTT-S International Microwave Symposium. 2005.

[12] Radisic V, Mei X, Sarkozy S, Yoshida W, Liu P-H, Uyeda J, Lai R, Deal WR. A 50 mW 220 GHz power amplifier module. In: IEEE MTT-S International Microwave Symposium.

[13] Tessmann A, Leuther A, Hurm V, Massler H, Zink M, Kuri M, Riessle M, Losch R, Schlechtweg M, Ambacher O. A 300 GHz mHEMT amplifier module. In: IEEE International Conference on Indium Phosphide & Related Materials. 2009. pp. 196-199. DOI:

[14] Samoska L, Church S, Cleary K, Fung A, Gaier TC, Kangaslahti P, Voll P. Cryogenic MMIC low noise amplifiers for W-band and beyond. In: International Symposium on

[15] Lee YC. Waveguide integrated high-gain amplifier module for millimeter-wave applica-

[16] Avago Technologies. AMMP-6441 36-40 GHz, 0.4 W Power Amplifier in SMT Package [Internet]. Available from: http://www.datasheetlib.com/datasheet/168419/ammp-6441-

tions. Progress in Electromagnetics Research Letters. 2015;**57**:125-130

Microwave Theory and Techniques. 1996;**44**(2):403-410. DOI: 10.1109/22.554569

Professional. 2017;**19**(1):12-20. DOI: 10.1109/MITP.2017.9

Progress in Electromagnetics Research C. 2011;**19**:163-177

pp. 1789-1792. DOI: 10.1109/MWSYM.1997.596891

pp. 1015-1018. DOI: 10.1109/MWSYM.2005.1516839

2010. pp. 45-48. DOI: 10.1109/MWSYM.2010.5515248

10.1109/ICIPRM.2009.5012477

Space Terahertz Technology. 2011

tr2g\_avago-technologies.html

### **5. Conclusion**

The 40 dB high-gain amplifier module with the isolation waveguide (WG) has been presented for millimeter wave applications. For the purpose of suppressing the oscillation due to the feedback effect, the isolation WG was integrated into the metal case of the amplifier module. In addition to input/output WG of the amplifier module, additional MSL-to-WR22 WG transitions are required due to the isolation WG and low-loss and wide-band transition is designed and manufactured on the 5 mil thick RT5880 substrate. Its measured loss and operational bandwidth were less than −0.44 dB/a transition and 15.9 GHz, respectively at 40 GHz. The high-gain amplifier module was designed and fabricated for the purpose of demonstration of the isolation WG. The amplifier module operated stably without oscillation at high gain over 40 dB. The fabricated high-gain amplifier module showed a high gain over 39.7 dB from 38 to 41 GHz. Its maximum gain of 44.25 dB was obtained at 38.7 GHz.

### **Author details**

Young Chul Lee

Address all correspondence to: rfleeyc@gmail.com

Division of Marine Mechatronics, Mokpo National Maritime University, Republic of Korea

### **References**


[4] Yaakob S, Samsuri NM, Mohamad R, Farid NE, Azmi IM, Hassan SMM, Khushairi N, Rahim SAEA, Rahim AIA, Rasmi A, Zamzuri AK, Idrus SM, Fan S. Live HD video transmission using 40GHz radio over fibre downlink system. In: IEEE 3rd International Conference on Photonics (ICP). 2012. pp. 246-249. DOI: 10.1109/ICP.2012.6379854

noticeably improved, but the return loss (|S11| meas) at its input connection part is degraded. In general, the return loss of an assembled amplifier module follows the return loss characteristic of the transition. Therefore, the improvement of |S22| meas is due to the characteristics of the transition. However, the deterioration of |S11| meas at the input stage is caused by alignment and PCB fabrication process problems during assembly of the WG and PCB transitions.

The 40 dB high-gain amplifier module with the isolation waveguide (WG) has been presented for millimeter wave applications. For the purpose of suppressing the oscillation due to the feedback effect, the isolation WG was integrated into the metal case of the amplifier module. In addition to input/output WG of the amplifier module, additional MSL-to-WR22 WG transitions are required due to the isolation WG and low-loss and wide-band transition is designed and manufactured on the 5 mil thick RT5880 substrate. Its measured loss and operational bandwidth were less than −0.44 dB/a transition and 15.9 GHz, respectively at 40 GHz. The high-gain amplifier module was designed and fabricated for the purpose of demonstration of the isolation WG. The amplifier module operated stably without oscillation at high gain over 40 dB. The fabricated high-gain amplifier module showed a high gain over 39.7 dB from 38 to

Division of Marine Mechatronics, Mokpo National Maritime University, Republic of Korea

[1] Tessmann A, Kudszus S, Feltgen T, Riessle M, Sklarczyk C, Haydl WH. A 94 GHz single-chip FMCW radar module for commercial sensor. In: IEEE MTT-S International Microwave Symposium. Vol. 3. 2002. pp. 1851-1854. DOI: 10.1109/MWSYM. 2002.1012223

[2] Tessmann A, Leuther A, Kuri M, Massler H, Riessle M, Essen H, Stanko H, Sommer R, Zink M, Stibal R, Reinert W, Schlechtweg M. 220 GHz low-noise amplifier modules for radiometric imaging applications. In: The 1st European Microwave Integrated Circuits

[3] Kim J-G, Kang D-W, Min B-W, Rebeiz GM. A single-chip 36-38 GHz 4-element transmit/ receive phased-array with 5-bit amplitude and phase control. In: IEEE MTT-S International

Microwave Symposium. 2009. pp. 561-564. DOI: 10.1109/MWSYM.2009.5165758

Conference. 2006. pp. 137-140. DOI: 10.1109/EMICC.2006.282770

41 GHz. Its maximum gain of 44.25 dB was obtained at 38.7 GHz.

Address all correspondence to: rfleeyc@gmail.com

**5. Conclusion**

56 Emerging Waveguide Technology

**Author details**

Young Chul Lee

**References**


[17] Leong Y-C, Weinreb S. Full band waveguide-to-microstrip probe transitions. In: IEEE MTT-S International Microwave Symposium; 1999. pp. 1435-1438. DOI: 10.1109/MWSYM. 1999.780219

**Chapter 4**

Provisional chapter

**Mathematical Analysis of Electrical Breakdown Effects**

DOI: 10.5772/intechopen.76973

The designers of microwave devices in the industry use the analytical solutions of the corona discharge equation to determine the minimum power breakdown threshold, in a particular device, such as waveguides and filters, and know whether it is in the established margins. There are two main ways to determine the breakdown threshold of a waveguide analytically; the most commonly used describes the plasma density generation completely as a function of the geometry by using the characteristic diffusion length, while the second is a more thorough method that involves the use of the effective diffusion length which considers collision frequency and electric field into the equations. Hence the aim of the designers is to obtain the closest results to experimental results, both methods must be considered in addition to the environmental changes so that they know the operational limits. This chapter describes the different methods to obtain analytical results for the breakdown threshold in any rectangular waveguide device, the influence of environmental conditions in the analysis and the inhomogeneous electric field effect

Keywords: electrical breakdown threshold, corona, characteristic diffusion length,

Traffic capacity in a high-frequency scenario where waveguides are involved is generally limited by two main factors: bandwidth and input power [1]. In cases where the atmospheric pressure is a factor, such as satellite communications, the maximum input power of the signal is determined by several factors, among which are the geometry of the device, the collision

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Mathematical Analysis of Electrical Breakdown Effects

**in Waveguides**

in Waveguides

Abstract

inside the devices.

1. Introduction

waveguide filters, effective diffusion length

Isaac Medina and Primo-Alberto Calva

Isaac Medina and Primo-Alberto Calva

http://dx.doi.org/10.5772/intechopen.76973

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter


#### **Mathematical Analysis of Electrical Breakdown Effects in Waveguides** Mathematical Analysis of Electrical Breakdown Effects in Waveguides

DOI: 10.5772/intechopen.76973

Isaac Medina and Primo-Alberto Calva Isaac Medina and Primo-Alberto Calva

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.76973

#### Abstract

[17] Leong Y-C, Weinreb S. Full band waveguide-to-microstrip probe transitions. In: IEEE MTT-S International Microwave Symposium; 1999. pp. 1435-1438. DOI: 10.1109/MWSYM.

[18] Shireen R, Shi S, Prather DW. W-band microstrip-to-waveguide transition using via

[21] Avago Technologies. Application note 5520-AMxP-XXXX Production Assembly process (Land Pattern A) [Internet]. Available from: http://www.avagotech.com/docs/AV02-

fences. Progress in Electromagnetics Research Letters. 2010;**16**:151-160

[19] Rogers Corporation. Available from: http://www.rogerscorp.com

[20] CST Microwave Studio. Available from: https://www.cst.com

1999.780219

58 Emerging Waveguide Technology

2954EN

The designers of microwave devices in the industry use the analytical solutions of the corona discharge equation to determine the minimum power breakdown threshold, in a particular device, such as waveguides and filters, and know whether it is in the established margins. There are two main ways to determine the breakdown threshold of a waveguide analytically; the most commonly used describes the plasma density generation completely as a function of the geometry by using the characteristic diffusion length, while the second is a more thorough method that involves the use of the effective diffusion length which considers collision frequency and electric field into the equations. Hence the aim of the designers is to obtain the closest results to experimental results, both methods must be considered in addition to the environmental changes so that they know the operational limits. This chapter describes the different methods to obtain analytical results for the breakdown threshold in any rectangular waveguide device, the influence of environmental conditions in the analysis and the inhomogeneous electric field effect inside the devices.

Keywords: electrical breakdown threshold, corona, characteristic diffusion length, waveguide filters, effective diffusion length

#### 1. Introduction

Traffic capacity in a high-frequency scenario where waveguides are involved is generally limited by two main factors: bandwidth and input power [1]. In cases where the atmospheric pressure is a factor, such as satellite communications, the maximum input power of the signal is determined by several factors, among which are the geometry of the device, the collision

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

frequency of the molecules and free electrons and the intrinsic characteristics of the propagating medium itself, that is, air or nitrogen [2].

Waveguides are conductor hollow tubes, generally consisting of a circular, elliptical, or rectangular cross-section. The cross-section dimensions are chosen by designers in such a way that electromagnetic waves propagate inside the guide. A waveguide can have several shapes and sizes, and frequently its performance is a function of the radiofrequency (RF) routing signals; this means that the way the wave is propagating inside the guide. Rectangular waveguides are the most commonly used; this is because they are easily fabricated, they have a very broad bandwidth and they present low losses within their operating frequencies [3].

Rectangular waveguides operate only in certain frequency bands, depending on the crosssection dimensions. Waveguide geometry determines the highest operating wavelength, this means, higher waveguide sizes operate at lower frequencies.

Waveguide filters are responsible of eliminating unwanted radiations and interferences in a communication scheme. These devices are also hollow conductor tubes, made generally of aluminium, with the difference that inside them are distance variations or obstructions that generate the wanted filtering effect. Figure 1 shows different types of waveguide filters.

High-performance RF filters are widely used in communications systems, where it is necessary to know its capabilities for input power handling. Since increasing the power levels is the simpler way to impulse the reach of the system and its capability for data transmission, the design for the high-power operation filters must consider the next effects: electrical breakdown by ionization (corona effect), multipactor effect and passive intermodulation interferences (PIM). Multipactor is a breakdown mechanism in vacuum, in which a resonant increase of freeelectron space charge develops between two surfaces. The applied field intensity is such that the electrons collide at ultra-high speeds against the walls of the device, causing the continuous release of secondary electrons in the medium and leading to a breakdown. However, this

Mathematical Analysis of Electrical Breakdown Effects in Waveguides

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61

Figure 2. Low-pass corrugated filter after breakdown has occurred, as reported in [4].

Ionization breakdown is a phenomenon that occurs in gases where the normally low electron density increases in a way similar to an avalanche, turning the isolating gas into conducting plasma; this happens at higher pressures than multipactor. In satellite communications, breakdown analysis must be considered for the components located on Earth and for the ones destined for space operations, since the RF components that are designed to operate in space must be tested frequently on Earth at their highest power and are fully operational during the launching stage for telemetry purposes. This is the reason the analysis is made for low-

Waveguide breakdown analysis follows the next three stages: breakdown threshold determination, circuit and field analysis to determine the maximum voltage or field values and comparison

Air ionization is caused because of the electrons' impact against air molecules. These electrons are accelerated by an RF field. If the energy level (provided by the RF field) is enough to cause ionization of neutral molecules, and the free electron total created by ionization exceeds the total losses of electrons due to attachment and recombination, the exponential growth of the

In low pressures, particles have a higher mean free path. Eventually, the mean free path increases until it reaches d (gap distance), where the multipactor effect takes place. By reducing

electron density generates electron plasma and, eventually, leads to breakdown.

is not the critical stage for design implications.

of the experimental worst case with breakdown threshold.

pressure applications.

The reason waveguide filters are analysed in terms of breakdown power is because this device is with the shortest cross-section of all the communications system, reaching even distances of 1 mm. In these sections, electric field density can be so high that it leads to electrical breakdown, rendering the components useless. Figure 2 shows a low-pass corrugated filter after breakdown has occurred [4].

The continuous miniaturization tendency in electronic devices and the increasing demand of services lead to higher component integration. In a transponder system, passive components are allocated, such as filters and waveguides, where, due to the used power, high densities of electric field are reached, presenting mainly the corona and multipactor effects [5].

Figure 1. (a) Coupled iris filter, (b) corrugated waveguide filter, (c) waffle iron filter, (d) posts filter, and (e) guarded waveguide filter.

Figure 2. Low-pass corrugated filter after breakdown has occurred, as reported in [4].

frequency of the molecules and free electrons and the intrinsic characteristics of the propagat-

Waveguides are conductor hollow tubes, generally consisting of a circular, elliptical, or rectangular cross-section. The cross-section dimensions are chosen by designers in such a way that electromagnetic waves propagate inside the guide. A waveguide can have several shapes and sizes, and frequently its performance is a function of the radiofrequency (RF) routing signals; this means that the way the wave is propagating inside the guide. Rectangular waveguides are the most commonly used; this is because they are easily fabricated, they have a very broad

Rectangular waveguides operate only in certain frequency bands, depending on the crosssection dimensions. Waveguide geometry determines the highest operating wavelength, this

Waveguide filters are responsible of eliminating unwanted radiations and interferences in a communication scheme. These devices are also hollow conductor tubes, made generally of aluminium, with the difference that inside them are distance variations or obstructions that generate the wanted filtering effect. Figure 1 shows different types of waveguide filters.

The reason waveguide filters are analysed in terms of breakdown power is because this device is with the shortest cross-section of all the communications system, reaching even distances of 1 mm. In these sections, electric field density can be so high that it leads to electrical breakdown, rendering the components useless. Figure 2 shows a low-pass corrugated filter after

The continuous miniaturization tendency in electronic devices and the increasing demand of services lead to higher component integration. In a transponder system, passive components are allocated, such as filters and waveguides, where, due to the used power, high densities of electric field are reached, presenting mainly the corona and multipactor effects [5].

Figure 1. (a) Coupled iris filter, (b) corrugated waveguide filter, (c) waffle iron filter, (d) posts filter, and (e) guarded

bandwidth and they present low losses within their operating frequencies [3].

means, higher waveguide sizes operate at lower frequencies.

ing medium itself, that is, air or nitrogen [2].

60 Emerging Waveguide Technology

breakdown has occurred [4].

waveguide filter.

High-performance RF filters are widely used in communications systems, where it is necessary to know its capabilities for input power handling. Since increasing the power levels is the simpler way to impulse the reach of the system and its capability for data transmission, the design for the high-power operation filters must consider the next effects: electrical breakdown by ionization (corona effect), multipactor effect and passive intermodulation interferences (PIM).

Multipactor is a breakdown mechanism in vacuum, in which a resonant increase of freeelectron space charge develops between two surfaces. The applied field intensity is such that the electrons collide at ultra-high speeds against the walls of the device, causing the continuous release of secondary electrons in the medium and leading to a breakdown. However, this is not the critical stage for design implications.

Ionization breakdown is a phenomenon that occurs in gases where the normally low electron density increases in a way similar to an avalanche, turning the isolating gas into conducting plasma; this happens at higher pressures than multipactor. In satellite communications, breakdown analysis must be considered for the components located on Earth and for the ones destined for space operations, since the RF components that are designed to operate in space must be tested frequently on Earth at their highest power and are fully operational during the launching stage for telemetry purposes. This is the reason the analysis is made for lowpressure applications.

Waveguide breakdown analysis follows the next three stages: breakdown threshold determination, circuit and field analysis to determine the maximum voltage or field values and comparison of the experimental worst case with breakdown threshold.

Air ionization is caused because of the electrons' impact against air molecules. These electrons are accelerated by an RF field. If the energy level (provided by the RF field) is enough to cause ionization of neutral molecules, and the free electron total created by ionization exceeds the total losses of electrons due to attachment and recombination, the exponential growth of the electron density generates electron plasma and, eventually, leads to breakdown.

In low pressures, particles have a higher mean free path. Eventually, the mean free path increases until it reaches d (gap distance), where the multipactor effect takes place. By reducing the atmospheric pressure, there is a lower input power handling capacity until it reaches a minimum operation power; this is called critical pressure [4].

a homogeneous field, due to the similar geometry among the parallel walls of the filter and

<sup>∂</sup><sup>t</sup> <sup>¼</sup> ð Þ vi � va <sup>n</sup> (3)

Mathematical Analysis of Electrical Breakdown Effects in Waveguides

dt <sup>¼</sup> <sup>0</sup>: (5)

n þ ð Þ vi � va n ¼ 0: (6)

<sup>D</sup> <sup>n</sup> <sup>¼</sup> <sup>0</sup> (7)

<sup>D</sup> <sup>n</sup> <sup>¼</sup> <sup>0</sup> (8)

<sup>D</sup> XY <sup>¼</sup> <sup>0</sup> (10)

<sup>D</sup> <sup>¼</sup> <sup>0</sup>: (11)

n xð Þ¼ ; y X xð ÞY yð Þ (9)

: (4)

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63

∂n

n tðÞ¼ n0e

dn

As a consequence, the general equation to solve the breakdown threshold stage is:

the devices analysed are rectangular waveguides and filters, the equation leads to:

∂2 n ∂y<sup>2</sup> þ

∂2 n ∂x<sup>2</sup> þ

Y d2 X dx<sup>2</sup> <sup>þ</sup> <sup>X</sup> <sup>d</sup><sup>2</sup>

> 1 X d2 X dx<sup>2</sup> þ

Establishing the solution as the product of two functions:

D∇<sup>2</sup>

∂2 n ∂x<sup>2</sup> þ

When ν<sup>i</sup> > νa, there is an electron avalanche, and the breakdown condition for the continuous

Solving the Laplacian term from Eq. (6), and considering a Cartesian coordinate system, since

∂2 n ∂z<sup>2</sup> þ

where ν ¼ ν<sup>i</sup> � ν<sup>a</sup> is the effective ionization frequency. The rectangular waveguide device must be analysed from the cross-section, so the third term can be discarded, considering only the

Substituting (9) in (8), knowing that the equation components are independent between them,

Y dy<sup>2</sup> þ

1 Y d2 Y dy<sup>2</sup> þ

among terms, the first term can be solved proposing a negative constant as a result:

This equation can be solved by proposing exponential solutions. Due to the independency

ν

ν

ν

∂2 n ∂y<sup>2</sup> þ ν

ð Þ vi�va t

waveguides. Then, Eq. (2) results in:

and solving the derivative results in:

waves can be simplified as:

width and height of the guide.

and dividing by (9) we get:

it results in:

This chapter covers the ionization breakdown in atmospheric air analysis; the equations that describe this consider different processes, such as the ionization, attachment and collision frequencies. Also, the analysis considers as variables the atmospheric pressure and electric field intensity, considering contaminant-free dry air as the propagating medium. Nevertheless, due to the breakdown variability, the design of filters and waveguides is a controversial topic for designers, who consider a wide tolerance range from 0 dB to 3 dB according to minimum breakdown power [4, 5].

Corona breakdown is the process when electron plasma is created due to the ionization of the gas in areas where the electrical fields are high. Electrical fields in filters and waveguides can lead to corona effects at relatively low pressures (from 1 to 100 Torr), which, in atmospheric terms, are reached in the ionosphere (from 80 to 800 km). This phenomenon cannot occur in vacuum conditions, since it is necessary for the presence of a gas to ionize [4].

#### 2. Breakdown power threshold

There are different processes that can generate ions; these are by electronic impact, field effect, photo-ionization and thermos-ionization. For the analysis of filters and waveguides, the most relevant is by electronic impact, being directly proportional to the collision frequency between electrons and molecules. The equation that describes the time evolution of free-electron generation is [4, 6–8]:

$$\frac{\partial n}{\partial t} = \nabla(D\nabla n) - \vec{v} \cdot \nabla n + (v\_i - v\_a)n - \beta n^2 + P \tag{1}$$

where ν<sup>i</sup> and ν<sup>a</sup> are the ionization and attachment frequencies, respectively, D is the diffusion coefficient, β is the recombination coefficient and P is the electron production rate by external sources; ∇ð Þ D∇n is the term that defines the diffusion of the electron cloud, from a highdensity area to a lower-density area, and is entirely space dependent; and v ! ∙∇n is the convective term that considers the possible movement of the gas.

For the corona effect analysis and from the pre-breakdown stage point of view, the recombination term is discarded, since it is only relevant once the electron density is high enough, which only occurs when the electrical discharge has already begun. Also, the convective term has to be discarded, since a stationary medium is assumed inside the waveguide devices, that is, there is no relevant movement of the gas molecules. Additionally, the diffusion coefficient is considered as space independent, since it is electric field independent [4]. The simplified equation is:

$$\frac{\partial n}{\partial t} = D\nabla^2 n + (\upsilon\_i - \upsilon\_a)n \tag{2}$$

Breakdown criteria are based on the fact that the electron density grows very fast once there are more freed electrons than captured. Considering a scenario where there is no diffusion, and a homogeneous field, due to the similar geometry among the parallel walls of the filter and waveguides. Then, Eq. (2) results in:

$$\frac{\partial n}{\partial t} = (\upsilon\_i - \upsilon\_a)n \tag{3}$$

and solving the derivative results in:

the atmospheric pressure, there is a lower input power handling capacity until it reaches a

This chapter covers the ionization breakdown in atmospheric air analysis; the equations that describe this consider different processes, such as the ionization, attachment and collision frequencies. Also, the analysis considers as variables the atmospheric pressure and electric field intensity, considering contaminant-free dry air as the propagating medium. Nevertheless, due to the breakdown variability, the design of filters and waveguides is a controversial topic for designers, who consider a wide tolerance range from 0 dB to 3 dB according to minimum

Corona breakdown is the process when electron plasma is created due to the ionization of the gas in areas where the electrical fields are high. Electrical fields in filters and waveguides can lead to corona effects at relatively low pressures (from 1 to 100 Torr), which, in atmospheric terms, are reached in the ionosphere (from 80 to 800 km). This phenomenon cannot occur in

There are different processes that can generate ions; these are by electronic impact, field effect, photo-ionization and thermos-ionization. For the analysis of filters and waveguides, the most relevant is by electronic impact, being directly proportional to the collision frequency between electrons and molecules. The equation that describes the time evolution of free-electron gener-

where ν<sup>i</sup> and ν<sup>a</sup> are the ionization and attachment frequencies, respectively, D is the diffusion coefficient, β is the recombination coefficient and P is the electron production rate by external sources; ∇ð Þ D∇n is the term that defines the diffusion of the electron cloud, from a high-

For the corona effect analysis and from the pre-breakdown stage point of view, the recombination term is discarded, since it is only relevant once the electron density is high enough, which only occurs when the electrical discharge has already begun. Also, the convective term has to be discarded, since a stationary medium is assumed inside the waveguide devices, that is, there is no relevant movement of the gas molecules. Additionally, the diffusion coefficient is considered as space independent, since it is electric field independent [4]. The simplified equation is:

Breakdown criteria are based on the fact that the electron density grows very fast once there are more freed electrons than captured. Considering a scenario where there is no diffusion, and

! <sup>∙</sup>∇<sup>n</sup> <sup>þ</sup> ð Þ vi � va <sup>n</sup> � <sup>β</sup>n<sup>2</sup> <sup>þ</sup> <sup>P</sup> (1)

n þ ð Þ vi � va n (2)

! ∙∇n is the convec-

vacuum conditions, since it is necessary for the presence of a gas to ionize [4].

minimum operation power; this is called critical pressure [4].

breakdown power [4, 5].

62 Emerging Waveguide Technology

ation is [4, 6–8]:

2. Breakdown power threshold

∂n

tive term that considers the possible movement of the gas.

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>∇</sup>ð Þ� <sup>D</sup>∇<sup>n</sup> <sup>v</sup>

density area to a lower-density area, and is entirely space dependent; and v

∂n <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>D</sup>∇<sup>2</sup>

$$m(t) = n\_0 e^{(v\_i - v\_a)t}.\tag{4}$$

When ν<sup>i</sup> > νa, there is an electron avalanche, and the breakdown condition for the continuous waves can be simplified as:

$$\frac{dn}{dt} = 0.\tag{5}$$

As a consequence, the general equation to solve the breakdown threshold stage is:

$$D\nabla^2 n + (v\_i - v\_a)n = 0.\tag{6}$$

Solving the Laplacian term from Eq. (6), and considering a Cartesian coordinate system, since the devices analysed are rectangular waveguides and filters, the equation leads to:

$$\frac{\partial^2 n}{\partial x^2} + \frac{\partial^2 n}{\partial y^2} + \frac{\partial^2 n}{\partial z^2} + \frac{\nu}{D} n = 0 \tag{7}$$

where ν ¼ ν<sup>i</sup> � ν<sup>a</sup> is the effective ionization frequency. The rectangular waveguide device must be analysed from the cross-section, so the third term can be discarded, considering only the width and height of the guide.

$$
\frac{
\partial^2 n
}{
\partial x^2
} + \frac{
\partial^2 n
}{
\partial y^2
} + \frac{
\nu}{D} n = 0
\tag{8}
$$

Establishing the solution as the product of two functions:

$$m(\mathbf{x}, y) = \mathbf{X}(\mathbf{x})\mathbf{Y}(y) \tag{9}$$

Substituting (9) in (8), knowing that the equation components are independent between them, it results in:

$$Y\frac{d^2X}{dx^2} + X\frac{d^2Y}{dy^2} + \frac{\nu}{D}XY = 0\tag{10}$$

and dividing by (9) we get:

$$\frac{1}{X}\frac{d^2X}{dx^2} + \frac{1}{Y}\frac{d^2Y}{dy^2} + \frac{\nu}{D} = 0.\tag{11}$$

This equation can be solved by proposing exponential solutions. Due to the independency among terms, the first term can be solved proposing a negative constant as a result:

$$
\frac{1}{X}\frac{d^2X}{d\mathbf{x}^2} = -\sigma
$$

$$
\frac{d^2X}{d\mathbf{x}^2} + \sigma X = 0.\tag{12}
$$

This analysis considers the first harmonic. Then, Eq. (8) results in:

filters, the shortest height area must be considered for the analysis.

of effective diffusion length is described by Ulf Jordan et al. [9].

values in space, the free-electron density diffusion curve varies [9].

characteristic diffusion length as [8]:

3. Effective diffusion length

inhomogeneous value being ν<sup>i</sup> ¼ νið Þx .

obtained using computational methods:

where

1 Λ2 eff

<sup>¼</sup> <sup>a</sup>�<sup>2</sup>

r

<sup>q</sup> <sup>¼</sup> <sup>a</sup> La � �<sup>2</sup>

π a � �<sup>2</sup>

þ π b � �<sup>2</sup>

ν <sup>D</sup> <sup>¼</sup> <sup>1</sup>

where a and b are the width and height of the rectangular waveguide. MacDonald defined the

This proves that diffusion processes are entirely dependent on the geometry. If one of the dimensions is much bigger than the other, as in a parallel plates experiment, the characteristic diffusion length is π divided by the squared gap distance. This is valid for rectangular waveguides, due to the fact that they have a constant width and height. For the case of waveguide

Nevertheless, a more realistic approach implies the presence of non-homogeneous fields, which renders the ionization frequency also non-homogeneous, and, consequently, the free electron density is also affected. To characterize diffusion losses in these situations, the concept

For homogeneous values of D, vi and νa, the electron density increasing curve is constant, determined only by the border conditions of the analysed geometry. For non-homogeneous

The inhomogeneity of these parameters occurs because of the inhomogeneity of the microwave electric field, which implies that D and ν<sup>a</sup> are approximately constant in space, the only

Ulf Jordan et al. [9] determined that the diffusion length in the presence of non-homogeneous fields also depends on the atmospheric pressure, as shown in Eq. (23). This equation was

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>π</sup><sup>4</sup> <sup>1</sup> <sup>þ</sup> <sup>0</sup>:3677<sup>β</sup> � � <sup>þ</sup>

> > þ πa b � �<sup>2</sup>

π<sup>2</sup>βq 2

þ π b � �<sup>2</sup>

(23)

¼ ν

<sup>D</sup> (21)

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65

Mathematical Analysis of Electrical Breakdown Effects in Waveguides

<sup>Λ</sup><sup>2</sup> (22)

An exponential solution for (12) is proposed and derived twice:

$$X(\mathfrak{x}) = e^{\mathfrak{y}\mathfrak{x}} \tag{13}$$

$$X''(\mathbf{x}) = \gamma^2 \mathbf{e}^{\nu \mathbf{x}}.\tag{14}$$

Substituting (13) and (14) in (12), we get:

$$
\gamma^2 e^{\gamma x} + \sigma e^{\gamma x} = 0 \tag{15}
$$

It can be determined that:

$$\begin{aligned} \gamma^2 + \sigma &= 0\\ \gamma &= \pm \sqrt{-\sigma} \end{aligned} \tag{16}$$

So, the general equation for X xð Þ is:

$$A\mathcal{e}^{i\sqrt{\alpha}\infty} + B\mathcal{e}^{-i\sqrt{\alpha}\infty}.\tag{17}$$

By Euler, Eq. (17) can be rewritten as:

$$\begin{aligned} A[\cos\sqrt{\sigma}\mathbf{x} + i\sin\sqrt{\sigma}\mathbf{x}] + B[\cos\sqrt{\sigma}\mathbf{x} - i\sin\sqrt{\sigma}\mathbf{x}] \\ (A+B)\cos\sqrt{\sigma}\mathbf{x} + (A-B)\sin\sqrt{\sigma}\mathbf{x} \\ k\_1\cos\sqrt{\sigma}\mathbf{x} + k\_2\sin\sqrt{\sigma}\mathbf{x}. \end{aligned} \tag{18}$$

Considering the next border conditions, since there are no free electrons on the walls of the waveguide:

$$X(0) = 0, \qquad \qquad X(a) = 0 \tag{19}$$

where a is the waveguide width distance. Then:

$$\begin{aligned} X(0) &= k\_1 = 0\\ X(\mathbf{x}) &= k\_2 \sin \sqrt{\sigma} \mathbf{x} \\ X(a) &= k\_2 \sin \sqrt{\sigma} b = 0 \end{aligned} \tag{20}$$

The only possibility for a non-trivial solution is: sin ffiffiffi <sup>σ</sup> <sup>p</sup> <sup>b</sup> <sup>¼</sup> <sup>0</sup> ) ffiffiffi <sup>σ</sup> <sup>p</sup> <sup>b</sup> <sup>¼</sup> <sup>m</sup>π; resulting in <sup>σ</sup> <sup>¼</sup> <sup>m</sup><sup>π</sup> a � �<sup>2</sup> where <sup>m</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>3</sup>, …: By the same method, the second component of Eq. (11) is: <sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>n</sup><sup>π</sup> b � �<sup>2</sup> where <sup>n</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>3</sup>, <sup>4</sup>, …

This analysis considers the first harmonic. Then, Eq. (8) results in:

$$
\left(\frac{\pi}{a}\right)^2 + \left(\frac{\pi}{b}\right)^2 = \frac{\nu}{D} \tag{21}
$$

where a and b are the width and height of the rectangular waveguide. MacDonald defined the characteristic diffusion length as [8]:

$$\frac{\nu}{D} = \frac{1}{\Lambda^2} \tag{22}$$

This proves that diffusion processes are entirely dependent on the geometry. If one of the dimensions is much bigger than the other, as in a parallel plates experiment, the characteristic diffusion length is π divided by the squared gap distance. This is valid for rectangular waveguides, due to the fact that they have a constant width and height. For the case of waveguide filters, the shortest height area must be considered for the analysis.

Nevertheless, a more realistic approach implies the presence of non-homogeneous fields, which renders the ionization frequency also non-homogeneous, and, consequently, the free electron density is also affected. To characterize diffusion losses in these situations, the concept of effective diffusion length is described by Ulf Jordan et al. [9].

#### 3. Effective diffusion length

For homogeneous values of D, vi and νa, the electron density increasing curve is constant, determined only by the border conditions of the analysed geometry. For non-homogeneous values in space, the free-electron density diffusion curve varies [9].

The inhomogeneity of these parameters occurs because of the inhomogeneity of the microwave electric field, which implies that D and ν<sup>a</sup> are approximately constant in space, the only inhomogeneous value being ν<sup>i</sup> ¼ νið Þx .

Ulf Jordan et al. [9] determined that the diffusion length in the presence of non-homogeneous fields also depends on the atmospheric pressure, as shown in Eq. (23). This equation was obtained using computational methods:

$$\frac{1}{\Lambda\_{\text{eff}}^2} = a^{-2} \sqrt{\pi^4 \left(1 + 0.3677 \beta\right) + \frac{\pi^2 \beta q}{2}} + \left(\frac{\pi}{b}\right)^2 \tag{23}$$

where

1 X d2 X dx2 ¼ �<sup>σ</sup>

d2 X

X00

γ2 e <sup>γ</sup><sup>x</sup> <sup>þ</sup> <sup>σ</sup><sup>e</sup>

Ae<sup>i</sup> ffiffi

<sup>σ</sup> <sup>p</sup> <sup>x</sup> <sup>þ</sup> <sup>i</sup> sin ffiffiffi <sup>σ</sup> <sup>p</sup> <sup>½</sup> <sup>x</sup>� þ <sup>B</sup> cos ffiffiffi

ð Þ <sup>A</sup> <sup>þ</sup> <sup>B</sup> cos ffiffiffi

k<sup>1</sup> cos ffiffiffi

X xð Þ¼ e

ð Þ¼ <sup>x</sup> <sup>γ</sup><sup>2</sup><sup>e</sup>

<sup>γ</sup><sup>2</sup> <sup>þ</sup> <sup>σ</sup> <sup>¼</sup> <sup>0</sup> <sup>γ</sup> ¼ � ffiffiffiffiffiffiffi

<sup>σ</sup> <sup>p</sup> <sup>x</sup> <sup>þ</sup> Be�<sup>i</sup> ffiffi

<sup>σ</sup> <sup>p</sup> <sup>x</sup>

<sup>σ</sup> <sup>p</sup> <sup>x</sup> <sup>þ</sup> ð Þ <sup>A</sup> � <sup>B</sup> sin ffiffiffi

<sup>σ</sup> <sup>p</sup> <sup>x</sup>:

<sup>σ</sup> <sup>p</sup> <sup>x</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> sin ffiffiffi

Considering the next border conditions, since there are no free electrons on the walls of the

X 0ð Þ¼ k1 ¼ 0 X xð Þ¼ <sup>k</sup><sup>2</sup> sin ffiffiffi

� �<sup>2</sup> where <sup>m</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>3</sup>, …: By the same method, the second component of Eq. (11) is:

X að Þ¼ <sup>k</sup><sup>2</sup> sin ffiffiffi

<sup>σ</sup> <sup>p</sup> <sup>x</sup>

<sup>σ</sup> <sup>p</sup> <sup>b</sup> <sup>¼</sup> <sup>0</sup>

<sup>σ</sup> <sup>p</sup> <sup>x</sup> � <sup>i</sup> sin ffiffiffi <sup>σ</sup> <sup>p</sup> ½ � <sup>x</sup>

<sup>σ</sup> <sup>p</sup> <sup>x</sup>

Xð Þ¼ 0 0, Xað Þ¼ 0 (19)

<sup>σ</sup> <sup>p</sup> <sup>b</sup> <sup>¼</sup> <sup>0</sup> ) ffiffiffi

γx

An exponential solution for (12) is proposed and derived twice:

Substituting (13) and (14) in (12), we get:

It can be determined that:

64 Emerging Waveguide Technology

waveguide:

<sup>σ</sup> <sup>¼</sup> <sup>m</sup><sup>π</sup> a

<sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>n</sup><sup>π</sup> b

So, the general equation for X xð Þ is:

By Euler, Eq. (17) can be rewritten as:

A cos ffiffiffi

where a is the waveguide width distance. Then:

� �<sup>2</sup> where <sup>n</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>3</sup>, <sup>4</sup>, …

The only possibility for a non-trivial solution is: sin ffiffiffi

dx<sup>2</sup> <sup>þ</sup> <sup>σ</sup><sup>X</sup> <sup>¼</sup> <sup>0</sup>: (12)

<sup>γ</sup><sup>x</sup> (13)

<sup>γ</sup><sup>x</sup> <sup>¼</sup> <sup>0</sup> (15)

�<sup>σ</sup> <sup>p</sup> (16)

: (17)

(18)

(20)

<sup>σ</sup> <sup>p</sup> <sup>b</sup> <sup>¼</sup> <sup>m</sup>π; resulting in

: (14)

$$q = \left(\frac{a}{L\_a}\right)^2 + \left(\frac{\pi a}{b}\right)^2$$

a and b are the width and height of the waveguide, respectively, β is a parameter that depends on the used gas (for air β ¼ 5:33) and the value of La is:

$$L\_a = \sqrt{\frac{D}{\nu\_a}}$$

Consequently, as the pressure increases, the effective diffusion length decreases, and the calculated breakdown thresholds are the same as the ones obtained by using the characteristic diffusion length.

#### 4. Ionization, attachment and diffusion in air

When a microwave field is applied, the energy transfer depends on the field's frequency and the environmental conditions (atmospheric pressure and humidity). An effective electric field is defined as [8]:

$$E\_{\rm eff} = \frac{E\_{rms}}{\left(1 + \frac{\alpha^2}{v\_c^2}\right)^{\frac{1}{2}}} \tag{24}$$

α is known as the reduced electric field. Eq. (27) is valid in the range of 32 < α < 100 [4, 6]. The effective field term is very useful, since it relates the properties of the corona in direct current (DC) and alternating current (AC). The effective field produces the same energy transfer as in a DC field, so experimental data can be analysed in DC instead of AC [4, 10, 11]. The attachment

pα<sup>2</sup>

Eb <sup>¼</sup> <sup>V</sup>

where d is the structure width and V is the breakdown voltage. Finally, the breakdown power

<sup>P</sup> <sup>¼</sup> <sup>V</sup><sup>2</sup>

Z is the characteristic impedance of the device. For waveguides and filters, Z is determined

ffiffiffiffiffiffiffiffiffiffiffiffi μ0=ε<sup>0</sup> p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>1</sup> <sup>2</sup>fa ffiffiffiffiffiffiffi μ0ε<sup>0</sup> p

μ<sup>0</sup> and ε<sup>0</sup> are the magnetic permeability and the electrical permittivity on vacuum, respec-

Commonly, the analytical results obtained by using the characteristic diffusion length are considerably lower than the experimental results at the critical pressure; this minimum power breakdown is known as Paschen minimum. Figure 3 shows the experimental values obtained by Carlos et al. [4] compared to the analytical results using the characteristic diffusion length, for a low-pass Ku band filter at 12.5 GHz. The experimental and analytical

Z

<sup>ν</sup>a<sup>3</sup> <sup>¼</sup> 102

ð Þ <sup>α</sup> <sup>þ</sup> <sup>218</sup> <sup>2</sup>

ν<sup>a</sup> ¼ νa<sup>2</sup> þ νa<sup>3</sup> (29)

Mathematical Analysis of Electrical Breakdown Effects in Waveguides

s�<sup>1</sup> � � (30)

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67

p<sup>2</sup> s�<sup>1</sup> � � (31)

<sup>d</sup> (32)

<sup>Z</sup> (33)

� �<sup>2</sup> <sup>r</sup> (34)

frequency is two- and three-body phenomena.

For electrostatic homogeneous fields:

tively, and f is the operating frequency.

5. Analytical results' comparison

results differ by 16%.

can be obtained by:

by:

νa<sup>2</sup> is only valid in the range 0 < α < 60 and is defined as [4, 6]:

<sup>ν</sup>a<sup>2</sup> <sup>≈</sup> <sup>7</sup>:<sup>6</sup> � <sup>10</sup>�<sup>4</sup>

The three-body attachment is field independent and is obtained as follows [4]:

where Erms is the root mean square electric field, ω is the angular frequency (2πf) and ν<sup>c</sup> is the collision frequency between electrons and molecules. For air, the general equation for the collision frequency is [8]:

$$
\nu\_c = 5 \times 10^9 p \text{ [s}^{-1}\text{]}\tag{25}
$$

p is the atmospheric pressure in Torr. From Eq. (24) it can be deduced that in high-pressure cases, the effective field is equal to the RMS field, since the collision frequency increases along with the atmospheric pressure.

The diffusion coefficient in air is determined by [8]:

$$D = \frac{10^6}{p} \left[ \text{cm}^2 \text{s}^{-1} \right]. \tag{26}$$

The ionization frequency can be obtained by [6]:

$$w\_i = 5.14 \times 10^{11} p \exp\left(-73\alpha^{-0.44}\right) \text{[s}^{-1}\text{]}\tag{27}$$

with

$$\alpha = \frac{E\_{rms}}{p\left(1 + \frac{\alpha^2}{v\_c^2}\right)^{\frac{1}{2}}} \equiv \frac{E\_{\text{eff}}}{p} \tag{28}$$

α is known as the reduced electric field. Eq. (27) is valid in the range of 32 < α < 100 [4, 6]. The effective field term is very useful, since it relates the properties of the corona in direct current (DC) and alternating current (AC). The effective field produces the same energy transfer as in a DC field, so experimental data can be analysed in DC instead of AC [4, 10, 11]. The attachment frequency is two- and three-body phenomena.

$$\nu\_a = \nu\_{a2} + \nu\_{a3} \tag{29}$$

νa<sup>2</sup> is only valid in the range 0 < α < 60 and is defined as [4, 6]:

$$
\Delta\nu\_{d2} \approx 7.6 \times 10^{-4} \text{pa}^2 \text{( $\alpha + 218$ )}^2 \text{ [ $\text{s}^{-1}$ ]} \tag{30}
$$

The three-body attachment is field independent and is obtained as follows [4]:

$$\nu\_{\text{d3}} = 10^2 p^2 \left[ \text{s}^{-1} \right] \tag{31}$$

For electrostatic homogeneous fields:

a and b are the width and height of the waveguide, respectively, β is a parameter that depends

Consequently, as the pressure increases, the effective diffusion length decreases, and the calculated breakdown thresholds are the same as the ones obtained by using the characteristic

When a microwave field is applied, the energy transfer depends on the field's frequency and the environmental conditions (atmospheric pressure and humidity). An effective electric field

Eeff <sup>¼</sup> Erms

where Erms is the root mean square electric field, ω is the angular frequency (2πf) and ν<sup>c</sup> is the collision frequency between electrons and molecules. For air, the general equation for the

p is the atmospheric pressure in Torr. From Eq. (24) it can be deduced that in high-pressure cases, the effective field is equal to the RMS field, since the collision frequency increases along

cm<sup>2</sup>

<sup>ν</sup><sup>c</sup> <sup>¼</sup> <sup>5</sup> � 109

<sup>D</sup> <sup>¼</sup> <sup>106</sup> p

<sup>α</sup> <sup>¼</sup> Erms <sup>p</sup> <sup>1</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup> v2 c � �<sup>1</sup> 2 � Eeff

<sup>1</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup> v2 c � �<sup>1</sup> 2 (24)

p s�<sup>1</sup> � � (25)

s�<sup>1</sup> � �: (26)

<sup>p</sup> (28)

vi <sup>¼</sup> <sup>5</sup>:<sup>14</sup> � 1011<sup>p</sup> exp �73α�0:<sup>44</sup> � � <sup>s</sup>�<sup>1</sup> � � (27)

ffiffiffiffi D νa

s

La ¼

on the used gas (for air β ¼ 5:33) and the value of La is:

4. Ionization, attachment and diffusion in air

diffusion length.

66 Emerging Waveguide Technology

is defined as [8]:

collision frequency is [8]:

with the atmospheric pressure.

with

The diffusion coefficient in air is determined by [8]:

The ionization frequency can be obtained by [6]:

$$E\_b = \frac{V}{d} \tag{32}$$

where d is the structure width and V is the breakdown voltage. Finally, the breakdown power can be obtained by:

$$P = \frac{V^2}{Z} \tag{33}$$

Z is the characteristic impedance of the device. For waveguides and filters, Z is determined by:

$$Z \frac{\sqrt{\mu\_0/\varepsilon\_0}}{\sqrt{1 - \left(\frac{1}{2^{\sharp\_0}\sqrt{\mu\_0\varepsilon\_0}}\right)^2}}\tag{34}$$

μ<sup>0</sup> and ε<sup>0</sup> are the magnetic permeability and the electrical permittivity on vacuum, respectively, and f is the operating frequency.

#### 5. Analytical results' comparison

Commonly, the analytical results obtained by using the characteristic diffusion length are considerably lower than the experimental results at the critical pressure; this minimum power breakdown is known as Paschen minimum. Figure 3 shows the experimental values obtained by Carlos et al. [4] compared to the analytical results using the characteristic diffusion length, for a low-pass Ku band filter at 12.5 GHz. The experimental and analytical results differ by 16%.

Figure 3. Low-pass Ku band filter operating at 12.5 GHz [4].

The results shown imply that it is necessary to consider the inhomogeneity of the electric fields, not due to geometry but due to the diffusion process that occurs at low pressures [12]. Then, instead of using the characteristic diffusion length, the effective diffusion length is used. Figure 4 shows the experimental and analytical results using the characteristic diffusion length compared to the analytical results using the effective diffusion length for the same low-pass filter.

threshold of a device with high amount of irregularities, such as Figure 6 (b), requires modifications to the equations, more specifically, the use of an effective field-dependent collision frequency equation. The large number of irises generates a much higher space charge density than for a conventional obstruction-free waveguide. Further analysis of space

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charge density and the correct equations for these cases are considered in Section 6.

Figure 5. Experimental and analytical results using Λ and Λeff for a Ku band filter operating at 12.2 GHz.

Figure 4. Experimental and analytical results using Λ and Λeff for a Ku band filter operating at 12.5 GHz.

Figure 5 shows another result comparison but for a corrugated waveguide filter operating at 12.2 GHz.

Figures 4 and 5 show a slight increase in breakdown power, proving that the effective diffusion length from Eq. (23) is a more suitable equation for the cases of waveguide devices where the field inhomogeneity is greater.

These results can be explained because microwave breakdown in an RF device is manifested by an avalanche-like growth in time of the free-electron density in the gas filling the device. The difference between these power threshold results resides not only in their operating frequency but in their geometries and the number of irregularities the filter contains. A bigger amount of irregularities, or irises (steps that help in the filtering process), contributes to generating more inhomogeneity on the electric field.

Figure 6 shows the transversal configuration and measures of each filter [4].

For the analysis of each filter, the minimum length located in the middle is considered.

The filter operating at 12.5 GHz is affected by the electric field inhomogeneity more than the other because of its high number of irises. Predicting mathematically the breakdown Mathematical Analysis of Electrical Breakdown Effects in Waveguides http://dx.doi.org/10.5772/intechopen.76973 69

Figure 4. Experimental and analytical results using Λ and Λeff for a Ku band filter operating at 12.5 GHz.

The results shown imply that it is necessary to consider the inhomogeneity of the electric fields, not due to geometry but due to the diffusion process that occurs at low pressures [12]. Then, instead of using the characteristic diffusion length, the effective diffusion length is used. Figure 4 shows the experimental and analytical results using the characteristic diffusion length compared

Figure 5 shows another result comparison but for a corrugated waveguide filter operating at

Figures 4 and 5 show a slight increase in breakdown power, proving that the effective diffusion length from Eq. (23) is a more suitable equation for the cases of waveguide devices where

These results can be explained because microwave breakdown in an RF device is manifested by an avalanche-like growth in time of the free-electron density in the gas filling the device. The difference between these power threshold results resides not only in their operating frequency but in their geometries and the number of irregularities the filter contains. A bigger amount of irregularities, or irises (steps that help in the filtering process), contributes to

to the analytical results using the effective diffusion length for the same low-pass filter.

12.2 GHz.

68 Emerging Waveguide Technology

the field inhomogeneity is greater.

generating more inhomogeneity on the electric field.

Figure 3. Low-pass Ku band filter operating at 12.5 GHz [4].

Figure 6 shows the transversal configuration and measures of each filter [4].

For the analysis of each filter, the minimum length located in the middle is considered.

The filter operating at 12.5 GHz is affected by the electric field inhomogeneity more than the other because of its high number of irises. Predicting mathematically the breakdown threshold of a device with high amount of irregularities, such as Figure 6 (b), requires modifications to the equations, more specifically, the use of an effective field-dependent collision frequency equation. The large number of irises generates a much higher space charge density than for a conventional obstruction-free waveguide. Further analysis of space charge density and the correct equations for these cases are considered in Section 6.

Figure 5. Experimental and analytical results using Λ and Λeff for a Ku band filter operating at 12.2 GHz.

Figure 6. Transversal configuration of waveguide filters. (a) Operating at 12.2 GHz. (b) Operating at 12.5 GHz [4].


Table 1. Calculated bit rate from power obtained by characteristic diffusion length and effective diffusion length for two different Ku band filters.

According to Witting [13], the transmission capacity of a communication network in terms of the number of users, power and data rate is:

$$\frac{R\_B \cdot N}{P} \le 20 \times 10^{18} \left[ \text{W}^{-1} \text{s}^{-1} \right] \tag{35}$$

The two main processes responsible for the electron losses during the breakdown stages are the diffusion from high-density regions towards lower-density regions and the attachment by neutral molecules, forming essentially negatively charged unmovable ions. For sufficiently enough electron density, at breakdown threshold, the region saturates and the electric field

The avalanche evolution can be affected by any agent that alters the space charge electronic density. Figure 7 shows the electric field Er around the avalanche and the resulting modification of the applied field E0. The space charge at the head of the avalanche is assumed as concentrated within a spherical volume, with the negative charge ahead because of the higher electron mobility. The field is enhanced in front of the head of the avalanche with field lines from the

Figure 7. Schematic representation of electric field distortion in a gap caused by space charge of an electron avalanche [16].

<sup>2</sup> and O� and the most important

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Mathematical Analysis of Electrical Breakdown Effects in Waveguides

<sup>2</sup> , N<sup>þ</sup> <sup>4</sup> and 71

propagation is affected by its reflection or absorption in the device walls.

positive ions formed during electrical breakdowns at atmospheric pressure are N<sup>þ</sup>

<sup>2</sup> . There have not been negative ions detected for nitrogen experimentally [15].

The most important negative ions present in air are O�

O<sup>þ</sup>

where RB is the data rate in bits/s, N is the number of users and P is the power in watts. Table 1 shows the difference of bit rate obtained with the power from characteristic diffusion length calculation and the one from effective diffusion length calculation for the two different Ku band filters.

By using (35), the resulting increase on the bit rate of the filters, when using the effective diffusion length, is of 4.3% in the case of the 12.5 GHz low-pass filter and of 3.1% for the 12.2 GHz low-pass filter. Therefore, a small raise in the power, even of 3 or 4 W, is heavily reflected on the data rate and an increase of almost 200 Gbps is achieved.

#### 6. Space charge density effects

The microwave devices' designers use the analytical solution of the corona discharge to determine if the operating power is within the established margins. As shown previously, the experimental results differ considerably from the analytical when the characteristic diffusion length is considered. It has been proved by some authors [2, 9, 12, 14] that the criteria used until now for the design of waveguide filters can be improved if the effective diffusion length is used instead of the characteristic diffusion length.

The two main processes responsible for the electron losses during the breakdown stages are the diffusion from high-density regions towards lower-density regions and the attachment by neutral molecules, forming essentially negatively charged unmovable ions. For sufficiently enough electron density, at breakdown threshold, the region saturates and the electric field propagation is affected by its reflection or absorption in the device walls.

The most important negative ions present in air are O� <sup>2</sup> and O� and the most important positive ions formed during electrical breakdowns at atmospheric pressure are N<sup>þ</sup> <sup>2</sup> , N<sup>þ</sup> <sup>4</sup> and O<sup>þ</sup> <sup>2</sup> . There have not been negative ions detected for nitrogen experimentally [15].

The avalanche evolution can be affected by any agent that alters the space charge electronic density. Figure 7 shows the electric field Er around the avalanche and the resulting modification of the applied field E0. The space charge at the head of the avalanche is assumed as concentrated within a spherical volume, with the negative charge ahead because of the higher electron mobility. The field is enhanced in front of the head of the avalanche with field lines from the

According to Witting [13], the transmission capacity of a communication network in terms of

Table 1. Calculated bit rate from power obtained by characteristic diffusion length and effective diffusion length for two

Λeff 86.1 4305

Λeff 101.4 5070

Figure 6. Transversal configuration of waveguide filters. (a) Operating at 12.2 GHz. (b) Operating at 12.5 GHz [4].

Analysed filter Minimum breakdown power [W] Data rate [Gb/s]

12.2 GHz Λ 83.5 4175

12.5 GHz Λ 97.2 4860

<sup>P</sup> <sup>≤</sup> <sup>20</sup> <sup>1018</sup> <sup>W</sup><sup>1</sup>

where RB is the data rate in bits/s, N is the number of users and P is the power in watts. Table 1 shows the difference of bit rate obtained with the power from characteristic diffusion length calculation and the one from effective diffusion length calculation for the two different Ku

By using (35), the resulting increase on the bit rate of the filters, when using the effective diffusion length, is of 4.3% in the case of the 12.5 GHz low-pass filter and of 3.1% for the 12.2 GHz low-pass filter. Therefore, a small raise in the power, even of 3 or 4 W, is heavily reflected on the data rate

The microwave devices' designers use the analytical solution of the corona discharge to determine if the operating power is within the established margins. As shown previously, the experimental results differ considerably from the analytical when the characteristic diffusion length is considered. It has been proved by some authors [2, 9, 12, 14] that the criteria used until now for the design of waveguide filters can be improved if the effective diffusion length is

s

<sup>1</sup> (35)

RB∙N

the number of users, power and data rate is:

and an increase of almost 200 Gbps is achieved.

used instead of the characteristic diffusion length.

6. Space charge density effects

band filters.

different Ku band filters.

70 Emerging Waveguide Technology

Figure 7. Schematic representation of electric field distortion in a gap caused by space charge of an electron avalanche [16].

anode terminating at the head, the region III. Further back in the avalanche, the field between the electrons and the ions left behind reduced the applied field (E0), the region II. Still, further back, the field between the cathode and the positive ions is enhanced again, the region I [16].

The resultant field strength in front of the avalanche is thus (E<sup>0</sup> þ Er), whereas in the positive ion region just behind the head the field is reduced to a value (E<sup>0</sup> � Er).

According to the results exhibited in Figures 4 and 5, where the analytical values of the breakdown power are lower than the experimental ones, this is an indication that the avalanche is mainly affected by the presence of positively charged ions instead of the negatively charged ions. The radial field produced by positive ions immediately behind the head of the avalanche can be calculated using the expression from [16]:

$$E\_r = 5.3 \cdot 10^{-7} \frac{\alpha e^{a\_{\Gamma} \chi}}{\binom{\mathfrak{z}}{p}^{\frac{1}{2}}} \left[\frac{\text{Volts}}{\text{cm}}\right] \tag{36}$$

It can be appreciated that the influence of the positive ionic space charge field is greater as a

For a more correct approach of the analytical results, Woo et al. [6] propose the collision

p α α þ 8 <sup>1</sup>=<sup>2</sup>

Figures 9 and 10 show the results of using this energy-dependent collision frequency equation

Figure 9. Breakdown power of a Ku band filter operating at 12.5 GHz using different collision frequency values.

Figure 10. Breakdown power of a Ku band filter operating at 12.2 GHz using different collision frequency values.

: (38)

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Mathematical Analysis of Electrical Breakdown Effects in Waveguides

<sup>ν</sup><sup>c</sup> <sup>¼</sup> <sup>5</sup> � 109

function of the development in the space of the avalanche.

and the effective diffusion length.

frequency equation dependent on the reduced electric field as:

where x is the distance in cm in which the avalanche has progressed, p is the gas pressure in Torr and α is the Townsend first coefficient of ionization, denoted by:

$$
\alpha\_T \mathbf{x} = \mathbf{1}7.7 + \ln \mathbf{x} \tag{37}
$$

The Townsend first ionization coefficient indicates the number of ions generated by the electron collision by length unity. Figure 8 shows the behaviour of Er and E<sup>0</sup> as a function of the pressure for different values of x, x ¼ a ¼ 0:25 cm, as this is the maximum height of the analysed Ku band filter.

Figure 8. Applied electric breakdown E<sup>0</sup> versus space charge electric field Er for different values of x.

It can be appreciated that the influence of the positive ionic space charge field is greater as a function of the development in the space of the avalanche.

anode terminating at the head, the region III. Further back in the avalanche, the field between the electrons and the ions left behind reduced the applied field (E0), the region II. Still, further back,

The resultant field strength in front of the avalanche is thus (E<sup>0</sup> þ Er), whereas in the positive

According to the results exhibited in Figures 4 and 5, where the analytical values of the breakdown power are lower than the experimental ones, this is an indication that the avalanche is mainly affected by the presence of positively charged ions instead of the negatively charged ions. The radial field produced by positive ions immediately behind the head of the avalanche can be

> x p <sup>1</sup> 2

where x is the distance in cm in which the avalanche has progressed, p is the gas pressure in

The Townsend first ionization coefficient indicates the number of ions generated by the electron collision by length unity. Figure 8 shows the behaviour of Er and E<sup>0</sup> as a function of the pressure for different values of x, x ¼ a ¼ 0:25 cm, as this is the maximum height of the

Volts cm 

αTx ¼ 17:7 þ ln x (37)

(36)

the field between the cathode and the positive ions is enhanced again, the region I [16].

Er <sup>¼</sup> <sup>5</sup>:3∙10�<sup>7</sup> <sup>α</sup>e<sup>α</sup><sup>T</sup> <sup>x</sup>

ion region just behind the head the field is reduced to a value (E<sup>0</sup> � Er).

Torr and α is the Townsend first coefficient of ionization, denoted by:

Figure 8. Applied electric breakdown E<sup>0</sup> versus space charge electric field Er for different values of x.

calculated using the expression from [16]:

analysed Ku band filter.

72 Emerging Waveguide Technology

For a more correct approach of the analytical results, Woo et al. [6] propose the collision frequency equation dependent on the reduced electric field as:

$$
\omega\_c = 5 \times 10^9 p \left[ \frac{\alpha}{\alpha + 8} \right]^{1/2}. \tag{38}
$$

Figures 9 and 10 show the results of using this energy-dependent collision frequency equation and the effective diffusion length.

Figure 9. Breakdown power of a Ku band filter operating at 12.5 GHz using different collision frequency values.

Figure 10. Breakdown power of a Ku band filter operating at 12.2 GHz using different collision frequency values.

It is shown that considering the diffusion length and the electric field-dependent collision frequency altogether, the results are far more similar to the experimental results, proving this to be an important approach towards the experimental results.

As minimal as these increases result, Table 1 shows the importance of power, and the regulations for the design of these devices can be increased in terms of input power tolerance.

### 7. Plasmonic waveguide filters for increased data rate transmission

As the actual waveguide devices reach the technological limit, in terms of their data rate, it is necessary to develop new alternatives to overcome the continuously increasing demand of services [12]. By using encoding techniques, it is possible to send up to 16 bits of information per each Hertz sent [17]; the current Ku band analysed devices operate generally around 12.2 GHz, so the data rate is only of 195.2 Gbps. Much higher frequencies, such as those provided by optical communication, of about 350 THz, show a much promising environment, delivering up to 5600 Tbps.

Optical wireless communications demand different multiplexing and de-multiplexing techniques than traditional RF communication. For this, some proposals include a wavelength divisor multiplexer (WDM), this can be a polymer substrate mode for photonic interconnections and is used even for satellite communications [18]. This helps in a way that incoming signals are directly coupled with the system chip, leaving out any optical-electrical and electrical-optical conversions. This is a partial solution since the system needs power and wavelength management; for this, digital grating processors (DGPs) are implemented. There are many advantages that these photonic interconnections provide, among them are introducing a planar platform for space-saving purposes, efficiency against any external perturbation, low propagation losses, compatibility with other surface mount technologies and low cost. Nevertheless, DGPs are components that demand energy from the system to operate and generate interruptions in the transmission due to electronic processing. Other components can be responsible for the filtering of signals; as seen by Calva et al. [2], a plasmonic waveguide filter is a viable option.

The configuration of these filters is formed by stacking nanometric waveguides of the same gap length. Multiple resonance modes are formed inside the devices; only the first and second mode can traverse through the next waveguides, the one in the middle of Figure 11 and the port 1 and 2 vertical waveguides. The SPPs travel through the principal plasmonic waveguide; resonance happens if the SPPs are enclosed in the middle cavity. This mid-section is very important, since its size is responsible for the filtering effect; modifications of its length alter the delivered wavelength through ports 1 and 2. A wide range of wavelengths can be covered by using these filters, from 500 to 10,000 nm. However, some optimal configurations have been suggested; for distances of L=280, 320, 350, 360, and 390 nm, the transmitted resultant wavelengths are λ=575, 850, 1060, 1310, and 1550 nm, respectively [2]. As mentioned before, it is imperative to consider the field skin deep so that light can travel through the metal of these waveguide structures; d<sup>1</sup> and d<sup>2</sup> distances must be smaller than 30 nm in the case of silver [21]. The field skin deep distance describes the length of a given material, silver in this case, that an electromagnetic signal can penetrate, this distance depends on the signal frequency and the material properties. So, in this case, the distances are 10 and 15 nm, respectively, with a gap distance for all the waveguides of W ¼ 50 nm. These filters have very good transmission spectra and can easily operate in wavelengths from 575 to 1500 nm, optimal for using them

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The analysis of electromagnetic waves through a surface already excited contemplates that the electrons are in a non-equilibrium state and that they are generated because of light absorption, not only due to collisions. The absorption can be linear or multiple, resulting in many nonequilibrium electrons; then, considering the diffusion effect, electron–electron collisions occur and there is an energy exchange between the photon-excited electrons and the non-equilibrium electrons. The evolution in time of the free-electron density generated by excited photons and

for an optical communication scheme.

Figure 11. Two-channel plasmonic waveguide filter structure.

electron–electron collisions is [22]:

Since the interconnection is very important, as the planar configuration of the devices, plasmonic waveguide filter proves a viable solution due to their capability of transfer information operating at different frequencies at the same time. Surface plasmons' inherent properties permit the signal to travel at the speed of light and also transport electrical and optical signals simultaneously [19]. The disadvantage of using these devices is that electrical breakdown due to ionization phenomena can occur.

These particular devices' operating principle is based on the light capability to penetrate some materials; for metals this can be up to 30 nm deep, helping in the generation of surface plasmons, which are oscillating free electrons in a coherent state that generate at the interface between any two materials. In some cases, incident light couples with the surface plasmon to generate self-sustaining propagating electromagnetic waves; these are known as surface plasmon polaritons (SPPs) [19]. A plasmonic waveguide filter example is shown in Figure 11; this is based on a metal–insulator–metal (MIM) structure [20].

Mathematical Analysis of Electrical Breakdown Effects in Waveguides http://dx.doi.org/10.5772/intechopen.76973 75

Figure 11. Two-channel plasmonic waveguide filter structure.

It is shown that considering the diffusion length and the electric field-dependent collision frequency altogether, the results are far more similar to the experimental results, proving this

As minimal as these increases result, Table 1 shows the importance of power, and the regulations for the design of these devices can be increased in terms of input power tolerance.

As the actual waveguide devices reach the technological limit, in terms of their data rate, it is necessary to develop new alternatives to overcome the continuously increasing demand of services [12]. By using encoding techniques, it is possible to send up to 16 bits of information per each Hertz sent [17]; the current Ku band analysed devices operate generally around 12.2 GHz, so the data rate is only of 195.2 Gbps. Much higher frequencies, such as those provided by optical communication,

Optical wireless communications demand different multiplexing and de-multiplexing techniques than traditional RF communication. For this, some proposals include a wavelength divisor multiplexer (WDM), this can be a polymer substrate mode for photonic interconnections and is used even for satellite communications [18]. This helps in a way that incoming signals are directly coupled with the system chip, leaving out any optical-electrical and electrical-optical conversions. This is a partial solution since the system needs power and wavelength management; for this, digital grating processors (DGPs) are implemented. There are many advantages that these photonic interconnections provide, among them are introducing a planar platform for space-saving purposes, efficiency against any external perturbation, low propagation losses, compatibility with other surface mount technologies and low cost. Nevertheless, DGPs are components that demand energy from the system to operate and generate interruptions in the transmission due to electronic processing. Other components can be responsible for the filtering

7. Plasmonic waveguide filters for increased data rate transmission

of about 350 THz, show a much promising environment, delivering up to 5600 Tbps.

of signals; as seen by Calva et al. [2], a plasmonic waveguide filter is a viable option.

to ionization phenomena can occur.

this is based on a metal–insulator–metal (MIM) structure [20].

Since the interconnection is very important, as the planar configuration of the devices, plasmonic waveguide filter proves a viable solution due to their capability of transfer information operating at different frequencies at the same time. Surface plasmons' inherent properties permit the signal to travel at the speed of light and also transport electrical and optical signals simultaneously [19]. The disadvantage of using these devices is that electrical breakdown due

These particular devices' operating principle is based on the light capability to penetrate some materials; for metals this can be up to 30 nm deep, helping in the generation of surface plasmons, which are oscillating free electrons in a coherent state that generate at the interface between any two materials. In some cases, incident light couples with the surface plasmon to generate self-sustaining propagating electromagnetic waves; these are known as surface plasmon polaritons (SPPs) [19]. A plasmonic waveguide filter example is shown in Figure 11;

to be an important approach towards the experimental results.

74 Emerging Waveguide Technology

The configuration of these filters is formed by stacking nanometric waveguides of the same gap length. Multiple resonance modes are formed inside the devices; only the first and second mode can traverse through the next waveguides, the one in the middle of Figure 11 and the port 1 and 2 vertical waveguides. The SPPs travel through the principal plasmonic waveguide; resonance happens if the SPPs are enclosed in the middle cavity. This mid-section is very important, since its size is responsible for the filtering effect; modifications of its length alter the delivered wavelength through ports 1 and 2. A wide range of wavelengths can be covered by using these filters, from 500 to 10,000 nm. However, some optimal configurations have been suggested; for distances of L=280, 320, 350, 360, and 390 nm, the transmitted resultant wavelengths are λ=575, 850, 1060, 1310, and 1550 nm, respectively [2]. As mentioned before, it is imperative to consider the field skin deep so that light can travel through the metal of these waveguide structures; d<sup>1</sup> and d<sup>2</sup> distances must be smaller than 30 nm in the case of silver [21]. The field skin deep distance describes the length of a given material, silver in this case, that an electromagnetic signal can penetrate, this distance depends on the signal frequency and the material properties. So, in this case, the distances are 10 and 15 nm, respectively, with a gap distance for all the waveguides of W ¼ 50 nm. These filters have very good transmission spectra and can easily operate in wavelengths from 575 to 1500 nm, optimal for using them for an optical communication scheme.

The analysis of electromagnetic waves through a surface already excited contemplates that the electrons are in a non-equilibrium state and that they are generated because of light absorption, not only due to collisions. The absorption can be linear or multiple, resulting in many nonequilibrium electrons; then, considering the diffusion effect, electron–electron collisions occur and there is an energy exchange between the photon-excited electrons and the non-equilibrium electrons. The evolution in time of the free-electron density generated by excited photons and electron–electron collisions is [22]:

$$\frac{\partial n}{\partial t} = D\nabla^2 n + \frac{(1-R)a\_1 I}{\hbar \omega} + \frac{(1-R)^2 a\_2 I^2}{2\hbar \omega} - \frac{n}{\langle \tau\_{\text{ee}} \rangle} \tag{39}$$

where D is the electronic diffusion coefficient, h i τee is the time between the electron–electron collisions, I is the irradiance of light in watts per square meter, R is the reflection coefficient, α<sup>1</sup> is the linear photonic absorption coefficient and α<sup>2</sup> is the two-photon absorption coefficient.

According to Bhushan et al. [23], there is no two-photon absorption for the cases where the plasmon has an angular momentum of <sup>l</sup> <sup>&</sup>gt; 1 kgm<sup>2</sup>=s. <sup>l</sup> <sup>¼</sup> 1 corresponds to the bipolar resonance of the plasmon, which is the one that occurs in these types of filters [20]. Then, Eq. (39) is reduced to the following:

$$\frac{1}{\Lambda\_{\text{eff}}} = \left(\frac{(1-R)\alpha\_1 I}{\hbar \omega} - \frac{n}{\langle \tau\_{\text{ee}} \rangle}\right) / nD \tag{40}$$

Substituting <sup>1</sup> h i <sup>τ</sup>ee <sup>¼</sup> <sup>ν</sup>c:

$$\frac{1}{\Lambda\_{\text{eff}}} = \frac{(1 - R)\alpha\_1 I}{\hbar \omega n D} - \frac{\nu\_c}{D} \tag{41}$$

The lineal photonic absorption is obtained using the following [24]:

$$
\alpha\_1 = 4\pi \overline{k} / \lambda \left[ \text{cm}^{-1} \right], \tag{42}
$$

These extremely low power values are not a problem in the data transmission, according to Radek Kvicala et al. [26]; the optical communication systems are capable of receiving very low

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Figure 12. Power breakdown threshold of a two-channel plasmonic waveguide filter at different wavelengths.

The suggested modifications to the waveguide devices breakdown threshold analysis change the operating power in terms of the continuously increasing bandwidths and component integration. Increasing power handling in these devices by just a few watts have a considerable effect in the data rate, increasing its value, whereas avoiding the risk of break-

Waveguide designers use the free electrons in the time equation to obtain the lowest possible breakdown thresholds, which implies that homogeneous electric fields as a function of the geometry are considered. However, the presence of space charge inside the devices causes inhomogeneities in the electric field; therefore, it is important to determine the device structure for a correct analysis. When analysing a waveguide filter, the substructures inside it that generate the filtering effect, highly non-homogeneous areas are located. In these cases, the use of the effective diffusion length, along with the collision frequency equation that highly depends on the electric field, must be imperative for a correct approximation of the real values. Plasmonic waveguide filters are a good proposal for the implementation of higher-frequency technologies. For wavelengths from 575 to 1500 nm, the power breakdown threshold is located between 0.1 and 0.4 Watts at 1 Torr atmospheric pressure. These power thresholds are

optical powers of about <sup>P</sup> <sup>¼</sup> <sup>4</sup> � <sup>10</sup>�<sup>14</sup> Watts.

8. Conclusions

down to occur.

where k is the extinction coefficient. Table 2 shows the experimentally obtained values in [24] for the extinction coefficients at a specific wavelength and the corresponding absorption coefficients.

The electron density in the electrical breakdown threshold is <sup>n</sup> <sup>¼</sup> <sup>1</sup>:<sup>1</sup> � 1015cm�<sup>3</sup> [25]. The reflection coefficient of silver is R ¼ 0:95. The ℏω term is the energy of a photon, where ℏ is the Planck's constant divided by 2π:

$$h\hbar\omega = \frac{h2\pi f}{2\pi} = hf = 6.62 \times 10^{-34} f \text{[J]}\tag{43}$$

Using these equations and the effective diffusion length, as discussed before, in (39) the power breakdown threshold of a plasmonic waveguide filter can be obtained. Figure 12 shows the power breakdown threshold of a plasmonic waveguide filter at different wavelengths.


Table 2. Experimental values for the extinction and linear photonic absorption coefficients.

Mathematical Analysis of Electrical Breakdown Effects in Waveguides http://dx.doi.org/10.5772/intechopen.76973 77

Figure 12. Power breakdown threshold of a two-channel plasmonic waveguide filter at different wavelengths.

These extremely low power values are not a problem in the data transmission, according to Radek Kvicala et al. [26]; the optical communication systems are capable of receiving very low optical powers of about <sup>P</sup> <sup>¼</sup> <sup>4</sup> � <sup>10</sup>�<sup>14</sup> Watts.

#### 8. Conclusions

∂n <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>D</sup>∇<sup>2</sup>

to the following:

76 Emerging Waveguide Technology

Substituting <sup>1</sup>

h i <sup>τ</sup>ee <sup>¼</sup> <sup>ν</sup>c:

Planck's constant divided by 2π:

<sup>n</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>R</sup> <sup>α</sup>1<sup>I</sup> ℏω

<sup>¼</sup> ð Þ <sup>1</sup> � <sup>R</sup> <sup>α</sup>1<sup>I</sup>

1 Λeff <sup>ℏ</sup><sup>ω</sup> � <sup>n</sup>

<sup>¼</sup> ð Þ <sup>1</sup> � <sup>R</sup> <sup>α</sup>1<sup>I</sup> <sup>ℏ</sup>ωnD � <sup>ν</sup><sup>c</sup>

where k is the extinction coefficient. Table 2 shows the experimentally obtained values in [24] for the extinction coefficients at a specific wavelength and the corresponding absorption coefficients.

The electron density in the electrical breakdown threshold is <sup>n</sup> <sup>¼</sup> <sup>1</sup>:<sup>1</sup> � 1015cm�<sup>3</sup> [25]. The reflection coefficient of silver is R ¼ 0:95. The ℏω term is the energy of a photon, where ℏ is the

Using these equations and the effective diffusion length, as discussed before, in (39) the power breakdown threshold of a plasmonic waveguide filter can be obtained. Figure 12 shows the

power breakdown threshold of a plasmonic waveguide filter at different wavelengths.

Wavelength nm½ � Extinction coefficient <sup>k</sup> Absorption coefficient cm�<sup>1</sup>

h i τee

1 Λeff

The lineal photonic absorption is obtained using the following [24]:

<sup>ℏ</sup><sup>ω</sup> <sup>¼</sup> <sup>h</sup>2π<sup>f</sup>

 3.45 7.54E + 05 5.70 8.43E + 05 7.33 8.69E + 05 9.10 8.73E + 05 10.60 8.59E + 05

Table 2. Experimental values for the extinction and linear photonic absorption coefficients.

where D is the electronic diffusion coefficient, h i τee is the time between the electron–electron collisions, I is the irradiance of light in watts per square meter, R is the reflection coefficient, α<sup>1</sup> is the linear photonic absorption coefficient and α<sup>2</sup> is the two-photon absorption coefficient. According to Bhushan et al. [23], there is no two-photon absorption for the cases where the plasmon has an angular momentum of <sup>l</sup> <sup>&</sup>gt; 1 kgm<sup>2</sup>=s. <sup>l</sup> <sup>¼</sup> 1 corresponds to the bipolar resonance of the plasmon, which is the one that occurs in these types of filters [20]. Then, Eq. (39) is reduced

<sup>þ</sup> ð Þ <sup>1</sup> � <sup>R</sup> <sup>2</sup>

α2I 2 <sup>2</sup>ℏ<sup>ω</sup> � <sup>n</sup>

h i τee

=nD (40)

<sup>D</sup> (41)

<sup>α</sup><sup>1</sup> <sup>¼</sup> <sup>4</sup>πk=<sup>λ</sup> cm�<sup>1</sup> , (42)

<sup>2</sup><sup>π</sup> <sup>¼</sup> hf <sup>¼</sup> <sup>6</sup>:<sup>62</sup> � <sup>10</sup>�<sup>34</sup><sup>f</sup> ½ �<sup>J</sup> (43)

(39)

The suggested modifications to the waveguide devices breakdown threshold analysis change the operating power in terms of the continuously increasing bandwidths and component integration. Increasing power handling in these devices by just a few watts have a considerable effect in the data rate, increasing its value, whereas avoiding the risk of breakdown to occur.

Waveguide designers use the free electrons in the time equation to obtain the lowest possible breakdown thresholds, which implies that homogeneous electric fields as a function of the geometry are considered. However, the presence of space charge inside the devices causes inhomogeneities in the electric field; therefore, it is important to determine the device structure for a correct analysis. When analysing a waveguide filter, the substructures inside it that generate the filtering effect, highly non-homogeneous areas are located. In these cases, the use of the effective diffusion length, along with the collision frequency equation that highly depends on the electric field, must be imperative for a correct approximation of the real values.

Plasmonic waveguide filters are a good proposal for the implementation of higher-frequency technologies. For wavelengths from 575 to 1500 nm, the power breakdown threshold is located between 0.1 and 0.4 Watts at 1 Torr atmospheric pressure. These power thresholds are sufficient for the electronic processing required in an optical environment, since optical systems are capable of fully operating while receiving very low power, <sup>P</sup> <sup>¼</sup> <sup>4</sup> � <sup>10</sup>�<sup>14</sup> <sup>W</sup>.

[11] Ali AW. Intense and Short Pulse Electric Field (DC and Microwave) Air Breakdown Param-

Mathematical Analysis of Electrical Breakdown Effects in Waveguides

http://dx.doi.org/10.5772/intechopen.76973

79

[12] Calva PA, Medina I. New solutions of the corona discharge equation for applications in waveguide filters in SAT-COM. IEEE Transactions on Plasma Science. April, 2013;41(4)

[13] Witting M. Satellite onboard processing for multimedia applications. IEEE Communica-

[14] Jordan U, Anderson D, Semenov V, Puech J. Discussion on the effective diffusion length

[17] International Telecommunications Union. Handbook on Satellite Communications. New

[18] Liu J, Gu L, Chen R, Craig D. WDM polymer substrate mode photonic interconnects for satellite communications. Photonics packaging and integration IV, Proceedings of SPIE,

[19] Novotny L, Hecht B. Principles of Nano-Optics. Cambridge: Cambridge University Press;

[20] Wen K, Yan L, Pan W, Luo B, Guo Z, Guo Y. Wavelength demultiplexing structure based on a plasmonic metal-insulator-metal waveguide. Journal of Optics, IOP Publishing. 2012;

[21] Dionne JA, Sweatlock LA, Atwater HA. Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization. Physical Review B. 2006;73(035407)

[22] Martsinovsky GA, et al. The role of plasmon-polaritons and waveguide modes in surface modification of semiconductors by ultrashort laser pulses, Fundamentals of laser assisted

[23] Bhushan B, Kundu T, Singh BP. Two photon absorption spectrum of silver nanoparticles.

[25] Unnikrishnan VK, Kamlesh A, et al. Measurements of plasma temperature and electron temperature in laser-induced copper plasma by time-resolved spectroscopy of neutral

[26] Kvicala R, Hampl M, P Kucera. Satellite terrestrial (Earth) station optical communication.

micro- and nanotechnologies. Proceedings of SPIE, Vol. 6985; 2008

atom and ion emissions. Pramana-Journal of Physics. 2010;74

Northern Optics, Bergen, IEEE; 2006. pp. 83-85

[24] Palik ED. Handbook of Optical Constants of Solids. New York: Academic; 1991

[15] Badaloni S, Gallimberti I. Basic data of air discharges: UPee - 72/05 Report, June 1972

eters. Washington, DC: Naval Research Laboratory; August 29, 1986. pp. 1-34

for microwave breakdown. Institute of Applied Physics RAS. pp. 1-2

[16] Kuffel E, Zaengl WS. High Voltage Engineering. London UK: Newnes; 2000

tions Magazine. June 2000;38(6):134-140

vol. 5358 (SPIE, Bellingham, WA); 2004

Optic Communication; 285:5420-5424

York: Wiley; 2002

2006

14(7):1-5

### Author details

Isaac Medina1,2\* and Primo-Alberto Calva1,2

\*Address all correspondence to: ismesa@gmail.com

1 Centro de Desarrollo Aeroespacial del Instituto Politécnico Nacional, Delegación Cuauhtémoc, México, Ciudad de México

2 Instituto Politécnico Nacional, Colonia Barrio la Laguna Ticomán, Delegación Gustavo A. Madero, México, Ciudad de México

### References


[11] Ali AW. Intense and Short Pulse Electric Field (DC and Microwave) Air Breakdown Parameters. Washington, DC: Naval Research Laboratory; August 29, 1986. pp. 1-34

sufficient for the electronic processing required in an optical environment, since optical sys-

tems are capable of fully operating while receiving very low power, <sup>P</sup> <sup>¼</sup> <sup>4</sup> � <sup>10</sup>�<sup>14</sup> <sup>W</sup>.

1 Centro de Desarrollo Aeroespacial del Instituto Politécnico Nacional, Delegación

2 Instituto Politécnico Nacional, Colonia Barrio la Laguna Ticomán, Delegación Gustavo A.

[1] Elbert BR. Introduction to Satellite Communication. London: Artech House; 2008

[2] Calva PA, Medina I. Power Breakdown Threshold of a Plasmonic Waveguide Filter.

[3] Marcuvitz N. Waveguide Handbook. London, United Kingdom: Peter Peregrinus Ltd.;

[4] Vicente Quiles CP. Passive intermodulation and corona discharge for microwave structures in communications satellites. Dissertation PhD Thesis, Germany: Technischen Universitat

[5] Ming Y. Power- handling capability for RF filters. IEEE Microwave Magazine. October

[6] Woo W, DeGroot J. Microwave absorption and plasma heating due to microwave break-

[7] MacDonald AD. Microwave Breakdown in Gases. New Jersey, United States: John Wiley

[8] MacDonald AD, Gaskell DU, Gitterman HN. Microwave breakdown in air, oxygen and

[9] Jordan U, Anderson D, et al. On the effective diffusion length for microwave breakdown.

[10] Scharfman WE, Morita T. Voltage Breakdown of Antennas at High Altitude. Proceedings

down in the atmosphere. IEEE Physical Fluids. 1984;27(2):475-487

nitrogen. Physical Review. June 1963;130:1841-1850

of the IRE, November 1960, pp. 1881-1887

IEEE Transactions on Plasma Science. 2006;34(2):421-430

Author details

78 Emerging Waveguide Technology

References

1985

2007:88-97

& Sons; 1966

Isaac Medina1,2\* and Primo-Alberto Calva1,2

Cuauhtémoc, México, Ciudad de México

Madero, México, Ciudad de México

\*Address all correspondence to: ismesa@gmail.com

Plasmonics, Springer US; January 2014

Darmstadt zur Erlangung der Wurde; 2005


**Chapter 5**

**Provisional chapter**

**Optical Waveguide for Measurement Application**

**Optical Waveguide for Measurement Application**

DOI: 10.5772/intechopen.76781

The chapter provides the analysis of the behaviour of Mach Zehnder interferometer waveguide (MZIW) sensing structure and establishes the general design principles. Photonics interferometers have been widely used because of their highly sensitive detection technique. The present study is based on the MZIW structure for sensing application and deals with interferometer single-mode transmission. Theoretically, short wavelength and high difference in index (Δη) results in the low depth of the evanescence wave and increase in sensitivity. MZIW under consideration is very small in size hence it is very difficult to guide the light into waveguide. The output monitor detection sensitivity of the entire MZI structure depends on light-guiding efficiency. To maintain minimum losses at various micro-branches of the entire MZIW structure, effective light propagation is important and it is a critical parameter of the entire interferometer. Various tests have been carried out to study the effects of the *Y* branch angle variation on light guiding into the MZIW structure especially in measurement

Guided wave optics has revolutionized the photonic sensing technology. It covers both fibre and integrated optics technology. Photonics technology improves optical communication and minimizes the optical components used for communication as well as measurement applications [1]. Silicon nano-photonics waveguides strongly confine light in a submicron waveguide

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Prashant Bansilal Patel and Satish T. Hamde

Prashant Bansilal Patel and Satish T. Hamde

Additional information is available at the end of the chapter

**Keywords:** interferometer, light propagation, waveguide

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.76781

**Abstract**

application.

**1. Introduction**

structure which has following advantages.

#### **Optical Waveguide for Measurement Application Optical Waveguide for Measurement Application**

DOI: 10.5772/intechopen.76781

Prashant Bansilal Patel and Satish T. Hamde Prashant Bansilal Patel and Satish T. Hamde

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.76781

#### **Abstract**

The chapter provides the analysis of the behaviour of Mach Zehnder interferometer waveguide (MZIW) sensing structure and establishes the general design principles. Photonics interferometers have been widely used because of their highly sensitive detection technique. The present study is based on the MZIW structure for sensing application and deals with interferometer single-mode transmission. Theoretically, short wavelength and high difference in index (Δη) results in the low depth of the evanescence wave and increase in sensitivity. MZIW under consideration is very small in size hence it is very difficult to guide the light into waveguide. The output monitor detection sensitivity of the entire MZI structure depends on light-guiding efficiency. To maintain minimum losses at various micro-branches of the entire MZIW structure, effective light propagation is important and it is a critical parameter of the entire interferometer. Various tests have been carried out to study the effects of the *Y* branch angle variation on light guiding into the MZIW structure especially in measurement application.

**Keywords:** interferometer, light propagation, waveguide

#### **1. Introduction**

Guided wave optics has revolutionized the photonic sensing technology. It covers both fibre and integrated optics technology. Photonics technology improves optical communication and minimizes the optical components used for communication as well as measurement applications [1]. Silicon nano-photonics waveguides strongly confine light in a submicron waveguide structure which has following advantages.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The fabrication of photonics circuits can be done on the similar line of complementary metaloxide-semiconductor (CMOS) circuits [2]. Due to the similarity, CMOS compatibility opens up options for the interface of photonics functions with electronics functions. Photonics sensing technology has now gained a place in the vast portfolio of practical measurement technologies [3, 4].

However, current innovations in the photonics technique to manipulate the light are continue to provide both opportunity and challenge to the micro-optical components used in rapid measurement and sensing technologies. The development in the modern photonics sensors is due to the advances made in the LASER and optical fibre technology. The progress in the microelectronics field accelerated the growth in silicon as well as polymer-based photonics devices.

Photonics technology also enhances precision as well as accuracy of the measurement. Photonics-based sensors reduce the measurement time which is not possible using conventional available techniques. The application of nanotechnology in the field of biology and biomedical field is known as bio-nano-technology or nano-bio-technology [5]. This technology gives rise to new devices and systems having improved sensitivity and accuracy for measurement application. Interferometer analysis using the *Y* branch is highlighted in this chapter for the MZIW structure and the general conclusions on optimization are drawn [6].

### **2. Interferometer technique**

In interferometer phenomenon, two similar input waves are superimposed at an output waveguide to detect the phase difference between them. If the two waves are in phase, their electrical fields gets added (this is called as constructive interference). If they are out of phase (phase shift is 180° between them), the electric field received at the output waveguide gets cancelled (this is called as destructive interference).

waveguide as shown in **Figure 1**. One branch is 1-2 and other is the 2-1 branch. The sensing reaction changes the refractive index of sensing waveguide and it changes the speed (phase) of light [9]. The light intensity changes due to the optical interference at the output waveguide (2-1 branch) [10]. **Figure 1** demonstrates one-input and two-output (1-2 branches) and twoinput one-output (2-1 branch)-type micro-*Y* branch for the MZIW structure. Many optical evanescent wave sensors in various forms have been used for highly sensitive sensing applications and MZIW is one class of such sensors. MZIW has been designed using one *Y* and one

Optical Waveguide for Measurement Application http://dx.doi.org/10.5772/intechopen.76781 83

The simplest light waveguide component is the *Y* branch and is a three-port device that acts as a light divider, *Y* branch (1-2) and light collector and inverted *Y* branch (2-1) [12]. **Figure 1** shows a *Y* branch made by splitting a planer waveguide into two branches bifurcating at some angle. These components are very much similar to a fibre optic coupler which can also act as a power splitter except that it has only three ports. Conceptually it differs considerably from a fibre coupler since there is no coupling region in which modes of two different waveguides overlap. Function of the *Y* branch is very simple. With reference to **Figure 1**, in the branch region the waveguide is thicker and supports higher-order modes. However the geometrical symmetry forbids the excitation of asymmetric modes. If the thickness is changed gradually in an adiabatic manner, even higher-order symmetric modes are not excited, and power is

inverted *Y* branch to form an entire waveguide structure [11].

**Figure 1.** *Y* branches (a) Branch [1-2] (b) Branch [2-1].

Various interferometer configurations like Mach Zehnder interferometer, Fabry Perot interferometer and Michelson interferometer have been realized using optical methods. Out of all above techniques, the main consideration in this chapter is given to Mach Zehnder interferometer [7, 8].

### **3. Mach Zehnder interferometer waveguide (MZIW)**

Interferometer is most suitable technique for analytical measurement with real-time interaction monitoring. Our main purpose is to provide the optimization and testing of MZIW. Interferometer based on MZIW consists of input waveguide structure (left *Y* branch) and output waveguide structure (right *Y* branch) [8]. In interferometer measurement, input light is equally guided into the two waveguides and light is recollected into the single

**Figure 1.** *Y* branches (a) Branch [1-2] (b) Branch [2-1].

(i) allows sharp bends due to which compact and tiny components can be analysed and

(ii) gives tremendous reduction in footprints, which in turns open up new areas for the

The fabrication of photonics circuits can be done on the similar line of complementary metaloxide-semiconductor (CMOS) circuits [2]. Due to the similarity, CMOS compatibility opens up options for the interface of photonics functions with electronics functions. Photonics sensing technology has now gained a place in the vast portfolio of practical measurement

However, current innovations in the photonics technique to manipulate the light are continue to provide both opportunity and challenge to the micro-optical components used in rapid measurement and sensing technologies. The development in the modern photonics sensors is due to the advances made in the LASER and optical fibre technology. The progress in the microelectronics field accelerated the growth in silicon as well as polymer-based photonics devices. Photonics technology also enhances precision as well as accuracy of the measurement. Photonics-based sensors reduce the measurement time which is not possible using conventional available techniques. The application of nanotechnology in the field of biology and biomedical field is known as bio-nano-technology or nano-bio-technology [5]. This technology gives rise to new devices and systems having improved sensitivity and accuracy for measurement application. Interferometer analysis using the *Y* branch is highlighted in this chapter for

the MZIW structure and the general conclusions on optimization are drawn [6].

In interferometer phenomenon, two similar input waves are superimposed at an output waveguide to detect the phase difference between them. If the two waves are in phase, their electrical fields gets added (this is called as constructive interference). If they are out of phase (phase shift is 180° between them), the electric field received at the output waveguide gets

Various interferometer configurations like Mach Zehnder interferometer, Fabry Perot interferometer and Michelson interferometer have been realized using optical methods. Out of all above techniques, the main consideration in this chapter is given to Mach Zehnder interferometer [7, 8].

Interferometer is most suitable technique for analytical measurement with real-time interaction monitoring. Our main purpose is to provide the optimization and testing of MZIW. Interferometer based on MZIW consists of input waveguide structure (left *Y* branch) and output waveguide structure (right *Y* branch) [8]. In interferometer measurement, input light is equally guided into the two waveguides and light is recollected into the single

large-scale integration of photonics component circuits.

characterized;

82 Emerging Waveguide Technology

technologies [3, 4].

**2. Interferometer technique**

cancelled (this is called as destructive interference).

**3. Mach Zehnder interferometer waveguide (MZIW)**

waveguide as shown in **Figure 1**. One branch is 1-2 and other is the 2-1 branch. The sensing reaction changes the refractive index of sensing waveguide and it changes the speed (phase) of light [9]. The light intensity changes due to the optical interference at the output waveguide (2-1 branch) [10]. **Figure 1** demonstrates one-input and two-output (1-2 branches) and twoinput one-output (2-1 branch)-type micro-*Y* branch for the MZIW structure. Many optical evanescent wave sensors in various forms have been used for highly sensitive sensing applications and MZIW is one class of such sensors. MZIW has been designed using one *Y* and one inverted *Y* branch to form an entire waveguide structure [11].

The simplest light waveguide component is the *Y* branch and is a three-port device that acts as a light divider, *Y* branch (1-2) and light collector and inverted *Y* branch (2-1) [12]. **Figure 1** shows a *Y* branch made by splitting a planer waveguide into two branches bifurcating at some angle. These components are very much similar to a fibre optic coupler which can also act as a power splitter except that it has only three ports. Conceptually it differs considerably from a fibre coupler since there is no coupling region in which modes of two different waveguides overlap. Function of the *Y* branch is very simple. With reference to **Figure 1**, in the branch region the waveguide is thicker and supports higher-order modes. However the geometrical symmetry forbids the excitation of asymmetric modes. If the thickness is changed gradually in an adiabatic manner, even higher-order symmetric modes are not excited, and power is divided into two branches without much loss. In practice, a sudden opening of the gap violates the adiabatic condition, resulting in insertion losses associated with any *Y* branch. These losses depend on the branching angle θ and increases as angle *θ* increases.

In practice, what attracts our attention is the presence of a number of relatively primitive, that is, straight and curved waveguides. As it was mentioned earlier, the layout is very coarse and the light interactions actually occur just on a small fraction of the layout. For this reason waveguides are also called as micro-waveguides and the branches are also called as microbranches. Various types of MZIW structures have been designed for sensing applications.

We have used beam propagation method (BPM) for the analysis of MZIW [13]. The physical propagation requires important information about the distribution of refractive index η(x, y, z) and input wave field, η(x, y, z = 0). From this we can detect the wave field throughout the rest of the domain u (x, y, z > 0).

In addition to above data, the BPM algorithm requires additional information in the form of numerical parameters like:


Generally, smaller grid sizes give results with more accuracy. But due to a small grid size, simulation time increases [14]. It is very important and critical to perform a convergence study on the X and Y grid sizes to provide optimization and the tradeoff between speed and accuracy.

We have analysed monitor output behaviour with respect to variations in *Y* branch angle.

From this, it is shown to split light and to combine light at *Y* branches already shown in **Figure 2**. The *Y* branch angle should be optimized. The angle should be less than 18°; above this, light propagation will not be appropriate through the MZIW structure. This reduces the

Also when light is guided through the bend structure, substantial radiation losses take place and significant distortion of the optical input launch field occurs when light proceeds through the MZIW structure [19]. As shown in **Figure 5(a)**, as we vary angle between 0° and 25°, monitor output value decreases, and as we increase angle variation beyond 25° that is up to 45°, monitor output value further decreases. This is represented in the graph shown in **Figure 5(b)**. In **Figure 5(c)** as we increase angle variation beyond 45° that is up to 75°, the

∆P

where ∆P is the monitor output power and ∆n is the refractive index change at measuring

∆n (1)

Optical Waveguide for Measurement Application http://dx.doi.org/10.5772/intechopen.76781 85

**Figure 5(a–c)** shows the graph of *Y* branch angle variation versus output value.

output monitor value [18].

**Figure 2.** MZIW Structure in 3D format.

branch.

monitor output value further goes on reducing.

*S* = \_\_\_

The measuring sensitivity of the MZI structure is given by Eq. (1):

## **4. Waveguide configuration and analysis**

**Figure 2** shows the MZIW structure in 3D (XYZ) format with reference to this proposed structure. We can join two *Y* branches (refer **Figure 1**) to form the entire MZIW structure (refer **Figure 2**) [15]. To perform this convergence study, we have used the scanning capabilities of beam propagation method, scanning and optimization tools. The MZI layout under the test is shown in **Figures 3** and **4**.

This scanning tool gives very good results. There are various configurations of photonic MZIW, which are used for sensing various physical parameters [16]. **Figure 3(a–d)** shows the experimental data for light output variation as a function of changes in the branch angle for *Y* branch (1-2).

We can use the proper numerical model for the design of the structure with appropriate characteristics.

**Figure 4(a–d)** shows experimental data for output variation as a function of changes in the branch angle for inverted *Y* branch (2-1) [17]. The performance of photonics interferometer depends upon various fibre geometry and fibre parameters.

For the successful design and working of sensor, the process of the parameter optimization is very critical and important.

**Figure 2.** MZIW Structure in 3D format.

divided into two branches without much loss. In practice, a sudden opening of the gap violates the adiabatic condition, resulting in insertion losses associated with any *Y* branch. These

In practice, what attracts our attention is the presence of a number of relatively primitive, that is, straight and curved waveguides. As it was mentioned earlier, the layout is very coarse and the light interactions actually occur just on a small fraction of the layout. For this reason waveguides are also called as micro-waveguides and the branches are also called as microbranches. Various types of MZIW structures have been designed for sensing applications.

We have used beam propagation method (BPM) for the analysis of MZIW [13]. The physical propagation requires important information about the distribution of refractive index η(x, y, z) and input wave field, η(x, y, z = 0). From this we can detect the wave field throughout the

In addition to above data, the BPM algorithm requires additional information in the form of

Generally, smaller grid sizes give results with more accuracy. But due to a small grid size, simulation time increases [14]. It is very important and critical to perform a convergence study on the X and Y grid sizes to provide optimization and the tradeoff between speed and accuracy.

**Figure 2** shows the MZIW structure in 3D (XYZ) format with reference to this proposed structure. We can join two *Y* branches (refer **Figure 1**) to form the entire MZIW structure (refer **Figure 2**) [15]. To perform this convergence study, we have used the scanning capabilities of beam propagation method, scanning and optimization tools. The MZI layout under the test

This scanning tool gives very good results. There are various configurations of photonic MZIW, which are used for sensing various physical parameters [16]. **Figure 3(a–d)** shows the experimental data for light output variation as a function of changes in the branch angle for *Y* branch (1-2). We can use the proper numerical model for the design of the structure with appropriate

**Figure 4(a–d)** shows experimental data for output variation as a function of changes in the branch angle for inverted *Y* branch (2-1) [17]. The performance of photonics interferometer

For the successful design and working of sensor, the process of the parameter optimization is

(i) finite computation domain {X ∈ (xmin, xmax)}, {Y ∈ (ymin, ymax)}, {Z∈ (zmin, zmax)},

losses depend on the branching angle θ and increases as angle *θ* increases.

rest of the domain u (x, y, z > 0).

(ii) transverse grid size Δx and Δy,

**4. Waveguide configuration and analysis**

depends upon various fibre geometry and fibre parameters.

(iii) longitudinal step size Δz.

is shown in **Figures 3** and **4**.

very critical and important.

characteristics.

numerical parameters like:

84 Emerging Waveguide Technology

We have analysed monitor output behaviour with respect to variations in *Y* branch angle. **Figure 5(a–c)** shows the graph of *Y* branch angle variation versus output value.

From this, it is shown to split light and to combine light at *Y* branches already shown in **Figure 2**. The *Y* branch angle should be optimized. The angle should be less than 18°; above this, light propagation will not be appropriate through the MZIW structure. This reduces the output monitor value [18].

Also when light is guided through the bend structure, substantial radiation losses take place and significant distortion of the optical input launch field occurs when light proceeds through the MZIW structure [19]. As shown in **Figure 5(a)**, as we vary angle between 0° and 25°, monitor output value decreases, and as we increase angle variation beyond 25° that is up to 45°, monitor output value further decreases. This is represented in the graph shown in **Figure 5(b)**. In **Figure 5(c)** as we increase angle variation beyond 45° that is up to 75°, the monitor output value further goes on reducing.

The measuring sensitivity of the MZI structure is given by Eq. (1):

$$\mathbf{S} = \frac{\Delta \mathbf{P}}{\Delta \mathbf{n}} \tag{1}$$

where ∆P is the monitor output power and ∆n is the refractive index change at measuring branch.

**Figure 3.** (a) Light guiding variation for *Y* branch at an angle of 18°; (b) light guiding variation for *Y* branch at an angle of 25°; (c) light guiding variation for *Y* branch at an angle of 45° and (c) light guiding variation for *Y* branch at an angle of more than 75°.

samples under measurement. The validation of the innovative approach is achieved by the characterization of the above MZIW structure. The sensitivity of the sensors is the most dominating and demanding parameter as it directly relates to how early the sensing parameters

**Figure 4.** (a–d) Experimental data for output variation as a function of changes in branch angle for inverted *Y* branch (2-1). (a) Light guiding variation for *Y* branch at an angle of 18°; (b) light guiding variation for *Y* branch at an angle of 60°; (c) light guiding variation for *Y* branch at an angle of 75°; and (d) light guiding variation for *Y* branch at an angle

Optical Waveguide for Measurement Application http://dx.doi.org/10.5772/intechopen.76781 87

According to the theory of interferometer, the intensity modulation scheme should be characterized by an output intensity of MZI behaving as a cosine function of the phase variation as shown in **Figure 5**. Indeed the detected light output power Iout at the output of the *Y* branch

Iout = Ir + Is+2<sup>√</sup> Is Ir cos (Δφ) (2)

(2-1) of the interferometer can be detected as given by Eq. (2).

are detected [20].

more than 75°.

From **Figure 5(a–c)**, it is easily interpreted that beyond 18° it is difficult to guide light to the waveguide. These losses depend on the branching angle and increase as the angle increases.

**Figure 6(a** and **b)** shows the complete MZIW structure after optimum selection of *Y* branch angle (18°) for both the branches. This waveguide has a width of 3 μm and length of 40 μm. This structure is used for sensing applications like refractive index measurement of small

**Figure 4.** (a–d) Experimental data for output variation as a function of changes in branch angle for inverted *Y* branch (2-1). (a) Light guiding variation for *Y* branch at an angle of 18°; (b) light guiding variation for *Y* branch at an angle of 60°; (c) light guiding variation for *Y* branch at an angle of 75°; and (d) light guiding variation for *Y* branch at an angle more than 75°.

samples under measurement. The validation of the innovative approach is achieved by the characterization of the above MZIW structure. The sensitivity of the sensors is the most dominating and demanding parameter as it directly relates to how early the sensing parameters are detected [20].

According to the theory of interferometer, the intensity modulation scheme should be characterized by an output intensity of MZI behaving as a cosine function of the phase variation as shown in **Figure 5**. Indeed the detected light output power Iout at the output of the *Y* branch (2-1) of the interferometer can be detected as given by Eq. (2).

From **Figure 5(a–c)**, it is easily interpreted that beyond 18° it is difficult to guide light to the waveguide. These losses depend on the branching angle and increase as the angle increases. **Figure 6(a** and **b)** shows the complete MZIW structure after optimum selection of *Y* branch angle (18°) for both the branches. This waveguide has a width of 3 μm and length of 40 μm. This structure is used for sensing applications like refractive index measurement of small

**Figure 3.** (a) Light guiding variation for *Y* branch at an angle of 18°; (b) light guiding variation for *Y* branch at an angle of 25°; (c) light guiding variation for *Y* branch at an angle of 45° and (c) light guiding variation for *Y* branch at an angle

of more than 75°.

86 Emerging Waveguide Technology

$$\mathbf{I}\_{\rm out} = \mathbf{I}\_{\rm r} + \mathbf{I}\_{\rm s \star 2, \rm y} \mathbf{I}\_{\rm s} \mathbf{I}\_{\rm r} \cos \left( \Delta \boldsymbol{\upmu} \right) \tag{2}$$

**Figure 5.** (a–c) Graph of angle variation between 0 and 75°. (a) Output value for angle variation between 0 and 25°; (b) output value for angle variation between 0 and 45°; (c) output value for angle variation between 0 and 75°.

where Ir and Is are the optional powers of the reference and sensing waveguide observed in each arm of MZIW and Δφ is the phase difference between both waveguides.

The above result and study elucidates the influence of the *Y* branch angle on light guiding the MZIW structure and the MZIW variation in the angle changes light guiding efficiency, so the angle must be properly selected for sensing applications. From the above analysis and experimentation, it is observed that the optimum value for the angle is required to be below 18° (for 1-2 branch). In case of the MZIW interferometer, we know that one arm of the waveguide structure is acting as a reference arm and other arm is acting as a measurement arm. According to the graphs shown in **Figure 5(a–c)**, if light is not properly guided into the reference arm and other measurement arms of waveguides, it will produce considerable variations in the output monitor value. Due to these variations, the physical parameter that is to be

sensed using the interferometer will not be detected correctly. Output measurement becomes difficult. These types of waveguide structures are applicable to any measurement application that changes phase and amplitude of light passing through the waveguide. Generally branch angle should be maintained below 1 radian to reduce insertion losses below 1 dB and 18° for

Optical Waveguide for Measurement Application http://dx.doi.org/10.5772/intechopen.76781 89

**Figure 7.** Graph of variation in phase shift and monitor output value for MZIW structure shown in **Figure 6**.

Analysis and experimental characterization of MZIW is performed using a "beam propagation method" algorithm. Measurement is carried out by using an MZIW structure having branch angle as 18° (for the *Y* branch); however, the insertion loss of power divided also

After the optimization of MZIW for the measurement application is completed, the next part is to analyse the structure for the refractive index (RI) measurement. **Figure 7** shows the phase shift variation due to refractive index variation and corresponding changes in the output monitor value. Due to proper light splitting and combining, changes in phase shift produced

Measurements are carried out by using an MZI structure having branch angle of 18° (for the *Y* branch); however, the insertion loss of power divided also increases rapidly and often becomes intolerable after three or four bifurcation stages. The next step of this chapter con-

increases rapidly and often becomes intolerable after three or four bifurcation stages.

due to variations in the refractive index of the sample can be measured.

sists of the characterization of rib waveguide-based MZI optical sensor.

measurement applications.

**5. Conclusion**

**Figure 6.** (a) MZIW structure using two *Y* branches and (b) complete MZIW structure used for sensing application.

**Figure 7.** Graph of variation in phase shift and monitor output value for MZIW structure shown in **Figure 6**.

sensed using the interferometer will not be detected correctly. Output measurement becomes difficult. These types of waveguide structures are applicable to any measurement application that changes phase and amplitude of light passing through the waveguide. Generally branch angle should be maintained below 1 radian to reduce insertion losses below 1 dB and 18° for measurement applications.

Analysis and experimental characterization of MZIW is performed using a "beam propagation method" algorithm. Measurement is carried out by using an MZIW structure having branch angle as 18° (for the *Y* branch); however, the insertion loss of power divided also increases rapidly and often becomes intolerable after three or four bifurcation stages.

After the optimization of MZIW for the measurement application is completed, the next part is to analyse the structure for the refractive index (RI) measurement. **Figure 7** shows the phase shift variation due to refractive index variation and corresponding changes in the output monitor value. Due to proper light splitting and combining, changes in phase shift produced due to variations in the refractive index of the sample can be measured.

#### **5. Conclusion**

where Ir

and Is

88 Emerging Waveguide Technology

are the optional powers of the reference and sensing waveguide observed in

The above result and study elucidates the influence of the *Y* branch angle on light guiding the MZIW structure and the MZIW variation in the angle changes light guiding efficiency, so the angle must be properly selected for sensing applications. From the above analysis and experimentation, it is observed that the optimum value for the angle is required to be below 18° (for 1-2 branch). In case of the MZIW interferometer, we know that one arm of the waveguide structure is acting as a reference arm and other arm is acting as a measurement arm. According to the graphs shown in **Figure 5(a–c)**, if light is not properly guided into the reference arm and other measurement arms of waveguides, it will produce considerable variations in the output monitor value. Due to these variations, the physical parameter that is to be

**Figure 6.** (a) MZIW structure using two *Y* branches and (b) complete MZIW structure used for sensing application.

**Figure 5.** (a–c) Graph of angle variation between 0 and 75°. (a) Output value for angle variation between 0 and 25°; (b)

each arm of MZIW and Δφ is the phase difference between both waveguides.

output value for angle variation between 0 and 45°; (c) output value for angle variation between 0 and 75°.

Measurements are carried out by using an MZI structure having branch angle of 18° (for the *Y* branch); however, the insertion loss of power divided also increases rapidly and often becomes intolerable after three or four bifurcation stages. The next step of this chapter consists of the characterization of rib waveguide-based MZI optical sensor.

### **Acknowledgements**

The authors would like to thank University of Pune and also appreciate the help extended by the University for providing funding under the Research and Development Program for this proposed research study work [Reference: BCUD/14, 2008–2010].

[8] Hu MH, Huang JZ, Scarmozzino R, Levy M, Osgood RM. Tunable Mach-Zehnder polarization splitter using height-tapered Y-branches. IEEE Photonics Technology Letters.

Optical Waveguide for Measurement Application http://dx.doi.org/10.5772/intechopen.76781 91

[9] Greivenkamp JE, editor. Handbook of Optics. Vol. 1. 2nd ed. University of Arizona,

[10] Estevez MC, Alvarez M, Lechuga LM. Integrated optical devices for lab-on-a-chip bio-

[11] Sasaki H, Shiki E, Mikoshiba N. Propagation characteristic of optical guided waves in asymmetric branching waveguides. IEEE Journal of Quantum Electronics. 1981;**QE-17**(6):

[12] Thylen L. The beam propagation method: An analysis of it applicability. Optical and

[13] Neyer A, Mevenkamp W, Thylen L, Bo L. A beam propagation method analysis of active and Pasive waveguide crossing. IEEE Journal of Lightwave Technology. 1985;**LT-3**(3):

[14] Doerr CR. Beam propagation method tailored for step-index waveguides. IEEE Photonics

[15] Patel PB, Hamde ST. Photonics sensing techniques based on various Mach Zehnder Interferometer Waveguide (MZIW) structures. In: International Conference on Biomedical

[16] Qi Z-m, Matsuda N, Itoh K, Murabayashi M, Lavers CR. A design for improving the sensitivity of a Mach-Zehnder interferometer to chemical and biological measurements.

[17] Sethi RS. Transducer aspects of biosensors. Biosensors & Bioelectronics. 1994;**9**:243-264 [18] Iqbal M, Gleeson MA, Spaugh B, Tybor F, Gunn WG, Hochberg M, Baehr-Jones T, Bailey RC, Gunn LC. Label-free biosensor arrays based on silicon ring resonators and highspeed optical scanning instrumentation. IEEE Journal of Selected Topics in Quantum

[19] Adams MJ. An Introduction to Optical Wave Guides. Chapter 7. New York: Wiley;

[20] Levy R, Peled A, Ruschin S. Waveguided SPR sensor using a Mach-Zehnder interferometer with variable power splitter ratio. Sensors and Actuators B. 2006;**119**:20-26

Tucson, Arizona: Interference, Optical Science Centre. Chapter 2

sensing applications. Laser & Photonics Reviews. 2012;**6**:463

Quantum Electronics. 1983;**15**:433-439

Engineering; NIT Jalandhar; 17-19th Dec, 2010

Sensors and Actuators, B. 2002;**81**:254-258

Electronics. 2010;**16**:654-661

2001;**6**:09

1051-1057

635-642

Letters. 2001;**13**(2):2-3

We also thank the Management of Dr. D.Y. Patil Institute of Technology, Pune, and S.G.G.S. Institute of Engineering and Technology, Nanded, for providing the necessary facilities for carrying out this work.

### **Author details**

Prashant Bansilal Patel1 \* and Satish T. Hamde2

\*Address all correspondence to: prashantbgpatel@gmail.com

1 Dr. D.Y. Patil Institute of Technology (DIT), Pune, Maharashtra, India

2 S.G.G.S. Institute of Engineering and Technology (SGGSIET), Nanded, Maharashtra, India

#### **References**


[8] Hu MH, Huang JZ, Scarmozzino R, Levy M, Osgood RM. Tunable Mach-Zehnder polarization splitter using height-tapered Y-branches. IEEE Photonics Technology Letters. 2001;**6**:09

**Acknowledgements**

90 Emerging Waveguide Technology

ties for carrying out this work.

**Author details**

**References**

21500-21517

**10**:209-217

sors. Sensors. 2007;*7*:797-859

Prashant Bansilal Patel1

The authors would like to thank University of Pune and also appreciate the help extended by the University for providing funding under the Research and Development Program for this

We also thank the Management of Dr. D.Y. Patil Institute of Technology, Pune, and S.G.G.S. Institute of Engineering and Technology, Nanded, for providing the necessary facili-

2 S.G.G.S. Institute of Engineering and Technology (SGGSIET), Nanded, Maharashtra, India

[1] Patel PB, Hamde ST. Analysis of the Mach-Zehnder interferometer waveguide structure

[2] Yuan D, Dong Y, Liuand Y, Li T. Mach-Zehnder interferometer biochemical sensor based on silicon-on-insulator rib waveguide with large cross section. Sensors. 2015;**15**:

[3] Weisser M, Tovar G, Mittler-Neher S, Knoll W, Brosinger F, Freimuth H, Lacher M, Ehrfeld W. Specific bio-recognition reactions observed with an integrated Mach-Zehnder

[4] Bosch ME, Sánchez AJR, Rojas FS, Ojeda CB. Recent development in optical fiber biosen-

[5] Patel PB, Hamde ST. Characterization of the properties of optical wavefields—An investigations. Journal of Environmental Research and Development (JERAD). 2009;**4**(2):1-2

[6] Patel PB, Hamde ST. Miniaturized Fiber Optic Integrated Biosensor Platforms for Human Health Monitoring, International Conference on Emerging Technologies and Application in Engineering, Technology and Sciences (ICETAETS-08) at Saurashtra

[7] Heideman RG, Kooyman RPH, Greve J. Performance of a highly sensitive optical waveguide Mach-Zehnder interferometer immunosensor. Sensors & Actuators B. 1993;

for refractive index measurement. Journal of Optics. 2017;**46**(4):398-402

interferometer. Biosensors and Bioelectronics. 1999;**14**:409-415

University; 13 and 14th January-2008; Rajkot, Gujrat

proposed research study work [Reference: BCUD/14, 2008–2010].

\* and Satish T. Hamde2

1 Dr. D.Y. Patil Institute of Technology (DIT), Pune, Maharashtra, India

\*Address all correspondence to: prashantbgpatel@gmail.com


**Section 2**

**Photonic and Optical Waveguides**

**Photonic and Optical Waveguides**

**Chapter 6**

Provisional chapter

**Review on Optical Waveguides**

Review on Optical Waveguides

Shankar Kumar Selvaraja and Purnima Sethi

Shankar Kumar Selvaraja and Purnima Sethi

Additional information is available at the end of the chapter

Optical devices are necessary to meet the anticipated future requirements for ultrafast and ultrahigh bandwidth communication and computing. All optical information processing can overcome optoelectronic conversions that limit both the speed and bandwidth and are also power consuming. The building block of an optical device/circuit is the optical waveguide, which enables low-loss light propagation and is thereby used to connect components and devices. This chapter reviews optical waveguides and their classification on the basis of geometry (Non-Planar (Slab/Optical Fiber)/Planar (Buried Channel, Strip-Loaded, Wire, Rib, Diffused, Slot, etc.)), refractive index (Step/Gradient Index), mode propagation (Single/Multimode), and material platform (Glass/Polymer/Semiconductor, etc.). A comparative analysis of waveguides realized in different material platforms along

DOI: 10.5772/intechopen.77150

Keywords: optical waveguides, integrated optics, optical devices, optical materials,

Waveguides are indispensable for communication and computing applications as they are immune to electromagnetic interference and induced cross talk and also counter diffraction. Next-generation high-end information processing (bandwidths >1 Tb/s and speed >10 Gb/s) is immensely challenging using copper-based interconnects. Optical interconnects transmit data through an optical waveguide and offer a potential solution to improve the data transmission [1, 2]. There are predominantly two classes of optical waveguide: those in which "classical optical elements, placed periodically along the direction of propagation of the wave, serve to confine the wave by successive refocusing in the vicinity of the optical axis (laser resonators and multiple lens waveguides); and those in which the guiding mechanism is that of multiple

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Additional information is available at the end of the chapter

with the propagation loss is also presented.

photonics integrated circuits

http://dx.doi.org/10.5772/intechopen.77150

Abstract

1. Introduction

#### **Review on Optical Waveguides** Review on Optical Waveguides

Shankar Kumar Selvaraja and Purnima Sethi Shankar Kumar Selvaraja and Purnima Sethi

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.77150

#### Abstract

Optical devices are necessary to meet the anticipated future requirements for ultrafast and ultrahigh bandwidth communication and computing. All optical information processing can overcome optoelectronic conversions that limit both the speed and bandwidth and are also power consuming. The building block of an optical device/circuit is the optical waveguide, which enables low-loss light propagation and is thereby used to connect components and devices. This chapter reviews optical waveguides and their classification on the basis of geometry (Non-Planar (Slab/Optical Fiber)/Planar (Buried Channel, Strip-Loaded, Wire, Rib, Diffused, Slot, etc.)), refractive index (Step/Gradient Index), mode propagation (Single/Multimode), and material platform (Glass/Polymer/Semiconductor, etc.). A comparative analysis of waveguides realized in different material platforms along with the propagation loss is also presented.

DOI: 10.5772/intechopen.77150

Keywords: optical waveguides, integrated optics, optical devices, optical materials, photonics integrated circuits

#### 1. Introduction

Waveguides are indispensable for communication and computing applications as they are immune to electromagnetic interference and induced cross talk and also counter diffraction. Next-generation high-end information processing (bandwidths >1 Tb/s and speed >10 Gb/s) is immensely challenging using copper-based interconnects. Optical interconnects transmit data through an optical waveguide and offer a potential solution to improve the data transmission [1, 2]. There are predominantly two classes of optical waveguide: those in which "classical optical elements, placed periodically along the direction of propagation of the wave, serve to confine the wave by successive refocusing in the vicinity of the optical axis (laser resonators and multiple lens waveguides); and those in which the guiding mechanism is that of multiple

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

total internal reflection from interfaces parallel to the optical axis" (fiber optical waveguides, slab waveguides, and resonators) [3].

Historically, high-frequency microwave sources had created a furore on guided wave photonics pioneered by Rayleigh and Sommerfeld. The first theoretical description of mode propagation along a dielectric guide was done by Hondros and Debye in 1910 [3]. The first dielectric waveguide to be examined at optical frequencies was the glass fiber used primarily for fiber optics imaging applications [4].

A waveguide can be defined as any structure (usually cylindrical) used for guiding the flow of electromagnetic wave in a direction parallel to its axis, confining it to a region either within or adjacent to its surfaces. In order to understand the propagation of light in a waveguide, it is imperative to derive the wave equation. The electromagnetic wave equation can be derived from the Maxwell's equation, assuming that we are operating in a source free ð Þ r ¼ 0; J ¼ 0 , linear <sup>ε</sup> and <sup>μ</sup> are independent of <sup>E</sup> and <sup>H</sup>Þ, and an isotropic medium. <sup>E</sup> and <sup>H</sup> are the electric and magnetic field amplitudes, respectively, ε is the electric permittivity of the medium, and μ is the magnetic permeability of the medium. The equations are:

$$
\nabla \times \overline{E} = -\frac{\partial \overline{B}}{\partial t} \tag{1}
$$

extended high-index medium called the Core, which is transversely surrounded by a low-index medium, called the Cladding. A guided optical wave propagates in the waveguide along the longitudinal direction. The characteristics of a waveguide are determined by the transverse profile of its dielectric constant (x, y), which is independent of the z coordinate. For a waveguide made of optically isotropic media, the waveguide can be characterized merely with a single spatially dependent transverse profile of the index of refraction, n(x, y). Broadly, the waveguides

Review on Optical Waveguides

97

http://dx.doi.org/10.5772/intechopen.77150

• Planar/2-D waveguides: Optical confinement is only in one transverse direction, the core is sandwiched between cladding layers in only one direction (Figure 1(a)). Optical confinement is only in the x-direction with index profile n(x). They are primarily used for

• Non-planar/3-D/channel optical waveguide: Comprises of two-dimensional transverse optical confinement, the core is surrounded by cladding in all transverse directions, and n(x, y) is a function of both x and y coordinates as shown in Figure 1(b). A channel waveguide (with guidance in both directions) has a guiding structure in the form of a stripe with a finite width. Examples: channel waveguides (Section 2.3.II) and circular

A waveguide in which the index profile changes abruptly between the core and the cladding is called a step-index waveguide, while one in which the index profile varies gradually is called a graded-index waveguide as shown in Figure 2. Recently, hybrid index profile waveguide was shown combining both inverse-step index waveguide and graded index waveguides for high-

A waveguide mode is an electromagnetic wave that propagates along a waveguide with a distinct phase velocity, group velocity, cross-sectional intensity distribution, and polarization. Each component of its electric and magnetic field is of the form f xð Þ ; <sup>y</sup> eiωt�ihz, where <sup>z</sup> is the axis of the waveguide. Modes are referred to as the "characteristic waves" of the structures because their field vector satisfies the homogenous wave equation in all the media that make up the

Figure 1. (a) Planar optical waveguide of 1-d transverse (x) optical confinement, (b) non-planar optical waveguide of 2-D

can be classified as [5]:

optical fibers [6].

2.1. Waveguide mode

transverse (x, y) optical confinement.

high-power waveguide lasers and amplifiers.

power amplification of a Gaussian single-mode beam [7].

$$
\nabla \times \overline{H} = \frac{\partial \overline{D}}{\partial t} \tag{2}
$$

$$
\nabla \overline{D} = 0 \tag{3}
$$

$$
\nabla \overline{B} = 0 \tag{4}
$$

Here, B and D are magnetic and electric fluxes, respectively. The wave equation derived from the above expressions is:

$$
\nabla^2 \overline{E} - \mu \varepsilon \frac{\partial^2 \overline{E}}{\partial t^2} = -\nabla \left( \overline{E}. \frac{\nabla \varepsilon}{\varepsilon} \right) \tag{5}
$$

The right-hand side of Eq. (5) is nonzero when there is a gradient in permittivity of the medium. Guided wave medium has a graded permittivity; however, in most structures, the term is negligible. Thus, the wave equation can be written as:

$$
\nabla^2 \overline{E} - \mu \varepsilon \frac{\partial^2 \overline{E}}{\partial t^2} = 0,\\
\nabla^2 \overline{H} - \mu \varepsilon \frac{\partial^2 \overline{H}}{\partial t^2} = 0 \tag{6}
$$

for electric and magnetic field amplitudes, respectively.

#### 2. Classification of waveguides

Optical waveguides can be classified according to their geometry, mode structure, refractive index (RI) distribution, and material. A dielectric optical waveguide comprises a longitudinally extended high-index medium called the Core, which is transversely surrounded by a low-index medium, called the Cladding. A guided optical wave propagates in the waveguide along the longitudinal direction. The characteristics of a waveguide are determined by the transverse profile of its dielectric constant (x, y), which is independent of the z coordinate. For a waveguide made of optically isotropic media, the waveguide can be characterized merely with a single spatially dependent transverse profile of the index of refraction, n(x, y). Broadly, the waveguides can be classified as [5]:


A waveguide in which the index profile changes abruptly between the core and the cladding is called a step-index waveguide, while one in which the index profile varies gradually is called a graded-index waveguide as shown in Figure 2. Recently, hybrid index profile waveguide was shown combining both inverse-step index waveguide and graded index waveguides for highpower amplification of a Gaussian single-mode beam [7].

#### 2.1. Waveguide mode

total internal reflection from interfaces parallel to the optical axis" (fiber optical waveguides,

Historically, high-frequency microwave sources had created a furore on guided wave photonics pioneered by Rayleigh and Sommerfeld. The first theoretical description of mode propagation along a dielectric guide was done by Hondros and Debye in 1910 [3]. The first dielectric waveguide to be examined at optical frequencies was the glass fiber used primarily for fiber

A waveguide can be defined as any structure (usually cylindrical) used for guiding the flow of electromagnetic wave in a direction parallel to its axis, confining it to a region either within or adjacent to its surfaces. In order to understand the propagation of light in a waveguide, it is imperative to derive the wave equation. The electromagnetic wave equation can be derived from the Maxwell's equation, assuming that we are operating in a source free ð Þ r ¼ 0; J ¼ 0 , linear <sup>ε</sup> and <sup>μ</sup> are independent of <sup>E</sup> and <sup>H</sup>Þ, and an isotropic medium. <sup>E</sup> and <sup>H</sup> are the electric and magnetic field amplitudes, respectively, ε is the electric permittivity of the medium, and μ

<sup>∇</sup> � <sup>E</sup> ¼ � <sup>∂</sup><sup>B</sup>

<sup>∇</sup> � <sup>H</sup> <sup>¼</sup> <sup>∂</sup><sup>D</sup>

Here, B and D are magnetic and electric fluxes, respectively. The wave equation derived from

The right-hand side of Eq. (5) is nonzero when there is a gradient in permittivity of the medium. Guided wave medium has a graded permittivity; however, in most structures, the

<sup>2</sup> <sup>¼</sup> <sup>0</sup>, <sup>∇</sup><sup>2</sup>

Optical waveguides can be classified according to their geometry, mode structure, refractive index (RI) distribution, and material. A dielectric optical waveguide comprises a longitudinally

<sup>2</sup> ¼ �∇ E:

H � με

∇ε ε 

> ∂2 H ∂t

∂2 E ∂t

∂2 E ∂t

<sup>∂</sup><sup>t</sup> (1)

<sup>∂</sup><sup>t</sup> (2)

<sup>2</sup> ¼ 0 (6)

(5)

∇:D ¼ 0 (3)

∇:B ¼ 0 (4)

is the magnetic permeability of the medium. The equations are:

∇2 E � με

term is negligible. Thus, the wave equation can be written as:

∇2 E � με

for electric and magnetic field amplitudes, respectively.

2. Classification of waveguides

slab waveguides, and resonators) [3].

96 Emerging Waveguide Technology

optics imaging applications [4].

the above expressions is:

A waveguide mode is an electromagnetic wave that propagates along a waveguide with a distinct phase velocity, group velocity, cross-sectional intensity distribution, and polarization. Each component of its electric and magnetic field is of the form f xð Þ ; <sup>y</sup> eiωt�ihz, where <sup>z</sup> is the axis of the waveguide. Modes are referred to as the "characteristic waves" of the structures because their field vector satisfies the homogenous wave equation in all the media that make up the

Figure 1. (a) Planar optical waveguide of 1-d transverse (x) optical confinement, (b) non-planar optical waveguide of 2-D transverse (x, y) optical confinement.

Figure 2. (a) Step-index type waveguide, (b) Graded-index waveguide, and (c) Hybrid waveguide.

guide, as well as the boundary conditions at the interfaces. The electric and magnetic fields of a mode can be written as Evð Þ¼ <sup>r</sup>; <sup>t</sup> Evð Þ <sup>x</sup>; <sup>y</sup> exp <sup>i</sup>βv<sup>z</sup> � <sup>i</sup>ω<sup>t</sup> and Hvð Þ¼ <sup>r</sup>; <sup>t</sup> Hvð Þ <sup>x</sup>; <sup>y</sup> exp <sup>i</sup>βv<sup>z</sup> � <sup>i</sup>ω<sup>t</sup> , where ν is the mode index, Evð Þ x; y and Hvð Þ x; y are the mode field profiles, and β<sup>v</sup> is the propagation constant of the mode.

A mode is characterized by an invariant transversal intensity profile and an effective index neff ). Each mode propagates through the waveguide with a phase velocity of c=neff , where c denotes the speed of light in vacuum and neff is the effective refractive index of that mode. It signifies how strongly the optical power is confined to the waveguide core. In order to understand modes intuitively, consider a simple step-index 2-D waveguide and an incident coherent light at an angle θ between the wave normal and the normal to the interface as shown in Figure 3. The critical angle at the upper interface is <sup>θ</sup><sup>c</sup> <sup>¼</sup> sin�<sup>1</sup> nc=nf and lower interface <sup>θ</sup><sup>s</sup> <sup>¼</sup> sin�<sup>1</sup> ns=nf and n<sup>s</sup> < n<sup>c</sup> ðθ<sup>s</sup> < θcÞ.

the fundamental waveguide modes become radiating. Fields in the waveguide can be classified based on the characteristics of the longitudinal field components, namely (1) Transverse electric and magnetic mode (TEM mode): Ez ¼ 0, and Hz ¼ 0. Dielectric waveguides do not support TEM modes, (2) Transverse electric mode (TE mode): Ez ¼ 0 and Hz 6¼ 0, (3) Transverse magnetic mode (TM mode): Hz ¼ 0 and Ez 6¼ 0, and (4) Hybrid mode: Ez 6¼ 0 and

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Homogeneous wave equations exist for planar slab waveguides of any index profile n(x). For a

Infinite slab waveguide: The slab waveguide is a step-index waveguide, comprising a high-index dielectric layer surrounded on either side by lower-index material (Figure 4). The slab is infinite in

Figure 4. Planar slab waveguide and transverse electric (TE) and transverse magnetic configuration (TM).

Hz 6¼ 0. Hybrid modes exist only in non-planar waveguide.

Figure 3. Ray-optical picture of modes propagating in an optical waveguide.

planar waveguide, the modes are either TE or TM.

2.2. Planar waveguide

Optical modes with an effective index higher than the largest cladding index are (1) Guided modes (θ<sup>s</sup> < θ < 90� Þ: As the wave is reflected back and forth between the two interfaces, it interferes with itself. A guided mode can exist only when a transverse resonance condition is satisfied so that the repeatedly reflected wave has constructive interference with itself. Modes with lower index are radiating and the optical power will leak to the cladding regions. They can be categorized as (2) Substrate radiation modes (θ<sup>c</sup> < θ < θs): Total reflection occurs only at the upper interface resulting in refraction of the incident wave at the lower interface from either the core or the substrate, (3) Substrate-cover radiation modes (θ < θcÞ: No total reflection at either interface. Incident wave is refracted at both interfaces, and it can transversely extend to infinity on both sides of the waveguide, and (4) Evanescent modes: Their fields decay exponentially along the z direction. For a lossless waveguide, the energy of an evanescent mode radiates away from the waveguide transversely.

The waveguide dimensions determine which modes can exist. Most waveguides support modes of two independent polarizations, with either the dominant magnetic (quasi-TM) or electric (quasi-TE) field component along the transverse (horizontal) direction. For most applications, it is preferable that the waveguides operate in a single-mode regime for each polarization. This single-mode regime is obtained by reducing the waveguide dimensions until all but

Figure 3. Ray-optical picture of modes propagating in an optical waveguide.

the fundamental waveguide modes become radiating. Fields in the waveguide can be classified based on the characteristics of the longitudinal field components, namely (1) Transverse electric and magnetic mode (TEM mode): Ez ¼ 0, and Hz ¼ 0. Dielectric waveguides do not support TEM modes, (2) Transverse electric mode (TE mode): Ez ¼ 0 and Hz 6¼ 0, (3) Transverse magnetic mode (TM mode): Hz ¼ 0 and Ez 6¼ 0, and (4) Hybrid mode: Ez 6¼ 0 and Hz 6¼ 0. Hybrid modes exist only in non-planar waveguide.

#### 2.2. Planar waveguide

guide, as well as the boundary conditions at the interfaces. The electric and magnetic fields of a mode can be written as Evð Þ¼ <sup>r</sup>; <sup>t</sup> Evð Þ <sup>x</sup>; <sup>y</sup> exp <sup>i</sup>βv<sup>z</sup> � <sup>i</sup>ω<sup>t</sup> and Hvð Þ¼ <sup>r</sup>; <sup>t</sup> Hvð Þ <sup>x</sup>; <sup>y</sup> exp <sup>i</sup>βv<sup>z</sup> � <sup>i</sup>ω<sup>t</sup> , where ν is the mode index, Evð Þ x; y and Hvð Þ x; y are the mode field profiles, and β<sup>v</sup> is the

Figure 2. (a) Step-index type waveguide, (b) Graded-index waveguide, and (c) Hybrid waveguide.

A mode is characterized by an invariant transversal intensity profile and an effective index

 ). Each mode propagates through the waveguide with a phase velocity of c=neff , where c denotes the speed of light in vacuum and neff is the effective refractive index of that mode. It signifies how strongly the optical power is confined to the waveguide core. In order to understand modes intuitively, consider a simple step-index 2-D waveguide and an incident coherent light at an angle θ between the wave normal and the normal to the interface as shown in

Optical modes with an effective index higher than the largest cladding index are (1) Guided

interferes with itself. A guided mode can exist only when a transverse resonance condition is satisfied so that the repeatedly reflected wave has constructive interference with itself. Modes with lower index are radiating and the optical power will leak to the cladding regions. They can be categorized as (2) Substrate radiation modes (θ<sup>c</sup> < θ < θs): Total reflection occurs only at the upper interface resulting in refraction of the incident wave at the lower interface from either the core or the substrate, (3) Substrate-cover radiation modes (θ < θcÞ: No total reflection at either interface. Incident wave is refracted at both interfaces, and it can transversely extend to infinity on both sides of the waveguide, and (4) Evanescent modes: Their fields decay exponentially along the z direction. For a lossless waveguide, the energy of an evanes-

The waveguide dimensions determine which modes can exist. Most waveguides support modes of two independent polarizations, with either the dominant magnetic (quasi-TM) or electric (quasi-TE) field component along the transverse (horizontal) direction. For most applications, it is preferable that the waveguides operate in a single-mode regime for each polarization. This single-mode regime is obtained by reducing the waveguide dimensions until all but

Þ: As the wave is reflected back and forth between the two interfaces, it

nc=nf and lower interface

Figure 3. The critical angle at the upper interface is <sup>θ</sup><sup>c</sup> <sup>¼</sup> sin�<sup>1</sup>

cent mode radiates away from the waveguide transversely.

ns=nf and n<sup>s</sup> < n<sup>c</sup> ðθ<sup>s</sup> < θcÞ.

propagation constant of the mode.

98 Emerging Waveguide Technology

neff

<sup>θ</sup><sup>s</sup> <sup>¼</sup> sin�<sup>1</sup>

modes (θ<sup>s</sup> < θ < 90�

Homogeneous wave equations exist for planar slab waveguides of any index profile n(x). For a planar waveguide, the modes are either TE or TM.

Infinite slab waveguide: The slab waveguide is a step-index waveguide, comprising a high-index dielectric layer surrounded on either side by lower-index material (Figure 4). The slab is infinite in

Figure 4. Planar slab waveguide and transverse electric (TE) and transverse magnetic configuration (TM).

the y-z plane and finite in x direction and the refractive index of ncore > ncladding, nsubstrate to ensure total internal reflection at the interface. For case (1): ncladding ¼ nsubstrate, the waveguide is denoted as Symmetric and for case (2): ncladding ¼6 nsubstrate, waveguide is Asymmetric.

For the electromagnetic analysis of the planar slab waveguide (infinite width), assuming ncore > nsubstrate > ncladding, we consider two possible electric field polarizations—TE or TM. The axis of waveguide is oriented in z-direction: k vector of the guided wave will propagate down the z-axis, striking the interfaces and angles greater than critical angle. The field could be TE which has no longitudinal component along z-axis (electric field is transverse to the plane of incidence established by the normal to the interface, and the k vector) or TM depending on the orientation of the electric field.

I. For TE Asymmetric waveguide: E field is polarized along the y-axis, and assuming that waveguide is excited by a source with frequency ω<sup>o</sup> and a vacuum wave vector of magnitude ωo <sup>c</sup> , the allowed modes can be evaluated by solving the wave equation in each dielectric region through boundary conditions. For a sinusoidal wave with angular frequency ωo, the wave equation for the electric field components in each region can be written as (j j k ¼ ω ffiffiffiffiffi με <sup>p</sup> <sup>¼</sup> <sup>k</sup>),

$$
\nabla^2 E\_y + k\_0^{\;\;2} n\_i^{\;2} E\_y = 0 \tag{7}
$$

The longitudinal wave vector β (z component of k) must satisfy k0nsubstrate < β < k0ncore (ncladding ≤ ncore) in order to be guided inside the waveguide. Eigen values for the waveguide can be derived using transverse components of electric field amplitudes in three regions as Eyð Þ¼ x Ae�γcladding<sup>x</sup> for 0 <sup>&</sup>lt; x; Eyð Þ¼ <sup>x</sup> Bcosð Þþ <sup>κ</sup>corex Csinð Þ <sup>κ</sup>corex for � <sup>h</sup> <sup>&</sup>lt; <sup>x</sup> <sup>&</sup>lt; 0; Eyð Þ¼ <sup>x</sup> De<sup>γ</sup>substrateð Þ <sup>x</sup>þ<sup>h</sup> for x < �h, where A, B, C, and D are amplitude coefficients to be derived using boundary

kcore sinð � kcoreh , and thus:

¼ A kcoresinð Þ� kcoreh γcladding cos kð Þ coreh

kcore

II. TM Asymmetric waveguide: The field components of the waveguide can be written as:

Figure 5. Plot for Eigen value equation for (a) Asymmetric TE mode slab waveguide, (b) Asymmetric TM mode slab

ωε Hmð Þx e

<sup>¼</sup> A cos kð Þþ coreh <sup>γ</sup>cladding

tanð Þ¼ hkcore

, Exð Þ¼ <sup>x</sup>; <sup>z</sup>; <sup>t</sup> <sup>β</sup>

h iðCore<sup>Þ</sup>

<sup>∂</sup><sup>x</sup> ð Þ at x ¼ 0 are continuous). Solving the equation, we get A ¼ B, C ¼

sinð Þ kcoreh � �γsubstrateðSubstrateÞ:

> γcladding þ γsubtrate kcore <sup>1</sup> � <sup>γ</sup>claddingγsubstrate k 2

> > �<sup>i</sup>ð Þ <sup>ω</sup>t�β<sup>z</sup> , and Ezð Þ¼ <sup>x</sup>; <sup>z</sup>; <sup>t</sup> �<sup>i</sup>

core h i (12)

ωε : The Eigen value for

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conditions (Ey, Hz, <sup>∂</sup>Ey

HYð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>t</sup> Hmð Þ<sup>x</sup> <sup>e</sup>�iω<sup>t</sup>

Waveguide.

β (Figure 5(b)) is given by:

<sup>κ</sup>core , <sup>D</sup> <sup>¼</sup> <sup>A</sup>½cosð Þþ <sup>κ</sup>coreh <sup>γ</sup>cladding

The Eigenvalue equation (Figure 5(a)) is given by:

∂Ey ∂x � � � � x¼�h

� <sup>A</sup>ϓcladding

here, ni can be the refractive index of either core, cladding, or the substrate. The solution to Equation (7) can be written as:

$$E\_y(\mathbf{x}, z) = E\_y(\mathbf{x})e^{-j\theta z} \tag{8}$$

due to the translational invariance of the waveguide in z-direction. β is the propagation constant along the <sup>z</sup>-direction (longitudinal). From Equation (8) and since <sup>d</sup>2<sup>y</sup> dx<sup>2</sup> ¼ 0, we can write:

$$\frac{\partial^2 E\_y}{\partial x^2} + \left(k\_0{}^2 n\_i{}^2 - \beta^2\right) E\_y = 0 \tag{9}$$

The solution to the wave equation can be deduced by considering Case (1) β > k0ni and E<sup>0</sup> is field amplitude at x = 0, solution is exponentially decaying and can be written as:

$$E\_y(\mathbf{x}) = E\_0 \mathbf{e}^{\pm \sqrt{\beta^2 - k\_0^2 n\_i^2 \mathbf{x}}} \tag{10}$$

The attenuation constant ϓ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>β</sup><sup>2</sup> � <sup>k</sup><sup>0</sup> 2 ni 2 q . Case (2) β < k0ni, solution has an oscillatory nature and is given by:

$$E\_{\mathbf{y}}(\mathbf{x}) = E\_0 e^{\pm \sqrt{k\_0^2 n\_i^2 - \beta^2} \mathbf{x}} \tag{11}$$

The transverse wave vector κ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k0 2 ni <sup>2</sup> � <sup>β</sup><sup>2</sup> q and the relation between β, κ and k are given by k <sup>2</sup> <sup>¼</sup> <sup>β</sup><sup>2</sup> <sup>þ</sup> <sup>κ</sup>2.

The longitudinal wave vector β (z component of k) must satisfy k0nsubstrate < β < k0ncore (ncladding ≤ ncore) in order to be guided inside the waveguide. Eigen values for the waveguide can be derived using transverse components of electric field amplitudes in three regions as Eyð Þ¼ x Ae�γcladding<sup>x</sup> for 0 <sup>&</sup>lt; x; Eyð Þ¼ <sup>x</sup> Bcosð Þþ <sup>κ</sup>corex Csinð Þ <sup>κ</sup>corex for � <sup>h</sup> <sup>&</sup>lt; <sup>x</sup> <sup>&</sup>lt; 0; Eyð Þ¼ <sup>x</sup> De<sup>γ</sup>substrateð Þ <sup>x</sup>þ<sup>h</sup> for x < �h, where A, B, C, and D are amplitude coefficients to be derived using boundary conditions (Ey, Hz, <sup>∂</sup>Ey <sup>∂</sup><sup>x</sup> ð Þ at x ¼ 0 are continuous). Solving the equation, we get A ¼ B, C ¼ � <sup>A</sup>ϓcladding <sup>κ</sup>core , <sup>D</sup> <sup>¼</sup> <sup>A</sup>½cosð Þþ <sup>κ</sup>coreh <sup>γ</sup>cladding kcore sinð � kcoreh , and thus:

$$\left. \frac{\partial E\_y}{\partial \mathbf{x}} \right|\_{\mathbf{x} = -h} = A \left[ k\_{core} \sin(k\_{core}h) - \mathcal{V}\_{cladding} \cos(k\_{core}h) \right] \text{(Core)}$$

$$= A \left[ \cos \left( k\_{core}h \right) + \frac{\mathcal{V}\_{cladding}}{k\_{core}} \sin(k\_{core}h) \right] \mathcal{V}\_{substrate} \text{(Substrate)}.$$

The Eigenvalue equation (Figure 5(a)) is given by:

the y-z plane and finite in x direction and the refractive index of ncore > ncladding, nsubstrate to ensure total internal reflection at the interface. For case (1): ncladding ¼ nsubstrate, the waveguide is denoted as

For the electromagnetic analysis of the planar slab waveguide (infinite width), assuming ncore > nsubstrate > ncladding, we consider two possible electric field polarizations—TE or TM. The axis of waveguide is oriented in z-direction: k vector of the guided wave will propagate down the z-axis, striking the interfaces and angles greater than critical angle. The field could be TE which has no longitudinal component along z-axis (electric field is transverse to the plane of incidence established by the normal to the interface, and the k vector) or TM depending on the

I. For TE Asymmetric waveguide: E field is polarized along the y-axis, and assuming that waveguide is excited by a source with frequency ω<sup>o</sup> and a vacuum wave vector of magnitude

<sup>c</sup> , the allowed modes can be evaluated by solving the wave equation in each dielectric region through boundary conditions. For a sinusoidal wave with angular frequency ωo, the wave

here, ni can be the refractive index of either core, cladding, or the substrate. The solution to

Eyð Þ¼ x; z Eyð Þx e

2 ni

field amplitude at x = 0, solution is exponentially decaying and can be written as:

Eyð Þ¼ x E0e

Eyð Þ¼ x E0e

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k0 2 ni <sup>2</sup> � <sup>β</sup><sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>β</sup><sup>2</sup> � <sup>k</sup><sup>0</sup> 2 ni 2

q

q

due to the translational invariance of the waveguide in z-direction. β is the propagation constant

The solution to the wave equation can be deduced by considering Case (1) β > k0ni and E<sup>0</sup> is

�

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>β</sup>2�k<sup>0</sup> 2 ni 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k0 2ni

με <sup>p</sup> <sup>¼</sup> <sup>k</sup>),

Ey ¼ 0 (7)

�jβ<sup>z</sup> (8)

dx<sup>2</sup> ¼ 0, we can write:

<sup>2</sup> � <sup>β</sup><sup>2</sup> � �Ey <sup>¼</sup> <sup>0</sup> (9)

<sup>p</sup> <sup>x</sup> (10)

<sup>2</sup>�β<sup>2</sup> <sup>p</sup> <sup>x</sup> (11)

and the relation between β, κ and k are given by

. Case (2) β < k0ni, solution has an oscillatory

equation for the electric field components in each region can be written as (j j k ¼ ω ffiffiffiffiffi

Ey þ k<sup>0</sup> 2 ni 2

∇2

along the <sup>z</sup>-direction (longitudinal). From Equation (8) and since <sup>d</sup>2<sup>y</sup>

∂2 Ey <sup>∂</sup>x<sup>2</sup> <sup>þ</sup> <sup>k</sup><sup>0</sup>

Symmetric and for case (2): ncladding ¼6 nsubstrate, waveguide is Asymmetric.

orientation of the electric field.

100 Emerging Waveguide Technology

Equation (7) can be written as:

The attenuation constant ϓ ¼

The transverse wave vector κ ¼

nature and is given by:

k

<sup>2</sup> <sup>¼</sup> <sup>β</sup><sup>2</sup> <sup>þ</sup> <sup>κ</sup>2.

ωo

$$\tan(hk\_{core}) = \frac{\gamma\_{cladding} + \gamma\_{substrate}}{k\_{core} \left[1 - \frac{\gamma\_{clading}\gamma\_{substrate}}{k\_{core}^2}\right]} \tag{12}$$

#### II. TM Asymmetric waveguide: The field components of the waveguide can be written as:

HYð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>t</sup> Hmð Þ<sup>x</sup> <sup>e</sup>�iω<sup>t</sup> , Exð Þ¼ <sup>x</sup>; <sup>z</sup>; <sup>t</sup> <sup>β</sup> ωε Hmð Þx e �<sup>i</sup>ð Þ <sup>ω</sup>t�β<sup>z</sup> , and Ezð Þ¼ <sup>x</sup>; <sup>z</sup>; <sup>t</sup> �<sup>i</sup> ωε : The Eigen value for β (Figure 5(b)) is given by:

Figure 5. Plot for Eigen value equation for (a) Asymmetric TE mode slab waveguide, (b) Asymmetric TM mode slab Waveguide.

$$\tan(hk\_{core}) = \frac{k\_{core} \left[ \frac{n\_{core}^2}{n\_{substate}^2} \mathcal{V}\_{substrate} + \frac{n\_{core}^2}{n\_{clading}^2} \mathcal{V}\_{cladding} \right]}{k\_{core}^2 - \frac{n\_{core}^4}{n\_{clading}^2 n\_{subrate}^2} \mathcal{V}\_{cladding} \mathcal{V}\_{substrate}} \tag{13}$$

waveguide dimensions; (iii) most modes will be unguided, and all modes are orthogonal to each other; (iv) some modes are degenerate. Degenerate modes will share the same value of β but will have distinguishable electric field distributions. The lower-order mode is expressed by βlowest order ≈ kncore and higher-order mode by βlowest order ≈ kncore cos θcritical ≈ knsubstrate. A wave-

> core � <sup>n</sup><sup>2</sup> subtrate � �1=<sup>2</sup>

The approximate number of modes (m) in the waveguide are given by m ≈V=π. Graphical

claddding�

where a is asymmetry parameter (ranges from 0 (symmetric waveguide) to infinity), b is normalized effective index (ranges from 0 (cutoff) to 1) and neff ¼ β=ko is the effective index of

where ν is an integer. The cut-off condition (b = 0) for modes in a step-index waveguide is given

Figure 7. Normalized index b versus normalized frequency V for different values of asymmetry coefficient a (a = 0, a = 10,

core � <sup>n</sup><sup>2</sup> substrate � �<sup>1</sup>=<sup>2</sup>

> = <sup>n</sup><sup>2</sup> core�n<sup>2</sup>

.

<sup>b</sup>=ð Þ <sup>1</sup> � <sup>b</sup> <sup>p</sup> <sup>þ</sup> tan�<sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>a</sup> <sup>p</sup> <sup>þ</sup> <sup>v</sup>π. The numerical aperture is defined as the maximum angle that an

substrate ð Þ (17)

substrate ð Þ (18)

ð Þ <sup>b</sup> <sup>þ</sup> <sup>a</sup> <sup>=</sup>ð Þ <sup>1</sup> � <sup>b</sup> <sup>p</sup> (19)

(16)

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guide is generally characterized by its normalized frequency, given by,

a ¼ � n2 substrate�n<sup>2</sup>

> b ¼ � n2 eff �n<sup>2</sup> substrate� = <sup>n</sup><sup>2</sup> core�n<sup>2</sup>

the waveguide. The normalized dispersion relation is given by (Figure 7):

<sup>1</sup> � <sup>b</sup> <sup>p</sup> <sup>¼</sup> <sup>v</sup><sup>π</sup> <sup>þ</sup> tan�<sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

solution to the waveguide can be evaluated by:

V ffiffiffiffiffiffiffiffiffiffiffi

by <sup>V</sup> <sup>¼</sup> tan �<sup>1</sup> ffiffi

a = ∞).

<sup>V</sup> <sup>¼</sup> hk n<sup>2</sup>

<sup>V</sup> <sup>¼</sup> <sup>k</sup>0h n<sup>2</sup>

III. TE Symmetric waveguide: The field equation of a TE mode within the symmetric waveguide is given by:

$$\begin{aligned} E\_y &= A e^{-\gamma \left(\mathbf{x} - \mathbf{\hat{v}}\_2\right)} & \text{for } \mathbf{x} \succeq \mathbf{\hat{v}}\_2\\ E\_y &= A \frac{\cos \kappa \mathbf{x}}{\cos^{\kappa \mathbf{\hat{v}}} \mathbf{\hat{z}}} \text{ or } \ A \frac{\sin \kappa \mathbf{x}}{\sin^{\kappa \mathbf{\hat{v}}} \mathbf{\hat{z}}} & \text{for } -\mathbf{\hat{v}}\_2 \le \mathbf{x} \le \mathbf{\hat{v}}\_2\\ E\_y &= \pm A e^{\gamma \left(\mathbf{x} + \boldsymbol{\psi}\_2\right)} & \text{for } \mathbf{x} \le -\mathbf{\hat{v}}\_2 \end{aligned} \tag{14}$$

The characteristic Eigen value equation for the TE modes in a symmetric waveguide is given by:

$$\begin{aligned} \tan \frac{kh}{2} &= \text{\textquotedblleft for even (cos) modes} \\ &= -k/\text{\textquotedblright} \text{ for odd (sin) modes} \end{aligned} \tag{15}$$

In order to plot the Eigen values of the TE modes of the symmetric waveguide, solutions of Eq. (15) are plotted for a wavelength of 1.55 μm and different "h" values (15 μm and 3 μm respectively) as shown in Figure 6.

The longitudinal wave vector β is quintessential to describe the field amplitudes in all regions of the waveguide. (i) Every Eigen value β corresponds to a distinct confined mode of the system. The amplitude of the mode is established by the power carried in the mode; (ii) only a finite number of modes will be guided depending on the wavelength, index contrast, and

Figure 6. Plot for Eigen value equation for Symmetric TE mode slab waveguide at a wavelength of 1.55 μm for waveguide width of (a) 15 μm and (b) 3 μm. Thick waveguide supports multimode transmission.

waveguide dimensions; (iii) most modes will be unguided, and all modes are orthogonal to each other; (iv) some modes are degenerate. Degenerate modes will share the same value of β but will have distinguishable electric field distributions. The lower-order mode is expressed by βlowest order ≈ kncore and higher-order mode by βlowest order ≈ kncore cos θcritical ≈ knsubstrate. A waveguide is generally characterized by its normalized frequency, given by,

tanð Þ¼ hkcore

Ey <sup>¼</sup> <sup>A</sup> cosκ<sup>x</sup> cosκ<sup>h</sup>=<sup>2</sup>

> tan κh

respectively) as shown in Figure 6.

¼ �<sup>k</sup> y

guide is given by:

102 Emerging Waveguide Technology

by:

kcore <sup>n</sup><sup>2</sup> core n2 substrate

core � <sup>n</sup><sup>4</sup>

Ey <sup>¼</sup> Ae�<sup>γ</sup> <sup>x</sup>�h<sup>=</sup> ð Þ<sup>2</sup> for x <sup>≥</sup> <sup>h</sup>=<sup>2</sup>

or A sinκ<sup>x</sup> sinκ<sup>h</sup>=<sup>2</sup>

Ey ¼ �Ae<sup>γ</sup> <sup>x</sup>þh<sup>=</sup> ð Þ<sup>2</sup> for x <sup>≤</sup> � <sup>h</sup>=<sup>2</sup>

The characteristic Eigen value equation for the TE modes in a symmetric waveguide is given

<sup>2</sup> <sup>¼</sup> <sup>γ</sup>=<sup>k</sup> for even <sup>ð</sup>cos<sup>Þ</sup> modes

In order to plot the Eigen values of the TE modes of the symmetric waveguide, solutions of Eq. (15) are plotted for a wavelength of 1.55 μm and different "h" values (15 μm and 3 μm

The longitudinal wave vector β is quintessential to describe the field amplitudes in all regions of the waveguide. (i) Every Eigen value β corresponds to a distinct confined mode of the system. The amplitude of the mode is established by the power carried in the mode; (ii) only a finite number of modes will be guided depending on the wavelength, index contrast, and

Figure 6. Plot for Eigen value equation for Symmetric TE mode slab waveguide at a wavelength of 1.55 μm for

waveguide width of (a) 15 μm and (b) 3 μm. Thick waveguide supports multimode transmission.

for odd <sup>ð</sup>sin<sup>Þ</sup> modes

n2 claddingn<sup>2</sup> substrate

III. TE Symmetric waveguide: The field equation of a TE mode within the symmetric wave-

k 2 <sup>γ</sup>substrate <sup>þ</sup> <sup>n</sup><sup>2</sup>

core

core n2 cladding

γcladdingγsubstrate

for � <sup>h</sup>=<sup>2</sup> ≤ x ≤ <sup>h</sup>=<sup>2</sup>

γcladding

(13)

(14)

(15)

$$V = hk \left(n\_{core}^2 - n\_{subrate}^2\right)^{\upsilon\_2}.$$

The approximate number of modes (m) in the waveguide are given by m ≈V=π. Graphical solution to the waveguide can be evaluated by:

$$V = k\_0 h \left( n\_{core}^2 - n\_{substate}^2 \right)^{\iota\_2} \tag{16}$$

$$\mathfrak{a} = \left( n\_{\text{substr}tr}^2 - n\_{\text{claddling}}^2 \right) \Big/ \begin{pmatrix} n\_{\text{avg}}^2 - n\_{\text{substr}tr}^2 \end{pmatrix} \tag{17}$$

$$b = \left(n\_{\text{eff}}^2 - n\_{\text{substate}}^2\right) / \left(n\_{\text{core}}^2 - n\_{\text{substate}}^2\right) \tag{18}$$

where a is asymmetry parameter (ranges from 0 (symmetric waveguide) to infinity), b is normalized effective index (ranges from 0 (cutoff) to 1) and neff ¼ β=ko is the effective index of the waveguide. The normalized dispersion relation is given by (Figure 7):

$$V\sqrt{1-b} = v\pi + \tan^{-1}\sqrt{b/(1-b)} + \tan^{-1}\sqrt{(b+a)/(1-b)}\tag{19}$$

where ν is an integer. The cut-off condition (b = 0) for modes in a step-index waveguide is given by <sup>V</sup> <sup>¼</sup> tan �<sup>1</sup> ffiffi <sup>a</sup> <sup>p</sup> <sup>þ</sup> <sup>v</sup>π. The numerical aperture is defined as the maximum angle that an

Figure 7. Normalized index b versus normalized frequency V for different values of asymmetry coefficient a (a = 0, a = 10, a = ∞).

incident wave can have and still be guided within the waveguide. It is given by: NA ¼ sin θmax ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 core � <sup>n</sup><sup>2</sup> substrate=cladding <sup>q</sup> .

IV. TM Symmetric waveguide: The characteristic Eigen value equation for the TM modes in a symmetric waveguide is given by:

$$\begin{aligned} \tan \frac{\kappa h}{2} &= - (n\_{\text{core}} / n\_{\text{substrate}})^2 \rangle\_{\text{\textsuperscript{\text{ $\chi$ }}}} : \text{ even (cos) modes} \\ &= - (n\_{\text{core}} / n\_{\text{substrate}})^2 \rangle\_{\text{\textsuperscript{\text{ $\chi$ }}}} / \_{\text{\textsuperscript{\text{ $\chi$ }}}} \text{ for odd (sin) modes} \end{aligned} \tag{20}$$

Graphical solution to the waveguide can be evaluated using:

$$V = k\_0 \text{l} \left( n\_{core}^2 - n\_{substate}^2 \right)^{\circ \circ} \tag{21}$$

R <sup>00</sup>Φ<sup>Z</sup> <sup>þ</sup> 1 r R0 ΦZ þ 1

(κ<sup>2</sup> <sup>¼</sup> <sup>k</sup> 2 <sup>0</sup>n<sup>2</sup> � <sup>β</sup><sup>2</sup>

Er <sup>¼</sup> �j<sup>β</sup>

defined as:

decreasing function).

BκJ 0

jβ γ2 jv <sup>κ</sup><sup>2</sup> AκJ 0

<sup>v</sup>ð Þ� <sup>κ</sup><sup>r</sup> <sup>j</sup>ωEcorev

<sup>r</sup> CKvð Þ� <sup>γ</sup><sup>r</sup> ωμ

<sup>v</sup>ð Þþ <sup>κ</sup><sup>r</sup> <sup>j</sup>ωμ<sup>v</sup>

<sup>β</sup><sup>r</sup> AJvð Þ κr h iejv<sup>ϕ</sup>e�jβ<sup>z</sup> and <sup>H</sup><sup>ϕ</sup> <sup>¼</sup> �j<sup>β</sup>

<sup>β</sup> DγK<sup>0</sup>

<sup>β</sup><sup>r</sup> BJvð Þ κr h iejv<sup>ϕ</sup>e�jβ<sup>z</sup>, <sup>E</sup><sup>ϕ</sup> <sup>¼</sup> �j<sup>β</sup>

is core's radius. In the cladding (r>a) Er <sup>¼</sup> <sup>j</sup><sup>β</sup>

<sup>v</sup>ð Þ γr h iejv<sup>ϕ</sup>e�jβ<sup>z</sup> and Hr <sup>¼</sup> <sup>j</sup><sup>β</sup>

And solution to (2) is

r 2 ∂2 R <sup>∂</sup>r<sup>2</sup> <sup>þ</sup> <sup>r</sup>

) and (2) Kνð Þ ϓr when k

function (1) can be approximated by (κr is large) (Figure 9):

The solution is given by Bessel functions: (1),Jνð Þ κr when k

Jvð Þ κr ≈

<sup>r</sup><sup>2</sup> <sup>R</sup>Φ<sup>00</sup><sup>Z</sup> <sup>þ</sup> <sup>R</sup>Φ<sup>Z</sup>

The solution to the wave equation is deduced from separation of variables, and we obtain:

∂R ∂r þ r <sup>2</sup> k<sup>2</sup>

2

ffiffiffiffiffiffiffiffi 2 πκr

Kvð Þ <sup>γ</sup><sup>r</sup> <sup>≈</sup> <sup>e</sup>�γ<sup>r</sup>

κ2 jv

The V-number or the normalized frequency is used to characterize the waveguide and is

Figure 9. Bessel Function of the (a) first kind (behaves as a damped sine wave) and (b) second kind (monotonic

The equation for field distribution in the step-index fiber can be calculated through:

r

00 þ k 2 0n2

<sup>0</sup>n<sup>2</sup> � <sup>β</sup><sup>2</sup> � <sup>v</sup><sup>2</sup>

cos <sup>κ</sup><sup>r</sup> � <sup>v</sup>

ffiffiffiffiffiffiffiffiffiffi

κ2 jv

<sup>γ</sup><sup>2</sup> DγK<sup>0</sup>

πr � π 4

r2

� �<sup>R</sup> <sup>¼</sup> <sup>0</sup> (27)

� � (28)

<sup>2</sup>πγ<sup>r</sup> <sup>p</sup> (29)

<sup>β</sup> BκJ 0 <sup>v</sup>ð Þ κr

<sup>β</sup> AκJ 0 <sup>v</sup>ð Þ κr h iejv<sup>ϕ</sup>e�jβ<sup>z</sup> for (r<a); <sup>a</sup>

<sup>v</sup>ð Þþ <sup>γ</sup><sup>r</sup> <sup>j</sup>ωμ<sup>v</sup>

<sup>v</sup>ð Þ� <sup>γ</sup><sup>r</sup> <sup>j</sup>ωEcladv

h iejv<sup>ϕ</sup>e�jβ<sup>z</sup>

<sup>β</sup><sup>r</sup> DKvð Þ γr h iejv<sup>ϕ</sup>e�jβ<sup>z</sup>

<sup>β</sup><sup>r</sup> CKvð Þ γr h iejv<sup>ϕ</sup>e�jβ<sup>z</sup>

2

<sup>0</sup>n<sup>2</sup> � <sup>β</sup><sup>2</sup> � <sup>ν</sup><sup>2</sup>=r<sup>2</sup> is negative (γ<sup>2</sup> <sup>¼</sup> <sup>β</sup><sup>2</sup> � <sup>k</sup>

<sup>r</sup> AJvð Þ� <sup>κ</sup><sup>r</sup> ωμ

<sup>r</sup> BJvð Þ� <sup>κ</sup><sup>r</sup> <sup>ω</sup>Ecore

<sup>γ</sup><sup>2</sup> CγK<sup>0</sup>

RΦZ ¼ 0 (26)

http://dx.doi.org/10.5772/intechopen.77150

Review on Optical Waveguides

105

<sup>0</sup>n<sup>2</sup> � <sup>β</sup><sup>2</sup> � <sup>ν</sup><sup>2</sup>=r<sup>2</sup> is positive

2

<sup>0</sup>n2). Bessel

, Hr <sup>¼</sup> �j<sup>β</sup> κ2

, E<sup>ϕ</sup> ¼

.

$$a = \frac{n\_{core}^2 n\_{substrute}^2 - n\_{cladding}^2}{n\_{cladding}^2 n\_{core}^2 - n\_{substrate}^2} \tag{22}$$

$$\mathbf{b} = \left( n\_{\text{eff}}^2 - n\_{\text{substate}}^2 \right) \Big/ \begin{pmatrix} n\_{\text{core}}^2 - n\_{\text{substate}}^2 \end{pmatrix} \tag{23}$$

#### 2.3. Non-planar waveguide

The following section describes step-index circular and channel waveguides.

I. Step-index circular waveguide: The wave equation for the step-index circular waveguides in cylindrical coordinates is given by:

$$E\left(r,\phi,z\right) = \hat{r}E\_r\left(r,\phi,z\right) + \hat{\phi}E\_\phi\left(r,\phi,z\right) + \hat{z}E\_z\left(r,\phi,z\right) \tag{24}$$

At z = 0, field is purely radial (Figure 8).The Ez component of the electric field couples only to itself and the scalar wave equation for Ez is given by:

$$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial E\_z}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 E\_z}{\partial \phi^2} + \frac{\partial^2 E\_z}{\partial z^2} + k\_0^2 n^2 E\_z = 0\tag{25}$$

One can write Ez <sup>r</sup>; <sup>ϕ</sup>; <sup>z</sup> � � <sup>¼</sup> R rð Þϕ φð ÞZ zð Þ, Eq. (24) can be written as:

Figure 8. Schematic representation of step-index circular waveguide.

Review on Optical Waveguides http://dx.doi.org/10.5772/intechopen.77150 105

$$R''\Phi Z + \frac{1}{r}R'\Phi Z + \frac{1}{r^2}R\Phi'^\prime Z + R\Phi Z^\prime + k\_0^2 n^2 R\Phi Z = 0\tag{26}$$

The solution to the wave equation is deduced from separation of variables, and we obtain:

$$r^2\frac{\partial^2 R}{\partial r^2} + r\frac{\partial R}{\partial r} + r^2 \left(k\_0^2 n^2 - \beta^2 - \frac{\sigma^2}{r^2}\right) R = 0\tag{27}$$

The solution is given by Bessel functions: (1),Jνð Þ κr when k 2 <sup>0</sup>n<sup>2</sup> � <sup>β</sup><sup>2</sup> � <sup>ν</sup><sup>2</sup>=r<sup>2</sup> is positive (κ<sup>2</sup> <sup>¼</sup> <sup>k</sup> 2 <sup>0</sup>n<sup>2</sup> � <sup>β</sup><sup>2</sup> ) and (2) Kνð Þ ϓr when k 2 <sup>0</sup>n<sup>2</sup> � <sup>β</sup><sup>2</sup> � <sup>ν</sup><sup>2</sup>=r<sup>2</sup> is negative (γ<sup>2</sup> <sup>¼</sup> <sup>β</sup><sup>2</sup> � <sup>k</sup> 2 <sup>0</sup>n2). Bessel function (1) can be approximated by (κr is large) (Figure 9):

$$J\_v(\kappa r) \approx \sqrt{\frac{2}{\pi \kappa r}} \cos \left(\kappa r - \frac{v}{\pi r} - \frac{\pi}{4}\right) \tag{28}$$

And solution to (2) is

incident wave can have and still be guided within the waveguide. It is given by:

IV. TM Symmetric waveguide: The characteristic Eigen value equation for the TM modes in a

<sup>2</sup><sup>κ</sup> <sup>γ</sup>

core � <sup>n</sup><sup>2</sup>

I. Step-index circular waveguide: The wave equation for the step-index circular waveguides

At z = 0, field is purely radial (Figure 8).The Ez component of the electric field couples only to

∂2 Ez <sup>∂</sup>z<sup>2</sup> <sup>þ</sup> <sup>k</sup> 2 0n2

substrate � <sup>n</sup><sup>2</sup>

core � <sup>n</sup><sup>2</sup>

substrate � �<sup>1</sup>=<sup>2</sup>

cladding

substrate

E r; <sup>ϕ</sup>; <sup>z</sup> � � <sup>¼</sup> <sup>b</sup>rEr <sup>r</sup>; <sup>ϕ</sup>; <sup>z</sup> � � <sup>þ</sup> <sup>ϕ</sup>bE<sup>ϕ</sup> <sup>r</sup>; <sup>ϕ</sup>; <sup>z</sup> � � <sup>þ</sup> <sup>b</sup>zEz <sup>r</sup>; <sup>ϕ</sup>; <sup>z</sup> � � (24)

substrate ð Þ (23)

Ez ¼ 0 (25)

<sup>2</sup>γ=<sup>κ</sup> : even <sup>ð</sup>cos<sup>Þ</sup> modes

� for odd <sup>ð</sup>sin<sup>Þ</sup> modes

(20)

(21)

(22)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Graphical solution to the waveguide can be evaluated using:

substrate=cladding

<sup>2</sup> <sup>¼</sup> ð Þ ncore=nsubstrate

¼ �ð Þ ncore=nsubstrate

<sup>V</sup> <sup>¼</sup> <sup>k</sup>0h n<sup>2</sup>

n2 claddingn<sup>2</sup>

<sup>a</sup> <sup>¼</sup> <sup>n</sup><sup>2</sup> coren<sup>2</sup>

The following section describes step-index circular and channel waveguides.

þ 1 r2 ∂2 Ez <sup>∂</sup>ϕ<sup>2</sup> <sup>þ</sup>

One can write Ez <sup>r</sup>; <sup>ϕ</sup>; <sup>z</sup> � � <sup>¼</sup> R rð Þϕ φð ÞZ zð Þ, Eq. (24) can be written as:

b ¼ � n2 eff �n<sup>2</sup> substrate � = <sup>n</sup><sup>2</sup> core�n<sup>2</sup>

.

n2 core � <sup>n</sup><sup>2</sup>

> tan κh

q

symmetric waveguide is given by:

2.3. Non-planar waveguide

in cylindrical coordinates is given by:

itself and the scalar wave equation for Ez is given by:

1 r ∂ ∂r r ∂Ez ∂r � �

Figure 8. Schematic representation of step-index circular waveguide.

NA ¼ sin θmax ¼

104 Emerging Waveguide Technology

$$K\_v(\gamma r) \approx \frac{e^{-\gamma r}}{\sqrt{2\pi \gamma r}}\tag{29}$$

The equation for field distribution in the step-index fiber can be calculated through:

Er <sup>¼</sup> �j<sup>β</sup> <sup>κ</sup><sup>2</sup> AκJ 0 <sup>v</sup>ð Þþ <sup>κ</sup><sup>r</sup> <sup>j</sup>ωμ<sup>v</sup> <sup>β</sup><sup>r</sup> BJvð Þ κr h iejv<sup>ϕ</sup>e�jβ<sup>z</sup>, <sup>E</sup><sup>ϕ</sup> <sup>¼</sup> �j<sup>β</sup> κ2 jv <sup>r</sup> AJvð Þ� <sup>κ</sup><sup>r</sup> ωμ <sup>β</sup> BκJ 0 <sup>v</sup>ð Þ κr h iejv<sup>ϕ</sup>e�jβ<sup>z</sup> , Hr <sup>¼</sup> �j<sup>β</sup> κ2 BκJ 0 <sup>v</sup>ð Þ� <sup>κ</sup><sup>r</sup> <sup>j</sup>ωEcorev <sup>β</sup><sup>r</sup> AJvð Þ κr h iejv<sup>ϕ</sup>e�jβ<sup>z</sup> and <sup>H</sup><sup>ϕ</sup> <sup>¼</sup> �j<sup>β</sup> κ2 jv <sup>r</sup> BJvð Þ� <sup>κ</sup><sup>r</sup> <sup>ω</sup>Ecore <sup>β</sup> AκJ 0 <sup>v</sup>ð Þ κr h iejv<sup>ϕ</sup>e�jβ<sup>z</sup> for (r<a); <sup>a</sup> is core's radius. In the cladding (r>a) Er <sup>¼</sup> <sup>j</sup><sup>β</sup> <sup>γ</sup><sup>2</sup> CγK<sup>0</sup> <sup>v</sup>ð Þþ <sup>γ</sup><sup>r</sup> <sup>j</sup>ωμ<sup>v</sup> <sup>β</sup><sup>r</sup> DKvð Þ γr h iejv<sup>ϕ</sup>e�jβ<sup>z</sup> , E<sup>ϕ</sup> ¼ jβ γ2 jv <sup>r</sup> CKvð Þ� <sup>γ</sup><sup>r</sup> ωμ <sup>β</sup> DγK<sup>0</sup> <sup>v</sup>ð Þ γr h iejv<sup>ϕ</sup>e�jβ<sup>z</sup> and Hr <sup>¼</sup> <sup>j</sup><sup>β</sup> <sup>γ</sup><sup>2</sup> DγK<sup>0</sup> <sup>v</sup>ð Þ� <sup>γ</sup><sup>r</sup> <sup>j</sup>ωEcladv <sup>β</sup><sup>r</sup> CKvð Þ γr h iejv<sup>ϕ</sup>e�jβ<sup>z</sup> .

The V-number or the normalized frequency is used to characterize the waveguide and is defined as:

Figure 9. Bessel Function of the (a) first kind (behaves as a damped sine wave) and (b) second kind (monotonic decreasing function).

Figure 10. Schematic representation of various channel waveguides.

II. Rectangular dielectric Waveguide: Channel/rectangular waveguides are the most commonly used non-planar waveguides for device applications. Channel waveguides include buried waveguides, strip-loaded, ridge, rib, diffused, slot, ARROW, and so on. Figure 10 shows the schematic of few of the channel waveguides. The wave equation analysis of a rectangular waveguide can be done by writing the scalar wave equation:

$$\frac{\delta^2 E}{\delta \mathbf{x}^2} + \frac{\delta^2 E}{\delta y^2} + \left[k\_0^2 n^2(\mathbf{x}, y) - \beta^2\right] E = 0\tag{30}$$

V ¼ k<sup>0</sup> a 2

220-nm-high and 450-nm-wide silicon waveguide at wavelength of 1.55 μm [8].

or electric (quasi-TE) field component along the transverse (horizontal) direction.

The following section describes various types of channel waveguides.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 <sup>1</sup> � <sup>n</sup><sup>2</sup> 2

(32)

107

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q

1. Wire waveguide: The schematic of silicon photonic wire waveguide is shown in Figure 11(a). The waveguide consists of a silicon core and silica-based cladding. Since the single-mode condition is very important in constructing functional devices, the core dimension should be determined so that a single-mode condition is fulfilled. The primary requisite is single-mode guiding of the TE00 and TM00 mode. When the effective refractive index is larger than the cladding and smaller than the core, mode is guided in the waveguide, and guiding will be stronger for higher values of effective index neff : Thus, modes with effective indices above nSiO<sup>2</sup> will not be radiated into the buffer layer and thus will be guided. Figure 11(b) depicts the quasi-TE mode of a

Each mode propagates through the waveguide with a phase velocity of c=neff , where c denotes the speed of light in vacuum and neff is the effective refractive index felt by that mode. It signifies how strongly the optical power is confined to the waveguide core. Most waveguides support modes of two independent polarizations, with either the major magnetic (quasi-TM)

Figure 11(c) shows neff as a function of the width of the photonic wire. The neff depends on the waveguide cross-section, waveguide materials, and the cladding material. Higher-order modes travel with a different propagation constant compared to the lowest-order mode and are less confined in the waveguides. As a consequence of the dissimilar propagation constants, there is modal dispersion which reduces the distance-bandwidth product of the waveguide. Due to the low confinement, first, a large field decay outside the waveguide reduces the maximum density of the devices and, second, in the waveguide bends the higher-order modes

Figure 11. (a) Silicon-on-insulator wire waveguide, (b) quasi-TE mode of a 220-nm-high and 450-nm-wide silicon waveguide at wavelength of 1.55 μm, and (c) effective refractive index (neffÞ at 1550 nm for a 220-nm-high silicon photonic wire

waveguide. The left of the hashed line is the single-mode region.

$$\text{V-number} = ak\_0 \sqrt{n\_{core}^2 - n\_{clad}^2} = \frac{2\pi a}{\lambda} \sqrt{n\_{core}^2 - n\_{clad}^2} \tag{31}$$

The general representation of the dielectric waveguide along with the electromagnetic field distribution in the regions is shown below:


where ϕ<sup>x</sup> and ϕ<sup>y</sup> are phase constants. The characteristic equations are given by tan κyb ¼ <sup>κ</sup><sup>y</sup> <sup>γ</sup>4þ<sup>γ</sup> ð Þ<sup>5</sup> κ2 <sup>y</sup>�γ4γ<sup>5</sup> and tan <sup>κ</sup>xa <sup>¼</sup> <sup>n</sup><sup>2</sup> 1 κ<sup>x</sup> n<sup>2</sup> 2γ3þn<sup>2</sup> <sup>3</sup><sup>γ</sup> ð Þ<sup>2</sup> n2 2n2 3κ2 x�n<sup>2</sup> <sup>1</sup>γ2γ<sup>3</sup> (γi ) are exponential decay constants. The critical cut-off condition is given by:

$$V = k\_0 \frac{a}{2} \sqrt{n\_1^2 - n\_2^2} \tag{32}$$

The following section describes various types of channel waveguides.

II. Rectangular dielectric Waveguide: Channel/rectangular waveguides are the most commonly used non-planar waveguides for device applications. Channel waveguides include buried waveguides, strip-loaded, ridge, rib, diffused, slot, ARROW, and so on. Figure 10 shows the schematic of few of the channel waveguides. The wave equation analysis of a

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 core � <sup>n</sup><sup>2</sup> clad

The general representation of the dielectric waveguide along with the electromagnetic field

Cos κyy þ Φ<sup>y</sup> � � exp �γ3<sup>x</sup> � � <sup>3</sup>

Cosð Þ κxx þ Φ<sup>x</sup> Cos κyy þ Φ<sup>y</sup>

Cos κyy þ Φ<sup>y</sup> � � exp �γ2ð Þ <sup>x</sup> � <sup>a</sup> � � <sup>2</sup>

where ϕ<sup>x</sup> and ϕ<sup>y</sup> are phase constants. The characteristic equations are given by tan κyb ¼

� � 1

<sup>¼</sup> <sup>2</sup>π<sup>a</sup> λ

q

<sup>0</sup>n<sup>2</sup>ð Þ� <sup>x</sup>; <sup>y</sup> <sup>β</sup><sup>2</sup> � � <sup>E</sup> <sup>¼</sup> <sup>0</sup> (30)

exp �γ3<sup>x</sup> � � exp �γ4ð Þ <sup>y</sup> � <sup>b</sup> � �

Cosð Þ κxx þ Φ<sup>x</sup> exp �γ4ð Þ <sup>y</sup> � <sup>b</sup> � � <sup>4</sup>

exp �γ4ð Þ <sup>y</sup> � <sup>b</sup> � � exp �γ2ð Þ <sup>x</sup> � <sup>a</sup> � � (31)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 core � <sup>n</sup><sup>2</sup> clad

) are exponential decay constants. The critical cut-off

rectangular waveguide can be done by writing the scalar wave equation:

δ2 E <sup>δ</sup>y<sup>2</sup> <sup>þ</sup> <sup>k</sup> 2

q

δ2 E δx<sup>2</sup> þ

Figure 10. Schematic representation of various channel waveguides.

V-number ¼ ak<sup>0</sup>

distribution in the regions is shown below:

and tan <sup>κ</sup>xa <sup>¼</sup> <sup>n</sup><sup>2</sup>

1 κ<sup>x</sup> n<sup>2</sup> 2γ3þn<sup>2</sup> <sup>3</sup><sup>γ</sup> ð Þ<sup>2</sup>

n2 2n2 3κ2 x�n<sup>2</sup> <sup>1</sup>γ2γ<sup>3</sup> (γi

<sup>κ</sup><sup>y</sup> <sup>γ</sup>4þ<sup>γ</sup> ð Þ<sup>5</sup> κ2 <sup>y</sup>�γ4γ<sup>5</sup>

condition is given by:

exp �γ3<sup>x</sup> � � exp �γ5<sup>y</sup> � �

106 Emerging Waveguide Technology

Cosð Þ κxx þ Φ<sup>x</sup> exp <sup>γ</sup>5<sup>x</sup> � � <sup>5</sup>

exp �γ2ð Þ <sup>x</sup> � <sup>a</sup> � � exp <sup>γ</sup>5<sup>y</sup> � �

1. Wire waveguide: The schematic of silicon photonic wire waveguide is shown in Figure 11(a). The waveguide consists of a silicon core and silica-based cladding. Since the single-mode condition is very important in constructing functional devices, the core dimension should be determined so that a single-mode condition is fulfilled. The primary requisite is single-mode guiding of the TE00 and TM00 mode. When the effective refractive index is larger than the cladding and smaller than the core, mode is guided in the waveguide, and guiding will be stronger for higher values of effective index neff : Thus, modes with effective indices above nSiO<sup>2</sup> will not be radiated into the buffer layer and thus will be guided. Figure 11(b) depicts the quasi-TE mode of a 220-nm-high and 450-nm-wide silicon waveguide at wavelength of 1.55 μm [8].

Each mode propagates through the waveguide with a phase velocity of c=neff , where c denotes the speed of light in vacuum and neff is the effective refractive index felt by that mode. It signifies how strongly the optical power is confined to the waveguide core. Most waveguides support modes of two independent polarizations, with either the major magnetic (quasi-TM) or electric (quasi-TE) field component along the transverse (horizontal) direction.

Figure 11(c) shows neff as a function of the width of the photonic wire. The neff depends on the waveguide cross-section, waveguide materials, and the cladding material. Higher-order modes travel with a different propagation constant compared to the lowest-order mode and are less confined in the waveguides. As a consequence of the dissimilar propagation constants, there is modal dispersion which reduces the distance-bandwidth product of the waveguide. Due to the low confinement, first, a large field decay outside the waveguide reduces the maximum density of the devices and, second, in the waveguide bends the higher-order modes

Figure 11. (a) Silicon-on-insulator wire waveguide, (b) quasi-TE mode of a 220-nm-high and 450-nm-wide silicon waveguide at wavelength of 1.55 μm, and (c) effective refractive index (neffÞ at 1550 nm for a 220-nm-high silicon photonic wire waveguide. The left of the hashed line is the single-mode region.

Wire waveguides are advantageous as they provide a small bending radius and realization of ultra-dense photonic circuits. However, they have higher propagation losses. On the one hand, wire waveguide allows low-loss sharp bends in the order of a few micrometres, while, on the other hand, the device structures produced are susceptible to geometric fluctuations such as feature drift size (resulting in degradation of device performance) and waveguide sidewall roughness (resulting in propagation losses) [9, 10]. Rib waveguides typically require bend radii >50 μm in SOI to ensure low bend losses, which eventually result in a larger device/circuit footprint. Figure 13 shows the TE mode loss in silicon wire and rib waveguide for a bend of 90

Slot waveguides are used to confine light in a low-index material between two high-index strip waveguides by varying the gap and dimensions (width and height) of the strip waveguides (Figure 14(a)). The normal component of the electric field (quasi TE) undergoes very high discontinuity at the boundary between a high- and a low-index material, which results into higher amplitude in the low-index slot region. The amplitude is proportional to the square of the ratio between the refractive indices of the high-index material (Si, Ge, Si3N4) and the low-index slot material (air). On the other hand, the effect of the presence of the slot is minimal on quasi-TM mode, which is continuous at the boundary. When the width of the slot waveguides is comparable to the decay length of the field, electric field remains across the slot and the section has high-field confinement [11–14], which results into propagation of light in the slot section; unlike in a conventional strip waveguide, where the propagating light is confined mainly in the high-index medium. Figure 14(c) shows the variation in effective index with the waveguide width for different slot gaps. The advantage of a slot waveguide is the high-field confinement in the slot section, which normally cannot be achieved using a simple strip- or a ridge-based waveguide, making it a potential candidate for applications that require light-matter interaction such as sensing [12] and nonlinear photonics [13]. The launching of light into a slot waveguide is normally done by phase matching the propagation constant of the strip waveguide and the slot

Figure 14. (a) Silicon-on-insulator slot waveguide, (b) Quasi-TE mode of a 220-nm-high (Gap = 100 nm) slot waveguide, (b) variation of effective refractive index with waveguide width for slot gap of 100, 150, and 200 nm, respectively, at a

3. Slot waveguide

wavelength of 1.55 μm.

.

109

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Figure 12. (a) Silicon-on-insulator rib waveguide, (b) quasi-TE mode of a 220-nm-high and 700-nm-wide silicon rib waveguide at wavelength of 1.55 μm, and (c) effective refractive index (neffÞ at 1550 nm for 220-nm-high silicon rib waveguide for ridge height (r) = 70 nm.

become leaky resulting in propagation losses. It is desirable that the difference between neff of the fundamental quasi-TE and quasi-TM modes be large so that the coupling between the modes is limited due to difference in mode profiles and also the phase-mismatch. For widths below �550 nm, silicon photonic wire will be single mode for each polarization.

2. Rib waveguide: Figure 12(a) and (b) shows the schematic and the fundamental quasi-TE mode of a silicon photonic rib waveguide (H = 220 nm, r = 70 nm). Although a rib waveguide can never truly be single mode, by optimizing the design, the power carried by the higherorder modes will eventually leak out of the waveguide over a very short distance, thus leaving only the fundamental mode. Figure 12(c) shows the dispersion neff as a function of the width of the photonic rib waveguide. For widths below �800 nm, silicon photonic rib waveguide will be single mode for each polarization.

Figure 13. Mode loss for silicon wire (cross-section: 450 � 220 nm2 ) and rib (cross-section: 600 � 220 nm<sup>2</sup> ) waveguides for a 90� bend with increasing bending radii.

Wire waveguides are advantageous as they provide a small bending radius and realization of ultra-dense photonic circuits. However, they have higher propagation losses. On the one hand, wire waveguide allows low-loss sharp bends in the order of a few micrometres, while, on the other hand, the device structures produced are susceptible to geometric fluctuations such as feature drift size (resulting in degradation of device performance) and waveguide sidewall roughness (resulting in propagation losses) [9, 10]. Rib waveguides typically require bend radii >50 μm in SOI to ensure low bend losses, which eventually result in a larger device/circuit footprint. Figure 13 shows the TE mode loss in silicon wire and rib waveguide for a bend of 90 .

### 3. Slot waveguide

become leaky resulting in propagation losses. It is desirable that the difference between neff of the fundamental quasi-TE and quasi-TM modes be large so that the coupling between the modes is limited due to difference in mode profiles and also the phase-mismatch. For widths

Figure 12. (a) Silicon-on-insulator rib waveguide, (b) quasi-TE mode of a 220-nm-high and 700-nm-wide silicon rib waveguide at wavelength of 1.55 μm, and (c) effective refractive index (neffÞ at 1550 nm for 220-nm-high silicon rib

2. Rib waveguide: Figure 12(a) and (b) shows the schematic and the fundamental quasi-TE mode of a silicon photonic rib waveguide (H = 220 nm, r = 70 nm). Although a rib waveguide can never truly be single mode, by optimizing the design, the power carried by the higherorder modes will eventually leak out of the waveguide over a very short distance, thus leaving

of the photonic rib waveguide. For widths below �800 nm, silicon photonic rib waveguide will

as a function of the width

) waveguides for

) and rib (cross-section: 600 � 220 nm<sup>2</sup>

below �550 nm, silicon photonic wire will be single mode for each polarization.

only the fundamental mode. Figure 12(c) shows the dispersion neff

be single mode for each polarization.

Figure 13. Mode loss for silicon wire (cross-section: 450 � 220 nm2

bend with increasing bending radii.

a 90�

waveguide for ridge height (r) = 70 nm.

108 Emerging Waveguide Technology

Slot waveguides are used to confine light in a low-index material between two high-index strip waveguides by varying the gap and dimensions (width and height) of the strip waveguides (Figure 14(a)). The normal component of the electric field (quasi TE) undergoes very high discontinuity at the boundary between a high- and a low-index material, which results into higher amplitude in the low-index slot region. The amplitude is proportional to the square of the ratio between the refractive indices of the high-index material (Si, Ge, Si3N4) and the low-index slot material (air). On the other hand, the effect of the presence of the slot is minimal on quasi-TM mode, which is continuous at the boundary. When the width of the slot waveguides is comparable to the decay length of the field, electric field remains across the slot and the section has high-field confinement [11–14], which results into propagation of light in the slot section; unlike in a conventional strip waveguide, where the propagating light is confined mainly in the high-index medium.

Figure 14(c) shows the variation in effective index with the waveguide width for different slot gaps. The advantage of a slot waveguide is the high-field confinement in the slot section, which normally cannot be achieved using a simple strip- or a ridge-based waveguide, making it a potential candidate for applications that require light-matter interaction such as sensing [12] and nonlinear photonics [13]. The launching of light into a slot waveguide is normally done by phase matching the propagation constant of the strip waveguide and the slot

Figure 14. (a) Silicon-on-insulator slot waveguide, (b) Quasi-TE mode of a 220-nm-high (Gap = 100 nm) slot waveguide, (b) variation of effective refractive index with waveguide width for slot gap of 100, 150, and 200 nm, respectively, at a wavelength of 1.55 μm.

waveguide. However, efficient coupling still remains a challenge because of scattering loss and mode mismatch of the slot and strip waveguides, with a reported propagation loss between 2 and 10 dB/cm [14].

### 4. Strip-loaded waveguide

A strip-loaded waveguide is formed by loading a planar waveguide, which already provides optical confinement in the x direction, with a dielectric strip of index n<sup>3</sup> < n<sup>1</sup> or a metal strip to facilitate optical confinement in the y direction, as shown in Figure 15(a). Strip-loaded waveguides do not require half-etching in waveguide fabrication and is therefore easier to fabricate. Figure 15(a) shows the schematic of a hydrogenated amorphous silicon strip-loaded waveguide where a thermal oxide is inserted between the layers for passivation [15]. Figure 15(b) shows the optical field for the waveguide for a 75-nm-thick and 800-nm-wide strip-loaded waveguide and Figure 15(c) depicts the variation in effective index with the strip waveguide width.

from the substrate, for example, through surface micromachining [17]. Figure 16 shows the

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TriPleX waveguides are a family of waveguide geometries that is based on an alternating layer stack consisting of two materials: Si3N<sup>4</sup> and SiO2. The waveguide geometries are categorized as box shell, single stripe (propagation loss <0.03 dB/cm), symmetric double stripe (propagation loss <0.1 dB/cm), and asymmetric double stripe (propagation loss <0.1 dB/cm) as shown in Figure 17(a) [19]. Different confinement regimes can be optimized

Figure 17. (a) Schematic of different type of TriPleX waveguides, (b) Variation in waveguide size of the box-shaped

waveguide, and (c) its diffraction angle versus the index contrast.

schematic of a suspended waveguide [18].

Figure 16. Schematic of a suspended waveguide.

6. TriplexTM technology

Figure 15. (a) Hydrogenated amorphous strip-loaded waveguide, (b) Quasi-TE mode of a 220-nm-high, 800-nm-wide, 75-nm-thick strip Waveguide, (c) Variation of effective refractive index with strip width at a wavelength of 1.55 μm.

#### 5. Suspended waveguide

Suspended waveguides have enabled new types of integrated optical devices for applications in optomechanics, nonlinear optics, and electro-optics. Fabrication involves removing a sacrificial layer above or below a waveguide core layer to design these waveguides [16]. Increasing absorption loss of SiO<sup>2</sup> at longer wavelengths makes it challenging to utilize SOI for low-loss components in the mid-infrared (MIR) [17]. Removing the SiO<sup>2</sup> layer opens the possibility of extending the low-loss SOI wavelength range up to 8 μm. For MEMS, it is imperative to have waveguides that can be mechanically actuated. This requires waveguides that are released

Figure 16. Schematic of a suspended waveguide.

from the substrate, for example, through surface micromachining [17]. Figure 16 shows the schematic of a suspended waveguide [18].

## 6. TriplexTM technology

waveguide. However, efficient coupling still remains a challenge because of scattering loss and mode mismatch of the slot and strip waveguides, with a reported propagation loss between 2

A strip-loaded waveguide is formed by loading a planar waveguide, which already provides optical confinement in the x direction, with a dielectric strip of index n<sup>3</sup> < n<sup>1</sup> or a metal strip to facilitate optical confinement in the y direction, as shown in Figure 15(a). Strip-loaded waveguides do not require half-etching in waveguide fabrication and is therefore easier to fabricate. Figure 15(a) shows the schematic of a hydrogenated amorphous silicon strip-loaded waveguide where a thermal oxide is inserted between the layers for passivation [15]. Figure 15(b) shows the optical field for the waveguide for a 75-nm-thick and 800-nm-wide strip-loaded waveguide and

Suspended waveguides have enabled new types of integrated optical devices for applications in optomechanics, nonlinear optics, and electro-optics. Fabrication involves removing a sacrificial layer above or below a waveguide core layer to design these waveguides [16]. Increasing absorption loss of SiO<sup>2</sup> at longer wavelengths makes it challenging to utilize SOI for low-loss components in the mid-infrared (MIR) [17]. Removing the SiO<sup>2</sup> layer opens the possibility of extending the low-loss SOI wavelength range up to 8 μm. For MEMS, it is imperative to have waveguides that can be mechanically actuated. This requires waveguides that are released

Figure 15. (a) Hydrogenated amorphous strip-loaded waveguide, (b) Quasi-TE mode of a 220-nm-high, 800-nm-wide, 75-nm-thick strip Waveguide, (c) Variation of effective refractive index with strip width at a wavelength of 1.55 μm.

Figure 15(c) depicts the variation in effective index with the strip waveguide width.

and 10 dB/cm [14].

110 Emerging Waveguide Technology

4. Strip-loaded waveguide

5. Suspended waveguide

TriPleX waveguides are a family of waveguide geometries that is based on an alternating layer stack consisting of two materials: Si3N<sup>4</sup> and SiO2. The waveguide geometries are categorized as box shell, single stripe (propagation loss <0.03 dB/cm), symmetric double stripe (propagation loss <0.1 dB/cm), and asymmetric double stripe (propagation loss <0.1 dB/cm) as shown in Figure 17(a) [19]. Different confinement regimes can be optimized

Figure 17. (a) Schematic of different type of TriPleX waveguides, (b) Variation in waveguide size of the box-shaped waveguide, and (c) its diffraction angle versus the index contrast.

for specific applications for these waveguides and tunable birefringence- and polarization dependent loss (PDL) can be achieved. Propagation losses (<0.1 dB/cm), very low PDL (< 0.1 dB/cm), and easy interconnection with optical fibers (<0.15 dB/facet) have been demonstrated in single-mode box-shaped waveguides [20]. Moreover, fabrication of the waveguide is a low-cost and simple process.

for the transverse component of the wave vector at the desired wavelength. Even though the ARROW mode is leaky, low-loss propagation over large distances can be achieved. Yin et al. have designed an ARROW waveguide exhibiting single-mode confinement and low-loss light propagation in a hollow air core on a semiconductor chip [24]. ARROW waveguides with non-solid lowindex cores have applications in gas and liquid sensing, quantum computing, quantum communications, and Raman scattering spectroscopy. Chalcogenide rib ARROW structures have also been shown with propagation loss 6 dB/cm to design opto-chemical sensors in the near- and mid-IR

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Light confinement in a low-index media has been shown in ARROW, slot, and plasmonic waveguides. However, ARROW waveguide has low confinement and is thus leaky. Strong light confinement in the low-index medium can be achieved by using silicon slot and plasmonic waveguide. Fabrication of the slot waveguide is cumbersome and hybrid plasmonic waveguide suffers from additional propagation losses due to the presence of metal. Augmented waveguide confines light efficiently in the low-index region by reducing the reflection at the high index-low index interface in a high-index contrast waveguide, which results in enhancement of light confinement in the low-index region [26]. Figure 18 shows the schematic

Waveguides can be classified on the basis of different material platforms. Wavelength range, ease of fabrication, compactness, and CMOS compatibility are few of the determining factors when selecting a material for a specific application. Table 1 compares various waveguide platforms along with their propagation losses [27]. Figure 19 shows variation of index contrast

region [25].

10. Augmented waveguide

of an augmented low-index waveguide.

with footprint for few material platforms.

Figure 18. Schematic of an augmented waveguide.

LioniX TriPleX technology is a versatile photonics platform suited for applications such as communications, biomedicine, sensing, and so on, over a broadband range of 0.4 to 2.35 μm [21]. Figure 17(b) and (c) depicts the variation in waveguide size and diffraction angle with index contrast of the box-shaped geometry for a wavelength of 1.55 μm.

### 7. Photonic crystal waveguide

Photonic crystal waveguides guiding mechanism is different from that of a traditional waveguide, which is based on internal reflection. A photonic crystal is a periodic dielectric structure with a photonic band gap, that is, a frequency range over which there is no propagation of light. The introduction of line defects into a photonic crystal structure creates an optical channel for propagation of light. If the line defect is properly designed, the resulting guiding mode falls within a photonic band gap, is highly confined, and can be used for guiding light. The guiding mode can also be designed to be broadband and thus gives rise to a compact, broadband photonic crystal waveguide [22]. Application of these waveguides includes nanofluidic tuning, RI measurements, optical characterization of molecule orientation, and biosensing.

### 8. Diffused waveguide

A diffused waveguide is formed by creating a high-index region in a substrate through diffusion of dopants, such as a LiNbO3 waveguide with a core formed by Titanium (Ti) diffusion. Due to the diffusion process, the core boundaries in the substrate are not sharply defined. A diffused waveguide has a thickness defined by the diffusion depth of the dopant and a width defined by the distribution of the dopant. Alternatively, the material can be exchanged with the substrate. Ion-exchanged glass waveguide is fabricated by diffusing mobile ions originally in glass with other ions of different size and polarizability [23].The additional impurities cause a change in refractive index that is approximately proportional to their concentration. A material can also be implanted using an ion implanter within the waveguide. However, this process damages the lattice and is therefore followed by annealing.

### 9. ARROW waveguide

In anti-resonant reflecting optical (ARROW) waveguides, light confinement is realized by choosing the cladding layer thicknesses accordingly to create an anti-resonant Fabry-Perot reflector for the transverse component of the wave vector at the desired wavelength. Even though the ARROW mode is leaky, low-loss propagation over large distances can be achieved. Yin et al. have designed an ARROW waveguide exhibiting single-mode confinement and low-loss light propagation in a hollow air core on a semiconductor chip [24]. ARROW waveguides with non-solid lowindex cores have applications in gas and liquid sensing, quantum computing, quantum communications, and Raman scattering spectroscopy. Chalcogenide rib ARROW structures have also been shown with propagation loss 6 dB/cm to design opto-chemical sensors in the near- and mid-IR region [25].

### 10. Augmented waveguide

for specific applications for these waveguides and tunable birefringence- and polarization dependent loss (PDL) can be achieved. Propagation losses (<0.1 dB/cm), very low PDL (< 0.1 dB/cm), and easy interconnection with optical fibers (<0.15 dB/facet) have been demonstrated in single-mode box-shaped waveguides [20]. Moreover, fabrication of the wave-

LioniX TriPleX technology is a versatile photonics platform suited for applications such as communications, biomedicine, sensing, and so on, over a broadband range of 0.4 to 2.35 μm [21]. Figure 17(b) and (c) depicts the variation in waveguide size and diffraction angle

Photonic crystal waveguides guiding mechanism is different from that of a traditional waveguide, which is based on internal reflection. A photonic crystal is a periodic dielectric structure with a photonic band gap, that is, a frequency range over which there is no propagation of light. The introduction of line defects into a photonic crystal structure creates an optical channel for propagation of light. If the line defect is properly designed, the resulting guiding mode falls within a photonic band gap, is highly confined, and can be used for guiding light. The guiding mode can also be designed to be broadband and thus gives rise to a compact, broadband photonic crystal waveguide [22]. Application of these waveguides includes nanofluidic tuning,

A diffused waveguide is formed by creating a high-index region in a substrate through diffusion of dopants, such as a LiNbO3 waveguide with a core formed by Titanium (Ti) diffusion. Due to the diffusion process, the core boundaries in the substrate are not sharply defined. A diffused waveguide has a thickness defined by the diffusion depth of the dopant and a width defined by the distribution of the dopant. Alternatively, the material can be exchanged with the substrate. Ion-exchanged glass waveguide is fabricated by diffusing mobile ions originally in glass with other ions of different size and polarizability [23].The additional impurities cause a change in refractive index that is approximately proportional to their concentration. A material can also be implanted using an ion implanter within the waveguide. However, this process damages the lattice and is therefore followed by annealing.

In anti-resonant reflecting optical (ARROW) waveguides, light confinement is realized by choosing the cladding layer thicknesses accordingly to create an anti-resonant Fabry-Perot reflector

with index contrast of the box-shaped geometry for a wavelength of 1.55 μm.

RI measurements, optical characterization of molecule orientation, and biosensing.

guide is a low-cost and simple process.

112 Emerging Waveguide Technology

7. Photonic crystal waveguide

8. Diffused waveguide

9. ARROW waveguide

Light confinement in a low-index media has been shown in ARROW, slot, and plasmonic waveguides. However, ARROW waveguide has low confinement and is thus leaky. Strong light confinement in the low-index medium can be achieved by using silicon slot and plasmonic waveguide. Fabrication of the slot waveguide is cumbersome and hybrid plasmonic waveguide suffers from additional propagation losses due to the presence of metal. Augmented waveguide confines light efficiently in the low-index region by reducing the reflection at the high index-low index interface in a high-index contrast waveguide, which results in enhancement of light confinement in the low-index region [26]. Figure 18 shows the schematic of an augmented low-index waveguide.

Waveguides can be classified on the basis of different material platforms. Wavelength range, ease of fabrication, compactness, and CMOS compatibility are few of the determining factors when selecting a material for a specific application. Table 1 compares various waveguide platforms along with their propagation losses [27]. Figure 19 shows variation of index contrast with footprint for few material platforms.

Figure 18. Schematic of an augmented waveguide.


Silicon-oninsulator waveguides Semiconductor nanomaterials

Silicon-on-Silica

Silicon-on-Sapphire

Silicon-onnitride

Thalliumdoped SOI Rib/Indiumdoped SOI Rib

Glass waveguides Mid-IR and Near-IR

Waveguides Range Configuration Propagation Loss

Strip-loaded Waveguide on Silicon

Near-IR Amorphous silicon nanowire �4:5 dB/cm [50] (�1550 nm)

Near-IR Silicon-on-silica strip waveguide �0:6 dB/cm [52] (�1550 and

Near-IR Silicon-on-silica slot waveguide �2:28 � 0:03 dB/cm [54] (�1064

Near-IR Silicon-on-Silica slot waveguide �3:7 dB/cm [55] (�1550 nm) Mid-IR Suspended silicon waveguide �3.1 dB/cm (7.67 μm) [56]

Mid-IR Silicon-on-sapphire slot waveguide �11 dB/cm [60] (�3.4 μm)

Near-IR Thallium-doped silicon waveguide �3 dB/cm [63] (�1.55 μm)

Silicon-on-nitride ridge waveguide �5:2 � 0:6dB/cm [62]

subwavelength waveguide

nanowires on a GaAs substrate

�4:1 dB/cm [47] (�1535 nm)

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�1 � 0:3 dB/cm [48] (�4 μm)

�10 dB/cm [49] (�400-550 nm)

�0:8 dB/mm [51] (�1550 nm)

�0:1 � 0:2 dB/cm [52] (�1550

�1:5 dB/cm [53] (�3800 nm)

�2 � 1 dB/cm [57] (�1550 nm)

�4:0 � 0:7 dB/cm [58] (�5.4-5.6

�0:8 dB/cm [61] (�1550 nm). �1:1 � 1:4 dB/cm [61] (�2080 nm)

�<2 dB/cm [61] (�5.18 μm)

�0:06=0:1 dB/cm [65] (�777 nm)

�1:3 dB/cm<sup>3</sup> [66] (�3.39 <sup>μ</sup>m)

�4:3 � 0:6 dB/cm [59]

2000 nm)

nm)

μm)

(�4.5 μm)

(�3.39 μm)

and 2000 nm)

Near-IR Erbium-Doped Phosphate Glass

Mid-IR Silicon-on-Sapphire Suspended Nanowire

Vis Tin-oxide nanoribbons-based

Near-IR InP/benzocyclobutene optical

Near-IR Silicon-on-silica rib (70-nm etch depth) waveguide

Mid-IR Silicon-on-silica rib (70-nm etch depth) waveguide

Near-IR Silicon-on-silica strip waveguide

Mid-IR Silicon-on-sapphire ridge waveguide

Mid-IR Silicon-on-sapphire ridge waveguide

Near/Mid-IR Silicon-on-sapphire nanowire waveguide

Near-IR Indium-doped silicon waveguide,

Mid-IR 3D laser-written silica glass stepindex high-contrast (HIC)

waveguide

μm).

Silica glass Vis Laser-written waveguide in fused

decrease in absorption coefficient �16 dB/cm [64] (wavelength �1.55

silica for vertical polarization (VP)/ horizontal polarization (HP) beam

coated with amorphous TiO ð Þ<sup>2</sup>


Semiconductor materials

114 Emerging Waveguide Technology

Gallium arsenide (GaAS)

Indium phosphide (InP)

Gallium antimony

Doped semiconductor

Waveguides Range Configuration Propagation Loss

Silicon Mid-IR Silicon nanophotonic waveguide 4 dB/cm (2030 nm)

Mid/Near-IR Silicon on porous silicon (SiPSi) 2:1 0:2 dB/cm (1.55 μm)

Mid/Near-IR Suspended silicon-membrane ridge waveguide (TM mode)

waveguide

Far-IR Germanium on silicon strip waveguide

Mid-IR Germanium on silicon rib waveguide

Mid-IR Germanium-rich silicon

waveguide

Mid-IR Silicon germanium ridge

Far-IR Germanium on GaAs ridge waveguide

Near-IR GaAS/Al0.3 Ga0.7As ridge

Near-IR GaAs-based single-line defect

Near-IR InP waveguides based on localized

Near-IR Suspended InP dual-waveguide

optical buffering

Mid-IR GaSb waveguides based on quasiphase matching (QPM)

Vis/Near-IR Rare-earth-doped GaN (gallium

Quantum dotes Near-IR Polymer waveguides containing

states

Mid IR/Far-IR Silicon germanium/silicon-based graded index waveguides

Mid-IR Germanium-on-silicon-on-insulator waveguides

germanium platform-based rib

waveguide on a silicon substrate

waveguide for manipulation of single-photon and two-photon

photonic crystal slab waveguide

Zn-diffusion process (MOVPE) to mitigate passive loss by p-dopants

structures for MEMS-actuated

infrared-emitting nanocrystal quantum dots (PbSe and InAs)

nitride) channel waveguide

Near-IR Suspended GaAs waveguide 0:4 dB/mm (TE) (1.55 μm) and

Germanium Mid-IR Germanium on silicon strip

10 dB/cm (2500 nm) [28]

2:8 0:5 dB/cm (3.39 μm) 5:6 0:3 dB/cm (1.53 μm) [29]

2:5 3 dB dB/cm [31] (5.15-5.4 μm)

2:4 0:2 dB/cm [33]

(3.8 μm)

(7.4 μm) [35]

3:9 0:2 dB/cm (3.39 μm) [30]

2:5 dB dB/cm [32] (5.8 μm)

1:5 0:5 dB/cm [34] (4.6 μm)

1 dB/cm (4.5 μm) 2 dB/cm

3:5 dB/cm (3.682 μm) [36]

0:5 dB/cm (4.75 μm) [37]

4:2 dB/cm (10 μm) [38]

1:6 dB/cm (1.55 μm) [39]

6 dB/mm (TM) (1.03 μm) [40]

0:76 dB/mm (1050–1580 nm)

0.4 dB/cm (1.55 μm) [42]

2.2 dB/cm (1.5–1.6 μm) [43]

0:7=1:1 dB/cm [44] (2/4 μm)

5:4=4:1 dB/cm [46] (633/1550

5 dB/cm (inclusive of fiber coupling loss) [45] (1550 nm)

[41]

nm)


> Lithium tantalate

Barium titanate (BTO)

Electro-optic polymer

Novel optical polymers

Polymer-based Conventional optical polymers

Waveguides Range Configuration Propagation Loss

Near-IR Lithium Niobate on Insulator Ridge Waveguide

Near-IR Thin film strip-loaded (SiN) lithium niobate waveguide

Near-IR Thin film strip-loaded (a-Si) lithium niobate waveguide

Near-IR Thin film lithium niobate ridge waveguide

Near-IR Ga3+-diffused lithium tantalate waveguide

Mid-IR BTO thin films on Lanthanum

waveguide

waveguide

Vis Liquid-crystal core channel

substrate

Near-IR Liquid crystal clad shallow-etched SOI waveguide

Vis PMMA (poly(methyl methacrylate) )-based optical waveguide

Vis/Near-IR Polyurethane (PU)-based optical waveguide

Vis Polymer PMMA-based waveguide using femtosecond laser

photopolymer) with embedded

using photo exposure/laser ablation

perfluorocyclobutane(XU 35121)-

Vis/Near-IR Polymeric waveguides (WIR30

micro-mirror

Near-IR Dow chemical

Vis/Near-IR Acrylate-based waveguide pattern

Vis/Near-IR Telephotonics-OASIC-based optical waveguide

based waveguide

Vis/Near-IR Epoxy resin-based optical waveguide

Liquid crystal Near-IR PDMS (poly (dimethyl siloxane))-

Vis BTO thin films on MgO-based ridge waveguide

Near-IR Hybrid Electro-Optic Polymer and TiO2 Slot Waveguide

liquid crystal-based optical

waveguide encapsulated in semicircular grooves with glass

Aluminate (LAO) with a-Si ridge

1 dB/cm [84] (1.55 μm)

[85] (1.55 μm)

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[86] (1.55 μm)

(TM) [87] (1.55 μm)

5:8 dB/cm (TE) 14dB/cm (TM)

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117

42 dB/cm (TE) 20dB/cm (TM)

0:268 dB/cm (TE) 1:3dB/cm

0:2 dB/cm (TE) 0:4 dB/cm (TM) [88] (1.55 μm)

4:2 dB/cm [90] (3.0 μm)

5 dB/cm [91] (1550 nm)

8dB/cm [92] (1550 nm)

1:3 dB/cm [93] (632.8 nm)

4:5 dB/cm [94] (1570 nm)

0:2 dB/cm [95] (850 nm)

0:18 dB/cm [97] (850 nm )

0:02=0:3 and 0.8 dB/cm [95] (840/1300 and 1550 nm)

< 0:01, 0:03, 0:1 dB/cm [95] (840/1300/1550 nm)

0:8 dB/cm [95] (633 and 1064 nm)

0:3=0:8 dB/cm [95] (633/1064 nm)

0:3 dB/cm [96] (638 and 850 mm)

0:25 dB/cm [95] (1300/1550 nm)

2 dB/cm (TE) [89] (633 nm)


116 Emerging Waveguide Technology

Silicon oxynitride (SiON)

Ion exchanged glass

Tungsten tellurite glass

Lithium niobate

Electro-optic waveguides

Waveguides Range Configuration Propagation Loss

material (Hydex) waveguides

coupled PECVD-based waveguides

/Na<sup>+</sup> ion-exchanged single-mode waveguides on silicate glass

glasses for thermal ion-exchanged

1:5 dB/cm [67] (1.55 μm)

0:06 dB/cm [68] (1.55 μm)

0:5 0:05 dB/cm,1:6 0:2 dB/cm and 0:5 0:06 dB/cm [69] (1330, 1550 and 1600 nm)

9 1 dB/cm [70] (0.405 μm)

0:53 dB/cm [71] (1.534 μm)

0:1 dB/cm [72] (1.55 μm)

0:2 dB/cm [73] ( 677 nm)

0:44 dB/cm [75] (1.53 μm)

0:35 dB/cm [76] (1.55 μm)

0:06=0:1 dB/cm [65](777 nm)

2:64 dB/cm [77] (1530 nm)

2:0 dB/cm [78] (632 nm)

<0:15 dB/cm [79] (1550 nm)

0:4 dB/cm [80] (1.55 μm)

<1 dB/cm [82] (1.55 μm)

< 0:2 0:4 dB/cm [83] (1.54

(TM) [81] (1.55 μm)

μm)

0:5 0:2 dB/cm [74]

(635 nm)

Near-IR Graded-index (GRIN) Cladding in HIC glass waveguides

Near-IR High-index, doped silica glass

Near-IR SiON deposited by inductively

Near-IR Alkaline aluminum phosphate

gel ridge waveguides

photorefractive films-based

based on tungsten-tellurite glass fabricated by RF Sputtering

light-wave circuit (PLC) glass doped with Boron and Phosphorous

silica for vertical polarization (VP)/ horizontal polarization (HP) beam

line waveguides in germanate and

in flexible As2S3 chalcogenide glass

Near-IR Lithium niobate ridge waveguide 0:3 dB/cm(TE) and 0:9 dB/cm

Vis Sol-gel derived silicon titania slab waveguides films

Near-IR Optical planar channel waveguide-

Vis Laser-written waveguide in fused

Near-IR Laser-written ferroelectric crystal in glass waveguide

Vis Femtosecond laser-written double-

Near-IR Ultrafast laser-written waveguides

Near-IR Lithium niobate on insulator rib waveguide

Near-IR Periodically poled lithium niobate waveguide

Near-IR Heterogeneous lithium niobate on silicon nitride waveguide

tellurite glasses

tape

Vis Sol–gel-derived glass-ceramic

waveguide

waveguide

Sol-gel glass Near-IR Hybrid organic-inorganic glass sol-

Laser-written Near-IR Laser-written waveguide in planar

Vis Ti<sup>+</sup>


Metamaterial optical waveguides

Titanium dioxide TiO ð Þ<sup>2</sup>

Silicon nitride on silica<sup>30</sup>

Tantalum pentoxide (Ta2O5) core/silica-clad/ silicon substrate

Suspended silicon-oninsulator waveguide

Photonic crystal fibers based waveguides

Table 1. Material platforms.

Waveguides Range Configuration Propagation Loss

�0:6 dB/cm ( 632.8 nm) [114]

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119

http://dx.doi.org/10.5772/intechopen.77150

�0.82 dB/cm (3.8μm) [115]

�2.0-3.5 dB/cm (633 nm) [116] <1 dB/cm (1530 nm) [65]

�23:9 � 1:2 dB/cm (1.3 μm)-TM

�0:04 dB/cm. [121] (�1550 nm)

�0:02 dB/cm. [122] (�1550 nm)

�0:37 dB/cm. [123] (�1550 nm)

�0:60 dB/cm. [124] (�2600 nm)

�0:16 dB/cm. [125] (�2650 nm)/ 2:10 dB/cm. [126] (�3700 nm)

�3:4 dB/cm. [17] (�3.8 μm).

�0:1/0.04 dB/cm. [128] (�905.7/

�0:1/0.04 dB/cm. [129] (�905.7/

�0:875 dB/cm [129] (�400 μm).

908 μm).

908 μm).

mode [118]

Vis Nano porous solid liquid core Waveguide

Mid-IR Suspended silicon waveguide with

Vis/Near-IR Atomic layer deposition (ALD) TiO2 slab waveguide

Near-IR LPCVD Silicon-nitride strip waveguide

Near-IR TripleX TM LPCVD silicon-nitride planar waveguide

Near-IR 900-nm-thick LPCVD Siliconnitride strip waveguide

Mid—IR LPVCD silicon-nitride strip waveguide

Mid-IR Silicon-rich LPVCD silicon-nitride strip waveguide

Mid-IR Waveguide with Subwavelength grating

THz Semiconductor silicon photonic crystal slab waveguides

structure

waveguide

THz 3-D printed THz waveguide based on Kagome photonic crystal

THZ Kagome-lattice hollow-core silicon photonic crystal slab-based

Silicon carbide Near-IR PECVD silicon-carbide-silicon

lateral cladding (subwavelength grating metamaterial)

oxide horizontal slot waveguide

Near-IR Amorphous TiO2 strip waveguide �2.4-0.2 dB/cm (1.55μm) [117]

Vis Silicon-nitride strip waveguide �2:25 dB/cm. [119] (�532 nm) Vis Silicon-nitride strip waveguide �0:51 dB/cm. [120] (�600 nm) Vis Silicon-nitride strip waveguide �1:30 dB/cm. [119] (�780 nm)

Near IR Planar waveguide �0:03 dB/cm. [127] (�1550 nm)


Table 1. Material platforms.

Material platforms

118 Emerging Waveguide Technology

Hollow waveguides Surface plasmon polariton waveguide

Metal/ Dielectric Coated

Waveguides Range Configuration Propagation Loss

0:79 dB/cm [98] (1550 nm)

0:05 dB/cm [99] (850 nm)/ 0:5=1 dB/cm [99] (1.31/

1:72 dB/cm [101] (1.55 μm).

1:98=1:89 dB/cm [102] (215/

1:77=1:62 dB/cm [102] (215/

0:037dB/cm [103] (2.94 μm)

0:3 dB/cm [104] (300 μm)

1:9 dB/m [105] (300 μm)

0:549=0:095 and 0:298 dB/m

4 dB/cm [109] (820-880 nm)

<1 dB/cm (0.3dB at 2000 cm<sup>1</sup> )

<0.42 dB/cm<sup>1</sup> (1550 nm) [112]

0:14 dB/cm ( 464-596 nm) [113]

) [111]

(808/1064 and 2940 nm)

1:7 dB/cm [107] (1.52-1.62 μm)

2-6 /<0.5 dB/cm [110] (1550 nm)

(1500-4000 cm<sup>1</sup>

0:5 1 dB/mm [100] (1.55 μm)

1.55μm)

513 μm)

513 μm)

[106]

Near-IR Circular-core UV-curable epoxiesbased optical waveguide

Near-IR Polymer-Silicones-based long range surface plasmon polariton waveguide (LRSPPW)

> range surface plasmon polariton waveguide (LRSPPW)

liquid phase chemical deposition

liquid phase chemical deposition

(Lead Sulfide) on Ag (Silver)-coated

photonics band-gap Bragg fiber

Far-IR Tapered hollow-air core waveguide 1:27 dB/cm [108] (6.2 μm)

Near-IR Exguide ZPU/LFR-based long

THz Gold-coated waveguide using

THz Silver-coated waveguide using

Mid-IR Cadmium sulfide CdS) and PbS

THz High-refractive index composite

THz Silver/polystyrene-coated hollow glass waveguides

Vis/Near-IR Silver/cyclic olefin hollow glass waveguide

(Si Substrate)

Vis Air-core anti-resonant reflecting

Strip/rib

Mid-IR Chalcogenide waveguides (Ge<sup>11</sup>:<sup>5</sup>As24Se<sup>64</sup>:5)

Near-IR Chalcogenide waveguides

etching)

glycol) air-cladding waveguide

Optical Waveguide (ARROW)

(Ge23Sb7S70) fabricated through CMOS-compatible lift-off process:

Ge23Sb7S<sup>70</sup> (chlorine-based plasma

Hollow glass Near-IR Hollow-core optical waveguide

Chalcogenide Near-IR Chalcogenide waveguides

Liquid core Vis Liquid core/(ethylene

hollow glass waveguide

process

Process

Vis/Near-IR Multi-mode Siloxane-based polymer waveguide, single-mode siloxane-based polymer waveguide

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Figure 19. Comparison of different waveguide platforms as a function of index contrast and compactness.

### 11. Conclusion

Classification of waveguides on the basis of geometry (planar/non-planar), mode propagation (Single/Multi-Mode), refractive index distribution (Step/Gradient Index), and material platform is described briefly. An overview of different kinds of channel waveguides, namely wire, rib, slot, strip-loaded, diffused, TriPleX, suspended, photonic crystal, ARROW, and augmented waveguide is given. A comparative analysis of material platforms used along with their propagation losses and wavelength range is also shown.

### Author details

Shankar Kumar Selvaraja\* and Purnima Sethi

\*Address all correspondence to: shankarks@iisc.ac.in

Centre for Nano Science and Engineering, Indian Institute of Science, Bangalore

### References


[6] Pollock CR, Lipson M. Integrated photonics. 2003;20(25)

11. Conclusion

120 Emerging Waveguide Technology

Author details

References

Classification of waveguides on the basis of geometry (planar/non-planar), mode propagation (Single/Multi-Mode), refractive index distribution (Step/Gradient Index), and material platform is described briefly. An overview of different kinds of channel waveguides, namely wire, rib, slot, strip-loaded, diffused, TriPleX, suspended, photonic crystal, ARROW, and augmented waveguide is given. A comparative analysis of material platforms used along with their

Figure 19. Comparison of different waveguide platforms as a function of index contrast and compactness.

propagation losses and wavelength range is also shown.

Shankar Kumar Selvaraja\* and Purnima Sethi

\*Address all correspondence to: shankarks@iisc.ac.in

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**Chapter 7**

Provisional chapter

**Graphene Based Waveguides**

Graphene Based Waveguides

Xianglian Song, Xiaoyu Dai and Yuanjiang Xiang

Graphene, which is well known as a one-atom thick carbon allotrope, has drawn lots of attention since its first announcement due to remarkable performance in mechanical, electrical, magnetic, thermal, and optical areas. In particular, unique properties of graphene such as low net absorption in broadband optical band, notably high nonlinear optical effects, and gate-variable optical conductivity make it an excellent candidate for high speed, high performance, and broadband electronic and photonics devices. Embedding graphene into optical devices longitudinally would enhance the light-graphene interaction, which shows great potential in photonic components. Since the carrier density of graphene could be tuned by external gate voltage, chemical doping, light excitation, graphene-based waveguide modulator could be designed to have high flexibility in controlling the absorption and modulation depth. Furthermore, graphene-based waveguides could take advantages in

DOI: 10.5772/intechopen.76796

With the increasing demand in data storage, high-performance computing, and broadband networks for communication, the requirement for high-performance optical devices with broadband working bandwidth could be imaged [1–3]. Among the whole process, integrating telecom network onto chips has irreplaceable importance [4]. Since waveguide is one of the most indispensable components in modern communication, its development surely means a lot, which would otherwise impede the whole progress of optical technology [5]. Silicon photonics could provide broad bandwidths, which have been applied to low-loss optical waveguides [6, 7]. Other photonic substrates such as germanium or compound semiconductors are required to achieve high performance at the same time [4, 8]. However, the common

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Xianglian Song, Xiaoyu Dai and Yuanjiang Xiang

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.76796

detection, sensing, polarizer, and so on.

Keywords: graphene, waveguide, photonics, tunable, optical

Abstract

1. Introduction

#### **Graphene Based Waveguides** Graphene Based Waveguides

Xianglian Song, Xiaoyu Dai and Yuanjiang Xiang Xianglian Song, Xiaoyu Dai and Yuanjiang Xiang

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.76796

#### Abstract

Graphene, which is well known as a one-atom thick carbon allotrope, has drawn lots of attention since its first announcement due to remarkable performance in mechanical, electrical, magnetic, thermal, and optical areas. In particular, unique properties of graphene such as low net absorption in broadband optical band, notably high nonlinear optical effects, and gate-variable optical conductivity make it an excellent candidate for high speed, high performance, and broadband electronic and photonics devices. Embedding graphene into optical devices longitudinally would enhance the light-graphene interaction, which shows great potential in photonic components. Since the carrier density of graphene could be tuned by external gate voltage, chemical doping, light excitation, graphene-based waveguide modulator could be designed to have high flexibility in controlling the absorption and modulation depth. Furthermore, graphene-based waveguides could take advantages in detection, sensing, polarizer, and so on.

DOI: 10.5772/intechopen.76796

Keywords: graphene, waveguide, photonics, tunable, optical

#### 1. Introduction

With the increasing demand in data storage, high-performance computing, and broadband networks for communication, the requirement for high-performance optical devices with broadband working bandwidth could be imaged [1–3]. Among the whole process, integrating telecom network onto chips has irreplaceable importance [4]. Since waveguide is one of the most indispensable components in modern communication, its development surely means a lot, which would otherwise impede the whole progress of optical technology [5]. Silicon photonics could provide broad bandwidths, which have been applied to low-loss optical waveguides [6, 7]. Other photonic substrates such as germanium or compound semiconductors are required to achieve high performance at the same time [4, 8]. However, the common

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

use of such materials is restricted due to limited bandwidth, inevitable cross-talk, high-energy consumption, expensive cost, and so on [9, 10].

cladding materials could be achieved by transferring and the lamination of graphene to the surface of silicon photonic substrates. Integrating graphene with photonic devices not only promotes the emergence of novel optoelectronic properties but also opens a new versatile

Graphene Based Waveguides

133

http://dx.doi.org/10.5772/intechopen.76796

Three independent research groups, Mueller's group [28] at Vienna University of Technology and Johannes Kepler University (Australia), Englund's group [29] at Columbia University, the Massachusetts Institute of Technology (MIT) and the IBM T.J. Watson Research Center (the USA), and Xu's group [30] at Chinese University of Hong Kong (China), reported almost at the same time, chip-integrated graphene photodetectors with high responsivities and speeds, for which the working wavelength is covered from 1.3 to 2.75 μm [31]. Graphene-based photodetectors have better performance than germanium-based devices, because germanium-based ones meet limitations to overcome the low efficiencies at wavelength above 1.5 μm. However, this problem does not exist in graphene due to the zero-bandgap intrinsic property. As a result, the optical absorption coefficient remains constant from visible to infrared range owing to the linear band structure. Thus, a nearly flat response covering almost the whole optical commu-

Pospischil and Mueller et al. [24] achieved a new kind of graphene-band optical interconnect, which owned an ultra-wideband operation from the O to the U band, as shown in Figure 2. Besides, the operation speed of graphene-based transition has been proved to be really high, which could be a perfect candidate for high-speed data transmission. Moreover, this device could overcome the biggest obstacles in conventional ones; the energy consumption in a graphene-based modulator is quite low. Due to the strong optical interaction in graphene, small devices in single chips were possible. The mechanical flexibility of graphene plays a role

Figure 2. (a) Colored scanning electron micrograph of a waveguide-integrated graphene photodetector. The violet region represents graphene sheet. (b) enlarged view of the section highlighted by the black dashed rectangle in a. (c) Schematic

platform to investigate more fundamental application of graphene.

in formulating active components in polymer-based optical circuits.

nication band would not be out of image.

illustration of the band diagram [24].

Two-dimensional (2D) materials, such as graphene, black phosphorus (BP), hexagonal boron nitride (hBN), and transition metal dichalcogenides (MX2, such as ReS2, MoS2, WS2, WSe2), have been of tremendous interest for applications in electronics, optoelectronics, and integrated photonics due to their unique and distinctive properties from bulk ones [11, 12]. Among all these 2D materials, graphene plays a special role in leading the exploration of 2D materials, which was first isolated mechanically in 2004 [13, 14]. Graphene, which is well known for thinnest, strongest, and highest mobility, shows great potential in various applications. Besides, graphene absorbs only ~2.3% in normally incident waves in and optical range, as shown in Figure 1, and the interaction of graphene with electromagnetic wave covers a broadband from the visible to terahertz spectral range [16, 17]. Remarkably, the conduction and valence bands in a mono layer graphene meet at direct, leading to a gapless and semimetallic band structure, which could be adjusted by doping or some other external excitations [18–20].

The unique and extraordinary properties of graphene make it possible to be an ideal alternative in high-performance optoelectronic devices [21–25]. Hence, graphene, the unique 2D carbon atoms arranged in a honeycomb lattice, has been widely reported as an excellent plasmonic material for light-matter interactions from terahertz to the infrared (IR) region [18]. The gapless linear dispersion of Dirac fermions makes it possible for graphene integrated with other substrates to formulate modulators, polarizers, broadband waveguides, photodetectors, bio-sensors and so on [25–27]. Especially in the optical range, graphene-based waveguides play a critical role in photonic integrated circuits, optical fiber communication and sensing, as shown in Figure 1. It has been reported that coplanar integration which are planarized with

Figure 1. Different configurations for light-matter interaction in graphene. (A) For normal incident wave in optical range, graphene has the advantage of broadband absorption, the total absorption is quite small though. (B) When graphene is placed into an oical resonator, the absorption could be enhanced since the interaction between light and mater is enhanced. (C) When integrating graphene at the surface of photonics substrates, the interaction length would increase while the broadband optical bandwidth remains unchanged [15].

cladding materials could be achieved by transferring and the lamination of graphene to the surface of silicon photonic substrates. Integrating graphene with photonic devices not only promotes the emergence of novel optoelectronic properties but also opens a new versatile platform to investigate more fundamental application of graphene.

use of such materials is restricted due to limited bandwidth, inevitable cross-talk, high-energy

Two-dimensional (2D) materials, such as graphene, black phosphorus (BP), hexagonal boron nitride (hBN), and transition metal dichalcogenides (MX2, such as ReS2, MoS2, WS2, WSe2), have been of tremendous interest for applications in electronics, optoelectronics, and integrated photonics due to their unique and distinctive properties from bulk ones [11, 12]. Among all these 2D materials, graphene plays a special role in leading the exploration of 2D materials, which was first isolated mechanically in 2004 [13, 14]. Graphene, which is well known for thinnest, strongest, and highest mobility, shows great potential in various applications. Besides, graphene absorbs only ~2.3% in normally incident waves in and optical range, as shown in Figure 1, and the interaction of graphene with electromagnetic wave covers a broadband from the visible to terahertz spectral range [16, 17]. Remarkably, the conduction and valence bands in a mono layer graphene meet at direct, leading to a gapless and semimetallic band structure, which could be adjusted by doping or some other external excitations

The unique and extraordinary properties of graphene make it possible to be an ideal alternative in high-performance optoelectronic devices [21–25]. Hence, graphene, the unique 2D carbon atoms arranged in a honeycomb lattice, has been widely reported as an excellent plasmonic material for light-matter interactions from terahertz to the infrared (IR) region [18]. The gapless linear dispersion of Dirac fermions makes it possible for graphene integrated with other substrates to formulate modulators, polarizers, broadband waveguides, photodetectors, bio-sensors and so on [25–27]. Especially in the optical range, graphene-based waveguides play a critical role in photonic integrated circuits, optical fiber communication and sensing, as shown in Figure 1. It has been reported that coplanar integration which are planarized with

Figure 1. Different configurations for light-matter interaction in graphene. (A) For normal incident wave in optical range, graphene has the advantage of broadband absorption, the total absorption is quite small though. (B) When graphene is placed into an oical resonator, the absorption could be enhanced since the interaction between light and mater is enhanced. (C) When integrating graphene at the surface of photonics substrates, the interaction length would increase

while the broadband optical bandwidth remains unchanged [15].

consumption, expensive cost, and so on [9, 10].

132 Emerging Waveguide Technology

[18–20].

Three independent research groups, Mueller's group [28] at Vienna University of Technology and Johannes Kepler University (Australia), Englund's group [29] at Columbia University, the Massachusetts Institute of Technology (MIT) and the IBM T.J. Watson Research Center (the USA), and Xu's group [30] at Chinese University of Hong Kong (China), reported almost at the same time, chip-integrated graphene photodetectors with high responsivities and speeds, for which the working wavelength is covered from 1.3 to 2.75 μm [31]. Graphene-based photodetectors have better performance than germanium-based devices, because germanium-based ones meet limitations to overcome the low efficiencies at wavelength above 1.5 μm. However, this problem does not exist in graphene due to the zero-bandgap intrinsic property. As a result, the optical absorption coefficient remains constant from visible to infrared range owing to the linear band structure. Thus, a nearly flat response covering almost the whole optical communication band would not be out of image.

Pospischil and Mueller et al. [24] achieved a new kind of graphene-band optical interconnect, which owned an ultra-wideband operation from the O to the U band, as shown in Figure 2. Besides, the operation speed of graphene-based transition has been proved to be really high, which could be a perfect candidate for high-speed data transmission. Moreover, this device could overcome the biggest obstacles in conventional ones; the energy consumption in a graphene-based modulator is quite low. Due to the strong optical interaction in graphene, small devices in single chips were possible. The mechanical flexibility of graphene plays a role in formulating active components in polymer-based optical circuits.

Figure 2. (a) Colored scanning electron micrograph of a waveguide-integrated graphene photodetector. The violet region represents graphene sheet. (b) enlarged view of the section highlighted by the black dashed rectangle in a. (c) Schematic illustration of the band diagram [24].

Gan et al. [29] achieved a photodetector which simultaneously exhibited high responsibility, high speed, and broadband spectral bandwidth by using a metal-doped graphene junction coupled evanescently with the waveguide, as shown in Figure 3. The absorption performance of

Figure 3. (a) Schematic of the waveguide-integrated graphene photodetector. (b) Optical microscopy top view of the device with a bilayer of graphene covering the waveguide. (c) SEM image showing the boxed region in (b) (false color), displaying the planarized waveguide (blue), graphene (purple) and metal electrodes (yellow) [29].

graphene is improved by extending the length of graphene or by coupling graphene with a transverse-magnetic (TM) mode with a stronger evanescent field. Besides, the internal quantum efficiency of the photodetector can be improved by electrically grating the graphene layer to reshape the depth and location of the potential difference. In their research, they proved that graphene could be integrated with complementary-metal-oxide-silicon (CMOS), which made possible the realization of scalable ultra-high bandwidth graphene-based optical interconnectors. The propagation of the electromagnetic field along the waveguide is summarized in two ways, which are known as transverse-electric (TE) and transverse-magnetic (TM) modes. Generally,

based plasma etching [33].

Figure 5. Schematic fabrication process flow to integrate chalcogenide glass photonic devices with graphene. Monolayer graphene had been grown on Cu foil by CVD method, which was then transferred onto the surface of target substrate by standard PMMA transformation process. Contract metals were then deposited and pattered on the surface of graphene. Subsequently, a standard electron-beam lithography process was adopted to patter graphene on the substrate. Then a glass film was deposited onto graphene surface by thermal evaporation, and the pattern of glass was defined by fluorine-

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Figure 4. Schematic sketch of the MGW setup viewed from above. A magnetic field B is applied everywhere in the graphene plane except for the waveguide region |x| < d/2, where the field is reversed [32].

Gan et al. [29] achieved a photodetector which simultaneously exhibited high responsibility, high speed, and broadband spectral bandwidth by using a metal-doped graphene junction coupled evanescently with the waveguide, as shown in Figure 3. The absorption performance of

134 Emerging Waveguide Technology

Figure 3. (a) Schematic of the waveguide-integrated graphene photodetector. (b) Optical microscopy top view of the device with a bilayer of graphene covering the waveguide. (c) SEM image showing the boxed region in (b) (false color),

Figure 4. Schematic sketch of the MGW setup viewed from above. A magnetic field B is applied everywhere in the

graphene plane except for the waveguide region |x| < d/2, where the field is reversed [32].

displaying the planarized waveguide (blue), graphene (purple) and metal electrodes (yellow) [29].

Figure 5. Schematic fabrication process flow to integrate chalcogenide glass photonic devices with graphene. Monolayer graphene had been grown on Cu foil by CVD method, which was then transferred onto the surface of target substrate by standard PMMA transformation process. Contract metals were then deposited and pattered on the surface of graphene. Subsequently, a standard electron-beam lithography process was adopted to patter graphene on the substrate. Then a glass film was deposited onto graphene surface by thermal evaporation, and the pattern of glass was defined by fluorinebased plasma etching [33].

graphene is improved by extending the length of graphene or by coupling graphene with a transverse-magnetic (TM) mode with a stronger evanescent field. Besides, the internal quantum efficiency of the photodetector can be improved by electrically grating the graphene layer to reshape the depth and location of the potential difference. In their research, they proved that graphene could be integrated with complementary-metal-oxide-silicon (CMOS), which made possible the realization of scalable ultra-high bandwidth graphene-based optical interconnectors.

The propagation of the electromagnetic field along the waveguide is summarized in two ways, which are known as transverse-electric (TE) and transverse-magnetic (TM) modes. Generally, in the TE mode, the electric lines of flux are perpendicular to the axis of the waveguides. While in TM mode, the magnetic lines of flux are perpendicular to the axis of the waveguides. Normally, for waveguides using a single conductor, no transverse-electromagnetic (TEM) mode could be transmitted. Most research focus on the transmission of the TE mode, while Cohnitz et al. [32] investigated a magnetic graphene waveguide, in which a clean graphene is exposed to a static inhomogeneous magnetic field along one of the planar directions. As shown in Figure 4, when applying magnetic fields to a monolayer graphene, quantum modes exhibited like classical snake orbits near the field switch lines. While in the other regions far from these region, only Landau-quantized cyclotron orbits could be detected.

Usually, the integration of graphene with photonic devices relies on the transformation of exfoliated or delaminated or chemical vapor deposition (CVD)-fabricated 2D material onto pretreated devices, as shown in Figure 5 [34]. However, the application of the transformation process is limited due to the shortcomings in uniformity and efficiency [35–37]. Most importantly, transferred 2D materials suffer from weak interaction with optical modes in pre-treated devices [37]. An atomic layer deposition (ALD) method has been adopted widely to obtain gate dielectric on graphene. Also, plasma-enhanced chemical vapor deposition (PECVD) could be another option for fabrication of silicon nitride on graphene. Last year, it was reported that the spin-coating process could be applied to directly fabricate the polymer waveguide modulator on graphene [38]. However, the fabrication technique of photonics devices integrating the graphene thin film needs to be improved for the difficulty of keeping the original properties of graphene after the following integrated progresses. Even though it's still a challenge to ensure the quality of integrated graphene up to date, the importance of optimized and the continuous study of graphene-based photonics devices, especially waveguides, are foreseen.

### 2. Electromagnetic properties of graphene-based waveguides

Most electromagnetic phenomena are governed by Maxwell equations, while the electromagnetic properties of materials are determined by two parameters, relative complex permittivity (ε) and relative complex magnetic permeability (μ), which describe the coupling of a material with incident electromagnetic energy. Normally, in the optical range, refraction index (n) is used as well to describe the macroscopic effective parameters of the material, and the refraction has as a relationship with relative complex permittivity (ε) and relative complex permeability (μ) the following:

$$m = \sqrt{\mu \varepsilon} \tag{1}$$

ε ωð Þ¼ 1 þ

σtotal ¼ σintra þ σ<sup>0</sup>

σintra ¼ σ<sup>0</sup>

� � � �

1 2π

tan �<sup>1</sup> <sup>ℏ</sup><sup>ω</sup> � <sup>2</sup>EF ℏΓ<sup>2</sup>

(σintra) and interband (σ<sup>0</sup>

σ0

as described in Figure 7(a):

ε3

ε<sup>4</sup> þ ε3tanhqd<sup>3</sup> ε<sup>3</sup> þ ε4tanhqd<sup>3</sup>

Figure 6. (a) Interband transition and (b) intraband transition in graphene [39].

þ ε<sup>2</sup>

inter ¼ σ<sup>0</sup> 1 þ

σ00

1 π

inter ¼ �σ<sup>0</sup>

where

inter +iσ<sup>00</sup>

where d is the thickness of the graphene layer, ε<sup>0</sup> is relative complex permittivity in vacuum (ε<sup>0</sup> ¼ 8:854 F=m), ω is the optical frequency, and σ is graphene's optical conductivity. By using the Kubo method, we can calculate the optical conductivity (σ) which consists of intraband

inter) [39], as shown in Figure 6, thus:

4EF π

In ð Þ <sup>2</sup>EF <sup>þ</sup> <sup>ℏ</sup><sup>ω</sup>

here, <sup>σ</sup><sup>0</sup> <sup>¼</sup> <sup>e</sup>24<sup>ℏ</sup> ffi <sup>60</sup>:<sup>8</sup> <sup>μ</sup>S is the universal optical conductance, EF is the Fermi level of graphene, <sup>ℏ</sup> is Planck's constant, and <sup>Γ</sup><sup>1</sup> <sup>¼</sup> <sup>8</sup>:<sup>3</sup> � 1011s�<sup>1</sup> and <sup>Γ</sup><sup>2</sup> <sup>¼</sup> <sup>10</sup>13s�<sup>1</sup> are relaxation rates at room temperature associated with the interband and intraband transitions, respectively.

Obviously, relative permittivity of this kind of waveguide is related to the conductivity of graphene, which would further influence the dispersion properties of the whole structure. When integrating graphene into the waveguide, the surface plasmon dispersion of graphene is strongly modified by the metal and other dielectric substrates; thus, the transmission of the incident electromagnetic wave in the waveguide is affected as well [40]. The characteristic plasmon dispersion relationship could be obtained by the following equation for the structure

> ε<sup>1</sup> þ ε2tanhqd<sup>2</sup> ε<sup>2</sup> þ ε1tanhqd<sup>2</sup>

¼ �i q ω

σð Þ q; ω (7)

ð Þ 2EF � ℏω

iσ ωð Þ

inter þ σ<sup>00</sup>

1

� 1 π

" #

<sup>2</sup> <sup>þ</sup> ð Þ <sup>ℏ</sup>Γ<sup>2</sup>

<sup>2</sup> <sup>þ</sup> ð Þ <sup>ℏ</sup>Γ<sup>2</sup>

ωε0<sup>d</sup> (2)

http://dx.doi.org/10.5772/intechopen.76796

Graphene Based Waveguides

137

inter (3)

� � (5)

(6)

<sup>ℏ</sup>ð Þ <sup>Γ</sup><sup>1</sup> � <sup>i</sup><sup>ω</sup> (4)

tan �<sup>1</sup> <sup>ℏ</sup><sup>ω</sup> <sup>þ</sup> <sup>2</sup>EF ℏΓ<sup>2</sup>

2

2

For most materials without magnetic properties, we treat μ = 1 here. A conventional waveguide consists of a high-index core surrounded by a lower-index cladding.

#### 2.1. Graphene's relative complex permittivity in vacuum

Graphene's optical properties can be determined by its relative complex permittivity. The equivalent in-plane component of graphene's relative permittivity is given by:

#### Graphene Based Waveguides http://dx.doi.org/10.5772/intechopen.76796 137

$$\varepsilon(\omega) = 1 + \frac{\mathrm{i}\sigma(\omega)}{\omega\varepsilon\_0 d} \tag{2}$$

where d is the thickness of the graphene layer, ε<sup>0</sup> is relative complex permittivity in vacuum (ε<sup>0</sup> ¼ 8:854 F=m), ω is the optical frequency, and σ is graphene's optical conductivity. By using the Kubo method, we can calculate the optical conductivity (σ) which consists of intraband (σintra) and interband (σ<sup>0</sup> inter +iσ<sup>00</sup> inter) [39], as shown in Figure 6, thus:

$$
\sigma\_{\text{total}} = \sigma\_{\text{intra}} + \sigma\_{\text{inter}}' + \sigma\_{\text{inter}}'' \tag{3}
$$

where

in the TE mode, the electric lines of flux are perpendicular to the axis of the waveguides. While in TM mode, the magnetic lines of flux are perpendicular to the axis of the waveguides. Normally, for waveguides using a single conductor, no transverse-electromagnetic (TEM) mode could be transmitted. Most research focus on the transmission of the TE mode, while Cohnitz et al. [32] investigated a magnetic graphene waveguide, in which a clean graphene is exposed to a static inhomogeneous magnetic field along one of the planar directions. As shown in Figure 4, when applying magnetic fields to a monolayer graphene, quantum modes exhibited like classical snake orbits near the field switch lines. While in the other regions far

Usually, the integration of graphene with photonic devices relies on the transformation of exfoliated or delaminated or chemical vapor deposition (CVD)-fabricated 2D material onto pretreated devices, as shown in Figure 5 [34]. However, the application of the transformation process is limited due to the shortcomings in uniformity and efficiency [35–37]. Most importantly, transferred 2D materials suffer from weak interaction with optical modes in pre-treated devices [37]. An atomic layer deposition (ALD) method has been adopted widely to obtain gate dielectric on graphene. Also, plasma-enhanced chemical vapor deposition (PECVD) could be another option for fabrication of silicon nitride on graphene. Last year, it was reported that the spin-coating process could be applied to directly fabricate the polymer waveguide modulator on graphene [38]. However, the fabrication technique of photonics devices integrating the graphene thin film needs to be improved for the difficulty of keeping the original properties of graphene after the following integrated progresses. Even though it's still a challenge to ensure the quality of integrated graphene up to date, the importance of optimized and the continuous

from these region, only Landau-quantized cyclotron orbits could be detected.

study of graphene-based photonics devices, especially waveguides, are foreseen.

2. Electromagnetic properties of graphene-based waveguides

guide consists of a high-index core surrounded by a lower-index cladding.

equivalent in-plane component of graphene's relative permittivity is given by:

2.1. Graphene's relative complex permittivity in vacuum

ability (μ) the following:

136 Emerging Waveguide Technology

Most electromagnetic phenomena are governed by Maxwell equations, while the electromagnetic properties of materials are determined by two parameters, relative complex permittivity (ε) and relative complex magnetic permeability (μ), which describe the coupling of a material with incident electromagnetic energy. Normally, in the optical range, refraction index (n) is used as well to describe the macroscopic effective parameters of the material, and the refraction has as a relationship with relative complex permittivity (ε) and relative complex perme-

n ¼ ffiffiffiffiffiffiffi

For most materials without magnetic properties, we treat μ = 1 here. A conventional wave-

Graphene's optical properties can be determined by its relative complex permittivity. The

με p (1)

$$
\sigma\_{\rm intra} = \sigma\_0 \frac{4E\_\rm F}{\pi} \frac{1}{\hbar (\Gamma\_1 - i\omega)}\tag{4}
$$

$$
\sigma\_{inter}' = \sigma\_0 \left[ 1 + \frac{1}{\pi} \tan^{-1} \left( \frac{\hbar \omega - 2E\_F}{\hbar \Gamma\_2} \right) \right] - \frac{1}{\pi} \tan^{-1} \left( \frac{\hbar \omega + 2E\_F}{\hbar \Gamma\_2} \right) \tag{5}
$$

$$
\sigma''\_{\rm inter} = -\sigma\_0 \frac{1}{2\pi} \text{Im} \left[ \frac{(2E\_F + \hbar \omega)^2 + (\hbar \Gamma\_2)^2}{\left(2E\_F - \hbar \omega\right)^2 + \left(\hbar \Gamma\_2\right)^2} \right] \tag{6}
$$

here, <sup>σ</sup><sup>0</sup> <sup>¼</sup> <sup>e</sup>24<sup>ℏ</sup> ffi <sup>60</sup>:<sup>8</sup> <sup>μ</sup>S is the universal optical conductance, EF is the Fermi level of graphene, <sup>ℏ</sup> is Planck's constant, and <sup>Γ</sup><sup>1</sup> <sup>¼</sup> <sup>8</sup>:<sup>3</sup> � 1011s�<sup>1</sup> and <sup>Γ</sup><sup>2</sup> <sup>¼</sup> <sup>10</sup>13s�<sup>1</sup> are relaxation rates at room temperature associated with the interband and intraband transitions, respectively.

Figure 6. (a) Interband transition and (b) intraband transition in graphene [39].

Obviously, relative permittivity of this kind of waveguide is related to the conductivity of graphene, which would further influence the dispersion properties of the whole structure. When integrating graphene into the waveguide, the surface plasmon dispersion of graphene is strongly modified by the metal and other dielectric substrates; thus, the transmission of the incident electromagnetic wave in the waveguide is affected as well [40]. The characteristic plasmon dispersion relationship could be obtained by the following equation for the structure as described in Figure 7(a):

$$
\varepsilon\_3 \frac{\varepsilon\_4 + \varepsilon\_3 \tanh qd\_3}{\varepsilon\_3 + \varepsilon\_4 \tanh qd\_3} + \varepsilon\_2 \frac{\varepsilon\_1 + \varepsilon\_2 \tanh qd\_2}{\varepsilon\_2 + \varepsilon\_1 \tanh qd\_2} = -\mathbf{i} \frac{q}{\omega} \sigma(q, \omega) \tag{7}
$$

Figure 7. (a) Planar waveguide, and (c) non-planar rectangular waveguide. Dispersion characteristics of planar (b) and non-planar (d) waveguides [41].

for which the explanation of permittivity could be found in the Figure 7(a). It should be noted here that the equation shows a good approximation when the thickness is much thicker than the skin depth [41]. As shown in Figure 7(b), an enhanced confinement and an increased propagation distance can be obtained by adopting a metal slab within a certain spectral region. However, only the plasmonic field in the direction perpendicular to the surface could be confined to the model of Figure 7(a). By adopting a non-planar structure as shown in Figure 7 (c), 2D confinement could be obtained due to the dielectric boundaries in the other direction, of which the dispersion relationship could be obtained by an effective index method.

However, in most conditions, graphene is regarded as a boundary condition when integrated into a waveguide due to the difficulty in a complex refraction index, while permittivity is regarded as a fundamental issue in graphene-based modern integrated photonics and devices. Yao's group [46] used a microfiber-based Mach-Zehnder interferometer to obtain the complex refraction index of the graphene waveguide from 1510 to 1590 nm, as shown in Figure 8. In this method, the microfiber acts as an effective mean to launch and collect the evanescent signal for the waveguide, for which, on the other hand, the contact length can adjusted if needed. As the results shown in Figure 8(b), the complex refraction index of graphene-based waveguide varies from 2:91 � i3:92 to 3:81 � i14:64 in the experimental-range wavelength

Figure 8. (a) Schematic diagram of the structure for the light propagating along the GMFW (red: Single mode fiber, white: Monolayer graphene, cyan: MgF2 substrate). The orange arrows show the transmitting direction of the evanescent waves. (b) neffRE of the microfiber (blue solid) and the microfiber on MgF2 (red dashed). (c) the experimental details of the

When model graphene is infinitely thin, local two-sided surfaces with conductivity σ could be

8

>>>>>>>><

>>>>>>>>:

1 ð Þ <sup>ω</sup> � <sup>j</sup>2<sup>Γ</sup> <sup>2</sup>

> � ð∞ 0

ð∞ 0 ε ∂f <sup>d</sup>ð Þε

<sup>∂</sup><sup>ε</sup> � <sup>∂</sup><sup>f</sup> <sup>d</sup>ð Þ �<sup>ε</sup> ∂ε � �d<sup>ε</sup>

dε

9

Graphene Based Waveguides

139

http://dx.doi.org/10.5772/intechopen.76796

>>>>>>>>=

(8)

>>>>>>>>;

f <sup>d</sup>ð Þ� �ε f <sup>d</sup>ð Þε ð Þ <sup>ω</sup> � <sup>j</sup>2<sup>Γ</sup> <sup>2</sup> � <sup>4</sup>ð Þ <sup>ε</sup>=<sup>ℏ</sup> <sup>2</sup> " #

from 1510 to 1590 nm.

2.3. The tunability of graphene permittivity

GMHW. (d) Geometry of the cross-section of the GMHW [46].

obtained based on Kubo function as the following:

σ ω; <sup>μ</sup>c; <sup>Γ</sup>; <sup>T</sup> � � <sup>¼</sup> je<sup>2</sup>ð Þ <sup>ω</sup> � <sup>j</sup>2<sup>Γ</sup>

<sup>π</sup>ℏ<sup>2</sup> �

Graphene's relative permittivity can be easily tuned by electrostatic gating or chemical doping, which makes it easier to be applied for Talbot effect than metal-based devices [42]. Plasmonic Talbot carpets were experimentally obtained by using surface plasmon polariton (SPP) launching gratings, and a sub-wavelength focal spot can obtained.

#### 2.2. The complex refraction index of graphene based waveguides

Borini's group estimated the optical index of graphene in visible range by dealing with universal optical conductivity and measured the optical spectrum within the framework of Fresnel's coefficient calculation [43]. Reflectometry is another method to acquire the reflection properties so as to obtain an average index over a broadband range by fitting the spectrum. Xu's group [44] calculated the complex refraction index of graphene at 1550 nm through reflectivity measurement on a SiO2/Si substrate. And as reported by Wang [45], Notle's group applied picometrology to measure the refraction of graphene on thermal oxide on silicon at 488, 532, and 633 nm, respectively, in which the strong dispersion of the graphene index was observed in an optical range.

Figure 8. (a) Schematic diagram of the structure for the light propagating along the GMFW (red: Single mode fiber, white: Monolayer graphene, cyan: MgF2 substrate). The orange arrows show the transmitting direction of the evanescent waves. (b) neffRE of the microfiber (blue solid) and the microfiber on MgF2 (red dashed). (c) the experimental details of the GMHW. (d) Geometry of the cross-section of the GMHW [46].

However, in most conditions, graphene is regarded as a boundary condition when integrated into a waveguide due to the difficulty in a complex refraction index, while permittivity is regarded as a fundamental issue in graphene-based modern integrated photonics and devices. Yao's group [46] used a microfiber-based Mach-Zehnder interferometer to obtain the complex refraction index of the graphene waveguide from 1510 to 1590 nm, as shown in Figure 8. In this method, the microfiber acts as an effective mean to launch and collect the evanescent signal for the waveguide, for which, on the other hand, the contact length can adjusted if needed. As the results shown in Figure 8(b), the complex refraction index of graphene-based waveguide varies from 2:91 � i3:92 to 3:81 � i14:64 in the experimental-range wavelength from 1510 to 1590 nm.

#### 2.3. The tunability of graphene permittivity

for which the explanation of permittivity could be found in the Figure 7(a). It should be noted here that the equation shows a good approximation when the thickness is much thicker than the skin depth [41]. As shown in Figure 7(b), an enhanced confinement and an increased propagation distance can be obtained by adopting a metal slab within a certain spectral region. However, only the plasmonic field in the direction perpendicular to the surface could be confined to the model of Figure 7(a). By adopting a non-planar structure as shown in Figure 7 (c), 2D confinement could be obtained due to the dielectric boundaries in the other direction, of

Figure 7. (a) Planar waveguide, and (c) non-planar rectangular waveguide. Dispersion characteristics of planar (b) and

Graphene's relative permittivity can be easily tuned by electrostatic gating or chemical doping, which makes it easier to be applied for Talbot effect than metal-based devices [42]. Plasmonic Talbot carpets were experimentally obtained by using surface plasmon polariton (SPP)

Borini's group estimated the optical index of graphene in visible range by dealing with universal optical conductivity and measured the optical spectrum within the framework of Fresnel's coefficient calculation [43]. Reflectometry is another method to acquire the reflection properties so as to obtain an average index over a broadband range by fitting the spectrum. Xu's group [44] calculated the complex refraction index of graphene at 1550 nm through reflectivity measurement on a SiO2/Si substrate. And as reported by Wang [45], Notle's group applied picometrology to measure the refraction of graphene on thermal oxide on silicon at 488, 532, and 633 nm, respectively, in which the strong dispersion of the graphene index was

which the dispersion relationship could be obtained by an effective index method.

launching gratings, and a sub-wavelength focal spot can obtained.

2.2. The complex refraction index of graphene based waveguides

observed in an optical range.

non-planar (d) waveguides [41].

138 Emerging Waveguide Technology

When model graphene is infinitely thin, local two-sided surfaces with conductivity σ could be obtained based on Kubo function as the following:

$$\sigma(\omega,\mu\_c,\Gamma,T) = \frac{j\varepsilon^2(\omega-j2\Gamma)}{\pi\hbar^2} \times \left\{ \begin{array}{c} \frac{1}{(\omega-j2\Gamma)^2} \int\_0^\infty \varepsilon \left[\frac{\partial f\_d(\varepsilon)}{\partial \varepsilon} - \frac{\partial f\_d(-\varepsilon)}{\partial \varepsilon}\right] d\varepsilon \\\\ -\int\_0^\infty \left[\frac{f\_d(-\varepsilon)-f\_d(\varepsilon)}{\left(\omega-j2\Gamma\right)^2-4\left(\varepsilon/\hbar\right)^2}\right] d\varepsilon \end{array} \right\} \tag{8}$$

where ω is radiated frequency, μ<sup>c</sup> is chemical potential, Γ refers to the phenomenological scattering rate that is assumed to be independent of energy, and T is temperature.

$$
\mu\_c = \hbar V\_F \sqrt{\frac{\pi \varepsilon\_{ox}}{ed\_{ox}} (V - V\_D)} \tag{9}
$$

<sup>ε</sup>sub <sup>¼</sup> <sup>ε</sup><sup>L</sup> <sup>þ</sup> <sup>α</sup>E<sup>2</sup> (10)

Graphene Based Waveguides

141

http://dx.doi.org/10.5772/intechopen.76796

H ¼ Hyy (11)

E ¼ Exx þ Ezz (12)

¼ iωμ0Hy þ iβEx (13)

iβHy ¼ �iωε0εEx (14)

dx <sup>¼</sup> <sup>i</sup>ωε0εEz (15)

=α (17)

H=m. When integrating graphene at the top

<sup>y</sup> (16)

where ε<sup>L</sup> corresponds to the relative complex permittivity of substrate under linear incidence and E corresponds to the external incident electric field. Besides, graphene with conductivity of

For TM polarization with propagation constant β, three field components, Ex, Ez, and Hy,

magnetic field H and electrical field E satisfy the following equations:

Figure 10. Schematic of waveguide constituted by graphene and nonlinear substrate [48].

exactly as <sup>ε</sup><sup>0</sup> <sup>¼</sup> <sup>8</sup>:<sup>854</sup> � <sup>10</sup>�12F=m, <sup>μ</sup><sup>0</sup> <sup>¼</sup> <sup>4</sup><sup>π</sup> � <sup>10</sup>�<sup>7</sup>

of the Kerr-type substrate, we get:

Thus, we further get:

dEz dx

dHy

ε2 E2 <sup>x</sup> <sup>¼</sup> <sup>β</sup><sup>2</sup> ω<sup>2</sup>ε<sup>2</sup> 0 H2

E2

where ε<sup>0</sup> and μ<sup>0</sup> correspond to electric permittivity and magnetic permeability in the vacuum,

<sup>x</sup> <sup>¼</sup> <sup>ε</sup> � <sup>ε</sup><sup>L</sup> � <sup>α</sup>E<sup>2</sup>

z

σ<sup>g</sup> is treated as boundary here considering a one-atom scale thickness (Figure 10).

where VF stands for Fermi velocity, εox, dox corresponds to the permittivity and thickness of the dielectric, respectively, VD is the Dirac voltage, and V is the externally applied voltage. Obviously, the chemical potential could be strongly tuned by the applied gate voltage, which, as a result, would impact the refraction index. Xu's group used the reflectivity measurement to obtain the complex refraction index of graphene on SiO2/Si, which could be tuned by gate electric voltage, which agreed well with the Kubo function [44].

As shown in Figure 9, the real and imaginary part of the conductivity of graphene display a relationship with wavelength in the mid-infrared range [47]. Besides, the chemical potential which attributes the carrier density in graphene plays a critical role in controlling the conductivity. When ℏω > 2 μ<sup>c</sup> � � � �, the optical absorption of graphene is related to the real part of conductivity, which comes from the interband transition. Obviously, photocarriers are generated during the transition process, which could be used in applications such as photo-detection or modulators. While for ℏω < 2 μ<sup>c</sup> � � � �, the conductivity of graphene could be explained by Pauli's blocking theory. Thus, an electrostatic grating is always applied to adjust the chemical potential, and thus to tune the absorption of graphene, which lies as the principle to design optical modulators. Moreover, it's displayed in Figure 9 that the imaginary part of intraband and interband conductivity has the opposite sign, which plays a critical role in determining whether the TE or TM mode could be propagated in graphene, which is always used in a polarizer.

#### 2.4. Graphene integrated with nonlinear substrates

When integrating graphene with a nonlinear substrate, the relative complex permittivity of the substrate (εsub) could be explained by the following equation named Kerr-type medium [48]:

Figure 9. Interband (solid lines) and intraband (dashed lines) contribution to the dynamic conductivity in graphene. The vertical black line marks the telecommunication wavelength of λ = 1.55 μm [47].

$$
\varepsilon\_{sub} = \varepsilon\_L + aE^2 \tag{10}
$$

where ε<sup>L</sup> corresponds to the relative complex permittivity of substrate under linear incidence and E corresponds to the external incident electric field. Besides, graphene with conductivity of σ<sup>g</sup> is treated as boundary here considering a one-atom scale thickness (Figure 10).

Figure 10. Schematic of waveguide constituted by graphene and nonlinear substrate [48].

For TM polarization with propagation constant β, three field components, Ex, Ez, and Hy, magnetic field H and electrical field E satisfy the following equations:

$$H = H\_y y \tag{11}$$

$$E = E\_x \mathbf{x} + E\_z \mathbf{z} \tag{12}$$

$$i\frac{dE\_z}{d\_x} = i\omega\mu\_0 H\_\mathcal{Y} + i\beta E\_x \tag{13}$$

$$
\dot{\imath}\beta H\_y = -i\omega\varepsilon\_0 \varepsilon E\_x \tag{14}
$$

$$\frac{dH\_y}{d\mathbf{x}} = i\omega\varepsilon\_0\varepsilon E\_z \tag{15}$$

where ε<sup>0</sup> and μ<sup>0</sup> correspond to electric permittivity and magnetic permeability in the vacuum, exactly as <sup>ε</sup><sup>0</sup> <sup>¼</sup> <sup>8</sup>:<sup>854</sup> � <sup>10</sup>�12F=m, <sup>μ</sup><sup>0</sup> <sup>¼</sup> <sup>4</sup><sup>π</sup> � <sup>10</sup>�<sup>7</sup> H=m. When integrating graphene at the top of the Kerr-type substrate, we get:

$$
\varepsilon^2 E\_x^2 = \frac{\beta^2}{\omega^2 \varepsilon\_0^2} H\_y^2 \tag{16}
$$

$$E\_\chi^2 = \left(\varepsilon - \varepsilon\_L - aE\_z^2\right)/a \tag{17}$$

Thus, we further get:

where ω is radiated frequency, μ<sup>c</sup> is chemical potential, Γ refers to the phenomenological

πεox edox

where VF stands for Fermi velocity, εox, dox corresponds to the permittivity and thickness of the dielectric, respectively, VD is the Dirac voltage, and V is the externally applied voltage. Obviously, the chemical potential could be strongly tuned by the applied gate voltage, which, as a result, would impact the refraction index. Xu's group used the reflectivity measurement to obtain the complex refraction index of graphene on SiO2/Si, which could be tuned by gate

As shown in Figure 9, the real and imaginary part of the conductivity of graphene display a relationship with wavelength in the mid-infrared range [47]. Besides, the chemical potential which attributes the carrier density in graphene plays a critical role in controlling the conductiv-

ity, which comes from the interband transition. Obviously, photocarriers are generated during the transition process, which could be used in applications such as photo-detection or modula-

theory. Thus, an electrostatic grating is always applied to adjust the chemical potential, and thus to tune the absorption of graphene, which lies as the principle to design optical modulators. Moreover, it's displayed in Figure 9 that the imaginary part of intraband and interband conductivity has the opposite sign, which plays a critical role in determining whether the TE or TM

When integrating graphene with a nonlinear substrate, the relative complex permittivity of the substrate (εsub) could be explained by the following equation named Kerr-type medium [48]:

Figure 9. Interband (solid lines) and intraband (dashed lines) contribution to the dynamic conductivity in graphene. The

mode could be propagated in graphene, which is always used in a polarizer.

r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð Þ V � VD

�, the optical absorption of graphene is related to the real part of conductiv-

�, the conductivity of graphene could be explained by Pauli's blocking

(9)

scattering rate that is assumed to be independent of energy, and T is temperature.

μ<sup>c</sup> ¼ ℏVF

electric voltage, which agreed well with the Kubo function [44].

ity. When ℏω > 2 μ<sup>c</sup>

140 Emerging Waveguide Technology

tors. While for ℏω < 2 μ<sup>c</sup>

� � �

> � � �

2.4. Graphene integrated with nonlinear substrates

vertical black line marks the telecommunication wavelength of λ = 1.55 μm [47].

$$
\varepsilon^3 - \left(\varepsilon\_L + \alpha E\_z^2\right)\varepsilon^2 - \frac{\alpha \beta^2}{\alpha^2 \varepsilon\_0^2} H\_y^2 = 0 \tag{18}
$$

such as unrivaled speed, low driving voltage, small physical footprints, and low power consumption [50], which can further be utilized in telecommunications and optoelectronics. While several graphene-based waveguides have been investigated, it still remains a challenge to combine graphene with plasmonic waveguides. How to control the intensity, phase, and polarization of the electromagnetic wave in an optical range through the waveguide attracts a lot of interest. When integrating graphene into a waveguide, the waveguide mode propagates along and is confined near a graphene sheet, which is regarded as the most promising for on-chip information processing. However, we have to admit here that even though the atomic thickness of graphene gives rise to lots of advantages, there are several challenges to deal with in such a device, such as

unavoidable power consumption, slowdown response, and lower modulation depth.

bellow 1 V, the Fermi level (EF) was lower than half photonic energy (<sup>1</sup>

resonance while the incident phonons were occupied.

Due to the tunable bandgap of graphene, waveguide modulators could be formed with broad flexibility [8, 50]. Besides, the carried density of graphene could be tuned manually through external gate voltage [51, 52], chemical doping [53, 54], and optical (laser) excitation [55]. As a result, the refraction index and the permittivity could be adjusted. It is worth mentioning here that the response of graphene in an optical range could be tuned by substrates as well [56], which may induce a bandgap opening in epitaxial graphene [57]. The transmission of 1.53 μm photons through the waveguide at a varied drive voltage is shown in Figure 11, which has been divided into three different regions from 6 to 6 V and the corresponding band structures are shown as insets. In the left region with drive voltage

were available for further interband transition. In the middle region for drive voltage ranging from 1 to 3.8 V, the Fermi level was close to Dirac point; thus, it is possible for electrons in occupied regions transiting to unoccupied regions. In other words, graphene sheets showed potential in phonon absorbing, indicating its modulation ability. In the third region from 3.8 to 6 V, the transition was blocked again since all the electron states which were in

Grigorenko's group reported a hybrid graphene-plasmon waveguide modulator for promising applications in telecom as shown in Figure 12, whose modulation depth was comparable with silicon-based waveguide modulators, showing a promising future for optical communication [50].

Figure 11. Displays the transmission of 1.53 μm photons through the waveguide at different drive voltages.

<sup>2</sup> ℏv), and no electrons

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By a mathematical transformation, we can finally get the discriminant of permittivity ε:

$$\Delta = -\left(\varepsilon\_{\rm L} + a\varepsilon\_{\rm x}^2\right)^3 \frac{\alpha \mathfrak{E}^2}{\alpha^2 \varepsilon\_0^2} \mathcal{H}\_{\rm y}^2 - 27 \frac{\alpha^2 \mathfrak{E}^4}{\alpha^4 \varepsilon\_0^4} \mathcal{H}\_{\rm y}^4 < 0 \tag{19}$$

if Δ < 0, then there is only one solution for ε. The equation is useful when we numerically calculate permittivity in the relaxation method. We can get permittivity through its real root. In particular, the nonlinear conductivity of graphene cannot be ignored any more under this condition, which can be expressed as:

$$
\sigma\_{\mathcal{S}} = \sigma\_L + \sigma^{NL} |E\_{\pi}|^2 \tag{20}
$$

where E<sup>τ</sup> is the tangential component of the electric field and σNL contributes to nonlinear conductivity:

$$\sigma^{NL} = -i \frac{3}{8} \frac{e^2}{\pi \hbar^2} \left(\frac{eV\_F}{\mu\_c \omega}\right)^2 \frac{\mu\_c}{\omega} \tag{21}$$

in which VF stands for Fermi velocity. It should be noted here that the conductivity of graphene is regarded as Drude type only in THz and far IR range.

#### 3. Applications

Graphene could be an ideal option to meet the increasing demand of high-performance optoelectronics or some other communication components when the incident wave is confined along the thin film surface [49]. The 2D structure of graphene and the planar configuration of silicon photonics are inherently compatible with each other [47]. Normally, the maximum absorption in the monolayer graphene integrating at the dielectric surface is about 10~20%, which would not make a big difference even applying highest practically achievable carrier concentrations [8]. This character could be enhanced when incorporating graphene on the surface of a passive silicon dielectric waveguide, and the modulation depth could be as high as 50% when applying voltage to the graphene sheet.

The application of graphene-based waveguides can be summarized to be modulators, detectors, sensors, polarizers and some other applications, as discussed in detail through the following.

#### 3.1. Graphene-based waveguide integrated modulators

Thanks to the outstanding properties of graphene in conductivity, current density, and charge mobility, graphene-based waveguides are supposed to have promising potentials in applications such as unrivaled speed, low driving voltage, small physical footprints, and low power consumption [50], which can further be utilized in telecommunications and optoelectronics. While several graphene-based waveguides have been investigated, it still remains a challenge to combine graphene with plasmonic waveguides. How to control the intensity, phase, and polarization of the electromagnetic wave in an optical range through the waveguide attracts a lot of interest. When integrating graphene into a waveguide, the waveguide mode propagates along and is confined near a graphene sheet, which is regarded as the most promising for on-chip information processing. However, we have to admit here that even though the atomic thickness of graphene gives rise to lots of advantages, there are several challenges to deal with in such a device, such as unavoidable power consumption, slowdown response, and lower modulation depth.

<sup>ε</sup><sup>3</sup> � <sup>ε</sup><sup>L</sup> <sup>þ</sup> <sup>α</sup>E<sup>2</sup>

<sup>Δ</sup> ¼ � <sup>ε</sup><sup>L</sup> <sup>þ</sup> <sup>α</sup>E<sup>2</sup>

condition, which can be expressed as:

conductivity:

142 Emerging Waveguide Technology

3. Applications

z <sup>ε</sup><sup>2</sup> � αβ<sup>2</sup>

By a mathematical transformation, we can finally get the discriminant of permittivity ε:

ω<sup>2</sup>ε<sup>2</sup> 0 H2

if Δ < 0, then there is only one solution for ε. The equation is useful when we numerically calculate permittivity in the relaxation method. We can get permittivity through its real root. In particular, the nonlinear conductivity of graphene cannot be ignored any more under this

where E<sup>τ</sup> is the tangential component of the electric field and σNL contributes to nonlinear

in which VF stands for Fermi velocity. It should be noted here that the conductivity of

Graphene could be an ideal option to meet the increasing demand of high-performance optoelectronics or some other communication components when the incident wave is confined along the thin film surface [49]. The 2D structure of graphene and the planar configuration of silicon photonics are inherently compatible with each other [47]. Normally, the maximum absorption in the monolayer graphene integrating at the dielectric surface is about 10~20%, which would not make a big difference even applying highest practically achievable carrier concentrations [8]. This character could be enhanced when incorporating graphene on the surface of a passive silicon dielectric waveguide, and the modulation depth could be as high

The application of graphene-based waveguides can be summarized to be modulators, detectors, sensors, polarizers and some other applications, as discussed in detail through the following.

Thanks to the outstanding properties of graphene in conductivity, current density, and charge mobility, graphene-based waveguides are supposed to have promising potentials in applications

eVF μcω <sup>2</sup> <sup>μ</sup><sup>c</sup>

3 8 e2 πℏ<sup>2</sup>

z <sup>3</sup> αβ<sup>2</sup>

<sup>σ</sup>NL ¼ �<sup>i</sup>

graphene is regarded as Drude type only in THz and far IR range.

as 50% when applying voltage to the graphene sheet.

3.1. Graphene-based waveguide integrated modulators

ω<sup>2</sup>ε<sup>2</sup> 0 H2

<sup>y</sup> � <sup>27</sup> <sup>α</sup><sup>2</sup>β<sup>4</sup> ω<sup>4</sup>ε<sup>4</sup> 0 H4

<sup>σ</sup><sup>g</sup> <sup>¼</sup> <sup>σ</sup><sup>L</sup> <sup>þ</sup> <sup>σ</sup>NLj j <sup>E</sup><sup>τ</sup> <sup>2</sup> (20)

<sup>y</sup> ¼ 0 (18)

<sup>y</sup> < 0 (19)

<sup>ω</sup> (21)

Due to the tunable bandgap of graphene, waveguide modulators could be formed with broad flexibility [8, 50]. Besides, the carried density of graphene could be tuned manually through external gate voltage [51, 52], chemical doping [53, 54], and optical (laser) excitation [55]. As a result, the refraction index and the permittivity could be adjusted. It is worth mentioning here that the response of graphene in an optical range could be tuned by substrates as well [56], which may induce a bandgap opening in epitaxial graphene [57]. The transmission of 1.53 μm photons through the waveguide at a varied drive voltage is shown in Figure 11, which has been divided into three different regions from 6 to 6 V and the corresponding band structures are shown as insets. In the left region with drive voltage bellow 1 V, the Fermi level (EF) was lower than half photonic energy (<sup>1</sup> <sup>2</sup> ℏv), and no electrons were available for further interband transition. In the middle region for drive voltage ranging from 1 to 3.8 V, the Fermi level was close to Dirac point; thus, it is possible for electrons in occupied regions transiting to unoccupied regions. In other words, graphene sheets showed potential in phonon absorbing, indicating its modulation ability. In the third region from 3.8 to 6 V, the transition was blocked again since all the electron states which were in resonance while the incident phonons were occupied.

Grigorenko's group reported a hybrid graphene-plasmon waveguide modulator for promising applications in telecom as shown in Figure 12, whose modulation depth was comparable with silicon-based waveguide modulators, showing a promising future for optical communication [50].

Figure 11. Displays the transmission of 1.53 μm photons through the waveguide at different drive voltages.

However, Murphy's group [8] designed a THz modulator formulated by setting large graphene sheets at the middle of two SiO2 layers (300 nm) with Si wafers on both their sides based on ridge waveguides, as shown in the figure above. When applying voltage to graphene sheets, light-matter interaction could be modulated. As a result, the modulation depth could be achieved as high as 50%. Obviously, the carrier concentration in the graphene sheet would be modified by adjusting gate voltage. When voltage was guided through the graphene sheet in the waveguide, the electric field would penetrate into the graphene sheet and lead to

Graphene Based Waveguides

145

http://dx.doi.org/10.5772/intechopen.76796

Conventionally, low band-gap semiconductors such as HgCdTe alloys or quantum-well and quantum-dot structures on III-V materials are adopted to formulate mid-infrared detectors [59]. With electrical tunability in light absorption and ultra-fast photo-response, graphene is regarded as a promising candidate for high-speed photo-detection applications [9]. It has been approved that graphene-based waveguide photodetectors could be applied from 300 nm to 6 μm or even longer [4]. It should be noted here that a dark current range may occur due to the gapless inherent properties of graphene, which should be avoided in practical application [60]. Thus, chemical potential must be tuned near the Dirac energy to ensure that incident field is

Wang et al. [30] integrated monolayer graphene into a silicon optical waveguide on a silicon-oninsulator (SOI) from near-to-mid-infrared operational range, which indicated that the combination of the graphene silicon structure made it possible to overcome the shortcomings of the traditional junction-less photodetectors. As a result, a much higher sensitivity could be expected in graphene-based waveguides. The transverse electric mode light (~10 μm spot size) was coupled into the waveguide via a focusing sub-wavelength grating. Avouris' group [60] reported an efficient photodetection of the waveguide based on graphene, as shown in Figure 14, which shows gate-dependent response, and the response is nearly linear on the entire device of 10 mV. And the measurement from network analyzer showed the relative A.C photo-response, which could be further improved by applying a bias within the photocurrent generation path.

Figure 14. Photocurrent generation, high-frequency characterization of the MGM photodetector, and operation of the

<sup>1</sup> with 1.55 μm light excitation [60].

absorption due to free carriers in the graphene sheet.

3.2. Graphene-based waveguide photodetectors

illuminated to the graphene thin film.

MGM photodetector at a data rate of 10 Gbit∙s

Figure 12. (a) The schematics of the hybrid graphene plasmonic waveguide modulators. (b) The optical micrograph of a typical hybrid graphene plasmonic modulator studied in this work. (c) Leakage radiation detection of wedge, upper panel, and flat, lower panel, plasmon-propagating modes. (d) A scanning electron micrograph of an area shown in b by the dotted box that shows corrugated waveguide and the semitransparent decoupling grating. (e) Optical Pauli blocking expressed in terms of graphene relative conductivity. (f) Sketches of three types of plasmonic modes under investigation —flat, corrugated and wedge plasmons. (g) 3D rendering of the experiment with the wedge plasmon mode. (h) The schematic of experiment where non-transparent grating couples light into plasmon modes [50].

The conductivity of graphene related to Fermi energy depends on the applied voltage between graphene layers and the thickness and permittivity of the dielectric layer located between two graphene layers. Asgari's group [58] applied bias voltage Vb between two graphene layers which leads to an equal increase in electron density in the top layer and hole density in the bottom layer, as shown in Figure 13. Therefore, the absolute values of Fermi energies (EF) in both graphene layers were identical but of opposite signs. Voltage application caused a potential difference between two graphene layers. Actually, when applying bias voltage between two graphene sheets, a capacitor effect could be observed. Besides, charge density in both sheets would increase with bias voltage, as well as Fermi energy. As a result, intraband conductivity would increase, while interband conductivity decreases.

Figure 13. (a) Schematic illustration of the two-graphene layer structure and direction of electric and magnetic field components in TM mode. (b) The band scheme of the structure at bias voltage Vb [58].

However, Murphy's group [8] designed a THz modulator formulated by setting large graphene sheets at the middle of two SiO2 layers (300 nm) with Si wafers on both their sides based on ridge waveguides, as shown in the figure above. When applying voltage to graphene sheets, light-matter interaction could be modulated. As a result, the modulation depth could be achieved as high as 50%. Obviously, the carrier concentration in the graphene sheet would be modified by adjusting gate voltage. When voltage was guided through the graphene sheet in the waveguide, the electric field would penetrate into the graphene sheet and lead to absorption due to free carriers in the graphene sheet.

#### 3.2. Graphene-based waveguide photodetectors

The conductivity of graphene related to Fermi energy depends on the applied voltage between graphene layers and the thickness and permittivity of the dielectric layer located between two graphene layers. Asgari's group [58] applied bias voltage Vb between two graphene layers which leads to an equal increase in electron density in the top layer and hole density in the bottom layer, as shown in Figure 13. Therefore, the absolute values of Fermi energies (EF) in both graphene layers were identical but of opposite signs. Voltage application caused a potential difference between two graphene layers. Actually, when applying bias voltage between two graphene sheets, a capacitor effect could be observed. Besides, charge density in both sheets would increase with bias voltage, as well as Fermi energy. As a result, intraband

Figure 13. (a) Schematic illustration of the two-graphene layer structure and direction of electric and magnetic field

Figure 12. (a) The schematics of the hybrid graphene plasmonic waveguide modulators. (b) The optical micrograph of a typical hybrid graphene plasmonic modulator studied in this work. (c) Leakage radiation detection of wedge, upper panel, and flat, lower panel, plasmon-propagating modes. (d) A scanning electron micrograph of an area shown in b by the dotted box that shows corrugated waveguide and the semitransparent decoupling grating. (e) Optical Pauli blocking expressed in terms of graphene relative conductivity. (f) Sketches of three types of plasmonic modes under investigation —flat, corrugated and wedge plasmons. (g) 3D rendering of the experiment with the wedge plasmon mode. (h) The

conductivity would increase, while interband conductivity decreases.

components in TM mode. (b) The band scheme of the structure at bias voltage Vb [58].

schematic of experiment where non-transparent grating couples light into plasmon modes [50].

144 Emerging Waveguide Technology

Conventionally, low band-gap semiconductors such as HgCdTe alloys or quantum-well and quantum-dot structures on III-V materials are adopted to formulate mid-infrared detectors [59]. With electrical tunability in light absorption and ultra-fast photo-response, graphene is regarded as a promising candidate for high-speed photo-detection applications [9]. It has been approved that graphene-based waveguide photodetectors could be applied from 300 nm to 6 μm or even longer [4]. It should be noted here that a dark current range may occur due to the gapless inherent properties of graphene, which should be avoided in practical application [60]. Thus, chemical potential must be tuned near the Dirac energy to ensure that incident field is illuminated to the graphene thin film.

Wang et al. [30] integrated monolayer graphene into a silicon optical waveguide on a silicon-oninsulator (SOI) from near-to-mid-infrared operational range, which indicated that the combination of the graphene silicon structure made it possible to overcome the shortcomings of the traditional junction-less photodetectors. As a result, a much higher sensitivity could be expected in graphene-based waveguides. The transverse electric mode light (~10 μm spot size) was coupled into the waveguide via a focusing sub-wavelength grating. Avouris' group [60] reported an efficient photodetection of the waveguide based on graphene, as shown in Figure 14, which shows gate-dependent response, and the response is nearly linear on the entire device of 10 mV. And the measurement from network analyzer showed the relative A.C photo-response, which could be further improved by applying a bias within the photocurrent generation path.

Figure 14. Photocurrent generation, high-frequency characterization of the MGM photodetector, and operation of the MGM photodetector at a data rate of 10 Gbit∙s <sup>1</sup> with 1.55 μm light excitation [60].

However, it's difficult to fabricate these materials which are challenging to operate at room temperature till now. Choi's group [61] integrated graphene with a Bi2Se3 heterostructure, in which graphene functioned as high mobility charge transport layers and Bi2Se3 functioned as a broadband IR absorber supplying holes in graphene. The graphene-Bi2Se3 structure showed broadband absorption and high-intensity response at room temperature.

3.4. Other waveguides

4. Conclusion

Author details

University, China

Xianglian Song, Xiaoyu Dai and Yuanjiang Xiang\*

\*Address all correspondence to: xiangyuanjiang@126.com

graphene and incident electromagnetics [70].

unchanged compared with the typical graphene waveguide [71, 72].

Usually, a fixed optical device works only on one particular polarization state, either the TE or the TM polarization state [68]. By coating silicon waveguides with graphene, a versatile polarizer works in two operation modes, which were based on the different effective mode index variations [69].

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Waveguide integrated with graphene has nonlinear parameters which depend strongly on the Fermi level of graphene. It has been demonstrated that Integrating graphene with slot waveguide would benefit non-linear properties, owing to the interaction enhancement between

When integrating graphene with nonlinear substrates on one or both sides, the surface plasmons' (SPs) localization length increased while their propagation length (PL) remained

In conclusion, due to its unique electric and electromagnetic properties, graphene acts as a promising candidate for photonics and communication component of high performance. Especially, integrating graphene with waveguides makes it possible to overcome shortcomings such as limited bandwidth, inevitable cross-talk, high energy consumption, and expensive cost in conventional devices. Furthermore, the gapless linear dispersion of Dirac fermions makes it possible for graphene integrated with other substrates to formulate modulators, polarizers, broadband waveguides, photodetectors, bio-sensors, and so on. And the permittivity of graphene-based waveguides could be calculated based on Maxwell's function, even integrating with nonlinear substrates. The application of graphene waveguides expands the broadband range, from 300 nm to 6 μm or even longer. By tuning the carrier density of graphene through external gate voltage, chemical doping, or optical excitation, the relative complex permittivity of graphene is tuned. Thus, graphene waveguide modulators could be formed which adjusts absorption and modulation depth and so on. Besides, graphene waveguides show potential in fast and high-response detection and chemical sensing. The recent development in graphene synthesis and photonics components' fabrication technique ensures the compatibility in the

integrated electronics platform, which shows a bright prospect in the near future.

International Collaborative Laboratory of 2D Materials for Optoelectronic Science and Technology of Ministry of Education, College of Optoelectronic Engineering, Shenzhen

#### 3.3. Graphene-based waveguide sensors

Graphene is used in sensors, thanks to its unique electric properties, which show great potential in chemical or biology sensors. Among all varieties of chemical gas sensors, photonics gas sensors have advantages because of their high sensitivity and stability. Graphene plays a critical role in gas sensing due to the sensitivity of carried density to environment [62]. Graphene's conductivity can be changed drastically by adsorbed gas molecules which serve as charge carrier donors or acceptors to modulate the local carrier concentration of graphene. Cheng and Goda [63] conducted a graphene-based waveguide to measure NO2 gas concentration based on germanium and silicon substrates, respectively, as shown in Figure 15, where sensitivity was about 20 times higher than that of the graphene-covered microfiber sensor. Li et al. [64] demonstrated a single graphene-based waveguide which simultaneously provides optical modulation and photodetection. For developing sensitive photonic gas sensors, it is important to consider the interaction of propagating light in the waveguide to the top graphene layer. Xiang's group [65, 66] conducted a series work on graphene waveguide bio-sensors, by coupling graphene surface plasmon polaritons (SPPs) and planar waveguides to realize the ultrasensitive response. The SPPs produced by graphene could be used for bio-sensors since the SPPs are extremely sensitive to changes in the dielectric constant; even small changes in molecular density could be detected [67]. Graphene-based waveguides could overcome the shortcomings in traditional bio-sensors such as low speed, more time, and insufficient sensitivity.

Figure 15. Design of graphene on silicon (GoG) and suspended membrane slot waveguide (SMSW) Bragg grating gas sensor in comparison with the graphene-covered microfiber Bragg grating gas sensor. The calculated 3 dB bandwidth of the proposed GoS-SMSW Bragg gratings as a function of the NO2 gas concentration was shown [63].

#### 3.4. Other waveguides

However, it's difficult to fabricate these materials which are challenging to operate at room temperature till now. Choi's group [61] integrated graphene with a Bi2Se3 heterostructure, in which graphene functioned as high mobility charge transport layers and Bi2Se3 functioned as a broadband IR absorber supplying holes in graphene. The graphene-Bi2Se3 structure showed

Graphene is used in sensors, thanks to its unique electric properties, which show great potential in chemical or biology sensors. Among all varieties of chemical gas sensors, photonics gas sensors have advantages because of their high sensitivity and stability. Graphene plays a critical role in gas sensing due to the sensitivity of carried density to environment [62]. Graphene's conductivity can be changed drastically by adsorbed gas molecules which serve as charge carrier donors or acceptors to modulate the local carrier concentration of graphene. Cheng and Goda [63] conducted a graphene-based waveguide to measure NO2 gas concentration based on germanium and silicon substrates, respectively, as shown in Figure 15, where sensitivity was about 20 times higher than that of the graphene-covered microfiber sensor. Li et al. [64] demonstrated a single graphene-based waveguide which simultaneously provides optical modulation and photodetection. For developing sensitive photonic gas sensors, it is important to consider the interaction of propagating light in the waveguide to the top graphene layer. Xiang's group [65, 66] conducted a series work on graphene waveguide bio-sensors, by coupling graphene surface plasmon polaritons (SPPs) and planar waveguides to realize the ultrasensitive response. The SPPs produced by graphene could be used for bio-sensors since the SPPs are extremely sensitive to changes in the dielectric constant; even small changes in molecular density could be detected [67]. Graphene-based waveguides could overcome the shortcomings in traditional bio-sensors

Figure 15. Design of graphene on silicon (GoG) and suspended membrane slot waveguide (SMSW) Bragg grating gas sensor in comparison with the graphene-covered microfiber Bragg grating gas sensor. The calculated 3 dB bandwidth of

the proposed GoS-SMSW Bragg gratings as a function of the NO2 gas concentration was shown [63].

broadband absorption and high-intensity response at room temperature.

3.3. Graphene-based waveguide sensors

146 Emerging Waveguide Technology

such as low speed, more time, and insufficient sensitivity.

Usually, a fixed optical device works only on one particular polarization state, either the TE or the TM polarization state [68]. By coating silicon waveguides with graphene, a versatile polarizer works in two operation modes, which were based on the different effective mode index variations [69].

Waveguide integrated with graphene has nonlinear parameters which depend strongly on the Fermi level of graphene. It has been demonstrated that Integrating graphene with slot waveguide would benefit non-linear properties, owing to the interaction enhancement between graphene and incident electromagnetics [70].

When integrating graphene with nonlinear substrates on one or both sides, the surface plasmons' (SPs) localization length increased while their propagation length (PL) remained unchanged compared with the typical graphene waveguide [71, 72].

### 4. Conclusion

In conclusion, due to its unique electric and electromagnetic properties, graphene acts as a promising candidate for photonics and communication component of high performance. Especially, integrating graphene with waveguides makes it possible to overcome shortcomings such as limited bandwidth, inevitable cross-talk, high energy consumption, and expensive cost in conventional devices. Furthermore, the gapless linear dispersion of Dirac fermions makes it possible for graphene integrated with other substrates to formulate modulators, polarizers, broadband waveguides, photodetectors, bio-sensors, and so on. And the permittivity of graphene-based waveguides could be calculated based on Maxwell's function, even integrating with nonlinear substrates. The application of graphene waveguides expands the broadband range, from 300 nm to 6 μm or even longer. By tuning the carrier density of graphene through external gate voltage, chemical doping, or optical excitation, the relative complex permittivity of graphene is tuned. Thus, graphene waveguide modulators could be formed which adjusts absorption and modulation depth and so on. Besides, graphene waveguides show potential in fast and high-response detection and chemical sensing. The recent development in graphene synthesis and photonics components' fabrication technique ensures the compatibility in the integrated electronics platform, which shows a bright prospect in the near future.

### Author details

Xianglian Song, Xiaoyu Dai and Yuanjiang Xiang\*

\*Address all correspondence to: xiangyuanjiang@126.com

International Collaborative Laboratory of 2D Materials for Optoelectronic Science and Technology of Ministry of Education, College of Optoelectronic Engineering, Shenzhen University, China

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**Chapter 8**

Provisional chapter

**Lithium Niobate Optical Waveguides and**

Lithium niobate has attracted much attention since the 1970s due to its capacity to modify the light by means of an electric control. In this chapter, we review the evolution of electrooptical (EO) lithium niobate waveguides throughout the years, from Ti-indiffused waveguides to photonic crystals. The race toward ever smaller EO components with ever-lower optical losses and power consumption has stimulated numerous studies, the challenge consisting of strongly confining the light while preserving low losses. We show how waveguides have evolved toward ridges or thin film-based microguides to increase the EO efficiency and reduce the driving voltage. In particular, a focus is made on an easy-toimplement technique using a circular precision saw to produce thin ridge waveguides or

DOI: 10.5772/intechopen.76798

Keywords: integrated optics, LiNbO3, electro-optics, optical grade dicing, photonics

The electric control of light has fascinated people since the advent of electricity. The advent of cleanroom technologies in the early 1940s and the advances in fabrication technologies in the 1950s and 1960s have progressively opened up pathways toward integrated optics, offering the possibility of guiding light in small devices while controlling its flux. In this context, lithium niobate, also named LiNbO3, has always played a prominent role. Indeed its refractive index—which governs the speed of light—is sensitive to electrical signals, thanks to its

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Lithium Niobate Optical Waveguides and

Nadège Courjal, Maria-Pilar Bernal, Alexis Caspar,

Nadège Courjal, Maria-Pilar Bernal, Alexis Caspar,

Ludovic Gauthier-Manuel and Miguel Suarez

Ludovic Gauthier-Manuel and Miguel Suarez

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

suspended membranes with low losses.

**Microwaveguides**

Microwaveguides

Gwenn Ulliac, Florent Bassignot,

Gwenn Ulliac, Florent Bassignot,

http://dx.doi.org/10.5772/intechopen.76798

Abstract

1. Introduction


#### **Chapter 8** Provisional chapter

#### **Lithium Niobate Optical Waveguides and Microwaveguides** Lithium Niobate Optical Waveguides and Microwaveguides

DOI: 10.5772/intechopen.76798

Nadège Courjal, Maria-Pilar Bernal, Alexis Caspar, Gwenn Ulliac, Florent Bassignot, Ludovic Gauthier-Manuel and Miguel Suarez Nadège Courjal, Maria-Pilar Bernal, Alexis Caspar, Gwenn Ulliac, Florent Bassignot, Ludovic Gauthier-Manuel and Miguel Suarez

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.76798

#### Abstract

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[65] Ruan B, Guo J, Wu L. Ultrasensitive terahertz biosensors based on Fano resonance of a graphene/waveguide hybrid structure. Sensors. 2017;17:1924. DOI: 10.3390/s17081924 [66] Wu L, Guo J, Xu H. Ultrasensitive biosensors based on long-range surface plasmon polariton and dielectric waveguide modes. Photonics Research. 2016;4:262-266. DOI:

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detectors. Nanoscale. 2016;8:13206-13211. DOI: 10.1039/C6NR03122F

of Applied Physics. 2017;122:083101. DOI: 10.1063/1.4991674

tions. Nature Photonics. 2010;4:297-301. DOI: 10.1038/nphoton.2010.40

488. DOI: 10.1021/acsphotonics.6b00972

snb.2012.11.092

152 Emerging Waveguide Technology

nl500712u

10.1364/PRJ.4.000262

10.1109/LPT.2015.2408375

083104. DOI: 10.1063/1.4865435

2015;117:213105. DOI: 10.1063/1.4922124

Lithium niobate has attracted much attention since the 1970s due to its capacity to modify the light by means of an electric control. In this chapter, we review the evolution of electrooptical (EO) lithium niobate waveguides throughout the years, from Ti-indiffused waveguides to photonic crystals. The race toward ever smaller EO components with ever-lower optical losses and power consumption has stimulated numerous studies, the challenge consisting of strongly confining the light while preserving low losses. We show how waveguides have evolved toward ridges or thin film-based microguides to increase the EO efficiency and reduce the driving voltage. In particular, a focus is made on an easy-toimplement technique using a circular precision saw to produce thin ridge waveguides or suspended membranes with low losses.

Keywords: integrated optics, LiNbO3, electro-optics, optical grade dicing, photonics

#### 1. Introduction

The electric control of light has fascinated people since the advent of electricity. The advent of cleanroom technologies in the early 1940s and the advances in fabrication technologies in the 1950s and 1960s have progressively opened up pathways toward integrated optics, offering the possibility of guiding light in small devices while controlling its flux. In this context, lithium niobate, also named LiNbO3, has always played a prominent role. Indeed its refractive index—which governs the speed of light—is sensitive to electrical signals, thanks to its

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

electro-optical (EO) properties [1]. Hence, LiNbO3 is often used when light modulation is required, as for example in fiber optic-based telecommunication systems.

from the intrinsic refractive index ni and the electrooptic tensor [r] by using Eq.(1) in the

The non-zero electrooptical coefficients r13, r51, r33, and r22 of the [r] tensor are summarized in

As r<sup>33</sup> is the highest electro-optic coefficient of the material, the most efficient EO configuration is achieved when both the electric field and the optical polarization are parallel with the third crystalline axis, that is, the Z-axis. This can be accomplished either by X-cut Y-propagating or by Z-cut waveguides. In what follows, we will mainly focus on X-cut waveguides (see

In an X-cut Y-propagating waveguide, the crystalline X-axis is vertical, the Z-axis is in the plan of the wafer, and a waveguide is implemented along the Y-axis. Coplanar electrodes are placed on both sides of the waveguide (see Figure 1), and the gap g between them is kept the same along the Y-axis. Therefore, a voltage applied on the electrodes generates an electric field over the optical guided mode, and this electric field is mainly horizontal and oriented along the

r<sup>13</sup> = 8.6 pm/V r<sup>51</sup> = 28 pm/V r<sup>33</sup> = 30.8 pm/V r<sup>22</sup> = 3.4 pm/V no = 2.210 ne = 2.138

Figure 1. Standard configurations of integrated LiNbO3 optical modulators in a X-cut substrate. (a) Phase modulator. (b) Mach-Zehnder modulator with push-pull electrodes: The signal electrode is placed between the two arms of the integrated interferometer. The buffer klayer helps preventing from optical leakage in the electrodes, and it also helps to

j¼1

rijEj (1)

Lithium Niobate Optical Waveguides and Microwaveguides

http://dx.doi.org/10.5772/intechopen.76798

155

1 n0 2 i

Table 1 at 1550 nm wavelength for LiNbO3 crystal substrates.

Figure 1), which show better thermal stability than Z-cut ones.

Z-axis, as represented by the horizontal white arrows in Figure 2(a).

Table 1. Electro-optical and optical properties of LiNbO3 at 1550 nm wavelength [13].

achieve index matching between the electrical and optical propagating waves.

ne and n<sup>o</sup> denote the extraordinary and ordinary index of the material.

¼ 1 n2 i þ<sup>X</sup> 3

crystalline coordinate system:

As compared with semiconductors modulating electrically the absorption of light through the Franz-Keldysh effect, LiNbO3-based modulators can change light intensity without any perturbation on the phase, that is, without chirp. Therefore, despite the great success of semiconductors for short-range telecommunication systems, the zero-chirp modulation provided by LiNbO3-based components is still privileged when high-bit rate optical signals have to propagate through thousands of kilometers [2].

In comparison with polymers [3] or other ferroelectrics such as SBN [4], KTN [5], BaTiO3 [6], which show even higher EO sensitivity than LiNbO3, the material benefits from technological maturity based on Czochralski's process, so that numerous suppliers can be found around the world. Moreover, its physical properties are compatible with cleanroom fabrication processes. For example, its high Curie temperature (~ 1200�C) preserves the EO properties even during annealing steps, which is not the case for materials like SBN or KTN. Additionally, LiNbO3 offers a wide transparency band [340–4.6 μm] which opens the range to applications from visible to mid-infrared. The low absorption losses (< 0.15%/cm at 1.06 μm) and its weak optical dispersion in the transparency band [7] also contribute to its success. As a result, lithium niobate has become indispensable for demanding applications such as broadband modulation for long-haul high-bit-rate optical telecommunication systems [2, 8], electromagnetic sensors [9], precision gyroscopes [10], and astronomy [11]. For each application, the challenge is to provide integrated configurations that are easy to implement, of low loss, low in energy consumption, and, if possible, compact. The basic element, namely the optical waveguide, is essential to meet these specifications.

First, we will provide some reminders about the classic electro-optical configurations. Then we will see the evolutions of LiNbO3 optical waveguides from their first appearance in the 1970s to recent evolutions. Finally, we will show how nanoscale structuring can open up new perspectives for the material.

### 2. Electro-optical configurations in lithium niobate

The technological evolutions concerning electrooptical LiNbO3 waveguides are oriented toward ever-lower propagation losses and lower coupling losses with fibers, while also seeking ever higher EO efficiency. The common specifications are summarized in Refs. [10, 12]. In what follows, we will take particular interest in optical losses and EO efficiency. Beforehand, we provide a quick reminder of the exploitation of the EO effect in lithium niobate.

LiNbO3 EO modulators exploit the classic Pockels effect, which corresponds to a linear change of the refractive index as a function of the applied voltage. This index modification is due to the

relative displacement of charges in the presence of an electric field E ! ð Þ E1; E2; E<sup>3</sup> , which induces a macroscopic polarization inside the material. The modified index n'<sup>i</sup> can be deduced from the intrinsic refractive index ni and the electrooptic tensor [r] by using Eq.(1) in the crystalline coordinate system:

$$\frac{1}{n\_i^{n\_i^2}} = \frac{1}{n\_i^2} + \sum\_{j=1}^3 r\_{ij} E\_j \tag{1}$$

The non-zero electrooptical coefficients r13, r51, r33, and r22 of the [r] tensor are summarized in Table 1 at 1550 nm wavelength for LiNbO3 crystal substrates.

As r<sup>33</sup> is the highest electro-optic coefficient of the material, the most efficient EO configuration is achieved when both the electric field and the optical polarization are parallel with the third crystalline axis, that is, the Z-axis. This can be accomplished either by X-cut Y-propagating or by Z-cut waveguides. In what follows, we will mainly focus on X-cut waveguides (see Figure 1), which show better thermal stability than Z-cut ones.

In an X-cut Y-propagating waveguide, the crystalline X-axis is vertical, the Z-axis is in the plan of the wafer, and a waveguide is implemented along the Y-axis. Coplanar electrodes are placed on both sides of the waveguide (see Figure 1), and the gap g between them is kept the same along the Y-axis. Therefore, a voltage applied on the electrodes generates an electric field over the optical guided mode, and this electric field is mainly horizontal and oriented along the Z-axis, as represented by the horizontal white arrows in Figure 2(a).


ne and n<sup>o</sup> denote the extraordinary and ordinary index of the material.

electro-optical (EO) properties [1]. Hence, LiNbO3 is often used when light modulation is

As compared with semiconductors modulating electrically the absorption of light through the Franz-Keldysh effect, LiNbO3-based modulators can change light intensity without any perturbation on the phase, that is, without chirp. Therefore, despite the great success of semiconductors for short-range telecommunication systems, the zero-chirp modulation provided by LiNbO3-based components is still privileged when high-bit rate optical signals have to

In comparison with polymers [3] or other ferroelectrics such as SBN [4], KTN [5], BaTiO3 [6], which show even higher EO sensitivity than LiNbO3, the material benefits from technological maturity based on Czochralski's process, so that numerous suppliers can be found around the world. Moreover, its physical properties are compatible with cleanroom fabrication processes. For example, its high Curie temperature (~ 1200�C) preserves the EO properties even during annealing steps, which is not the case for materials like SBN or KTN. Additionally, LiNbO3 offers a wide transparency band [340–4.6 μm] which opens the range to applications from visible to mid-infrared. The low absorption losses (< 0.15%/cm at 1.06 μm) and its weak optical dispersion in the transparency band [7] also contribute to its success. As a result, lithium niobate has become indispensable for demanding applications such as broadband modulation for long-haul high-bit-rate optical telecommunication systems [2, 8], electromagnetic sensors [9], precision gyroscopes [10], and astronomy [11]. For each application, the challenge is to provide integrated configurations that are easy to implement, of low loss, low in energy consumption, and, if possible, compact. The basic element, namely the optical waveguide, is

First, we will provide some reminders about the classic electro-optical configurations. Then we will see the evolutions of LiNbO3 optical waveguides from their first appearance in the 1970s to recent evolutions. Finally, we will show how nanoscale structuring can open up new

The technological evolutions concerning electrooptical LiNbO3 waveguides are oriented toward ever-lower propagation losses and lower coupling losses with fibers, while also seeking ever higher EO efficiency. The common specifications are summarized in Refs. [10, 12]. In what follows, we will take particular interest in optical losses and EO efficiency. Beforehand,

LiNbO3 EO modulators exploit the classic Pockels effect, which corresponds to a linear change of the refractive index as a function of the applied voltage. This index modification is due to the

induces a macroscopic polarization inside the material. The modified index n'<sup>i</sup> can be deduced

!

ð Þ E1; E2; E<sup>3</sup> , which

we provide a quick reminder of the exploitation of the EO effect in lithium niobate.

relative displacement of charges in the presence of an electric field E

required, as for example in fiber optic-based telecommunication systems.

propagate through thousands of kilometers [2].

154 Emerging Waveguide Technology

essential to meet these specifications.

2. Electro-optical configurations in lithium niobate

perspectives for the material.

Table 1. Electro-optical and optical properties of LiNbO3 at 1550 nm wavelength [13].

Figure 1. Standard configurations of integrated LiNbO3 optical modulators in a X-cut substrate. (a) Phase modulator. (b) Mach-Zehnder modulator with push-pull electrodes: The signal electrode is placed between the two arms of the integrated interferometer. The buffer klayer helps preventing from optical leakage in the electrodes, and it also helps to achieve index matching between the electrical and optical propagating waves.

Figure 2. Cross-sections of optical waveguides. The electric optical guided mode and the applied electric field are calculated by F.E.M. (comsol® software) and represented with a color map and arrows respectively. (a) Standard Tiindiffused waveguide with a gap g = 6 μm between electrodes, a thickness buffer layer t = 200 nm, and a w = 6 μm wide titanium rib. (b) Same waveguide as in (a), but with a ridge height h = 3 μm. (c) Adhered ridge waveguide with a width of 6 μm, a ridge height of 6 μm, a LiNbO3 layer thickness of 6 μm; and t = 200 nm. (d) High aspect ratio ridge waveguide done by Ti-indiffusion. The ridge height is 19 μm and the width is 6 μm; t = 200 nm (a) Ti-indiffused waveguide, (b) standard ridge waveguide, (c) adhered ridge waveguide, (d) high-aspect ratio ridge waveguide.

In these conditions, the index modification induced by the voltage on the ordinary index and extraordinary index is evaluated through Eq. (2):

$$n'\_o = n\_o - \frac{n\_o^3 \cdot r\_{13} \cdot \Gamma \cdot V}{2 \cdot \text{g}} \text{ and } n'\_e = n\_e - \frac{n\_e^3 \cdot r\_{33} \cdot \Gamma \cdot V}{2 \cdot \text{g}} \tag{2}$$

where Γ is the electro-optic overlap coefficient that takes into account the non-uniform behavior of the electric field over the optical mode cross-section:

$$
\Gamma = \frac{\iint \varepsilon\_Z^2 \cdot E\_3 \cdot d\mathcal{S}}{\iint \varepsilon\_Z^2 \cdot d\mathcal{S}} \cdot \frac{\mathcal{S}}{V} \tag{3}
$$

other words, Γ is the average electric field seen by the optical electric field and normalized by

Γ<sup>1</sup> and Γ<sup>2</sup> denote the electro-optic overlap coefficients in arm #1 and arm#2 of the Mach-Zehnder device, respectively, when the polarization is parallel with (OZ). In a symmetric X-cut standard Mach-Zehnder configuration, Γ<sup>1</sup> = � Γ2. Values

Configuration Definition of the phase induced shift Expression of V<sup>π</sup>

<sup>e</sup> <sup>∙</sup>r<sup>33</sup> <sup>L</sup> <sup>λ</sup>∙<sup>g</sup>

<sup>2</sup>∙<sup>g</sup> ð Þ <sup>Γ</sup><sup>1</sup> � <sup>Γ</sup><sup>2</sup> <sup>L</sup> <sup>λ</sup>∙<sup>g</sup>

Γ∙n<sup>3</sup> <sup>e</sup> ð Þ <sup>∙</sup>r<sup>33</sup> <sup>L</sup>

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157

n3 <sup>e</sup> ∙r<sup>33</sup> ∙ð Þ Γ1�Γ<sup>2</sup> L

Lithium Niobate Optical Waveguides and Microwaveguides

<sup>λ</sup>∙<sup>g</sup> <sup>Γ</sup>∙n<sup>3</sup>

λ n3 <sup>e</sup> ∙r<sup>33</sup> ∙V

Two basic configurations can be distinguished, the first one being the phase modulator for which only the phase of the optical signal is modified by the electric field (see Figure 1(a)) and the second one being the intensity modulator which exploits the interference between two signals having different paths in the material, such as the Mach-Zehnder illustrated schematically in Figure 1(b). A precise evaluation of the electro-optic efficiency is given by the halfwave voltage, which is the voltage needed to obtain an induced phase shift of π. In the case of a phase modulator, the phase shift φ denotes the phase accumulated through the electrode length L, and its expression is summarized in row #1 of Table 2. In a Mach-Zehnder intensity modulator, the phase shift Δφ corresponds to the phase difference induced between the two

From Table 2 we can infer that the critical parameters for obtaining a low driving voltage V<sup>π</sup> and therefore low power consumption are the gap g, the active length L, and the electro-optic overlap coefficient Γ. In standard modulators, g ranges from 6 to 30 μm, depending on the targeted performances in terms of impedance, bandwidth, and voltage, and Γ is usually lower than 0.5, which is due to the weak overlap between the electrical field and optical guided mode. As an example, Γ = 32% in the X-cut Ti-indiffused phase modulator with g = 6 μm and with a δ = 0.2 μm silica buffer layer thickness calculated in Figure 4(a). This means that an electrode longer than L = 2 cm is needed to obtain a driving voltage lower than 5 V. So the achievement of compact modulators with low driving voltage requires an increase of Γ.

This latter is controlled by the fabrication technologies enabling more or less tight light confinement. In the following paragraphs we describe the evolutions from the first LiNbO3

Titanium diffusion and proton exchange (PE) constitute the two main commercial techniques

V/g. Noteworthy, Γ can be larger than 1 in specific cases.

of Γ are strongly dependent on the electrode and buffer layer thickness.

Table 2. Evaluation of the electro-optic efficiency in classic configurations.

Phase modulation Figure 1(a) <sup>φ</sup> <sup>¼</sup> <sup>π</sup>∙<sup>V</sup>

Intensity modulation Figure 1(b) <sup>Δ</sup><sup>φ</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup>

arms at their output (see row #2 of Table 2).

guides to the recent confined ones with increased Γ.

for the manufacture of LiNbO3-based optical waveguides.

3. Standard waveguides

ε<sup>z</sup> denotes the third component of the electric optical field ε ! in the crystalline coordinate system, and E3 is the electric field induced by the voltage V applied on the electrodes. So in


Γ<sup>1</sup> and Γ<sup>2</sup> denote the electro-optic overlap coefficients in arm #1 and arm#2 of the Mach-Zehnder device, respectively, when the polarization is parallel with (OZ). In a symmetric X-cut standard Mach-Zehnder configuration, Γ<sup>1</sup> = � Γ2. Values of Γ are strongly dependent on the electrode and buffer layer thickness.

Table 2. Evaluation of the electro-optic efficiency in classic configurations.

other words, Γ is the average electric field seen by the optical electric field and normalized by V/g. Noteworthy, Γ can be larger than 1 in specific cases.

Two basic configurations can be distinguished, the first one being the phase modulator for which only the phase of the optical signal is modified by the electric field (see Figure 1(a)) and the second one being the intensity modulator which exploits the interference between two signals having different paths in the material, such as the Mach-Zehnder illustrated schematically in Figure 1(b). A precise evaluation of the electro-optic efficiency is given by the halfwave voltage, which is the voltage needed to obtain an induced phase shift of π. In the case of a phase modulator, the phase shift φ denotes the phase accumulated through the electrode length L, and its expression is summarized in row #1 of Table 2. In a Mach-Zehnder intensity modulator, the phase shift Δφ corresponds to the phase difference induced between the two arms at their output (see row #2 of Table 2).

From Table 2 we can infer that the critical parameters for obtaining a low driving voltage V<sup>π</sup> and therefore low power consumption are the gap g, the active length L, and the electro-optic overlap coefficient Γ. In standard modulators, g ranges from 6 to 30 μm, depending on the targeted performances in terms of impedance, bandwidth, and voltage, and Γ is usually lower than 0.5, which is due to the weak overlap between the electrical field and optical guided mode. As an example, Γ = 32% in the X-cut Ti-indiffused phase modulator with g = 6 μm and with a δ = 0.2 μm silica buffer layer thickness calculated in Figure 4(a). This means that an electrode longer than L = 2 cm is needed to obtain a driving voltage lower than 5 V. So the achievement of compact modulators with low driving voltage requires an increase of Γ.

This latter is controlled by the fabrication technologies enabling more or less tight light confinement. In the following paragraphs we describe the evolutions from the first LiNbO3 guides to the recent confined ones with increased Γ.

### 3. Standard waveguides

In these conditions, the index modification induced by the voltage on the ordinary index and

Figure 2. Cross-sections of optical waveguides. The electric optical guided mode and the applied electric field are calculated by F.E.M. (comsol® software) and represented with a color map and arrows respectively. (a) Standard Tiindiffused waveguide with a gap g = 6 μm between electrodes, a thickness buffer layer t = 200 nm, and a w = 6 μm wide titanium rib. (b) Same waveguide as in (a), but with a ridge height h = 3 μm. (c) Adhered ridge waveguide with a width of 6 μm, a ridge height of 6 μm, a LiNbO3 layer thickness of 6 μm; and t = 200 nm. (d) High aspect ratio ridge waveguide done by Ti-indiffusion. The ridge height is 19 μm and the width is 6 μm; t = 200 nm (a) Ti-indiffused waveguide,

and n<sup>0</sup>

where Γ is the electro-optic overlap coefficient that takes into account the non-uniform behav-

system, and E3 is the electric field induced by the voltage V applied on the electrodes. So in

<sup>Z</sup>∙E3∙dS ÐÐ ε<sup>2</sup> <sup>Z</sup>∙dS <sup>∙</sup>

ÐÐ ε<sup>2</sup>

<sup>e</sup> <sup>¼</sup> ne � <sup>n</sup><sup>3</sup>

g

<sup>e</sup> ∙r33∙Γ∙V

<sup>2</sup>∙<sup>g</sup> (2)

! in the crystalline coordinate

<sup>V</sup> (3)

<sup>o</sup> ∙r13∙Γ∙V 2∙g

(b) standard ridge waveguide, (c) adhered ridge waveguide, (d) high-aspect ratio ridge waveguide.

Γ ¼

extraordinary index is evaluated through Eq. (2):

156 Emerging Waveguide Technology

n0

<sup>o</sup> <sup>¼</sup> no � <sup>n</sup><sup>3</sup>

ε<sup>z</sup> denotes the third component of the electric optical field ε

ior of the electric field over the optical mode cross-section:

Titanium diffusion and proton exchange (PE) constitute the two main commercial techniques for the manufacture of LiNbO3-based optical waveguides.

#### 3.1. Titanium-indiffused waveguides

Since their discovery in 1974 by Schmidt et al. [14], Ti-indiffused optical waveguides have maintained continuous interest for commercial applications in high-bit-rate data-processing systems. More recently, they have attracted attention for mid-infrared applications dedicated to astrophysics [11, 15] or spectrometry [16].

proton exchange zones may consist of several phase multilayers, which deteriorate their electro-optical performance. Several techniques have been successfully developed to design the index profile and to restitute the phase and optical properties. The mostly used techniques

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• Annealed-PE [23], where annealing is performed subsequently to PE for obtaining mono-

• Soft-PE [24], where the proton exchange is done in soft conditions in buffered melts (i.e., lithium benzoate added to the benzoic acid) so that the optimal mono-phase waveguides

• Reversed-PE, where the PE waveguides are immersed in a eutectic mixture of LiNO3, KNO3, and NaNO3 to achieve buried waveguides with minimal loss connections with

Similar to Ti-indiffused waveguides, PE-based waveguides can yield very low insertion losses [25]. If light confinement is slightly tighter than their Ti-indiffused counterparts, PE-based waveguides are however not confined enough to reduce significantly the active length of EO

This observation led to strong efforts in the 1980s to obtain waveguides with both tight light confinement and low insertion losses. The first proposed solutions were based on ridge wave-

As represented in Figure 2(b)–(d), a ridge waveguide is an optical waveguide etched on both sides: the lateral confinement is ensured by a step index between LiNbO3 and air. This configuration was firstly proposed by Kaminow et al. more than 40 years ago [26] to achieve a lateral confinement in planar Ti-outdiffused waveguides. Ridge waveguides are still a hot topic in

The first generation of ridge waveguides was produced through Ti indiffusion or proton exchange techniques followed by dry or wet etching. The wafer thickness was about 500 μm and the ridge depth was typically lower than 10 μm. If their first apparition was in 1974, their attractiveness dates from the mid-1990s, when ridges were identified as the best solution to achieve both large bandwidth and impedance matching in EO Mach-Zehnder interferometers for long-haul high-bit-rate telecommunication systems [27]. Noteworthy, the moderate depth of the ridges was compatible with standard metal deposition techniques, which was convenient for the electrode deposition. Hence, ridge-based modulators with bandwidth as high as 100 GHz and driving voltage of 5.1 V were demonstrated [28] with 2-cm-long electrodes in

research, with applications and manufacturing techniques evolving over the years.

4.1. Ridge made in standard waveguides by wet etching or plasma etching

phase waveguides with restored EO properties;

are obtained in a one-step process;

optical fibers.

4. Ridge waveguides

modulators.

guides.

are:

Titanium-indiffused waveguides are fabricated through the diffusion of titanium ribs at a high temperature (~1000�C). The typical width of the Ti ribs is 6–7 μm and the thickness ranges from 80 to 100 nm for operations at 1550 nm wavelength. Extensive description of the process is given by Burns et al. [17]. Ti-indiffused waveguides have the advantages of low insertion losses (1 dB by Ramaswamy [18]). Their ability to guide two polarizations is also a specificity of interest, and they are good candidates over a very large spectral bandwidth from 1.0 to 4.7 μm.

The particularity of Ti-indiffused waveguides is their weak light confinement. Typically, the full width at half maximum (FWHM) of a standard X-cut Ti-indiffused waveguide working at 1550 nm is 7.5 μm in the horizontal direction and 4.8 μm in the vertical direction. This specificity allows a large η overlap integral with single-mode fibers (SMF), expressed by Eq. (4):

$$\eta = \frac{\int\_{\mathcal{S}} \varepsilon\_{\text{fiber}}(\mathbf{x}, z) \cdot \varepsilon\_{\text{guda}}^{\*}(\mathbf{x}, z) \cdot d\mathcal{S}}{\left(\iint\_{\mathcal{S}} \varepsilon\_{\text{fiber}}(\mathbf{x}, z) \cdot \varepsilon\_{\text{fiber}}^{\*}(\mathbf{x}, z) \cdot d\mathcal{S}\right) \left(\iint\_{\mathcal{S}} \varepsilon\_{\text{guda}}(\mathbf{x}, z) \cdot \varepsilon\_{\text{guda}}^{\*}(\mathbf{x}, z) \cdot d\mathcal{S}\right)} \tag{4}$$

where Efiber(x,z) denotes the spatial distribution of the optical guided electric field within the fiber, and Eguide(x,z) is the spatial distribution of the optical guided electric field within the optical waveguide. η can be larger than 85% in Ti-indiffused waveguides, which explains their low coupling losses with fibers and consequently their low insertion losses.

Although attractive for obtaining low losses, the weak light confinement is limiting for EO interaction, which is illustrated by a small Γ of 32% in the example of Figure 2(a). Therefore, Tiindiffused waveguides are not the best candidates when compact and efficient EO modulators are required and alternative techniques have been considered.

#### 3.2. Proton exchange-based waveguides

Proton exchange guides are well appreciated for non-linear periodically poled LiNbO3 devices [19], due notably to a photorefractive threshold higher than Ti-indiffused waveguides [20]. The polarizing nature of PE waveguides also makes them prime waveguides for applications where polarization control is crucial. Fiber-optics gyroscopes [21] and polarizing phase modulators are particularly concerned [10].

Proton exchange [22] is a low-temperature process (� 120–250�C) whereby Li ions from the LiNbO3 wafer are exchanged with protons from an acid bath. Exchanged layers exhibit an increase in the extraordinary index and a slight decrease in the ordinary index, which is the origin of the polarizing nature of the PE-based waveguides. In the most general case, the proton exchange zones may consist of several phase multilayers, which deteriorate their electro-optical performance. Several techniques have been successfully developed to design the index profile and to restitute the phase and optical properties. The mostly used techniques are:


Similar to Ti-indiffused waveguides, PE-based waveguides can yield very low insertion losses [25]. If light confinement is slightly tighter than their Ti-indiffused counterparts, PE-based waveguides are however not confined enough to reduce significantly the active length of EO modulators.

This observation led to strong efforts in the 1980s to obtain waveguides with both tight light confinement and low insertion losses. The first proposed solutions were based on ridge waveguides.

### 4. Ridge waveguides

3.1. Titanium-indiffused waveguides

158 Emerging Waveguide Technology

η ¼

3.2. Proton exchange-based waveguides

lators are particularly concerned [10].

ÐÐ

<sup>S</sup>εfiberð Þ <sup>x</sup>; <sup>z</sup> <sup>∙</sup>ε<sup>∗</sup>

are required and alternative techniques have been considered.

4.7 μm.

Eq. (4):

to astrophysics [11, 15] or spectrometry [16].

Since their discovery in 1974 by Schmidt et al. [14], Ti-indiffused optical waveguides have maintained continuous interest for commercial applications in high-bit-rate data-processing systems. More recently, they have attracted attention for mid-infrared applications dedicated

Titanium-indiffused waveguides are fabricated through the diffusion of titanium ribs at a high temperature (~1000�C). The typical width of the Ti ribs is 6–7 μm and the thickness ranges from 80 to 100 nm for operations at 1550 nm wavelength. Extensive description of the process is given by Burns et al. [17]. Ti-indiffused waveguides have the advantages of low insertion losses (1 dB by Ramaswamy [18]). Their ability to guide two polarizations is also a specificity of interest, and they are good candidates over a very large spectral bandwidth from 1.0 to

The particularity of Ti-indiffused waveguides is their weak light confinement. Typically, the full width at half maximum (FWHM) of a standard X-cut Ti-indiffused waveguide working at 1550 nm is 7.5 μm in the horizontal direction and 4.8 μm in the vertical direction. This specificity allows a large η overlap integral with single-mode fibers (SMF), expressed by

<sup>S</sup>εfiberð Þ <sup>x</sup>; <sup>z</sup> <sup>∙</sup>ε<sup>∗</sup>

where Efiber(x,z) denotes the spatial distribution of the optical guided electric field within the fiber, and Eguide(x,z) is the spatial distribution of the optical guided electric field within the optical waveguide. η can be larger than 85% in Ti-indiffused waveguides, which explains their

Although attractive for obtaining low losses, the weak light confinement is limiting for EO interaction, which is illustrated by a small Γ of 32% in the example of Figure 2(a). Therefore, Tiindiffused waveguides are not the best candidates when compact and efficient EO modulators

Proton exchange guides are well appreciated for non-linear periodically poled LiNbO3 devices [19], due notably to a photorefractive threshold higher than Ti-indiffused waveguides [20]. The polarizing nature of PE waveguides also makes them prime waveguides for applications where polarization control is crucial. Fiber-optics gyroscopes [21] and polarizing phase modu-

Proton exchange [22] is a low-temperature process (� 120–250�C) whereby Li ions from the LiNbO3 wafer are exchanged with protons from an acid bath. Exchanged layers exhibit an increase in the extraordinary index and a slight decrease in the ordinary index, which is the origin of the polarizing nature of the PE-based waveguides. In the most general case, the

fiberð Þ x; z ∙dS

guideð Þ x; z ∙dS

<sup>S</sup>εguideð Þ <sup>x</sup>; <sup>z</sup> <sup>∙</sup>ε<sup>∗</sup>

guideð Þ x; z ∙dS � � (4)

ÐÐ

low coupling losses with fibers and consequently their low insertion losses.

� � ÐÐ

As represented in Figure 2(b)–(d), a ridge waveguide is an optical waveguide etched on both sides: the lateral confinement is ensured by a step index between LiNbO3 and air. This configuration was firstly proposed by Kaminow et al. more than 40 years ago [26] to achieve a lateral confinement in planar Ti-outdiffused waveguides. Ridge waveguides are still a hot topic in research, with applications and manufacturing techniques evolving over the years.

#### 4.1. Ridge made in standard waveguides by wet etching or plasma etching

The first generation of ridge waveguides was produced through Ti indiffusion or proton exchange techniques followed by dry or wet etching. The wafer thickness was about 500 μm and the ridge depth was typically lower than 10 μm. If their first apparition was in 1974, their attractiveness dates from the mid-1990s, when ridges were identified as the best solution to achieve both large bandwidth and impedance matching in EO Mach-Zehnder interferometers for long-haul high-bit-rate telecommunication systems [27]. Noteworthy, the moderate depth of the ridges was compatible with standard metal deposition techniques, which was convenient for the electrode deposition. Hence, ridge-based modulators with bandwidth as high as 100 GHz and driving voltage of 5.1 V were demonstrated [28] with 2-cm-long electrodes in Mach-Zehnder modulators. The reported ridges were achieved through Ti indiffusion followed by selective dry etching with electron cyclotron resonance with argon and C2F6 gases.

waveguides are made simply through standard techniques (Ti indiffusion, proton exchange, ion implantation), and in a second step two grooves are diced along waveguide sides. If the machining parameters—such as translation and rotation speeds and nature and size of the blade—are properly chosen, the ridge patterns are diced and polished at the same time, which yield roughness lower than 20 nm and propagation losses lower than 0.1 dB/cm [37]. The resulting depths can be higher than 500 (see Figure 3(a)), but the preferred depth is between 10 and 50 μm [38] to obtain robust reproducible guided mode cross-sections that are indepen-

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The uniform deposition of a buffer layer over the vertical edges of the ridge can be accomplished by atomic layer deposition (ALD), followed by a two-step side deposition of electrodes [39]. A schematic overview of the resulting EO ridge is seen in Figure 4(a), while Figure 4(b) shows a standard electron microscope (SEM) image of the EO HAR ridge with ALD-deposited buffer layer.

Figure 3. SEM images of high aspect ratio ridge waveguides. Record aspect ratio with a depth of 526 μm for a width of

Figure 4. Ridge-based EO device. (a) Electro-optical ridge buffer layer with electrodes and spot-size-converters. (b) SEM

only 1 μm. (b) Optimal high aspect ratio waveguide, in term of robustness and losses [38].

image of an EO ridge with vertical buffer layer and electrodes.

dent on the ridge depth.

More generally, most of the reported dry etching techniques use fluorine-based gases to exploit their chemical reactivity with LiNbO3. Wet etching techniques associated with ion implantation, proton exchange, or domain inversion have also been widely studied [29, 30]. When followed by a subsequent step of annealing or Ti-in-diffusion at high temperature, the method is very efficient for the fabrication of ultra-smooth ridges with propagation losses as low as 0.05 dB/cm [31].

However, the etching step lasts for several hours to obtain depths of a few micrometers, and such small depths do not increase the EO efficiency significantly as compared with a standard waveguide. This is illustrated in Figure 2(a) and (b) where the increase of Γ is only 18% in the 3-μm-deep ridge, as compared with the Ti-indiffused waveguide. According to finite element method (FEM) calculations, the ridge depth needs to be higher than 8 μm (see Figure 2(d)) or the substrate needs to be thinned down to a few micrometers (see Figure 2(c)) to achieve an increase larger than 60% in EO efficiency. Therefore, alternative techniques have been proposed, based on mechanical machining, to produce more efficient ridges.

#### 4.2. Adhered ridge waveguides

Adhered ridge waveguides are made through mechanical processing. In a first step, the LiNbO3 wafer is bonded to another one (LiNbO3, LaTiO3, Si…), and then the wafer is thinned down to a few micrometers by lapping polishing. Finally grooves are inscribed inside the thinned wafer by precise dicing [32] or by dry etching [33]. The adhered ridge waveguide is the remaining matter between two grooves. The first adhered ridge waveguide [32] aimed at replacing PE-based waveguides whose mobile protons induce long-term degradation of the optical response. The main application was nonlinear frequency conversion.

Since then, adhered ridge waveguides have been widely employed still for nonlinear (NL) applications [34, 35]. The typical thickness of the LiNbO3 thinned layer ranges from 3 to 10 μm; the width of the ridge is typically 3–5 μm, and the depth is usually lower than the LiNbO3 layer thickness (see Figure 2(c)). The pigtailing is performed by using lens coupling and laser welding [34]. Such pigtailed modules are now commercialized [36].

As attractive as they may be for NL applications, adhered ridge waveguides do not attract much attention for electro-optical modulation. One of the reasons is their multimode behavior which limits the modulation contrast. A step has been put forward with high-aspect ratio ridge waveguides having spot-size-converters, allowing for low insertion losses and for the filtering of optical modes.

#### 4.3. High-aspect ratio ridge waveguides made by optical grade dicing

High-aspect ratio (HAR) ridge waveguides are ridges with depths larger than 10 μm (see Figure 2(d) and 3). Such depths are achieved by optical grade dicing, meaning that the substrate is diced and polished at the same time [37]. In a first step, optical channels or planar waveguides are made simply through standard techniques (Ti indiffusion, proton exchange, ion implantation), and in a second step two grooves are diced along waveguide sides. If the machining parameters—such as translation and rotation speeds and nature and size of the blade—are properly chosen, the ridge patterns are diced and polished at the same time, which yield roughness lower than 20 nm and propagation losses lower than 0.1 dB/cm [37]. The resulting depths can be higher than 500 (see Figure 3(a)), but the preferred depth is between 10 and 50 μm [38] to obtain robust reproducible guided mode cross-sections that are independent on the ridge depth.

Mach-Zehnder modulators. The reported ridges were achieved through Ti indiffusion followed by selective dry etching with electron cyclotron resonance with argon and C2F6 gases. More generally, most of the reported dry etching techniques use fluorine-based gases to exploit their chemical reactivity with LiNbO3. Wet etching techniques associated with ion implantation, proton exchange, or domain inversion have also been widely studied [29, 30]. When followed by a subsequent step of annealing or Ti-in-diffusion at high temperature, the method is very efficient for the fabrication of ultra-smooth ridges with propagation losses as low as

However, the etching step lasts for several hours to obtain depths of a few micrometers, and such small depths do not increase the EO efficiency significantly as compared with a standard waveguide. This is illustrated in Figure 2(a) and (b) where the increase of Γ is only 18% in the 3-μm-deep ridge, as compared with the Ti-indiffused waveguide. According to finite element method (FEM) calculations, the ridge depth needs to be higher than 8 μm (see Figure 2(d)) or the substrate needs to be thinned down to a few micrometers (see Figure 2(c)) to achieve an increase larger than 60% in EO efficiency. Therefore, alternative techniques have been pro-

Adhered ridge waveguides are made through mechanical processing. In a first step, the LiNbO3 wafer is bonded to another one (LiNbO3, LaTiO3, Si…), and then the wafer is thinned down to a few micrometers by lapping polishing. Finally grooves are inscribed inside the thinned wafer by precise dicing [32] or by dry etching [33]. The adhered ridge waveguide is the remaining matter between two grooves. The first adhered ridge waveguide [32] aimed at replacing PE-based waveguides whose mobile protons induce long-term degradation of the

Since then, adhered ridge waveguides have been widely employed still for nonlinear (NL) applications [34, 35]. The typical thickness of the LiNbO3 thinned layer ranges from 3 to 10 μm; the width of the ridge is typically 3–5 μm, and the depth is usually lower than the LiNbO3 layer thickness (see Figure 2(c)). The pigtailing is performed by using lens coupling and laser

As attractive as they may be for NL applications, adhered ridge waveguides do not attract much attention for electro-optical modulation. One of the reasons is their multimode behavior which limits the modulation contrast. A step has been put forward with high-aspect ratio ridge waveguides having spot-size-converters, allowing for low insertion losses and for the filtering

High-aspect ratio (HAR) ridge waveguides are ridges with depths larger than 10 μm (see Figure 2(d) and 3). Such depths are achieved by optical grade dicing, meaning that the substrate is diced and polished at the same time [37]. In a first step, optical channels or planar

posed, based on mechanical machining, to produce more efficient ridges.

optical response. The main application was nonlinear frequency conversion.

welding [34]. Such pigtailed modules are now commercialized [36].

4.3. High-aspect ratio ridge waveguides made by optical grade dicing

0.05 dB/cm [31].

160 Emerging Waveguide Technology

of optical modes.

4.2. Adhered ridge waveguides

The uniform deposition of a buffer layer over the vertical edges of the ridge can be accomplished by atomic layer deposition (ALD), followed by a two-step side deposition of electrodes [39]. A schematic overview of the resulting EO ridge is seen in Figure 4(a), while Figure 4(b) shows a standard electron microscope (SEM) image of the EO HAR ridge with ALD-deposited buffer layer.

Figure 3. SEM images of high aspect ratio ridge waveguides. Record aspect ratio with a depth of 526 μm for a width of only 1 μm. (b) Optimal high aspect ratio waveguide, in term of robustness and losses [38].

Figure 4. Ridge-based EO device. (a) Electro-optical ridge buffer layer with electrodes and spot-size-converters. (b) SEM image of an EO ridge with vertical buffer layer and electrodes.

Finite element method (FEM) simulations performed on HAR waveguides show that they are optimal in terms of EO efficiency, since the electric field is uniform over the optical waveguide. This is illustrated in Figure 3(d) with the white arrows representing the applied electric field. As a consequence, the electro-optical coefficient can be evaluated simply as a function of g:

$$\Gamma = \frac{\text{g} \cdot \varepsilon\_{diel}}{2 \cdot \varepsilon\_{LN} \cdot \delta + (\text{g} - \text{2} \cdot \delta) \cdot \varepsilon\_{diel}} \tag{5}$$

treatment with temperature ramp is employed to split the samples along the implanted He layer. Hence, a thin LiNbO3 layer bonded on another substrate remains. Afterward, annealing and chemical mechanical polishing are used to obtain a roughness of 0.5 nm. The technique is well mastered and thin-film LiNbO3 layers are now commercialized [43, 44], which have boosted the development of compact LiNbO3 components from low-loss ridge waveguides [45] to electro-optic microring resonators [46, 47], wire waveguide-based modulators [48], and nonlinear nanowires or ridges [49]. The lateral confinement can be ensured by proton exchangebased techniques or by direct etching of the thin layer. However, the insertion losses are often higher than 10 dB due to mode mismatch between the confined waveguides and the single mode fibers (SMFs) (see Figure 5(b): the vertical confinement is significantly tighter than a standard

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One approach to reduce the coupling losses is to guide the light in another material than LiNbO3. As an example, as studied by Chen et al. [50], the light is guided in a silicon-oninsulator microring bonded to an ion-sliced LiNbO3 film. The resulting low insertion losses (4.3 dB) are mitigated by a figure of merit of V<sup>π</sup>L = 9.1 Vcm, which is no better than what is reported in standard Ti-indiffused waveguides. The compacity is however made possible by

Another approach has been proposed recently, which relies on an easy-to-implement tech-

Figure 6 shows how free-suspended waveguides can be made by optical grade dicing. A monomode optical waveguide is firstly fabricated through the diffusion of 6-μm-wide and 90 nm-thick Ti stripes at 1030C for 10 h. Then, gold electrodes are sputtered over the substrate. Noteworthy, the electro-optical patterns are achieved by UV lithography, so that many other patterns can be envisioned with the same technology. Then, a curved slot is inscribed into

Figure 5. Thin film-based wire waveguides. (a) SEM image of a wire waveguide made by ion slicing. The dielectric bond layer is a BCB. (b) Amplitude of the electric field, TE optical mode calculated F.E.M. The overlap coefficient with a SMF

using a microring resonator with small radius curvature and low radiation losses.

nique and allows low-loss free-suspended waveguides with calibrated thickness.

5.2. Suspended membranes fabricated by optical grade dicing

mode is lower than – 10 dB.

SMF fiber, which leads to η coefficient lower than 10%).

where δ denotes the buffer layer thickness and εLN and εdiel are the dielectric permittivities of LiNbO3 and of the buffer layer, respectively. A quantitative comparison between a HAR ridge waveguide and a standard Ti-indiffused waveguide with the same gap and the same buffer layer shows that Γ is increased twofold in favor of the HAR ridge waveguide. This was confirmed experimentally by Caspar et al. [39].

However, the counterpart of the strong lateral confinement is a weaker mode matching with SMFs: the η coefficient is reduced down to 50%, meaning that 50% of the optical energy is lost per facet and consequently the insertion losses are ineluctably higher than 6 dB. This issue can be circumvented by using vertical spot-size converters such as the ones schematically depicted in Figure 4(a). They are made simply by progressively decreasing the ridge depth, which can be done by lifting the blade before the end of the ridge. In this case, the total insertion losses are measured to be lower than 3 dB for a 2-cm-long ridge, and only the fundamental mode is allowed to propagate in the ridge [40].

If the high-aspect ratio ridge waveguide is the best configuration in terms of EO efficiency, there is however an interest in developing confined waveguides allowing more complex patterns than straight waveguides. This is the reason why thin films and membranes have also known to be a great success over the years.

### 5. Thin film-based and membrane-based waveguides

Thin film-based waveguides can be described as rib waveguides or gradient index-based waveguides integrated in LiNbO3 thin films with a thickness lower than 5 mm. Ion slicing [41] is currently the prevailing manufacturing technique. These waveguides are also called microwaveguides.

#### 5.1. Thin films fabricated by ion slicing

By combining wafer bonding and ion implantation, the ion slicing technique allows the waferscale production of thin layers of Z-cut and X-cut substrates with sub-micrometric thickness, and the roughness is lower than 0.5 nm [42]. More precisely, the fabrication process begins by He+ ion implantation with a dose of about 4�1016 ions/cm<sup>2</sup> to form an amorphous layer at a depth dependent on the implantation energy (typically a few hundreds of keV for submicronic LiNbO3 layers). Then the implanted wafer is bonded to another one by using an intermediate layer with low refractive index, such as benzocyclobutene (BCB) or SiO2 [42]. A thermal treatment with temperature ramp is employed to split the samples along the implanted He layer. Hence, a thin LiNbO3 layer bonded on another substrate remains. Afterward, annealing and chemical mechanical polishing are used to obtain a roughness of 0.5 nm. The technique is well mastered and thin-film LiNbO3 layers are now commercialized [43, 44], which have boosted the development of compact LiNbO3 components from low-loss ridge waveguides [45] to electro-optic microring resonators [46, 47], wire waveguide-based modulators [48], and nonlinear nanowires or ridges [49]. The lateral confinement can be ensured by proton exchangebased techniques or by direct etching of the thin layer. However, the insertion losses are often higher than 10 dB due to mode mismatch between the confined waveguides and the single mode fibers (SMFs) (see Figure 5(b): the vertical confinement is significantly tighter than a standard SMF fiber, which leads to η coefficient lower than 10%).

One approach to reduce the coupling losses is to guide the light in another material than LiNbO3. As an example, as studied by Chen et al. [50], the light is guided in a silicon-oninsulator microring bonded to an ion-sliced LiNbO3 film. The resulting low insertion losses (4.3 dB) are mitigated by a figure of merit of V<sup>π</sup>L = 9.1 Vcm, which is no better than what is reported in standard Ti-indiffused waveguides. The compacity is however made possible by using a microring resonator with small radius curvature and low radiation losses.

Another approach has been proposed recently, which relies on an easy-to-implement technique and allows low-loss free-suspended waveguides with calibrated thickness.

#### 5.2. Suspended membranes fabricated by optical grade dicing

Finite element method (FEM) simulations performed on HAR waveguides show that they are optimal in terms of EO efficiency, since the electric field is uniform over the optical waveguide. This is illustrated in Figure 3(d) with the white arrows representing the applied electric field. As a consequence, the electro-optical coefficient can be evaluated simply as a function of g:

2 � εLN � δ þ ð Þ� g � 2 � δ εdiel

where δ denotes the buffer layer thickness and εLN and εdiel are the dielectric permittivities of LiNbO3 and of the buffer layer, respectively. A quantitative comparison between a HAR ridge waveguide and a standard Ti-indiffused waveguide with the same gap and the same buffer layer shows that Γ is increased twofold in favor of the HAR ridge waveguide. This was

However, the counterpart of the strong lateral confinement is a weaker mode matching with SMFs: the η coefficient is reduced down to 50%, meaning that 50% of the optical energy is lost per facet and consequently the insertion losses are ineluctably higher than 6 dB. This issue can be circumvented by using vertical spot-size converters such as the ones schematically depicted in Figure 4(a). They are made simply by progressively decreasing the ridge depth, which can be done by lifting the blade before the end of the ridge. In this case, the total insertion losses are measured to be lower than 3 dB for a 2-cm-long ridge, and only the fundamental mode is

If the high-aspect ratio ridge waveguide is the best configuration in terms of EO efficiency, there is however an interest in developing confined waveguides allowing more complex patterns than straight waveguides. This is the reason why thin films and membranes have also

Thin film-based waveguides can be described as rib waveguides or gradient index-based waveguides integrated in LiNbO3 thin films with a thickness lower than 5 mm. Ion slicing [41] is currently the prevailing manufacturing technique. These waveguides are also called

By combining wafer bonding and ion implantation, the ion slicing technique allows the waferscale production of thin layers of Z-cut and X-cut substrates with sub-micrometric thickness, and the roughness is lower than 0.5 nm [42]. More precisely, the fabrication process begins by He+ ion implantation with a dose of about 4�1016 ions/cm<sup>2</sup> to form an amorphous layer at a depth dependent on the implantation energy (typically a few hundreds of keV for submicronic LiNbO3 layers). Then the implanted wafer is bonded to another one by using an intermediate layer with low refractive index, such as benzocyclobutene (BCB) or SiO2 [42]. A thermal

(5)

<sup>Γ</sup> <sup>¼</sup> <sup>g</sup> � <sup>ε</sup>diel

confirmed experimentally by Caspar et al. [39].

162 Emerging Waveguide Technology

allowed to propagate in the ridge [40].

known to be a great success over the years.

5.1. Thin films fabricated by ion slicing

microwaveguides.

5. Thin film-based and membrane-based waveguides

Figure 6 shows how free-suspended waveguides can be made by optical grade dicing. A monomode optical waveguide is firstly fabricated through the diffusion of 6-μm-wide and 90 nm-thick Ti stripes at 1030C for 10 h. Then, gold electrodes are sputtered over the substrate. Noteworthy, the electro-optical patterns are achieved by UV lithography, so that many other patterns can be envisioned with the same technology. Then, a curved slot is inscribed into

Figure 5. Thin film-based wire waveguides. (a) SEM image of a wire waveguide made by ion slicing. The dielectric bond layer is a BCB. (b) Amplitude of the electric field, TE optical mode calculated F.E.M. The overlap coefficient with a SMF mode is lower than – 10 dB.

Figure 6. Process flow for the production of low-loss free-standing electro-photonic waveguides. The bottom face is the face of the wafer opposite to the waveguides.

the bottom face as schematically depicted in Figure 6(c) by optical grade dicing. The rotation speed and the moving speed of the 3350 DISCO DAD dicing saw are, respectively, 10,000 rpm and 0.2 mm/s. A resinoid blade progressively enters the wafers to a depth δ<sup>b</sup> and inscribes a curved shape inside the bottom face of the sample. Then, the blade is translated along the waveguide and it is lifted before the end of the waveguide. The remaining matter is a membrane with a controlled thickness t = e-δ<sup>b</sup> where e is the wafer thickness. Thanks to this technique, vertical tapers are fabricated at the extremities of the membrane [51]. Indeed, they are created by the curved shape of the blade (see Figure 6 and the tapers in Figure 7). Finally, wedges are bonded by UV adhesive on the top face of the wafer, and the chips are separated by optical grade dicing, enabling polished facets with an enlarged surface for pigtailing with fibers.

Figure 7 shows a schematic diagram of a resulting membrane-based electro-optical waveguide. In the free-standing Section 1, the waveguide is thinned, and the thickness can take any value between 450 nm and 500 μm. This is achieved by a preliminar depth calibration in a dead zone of the wafer. The suspended waveguide is surrounded by electrodes allowing an electrical control of the effective refractive index. The cross-sections in Figure 7 represent the optical guided mode along the suspended waveguide. In Section 1, the strongly confined mode allows an electro-optical coefficient twofold higher than in Section 3. On the other hand, the weakly confined mode in Section 3 allows mode matching and low coupling losses between the waveguide and the SMFs. Hence, the overlap coefficient η can be as high as 85%. Additionally, to mode matching with fibers, the vertical tapers allow filtering in favor of the fundamental mode, which avoids parasitical beatings in the spectral transmission response [51].

The experimental measurements of losses are summarized in Table 3 for both TE and TM polarizations and for different thicknesses. They are also reported by Courjal et al. [51]. They are compared with the average propagation losses of a non-thinned Ti-indiffused waveguide (t = 500 μm) fabricated in the same conditions with the same total length. The measurements were repeated five times for each waveguide and polarization. When the membrane has a thickness t ≥ 7 μm, there is no difference observed between a membrane-based waveguide and non-thinned one. When the waveguide is thinned below 4.5 μm, the guided wave undergoes increased propagation losses, up to 5.1 dB/cm for the TM wave and 3.2 dB/cm for the TE wave when t = 450 nm. Overall, the losses are lower for the TE waves than for TM waves, which is

Table 3. Experimental assessment of losses for X-cut membrane-based waveguides with a total length of L = 1.1 cm at

Figure 7. Schematic diagram of a free-suspended waveguide made by optical-grade-diving. The waveguide is thinned locally to get high EO efficiency. It is surrounded with tapers allowing mode matching. The inserts show the optical guided mode in different sections. Section#1: The layer thickness is 1 μm. Section#2: The layer thickness is 4 μm.

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T (μm) Pol. α (dB/cm) Total IL (dB) ξ (dB) Δng/ΔV (R.I.U./V) Γ (%) 7500 TE 0.20 0.06 2.8 0.2 1.1 0.3 7.8 1.0.10<sup>6</sup> <sup>34</sup>

4.50 TE 0.20 0.06 2.8 0.2 1.1 0.3 13.4 <sup>1</sup><sup>10</sup><sup>6</sup> <sup>58</sup>

0.45 TE 3.20 0.06 6.6 0.5 1.1 0.3 — —

thick waveguides did not have electrodes so that the EO measurements could not be performed.

TM 0.22 0.06 2.8 0.2 1.1 0.3 3.2 1.0<sup>10</sup><sup>6</sup> <sup>35</sup>

TM 0.32 0.06 2.9 0.2 1.1 0.3 15.8 <sup>1</sup><sup>10</sup><sup>6</sup> <sup>175</sup>

TM 5.12 0.06 8.1 0.8 1.1 0.3 — —

t, α, IL,ξ are respectively the membrane thickness, the average propagation losses, the insertion losses, the reflection coefficient and the coupling losses per facet. Γ is the overlap coefficient deduced from the measured Δng/ΔV. The 450 nm-

Section#3: The layer thickness is 500 μm.

1550 nm wavelength.

Figure 7. Schematic diagram of a free-suspended waveguide made by optical-grade-diving. The waveguide is thinned locally to get high EO efficiency. It is surrounded with tapers allowing mode matching. The inserts show the optical guided mode in different sections. Section#1: The layer thickness is 1 μm. Section#2: The layer thickness is 4 μm. Section#3: The layer thickness is 500 μm.

the bottom face as schematically depicted in Figure 6(c) by optical grade dicing. The rotation speed and the moving speed of the 3350 DISCO DAD dicing saw are, respectively, 10,000 rpm and 0.2 mm/s. A resinoid blade progressively enters the wafers to a depth δ<sup>b</sup> and inscribes a curved shape inside the bottom face of the sample. Then, the blade is translated along the waveguide and it is lifted before the end of the waveguide. The remaining matter is a membrane with a controlled thickness t = e-δ<sup>b</sup> where e is the wafer thickness. Thanks to this technique, vertical tapers are fabricated at the extremities of the membrane [51]. Indeed, they are created by the curved shape of the blade (see Figure 6 and the tapers in Figure 7). Finally, wedges are bonded by UV adhesive on the top face of the wafer, and the chips are separated by optical grade dicing, enabling polished facets with an enlarged surface for pigtailing with fibers. Figure 7 shows a schematic diagram of a resulting membrane-based electro-optical waveguide. In the free-standing Section 1, the waveguide is thinned, and the thickness can take any value between 450 nm and 500 μm. This is achieved by a preliminar depth calibration in a dead zone of the wafer. The suspended waveguide is surrounded by electrodes allowing an electrical control of the effective refractive index. The cross-sections in Figure 7 represent the optical guided mode along the suspended waveguide. In Section 1, the strongly confined mode allows an electro-optical coefficient twofold higher than in Section 3. On the other hand, the weakly confined mode in Section 3 allows mode matching and low coupling losses between the waveguide and the SMFs. Hence, the overlap coefficient η can be as high as 85%. Additionally, to mode matching with fibers, the vertical tapers allow filtering in favor of the fundamen-

Figure 6. Process flow for the production of low-loss free-standing electro-photonic waveguides. The bottom face is the

face of the wafer opposite to the waveguides.

164 Emerging Waveguide Technology

tal mode, which avoids parasitical beatings in the spectral transmission response [51].

The experimental measurements of losses are summarized in Table 3 for both TE and TM polarizations and for different thicknesses. They are also reported by Courjal et al. [51]. They are compared with the average propagation losses of a non-thinned Ti-indiffused waveguide (t = 500 μm) fabricated in the same conditions with the same total length. The measurements


t, α, IL,ξ are respectively the membrane thickness, the average propagation losses, the insertion losses, the reflection coefficient and the coupling losses per facet. Γ is the overlap coefficient deduced from the measured Δng/ΔV. The 450 nmthick waveguides did not have electrodes so that the EO measurements could not be performed.

Table 3. Experimental assessment of losses for X-cut membrane-based waveguides with a total length of L = 1.1 cm at 1550 nm wavelength.

were repeated five times for each waveguide and polarization. When the membrane has a thickness t ≥ 7 μm, there is no difference observed between a membrane-based waveguide and non-thinned one. When the waveguide is thinned below 4.5 μm, the guided wave undergoes increased propagation losses, up to 5.1 dB/cm for the TM wave and 3.2 dB/cm for the TE wave when t = 450 nm. Overall, the losses are lower for the TE waves than for TM waves, which is due to an increased sensitivity of the TM waves to membrane roughness. It is noteworthy that the coupling losses remain the same regardless of the membrane thickness, which confirms the efficiency of the tapers to mode match with the fibers.

The EO overlap coefficient is deduced from the Fourier transform of the reflected spectral density (see Figure 8), which is also the autocorrelation of the impulse response. Due to the Fabry-Perot oscillations inside the cavity formed by the waveguide, a peak appears in the Fourier transform, which coincides with a round trip of the light between the two facets of the waveguide. From this peak, we can deduce the global effective group index: ngeff = t2�c0/(2�Ltot), c0 being the speed of light and Ltot denoting the waveguide length. The resulting effective group indexes are neffTE = 2.189�0.005 and neffTM = 2.269�0.005 for TE and TM polarizations, respectively, in an X-cut Y-propagating waveguide with a membrane thickness of 4.5 μm. The effective group index is measured voltage by voltage from Figure 8, for the assessment of the group index variation per voltage: Δng/ΔV. The results are exposed in Table 2. Γ is calculated from Δng/ΔV by using expressions (6) and (7) for the TE and TM waves, respectively:

$$
\Gamma\_{TE} = \frac{\Delta n\_{\text{eg}}}{\Delta V} \cdot \frac{\text{2} \cdot \text{g}}{n\_{\text{e}}^3 \cdot r\_{33}} \tag{6}
$$

So membrane-based waveguides are good candidates to reduce the active length, without impacting the other parameters such as the losses. It is now tempting to go one step further

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It is well established that the machining of materials at the wavelength scale can yield a specific control of the light flux, which is of great interest to enhance significantly the electro-optical efficiency. In particular, photonic crystals (PhCs) are periodic structures that are designed to affect the motion of photons in a similar way that periodicity of a semiconductor crystal affects the behavior of electrons. The non-existence of propagating electromagnetic modes inside the structures at certain frequencies introduces unique optical phenomena such as tight light confinement. The part of the spectrum for which wave propagation is not possible is called

The first reported PhC-based EO LiNbO3 modulator [52] was configured to behave as an electro-absorbent modulator: it was designed to have a photonic bandgap, and the spectral edge was exploited for intensity modulation. Hence, an active length of 11 μm was sufficient to modulate the light with a driving voltage of 13 V [52]. A schematic diagram of such a device is seen in Figure 9. It consists of a square lattice PhC integrated on an annealed proton exchanged optical waveguide and surrounded by capacitive coplanar electrodes. The LiNbO3 PhC was fabricated through focused ion beam (FIB) milling The refractive index was modified by the electric field Ez generated between the electrodes, which moved the spectral position of the PBG and consequently the transmitted intensity. The geometric properties of the PhC were

This compact modulator showed an EO interaction 312 times higher than the one predicted from Eq. (2): the extraordinary effect is shown in Figure 10 where the PBG is spectrally shifted

Figure 9. Image of the first reported PhC-based EO modulator [53]. (a) Schematic view of the ultra-compact modulator.

toward compacity by inscribing nanostructures inside the membranes.

6. LiNbO3 nanostructures and photonic crystals

the photonic bandgap (PBG).

chosen to benefit from slow light effects.

(b) SEM image of the photonic crystal surrounded by electrodes.

$$
\Gamma\_{\rm TM} = \frac{\Delta n\_{\rm og}}{\Delta V} \cdot \frac{\mathbf{2} \cdot \mathbf{g}}{n\_{\rm o}^3 \cdot r\_{\rm 13}} \tag{7}
$$

Table 2 confirms the twofold enhancement of the EO interaction when the membrane is thinned down to 4.5 μm, and it shows that this enhancement is even higher for the TMpropagating wave, although this was not anticipated from the FEM calculations. This latter result can be of great interest to seek for isotropic EO behavior of the guided wave in the presence of an applied voltage.

Figure 8. Zoom view of the Fourier transform of the reflected optical density spectrum. These measurements are achieved as a function of the applied voltage through a 12.0 mm long tapered-membrane-based waveguide with a width of 6 μm and a thickness of 4.5 μm. The total length was Ltot = 1.2 mm.

So membrane-based waveguides are good candidates to reduce the active length, without impacting the other parameters such as the losses. It is now tempting to go one step further toward compacity by inscribing nanostructures inside the membranes.
