**5. Digital transmission line differential relays using digital communication**

Natural disasters may be of significant impact on overhead transmission lines and cause communication outage related to pilot protection.

A dedicated fiber connection relies solely on optics residing within the relays to send IEEE C37.94 signaling bi-directionally, usually on a pair of single-mode strands of fiber from one relay to another. Potential sources of trouble for direct-connected current differential relay schemes normally can be traced to one of the following:

Most line-current differential relays use a 64 kbps communication interface even though designs with higher bandwidths (n × 64 kbps) are more common. The data frame length depends on the different relay designs. It varies between 15 bits and 200 or 400 bits. Sometimes it could be out of these bounds. The current data transmission has a large range also—from one to four per cycle, sometimes even more. There are some line current differential relays generally used for distribution line protection that uses slower-speed asynchronous serial communications. Presently, the use of Ethernet communications has not been widely implemented for line current differential relaying but is expected in future designs.

Longitudinal Differential Protection of Power Systems Transmission Lines Using Optical… http://dx.doi.org/10.5772/intechopen.76621 251

**Figure 12.** Protective relaying communications.

The communication channel can be over a dedicated fiber or over a multiplexed network, as shown in **Figure 12**. The dedicated fiber connection typically deploys LED or laser depending on the fiber's distance. The laser option can typically be applied for up to 100 km. The longer fiber lengths may need repeaters [12].

In order to guarantee the safe and stable operation of high-voltage transmission lines, differential protection is adopted as the main protection for the benefits of its phase-selection function and immunity to power swings and operation modes.

Many various principles for realization of the differential protection have been published in the recent literatures. The fast communication between relays installed at the opposite ends of the line is necessary [12].

There are a number of different relay measuring principles used by current differential relays:


occur. The main problem of the pilot wires is its length. It depends on loop resistance. There is a maximum value required. It is 2000 Ω. Shunt capacitance also has a limiting value. It is

Connection with the d-c wire pilot means that a lot of elements have to be used. This is one of the disadvantages. An a-c connection does not have these problems. Besides that, a-c connection also is immune to power swings. The good thing about d-c connection is the existence of

Natural disasters may be of significant impact on overhead transmission lines and cause com-

A dedicated fiber connection relies solely on optics residing within the relays to send IEEE C37.94 signaling bi-directionally, usually on a pair of single-mode strands of fiber from one relay to another. Potential sources of trouble for direct-connected current differential relay

Most line-current differential relays use a 64 kbps communication interface even though designs with higher bandwidths (n × 64 kbps) are more common. The data frame length depends on the different relay designs. It varies between 15 bits and 200 or 400 bits. Sometimes it could be out of these bounds. The current data transmission has a large range also—from one to four per cycle, sometimes even more. There are some line current differential relays generally used for distribution line protection that uses slower-speed asynchronous serial communications. Presently, the use of Ethernet communications has not been widely imple-

recommended that this capacitance be less than 1.5 microfarads.

**5. Digital transmission line differential relays using digital** 

mented for line current differential relaying but is expected in future designs.

these wires because of the telephone companies [10, 11].

schemes normally can be traced to one of the following:

munication outage related to pilot protection.

**Figure 11.** Pilot wire simplified arrangement.

250 Emerging Waveguide Technology

**communication**

These principles are briefly described in the following subsections.

#### **5.1. Percentage differential relays**

This is most like a classic approach. **Figures 1** and **2** show a basic arrangement. At each terminal, an evaluation of the sum of the local and remote current values is made in order to calculate a differential current. Under normal operating conditions or external faults, the current entering at one end of the protected circuit is practically the same as that leaving at the other end. Hence, the differential current value is practically zero and operation of the protection will not occur. For a fault on the protected power line, the differential current value will exceed the operation threshold value and the protection will operate to clear the fault. There are so many modifications of the classic approach. Some of them use negative or zerosequence current components.

There are two types of algorithms for differential protection: those that use phasors [13–16] and those that use instantaneous values of electrical quantities [17, 18].

#### **5.2. Charge comparison operating principle**

Charging current is a capacitive leakage current on the transmission line. The operating principles of charge comparison are similar to those of the more common percentage restraint current differential type of protective relay. Current differential relays compare the total currents entering and leaving the primary protection zone. They will trip if the difference between these currents exceeds some pre-defined restraint limit. For this comparison to be made, the current differential relay at the local station has to know the identical phase current recorded at the remote station(s) for the same interval being considered at the local station. This requires precise communications delay measurement and compensation. With current differential relays that compare instantaneous values, any error in compensation causes an error in the comparison and results in a variation of the pickup point. In addition, many samples per cycle are sent to the remote station, placing a burden on the communications channel [12].

The simple system shown in **Figure 13** is assumed.

Charging current compensation is a solution which removes charging current from the measured current and hence excludes the charging current from the differential current calculation. The charging current (*I c* ) can be estimated on the basis of operational capacitance:

$$I\_{\mathbb{C}} = \frac{2 \times \pi}{\sqrt{3}} \times U\_n \times f\_n \times c\_d \times L \times 10^{-6} \tag{3}$$

Current that has a higher value should be compensated. It depends on the load flow direction.

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Assuming a perfect alignment between local and remote currents, there is ±4 ms of error tolerance in the compensation. Given that line capacitance will cause a slight misalignment of the currents and that any compensation is going to have some resolution, it is practical to use ±3 ms as a limit for how much the communication channel delay can vary during normal operation without affecting the relaying. This is well within the normal operational limits of most communication channels in a dedicated fiber environment, even under adverse

In addition to the classic approach, which uses current signals exclusively, there are solutions

Charging current is treated as the main cause of differential relay wrong operation in many papers. This current could be compensated with the power differential principle. This method proposes measuring currents and voltages at both ends of the transmission line. The original differential principle is saved. Operate and restrain values are compared but instead of cur-

In this case, calculation time is reduced and length of line is not a limiting factor. These are the great advantages. Wrong operation during fault with low resistance and no faulted phase

There is a different method for realization of this protection based on power losses determination. It requires measuring currents and voltages at each end of the line too. Fault location could be determined in an easy way. This method compares difference in the real power at each end of the transmission line with the maximum power losses in the protected element. Every fault could be detected and the protection is secure and dependable. Fast communica-

The higher value has current measured in the terminal where energy flows in.

conditions.

**5.3. Power differential relays**

**Figure 13.** Charging current compensation.

which require voltage inputs too [19–21].

rents, active powers at both ends are considered.

tion between relays is very important [19–21].

selection are the disadvantages of the mentioned technique.

where:

*Ic* —charging current (A),

*Un* —rated network voltage (kV),

*f n* —rated frequency (Hz),

*cd* —longitudinal operational capacitance (nF/km),

 $L-$  line length (km).

Then the compensated current can be calculated as follows:

$$I\_x = I\_x' - I\_c \tag{4}$$

where:

*I x* —is compensated current at terminal x,

*Ix '*—is measured current at terminal x. Longitudinal Differential Protection of Power Systems Transmission Lines Using Optical… http://dx.doi.org/10.5772/intechopen.76621 253

**Figure 13.** Charging current compensation.

will exceed the operation threshold value and the protection will operate to clear the fault. There are so many modifications of the classic approach. Some of them use negative or zero-

There are two types of algorithms for differential protection: those that use phasors [13–16]

Charging current is a capacitive leakage current on the transmission line. The operating principles of charge comparison are similar to those of the more common percentage restraint current differential type of protective relay. Current differential relays compare the total currents entering and leaving the primary protection zone. They will trip if the difference between these currents exceeds some pre-defined restraint limit. For this comparison to be made, the current differential relay at the local station has to know the identical phase current recorded at the remote station(s) for the same interval being considered at the local station. This requires precise communications delay measurement and compensation. With current differential relays that compare instantaneous values, any error in compensation causes an error in the comparison and results in a variation of the pickup point. In addition, many samples per cycle are sent to the remote station, placing a burden on the communications channel [12].

Charging current compensation is a solution which removes charging current from the measured current and hence excludes the charging current from the differential current calcula-

> *<sup>x</sup>* = *I x* ′ − *I*

) can be estimated on the basis of operational capacitance:

*<sup>n</sup>* × *cd* × *L* × 10−<sup>6</sup> (3)

*<sup>C</sup>* (4)

and those that use instantaneous values of electrical quantities [17, 18].

sequence current components.

252 Emerging Waveguide Technology

**5.2. Charge comparison operating principle**

The simple system shown in **Figure 13** is assumed.

—longitudinal operational capacitance (nF/km),

*I*

—is compensated current at terminal x,

*'*—is measured current at terminal x.

Then the compensated current can be calculated as follows:

*c*

*<sup>C</sup>* <sup>=</sup> \_\_\_\_ <sup>2</sup> <sup>×</sup> *<sup>π</sup>* √ \_\_ <sup>3</sup> <sup>×</sup> *Un* <sup>×</sup> *<sup>f</sup>*

tion. The charging current (*I*

—charging current (A),

—rated frequency (Hz),

*L*—line length (km).

where:

*Ic*

*Un*

*f n*

*cd*

where:

*I x*

*Ix*

*I*

—rated network voltage (kV),

Current that has a higher value should be compensated. It depends on the load flow direction. The higher value has current measured in the terminal where energy flows in.

Assuming a perfect alignment between local and remote currents, there is ±4 ms of error tolerance in the compensation. Given that line capacitance will cause a slight misalignment of the currents and that any compensation is going to have some resolution, it is practical to use ±3 ms as a limit for how much the communication channel delay can vary during normal operation without affecting the relaying. This is well within the normal operational limits of most communication channels in a dedicated fiber environment, even under adverse conditions.

#### **5.3. Power differential relays**

In addition to the classic approach, which uses current signals exclusively, there are solutions which require voltage inputs too [19–21].

Charging current is treated as the main cause of differential relay wrong operation in many papers. This current could be compensated with the power differential principle. This method proposes measuring currents and voltages at both ends of the transmission line. The original differential principle is saved. Operate and restrain values are compared but instead of currents, active powers at both ends are considered.

In this case, calculation time is reduced and length of line is not a limiting factor. These are the great advantages. Wrong operation during fault with low resistance and no faulted phase selection are the disadvantages of the mentioned technique.

There is a different method for realization of this protection based on power losses determination. It requires measuring currents and voltages at each end of the line too. Fault location could be determined in an easy way. This method compares difference in the real power at each end of the transmission line with the maximum power losses in the protected element. Every fault could be detected and the protection is secure and dependable. Fast communication between relays is very important [19–21].

#### **5.4. Alpha plane relays**

The alpha plane current differential protection principle compares individual magnitudes and angles of the currents. Magnitude and phase angle of each current at the opposite line ends are measured. According to these values, vector *r* is determined (its magnitude and angle). The alpha plane depicts the complex ratio of *IR/IL*, and it is shown in **Figure 14**.

The horizontal axis variable *a* is the real part of the complex ratio of *IR/IL*, and the vertical axis variable *b* is the imaginary part of the complex ratio of *IR/IL*. In the alpha plane element, the angular setting *α* [°] and the radius setting *R* define restraining region. The first one allows the accommodation of current transformer and current alignment errors and the second one

Longitudinal Differential Protection of Power Systems Transmission Lines Using Optical…

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255

Data transmission delay has to be compensated. Current measuring and comparison depend on the communication channel. There are delays in two directions—transmit and receive. These delays sometimes are different. SONET/SDH systems could be the reason for this asymmetry. One technique for overcoming this problem, so-called ping-pong, involves measuring the round-trip channel delay. The communications path-delay differences are typically

This chapter describes using optical waveguide for communication between two differential relays on the opposite ends of the power systems' transmission line. Using the pilot wires limits utilizing this protection only for short lines. If optical protection ground wires are used, instead of pilot wires, the length of the line ceases to be a limiting factor. This chapter tells us more about constructions, assembly and utilization of the optical waveguides in differential

The sections present a classic approach of longitudinal differential protection of transmission lines, talk more about construction and installation of the OPGW cables and discuss digital protection algorithms. All the algorithms are difficult to be implemented without using

The papers that emphasize this method are authored by Almedia and Silva [22, 23].

protection. Also the newest algorithms of this protection are mentioned.

University of Belgrade-School of Electrical Engineering, Belgrade, Serbia

[1] Ðurić M, Stojanović Z. Relay Protection. Belgrade: KIZ 'CENTER'; 2014

[2] Ðurić M, Terzija V, Radojević Z, et al. Algorithms for Digital Relaying. Belgrade: ETA;

modifies sensitivity.

**6. Conclusion**

OPGW.

**Author details**

Address all correspondence to: rajic@etf.rs

Tomislav Rajić

**References**

2012

less than 2 ms. Delays of 3–5 ms are rare [12].

Operating and restraining regions are presented in **Figure 15**. Comparison of the amplitudes and the angles lead to one of these decisions.

**Figure 14.** Complex current ratio plane (*α*-plane).

**Figure 15.** Current ratio plane.

The horizontal axis variable *a* is the real part of the complex ratio of *IR/IL*, and the vertical axis variable *b* is the imaginary part of the complex ratio of *IR/IL*. In the alpha plane element, the angular setting *α* [°] and the radius setting *R* define restraining region. The first one allows the accommodation of current transformer and current alignment errors and the second one modifies sensitivity.

Data transmission delay has to be compensated. Current measuring and comparison depend on the communication channel. There are delays in two directions—transmit and receive. These delays sometimes are different. SONET/SDH systems could be the reason for this asymmetry. One technique for overcoming this problem, so-called ping-pong, involves measuring the round-trip channel delay. The communications path-delay differences are typically less than 2 ms. Delays of 3–5 ms are rare [12].

The papers that emphasize this method are authored by Almedia and Silva [22, 23].

## **6. Conclusion**

**5.4. Alpha plane relays**

254 Emerging Waveguide Technology

and the angles lead to one of these decisions.

**Figure 14.** Complex current ratio plane (*α*-plane).

**Figure 15.** Current ratio plane.

The alpha plane current differential protection principle compares individual magnitudes and angles of the currents. Magnitude and phase angle of each current at the opposite line ends are measured. According to these values, vector *r* is determined (its magnitude and angle). The alpha plane depicts the complex ratio of *IR/IL*, and it is shown in **Figure 14**.

Operating and restraining regions are presented in **Figure 15**. Comparison of the amplitudes

This chapter describes using optical waveguide for communication between two differential relays on the opposite ends of the power systems' transmission line. Using the pilot wires limits utilizing this protection only for short lines. If optical protection ground wires are used, instead of pilot wires, the length of the line ceases to be a limiting factor. This chapter tells us more about constructions, assembly and utilization of the optical waveguides in differential protection. Also the newest algorithms of this protection are mentioned.

The sections present a classic approach of longitudinal differential protection of transmission lines, talk more about construction and installation of the OPGW cables and discuss digital protection algorithms. All the algorithms are difficult to be implemented without using OPGW.

### **Author details**

Tomislav Rajić

Address all correspondence to: rajic@etf.rs

University of Belgrade-School of Electrical Engineering, Belgrade, Serbia

### **References**


[3] Ziegler G. Numerical Differential Protection: Principles and Applications. 2nd ed. Erlangen: Publicis Publishing; 2012. pp. 66-69 and 202-233

[17] Deng X, Yuan R, Li T, et al. Digital differential protection technique of transmission line using instantaneous active current: Theory, simulation and experiment. IET Generation

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[18] Rajić T, Stojanović Z. An algorithm for longitudinal differential protection of transmission lines. International Journal of Electrical Power & Energy Systems. 2018;**94**:276-286

[19] Wen M, Chen D, Yin X. An energy differential relay for long transmission lines. International Journal of Electrical Power & Energy Systems. 2014;**55**:497-502

[20] Kawady T, Talaab A, Ahmed E. Dynamic performance of the power differential relay for transmission line protection. International Journal of Electrical Power & Energy

[21] Aziz MMA, Zobaa AF, Ibrahim DK, et al. Transmission lines differential protection based on the energy conservation law. Electric Power Systems Research. 2008;**78**:1865-1872 [22] Almeida M, Silva K. Transmision lines differential protection based on an alternative incremental complex power alpha plane. IET Generation Transmission and Distribution.

[23] Silva K, Bainy R. Generalized alpha plane for numerical differential protection applica-

tions. IEEE Transactions on Power Delivery. 2016;**31**:2565-2566

Transmission and Distribution. 2015;**9**:996-1005

Systems. 2010;**32**:390-397

2017;**11**:10-17


[17] Deng X, Yuan R, Li T, et al. Digital differential protection technique of transmission line using instantaneous active current: Theory, simulation and experiment. IET Generation Transmission and Distribution. 2015;**9**:996-1005

[3] Ziegler G. Numerical Differential Protection: Principles and Applications. 2nd ed.

[4] Caledonian. Outdoor Fiber Cable [Internet]. Available from: http://www.caledoniancables.com/Fiber%20Cable/New%20Fiber%20Cables/OPGW.html [Accessed: December

[5] Elsewedy Electric. Fiber Optic Cables [Internet]. Available from: http://www.elsewedyelectric.com/fe/Common/ProductDivisions.aspx [Accessed: December 11, 2017]

[6] PRYSMIAN. Cables and Systems [Internet]. Available from: http://mcwadeproductions.co.za/wp-content/uploads/2015/08/Prysmian-Catalogue-Aug-2015.pdf [Accessed:

[7] W. T. Connect the World. Communication Cables [Internet]. Available from: https:// www.alibaba.com/product-detail/Double-Layer-Stranded-Optical-Ground-

[8] American Wire Group. OPGW Optical Ground Wire—Multi Stainless Steel Tube [Internet]. Available from: http://wire.buyawg.com/viewitems/opgw-optical-ground-

[9] GE Grid Solutions. 5wire-Pilot Relays [Internet]. Available from: http://www.gegridso-

[10] GE Grid Solutions. 15line Protection with Pilot Relays [Internet]. Available from: http:// www.gegridsolutions.com/multilin/notes/artsci/art15.pdf [Accessed: December 02, 2017]

[11] Voloh I, Johnson R. Applying digital line current differential relays over pilot wires. In: 2005 58th Annual Conference for Protective Relay Engineers; 5-7 April 2005; College

[12] C37.243-2015—IEEE Guide for Application of Digital Line Current Differential Relays Using Digital Communication. Date of Publication: 7 Aug. 2015. DOI: 10.1109/

[13] Krishnanand K, Dash P, Naeem M. Detection, classification, and location of faults in power transmission lines. International Journal of Electrical Power & Energy Systems.

[14] Hosny A, Sood VK. Transformer differential protection with phase angle difference

[15] Dambhare S, Soman SA, Chandorkar MC. Adaptive current differential protection schemes for transmission-line protection. IEEE Transactions on Power Delivery.

[16] Adly A, El Sehiemy R, Abdelaziz A. A directional protection scheme during single pole

based inrush restraint. Electric Power Systems Research. 2014;**115**:57-64

tripping. Electric Power Systems Research. 2017;**144**:197-207

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wire-2/opgw-optical-ground-wire? [Accessed: December 12, 2017]

lutions.com/multilin/notes/artsci/art05.pdf [Accessed: December 02, 2017]

01, 2017]

256 Emerging Waveguide Technology

December 05, 2017]

Station, TX, USA. USA: IEEE; 2005

IEEESTD.2015.7181615

2015;**67**:76-86

2009;**24**:1756-1762


**Chapter 14**

**Provisional chapter**

**End-Fire Mode Spectroscopy: A Measuring Technique**

**End-Fire Mode Spectroscopy: A Measuring Technique** 

End-fire mode spectroscopy technique provides reliable measurement of the whole mode spectrum of optical waveguides having arbitrary cross refractive index profile. The method is based on registration of light beams radiated from the abrupt output edge of the waveguide, with each beam corresponding to the individual waveguide mode. Due to different values of mode propagation constants, modes of different orders demonstrate different refraction angles at the output waveguide face when modes reach that face under the same nonzero inclination angle. Just this feature is used in the technique. Mode excitation is performed directly through the input waveguide face, and therefore the technique can be applied to analyze mode spectrum of arbitrary waveguides, including the ones with non-monotonic index profiles (particularly, symmetric step-index pro-

files or buried graded-index waveguides with any burying depths).

**Keywords:** waveguides, integrated optics, mode index, refractive index, optical

Optical waveguides are the basic elements of any photonic device, and their parameters define the operating performances of the whole photonic unit. Optimization of those parameters requires performing the choice of appropriate technology conditions. Development of fabrication technology and further designing the waveguide elements having pre-defined properties need performing a control of the characteristics of trial waveguide samples. Furthermore, planar optical waveguides are used intensively in determination of the basic properties of optical materials. And the problem of reliable determination of the main waveguide perfor-

> © 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.

DOI: 10.5772/intechopen.75558

**for Optical Waveguides**

**for Optical Waveguides**

http://dx.doi.org/10.5772/intechopen.75558

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Dmitry V. Svistunov

Dmitry V. Svistunov

**Abstract**

measurements

**1. Introduction**

mances is still actual.

#### **End-Fire Mode Spectroscopy: A Measuring Technique for Optical Waveguides End-Fire Mode Spectroscopy: A Measuring Technique for Optical Waveguides**

DOI: 10.5772/intechopen.75558

Dmitry V. Svistunov Dmitry V. Svistunov

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.75558

#### **Abstract**

End-fire mode spectroscopy technique provides reliable measurement of the whole mode spectrum of optical waveguides having arbitrary cross refractive index profile. The method is based on registration of light beams radiated from the abrupt output edge of the waveguide, with each beam corresponding to the individual waveguide mode. Due to different values of mode propagation constants, modes of different orders demonstrate different refraction angles at the output waveguide face when modes reach that face under the same nonzero inclination angle. Just this feature is used in the technique. Mode excitation is performed directly through the input waveguide face, and therefore the technique can be applied to analyze mode spectrum of arbitrary waveguides, including the ones with non-monotonic index profiles (particularly, symmetric step-index profiles or buried graded-index waveguides with any burying depths).

**Keywords:** waveguides, integrated optics, mode index, refractive index, optical measurements

#### **1. Introduction**

Optical waveguides are the basic elements of any photonic device, and their parameters define the operating performances of the whole photonic unit. Optimization of those parameters requires performing the choice of appropriate technology conditions. Development of fabrication technology and further designing the waveguide elements having pre-defined properties need performing a control of the characteristics of trial waveguide samples. Furthermore, planar optical waveguides are used intensively in determination of the basic properties of optical materials. And the problem of reliable determination of the main waveguide performances is still actual.

© 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.

Important part of examination of planar optical waveguides is measurement of the waveguide mode spectrum. Usually this procedure is performed by a well-known m-line spectroscopy technique (e.g., see [1–5]). The measured set of mode indices can be considered as initial data as for preliminary determination of the device operating performances as for reconstruction of cross refractive index profile in the formed trial sample. Traditional computing techniques allowing reconstruction of that profile are described in Refs. [6, 7]. However, in cases of planar waveguide structures with thick cover layers or so-called buried graded-index waveguides m-line spectroscopy does not provide reliable measurements. Thick cover layers or large burying depths do not allow tunneling the modes to the external prism and forming the corresponding spatial m-lines. In these cases, some modes (first of all, the lower-order modes) may be simply missed in examinations by m-line spectroscopy [8, 9]. The greater the burying depth the fewer number of modes can be measured by this method. To avoid missing the modes, a layer-by-layer etching of the sample surface could be applied [10, 11]. However, that procedure has many chances to cut a part of the refractive index profile occupied by the mode fields, and that should lead to distortion of the original waveguide mode spectrum. The use of nonlinear optical effects like second harmonic generation [9] can be successful only for limited number of optical materials demonstrating high values of the corresponding coefficients.

propagation of light wave having some cross field distribution. That description means that the waveguide mode is represented by the equivalent model of plane wave propagating in homogeneous uniform medium with the same light phase velocity as the original waveguide mode. So, we can consider a waveguide transmitting several modes as an imaginary set of superimposed layers with uniform refractive indices *N*m, each denoting the certain waveguide mode and transmitting the plane wave having corresponding cross field distribution as it is shown in **Figure 1**. Then our aim (to measure the mode spectrum) can be associated with the task of measuring the indices of different uniform media illuminated with plane light waves. That task can be performed by the technique like one of the usual methods of traditional bulk refractometry, for example by the goniometric technique that bases on the Snell's law and employs registration of the beam declination angle when the prism of tested optical material is illuminated with the collimated incident light. Thus we could try to provide similar experi-

End-Fire Mode Spectroscopy: A Measuring Technique for Optical Waveguides

http://dx.doi.org/10.5772/intechopen.75558

261

The content of the end-fire mode spectroscopy is registration of light beams radiated from the abrupt output edge of a planar waveguide, with each beam corresponding to the individual waveguide mode. Due to different values of mode propagation constants, modes of different orders demonstrate different refraction angles at the output sample face if they are directed to that face under nonzero incidence angle into the waveguide. Just this feature is exploited by the technique in procedures of mode spectrum measurements. Both excitation and output of waveguide modes are performed at the sample faces by the end-fire coupling method which allows reliable launching and output of the whole mode spectrum in any planar waveguide. Therefore, the end-fire mode spectroscopy technique can be applied to examination of planar waveguides having arbitrary cross refractive index profiles including symmetric step-index

The measuring block-scheme is shown in **Figure 2**. Collimated light beam is focused on the input waveguide face by the cylindrical lens. The whole mode spectrum can be launched in this manner in few-mode waveguides. Application of the input cylindrical lens provides obtaining collimated (in the sample planform) mode beams propagating into the examined waveguide. In the case of a thick multimode waveguide, a group of modes is excited simultaneously and can be registered. Further scanning the input sample face along the Y axis allows launching and registration of other mode groups until the whole waveguide mode spectrum is measured.

**Figure 1.** Equivalent representation of graded-index planar waveguide as a set of superimposed layers having uniform

mental conditions in order to examine the waveguide samples.

ones and deep-buried graded-index waveguides.

refractive indices.

Here we describe a measuring technique named the end-fire mode spectroscopy which is suitable for examination of planar waveguides having arbitrary refractive index profiles including the case of buried waveguide structures with any burying depths, and the presented results of examination of buried waveguides prove this advantage of the technique. Furthermore, here we show that this technique allows also conducting direct measurements of another important characteristic – the maximal refractive index in graded-index waveguides, unlike conventional techniques that involve the set of measured mode indices and employ computing of the maximal value in the refractive index profile using different approximations.

### **2. Method content**

#### **2.1. Mode spectrum measuring**

Usually waveguide mode spectrum is presented with a kit of mode indices *N*m (the value obtained from the mode propagation constant βm as *N*<sup>m</sup> = βm /k, where k = ω /c = 2π /λ; ω, c and λ – the light frequency, velocity and wavelength in vacuum correspondingly, m – the mode number). These values relate to the mode phase velocity, and they are involved into expressions describing the mode fields which are derived as solutions of wave equations. We can write a well-known general form of such expression for mode electric field vector (magnetic component of light wave is of the same view): **Em** (x,y,z,t) = **E0** (x,y) × exp[i(ωt – βm z)]. That representation is applied for all kinds of waveguides: planar and 3D (including stripe and channel waveguides, fibers, etc.) having as different step-like cross index distributions as graded-index ones. The expression written in cylindrical coordinates has a similar view (that approach is more convenient for optical fibers). Depending on the mode type, corresponding nonzero field projections are considered. Evidently, that expression describes unidirectional propagation of light wave having some cross field distribution. That description means that the waveguide mode is represented by the equivalent model of plane wave propagating in homogeneous uniform medium with the same light phase velocity as the original waveguide mode. So, we can consider a waveguide transmitting several modes as an imaginary set of superimposed layers with uniform refractive indices *N*m, each denoting the certain waveguide mode and transmitting the plane wave having corresponding cross field distribution as it is shown in **Figure 1**. Then our aim (to measure the mode spectrum) can be associated with the task of measuring the indices of different uniform media illuminated with plane light waves. That task can be performed by the technique like one of the usual methods of traditional bulk refractometry, for example by the goniometric technique that bases on the Snell's law and employs registration of the beam declination angle when the prism of tested optical material is illuminated with the collimated incident light. Thus we could try to provide similar experimental conditions in order to examine the waveguide samples.

Important part of examination of planar optical waveguides is measurement of the waveguide mode spectrum. Usually this procedure is performed by a well-known m-line spectroscopy technique (e.g., see [1–5]). The measured set of mode indices can be considered as initial data as for preliminary determination of the device operating performances as for reconstruction of cross refractive index profile in the formed trial sample. Traditional computing techniques allowing reconstruction of that profile are described in Refs. [6, 7]. However, in cases of planar waveguide structures with thick cover layers or so-called buried graded-index waveguides m-line spectroscopy does not provide reliable measurements. Thick cover layers or large burying depths do not allow tunneling the modes to the external prism and forming the corresponding spatial m-lines. In these cases, some modes (first of all, the lower-order modes) may be simply missed in examinations by m-line spectroscopy [8, 9]. The greater the burying depth the fewer number of modes can be measured by this method. To avoid missing the modes, a layer-by-layer etching of the sample surface could be applied [10, 11]. However, that procedure has many chances to cut a part of the refractive index profile occupied by the mode fields, and that should lead to distortion of the original waveguide mode spectrum. The use of nonlinear optical effects like second harmonic generation [9] can be successful only for limited number of optical materials demonstrating high values of the corresponding coefficients. Here we describe a measuring technique named the end-fire mode spectroscopy which is suitable for examination of planar waveguides having arbitrary refractive index profiles including the case of buried waveguide structures with any burying depths, and the presented results of examination of buried waveguides prove this advantage of the technique. Furthermore, here we show that this technique allows also conducting direct measurements of another important characteristic – the maximal refractive index in graded-index waveguides, unlike conventional techniques that involve the set of measured mode indices and employ computing of the maximal value in the refractive index profile using different approximations.

Usually waveguide mode spectrum is presented with a kit of mode indices *N*m (the value obtained from the mode propagation constant βm as *N*<sup>m</sup> = βm /k, where k = ω /c = 2π /λ; ω, c and λ – the light frequency, velocity and wavelength in vacuum correspondingly, m – the mode number). These values relate to the mode phase velocity, and they are involved into expressions describing the mode fields which are derived as solutions of wave equations. We can write a well-known general form of such expression for mode electric field vector (mag-

That representation is applied for all kinds of waveguides: planar and 3D (including stripe and channel waveguides, fibers, etc.) having as different step-like cross index distributions as graded-index ones. The expression written in cylindrical coordinates has a similar view (that approach is more convenient for optical fibers). Depending on the mode type, corresponding nonzero field projections are considered. Evidently, that expression describes unidirectional

(x,y) × exp[i(ωt – βm z)].

netic component of light wave is of the same view): **Em** (x,y,z,t) = **E0**

**2. Method content**

260 Emerging Waveguide Technology

**2.1. Mode spectrum measuring**

The content of the end-fire mode spectroscopy is registration of light beams radiated from the abrupt output edge of a planar waveguide, with each beam corresponding to the individual waveguide mode. Due to different values of mode propagation constants, modes of different orders demonstrate different refraction angles at the output sample face if they are directed to that face under nonzero incidence angle into the waveguide. Just this feature is exploited by the technique in procedures of mode spectrum measurements. Both excitation and output of waveguide modes are performed at the sample faces by the end-fire coupling method which allows reliable launching and output of the whole mode spectrum in any planar waveguide. Therefore, the end-fire mode spectroscopy technique can be applied to examination of planar waveguides having arbitrary cross refractive index profiles including symmetric step-index ones and deep-buried graded-index waveguides.

The measuring block-scheme is shown in **Figure 2**. Collimated light beam is focused on the input waveguide face by the cylindrical lens. The whole mode spectrum can be launched in this manner in few-mode waveguides. Application of the input cylindrical lens provides obtaining collimated (in the sample planform) mode beams propagating into the examined waveguide. In the case of a thick multimode waveguide, a group of modes is excited simultaneously and can be registered. Further scanning the input sample face along the Y axis allows launching and registration of other mode groups until the whole waveguide mode spectrum is measured.

**Figure 1.** Equivalent representation of graded-index planar waveguide as a set of superimposed layers having uniform refractive indices.

inclination of a certain output beam to the output waveguide face: the lower the mode order, the bigger the output angle. The fundamental mode forms the light beam having maximal value of the output angle *ψmax*. In the upright YZ projection, the output beams have large divergence (see **Figure 2**). Therefore, these beams appear on the cross screen apart of the

Application of the Snell's law both to input and output sample faces leads to following expres-

where *m* is the mode order, *i* and *ψm* are the incident and output angles of the spatial light beams measured in the XZ plane, and α is the angle between the input and output waveguide

Evidently, normal incidence of the probe beam to the input waveguide face (i.e., the condition *i* = 0) is the simplest scheme variant that is most suitable for measuring. For such scheme, Eq.

*Nm* = sin *ψm*/sin *α* (2)

When graded-index waveguides are examined, the pattern of output beams has a specific view as a set of slightly curved light strips on the cross screen. The simplest way to explain that pattern is application of the known ray approximation of waveguide light propagation. Following to that approach, **Figure 4** demonstrates the rays into shallow graded-index planar

One can see that the ray 3 is radiated being parallel to the sample surface from the output face point with the depth of so-called turning point which corresponds to the depth in the refractive index cross distribution where the mode index is equal to the refractive index value. So, just the direction of that ray in the planform XZ plane should be registered in measuring the waveguide mode spectrum. The ray scheme demonstrates also the rays 2 and 4 radiated from arbitrary point of the output face under equal opposite tilts. The presence of those rays means

**Figure 4.** Ray approximation of mode propagation into graded-index planar waveguide. All plotted rays represent the

waveguide. All drawn rays represent the same waveguide mode.

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

(sin *<sup>ψ</sup>m*/sin *<sup>α</sup>* <sup>+</sup> sin *<sup>i</sup>*/tan *<sup>α</sup>*)<sup>2</sup> <sup>+</sup> (sin *<sup>i</sup>*)2 (1)

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263

sample as separate light strips.

*Nm* = √

faces.

(1) transforms to.

same mode.

sion for calculating the mode indices *Nm*:

**Figure 2.** Measuring block-scheme. 1 – Laser with collimator, 2 – Polarizer, 3 and 6 – Cylindrical lenses, 4 – Examined waveguide sample, 5 – Goniometer sample mount, 7 – Goniometer telescope.

Cylindrical lens in the recording block is not necessary for measuring the mode spectrum. That lens should be installed in procedures of complex measuring as described below.

Two scheme variants providing skew incidence of waveguide beams to the sample output face have been proposed [12]. The former one uses the trapezoidal (in planform section) samples having non-parallel opposite (input and output) waveguide faces as it is shown in **Figure 3**. The latter one allows testing the samples having usual rectangular form with mode launching performed under special procedure by focusing the probe beam on the polished side face of the examined waveguide. Application of a cylindrical lens for mode excitation enables obtaining collimated (in planform XZ section) waveguide beams in both scheme variants. The latter variant is very attractive because of its non-destructive character, but the alternative former scheme is more convenient for conducting measurements.

The optical scheme has been built according to the former measuring variant in our experiments, and **Figure 3** demonstrates light ray paths in planform XZ section of the sample. Whereas the directions of simultaneously excited modes slightly differ due to different refraction at the input sample face, the rays of only one mode beam are shown into the waveguide in order to simplify the drawing. Each output light beam is associated with the individual waveguide mode, and the mode orders of the corresponding modes are identified by the

**Figure 3.** Planform of the sample on the goniometer mount. Two-sided arrow denotes here the cylindrical lens generatrix; other designations are the same as in **Figure 2**.

inclination of a certain output beam to the output waveguide face: the lower the mode order, the bigger the output angle. The fundamental mode forms the light beam having maximal value of the output angle *ψmax*. In the upright YZ projection, the output beams have large divergence (see **Figure 2**). Therefore, these beams appear on the cross screen apart of the sample as separate light strips.

Application of the Snell's law both to input and output sample faces leads to following expression for calculating the mode indices *Nm*:

$$\begin{aligned} \text{Solution for calculating the mode indices } N\_n; \\\\ N\_n &= \sqrt{(\sin \psi\_n / \sin a + \sin i / \tan a)^2 + (\sin \beta)^2} \end{aligned} \tag{1}$$

where *m* is the mode order, *i* and *ψm* are the incident and output angles of the spatial light beams measured in the XZ plane, and α is the angle between the input and output waveguide faces.

Cylindrical lens in the recording block is not necessary for measuring the mode spectrum. That

**Figure 2.** Measuring block-scheme. 1 – Laser with collimator, 2 – Polarizer, 3 and 6 – Cylindrical lenses, 4 – Examined

Two scheme variants providing skew incidence of waveguide beams to the sample output face have been proposed [12]. The former one uses the trapezoidal (in planform section) samples having non-parallel opposite (input and output) waveguide faces as it is shown in **Figure 3**. The latter one allows testing the samples having usual rectangular form with mode launching performed under special procedure by focusing the probe beam on the polished side face of the examined waveguide. Application of a cylindrical lens for mode excitation enables obtaining collimated (in planform XZ section) waveguide beams in both scheme variants. The latter variant is very attractive because of its non-destructive character, but the alternative former

The optical scheme has been built according to the former measuring variant in our experiments, and **Figure 3** demonstrates light ray paths in planform XZ section of the sample. Whereas the directions of simultaneously excited modes slightly differ due to different refraction at the input sample face, the rays of only one mode beam are shown into the waveguide in order to simplify the drawing. Each output light beam is associated with the individual waveguide mode, and the mode orders of the corresponding modes are identified by the

**Figure 3.** Planform of the sample on the goniometer mount. Two-sided arrow denotes here the cylindrical lens generatrix;

lens should be installed in procedures of complex measuring as described below.

scheme is more convenient for conducting measurements.

other designations are the same as in **Figure 2**.

waveguide sample, 5 – Goniometer sample mount, 7 – Goniometer telescope.

262 Emerging Waveguide Technology

Evidently, normal incidence of the probe beam to the input waveguide face (i.e., the condition *i* = 0) is the simplest scheme variant that is most suitable for measuring. For such scheme, Eq. (1) transforms to.

$$N\_m = \sin\psi\_m / \sin\ a \tag{2}$$

When graded-index waveguides are examined, the pattern of output beams has a specific view as a set of slightly curved light strips on the cross screen. The simplest way to explain that pattern is application of the known ray approximation of waveguide light propagation. Following to that approach, **Figure 4** demonstrates the rays into shallow graded-index planar waveguide. All drawn rays represent the same waveguide mode.

One can see that the ray 3 is radiated being parallel to the sample surface from the output face point with the depth of so-called turning point which corresponds to the depth in the refractive index cross distribution where the mode index is equal to the refractive index value. So, just the direction of that ray in the planform XZ plane should be registered in measuring the waveguide mode spectrum. The ray scheme demonstrates also the rays 2 and 4 radiated from arbitrary point of the output face under equal opposite tilts. The presence of those rays means

**Figure 4.** Ray approximation of mode propagation into graded-index planar waveguide. All plotted rays represent the same mode.

that the output light beam is symmetric relatively to the ray 3. Furthermore, the rays 1–3 are radiated from the sample points at different depths where the refractive index has different values. Therefore, the projections of those output rays to the planform XZ should have different directions in that plane. Thus the pattern of the output light beam (corresponding to the certain waveguide mode) on the cross screen apart the sample looks like a curved light strip which is symmetric relatively to the waveguide surface as it is shown in **Figure 5**. As we must register the rays that are analogous to the ray 3, we find those rays at the top of parabolic-like light strip.

Reliability of the results of mode spectrum measuring by the end-fire mode spectroscopy was proved in comparative examinations of graded-index planar waveguides fabricated in optical glasses. The mode spectrum of the same waveguides had been measured independently with the described technique and also by the traditional m-line spectroscopy method. Whereas the main advantage of the end-fire spectroscopy is its capability to analyze buried waveguides, several shallow graded-index samples had been chosen for comparative measuring because the traditional technique provides reliable results only for that type of waveguides.

**Figure 6** presents the photo of the typical pattern formed by output light beams on the cross screen apart from the sample in examinations by the end-fire mode spectroscopy.

The whole spectrum of TE modes was launched simultaneously in this experiment, and the light strips really demonstrate some curvature due to the cross graded-index profile in the examined waveguide layer (see **Figure 6a**). The centers of the parabolic-like light curves were used for measuring the output angles *ψm*. The enlarged view of those central parts of the light strips (**Figure 6b**) demonstrates good separation of the strips. So, they can be easily registered in measuring.

As an example, **Table 1** presents the results of comparative examinations of the planar waveguide fabricated on the substrate of commercial sodium-containing glass K8 by ion exchange in a potassium nitrate melt at 400°C. The probe light of 633 nm wavelength was used, and the bevel angle between the opposite sample faces was measured with the goniometer by the autocollimation method as α = 38<sup>0</sup> 57′ 08′′± 5′′.

> A good agreement between the mode index values measured by both techniques is evident. The difference between the results obtained by those methods does not exceed 10−4, which is similar to the errors considered quite acceptable in traditional mode index measurements.

waveguide, TE modes, λ = 633 nm.

**Figure 6.** Photos of the patterns on the cross screen in procedures of mode spectrum measuring (a, b) and in complex

waveguide. (Here all photos – Negatives, colored positives – In online version).

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Furthermore, the samples of buried waveguide structures have been tested by the end-fire spectroscopy technique. Planar buried waveguides have been formed in optical glasses by two-staged ion exchange. First, the shallow planar waveguides having the maximal refractive index at the sample surface were fabricated. Then the samples were treated into another melt providing appearance of reverse direction of diffusion process into the sample. That procedure led to decreasing the surface refractive index while the maximum of the cross index distribution was shifted deeper to the sample depth. Performed choice of the stage's durations

Mode excitation was performed in examinations by focusing the light beam to the input sample face. However, no waveguide modes have been registered in examinations of those structures by traditional m-line spectroscopy while both direct watching and application of 40x

resulted in fabrication of buried waveguide structures.

measuring (c). λ = 633 nm, planar K8:K<sup>+</sup>

**Table 1.** Comparative examination of planar K8: K<sup>+</sup>

**Figure 5.** Spatial light beam radiated from the output waveguide face.

that the output light beam is symmetric relatively to the ray 3. Furthermore, the rays 1–3 are radiated from the sample points at different depths where the refractive index has different values. Therefore, the projections of those output rays to the planform XZ should have different directions in that plane. Thus the pattern of the output light beam (corresponding to the certain waveguide mode) on the cross screen apart the sample looks like a curved light strip which is symmetric relatively to the waveguide surface as it is shown in **Figure 5**. As we must register the rays that are analogous to the ray 3, we find those rays at the top of parabolic-like light strip. Reliability of the results of mode spectrum measuring by the end-fire mode spectroscopy was proved in comparative examinations of graded-index planar waveguides fabricated in optical glasses. The mode spectrum of the same waveguides had been measured independently with the described technique and also by the traditional m-line spectroscopy method. Whereas the main advantage of the end-fire spectroscopy is its capability to analyze buried waveguides, several shallow graded-index samples had been chosen for comparative measuring because

the traditional technique provides reliable results only for that type of waveguides.

screen apart from the sample in examinations by the end-fire mode spectroscopy.

57′ 08′′± 5′′.

in measuring.

264 Emerging Waveguide Technology

autocollimation method as α = 38<sup>0</sup>

**Figure 5.** Spatial light beam radiated from the output waveguide face.

**Figure 6** presents the photo of the typical pattern formed by output light beams on the cross

The whole spectrum of TE modes was launched simultaneously in this experiment, and the light strips really demonstrate some curvature due to the cross graded-index profile in the examined waveguide layer (see **Figure 6a**). The centers of the parabolic-like light curves were used for measuring the output angles *ψm*. The enlarged view of those central parts of the light strips (**Figure 6b**) demonstrates good separation of the strips. So, they can be easily registered

As an example, **Table 1** presents the results of comparative examinations of the planar waveguide fabricated on the substrate of commercial sodium-containing glass K8 by ion exchange in a potassium nitrate melt at 400°C. The probe light of 633 nm wavelength was used, and the bevel angle between the opposite sample faces was measured with the goniometer by the

**Figure 6.** Photos of the patterns on the cross screen in procedures of mode spectrum measuring (a, b) and in complex measuring (c). λ = 633 nm, planar K8:K<sup>+</sup> waveguide. (Here all photos – Negatives, colored positives – In online version).


**Table 1.** Comparative examination of planar K8: K<sup>+</sup> waveguide, TE modes, λ = 633 nm.

A good agreement between the mode index values measured by both techniques is evident. The difference between the results obtained by those methods does not exceed 10−4, which is similar to the errors considered quite acceptable in traditional mode index measurements.

Furthermore, the samples of buried waveguide structures have been tested by the end-fire spectroscopy technique. Planar buried waveguides have been formed in optical glasses by two-staged ion exchange. First, the shallow planar waveguides having the maximal refractive index at the sample surface were fabricated. Then the samples were treated into another melt providing appearance of reverse direction of diffusion process into the sample. That procedure led to decreasing the surface refractive index while the maximum of the cross index distribution was shifted deeper to the sample depth. Performed choice of the stage's durations resulted in fabrication of buried waveguide structures.

Mode excitation was performed in examinations by focusing the light beam to the input sample face. However, no waveguide modes have been registered in examinations of those structures by traditional m-line spectroscopy while both direct watching and application of 40x and 90x objectives proved the presence of the mode light spots at the output sample face. That means that the obtained burying depths were sufficient for the case when the "tails" of mode field distribution were so weak near the sample surface that their tunneling to the prism did not resulted in appearance of the output light beams which could be registered. Application of the end-fire mode spectroscopy technique allowed analyzing the waveguide structures that supported propagation of single TE mode. The results of performed examinations by the described technique are presented in **Table 2**.

in the recording scheme block in order to collimate the measured light beam. Considering the

where *f* is the focus length of the cylindrical lens, and *d* is the size of the output curved light strip measured along the Y axis behind that lens. Considering (4) and also a known relation

*N*max. Numerical solution of the obtained equation gives the desired value of the maximal

It should be noted that mentioned collimation of the output beam is needed only in measurements of the maximal refractive index when one must register high-divergent boundary rays. Measurement of the mode spectrum is performed by registering the central parts of the output light beams, and it does not matter is the output cylindrical lens applied or not in that case. As an example, let us consider the results of evaluation of the maximal refractive index in

output light beam obtained in measuring by the described technique is shown in **Figure 6c**. For comparison, the maximal refractive index had been measured directly by the end-fire mode spectroscopy and also computed according to conventional methods using the measured mode spectrum. The White-Heidrich method [6] gives the result as *N*max,WH = 1.52204, and the Chiang method [7] is resulted in *N*max,Ch = 1.52138. So, these widespread computing techniques give different results for the same waveguide. Basing on the results obtained for waveguides of that type (see, for example, Ref. [14]), and also taking into account our previous experience we can guess that application of the White-Heidrich technique is more appropriate for reconstruction of refractive index profile in the examined sample because the used fabrication technology results in graded-index layers demonstrating cross refractive index distributions which are well described with the *erfc* function. Direct measuring conducted by the end-fire mode spectroscopy resulted in the value *N*max = 1.5223 when the highest-order mode was registered. This value is evidently closer to the result of the White-Heidrich method than the solution of the Chiang method. It means that the described technique provides direct measuring of the maximal refractive index in graded-index waveguides with rather good accuracy. We must note that some imprecision occurs in measuring caused by diffractive character of real output light beams that defines smudgy ends of registered light strips and therefore impedes obtaining a high accuracy. However, the deviation of the measured result from the actual one is less than the difference (and, consequently, uncertainty) between the results computed according to traditional techniques. In any way, the obtained value is to be considered as the result of maximal index estimation and can be used as itself for further estimations of treatment conditions or waveguide unit performances, as for choosing between

the noted traditional computing techniques of index profile reconstruction.

The pattern shown in **Figure 6c** demonstrates that application of the cylindrical leans in the scheme recording block really does not affect the central parts of light strips and allows

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

][(*N*max /*Nm*)<sup>2</sup> − (cos *α*)2

End-Fire Mode Spectroscopy: A Measuring Technique for Optical Waveguides

*ϕ*, one can deduce from (3) the final equation containing the only unknown

waveguide whose mode spectrum is presented above. The photo of the pattern of

] /sin *<sup>α</sup>* (4)

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parameters of the lens, relation between the mentioned angles can be deduced as.

[1 + (*d*/2*f*)2

tan *<sup>ϕ</sup>* <sup>=</sup> tan *<sup>ϕ</sup>XZ* <sup>√</sup>

refractive index in the graded-index waveguide.

sec2

the K8: K<sup>+</sup>

*ϕ* = 1 + tan2

Thus, the end-fire mode spectroscopy technique has demonstrated successfully its advantageous feature that is capability of examination of planar structures of any type.

#### **2.2. Evaluation of maximal refractive index**

Besides mode spectrum measuring, the technique enables evaluating the maximal refractive index in graded-index waveguides. A principle of that procedure can also be explained involving ray approximation of mode propagation. One can conclude from **Figure 4** that we should register the ray 1 radiated from the waveguide at the sample point having maximal refractive index in the graded-index layer. Determination of that index is performed using the output angle *ϕ* of that boundary ray of the emitted light beam. That angle is marked in **Figure 5**, and we see that it must be measured in the plane formed by both the considered ray and the normal to the output sample face, and that plane is tilted to the planform XZ plane. Ray tracing performed for that boundary ray shows that the maximal refractive index *N*max can be determined by solving Eq. [13].

$$N\_{\text{max}} \cdot \cos\left\{ \arcsin\left[ (N\_w/N\_{\text{max}}) \cos a \right] \right\} = \sin \varphi \tag{3}$$

However, in our optical scheme the goniometer measures the angles lying in the XZ plane. So, we obtain in our measuring the values of the angle *ϕ*xz which is the projection of the angle *ϕ* to the XZ plane, and the relation between those angles should be used in calculations. Another circumstance to be considered is following: both angular and linear apertures of the goniometer telescope are limited, but the registered beam is high-divergent in the upright projection YZ. Therefore, we should apply the cylindrical lens (drawn by dash line in **Figure 2**)


**Table 2.** Examination of buried waveguides by end-fire mode spectroscopy, TE<sup>0</sup> , λ = 633 nm.

in the recording scheme block in order to collimate the measured light beam. Considering the parameters of the lens, relation between the mentioned angles can be deduced as. ] /sin *<sup>α</sup>* (4)

and 90x

266 Emerging Waveguide Technology

described technique are presented in **Table 2**.

**2.2. Evaluation of maximal refractive index**

determined by solving Eq. [13].

objectives proved the presence of the mode light spots at the output sample face. That

means that the obtained burying depths were sufficient for the case when the "tails" of mode field distribution were so weak near the sample surface that their tunneling to the prism did not resulted in appearance of the output light beams which could be registered. Application of the end-fire mode spectroscopy technique allowed analyzing the waveguide structures that supported propagation of single TE mode. The results of performed examinations by the

Thus, the end-fire mode spectroscopy technique has demonstrated successfully its advanta-

Besides mode spectrum measuring, the technique enables evaluating the maximal refractive index in graded-index waveguides. A principle of that procedure can also be explained involving ray approximation of mode propagation. One can conclude from **Figure 4** that we should register the ray 1 radiated from the waveguide at the sample point having maximal refractive index in the graded-index layer. Determination of that index is performed using the output angle *ϕ* of that boundary ray of the emitted light beam. That angle is marked in **Figure 5**, and we see that it must be measured in the plane formed by both the considered ray and the normal to the output sample face, and that plane is tilted to the planform XZ plane. Ray tracing performed for that boundary ray shows that the maximal refractive index *N*max can be

*N*max ⋅ cos{arcsin[(*Nm*/*N*max) cos *α*]} = sin *ϕ* (3)

However, in our optical scheme the goniometer measures the angles lying in the XZ plane. So, we obtain in our measuring the values of the angle *ϕ*xz which is the projection of the angle *ϕ* to the XZ plane, and the relation between those angles should be used in calculations. Another circumstance to be considered is following: both angular and linear apertures of the goniometer telescope are limited, but the registered beam is high-divergent in the upright projection YZ. Therefore, we should apply the cylindrical lens (drawn by dash line in **Figure 2**)

, λ = 633 nm.

**Table 2.** Examination of buried waveguides by end-fire mode spectroscopy, TE<sup>0</sup>

geous feature that is capability of examination of planar structures of any type.

$$\tan\,\varphi = \tan\,\varphi\_{\chi\chi}\sqrt{\left[1+\left(d/2f\right)^2\right]\left[\left(N\_{\text{max}}/N\_n\right)^2-\left(\cos\,\alpha\right)^2\right]}/\sin\,\alpha\tag{4}$$

where *f* is the focus length of the cylindrical lens, and *d* is the size of the output curved light strip measured along the Y axis behind that lens. Considering (4) and also a known relation sec2 *ϕ* = 1 + tan2 *ϕ*, one can deduce from (3) the final equation containing the only unknown *N*max. Numerical solution of the obtained equation gives the desired value of the maximal refractive index in the graded-index waveguide.

It should be noted that mentioned collimation of the output beam is needed only in measurements of the maximal refractive index when one must register high-divergent boundary rays. Measurement of the mode spectrum is performed by registering the central parts of the output light beams, and it does not matter is the output cylindrical lens applied or not in that case.

As an example, let us consider the results of evaluation of the maximal refractive index in the K8: K<sup>+</sup> waveguide whose mode spectrum is presented above. The photo of the pattern of output light beam obtained in measuring by the described technique is shown in **Figure 6c**. For comparison, the maximal refractive index had been measured directly by the end-fire mode spectroscopy and also computed according to conventional methods using the measured mode spectrum. The White-Heidrich method [6] gives the result as *N*max,WH = 1.52204, and the Chiang method [7] is resulted in *N*max,Ch = 1.52138. So, these widespread computing techniques give different results for the same waveguide. Basing on the results obtained for waveguides of that type (see, for example, Ref. [14]), and also taking into account our previous experience we can guess that application of the White-Heidrich technique is more appropriate for reconstruction of refractive index profile in the examined sample because the used fabrication technology results in graded-index layers demonstrating cross refractive index distributions which are well described with the *erfc* function. Direct measuring conducted by the end-fire mode spectroscopy resulted in the value *N*max = 1.5223 when the highest-order mode was registered. This value is evidently closer to the result of the White-Heidrich method than the solution of the Chiang method. It means that the described technique provides direct measuring of the maximal refractive index in graded-index waveguides with rather good accuracy. We must note that some imprecision occurs in measuring caused by diffractive character of real output light beams that defines smudgy ends of registered light strips and therefore impedes obtaining a high accuracy. However, the deviation of the measured result from the actual one is less than the difference (and, consequently, uncertainty) between the results computed according to traditional techniques. In any way, the obtained value is to be considered as the result of maximal index estimation and can be used as itself for further estimations of treatment conditions or waveguide unit performances, as for choosing between the noted traditional computing techniques of index profile reconstruction.

The pattern shown in **Figure 6c** demonstrates that application of the cylindrical leans in the scheme recording block really does not affect the central parts of light strips and allows conducting measurements of the mode spectrum also in that variant of optical scheme. So, the end-fire mode spectroscopy technique allows performing reliable direct complex measurements of the set of important optical characteristics of arbitrary planar waveguides (the mode spectrum and the maximal refractive index) in a single procedure.

**3.2. Requirements to the collimator**

small maladjustment of the collimator.

*ψ*av = (*ψ*<sup>m</sup> + *ψ*<sup>m</sup> <sup>+</sup> <sup>1</sup>

)/(*N*<sup>2</sup>

*N*<sup>m</sup> <sup>+</sup> <sup>1</sup>

and *N*av ≈ (*N*<sup>m</sup> + *N*<sup>m</sup> <sup>+</sup> <sup>1</sup>

max – *N*<sup>2</sup> av)

as *N*max = 1.53, *N*av = 1.51 and also (*N*m – *N*<sup>m</sup> <sup>+</sup> <sup>1</sup>

**3.3. Tolerance of incident beam direction**

Adjustment of the collimator must provide acceptable divergence of the collimated light beam. Let us first consider the axial section of that beam by the plane normal to the waveguide surface (the upright plane YZ in **Figure 2**). One can conclude from the drawn ray scheme that divergent (in that upright projection) beam causes a mismatch between apertures of waveguide mode and exciting focused spatial light beam. That mismatch results in decreasing the excitation efficiency. However, it does not influence on the registered angular values (they are measured into another planform projection XZ). Now let us consider the affect of beam divergence into that plane XZ (that coincides with the sample surface) on the results of angular measurements. One can conclude from **Figure 3** that if a divergence of the incident light beam is small, each output beam acquires a slight angular broadening, but it remains approximately axisymmetric relatively to the direction *ψ*m. Therefore, registration of the central direction of each weakly divergent output light beam eliminates the affect of possible

End-Fire Mode Spectroscopy: A Measuring Technique for Optical Waveguides

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Another requirement concerns performances of examined waveguides: the divergence of each output beam must be less than the angles between the output beams corresponding to waveguide modes of adjacent orders. Otherwise, overlapped different beams could not be distinguished and measured. Let us evaluate these angles basing on performances of gradedindex waveguides which mostly demonstrate the closer mode indices (and, hence, would form the least beam spatial separation in the considered scheme) for higher-order modes. Considering the presented condition for the bevel angle α, we can deduce following expres-

where *N*av – the value corresponding to the mean direction between considered output beams,

guide fabricated by ion-exchanged diffusion in commercial glass. Defining the model values

4 × 10−4 radian ≈ 1.4 arcmin. So, the divergence of the collimated light beam must not exceed

Thus we can expect that the source of the main affect on the results could be deviations from

As Eq. (1) involves the angle of incidence *i*, that angle can be considered as a factor capable of affecting the reliability of the results of mode index measurements. Being considered together, Eqs. (1) and (2) allow evaluating the occurred errors caused by deviation of the incident beam from the condition of normal incidence to the input waveguide face. Indeed, in the basic measuring regime calculations of mode indices are executed according to Eq. (2). However, if the mode launching unit is adjusted inaccurately (i.e. when *i* ≠ 0), substitution of

)/2. On the other hand, we obtain from Eq. (1) that cos *ψ*<sup>m</sup> = [1 – (*N*av /*N*max)2

)/2. Hence the angle between those light beams δ*ψ* = *ψ*m – *ψ*<sup>m</sup> <sup>+</sup> <sup>1</sup> ≈ (*N*m –

) = 10−4 for higher-order modes, we obtain δ*ψ* ≈

1/2. As an example, let us evaluate δ*ψ* for some planar graded-index wave-

) cos *ψ*av, where

]1/2

269

sion from Eq.(1) for adjacent higher-order modes: *N*m – *N*<sup>m</sup> <sup>+</sup> <sup>1</sup> ≈ *N*max × (*ψ*m – *ψ*<sup>m</sup> <sup>+</sup> <sup>1</sup>

that limit. That condition can be easily met in adjusting the optical scheme.

the conditions of mode excitation occurring in the launching scheme unit.

#### **3. Measuring conditions**

#### **3.1. Choice of the sample form**

For enhancing the technique sensitivity and accuracy, we must consider the conditions providing maximal variation δ*ψ*m of the light beam emission line related to mode index variation δ*N*m. Referring to Eq. (2), one can see that the value of the term ∂*ψ*m/∂*N*<sup>m</sup> = sin α/(1 − *N*<sup>2</sup> m sin2 α)1/2 increases with a rise in the bevel angle α between the opposite waveguide faces. So, in order to provide high method sensitivity we must maximize that angle. The same can be concluded from the dependence *ψ*m(α) that is plotted in **Figure 7**.

However, when α approaches the value αlim = arcsin(1/*N*m), total inner reflection appears at the output waveguide face, and the mode is reflected back to the waveguide (to the side sample face) instead of being emitted out through the output sample face. That limit value is marked in the graph with the dash line. Proper choice of the angle α should be performed considering the maximal refractive index value, which is expected to be obtained into the sample by the applied method of waveguide fabrication. Slightly overestimated value of that refractive index is substituted for *N*m to the expression for αlim. Being calculated in such a manner, angle limit represents the optimal value of the angle α. Indeed, as the mode indices are always less than the maximal refractive index in a waveguide, described manner of the bevel angle choice prevents the occurrence of total inner reflection at the output sample face for all modes of the examined waveguide.

**Figure 7.** Influence of the sample bevel angle on the sensitivity of measuring. *N*<sup>m</sup> = 1.55.

#### **3.2. Requirements to the collimator**

conducting measurements of the mode spectrum also in that variant of optical scheme. So, the end-fire mode spectroscopy technique allows performing reliable direct complex measurements of the set of important optical characteristics of arbitrary planar waveguides (the

For enhancing the technique sensitivity and accuracy, we must consider the conditions providing maximal variation δ*ψ*m of the light beam emission line related to mode index variation δ*N*m. Referring to Eq. (2), one can see that the value of the term ∂*ψ*m/∂*N*<sup>m</sup> = sin α/(1 − *N*<sup>2</sup>

α)1/2 increases with a rise in the bevel angle α between the opposite waveguide faces. So, in order to provide high method sensitivity we must maximize that angle. The same can be

However, when α approaches the value αlim = arcsin(1/*N*m), total inner reflection appears at the output waveguide face, and the mode is reflected back to the waveguide (to the side sample face) instead of being emitted out through the output sample face. That limit value is marked in the graph with the dash line. Proper choice of the angle α should be performed considering the maximal refractive index value, which is expected to be obtained into the sample by the applied method of waveguide fabrication. Slightly overestimated value of that refractive index is substituted for *N*m to the expression for αlim. Being calculated in such a manner, angle limit represents the optimal value of the angle α. Indeed, as the mode indices are always less than the maximal refractive index in a waveguide, described manner of the bevel angle choice prevents the occurrence of total inner reflection at the output sample face for all

m

mode spectrum and the maximal refractive index) in a single procedure.

concluded from the dependence *ψ*m(α) that is plotted in **Figure 7**.

**Figure 7.** Influence of the sample bevel angle on the sensitivity of measuring. *N*<sup>m</sup> = 1.55.

**3. Measuring conditions**

268 Emerging Waveguide Technology

**3.1. Choice of the sample form**

modes of the examined waveguide.

sin2

Adjustment of the collimator must provide acceptable divergence of the collimated light beam. Let us first consider the axial section of that beam by the plane normal to the waveguide surface (the upright plane YZ in **Figure 2**). One can conclude from the drawn ray scheme that divergent (in that upright projection) beam causes a mismatch between apertures of waveguide mode and exciting focused spatial light beam. That mismatch results in decreasing the excitation efficiency. However, it does not influence on the registered angular values (they are measured into another planform projection XZ). Now let us consider the affect of beam divergence into that plane XZ (that coincides with the sample surface) on the results of angular measurements. One can conclude from **Figure 3** that if a divergence of the incident light beam is small, each output beam acquires a slight angular broadening, but it remains approximately axisymmetric relatively to the direction *ψ*m. Therefore, registration of the central direction of each weakly divergent output light beam eliminates the affect of possible small maladjustment of the collimator.

Another requirement concerns performances of examined waveguides: the divergence of each output beam must be less than the angles between the output beams corresponding to waveguide modes of adjacent orders. Otherwise, overlapped different beams could not be distinguished and measured. Let us evaluate these angles basing on performances of gradedindex waveguides which mostly demonstrate the closer mode indices (and, hence, would form the least beam spatial separation in the considered scheme) for higher-order modes. Considering the presented condition for the bevel angle α, we can deduce following expression from Eq.(1) for adjacent higher-order modes: *N*m – *N*<sup>m</sup> <sup>+</sup> <sup>1</sup> ≈ *N*max × (*ψ*m – *ψ*<sup>m</sup> <sup>+</sup> <sup>1</sup> ) cos *ψ*av, where *ψ*av = (*ψ*<sup>m</sup> + *ψ*<sup>m</sup> <sup>+</sup> <sup>1</sup> )/2. On the other hand, we obtain from Eq. (1) that cos *ψ*<sup>m</sup> = [1 – (*N*av /*N*max)2 ]1/2 where *N*av – the value corresponding to the mean direction between considered output beams, and *N*av ≈ (*N*<sup>m</sup> + *N*<sup>m</sup> <sup>+</sup> <sup>1</sup> )/2. Hence the angle between those light beams δ*ψ* = *ψ*m – *ψ*<sup>m</sup> <sup>+</sup> <sup>1</sup> ≈ (*N*m – *N*<sup>m</sup> <sup>+</sup> <sup>1</sup> )/(*N*<sup>2</sup> max – *N*<sup>2</sup> av) 1/2. As an example, let us evaluate δ*ψ* for some planar graded-index waveguide fabricated by ion-exchanged diffusion in commercial glass. Defining the model values as *N*max = 1.53, *N*av = 1.51 and also (*N*m – *N*<sup>m</sup> <sup>+</sup> <sup>1</sup> ) = 10−4 for higher-order modes, we obtain δ*ψ* ≈ 4 × 10−4 radian ≈ 1.4 arcmin. So, the divergence of the collimated light beam must not exceed that limit. That condition can be easily met in adjusting the optical scheme.

Thus we can expect that the source of the main affect on the results could be deviations from the conditions of mode excitation occurring in the launching scheme unit.

#### **3.3. Tolerance of incident beam direction**

As Eq. (1) involves the angle of incidence *i*, that angle can be considered as a factor capable of affecting the reliability of the results of mode index measurements. Being considered together, Eqs. (1) and (2) allow evaluating the occurred errors caused by deviation of the incident beam from the condition of normal incidence to the input waveguide face. Indeed, in the basic measuring regime calculations of mode indices are executed according to Eq. (2). However, if the mode launching unit is adjusted inaccurately (i.e. when *i* ≠ 0), substitution of measured values *ψ*m to Eq. (2) leads to appearance of errors because the right way is application of Eq. (1) in this case.

Let us evaluate those errors. We can deduce from Eq. (2) that *∂Nm/∂ψ<sup>m</sup>* = (cos *ψm*/ sin α). Ray tracing that was performed for the scheme shown in **Figure 3** resulted in following *ψm(i)* dependence: *ψ<sup>m</sup>* = arcsin{*Nm* sin[α − arcsin(sin *i/ Nm*)]}. We could note that the operating variant of the measuring technique is the case *i* ≈ 0. Then, considering negligibility of the angle *i*, the term *∂ψm/ ∂i* can be written as *∂ψm/ ∂i* ≈ − cos α / cos *ψm*. Substituting that term into the expression for *∂Nm/ ∂ψm* and defining Δ*Nm* = *|∂Nm|*, we can evaluate the mode index error as Δ*Nm* = Δ*i* / tan α. Here Δ*i* denotes the increment of angle *i* from some value. Since we are considering the deviation from the condition of normal incidence of the input beam (i.e., from zero angle of incidence), just the small angle *i* plays itself the role of the increment Δ*i* here. Then, replacing the values in the derived expression, we obtain the desired dependence of the mode index error on the angle of incidence:

$$
\Delta N\_m(\mathbf{i}) = \mathbf{i}/\tan\alpha \tag{5}
$$

technique but also to reduce the influence of deviation from the condition of normal beam

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Besides the inclination of the incident beam in the XZ plane, occurrence of some turn of the cylindrical lens generatrix relatively to the waveguide plane in the launching scheme block is possible in measuring. Let us consider that case of the lens rotated around its optical axis assuming that the lens axis is directed along the normal to the input sample face. The ray

Let the lens generatrix be tilted toward the surface of the sample by angle γ. Then the plane of incidence of the focused light beam (that involves the two-sided arrow denoting the lens in the scheme) is turned to the vertical Y axis by the same angle as it is shown in **Figure 9**. Considering reversible character of light paths in optical systems containing the passive

paths in waveguide mode excitation procedure are shown in **Figure 9**.

**Figure 9.** Waveguide mode excitation under the turn of input cylindrical lens.

**Figure 8.** Errors caused by deviation from the condition of normal incidence of exciting beam.

incidence to the input waveguide face.

**3.4. Adjustment of input cylindrical lens**

where *i* is a small angle in radian measure. This expression indicates that the mentioned mode index errors caused by wrong application of Eq. (2) in the case of deviation from the condition of normal incidence of the input light beam will be identical for all modes in the examined waveguide.

To confirm that unobvious conclusion, we can consider the described sample of K8: K<sup>+</sup> waveguide and evaluate numerically the mentioned errors by another manner. In this case we assume that considered mode index error can be defined as Δ*N*m(*i*) = *N*<sup>m</sup> − *N*' m, where *N*' m is the incorrect value calculating according to Eq.(2) without involving the angle of incidence, and *N*m is the actual mode index value. The mode indices measured by the traditional m-line spectroscopy technique (see **Table 1**) are substituted for those actual values. Defining the angle of incidence *i* as a variable and using the actual mode index values, we calculated from Eq. (1) the values of the output angle *ψ*' m that would be obtained in measurements for the given angles *i*. Then we substitute those angles *ψ*' m into Eq. (2) and find the corresponding *N*' m values. The errors of mode index determination are obtained by comparing the actual mode indices with those calculated *N*' m values. The results of determination of Δ*N*m(*i*) values by that manner are shown in **Figure 8**.

It can be noted that the errors calculated for the modes of different orders are really identical in the most practical cases of small angles of incidence. The Δ*N*m(*i*) dependence obtained for that waveguide sample by calculation according to Eq. (5) coincides completely with the one shown in this figure. So, the obtained results confirm a validity of the derived Eq. (5), and it can be applied for evaluation of mode index errors. We can note here that precise adjustment of the used GS-5 goniometer sample mount in our experiments allowed limiting the values of the angle of incidence within the range of 5 arcsec. Therefore the mentioned errors do not exceed 3 × 10−5, and that is quite acceptable value corresponding to the usual level of errors occurred in measuring by the traditional m-line spectroscopy technique.

One can also see from Eq. (5) that the choice of the bevel angle α closer to the upper limit value makes it possible not only to increase the sensitivity of the end-fire mode spectroscopy

**Figure 8.** Errors caused by deviation from the condition of normal incidence of exciting beam.

technique but also to reduce the influence of deviation from the condition of normal beam incidence to the input waveguide face.

#### **3.4. Adjustment of input cylindrical lens**

measured values *ψ*m to Eq. (2) leads to appearance of errors because the right way is applica-

Let us evaluate those errors. We can deduce from Eq. (2) that *∂Nm/∂ψ<sup>m</sup>* = (cos *ψm*/ sin α). Ray tracing that was performed for the scheme shown in **Figure 3** resulted in following *ψm(i)* dependence: *ψ<sup>m</sup>* = arcsin{*Nm* sin[α − arcsin(sin *i/ Nm*)]}. We could note that the operating variant of the measuring technique is the case *i* ≈ 0. Then, considering negligibility of the angle *i*, the term *∂ψm/ ∂i* can be written as *∂ψm/ ∂i* ≈ − cos α / cos *ψm*. Substituting that term into the expression for *∂Nm/ ∂ψm* and defining Δ*Nm* = *|∂Nm|*, we can evaluate the mode index error as Δ*Nm* = Δ*i* / tan α. Here Δ*i* denotes the increment of angle *i* from some value. Since we are considering the deviation from the condition of normal incidence of the input beam (i.e., from zero angle of incidence), just the small angle *i* plays itself the role of the increment Δ*i* here. Then, replacing the values in the derived expression, we obtain the desired dependence of the mode index error on the

where *i* is a small angle in radian measure. This expression indicates that the mentioned mode index errors caused by wrong application of Eq. (2) in the case of deviation from the condition of normal incidence of the input light beam will be identical for all modes in the examined

guide and evaluate numerically the mentioned errors by another manner. In this case we

the incorrect value calculating according to Eq.(2) without involving the angle of incidence, and *N*m is the actual mode index value. The mode indices measured by the traditional m-line spectroscopy technique (see **Table 1**) are substituted for those actual values. Defining the angle of incidence *i* as a variable and using the actual mode index values, we calculated from

values. The errors of mode index determination are obtained by comparing the actual mode

It can be noted that the errors calculated for the modes of different orders are really identical in the most practical cases of small angles of incidence. The Δ*N*m(*i*) dependence obtained for that waveguide sample by calculation according to Eq. (5) coincides completely with the one shown in this figure. So, the obtained results confirm a validity of the derived Eq. (5), and it can be applied for evaluation of mode index errors. We can note here that precise adjustment of the used GS-5 goniometer sample mount in our experiments allowed limiting the values of the angle of incidence within the range of 5 arcsec. Therefore the mentioned errors do not exceed 3 × 10−5, and that is quite acceptable value corresponding to the usual level of errors

One can also see from Eq. (5) that the choice of the bevel angle α closer to the upper limit value makes it possible not only to increase the sensitivity of the end-fire mode spectroscopy

To confirm that unobvious conclusion, we can consider the described sample of K8: K<sup>+</sup>

assume that considered mode index error can be defined as Δ*N*m(*i*) = *N*<sup>m</sup> − *N*'

occurred in measuring by the traditional m-line spectroscopy technique.

(*i*) = *i*/tan *α* (5)

m that would be obtained in measurements for the

m values. The results of determination of Δ*N*m(*i*) values by that

m into Eq. (2) and find the corresponding *N*'

wave-

m is

m

m, where *N*'

tion of Eq. (1) in this case.

270 Emerging Waveguide Technology

angle of incidence:

waveguide.

Δ *Nm*

Eq. (1) the values of the output angle *ψ*'

indices with those calculated *N*'

manner are shown in **Figure 8**.

given angles *i*. Then we substitute those angles *ψ*'

Besides the inclination of the incident beam in the XZ plane, occurrence of some turn of the cylindrical lens generatrix relatively to the waveguide plane in the launching scheme block is possible in measuring. Let us consider that case of the lens rotated around its optical axis assuming that the lens axis is directed along the normal to the input sample face. The ray paths in waveguide mode excitation procedure are shown in **Figure 9**.

Let the lens generatrix be tilted toward the surface of the sample by angle γ. Then the plane of incidence of the focused light beam (that involves the two-sided arrow denoting the lens in the scheme) is turned to the vertical Y axis by the same angle as it is shown in **Figure 9**. Considering reversible character of light paths in optical systems containing the passive

**Figure 9.** Waveguide mode excitation under the turn of input cylindrical lens.

elements only, we can consider our case in the reverse direction as an emission of the waveguide mode from the output waveguide face. That approach allows using the reasons and the results described above.

It is shown that the output mode beam looks like a curved light strip on the cross screen when a skew incidence of collimated beam to the output face is performed into the waveguide, and the strip edges demonstrate maximal inclinations from the longitudinal upright section. Then our studied case can be considered as reversible one – the tilted boundary rays of the incident light beam formed with the turned cylindrical lens excite the mode which is directed at some angle to the normal to the input sample face (see **Figure 9**) while the paraxial rays of the incident beam still meet the condition of normal incidence. That means excitation of divergent (in the planform XZ plane) waveguide mode beam, and consequently the spatial light beam which is radiated from the opposite output sample face should demonstrate angular broadening into that XZ projection.

If the numerical aperture of the exciting lens, as well as the size and position of the light spot at the input sample face are matched with the corresponding parameters of the excited mode, the output light beam is approximately axisymmetric relatively to the direction of unperturbed output beam, and that produces no additional errors neither in measuring the output beam direction nor in further calculation of the mode index. However, a set of waveguide modes is excited usually with the end-fire technique, and there is a natural wish to use this circumstance by measuring the characteristics of several modes under the single adjustment procedure. For this purpose, one scans the input end with the focused input beam and chooses a beam position that leads to optimizing the visibility of the mode's set. Then only the part of the shifted incident beam actually excites some individual mode. As the exciting lens is turned in the considered manner, the output light beam is broadened asymmetrically (relatively to the output mode direction that could be in the case of zero lens turn) as can be seen in **Figure 9**. Just that reason leads to additional errors in measuring. The detailed procedure of determination of dependence Δ*N*m(γ) is presented in Ref. [15]. Here we show in **Figure 10** the example of that dependence calculated for the waveguide K8: K<sup>+</sup> considered above. Unlike the error type described in the preceding paragraph, one can see that the errors caused by lens turning differ one from another for modes of different orders, and the lowest-order mode indicate the minimum error. The noted difference between the errors decreases as the mode order increases, and for the two highest modes the mentioned errors almost coincide.

For tuning the lens turn, a rather simple technique associated with a variant of autocollimation method could be suggested. The performed procedures are illustrated by **Figures 11** and **12**. First the tilt of the incident light beam should be registered by watching through the goniometer telescope or with the photoreceiver matrix. Then the reference glass cube (given in the goniometer tool kit) is installed on the goniometer sample mount, and the reflected light beam is registered near the opposite direction according to the scheme in **Figure 11**. Registered light

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If the measuring system is misadjusted in parameter γ, the strips representing direct and reflected beams demonstrate opposite inclinations at the angle of 2γ to each other. Turning

**Figure 11.** Adjusting the orientation of input cylindrical lens under consecutive registration of direct and reflected light

strip patterns are shown in **Figure 12**.

beams.

**Figure 10.** Errors caused by the turn of input cylindrical lens.

Whereas the considered error Δ*N*m(γ) is substantially less than the error Δ*N*m(*i*) for identical values of the angles *i* and γ, the former error caused by the lens turning dominates in practice because the used standard equipment provides precise control of normal light incidence to the sample face, but does not enable accurate orienting the lens generatrix relatively to the sample surface. In those cases the expected errors of the lens orientation could be up to the order of arc minutes, and therefore just that range of angle γ variations is used in the graph shown in **Figure 10**. It is seen that mode index errors are rather significant for those angles γ, and, in order to keep the errors within acceptable limits (no greater than 10−4), one should try to control the orientation of the cylindrical lens with accuracy of about 2–3 arcmin.

End-Fire Mode Spectroscopy: A Measuring Technique for Optical Waveguides http://dx.doi.org/10.5772/intechopen.75558 273

**Figure 10.** Errors caused by the turn of input cylindrical lens.

elements only, we can consider our case in the reverse direction as an emission of the waveguide mode from the output waveguide face. That approach allows using the reasons and the

It is shown that the output mode beam looks like a curved light strip on the cross screen when a skew incidence of collimated beam to the output face is performed into the waveguide, and the strip edges demonstrate maximal inclinations from the longitudinal upright section. Then our studied case can be considered as reversible one – the tilted boundary rays of the incident light beam formed with the turned cylindrical lens excite the mode which is directed at some angle to the normal to the input sample face (see **Figure 9**) while the paraxial rays of the incident beam still meet the condition of normal incidence. That means excitation of divergent (in the planform XZ plane) waveguide mode beam, and consequently the spatial light beam which is radiated from the opposite output sample face should demonstrate angular broaden-

If the numerical aperture of the exciting lens, as well as the size and position of the light spot at the input sample face are matched with the corresponding parameters of the excited mode, the output light beam is approximately axisymmetric relatively to the direction of unperturbed output beam, and that produces no additional errors neither in measuring the output beam direction nor in further calculation of the mode index. However, a set of waveguide modes is excited usually with the end-fire technique, and there is a natural wish to use this circumstance by measuring the characteristics of several modes under the single adjustment procedure. For this purpose, one scans the input end with the focused input beam and chooses a beam position that leads to optimizing the visibility of the mode's set. Then only the part of the shifted incident beam actually excites some individual mode. As the exciting lens is turned in the considered manner, the output light beam is broadened asymmetrically (relatively to the output mode direction that could be in the case of zero lens turn) as can be seen in **Figure 9**. Just that reason leads to additional errors in measuring. The detailed procedure of determination of dependence Δ*N*m(γ) is presented in Ref. [15]. Here we show in **Figure 10** the example of that dependence calculated for the waveguide K8: K<sup>+</sup> considered above. Unlike the error type described in the preceding paragraph, one can see that the errors caused by lens turning differ one from another for modes of different orders, and the lowest-order mode indicate the minimum error. The noted difference between the errors decreases as the mode order increases, and for the two highest modes the mentioned

Whereas the considered error Δ*N*m(γ) is substantially less than the error Δ*N*m(*i*) for identical values of the angles *i* and γ, the former error caused by the lens turning dominates in practice because the used standard equipment provides precise control of normal light incidence to the sample face, but does not enable accurate orienting the lens generatrix relatively to the sample surface. In those cases the expected errors of the lens orientation could be up to the order of arc minutes, and therefore just that range of angle γ variations is used in the graph shown in **Figure 10**. It is seen that mode index errors are rather significant for those angles γ, and, in order to keep the errors within acceptable limits (no greater than 10−4), one should try

to control the orientation of the cylindrical lens with accuracy of about 2–3 arcmin.

results described above.

272 Emerging Waveguide Technology

ing into that XZ projection.

errors almost coincide.

For tuning the lens turn, a rather simple technique associated with a variant of autocollimation method could be suggested. The performed procedures are illustrated by **Figures 11** and **12**.

First the tilt of the incident light beam should be registered by watching through the goniometer telescope or with the photoreceiver matrix. Then the reference glass cube (given in the goniometer tool kit) is installed on the goniometer sample mount, and the reflected light beam is registered near the opposite direction according to the scheme in **Figure 11**. Registered light strip patterns are shown in **Figure 12**.

If the measuring system is misadjusted in parameter γ, the strips representing direct and reflected beams demonstrate opposite inclinations at the angle of 2γ to each other. Turning

**Figure 11.** Adjusting the orientation of input cylindrical lens under consecutive registration of direct and reflected light beams.

**Author details**

Dmitry V. Svistunov

**References**

Address all correspondence to: svistunov@mail.ru

p. DOI: 10.1007/978-1-4684-2082-1

DOI: 10.1088/1742-6596/909/1/012022

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guides. Opitcs Express. 2004;**12**(2):294-298. DOI: 10.1364/OPEX.12.000294

**Figure 12.** Patterns from the goniometer telescope in adjusting procedure: Direct (a) and reflected (b) light beams before, and also after (c) the procedure.

the lens around its ax and repeating the procedures one can obtain coincidence of orientations of those light strips. This will be the criterion for correct orientation of the cylindrical lens – the lens generatrix gets right orientation parallel to the sample mount surface. For example: using the micrometer ocular of the goniometer tool kit we were able to adjust the lens orientation with the accuracy of 2 arcmin. It can be concluded from the dependences shown in **Figure 10** that the corresponding mode index errors are reduced to an acceptable level. We can also note that our evaluation results in the maximum level of the measurement errors when the cylindrical lens is rotated around the axis, accompanied by the displacement of the input beam as it scans over the end of the waveguide. The mode index errors decrease as the incident beam is shifted toward the position that matches the region of localization of the measured mode, and the range of allowable lens tilts is broadened.

#### **4. Conclusion**

The presented materials prove applicability of the end-fire mode spectroscopy technique to analysis of planar optical waveguides with arbitrary cross refractive index profiles, and performed measurements of the characteristics of buried waveguides highlight this advantage of the technique. Furthermore, the technique allows conducting reliable direct complex measurements of the set of important optical characteristics of arbitrary planar waveguides (the mode spectrum and the maximal refractive index) in a single procedure. End-fire mode spectroscopy has a good potential for wide practical application in examinations of planar structures. Further developments should be aimed at modifying the measuring scheme in order to be able to analyze 3D optical guides. That could allow extending the area of technique applications by involving additional large group of waveguides including optical fibers.

## **Author details**

Dmitry V. Svistunov

Address all correspondence to: svistunov@mail.ru

Peter-the-Great St.-Petersburg Polytechnic University, St.-Petersburg, Russia

### **References**

the lens around its ax and repeating the procedures one can obtain coincidence of orientations of those light strips. This will be the criterion for correct orientation of the cylindrical lens – the lens generatrix gets right orientation parallel to the sample mount surface. For example: using the micrometer ocular of the goniometer tool kit we were able to adjust the lens orientation with the accuracy of 2 arcmin. It can be concluded from the dependences shown in **Figure 10** that the corresponding mode index errors are reduced to an acceptable level. We can also note that our evaluation results in the maximum level of the measurement errors when the cylindrical lens is rotated around the axis, accompanied by the displacement of the input beam as it scans over the end of the waveguide. The mode index errors decrease as the incident beam is shifted toward the position that matches the region of localization of

**Figure 12.** Patterns from the goniometer telescope in adjusting procedure: Direct (a) and reflected (b) light beams before,

The presented materials prove applicability of the end-fire mode spectroscopy technique to analysis of planar optical waveguides with arbitrary cross refractive index profiles, and performed measurements of the characteristics of buried waveguides highlight this advantage of the technique. Furthermore, the technique allows conducting reliable direct complex measurements of the set of important optical characteristics of arbitrary planar waveguides (the mode spectrum and the maximal refractive index) in a single procedure. End-fire mode spectroscopy has a good potential for wide practical application in examinations of planar structures. Further developments should be aimed at modifying the measuring scheme in order to be able to analyze 3D optical guides. That could allow extending the area of technique applications by involving additional large group of waveguides includ-

the measured mode, and the range of allowable lens tilts is broadened.

**4. Conclusion**

and also after (c) the procedure.

274 Emerging Waveguide Technology

ing optical fibers.


[11] Monir M, El-Refaei H, Khalil D. Single-mode refractive index reconstruction using an nm-line technique. Fiber and Integrated Optics. 2006;**25**(2):69-74. DOI: 10.1080/014680 30500466230

**Chapter 15**

**Provisional chapter**

**Polymer Resonant Waveguide Gratings**

**Polymer Resonant Waveguide Gratings**

DOI: 10.5772/intechopen.76917

This chapter deals with the advances in polymeric waveguide gratings for filtering and integrated optics applications. Optical polymer materials are widely used for planar and corrugated micro-optical waveguide grating structures ranging from down a micrometer to several hundred micrometers. Light in a polymeric waveguide is transmitted in discrete modes whose propagation orders depend on incident wavelength, waveguide dimensional parameters, and material properties. Diffracted optical structures are permittivity-modulated microstructures whose micro-relief surface profiles exhibit global/local periodicity. The resonant nature and location of such globally periodic structures (diffraction gratings) excite leaky waveguide modes which couple incident light into reflected/ transmitted plane wave diffraction orders. It describes design & analysis, fabrication, and characterization of sub-wavelength polymer grating structures replicated in different polymeric materials (polycarbonate, cyclic olefin copolymer, Ormocomp) by a simple, cost-effective, accurate, and large scale production method. The master stamp (mold) for polymer replication is fabricated with an etchless process with smooth surface profile.

**Keywords:** resonant waveguide gratings, polymeric materials, nanoimprint lithography

Conventional optical waveguides work on the principle to guide waves in a material surrounded by other material media, the refractive index of the material should be slightly higher than that of surrounded media such that light can bounce along the waveguide by means of total internal reflections at the boundaries between different media. The indefinite guiding progress the waves from successive boundaries which must interfere constructively to generate a continuous and stable interference pattern along the waveguide. If the interference pattern in not fully constructive, the waves cancel, owing to the self-destruction. The conventional

> © 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.

Muhammad Rizwan Saleem and Rizwan Ali

Muhammad Rizwan Saleem and Rizwan Ali

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.76917

**Abstract**

**1. Introduction**


#### **Polymer Resonant Waveguide Gratings Polymer Resonant Waveguide Gratings**

Muhammad Rizwan Saleem and Rizwan Ali Muhammad Rizwan Saleem and Rizwan Ali

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.76917

#### **Abstract**

[11] Monir M, El-Refaei H, Khalil D. Single-mode refractive index reconstruction using an nm-line technique. Fiber and Integrated Optics. 2006;**25**(2):69-74. DOI: 10.1080/014680

[12] Svistunov DV. End-fire mode spectroscopy technique of examination of planar waveguides. Journal of Optics A: Pure and Applied Optics. 2008;**10**(8):085301(4 pp). DOI:

[13] Svistunov DV. New measuring method of examination of planar optical waveguides. In: Proceedings of Progress in Electromagnetics Research Symposium (PIERS 2009 in Beijing); 23-27 March 2009; Beijing. 2009. pp. 1689-1693. DOI: 10.2529/PIERS080904135015

[14] Zolotov EM, Kiselyov VA, Pelekhaty VM. Determination of characteristics of optical diffuse waveguides. Soviet J Quant. Electr. 1978;**8**(11):2386-2382. DOI: 10.1070/QE1978

[15] Svistunov DV. Optimizing the mode-excitation conditions when a planar waveguide is being investigated by means of end-fire mode spectroscopy. Journal of Optical

Technology. 2014;**81**(1):1-5. DOI: 10.1364/JOT.81.000001

30500466230

276 Emerging Waveguide Technology

10.1088/1464-4258/10/8/085301

v008n11ABEH011261

This chapter deals with the advances in polymeric waveguide gratings for filtering and integrated optics applications. Optical polymer materials are widely used for planar and corrugated micro-optical waveguide grating structures ranging from down a micrometer to several hundred micrometers. Light in a polymeric waveguide is transmitted in discrete modes whose propagation orders depend on incident wavelength, waveguide dimensional parameters, and material properties. Diffracted optical structures are permittivity-modulated microstructures whose micro-relief surface profiles exhibit global/local periodicity. The resonant nature and location of such globally periodic structures (diffraction gratings) excite leaky waveguide modes which couple incident light into reflected/ transmitted plane wave diffraction orders. It describes design & analysis, fabrication, and characterization of sub-wavelength polymer grating structures replicated in different polymeric materials (polycarbonate, cyclic olefin copolymer, Ormocomp) by a simple, cost-effective, accurate, and large scale production method. The master stamp (mold) for polymer replication is fabricated with an etchless process with smooth surface profile.

DOI: 10.5772/intechopen.76917

**Keywords:** resonant waveguide gratings, polymeric materials, nanoimprint lithography

#### **1. Introduction**

Conventional optical waveguides work on the principle to guide waves in a material surrounded by other material media, the refractive index of the material should be slightly higher than that of surrounded media such that light can bounce along the waveguide by means of total internal reflections at the boundaries between different media. The indefinite guiding progress the waves from successive boundaries which must interfere constructively to generate a continuous and stable interference pattern along the waveguide. If the interference pattern in not fully constructive, the waves cancel, owing to the self-destruction. The conventional

© 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.

optical waveguides are primarily the most common type of thin film optical filters used widely as narrowband filters in laser cavities, optical telecommunications, and light modulators [1]. However, the realization of sub-nanometer narrowband filters with thin film technique require hundreds of optical thin-films stack with stringent tolerances over thicknesses and refractive index variations [2]. Resonant Waveguide Gratings (RWGs) are a new class of narrowband filters and are widely used in applications such as polarizers [3], laser cavity reflectors [3, 4], light modulators [5], biosensors [6], and wavelength division multiplexing [7]. Such narrowband reflectance/transmittance spectral characteristics can be observed by sub-wavelength grating structures in terms of resonance anomalies [8, 9] with numerous potential applications. RWG operates with resonance effects with relatively simpler structure of few layers. Owing to the resonant nature of the sub-wavelength grating, the leaky modes are supported by the structure (waveguide layer and a grating layer) [10]. In the absence of the grating layer, waveguide layer supports a true bound mode. This bound mode becomes leaky when a grating layer is added with the waveguide layer. Eventually, optical energy is coupled out of the waveguide into radiation modes. On the other hand, the incident plane wave energy is coupled to the waveguide. The incident plane wave energy is coupled into leaky modes and then back to one or more radiation modes. This coupling mostly depends on the wavelength, angle of incidence and other structural parameters of the grating layer. At resonance a sharp peak in reflected/ transmitted light might be observed at a specific combination of these parameters.

stamp by electron beam lithography and subsequent replication in polymer materials by NIL is presented to target a number of applications in thermoplastics and UV curable polymers.

The grating structure to enhance the resonance anomalies of a periodic profile (periodic modulation of refractive index) by coupling illuminating plane wave to the leaky modes of the waveguide of the grating is known as resonant waveguide gratings. Diffraction grating splits incident plane wave to propagate in different directions so-called diffraction orders. The periodic structure undergo complete interference and resonates with no transmission at a particular wavelength and incident angle [16]. As a result, light couples out of the waveguide, propagates up to smaller distances, and appears in the form of narrower reflectance peaks

whose power varies from 0 to 100% over a range of structural parameters [10, 16, 17].

*β* = *β*<sup>0</sup> + *i*, (1)

represents the propagation loss of the leaky modes [18]. Generally, the spectral width of a resonance curve is proportional to this propagation loss and the full width at half maximum

In 1994, researchers used Effective Medium Theory (EMT) to model stratified media as a thinfilm stack possessing some effective index. The attempts were made to achieve symmetric spectral response with low sidebands by varying thicknesses of thin-film stacks. This approach results in to design an effective thin-film layer to be antireflective at resonant wavelength [19, 20]. Several researchers considered thin-film model for the design of symmetric filters and suggested numerical solutions based on rigorous modeling methods [21]. To design grating layer, the effective index model was suggested for thin-film method [19]. In 1956, Rytov developed a transcendental equation based on EMT to correlate effective refractive index of a stratified medium to physical parameters and wavelength of light [22]. This equation can be

*<sup>λ</sup>*] = −(*n*<sup>L</sup>

where *n*H, *n*L, *n*eff are the high, low, and effective indices of the grating, respectively, *f* is the fill factor, *d* is the grating periodicity, and *λ* is the wavelength of the light. In the long wavelength

<sup>2</sup> − *n*eff

<sup>2</sup> )<sup>1</sup>/<sup>2</sup> tan[*π*(*n*<sup>L</sup>

<sup>2</sup> − *n*eff 2 ) \_\_1 <sup>2</sup> <sup>×</sup> (<sup>1</sup> <sup>−</sup> *<sup>f</sup>*)*<sup>d</sup>* \_\_\_\_\_

*<sup>λ</sup>* ], (3)

and *γ* are pure real numbers while the imaginary part (*γ*) of the propagation constant

*d* \_\_\_

*<sup>π</sup>* , (2)

Polymer Resonant Waveguide Gratings http://dx.doi.org/10.5772/intechopen.76917 279

The propagation constant of a leaky mode is complex quantity and expressed as,

**2. Theory of resonant waveguide gratings**

∆*λ*FWHM = *λ*<sup>0</sup>

is the resonant wavelength, *d* is the periodicity.

applied to grating problems and written as for a TE-polarized light:

<sup>2</sup> − *n*eff 2 ) \_\_1 <sup>2</sup> <sup>×</sup> *fd*\_\_

*<sup>λ</sup>* <sup>→</sup> <sup>0</sup>), (Eq. (3)) can be solved in terms of an analytic relation:

where *β*<sup>0</sup>

where *λ*<sup>0</sup>

(*n*<sup>H</sup>

limit ( \_\_*d* <sup>2</sup> − *n*eff

<sup>2</sup> )<sup>1</sup>/<sup>2</sup> tan[*π* (*n*<sup>H</sup>

is approximated by.

Large scale demands for cost-effective yet reliable and efficient photonic components have led many researchers to consider polymer materials. Polymeric materials become widely useful and increasingly attractive in the fabrication of various micro and nanostructures with the potential replacement of conventional inorganic materials such as SiO2 and LiNbO<sup>3</sup> or semiconductors. Novel polymers have been introduced for replication of nanostructure RWG through inexpensive and mass-production process [11]. The polymeric materials offer high thermal expansion coefficient (almost an order of magnitude) in comparison to traditional inorganic materials as well as thermo-optic coefficient which enable them to use as fast rate switches. Investigations to fabricate RWG in polymeric materials have been actively pursued throughout over the past two decades [12, 13].

The capability to fabricate precise novel structures at micro to nanoscale with a wide variety of materials imposes great challenges to the advancement of nanotechnology and the nanosciences. The semiconductor industry continues pushing to lower structural size and to manufacture smaller transistors and high density integrated circuits. The demanding industrial processes through newly developed lithographic methodologies need to address some critical issues such as speed, reliability, overlay accuracy, etc. Many alternative approaches have been used to manufacture nanostructures in past two decades, despite of using expensive tools such as deep-UV projection lithography and electron beam lithography techniques. These techniques include micro-contact printing, scanning probe based techniques, dip-pen lithography, and Nanoimprint Lithography (NIL) [14]. NIL can not only fabricate nanostructures in resists but can also imprint functional devices in many polymers through ease and cost-effective processes in a number of applications such as photonics, data storage, biotechnology, and electronics [15].

In this chapter we present details of design, fabrication, and characterization of polymeric RWG employing affordable techniques and mass production processes. The fabrication of master stamp by electron beam lithography and subsequent replication in polymer materials by NIL is presented to target a number of applications in thermoplastics and UV curable polymers.

#### **2. Theory of resonant waveguide gratings**

optical waveguides are primarily the most common type of thin film optical filters used widely as narrowband filters in laser cavities, optical telecommunications, and light modulators [1]. However, the realization of sub-nanometer narrowband filters with thin film technique require hundreds of optical thin-films stack with stringent tolerances over thicknesses and refractive index variations [2]. Resonant Waveguide Gratings (RWGs) are a new class of narrowband filters and are widely used in applications such as polarizers [3], laser cavity reflectors [3, 4], light modulators [5], biosensors [6], and wavelength division multiplexing [7]. Such narrowband reflectance/transmittance spectral characteristics can be observed by sub-wavelength grating structures in terms of resonance anomalies [8, 9] with numerous potential applications. RWG operates with resonance effects with relatively simpler structure of few layers. Owing to the resonant nature of the sub-wavelength grating, the leaky modes are supported by the structure (waveguide layer and a grating layer) [10]. In the absence of the grating layer, waveguide layer supports a true bound mode. This bound mode becomes leaky when a grating layer is added with the waveguide layer. Eventually, optical energy is coupled out of the waveguide into radiation modes. On the other hand, the incident plane wave energy is coupled to the waveguide. The incident plane wave energy is coupled into leaky modes and then back to one or more radiation modes. This coupling mostly depends on the wavelength, angle of incidence and other structural parameters of the grating layer. At resonance a sharp peak in reflected/

transmitted light might be observed at a specific combination of these parameters.

the potential replacement of conventional inorganic materials such as SiO2

throughout over the past two decades [12, 13].

278 Emerging Waveguide Technology

Large scale demands for cost-effective yet reliable and efficient photonic components have led many researchers to consider polymer materials. Polymeric materials become widely useful and increasingly attractive in the fabrication of various micro and nanostructures with

semiconductors. Novel polymers have been introduced for replication of nanostructure RWG through inexpensive and mass-production process [11]. The polymeric materials offer high thermal expansion coefficient (almost an order of magnitude) in comparison to traditional inorganic materials as well as thermo-optic coefficient which enable them to use as fast rate switches. Investigations to fabricate RWG in polymeric materials have been actively pursued

The capability to fabricate precise novel structures at micro to nanoscale with a wide variety of materials imposes great challenges to the advancement of nanotechnology and the nanosciences. The semiconductor industry continues pushing to lower structural size and to manufacture smaller transistors and high density integrated circuits. The demanding industrial processes through newly developed lithographic methodologies need to address some critical issues such as speed, reliability, overlay accuracy, etc. Many alternative approaches have been used to manufacture nanostructures in past two decades, despite of using expensive tools such as deep-UV projection lithography and electron beam lithography techniques. These techniques include micro-contact printing, scanning probe based techniques, dip-pen lithography, and Nanoimprint Lithography (NIL) [14]. NIL can not only fabricate nanostructures in resists but can also imprint functional devices in many polymers through ease and cost-effective processes in a number of applications such as photonics, data storage, biotechnology, and electronics [15]. In this chapter we present details of design, fabrication, and characterization of polymeric RWG employing affordable techniques and mass production processes. The fabrication of master

and LiNbO<sup>3</sup>

or

The grating structure to enhance the resonance anomalies of a periodic profile (periodic modulation of refractive index) by coupling illuminating plane wave to the leaky modes of the waveguide of the grating is known as resonant waveguide gratings. Diffraction grating splits incident plane wave to propagate in different directions so-called diffraction orders. The periodic structure undergo complete interference and resonates with no transmission at a particular wavelength and incident angle [16]. As a result, light couples out of the waveguide, propagates up to smaller distances, and appears in the form of narrower reflectance peaks whose power varies from 0 to 100% over a range of structural parameters [10, 16, 17].

The propagation constant of a leaky mode is complex quantity and expressed as,

$$
\beta = \beta\_0 + \text{i}\gamma.\tag{1}
$$

where *β*<sup>0</sup> and *γ* are pure real numbers while the imaginary part (*γ*) of the propagation constant represents the propagation loss of the leaky modes [18]. Generally, the spectral width of a resonance curve is proportional to this propagation loss and the full width at half maximum is approximated by.

$$
\Delta\lambda\_{\text{FWHM}} = \lambda\_0 \frac{d\eta}{\pi\nu} \tag{2}
$$

where *λ*<sup>0</sup> is the resonant wavelength, *d* is the periodicity.

In 1994, researchers used Effective Medium Theory (EMT) to model stratified media as a thinfilm stack possessing some effective index. The attempts were made to achieve symmetric spectral response with low sidebands by varying thicknesses of thin-film stacks. This approach results in to design an effective thin-film layer to be antireflective at resonant wavelength [19, 20]. Several researchers considered thin-film model for the design of symmetric filters and suggested numerical solutions based on rigorous modeling methods [21]. To design grating layer, the effective index model was suggested for thin-film method [19]. In 1956, Rytov developed a transcendental equation based on EMT to correlate effective refractive index of a stratified medium to physical parameters and wavelength of light [22]. This equation can be applied to grating problems and written as for a TE-polarized light:

$$\left(n\_{\rm H}^2 - n\_{\rm eff}^2\right)^{1/2} \tan\left[\pi \left(n\_{\rm H}^2 - n\_{\rm eff}^2\right)^{\frac{1}{2}} \times \frac{fd}{\lambda}\right] = -\left(n\_{\rm L}^2 - n\_{\rm eff}^2\right)^{1/2} \tan\left[\pi \left(n\_{\rm L}^2 - n\_{\rm eff}^2\right)^{\frac{1}{2}} \times \frac{(1-f)d}{\lambda}\right],\tag{3}$$

where *n*H, *n*L, *n*eff are the high, low, and effective indices of the grating, respectively, *f* is the fill factor, *d* is the grating periodicity, and *λ* is the wavelength of the light. In the long wavelength limit ( \_\_*d <sup>λ</sup>* <sup>→</sup> <sup>0</sup>), (Eq. (3)) can be solved in terms of an analytic relation:

$$m\_{\rm eff} = \left[f n\_{\rm H}^2 + \left(1 - f\right) n\_{\rm L}^2\right]^{1/2}.\tag{4}$$

Eq. (3) is referred as the exact effective index model whereas (Eq. (4)) as zeroth order effective index model.

The fundamental structure of a RWG is shown in **Figure 1**. The waveguide grating consist of a substrate material with refractive index *n*<sup>t</sup> , a coupled grating layer with refractive index distribution *n*<sup>2</sup> (x) along *x*-direction and a superstrate layer (generally air) with refractive index *n*i . When light of wavelength *λ* illuminates the grating at an incident angle *θ*<sup>i</sup> , it results in generation of various propagated diffraction orders through one-dimensional grating which can be calculated by fundamental grating equation, given as:

$$n\_{\rm n} \sin \Theta\_{\rm m} = n\_{\rm i} \sin \Theta\_{\rm in} + m \frac{\Lambda}{d} \tag{5}$$

*<sup>γ</sup>*<sup>0</sup> <sup>=</sup> *kx* <sup>+</sup> *<sup>n</sup>* \_\_\_ <sup>2</sup>*<sup>π</sup>*

values on this complex *γ*-plane with a separation of \_\_\_ <sup>2</sup>*<sup>π</sup>*

where *d* is periodicity of the grating, *<sup>γ</sup>* <sup>0</sup>

*k x*

*<sup>d</sup>* , (6)

Polymer Resonant Waveguide Gratings http://dx.doi.org/10.5772/intechopen.76917 281

*<sup>d</sup>* . Thus, the leaky waveguide modes are

is propagation constant of fundamental mode and

 is wave vector associated with the illuminating plane wave. At the resonance regime the rapidly varying phase of the secondary field with respect to the incident field (wave number) becomes similar in phase which gives rise resonance in the form of narrow reflected peak with wavelength or angle of incidence [23]. In **Figure 1**, the leaky waveguide modes in lateral direction are represented by propagation constant *γ*. Due to the leaky nature of propagated modes, they are shown to possess both real and imaginary parts and form a plane, so-called *complex γ-plane*. A leaky mode is described by a pole on this complex *γ*-plane. A planar waveguide supports at least one mode, the pole of which is represented by the real value on this *γ*-plane. Owing to the introduction of periodicity in the planar structure, such single mode splits into an infinite number of spatially diffracted orders whose poles are represented by complex

**Figure 2.** Schematic view of RWG with refractive index distributions and coated high index cover layer.

primarily associated with the periodicity of grating structure and much more closely spaced poles can be observed for sufficiently small periodic structure compared to incident wavelength. The magnitudes of real and imaginary parts of such complex poles show the extent of leaky modes excited by the input plane wave i.e., the coupling of the real part of modes

Polymeric materials have become potential candidate with versatility optical device performance and functionality. In comparison to inorganic materials, polymeric materials possess many attributable characteristics. The properties of polymer materials can be changed

(poles) with input filed and the associated coupling loss, respectively [24].

**3. Selection of polymer materials for optical waveguide**

where *λ* is the wavelength, *θ*in is the incident angle of light, *d* is the periodicity of the grating structure, *θ*m is the diffraction order, *m* = 0, ±1, ±2, ±3,… is the index of diffraction order, *n*<sup>i</sup> , and *n*<sup>2</sup> are the indices before and after the interface. For reflection gratings *n*<sup>2</sup> is replaced by *n*<sup>i</sup> and for transmission gratings by *n*<sup>t</sup> .

Narrow reflection or transmission peaks can be achieved by understanding the physics of the structure which depends on the excitation of leaky waveguide modes. Consider a reflection grating with periodicity smaller than the wavelength of light used to allow only zeroth-order diffraction under plane wave illumination as shown in **Figure 2**. The resulted reflecting fields from the gratings may be assumed to produce from two contributions, namely: a direct reflection and a scattered field reflection [22]. The inherent direct reflection from upper interface is primary reflection so-called *Fresnel reflection* whereas the secondary reflection from the grating structure is due to excitation and rescattering of leaky waveguide modes whose phase vary continuously to fulfill the coupling relation given below:

**Figure 1.** Schematic representation of resonant waveguide Grating's structure with forward and backward propagated diffraction orders.

**Figure 2.** Schematic view of RWG with refractive index distributions and coated high index cover layer.

*n*eff = [*f n*<sup>H</sup>

a substrate material with refractive index *n*<sup>t</sup>

and for transmission gratings by *n*<sup>t</sup>

can be calculated by fundamental grating equation, given as:

*n*<sup>2</sup> sin *θ*<sup>m</sup> = *n*<sup>i</sup> sin *θ*in + *m* \_\_

index model.

280 Emerging Waveguide Technology

tribution *n*<sup>2</sup>

*n*i

and *n*<sup>2</sup>

diffraction orders.

<sup>2</sup> + (1 − *f*) *n*<sup>L</sup>

Eq. (3) is referred as the exact effective index model whereas (Eq. (4)) as zeroth order effective

The fundamental structure of a RWG is shown in **Figure 1**. The waveguide grating consist of

generation of various propagated diffraction orders through one-dimensional grating which

where *λ* is the wavelength, *θ*in is the incident angle of light, *d* is the periodicity of the grating structure, *θ*m is the diffraction order, *m* = 0, ±1, ±2, ±3,… is the index of diffraction order, *n*<sup>i</sup>

Narrow reflection or transmission peaks can be achieved by understanding the physics of the structure which depends on the excitation of leaky waveguide modes. Consider a reflection grating with periodicity smaller than the wavelength of light used to allow only zeroth-order diffraction under plane wave illumination as shown in **Figure 2**. The resulted reflecting fields from the gratings may be assumed to produce from two contributions, namely: a direct reflection and a scattered field reflection [22]. The inherent direct reflection from upper interface is primary reflection so-called *Fresnel reflection* whereas the secondary reflection from the grating structure is due to excitation and rescattering of leaky waveguide modes whose phase

**Figure 1.** Schematic representation of resonant waveguide Grating's structure with forward and backward propagated

. When light of wavelength *λ* illuminates the grating at an incident angle *θ*<sup>i</sup>

are the indices before and after the interface. For reflection gratings *n*<sup>2</sup>

.

vary continuously to fulfill the coupling relation given below:

2 ] 1/2

(x) along *x*-direction and a superstrate layer (generally air) with refractive index

*λ d*

. (4)

, (5)

, it results in

is replaced by *n*<sup>i</sup>

,

, a coupled grating layer with refractive index dis-

$$\gamma\_0 = k\_v + n\frac{2\pi}{d},\tag{6}$$

where *d* is periodicity of the grating, *<sup>γ</sup>* <sup>0</sup> is propagation constant of fundamental mode and *k x* is wave vector associated with the illuminating plane wave. At the resonance regime the rapidly varying phase of the secondary field with respect to the incident field (wave number) becomes similar in phase which gives rise resonance in the form of narrow reflected peak with wavelength or angle of incidence [23]. In **Figure 1**, the leaky waveguide modes in lateral direction are represented by propagation constant *γ*. Due to the leaky nature of propagated modes, they are shown to possess both real and imaginary parts and form a plane, so-called *complex γ-plane*. A leaky mode is described by a pole on this complex *γ*-plane. A planar waveguide supports at least one mode, the pole of which is represented by the real value on this *γ*-plane. Owing to the introduction of periodicity in the planar structure, such single mode splits into an infinite number of spatially diffracted orders whose poles are represented by complex values on this complex *γ*-plane with a separation of \_\_\_ <sup>2</sup>*<sup>π</sup> <sup>d</sup>* . Thus, the leaky waveguide modes are primarily associated with the periodicity of grating structure and much more closely spaced poles can be observed for sufficiently small periodic structure compared to incident wavelength. The magnitudes of real and imaginary parts of such complex poles show the extent of leaky modes excited by the input plane wave i.e., the coupling of the real part of modes (poles) with input filed and the associated coupling loss, respectively [24].

#### **3. Selection of polymer materials for optical waveguide**

Polymeric materials have become potential candidate with versatility optical device performance and functionality. In comparison to inorganic materials, polymeric materials possess many attributable characteristics. The properties of polymer materials can be changed

where solution of Maxwell's equations is determined at each slab. Such solutions appear in the form of forward and backward propagated fields consisting of modal fields. These

mode. The eigenvalue problem is shown in matrix form which expresses a set of allowed *β* values and transverse field distributions for each polarization of light. The emerging fields from each slab are combined at each interface by applying boundary values. This computation shows an overall field inside the modulated region which is then matched with the fields in homogeneous regions surrounding the modulated region. At the end the problem is expressed in a matrix form to calculate complex transmission and reflection field ampli-

**5. Cost-effective master stamp fabrication process by electron beam** 

The properties of stamping material play a significant role in replication process to achieve a well-defined replicated features. In this section, patterns are defined on a resist material

**Figure 4.** Schematic representation of fabrication and replication process of HSQ mold and polymeric binary grating structures with nanoscale surface-relief features. Reproduced with permission from [11]. Copyright 2012, SPIE.

**lithography (EBL) and hydrogen silsesquioxane (HSQ) resist**

, where *β* is the eigenvalue of a

Polymer Resonant Waveguide Gratings http://dx.doi.org/10.5772/intechopen.76917 283

fields are pseudoperiodic in nature and expressed in the *e* <sup>±</sup>*i<sup>z</sup>*

tudes [31].

**Figure 3.** (a) Schematic representation of originally proposed NIL process by Chou. (b) Scanning Electron Microscopy (SEM) image of a mold possessing pillar array diameter of 10 nm. (c) Replicated structure of mold in PMMA polymer material with hole array of size 10 nm. Reproduced with permission from [27]. Copyright 1997, American Institute of Physics.

chemically after modifying the chemical structure of the monomers, polymer backbones, addition of functional groups or chromophores. Polymer materials can be made to manipulate easily by many conventional or unconventional fabrication methods such as reactive ion etching (dry etching), wet etching, soft lithography etc. [25]. Polymer materials offer a simple, low-cost, and reliable fabrication process irrespective to fragile silica or expensive semiconductor materials. Functional polymeric materials provide interesting properties for integrating several diversified materials with different functionalities.

Optical waveguide structures can be fabricated directly by electron beam lithography which is the most effective method to fabricate micro- and nanostructures [25]. Alternatively, soft lithography technique has been extensively developed during past 20 years and improving optical waveguide manufacturing by the use of a master stamp to generate several soft molds to reproduce its replicas [25, 26]. **Figure 3** shows schematic of the originally NIL process proposed by Chao almost two decades before [27, 28]. The master stamp or mold containing nanoscale surface relief features is pressed against a polymeric material on a substrate with tightly controlled temperature and pressure to create a thickness contrast in polymer material. Furthermore, a thin residual layer is made beneath the stamp protrusions as a cushioning layer to protect nanoscale structure on mold surface from a direct impact of mold on the substrate. However, this residual layer can be removed at the end of the process by an anisotropic O<sup>2</sup> plasma etching. **Figure 3b** and **c** shows Scanning Electron Microscopy (SEM) images of a mold with pillar array of diameter 10 nm and replicated hole array in poly (methyl methacrylate) (PMMA) [28].

### **4. RWG modeling tool**

In this chapter the RWG structures are designed and modeled using most efficient method which is based on the Fourier expansion, commonly known as Fourier Modal Method (FMM) or the coupled wave method (CWM) [29]. FMM determines eigen-solution values of Maxwell's equations in a periodic or piecewise continuous medium by expanding the electromagnetic fields and permittivity functions to Fourier series and applying the boundary conditions to show fields inside the grating by an algebraic eigenvalue problem [30]. Employing FMM to periodic-modulated region, the modulated region sections in slabs where solution of Maxwell's equations is determined at each slab. Such solutions appear in the form of forward and backward propagated fields consisting of modal fields. These fields are pseudoperiodic in nature and expressed in the *e* <sup>±</sup>*i<sup>z</sup>* , where *β* is the eigenvalue of a mode. The eigenvalue problem is shown in matrix form which expresses a set of allowed *β* values and transverse field distributions for each polarization of light. The emerging fields from each slab are combined at each interface by applying boundary values. This computation shows an overall field inside the modulated region which is then matched with the fields in homogeneous regions surrounding the modulated region. At the end the problem is expressed in a matrix form to calculate complex transmission and reflection field amplitudes [31].

### **5. Cost-effective master stamp fabrication process by electron beam lithography (EBL) and hydrogen silsesquioxane (HSQ) resist**

chemically after modifying the chemical structure of the monomers, polymer backbones, addition of functional groups or chromophores. Polymer materials can be made to manipulate easily by many conventional or unconventional fabrication methods such as reactive ion etching (dry etching), wet etching, soft lithography etc. [25]. Polymer materials offer a simple, low-cost, and reliable fabrication process irrespective to fragile silica or expensive semiconductor materials. Functional polymeric materials provide interesting properties for

**Figure 3.** (a) Schematic representation of originally proposed NIL process by Chou. (b) Scanning Electron Microscopy (SEM) image of a mold possessing pillar array diameter of 10 nm. (c) Replicated structure of mold in PMMA polymer material with hole array of size 10 nm. Reproduced with permission from [27]. Copyright 1997, American Institute of Physics.

Optical waveguide structures can be fabricated directly by electron beam lithography which is the most effective method to fabricate micro- and nanostructures [25]. Alternatively, soft lithography technique has been extensively developed during past 20 years and improving optical waveguide manufacturing by the use of a master stamp to generate several soft molds to reproduce its replicas [25, 26]. **Figure 3** shows schematic of the originally NIL process proposed by Chao almost two decades before [27, 28]. The master stamp or mold containing nanoscale surface relief features is pressed against a polymeric material on a substrate with tightly controlled temperature and pressure to create a thickness contrast in polymer material. Furthermore, a thin residual layer is made beneath the stamp protrusions as a cushioning layer to protect nanoscale structure on mold surface from a direct impact of mold on the substrate. However,

ing. **Figure 3b** and **c** shows Scanning Electron Microscopy (SEM) images of a mold with pillar array of diameter 10 nm and replicated hole array in poly (methyl methacrylate) (PMMA) [28].

In this chapter the RWG structures are designed and modeled using most efficient method which is based on the Fourier expansion, commonly known as Fourier Modal Method (FMM) or the coupled wave method (CWM) [29]. FMM determines eigen-solution values of Maxwell's equations in a periodic or piecewise continuous medium by expanding the electromagnetic fields and permittivity functions to Fourier series and applying the boundary conditions to show fields inside the grating by an algebraic eigenvalue problem [30]. Employing FMM to periodic-modulated region, the modulated region sections in slabs

plasma etch-

integrating several diversified materials with different functionalities.

this residual layer can be removed at the end of the process by an anisotropic O<sup>2</sup>

**4. RWG modeling tool**

282 Emerging Waveguide Technology

The properties of stamping material play a significant role in replication process to achieve a well-defined replicated features. In this section, patterns are defined on a resist material

**Figure 4.** Schematic representation of fabrication and replication process of HSQ mold and polymeric binary grating structures with nanoscale surface-relief features. Reproduced with permission from [11]. Copyright 2012, SPIE.

which is coated on a silicon substrate and written by electron beam lithography (EBL) without reactive ion etching (RIE) of silicon. Moreover, accurate control to pattern depth is challenging and inaccuracies in depth profiles are inherent with different width structures. Furthermore, the associated EBL proximity effect increases with the pattern depths and become more pronounced when beam size becomes comparable to the pattern size. The line edge variations occurred due to incomplete suppression of resist after development process resulted in polymer molecule agglomerate formation at pattern line edges [20]. Hydrogen silsesquioxane (HSQ) is a high resolution, inorganic, negative tone EB resist material with small linewidth variations in comparison to positive EB resists such as PMMA and ZEP. To fabricate structures with high resolutions, the molecular size of resist material need to be smaller than the nanoscale features to be replicated for which HSQ resist possesses dominating properties with slight line roughness and high etch resistance in addition [32].

the formed structure is heat treated to improve mechanical properties of the resist for 180 min at 300°C temperature. Such thermal treatment improves density and hardness of HSQ resist to enable it for the use of hard stamp with high imprint pattern fidelity. The heat treated mold(s) are surface treated in nitrogen environment to deposit a silane layer to act as an antiadhesive layer for imprinting. Finally, the imprinted polymeric gratings with several periodicities are coated

the replicated structures are investigated by a variable angle spectroscopic ellipsometer. **Figure 4** depicts schematic representation of complete process flow of HSQ mold fabrication and imprint-

**Figure 5** shows SEM images of top view of HSQ resist molds (grating structures) at different magnifications on silicon substrate with period *d* of 325 nm. **Figure 6a** and **b** shows crosssectional view of SEM images of HSQ binary molds with period 425 nm and **Figure 6c** and **d**

**Figure 7** shows imprinted sub-wavelength grating structures in thermoplastic and UV curable

polycarbonate, cyclic olefin copolymer and UV curable Ormocomp by atomic layer deposition

**Figure 6.** SEM images of cross-sectional view of HSQ resist master stamp on silicon with periodicities: (a and b)

*d* = 425 nm and (c and d) *d* = 325 nm. Reproduced with permission from [33]. Copyright 2013, NUST.

ing into polymeric materials (thermoplastics and UV curable) with high index TiO<sup>2</sup>

plastic materials by NIL tool. **Figure 8** shows various thin films of amorphous TiO<sup>2</sup>

thin films by atomic layer deposition. Spectral characteristics of

thin layer.

285

Polymer Resonant Waveguide Gratings http://dx.doi.org/10.5772/intechopen.76917

coated on

with high index amorphous TiO<sup>2</sup>

with period 325 nm [33].

In this Chapter we show replication of nanoscale structures in thermoplastic thin films and UV curable polymers by using an HSQ mold. The mold is fabricated by spin coating HSQ resist layer on silicon substrate, direct e-beam writing followed by development process without reactive ion etching. The HSQ resist thickness is adjusted to obtain structure design height *h*. Additionally,

**Figure 5.** SEM images of top view of HSQ resist master stamp on silicon substrate with periodicity (*d* = 325 nm) at different magnifications: (a) 100.00 KX, (b) 150.00 KX, (c) 200.00 KX, and (d) 250.00 KX.

the formed structure is heat treated to improve mechanical properties of the resist for 180 min at 300°C temperature. Such thermal treatment improves density and hardness of HSQ resist to enable it for the use of hard stamp with high imprint pattern fidelity. The heat treated mold(s) are surface treated in nitrogen environment to deposit a silane layer to act as an antiadhesive layer for imprinting. Finally, the imprinted polymeric gratings with several periodicities are coated with high index amorphous TiO<sup>2</sup> thin films by atomic layer deposition. Spectral characteristics of the replicated structures are investigated by a variable angle spectroscopic ellipsometer. **Figure 4** depicts schematic representation of complete process flow of HSQ mold fabrication and imprinting into polymeric materials (thermoplastics and UV curable) with high index TiO<sup>2</sup> thin layer.

which is coated on a silicon substrate and written by electron beam lithography (EBL) without reactive ion etching (RIE) of silicon. Moreover, accurate control to pattern depth is challenging and inaccuracies in depth profiles are inherent with different width structures. Furthermore, the associated EBL proximity effect increases with the pattern depths and become more pronounced when beam size becomes comparable to the pattern size. The line edge variations occurred due to incomplete suppression of resist after development process resulted in polymer molecule agglomerate formation at pattern line edges [20]. Hydrogen silsesquioxane (HSQ) is a high resolution, inorganic, negative tone EB resist material with small linewidth variations in comparison to positive EB resists such as PMMA and ZEP. To fabricate structures with high resolutions, the molecular size of resist material need to be smaller than the nanoscale features to be replicated for which HSQ resist possesses dominating properties with slight line roughness and high etch resistance in addition [32].

284 Emerging Waveguide Technology

In this Chapter we show replication of nanoscale structures in thermoplastic thin films and UV curable polymers by using an HSQ mold. The mold is fabricated by spin coating HSQ resist layer on silicon substrate, direct e-beam writing followed by development process without reactive ion etching. The HSQ resist thickness is adjusted to obtain structure design height *h*. Additionally,

**Figure 5.** SEM images of top view of HSQ resist master stamp on silicon substrate with periodicity (*d* = 325 nm) at

different magnifications: (a) 100.00 KX, (b) 150.00 KX, (c) 200.00 KX, and (d) 250.00 KX.

**Figure 5** shows SEM images of top view of HSQ resist molds (grating structures) at different magnifications on silicon substrate with period *d* of 325 nm. **Figure 6a** and **b** shows crosssectional view of SEM images of HSQ binary molds with period 425 nm and **Figure 6c** and **d** with period 325 nm [33].

**Figure 7** shows imprinted sub-wavelength grating structures in thermoplastic and UV curable plastic materials by NIL tool. **Figure 8** shows various thin films of amorphous TiO<sup>2</sup> coated on polycarbonate, cyclic olefin copolymer and UV curable Ormocomp by atomic layer deposition

**Figure 6.** SEM images of cross-sectional view of HSQ resist master stamp on silicon with periodicities: (a and b) *d* = 425 nm and (c and d) *d* = 325 nm. Reproduced with permission from [33]. Copyright 2013, NUST.

**Figure 7.** SEM images of cross-sectional view of imprinted structures in: (a and b) polycarbonate with period *d* = 368 nm, (c) cyclic olefin copolymer with period *d* = 325 nm and (d) UV curable material Ormocomp with period *d* = 325 nm. Reproduced with permission from [33]. Copyright 2013, NUST.

**Figure 8.** SEM images of amorphous TiO<sup>2</sup>

with period *d* = 325 nm and TiO2

and TiO2

thickness *t* = 80 nm, (b) polycarbonate with period *d* = 368 nm and TiO2

thickness *t* = 50 nm. Reproduced with permission from [33]. Copyright 2013, NUST.

**Figure 9.** Experimental setup of an ellipsometer to measure specular reflectance or transmittance.

coated replicated gratings: (a) polycarbonate with period *d* = 368 nm and TiO2

thickness *t* = 50 nm, and (d) UV curable material Ormocomp with period *d* = 325 nm

thickness *t* = 60 nm, (c) cyclic olefin copolymer

Polymer Resonant Waveguide Gratings http://dx.doi.org/10.5772/intechopen.76917 287

as a waveguide layer [30]. The details of conformal growth of amorphous TiO2 thin films by atomic layer deposition is described in Refs. [34, 35].

**Figure 9** shows the ellipsometric measurement setup when a linearly polarized plane wave (electric field vector is parallel called TE or perpendicular called TM to the grating lines) incident on the sample at an incident angle Φ with respect to normal of the RWG sample. The light-matter interaction results in specular reflectance/transmittance of the resonant gratings. The polarization state (TE or TM) of the illuminated light is selected by a polarizer stage which transforms the unpolarized light beam into a linearly polarized light beam. The polarization stage composed of a polarizer mounted on a continuously rotated stepper motor with high accuracy. The rotating polarizer changes the intensity of the light. The phase and amplitude of the modulated light represents the polarization state of the beam entering the analyzer/detector. In general, ellipsometer predicts the ellipticity of the polarization state of the light, optical constants (*n* and *k*) of optical materials, and the thickness of the thin film. The ellipsometric measurement uses two parameters which are connected by Eq. (7) [36].

**Figure 8.** SEM images of amorphous TiO<sup>2</sup> coated replicated gratings: (a) polycarbonate with period *d* = 368 nm and TiO2 thickness *t* = 80 nm, (b) polycarbonate with period *d* = 368 nm and TiO2 thickness *t* = 60 nm, (c) cyclic olefin copolymer with period *d* = 325 nm and TiO2 thickness *t* = 50 nm, and (d) UV curable material Ormocomp with period *d* = 325 nm and TiO2 thickness *t* = 50 nm. Reproduced with permission from [33]. Copyright 2013, NUST.

**Figure 9.** Experimental setup of an ellipsometer to measure specular reflectance or transmittance.

as a waveguide layer [30]. The details of conformal growth of amorphous TiO2

measurement uses two parameters which are connected by Eq. (7) [36].

**Figure 9** shows the ellipsometric measurement setup when a linearly polarized plane wave (electric field vector is parallel called TE or perpendicular called TM to the grating lines) incident on the sample at an incident angle Φ with respect to normal of the RWG sample. The light-matter interaction results in specular reflectance/transmittance of the resonant gratings. The polarization state (TE or TM) of the illuminated light is selected by a polarizer stage which transforms the unpolarized light beam into a linearly polarized light beam. The polarization stage composed of a polarizer mounted on a continuously rotated stepper motor with high accuracy. The rotating polarizer changes the intensity of the light. The phase and amplitude of the modulated light represents the polarization state of the beam entering the analyzer/detector. In general, ellipsometer predicts the ellipticity of the polarization state of the light, optical constants (*n* and *k*) of optical materials, and the thickness of the thin film. The ellipsometric

**Figure 7.** SEM images of cross-sectional view of imprinted structures in: (a and b) polycarbonate with period *d* = 368 nm, (c) cyclic olefin copolymer with period *d* = 325 nm and (d) UV curable material Ormocomp with period *d* = 325 nm.

atomic layer deposition is described in Refs. [34, 35].

Reproduced with permission from [33]. Copyright 2013, NUST.

286 Emerging Waveguide Technology

thin films by

$$
\tan \psi \, e^{i\Lambda} = \frac{R\_p}{R\_i} \tag{7}
$$

ones with two TiO2

height *h* = 120 nm and TiO2

[37]. Copyright 2013, Elsevier.

above *T*<sup>g</sup>

fact, *T*<sup>g</sup>

transition temperature (*T*<sup>g</sup>

at room temperatures. Moreover, below *T*<sup>g</sup>

after imprinting process polymer is cooled down below *T*<sup>g</sup>

thicknesses. Both experimentally measured and calculated spectra are in

thickness t = 50 nm: a) polycarbonate with periodicity (*d* = 368 nm), b) cyclic olefin copolymer

), both Young's modulus

Polymer Resonant Waveguide Gratings http://dx.doi.org/10.5772/intechopen.76917 289

to preserve imprinted pattern. In

the value of Young's modulus of glassy polymers

agreement, provided few spectral shifts occur due to reasons described above.

For replication by thermal NIL, the temperature of the polymer materials are raised above glass

**Figure 10.** Theoretically calculated and experimentally measured specular reflectance of replicated gratings with grating

with periodicity (*d* = 325 nm), c) Ormocomp with periodicity (*d* = 325 nm); measured reflectance spectra of all three designed gratings: d) polycarbonate, e) cyclic olefin copolymer, and f) ormocomp. Reproduced with permission from

) of polymers. At such a condition (*T* ˃ *T*<sup>g</sup>

and viscosity of polymers reduce by several orders of magnitude in comparison to their values

remains constant for many polymers, approximately 3 × 10<sup>9</sup> Pa in comparison to their respective values at room temperature. In general practice, the temperature rise for thermal NIL is 60–90°C

so that polymer transform into a viscous flow to fill micro and nanocavities, however,

 is onset temperature for molecular motion in polymers. There are many factors which increase energy for molecular motion, such as, intermolecular forces, interchain steric hindrance (branching or cross-linking, bulky and stiff side groups). In some processes, it is desirable to use lower temperature values, which is then compensated by corresponding increase in the process pressure and time to obtain perfect imprinting. The requirement of high temperature and pressure for NIL process may restrict the production of NIL technology. Furthermore, the mismatch of thermal expansion coefficient between the mold material and substrate may impose limitations for pattern overlay for large substrates. Alternatively, liquid precursors having low Young's modulus and viscosity can be cured by UV light at ambient temperatures. Due to low viscosity of the fluid the imprinting process is facilitated and minimize pattern density effects.

Where *R*p and *R*<sup>s</sup> are the complex-amplitude reflectance coefficients for p- and s-polarization state of light, *Ψ* represents elliptical state of polarization and *Δ* is the relative phase of the vibrations along *x*- and *y*-directions which can vary from zero to 2*π*.

### **6. Results and discussion**

**Figure 10** shows designed and experimentally predicted spectral response (specular reflectance) of replicated grating structures in polycarbonate (PC), cyclic olefin copolymer (COC), and UV curable polymer Ormocomp [33]. The measured specular reflectance of PC, COC, and Ormocomp show reflectance peaks at 698.6 nm, 631.4 nm, and 630.4 nm with peak reflectance efficiencies 0.71, 0.94, and 0.65, respectively as shown in **Figure 10d–f**. The resonance peaks occur at different spectral positions with lower diffraction efficiencies than those calculated theoretically as shown in **Figure 10a–c**. The spectral shifts might occur due to inaccuracies in the dimensional profile of the replicated structures including rounding of grating edges rather completely rectangular as shown in ideal profile of **Figures 1** and **2**. The reduction of measured peak efficiencies are most likely caused by scattering of light from surface roughness, slight irregularities in the straightness of the grating lines, porosity and volume variations in polymers that cause refractive index changes in microscopic scale.

The observed variations may also be explained by molecular orientations of the polymer chains. The stress induction during mold filling may result in a partial orientation and configuration of polymeric chain along principal stress directions. Such molecular orientations may relax in thermal environment over a certain length of time. If however, temperature environment is kept constant, for example, for a UV curable material, the molecular orientations can be frozen up in the glassy state of the polymers. Such frozen-in-stresses in the newly molecular chain orientations may lead to generate an anisotropic behavior in the refractive index and cause peak shift.

**Figure 11a** and **b** shows specular reflectance of two designed replicated gratings in polycarbonate (with periodicities *d* = 425 nm and *d* = 368 nm), illuminated with TE-polarized light (electric field is parallel to grating lines) at three different angles of incidence (18°, 19°, and 20°) with Full Width Half Maximum (FWHM) of about 11 nm. **Figure 11c** and **d** shows measured spectral reflectance of designed gratings (with periodicities *d* = 425 nm and *d* = 368 nm) with FWHM of 13.5 nm and 11 nm, respectively. The experimentally predicted spectra is in close agreement to that of calculated, however, the wavelength shifts may be attributed due to slight variations of refractive indices of materials interacted with light. **Figure 11e** shows the simulated reflectance efficiency variations of two gratings with TiO<sup>2</sup> thicknesses of 60 nm and 75 nm. **Figure 10f** shows experimentally measured spectral efficiencies verses calculated

tan*<sup>ψ</sup> ei*<sup>Δ</sup> <sup>=</sup> *<sup>R</sup>*<sup>p</sup> \_\_\_

vibrations along *x*- and *y*-directions which can vary from zero to 2*π*.

Where *R*p and *R*<sup>s</sup>

288 Emerging Waveguide Technology

**6. Results and discussion**

changes in microscopic scale.

index and cause peak shift.

*R*s

state of light, *Ψ* represents elliptical state of polarization and *Δ* is the relative phase of the

**Figure 10** shows designed and experimentally predicted spectral response (specular reflectance) of replicated grating structures in polycarbonate (PC), cyclic olefin copolymer (COC), and UV curable polymer Ormocomp [33]. The measured specular reflectance of PC, COC, and Ormocomp show reflectance peaks at 698.6 nm, 631.4 nm, and 630.4 nm with peak reflectance efficiencies 0.71, 0.94, and 0.65, respectively as shown in **Figure 10d–f**. The resonance peaks occur at different spectral positions with lower diffraction efficiencies than those calculated theoretically as shown in **Figure 10a–c**. The spectral shifts might occur due to inaccuracies in the dimensional profile of the replicated structures including rounding of grating edges rather completely rectangular as shown in ideal profile of **Figures 1** and **2**. The reduction of measured peak efficiencies are most likely caused by scattering of light from surface roughness, slight irregularities in the straightness of the grating lines, porosity and volume variations in polymers that cause refractive index

The observed variations may also be explained by molecular orientations of the polymer chains. The stress induction during mold filling may result in a partial orientation and configuration of polymeric chain along principal stress directions. Such molecular orientations may relax in thermal environment over a certain length of time. If however, temperature environment is kept constant, for example, for a UV curable material, the molecular orientations can be frozen up in the glassy state of the polymers. Such frozen-in-stresses in the newly molecular chain orientations may lead to generate an anisotropic behavior in the refractive

**Figure 11a** and **b** shows specular reflectance of two designed replicated gratings in polycarbonate (with periodicities *d* = 425 nm and *d* = 368 nm), illuminated with TE-polarized light (electric field is parallel to grating lines) at three different angles of incidence (18°, 19°, and 20°) with Full Width Half Maximum (FWHM) of about 11 nm. **Figure 11c** and **d** shows measured spectral reflectance of designed gratings (with periodicities *d* = 425 nm and *d* = 368 nm) with FWHM of 13.5 nm and 11 nm, respectively. The experimentally predicted spectra is in close agreement to that of calculated, however, the wavelength shifts may be attributed due to slight variations of refractive indices of materials interacted with light. **Figure 11e** shows

and 75 nm. **Figure 10f** shows experimentally measured spectral efficiencies verses calculated

the simulated reflectance efficiency variations of two gratings with TiO<sup>2</sup>

are the complex-amplitude reflectance coefficients for p- and s-polarization

, (7)

thicknesses of 60 nm

**Figure 10.** Theoretically calculated and experimentally measured specular reflectance of replicated gratings with grating height *h* = 120 nm and TiO2 thickness t = 50 nm: a) polycarbonate with periodicity (*d* = 368 nm), b) cyclic olefin copolymer with periodicity (*d* = 325 nm), c) Ormocomp with periodicity (*d* = 325 nm); measured reflectance spectra of all three designed gratings: d) polycarbonate, e) cyclic olefin copolymer, and f) ormocomp. Reproduced with permission from [37]. Copyright 2013, Elsevier.

ones with two TiO2 thicknesses. Both experimentally measured and calculated spectra are in agreement, provided few spectral shifts occur due to reasons described above.

For replication by thermal NIL, the temperature of the polymer materials are raised above glass transition temperature (*T*<sup>g</sup> ) of polymers. At such a condition (*T* ˃ *T*<sup>g</sup> ), both Young's modulus and viscosity of polymers reduce by several orders of magnitude in comparison to their values at room temperatures. Moreover, below *T*<sup>g</sup> the value of Young's modulus of glassy polymers remains constant for many polymers, approximately 3 × 10<sup>9</sup> Pa in comparison to their respective values at room temperature. In general practice, the temperature rise for thermal NIL is 60–90°C above *T*<sup>g</sup> so that polymer transform into a viscous flow to fill micro and nanocavities, however, after imprinting process polymer is cooled down below *T*<sup>g</sup> to preserve imprinted pattern. In fact, *T*<sup>g</sup> is onset temperature for molecular motion in polymers. There are many factors which increase energy for molecular motion, such as, intermolecular forces, interchain steric hindrance (branching or cross-linking, bulky and stiff side groups). In some processes, it is desirable to use lower temperature values, which is then compensated by corresponding increase in the process pressure and time to obtain perfect imprinting. The requirement of high temperature and pressure for NIL process may restrict the production of NIL technology. Furthermore, the mismatch of thermal expansion coefficient between the mold material and substrate may impose limitations for pattern overlay for large substrates. Alternatively, liquid precursors having low Young's modulus and viscosity can be cured by UV light at ambient temperatures. Due to low viscosity of the fluid the imprinting process is facilitated and minimize pattern density effects.

**7. Conclusions**

ity to mass production.

TiO2

and precise nanoimprinting are highly desirable.

cyclic olefin copolymer and Ormocomp.

**Acknowledgements**

The replication of nanophotonic components with sub-wavelength features in polymeric materials is demonstrated and described as the most promising technology to produce narrow band-pass filters which are efficient, reliable, cost-effective, environmentally stable and effective at bulk scale production. Nanoimprint lithography is an economic process which initially requires the manufacturing of a master stamp (mold) which is fabricated commonly by EBL and reactive ion etching (RIE) processes. These processes enhance cost, inaccuracies and a reduction in efficiency and device performance. This work presented the manufacturing of master stamp by EBL using a negative tone binary electron beam resist HSQ without RIE process. The sub-wavelength replicated structures' profile height was adjusted by the thickness of resist layer on silicon substrate by spin coating process. A direct pattern writing on HSQ resist was performed by EBL followed by development for sufficient time. The RIE process step was replaced by HSQ pattern resist heat treatment to improve the mechanical and physical properties such as hardness and density of HSQ resist, respectively. The simple etchless process of mold formation brings fast prototyping of nano-optical devices with rapid processing time and high pattern fidelity, superior optical performance and wide applicabil-

Polymer Resonant Waveguide Gratings http://dx.doi.org/10.5772/intechopen.76917 291

In NIL two important steps performed are mold release and pattern transfer. The imprinting process lead strong adhesive forces between the mold and the resist at large contact area. A perfect mold release keeps both resist shape integrity and a complete mold-resist separation as well as suitable plasma-etching resistance for pattern transfer into substrate. This means, nanoimprint resists which give rise both mold-release, etch-resist properties and allow fast

The replicated grating structures in polymer materials further coat by thin dielectric films of

The author is thankful to all professors, researchers, engineers, technicians, and students who contributed to the continuous development of various different processes of the nanofabrication at the Department of Physics and Mathematics, University of Eastern Finland, Joensuu, Finland. We are greatly thankful to HoD Prof. Dr. Seppo Honkanen for providing the financial fundings to publish this Chapter from Department of Physics and Mathematics, University of Eastern Finland under Project code 931351, which is highly appreciated. We greatly appreciate the editorial support from InTech in preparing this

chapter. The chapter is dedicated to my Father-in Law Mr. Ehsan Elahi (late).

 as waveguide layer to support optical modes. Theoretically simulated results agree with the experimentally measured for the RWG in a number of polymers such as polycarbonate,

**Figure 11.** Theoretically calculated specular reflectance at three illuminating angles of replicated gratings: (a) with *d* = 425 nm, (b) with *d* = 368 nm. Experimentally calculated specular reflectance at three illuminating angles of replicated gratings: (c) with *d* = 425 nm, (d) with *d* = 368 nm, (e) variation in simulated spectral reflectance as a function of TiO<sup>2</sup> thickness *t* and wavelength of illuminating TE light, and (f) theoretically calculated and experimentally predicted specular reflectance as a function of wavelength. Blue curves show layer thickness *t* = 60 nm and brown curves *t* = 75 nm. Reproduced with permission from [11]. Copyright 2012, American Optical Society.

### **7. Conclusions**

The replication of nanophotonic components with sub-wavelength features in polymeric materials is demonstrated and described as the most promising technology to produce narrow band-pass filters which are efficient, reliable, cost-effective, environmentally stable and effective at bulk scale production. Nanoimprint lithography is an economic process which initially requires the manufacturing of a master stamp (mold) which is fabricated commonly by EBL and reactive ion etching (RIE) processes. These processes enhance cost, inaccuracies and a reduction in efficiency and device performance. This work presented the manufacturing of master stamp by EBL using a negative tone binary electron beam resist HSQ without RIE process. The sub-wavelength replicated structures' profile height was adjusted by the thickness of resist layer on silicon substrate by spin coating process. A direct pattern writing on HSQ resist was performed by EBL followed by development for sufficient time. The RIE process step was replaced by HSQ pattern resist heat treatment to improve the mechanical and physical properties such as hardness and density of HSQ resist, respectively. The simple etchless process of mold formation brings fast prototyping of nano-optical devices with rapid processing time and high pattern fidelity, superior optical performance and wide applicability to mass production.

In NIL two important steps performed are mold release and pattern transfer. The imprinting process lead strong adhesive forces between the mold and the resist at large contact area. A perfect mold release keeps both resist shape integrity and a complete mold-resist separation as well as suitable plasma-etching resistance for pattern transfer into substrate. This means, nanoimprint resists which give rise both mold-release, etch-resist properties and allow fast and precise nanoimprinting are highly desirable.

The replicated grating structures in polymer materials further coat by thin dielectric films of TiO2 as waveguide layer to support optical modes. Theoretically simulated results agree with the experimentally measured for the RWG in a number of polymers such as polycarbonate, cyclic olefin copolymer and Ormocomp.

### **Acknowledgements**

**Figure 11.** Theoretically calculated specular reflectance at three illuminating angles of replicated gratings: (a) with *d* = 425 nm, (b) with *d* = 368 nm. Experimentally calculated specular reflectance at three illuminating angles of replicated gratings: (c) with *d* = 425 nm, (d) with *d* = 368 nm, (e) variation in simulated spectral reflectance as a function of TiO<sup>2</sup> thickness *t* and wavelength of illuminating TE light, and (f) theoretically calculated and experimentally predicted specular reflectance as a function of wavelength. Blue curves show layer thickness *t* = 60 nm and brown curves *t* = 75 nm.

Reproduced with permission from [11]. Copyright 2012, American Optical Society.

290 Emerging Waveguide Technology

The author is thankful to all professors, researchers, engineers, technicians, and students who contributed to the continuous development of various different processes of the nanofabrication at the Department of Physics and Mathematics, University of Eastern Finland, Joensuu, Finland. We are greatly thankful to HoD Prof. Dr. Seppo Honkanen for providing the financial fundings to publish this Chapter from Department of Physics and Mathematics, University of Eastern Finland under Project code 931351, which is highly appreciated. We greatly appreciate the editorial support from InTech in preparing this chapter. The chapter is dedicated to my Father-in Law Mr. Ehsan Elahi (late).

### **Author details**

Muhammad Rizwan Saleem1,2\* and Rizwan Ali1

\*Address all correspondence to: rizwan@casen.nust.edu.pk

1 Institute of Photonics, University of Eastern Finland, Joensuu, Finland

2 US-Pakistan Center for Advance Studies in Energy (USPCAS-E), National University of Sciences and Technology (NUST), Islamabad, Pakistan

[12] Mai X, Moshrefzadeh R, Gibstingson UJ, Stegeman GI, Seaton CT. Simple versatile method for fabricating guided-wave gratings. Applied Optics. 1985;**24**:3155-3156 [13] Rochon P, Natansohmer A, Callender CL, Robitaille L. Guided mode resonance filters

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[14] Byron D, Gates QX, Stewart M, Ryan D, Grant Willson C, Whitesides GM. New approaches to nanofabrication: Molding, printing, and other techniques. Chemical

[15] Jay Guo L. Recent progress in nanoimprint technology and its application. Journal of

[16] Golubenko GA, Svakhin AS, Sychugov AV, Tishchenko AV. Total reflection of light from a corrugated surface of a dielectric waveguide. Soviet Journal of Quantum Electronics.

[17] Popov E, Mashev L, Maystre D. Theoretical study of the anomalies of coated dielectric

[18] Thurman ST, Michael Morris G. Controlling the spectral response in guided-mode reso-

[19] S.T. Thurman and G.M. Morris. Resonant-grating filter design: The appropriate effective-index model. In: Presented at the OSA Annual Meeting, Providence, R.I; 22-26 Oct;

[20] Hessel A, Oliner AA. A new theory of Wood's anomalies on optical gratings. Applied

[21] Hegedus Z, Netterfield R. Low sideband guided-mode resonant filters. Applied Optics.

[22] Rytov SM. Electromagnetic properties of a finally stratified medium. Soviet Physics,

[23] Rosenblatt D, Sharon A, Friesem AA. Resonant grating waveguide structures. Journal of

[24] Norton SM, Erdogan T, Morris GM. Coupled-mode theory of resonant-grating filters.

[25] Huang Y, Paloczi GT, Yariv A, Cheng Z, Dalton LR. Fabrication and replication of polymer integrated optical devices using electron-beam lithography and soft lithography.

[26] Eldada L, Shacklette LW. Advances in polymer integrated optics. IEEE Journal of

[27] Chou SY, Krauss PR, Zhang W, Guo L, Zhuang L. Sub-10 nm imprinted lithography and applications. Journal of Vacuum Science and Technology B. 1997;**15**:2897. DOI: 10.1116/

using polymer films. Applied Physics Letters. 1997;**71**:1008-1010

Reviews. 2005;**105**(4):1171-1196. DOI: 10.1021/cr030076o

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**Author details**

292 Emerging Waveguide Technology

**References**

Muhammad Rizwan Saleem1,2\* and Rizwan Ali1

\*Address all correspondence to: rizwan@casen.nust.edu.pk

Sciences and Technology (NUST), Islamabad, Pakistan

4094. SPIE; 2000. pp. 46-57

1992;**61**:1022-1024

1 Institute of Photonics, University of Eastern Finland, Joensuu, Finland

2 US-Pakistan Center for Advance Studies in Energy (USPCAS-E), National University of

[1] Macleod HA, editor. Thin-Film Optical Filters. New York: American Elsevier; 1969

[2] Macleod HA. Challenges in the design and production of narrow-band filters for optical fiber communications. In: Fulton ML, editor. Optical and Infrared Thin Films, Proc. SPIE

[3] Magnusson R, Wang SS. New principle for optical filters. Applied Physics Letters.

[4] Cox JA, Morgan RA, Wilke R, Ford CM. Guided-mode grating resonant filters for VCSEL applications. In: Cindrich I, Lee SH, editors. Diffractive and Holographic Device

[5] Sharon A, Rosenblatt D, Friesem AA, Weber HG, Engel H, Steingrueber R. Light modulation with resonant grating-waveguide structures. Optics Letters. 1996;**21**:1564-1566 [6] Norton SM. resonant grating structures: Theory, design, and applications [dissertation].

[7] Golubenko GA, Sychugov VA, Tishchenko AV. The phenomenon of full 'external' reflection of light from the surface of a corrugated dielectric waveguide and its use in narrow-

[8] Wang SS, Magnusson R. Design of waveguide-grating filters with symmetrical line

[9] Tibuleac S, Magnusson R. Reflection and transmission of guided-mode resonance filters.

[10] Muhammad Rizwan Saleem. Resonant waveguide gratings by replication and atomic layer deposition [dissertation]. University of Eastern Finland, Joensuu, Finland:

[11] Saleem MR, Stenberg PA, Khan MB, Khan ZM, Honkanen S, Turunen J. Hydrogen silsesquioxane resist stamp for replication of nanophotonic components in polymers. Journal

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**Waveguide Analytical Solutions**


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[29] Moharam MG, Pommet DA, Gran EB. Stable implementation of the rigorous coupledwave analysis for surface-relief gratings: Enhanced transmittance matrix approach.

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[32] Namatsu H et al. Three dimensional siloxane resist for the formation of nanopatterns with minimum linewidth fluctuations. Journal of Vacuum Science and Technology B.

[33] Saleem MR. Design, fabrication and analysis of photonic device nanostructures [dissertation]. Islamabad, Pakistan: National University of Sciences and Technology (NUST);

[36] Azzam RMA, Bashara NM. Ellipsometry and Polarized Light. Amsterdam: North

[37] Saleem MR, Honkanen S, Turunen J. Thermo-optic coefficient of Ormocomp and comparison of polymer materials in athermal replicated subwavelength resonant waveguide

films grown

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[35] Saleem MR, Ali R, Honkanen S, Turunen J. Thermal properties of thin Al<sup>2</sup>

by atomic layer deposition. Thin Solid Films. 2012;**520**:5442-5446

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2013

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**Chapter 16**

Provisional chapter

**Applications and Solving Techniques of Propagated**

DOI: 10.5772/intechopen.76793

This chapter presents techniques to solve problems of propagation along the straight rectangular and circular waveguides with inhomogeneous dielectric materials in the cross section. These techniques are very important to improve the methods that are based on Laplace and Fourier transforms and their inverse transforms also for the discontinuous rectangular and circular profiles in the cross section (and not only for the continuous profiles). The main objective of this chapter is to develop the techniques that enable us to solve problems with inhomogeneous dielectric materials in the cross section of the straight rectangular and circular waveguides. The second objective is to understand the influence of the inhomogeneous dielectric materials on the output fields. The method in this chapter is based on the Laplace and Fourier transforms and their inverse transforms. The proposed techniques together with the methods that are based on Laplace and Fourier transforms and their inverse transforms are important to improve the methods also for the discontinuous rectangular and circular profiles in the cross section. The applications are useful for straight waveguides in the microwave and the millimeter-wave regimes, for the straight hollow waveguide and for infrared field, also in the cases of inhomogeneous

Keywords: wave propagation, inhomogeneous dielectric materials, rectangular and

The methods of straight waveguides have been proposed in the literature. Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides with

> © 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.

Applications and Solving Techniques of Propagated

**Wave in Waveguides Filled with Inhomogeneous**

Wave in Waveguides Filled with Inhomogeneous

**Dielectric Materials**

Dielectric Materials

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

dielectric materials in the cross section.

circular waveguides, dielectric profiles

http://dx.doi.org/10.5772/intechopen.76793

Zion Menachem

Abstract

1. Introduction

Zion Menachem

#### **Applications and Solving Techniques of Propagated Wave in Waveguides Filled with Inhomogeneous Dielectric Materials** Applications and Solving Techniques of Propagated Wave in Waveguides Filled with Inhomogeneous Dielectric Materials

DOI: 10.5772/intechopen.76793

Zion Menachem Zion Menachem

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.76793

#### Abstract

This chapter presents techniques to solve problems of propagation along the straight rectangular and circular waveguides with inhomogeneous dielectric materials in the cross section. These techniques are very important to improve the methods that are based on Laplace and Fourier transforms and their inverse transforms also for the discontinuous rectangular and circular profiles in the cross section (and not only for the continuous profiles). The main objective of this chapter is to develop the techniques that enable us to solve problems with inhomogeneous dielectric materials in the cross section of the straight rectangular and circular waveguides. The second objective is to understand the influence of the inhomogeneous dielectric materials on the output fields. The method in this chapter is based on the Laplace and Fourier transforms and their inverse transforms. The proposed techniques together with the methods that are based on Laplace and Fourier transforms and their inverse transforms are important to improve the methods also for the discontinuous rectangular and circular profiles in the cross section. The applications are useful for straight waveguides in the microwave and the millimeter-wave regimes, for the straight hollow waveguide and for infrared field, also in the cases of inhomogeneous dielectric materials in the cross section.

Keywords: wave propagation, inhomogeneous dielectric materials, rectangular and circular waveguides, dielectric profiles

#### 1. Introduction

The methods of straight waveguides have been proposed in the literature. Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides with

© 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.

emphasis on recent developments has been published [1]. Examples of important methods have been proposed such as finite-difference method, the integral-equation method, and methods based on series expansion. Full-vectorial matched interface and boundary method for modal analysis of dielectric waveguides has been proposed [2]. The method distinguishes itself with other existing interface methods by avoiding the use of the Taylor series expansion and by introducing the concept of the iterative use of low-order jump conditions.

of arbitrary cross section shape has been proposed [19]. In this paper, the proposed techniques used to solve problems of scattering by irregularly shaped dielectric bodies, and in the static limit, for solving the problem of an irregular dielectric or permeable body in an external field.

Applications and Solving Techniques of Propagated Wave in Waveguides Filled with Inhomogeneous Dielectric…

http://dx.doi.org/10.5772/intechopen.76793

299

The rectangular dielectric waveguide technique for the determination of complex permittivity of a wide class of dielectric materials of various thicknesses and cross sections has been described [20]. In this paper, the technique has been presented to determine the dielectric constant of materials. The fields and propagation constants of dielectric waveguides have been determined within the scalar regime [21] by using two-dimensional Fourier series expansions. Propagation of modes in some rectangular waveguides using the finite-difference time-domain method has been proposed [22]. Analysis of rectangular waveguide using finite element

Wave propagation and dielectric permittivity reconstruction in the case of a rectangular waveguide have been studied [24]. According to this paper, we study the electromagnetic wave propagation in a rectangular waveguide filled with an inhomogeneous dielectric material in the longitudinal direction. Light propagation in a cylindrical waveguide with a complex, metallic and dielectric function has been proposed [25]. Advancement of algebraic function approximation in eigenvalue problems of lossless metallic waveguides to infinite dimensions has been investigated [26]. The method of algebraic function approximation in eigenvalue problems of lossless metallic waveguides such as a closed uniform cylindrical waveguides has been proposed [27]. Analysis of longitudinally inhomogeneous waveguides using Taylor's series expansion has been proposed [28]. Analysis of longitudinally inhomogeneous waveguides using the Fourier series expansion has been proposed [29]. The method of external excitation for analysis of

A circular metallic hollow waveguide with inner dielectric multilayers has been designed by Miyagi and Kawakami [31] with the emphasis on low-loss transmission of the HE<sup>11</sup> mode for the infrared. According to this paper, the transmission losses of the dielectric-coated metal waveguides are drastically reduced when a multiple dielectric layer is formed instead of a single dielectric layer. The simplest and most efficient multilayer structure is a three-dielectric-

A transfer matrix function for the analysis of electromagnetic wave propagation along the straight dielectric waveguide with arbitrary profiles has been proposed [32]. According to this paper, the method is based on the Laplace and Fourier transforms, and the inverse Laplace and Fourier transforms. A rigorous approach for the propagation of electromagnetic fields along a straight hollow waveguide with a circular cross section has been proposed [33]. The cross section is made of a metallic layer, and only one dielectric layer upon it. The separation of variables is obtained by using the orthogonal-relations. The longitudinal components of the fields are developed into the Fourier-Bessel series. The transverse components of the fields are expressed as functions of the longitudinal components in the Laplace plane and are obtained

In order to solve more complex problems of coatings in the cross section of the dielectric waveguides, such as rectangular and circular profiles, then it is important to develop in each

arbitrarily-shaped hollow conducting waveguides has been proposed [30].

by using the inverse Laplace transform by the residue method.

method has been proposed [23].

layer stack deposited on a metal layer.

A review of the hollow waveguide and the applications has been presented [3, 4]. A review of hollow waveguides, infrared transmitting, and fibers has been presented [5]. Hollow waveguides with metallic and dielectric layers have been proposed to reduce the transmission losses. A hollow waveguide can be made from any flexible or rigid tube, such as glass, plastic or glass, and the inner hollow surface is covered by a metallic layer and a dielectric overlayer. The structure of the layer enables to transmit both the TE and TM polarization with low attenuation [6, 7].

Selective suppression of electromagnetic modes in rectangular waveguides has been presented [8] by using distributed wall losses. Analytical model for the corrugated rectangular waveguide has been extended to compute the dispersion and interaction impedance [9].

A Fourier operator method has been used to derive for the first time an exact closed-form eigenvalue equation for the scalar mode propagation constants of a buried rectangular dielectric waveguide [10]. Wave propagation in an inhomogeneous transversely magnetized rectangular waveguide has been studied with the aid of a modified Sturm-Liouville differential equation [11]. A fundamental and accurate technique to compute the propagation constant of waves in a lossy rectangular waveguide has been proposed [12]. This method is based on matching the electric and magnetic fields at the boundary and allowing the wavenumbers to take complex values.

A method that relates to the propagation constant for the bound modes in the dielectric rectangular waveguides has been proposed [13]. An analysis of rectangular folded-waveguide slow-wave structure has been developed using conformal mapping by using Schwarz Christoffel transformation [14]. A simple closed form expression to compute the time-domain reflection coefficient for a transient TE<sup>10</sup> mode wave incident on a dielectric step discontinuity in a rectangular waveguide has been presented [15]. In this paper, an exponential series approximation was provided for efficient computation of the reflected and transmitted field waveforms.

The electromagnetic fields in rectangular conducting waveguides filled with uniaxial anisotropic media have been characterized [16]. In this paper, the electric type dyadic Green's function due to an electric source was derived by using eigenfunctions expansion and the Ohm-Rayleigh method. An improved generalized admittance matrix technique based on mode matching method has been proposed [17]. The generalized scattering matrix of waveguide structure and its discontinuity problems is obtained with relationship equations and reflection coefficients.

A full-vectorial boundary integral equation method for computing guided modes of optical waveguides has been proposed [18]. Method for the propagation constants of fiber waveguides of arbitrary cross section shape has been proposed [19]. In this paper, the proposed techniques used to solve problems of scattering by irregularly shaped dielectric bodies, and in the static limit, for solving the problem of an irregular dielectric or permeable body in an external field.

emphasis on recent developments has been published [1]. Examples of important methods have been proposed such as finite-difference method, the integral-equation method, and methods based on series expansion. Full-vectorial matched interface and boundary method for modal analysis of dielectric waveguides has been proposed [2]. The method distinguishes itself with other existing interface methods by avoiding the use of the Taylor series expansion

A review of the hollow waveguide and the applications has been presented [3, 4]. A review of hollow waveguides, infrared transmitting, and fibers has been presented [5]. Hollow waveguides with metallic and dielectric layers have been proposed to reduce the transmission losses. A hollow waveguide can be made from any flexible or rigid tube, such as glass, plastic or glass, and the inner hollow surface is covered by a metallic layer and a dielectric overlayer. The structure of the layer enables to transmit both the TE and TM polarization with low

Selective suppression of electromagnetic modes in rectangular waveguides has been presented [8] by using distributed wall losses. Analytical model for the corrugated rectangular wave-

A Fourier operator method has been used to derive for the first time an exact closed-form eigenvalue equation for the scalar mode propagation constants of a buried rectangular dielectric waveguide [10]. Wave propagation in an inhomogeneous transversely magnetized rectangular waveguide has been studied with the aid of a modified Sturm-Liouville differential equation [11]. A fundamental and accurate technique to compute the propagation constant of waves in a lossy rectangular waveguide has been proposed [12]. This method is based on matching the electric and magnetic fields at the boundary and allowing the wavenumbers to

A method that relates to the propagation constant for the bound modes in the dielectric rectangular waveguides has been proposed [13]. An analysis of rectangular folded-waveguide slow-wave structure has been developed using conformal mapping by using Schwarz Christoffel transformation [14]. A simple closed form expression to compute the time-domain reflection coefficient for a transient TE<sup>10</sup> mode wave incident on a dielectric step discontinuity in a rectangular waveguide has been presented [15]. In this paper, an exponential series approximation was provided for efficient computation of the reflected and transmitted field

The electromagnetic fields in rectangular conducting waveguides filled with uniaxial anisotropic media have been characterized [16]. In this paper, the electric type dyadic Green's function due to an electric source was derived by using eigenfunctions expansion and the Ohm-Rayleigh method. An improved generalized admittance matrix technique based on mode matching method has been proposed [17]. The generalized scattering matrix of waveguide structure and its discontinuity problems is obtained with relationship equations and

A full-vectorial boundary integral equation method for computing guided modes of optical waveguides has been proposed [18]. Method for the propagation constants of fiber waveguides

guide has been extended to compute the dispersion and interaction impedance [9].

and by introducing the concept of the iterative use of low-order jump conditions.

attenuation [6, 7].

298 Emerging Waveguide Technology

take complex values.

waveforms.

reflection coefficients.

The rectangular dielectric waveguide technique for the determination of complex permittivity of a wide class of dielectric materials of various thicknesses and cross sections has been described [20]. In this paper, the technique has been presented to determine the dielectric constant of materials. The fields and propagation constants of dielectric waveguides have been determined within the scalar regime [21] by using two-dimensional Fourier series expansions. Propagation of modes in some rectangular waveguides using the finite-difference time-domain method has been proposed [22]. Analysis of rectangular waveguide using finite element method has been proposed [23].

Wave propagation and dielectric permittivity reconstruction in the case of a rectangular waveguide have been studied [24]. According to this paper, we study the electromagnetic wave propagation in a rectangular waveguide filled with an inhomogeneous dielectric material in the longitudinal direction. Light propagation in a cylindrical waveguide with a complex, metallic and dielectric function has been proposed [25]. Advancement of algebraic function approximation in eigenvalue problems of lossless metallic waveguides to infinite dimensions has been investigated [26]. The method of algebraic function approximation in eigenvalue problems of lossless metallic waveguides such as a closed uniform cylindrical waveguides has been proposed [27]. Analysis of longitudinally inhomogeneous waveguides using Taylor's series expansion has been proposed [28]. Analysis of longitudinally inhomogeneous waveguides using the Fourier series expansion has been proposed [29]. The method of external excitation for analysis of arbitrarily-shaped hollow conducting waveguides has been proposed [30].

A circular metallic hollow waveguide with inner dielectric multilayers has been designed by Miyagi and Kawakami [31] with the emphasis on low-loss transmission of the HE<sup>11</sup> mode for the infrared. According to this paper, the transmission losses of the dielectric-coated metal waveguides are drastically reduced when a multiple dielectric layer is formed instead of a single dielectric layer. The simplest and most efficient multilayer structure is a three-dielectriclayer stack deposited on a metal layer.

A transfer matrix function for the analysis of electromagnetic wave propagation along the straight dielectric waveguide with arbitrary profiles has been proposed [32]. According to this paper, the method is based on the Laplace and Fourier transforms, and the inverse Laplace and Fourier transforms. A rigorous approach for the propagation of electromagnetic fields along a straight hollow waveguide with a circular cross section has been proposed [33]. The cross section is made of a metallic layer, and only one dielectric layer upon it. The separation of variables is obtained by using the orthogonal-relations. The longitudinal components of the fields are developed into the Fourier-Bessel series. The transverse components of the fields are expressed as functions of the longitudinal components in the Laplace plane and are obtained by using the inverse Laplace transform by the residue method.

In order to solve more complex problems of coatings in the cross section of the dielectric waveguides, such as rectangular and circular profiles, then it is important to develop in each modal an improved technique to calculate the dielectric profile, the elements of the matrix and its derivatives of the dielectric profile.

circular profile in the cross section of the straight rectangular waveguide. Figure 1(c) shows an example for three dielectric layers and a metallic layer in the cross section of the straight

Applications and Solving Techniques of Propagated Wave in Waveguides Filled with Inhomogeneous Dielectric…

In order to solve these inhomogeneous dielectric materials, we need to calculate the dielectric profile, the elements of the matrix and its derivatives of the dielectric profile. In the next section, we explain the techniques to solve problems with inhomogeneous dielectric materials in the cross section of the straight rectangular waveguide (Figure 1(a) and 1(b)) and also in the

The particular application is based on the ωε function [34]. The ωε function is used in order to solve discontinuous problems of the rectangular profile, and circular profile in the cross section of the straight waveguide. The ωε function is defined as ωεð Þ¼ <sup>r</sup> <sup>C</sup><sup>ε</sup> exp �ε<sup>2</sup><sup>=</sup> <sup>ε</sup><sup>2</sup> � j j<sup>r</sup> <sup>2</sup> h i � � for <sup>∣</sup>r<sup>∣</sup> <sup>&</sup>gt; <sup>ε</sup>,

The technique that based on ωε function is very effective to solve complex problems, in relation to the conventional methods, especially when we have a large numbers of dielectric layers and a metal layer, as shown in Figure 1(c). We will demonstrate how to use with the proposed

3.1. The technique based on ωε function for the rectangular profile in the cross section of

The elements of the matrix g(n, m) are calculated for an arbitrary profile in the cross section of the straight waveguide according to Figure 3(a) and 3(b). Figure 3(a) shows the arbitrary profile in the cross section of the straight waveguide. Figure 3(b) shows the rectangular profile

ωεð Þr dr ¼ 1. In the limit ε ! 0, the ωε function is shown in Figure 2.

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301

cross section of the straight circular waveguide (Figure 1(c)).

3. The technique to solve inhomogeneous dielectric profiles

technique for all the cases that are shown in the examples in Figure 1(a)–(c).

in the cross section of the straight waveguide, according to Figure 1(a).

Figure 2. The technique based on ωε function in the limit ε ! 0 to solve discontinuous problems.

hollow waveguide.

where C<sup>ε</sup> is a constant, and Ð

the straight rectangular waveguide

The main objective of this chapter is to develop the techniques that enable us to solve problems with inhomogeneous dielectric materials in the cross section of the straight rectangular and circular waveguides. The second objective is to understand the influence of the inhomogeneous dielectric materials on the output fields. Thus, we need to develop the technique and a particular application to calculate the profiles in the cross section. Namely, we need to calculate the dielectric profile, the elements of the matrix and its derivatives of the dielectric profile in the cases of the straight rectangular and circular waveguides. The proposed techniques are important to improve the methods that are based on Laplace and Fourier transforms and their inverse Laplace and Fourier transforms also for the discontinuous rectangular and circular profiles in the cross section (and not only for the continuous profiles).

#### 2. Formulation of the problem

In this chapter, we present techniques for solving discontinuous problems of dielectric materials in the cross section of the straight waveguide for applications in the microwave and millimeter-wave regimes and in the cases of infrared regime. The proposed techniques are very effective in relation to the conventional methods because they allow the development of expressions in the cross section only according to the specific discontinuous problem. In this way, the mode model method becomes an improved method to solve discontinuous problems in the cross section.

Three examples of inhomogeneous dielectric materials in the cross section of the straight waveguides are shown in Figure 1(a)–(c). Figure 1(a) shows an example of rectangular profile in the cross section of the straight rectangular waveguide. Figure 1(b) shows an example of

Figure 1. Three examples of inhomogeneous dielectric materials in the cross section of the straight waveguides. (a) Rectangular profile in the cross section of the straight rectangular waveguide; (b) Circular profile in the cross section of the straight rectangular waveguide; (c) Three dielectric layers and a metallic layer in the cross section of the hollow waveguide.

circular profile in the cross section of the straight rectangular waveguide. Figure 1(c) shows an example for three dielectric layers and a metallic layer in the cross section of the straight hollow waveguide.

In order to solve these inhomogeneous dielectric materials, we need to calculate the dielectric profile, the elements of the matrix and its derivatives of the dielectric profile. In the next section, we explain the techniques to solve problems with inhomogeneous dielectric materials in the cross section of the straight rectangular waveguide (Figure 1(a) and 1(b)) and also in the cross section of the straight circular waveguide (Figure 1(c)).

## 3. The technique to solve inhomogeneous dielectric profiles

modal an improved technique to calculate the dielectric profile, the elements of the matrix and

The main objective of this chapter is to develop the techniques that enable us to solve problems with inhomogeneous dielectric materials in the cross section of the straight rectangular and circular waveguides. The second objective is to understand the influence of the inhomogeneous dielectric materials on the output fields. Thus, we need to develop the technique and a particular application to calculate the profiles in the cross section. Namely, we need to calculate the dielectric profile, the elements of the matrix and its derivatives of the dielectric profile in the cases of the straight rectangular and circular waveguides. The proposed techniques are important to improve the methods that are based on Laplace and Fourier transforms and their inverse Laplace and Fourier transforms also for the discontinuous rectangular and circular

In this chapter, we present techniques for solving discontinuous problems of dielectric materials in the cross section of the straight waveguide for applications in the microwave and millimeter-wave regimes and in the cases of infrared regime. The proposed techniques are very effective in relation to the conventional methods because they allow the development of expressions in the cross section only according to the specific discontinuous problem. In this way, the mode model method becomes an improved method to solve discontinuous problems

Three examples of inhomogeneous dielectric materials in the cross section of the straight waveguides are shown in Figure 1(a)–(c). Figure 1(a) shows an example of rectangular profile in the cross section of the straight rectangular waveguide. Figure 1(b) shows an example of

Figure 1. Three examples of inhomogeneous dielectric materials in the cross section of the straight waveguides. (a) Rectangular profile in the cross section of the straight rectangular waveguide; (b) Circular profile in the cross section of the straight rectangular waveguide; (c) Three dielectric layers and a metallic layer in the cross section of the hollow

profiles in the cross section (and not only for the continuous profiles).

its derivatives of the dielectric profile.

300 Emerging Waveguide Technology

2. Formulation of the problem

in the cross section.

waveguide.

The particular application is based on the ωε function [34]. The ωε function is used in order to solve discontinuous problems of the rectangular profile, and circular profile in the cross section of the straight waveguide. The ωε function is defined as ωεð Þ¼ <sup>r</sup> <sup>C</sup><sup>ε</sup> exp �ε<sup>2</sup><sup>=</sup> <sup>ε</sup><sup>2</sup> � j j<sup>r</sup> <sup>2</sup> h i � � for <sup>∣</sup>r<sup>∣</sup> <sup>&</sup>gt; <sup>ε</sup>, where C<sup>ε</sup> is a constant, and Ð ωεð Þr dr ¼ 1. In the limit ε ! 0, the ωε function is shown in Figure 2.

The technique that based on ωε function is very effective to solve complex problems, in relation to the conventional methods, especially when we have a large numbers of dielectric layers and a metal layer, as shown in Figure 1(c). We will demonstrate how to use with the proposed technique for all the cases that are shown in the examples in Figure 1(a)–(c).

#### 3.1. The technique based on ωε function for the rectangular profile in the cross section of the straight rectangular waveguide

The elements of the matrix g(n, m) are calculated for an arbitrary profile in the cross section of the straight waveguide according to Figure 3(a) and 3(b). Figure 3(a) shows the arbitrary profile in the cross section of the straight waveguide. Figure 3(b) shows the rectangular profile in the cross section of the straight waveguide, according to Figure 1(a).

Figure 2. The technique based on ωε function in the limit ε ! 0 to solve discontinuous problems.

The dielectric profile g xð Þ ; y is given according to εð Þ¼ x; y ε0ð Þ 1 þ g xð Þ ; y . According to Figure 3(a) and 3(b) and for g xð Þ¼ ; y g0, we obtain

$$\begin{split} g(\mathbf{r},m) &= \frac{g\_0}{4ab} \int\_{-a}^{a} dx \int\_{-b}^{b} \exp\left(-j(k\_x \mathbf{x} + k\_y y)\right) dy \\ &= \frac{g\_0}{4ab} \left\{ \int\_{x\_{11}}^{x\_{12}} dx \int\_{y\_{11}}^{y\_{12}} \exp\left(-j(k\_x \mathbf{x} + k\_y y)\right) dy + \int\_{-x\_{12}}^{-x\_{11}} dx \int\_{y\_{11}}^{y\_{12}} \exp\left(-j(k\_x \mathbf{x} + k\_y y)\right) dy \right. \\ &\left. + \int\_{-x\_{12}}^{-x\_{11}} dx \int\_{-y\_{12}}^{-y\_{11}} \exp\left(-j(k\_x \mathbf{x} + k\_y y)\right) dy + \int\_{x\_{11}}^{x\_{12}} dx \int\_{-y\_{12}}^{-y\_{11}} \exp\left(-j(k\_x \mathbf{x} + k\_y y)\right) dy \right\}. \end{split} \tag{1}$$

If y<sup>11</sup> and y<sup>12</sup> are not functions of x, then the dielectric profile is given by

$$\mathrm{g}(n,m) = \frac{\mathcal{g}\_0}{ab} \int\_{x\_{11}}^{x\_{12}} \cos\left(k\_x x\right) dx \int\_{y\_{11}}^{y\_{12}} \cos\left(k\_y y\right) dy. \tag{2}$$

3.2. The technique based on ωε function for the circular profile in the cross section of the

Applications and Solving Techniques of Propagated Wave in Waveguides Filled with Inhomogeneous Dielectric…

located at (0.5 a, 0.5 b), as shown in Figure 1(b). We obtain two possibilities without this

The dielectric profile for the circle is given where the center is located at (0.5 a, 0.5 b) (Figure 1(b)) by

g<sup>0</sup> 0 ≤ r < r<sup>1</sup> � ε1=2

ε1 2

The derivatives of the dielectric profile for the circle are given where the center is located at (0.5 a,

gx <sup>¼</sup> �<sup>2</sup> <sup>g</sup><sup>0</sup> cos <sup>θ</sup>exp <sup>1</sup> � <sup>q</sup>εð Þ<sup>r</sup> � �½ � <sup>r</sup> � ð Þ <sup>r</sup><sup>1</sup> � <sup>ε</sup>1=<sup>2</sup> <sup>ε</sup><sup>1</sup>

gy <sup>¼</sup> �<sup>2</sup> <sup>g</sup><sup>0</sup> sin <sup>θ</sup>exp <sup>1</sup> � <sup>q</sup>εð Þ<sup>r</sup> � �½ � <sup>r</sup> � ð Þ <sup>r</sup><sup>1</sup> � <sup>ε</sup>1=<sup>2</sup> <sup>ε</sup><sup>1</sup>

The elements of the matrices for the circular profile are given where the center is located at (0.5 a,

a 2 mπ

<sup>b</sup> <sup>r</sup> sin <sup>θ</sup> <sup>þ</sup>

mπ

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>r</sup><sup>1</sup> � <sup>ε</sup>1=<sup>2</sup> <sup>2</sup> h i<sup>2</sup>

<sup>b</sup> <sup>r</sup> sin <sup>θ</sup> <sup>þ</sup>

mπ

<sup>b</sup> <sup>r</sup> sin <sup>θ</sup> <sup>þ</sup>

<sup>2</sup>½ � <sup>r</sup> � ð Þ <sup>r</sup><sup>1</sup> � <sup>ε</sup>1=<sup>2</sup> exp 1 � <sup>q</sup>εð Þ<sup>r</sup> � � cos <sup>θ</sup>

b 2 � � ���rdrdθ,

b 2 � � � � exp 1 � <sup>q</sup>εð Þ<sup>r</sup> � �)

<sup>g</sup><sup>0</sup> exp 1 � <sup>q</sup>εð Þ<sup>r</sup> � � <sup>r</sup><sup>1</sup> � <sup>ε</sup>1=<sup>2</sup> <sup>≤</sup> <sup>r</sup> <sup>&</sup>lt; <sup>r</sup><sup>1</sup> <sup>þ</sup> <sup>ε</sup>1=<sup>2</sup>

q

and y12ð Þ¼ x b=2 þ

. The center of the circle is

http://dx.doi.org/10.5772/intechopen.76793

,

.

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>r</sup><sup>1</sup> � <sup>ε</sup>1=<sup>2</sup> <sup>2</sup> , (5)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>x</sup> � <sup>a</sup>=<sup>2</sup> <sup>2</sup> <sup>þ</sup> ð Þ <sup>y</sup> � <sup>b</sup>=<sup>2</sup> <sup>2</sup>

2

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>r</sup><sup>1</sup> � <sup>ε</sup>1=<sup>2</sup> <sup>2</sup> h i<sup>2</sup> , (6)

2

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>r</sup><sup>1</sup> � <sup>ε</sup>1=<sup>2</sup> <sup>2</sup> h i<sup>2</sup> , (7)

b 2

<sup>1</sup> <sup>þ</sup> <sup>g</sup><sup>0</sup> exp 1 � <sup>q</sup>εð Þ<sup>r</sup> � � � �

, where y<sup>12</sup> � y<sup>11</sup> ¼

(4)

303

rdrdθ,

(8)

(9)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>r</sup><sup>2</sup> � ð Þ <sup>x</sup> � <sup>a</sup>=<sup>2</sup> <sup>2</sup>

q

The equation of the circle is given by ð Þ <sup>x</sup> � <sup>a</sup>=<sup>2</sup> <sup>2</sup> <sup>þ</sup> ð Þ <sup>y</sup> � <sup>b</sup>=<sup>2</sup> <sup>2</sup> <sup>¼</sup> <sup>r</sup><sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>r</sup><sup>2</sup> � ð Þ <sup>x</sup> � <sup>a</sup>=<sup>2</sup> <sup>2</sup>

g xð Þ¼ ; y

and y<sup>12</sup> þ y<sup>11</sup> ¼ b.

else g xð Þ ; y = 0. The radius of the circle is given by r ¼

�

qεð Þ¼ r

0.5 b), as shown in Figure 1(b), where r<sup>1</sup> � ε1=2 ≤ r < r<sup>1</sup> þ ε1=2 by

ε1

<sup>1</sup> <sup>þ</sup> <sup>g</sup><sup>0</sup> exp 1 � <sup>q</sup>εð Þ<sup>r</sup> � � � � <sup>ε</sup><sup>1</sup>

<sup>1</sup> <sup>þ</sup> <sup>g</sup><sup>0</sup> exp 1 � <sup>q</sup>εð Þ<sup>r</sup> � � � � <sup>ε</sup><sup>1</sup>

r cos θ þ

r cos θ þ

ð<sup>r</sup>1þε1=<sup>2</sup> r1�ε1=2

r cos θ þ

� � � � cos

� � � � cos

� � � � cos

a 2

ε1

ε1

a 2

( � � � �

straight rectangular waveguide

q

y11ð Þ¼ x b=2 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>r</sup><sup>2</sup> � ð Þ <sup>x</sup> � <sup>a</sup>=<sup>2</sup> <sup>2</sup>

else gx = 0, and gy = 0.

0.5 b) (Figure 1(b)) by

þ ð<sup>2</sup><sup>π</sup> 0

ab <sup>ð</sup><sup>2</sup><sup>π</sup> 0

gxð Þ¼� n; m

ð<sup>r</sup>1þε1=<sup>2</sup> r1�ε1=2

ð<sup>r</sup>1�ε1=<sup>2</sup> 0

> 2g<sup>0</sup> ab <sup>ð</sup><sup>2</sup><sup>π</sup> 0

cos nπ a

cos nπ a

> 8 ><

> >:

cos nπ a

g nð Þ¼ ; <sup>m</sup> <sup>g</sup><sup>0</sup>

2

q

where

The derivative of the dielectric profile in the case of y<sup>11</sup> and y<sup>12</sup> are functions of x, is given by

$$\mathcal{G}\_{\mathbf{x}}(\mathbf{n},m) = \frac{2}{am\pi\pi} \int\_{x\_{11}}^{x\_{12}} \mathcal{G}\_{\mathbf{x}}(\mathbf{x},y) \sin\left[\frac{k\_y}{2}(y\_{12}-y\_{11})\right] \cos\left[\frac{k\_y}{2}(y\_{12}+y\_{11})\right] \cos\left(k\_{\mathbf{x}}\mathbf{x}\right) d\mathbf{x},\tag{3}$$

where gxð Þ¼ x;y ð Þ 1=eð Þ x;y ð Þ deð Þ x;y =dx , eð Þ¼ x;y e0ð Þ 1þg xð Þ ;y , kx ¼ ð Þ nπx =a, and ky ¼ ð Þ mπy =b. Similarly, we can calculate the value of gyð Þ n; m , where gyð Þ¼ x; y ð Þ 1=eð Þ x; y ð Þ deð Þ x; y =dy .

For the cross section as shown in Figures 1(a) and 3(b), the center of the rectangle is located at (0.5 a, 0.5 b), y<sup>12</sup> = b/2 + c/2 and y<sup>11</sup> = b/2�c/2. Thus, for this case, y<sup>12</sup> � y<sup>11</sup> = c and y<sup>12</sup> þ y<sup>11</sup> = b.

Figure 3. (a). The arbitrary profile in the cross section of the straight waveguide. (b). The rectangular profile in the cross section of the straight waveguide, according to Figure 1(a).

#### 3.2. The technique based on ωε function for the circular profile in the cross section of the straight rectangular waveguide

The equation of the circle is given by ð Þ <sup>x</sup> � <sup>a</sup>=<sup>2</sup> <sup>2</sup> <sup>þ</sup> ð Þ <sup>y</sup> � <sup>b</sup>=<sup>2</sup> <sup>2</sup> <sup>¼</sup> <sup>r</sup><sup>2</sup> . The center of the circle is located at (0.5 a, 0.5 b), as shown in Figure 1(b). We obtain two possibilities without this y11ð Þ¼ x b=2 � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>r</sup><sup>2</sup> � ð Þ <sup>x</sup> � <sup>a</sup>=<sup>2</sup> <sup>2</sup> q and y12ð Þ¼ x b=2 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>r</sup><sup>2</sup> � ð Þ <sup>x</sup> � <sup>a</sup>=<sup>2</sup> <sup>2</sup> q , where y<sup>12</sup> � y<sup>11</sup> ¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>r</sup><sup>2</sup> � ð Þ <sup>x</sup> � <sup>a</sup>=<sup>2</sup> <sup>2</sup> q and y<sup>12</sup> þ y<sup>11</sup> ¼ b.

The dielectric profile for the circle is given where the center is located at (0.5 a, 0.5 b) (Figure 1(b)) by

$$\log(\mathbf{x}, y) = \begin{cases} \mathcal{g}\_0 & 0 \le r < r\_1 - \varepsilon\_1/2 \\ \mathcal{g}\_0 \exp\left[1 - q\_\varepsilon(r)\right] & r\_1 - \varepsilon\_1/2 \le r < r\_1 + \varepsilon\_1/2' \end{cases} \tag{4}$$

where

The dielectric profile g xð Þ ; y is given according to εð Þ¼ x; y ε0ð Þ 1 þ g xð Þ ; y . According to Figure 3(a)

cos ð Þ kxx dx

The derivative of the dielectric profile in the case of y<sup>11</sup> and y<sup>12</sup> are functions of x, is given by

<sup>2</sup> <sup>y</sup><sup>12</sup> � <sup>y</sup><sup>11</sup> � � � �

where gxð Þ¼ x;y ð Þ 1=eð Þ x;y ð Þ deð Þ x;y =dx , eð Þ¼ x;y e0ð Þ 1þg xð Þ ;y , kx ¼ ð Þ nπx =a, and ky ¼ ð Þ mπy =b. Similarly, we can calculate the value of gyð Þ n; m , where gyð Þ¼ x; y ð Þ 1=eð Þ x; y ð Þ deð Þ x; y =dy .

For the cross section as shown in Figures 1(a) and 3(b), the center of the rectangle is located at (0.5 a, 0.5 b), y<sup>12</sup> = b/2 + c/2 and y<sup>11</sup> = b/2�c/2. Thus, for this case, y<sup>12</sup> � y<sup>11</sup> = c and y<sup>12</sup> þ y<sup>11</sup> = b.

Figure 3. (a). The arbitrary profile in the cross section of the straight waveguide. (b). The rectangular profile in the cross

ky

ð�x<sup>11</sup> �x<sup>12</sup> dx ð<sup>y</sup><sup>12</sup> y11

ð<sup>x</sup><sup>12</sup> x<sup>11</sup> dx ð�y<sup>11</sup> �y<sup>12</sup>

ð<sup>y</sup><sup>12</sup> y11

> cos ky

<sup>2</sup> <sup>y</sup><sup>12</sup> <sup>þ</sup> <sup>y</sup><sup>11</sup> � � � �

exp �j kxx <sup>þ</sup> kyy � � � � dy

exp �j kxx <sup>þ</sup> kyy � � � � dy

cos kyy � �dy: (2)

) :

cos ð Þ kxx dx, (3)

(1)

exp �j kxx <sup>þ</sup> kyy � � � � dy

exp �j kxx <sup>þ</sup> kyy � � � � dy <sup>þ</sup>

exp �j kxx <sup>þ</sup> kyy � � � � dy <sup>þ</sup>

If y<sup>11</sup> and y<sup>12</sup> are not functions of x, then the dielectric profile is given by

ab ð<sup>x</sup><sup>12</sup> x<sup>11</sup>

g nð Þ¼ ; <sup>m</sup> <sup>g</sup><sup>0</sup>

gxð Þ x; y sin

and 3(b) and for g xð Þ¼ ; y g0, we obtain

ð<sup>x</sup><sup>12</sup> x<sup>11</sup> dx ð<sup>y</sup><sup>12</sup> y11

(

g nð Þ¼ ; <sup>m</sup> <sup>g</sup><sup>0</sup>

302 Emerging Waveguide Technology

4ab ða �a dx ðb �b

<sup>¼</sup> <sup>g</sup><sup>0</sup> 4ab

> þ ð�x<sup>11</sup> �x<sup>12</sup> dx ð�y<sup>11</sup> �y<sup>12</sup>

gxð Þ¼ n; m

2 amπ

section of the straight waveguide, according to Figure 1(a).

ð<sup>x</sup><sup>12</sup> x<sup>11</sup>

$$\eta\_{\varepsilon}(r) = \frac{\varepsilon\_1^{\,^2}}{\varepsilon\_1^{\,^2} - \left[r - (r\_1 - \varepsilon\_1/2)\right]^2},\tag{5}$$

else g xð Þ ; y = 0. The radius of the circle is given by r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>x</sup> � <sup>a</sup>=<sup>2</sup> <sup>2</sup> <sup>þ</sup> ð Þ <sup>y</sup> � <sup>b</sup>=<sup>2</sup> <sup>2</sup> q .

The derivatives of the dielectric profile for the circle are given where the center is located at (0.5 a, 0.5 b), as shown in Figure 1(b), where r<sup>1</sup> � ε1=2 ≤ r < r<sup>1</sup> þ ε1=2 by

$$\mathbf{g}\_x = \frac{-2 \begin{array}{c} \mathbf{g}\_0 \ \cos\theta \exp\left[1 - q\_\ell(r)\right] \left[r - (r\_1 - \varepsilon\_1/2)\right] \varepsilon\_1^2}{\left\{1 + \mathbf{g}\_0 \exp\left[1 - q\_\ell(r)\right]\right\} \left[\varepsilon\_1^2 - \left[r - (r\_1 - \varepsilon\_1/2)\right]^2\right]^{2'}} \tag{6}$$

$$\mathbf{g}\_y = \frac{-2 \begin{array}{c} \mathbf{g}\_0 \ \sin \theta \exp\left[1 - q\_\ell(r)\right] \left[r - (r\_1 - \varepsilon\_1/2)\right] \varepsilon\_1^2}{\left\{1 + \mathbf{g}\_0 \exp\left[1 - q\_\ell(r)\right]\right\} \left[\varepsilon\_1^2 - \left[r - (r\_1 - \varepsilon\_1/2)\right]^2\right]^2} \tag{7}$$

else gx = 0, and gy = 0.

The elements of the matrices for the circular profile are given where the center is located at (0.5 a, 0.5 b) (Figure 1(b)) by

$$\begin{split} g(n,m) &= \frac{g\_0}{ab} \left\{ \int\_0^{2\pi} \int\_0^{r\_1 - \varepsilon\_1/2} \cos\left[\frac{n\pi}{a}\left(r\cos\theta + \frac{a}{2}\right)\right] \cos\left[\frac{m\pi}{b}\left(r\sin\theta + \frac{b}{2}\right)\right] \\ &+ \int\_0^{2\pi} \int\_{r\_1 - \varepsilon\_1/2}^{r\_1 + \varepsilon\_1/2} \cos\left[\frac{m\pi}{a}\left(r\cos\theta + \frac{a}{2}\right)\right] \cos\left[\frac{m\pi}{b}\left(r\sin\theta + \frac{b}{2}\right)\right] \exp\left[1 - q\_\varepsilon(r)\right] \right\} r dr d\theta,\end{split} \tag{8}$$

$$g\_x(n,m) = -\frac{2g\_0}{ab} \left\{ \int\_0^{2\pi} \int\_{r\_1 - \varepsilon\_1/2}^{r\_1 + \varepsilon\_1/2} \frac{\varepsilon\_1^2 [r - (r\_1 - \varepsilon\_1/2)] \exp\left[1 - q\_\varepsilon(r)\right] \cos\theta}{\left[\varepsilon\_1^2 - \left[r - (r\_1 - \varepsilon\_1/2)\right]^2\right]^2 \left[1 + g\_0 \exp\left[1 - q\_\varepsilon(r)\right]\right]} \right. \tag{9}$$
 
$$\cos\left[\frac{n\pi}{a}\left(r\cos\theta + \frac{a}{2}\right)\right] \cos\left[\frac{m\pi}{b}\left(r\sin\theta + \frac{b}{2}\right)\right] \right\} r dr d\theta,$$

$$g\_y(n,m) = -\frac{2g\_0}{ab} \left\{ \int\_0^{2\pi} \int\_{r\_1 - \varepsilon\_1/2}^{r\_1 + \varepsilon\_1/2} \frac{\varepsilon\_1^2 [r - (r\_1 - \varepsilon\_1/2)] \exp\left[1 - q\_\varepsilon(r)\right] \sin\theta}{\left[\varepsilon\_1^2 - \left[r - (r\_1 - \varepsilon\_1/2)\right]^2\right]^2 \left[1 + g\_0 \exp\left[1 - q\_\varepsilon(r)\right]\right]} \right. \tag{10}$$
 
$$\cos\left[\frac{n\pi}{a}\left(r\cos\theta + \frac{a}{2}\right)\right] \cos\left[\frac{m\pi}{b}\left(r\sin\theta + \frac{b}{2}\right)\right] \right\} r dr d\theta,$$
 
$$= -\sqrt{\left(\mathbf{x} - \mathbf{a}/2\right)^2 + \left(\mathbf{y} - b/2\right)^2}$$

2b1, 2b2, 2a, and 2(a+δm), respectively, where δ<sup>m</sup> is the thickness of the metallic layer. In addition, we denote the thickness of the dielectric layers as d1, d2, and d3, respectively, where d<sup>1</sup> = b<sup>1</sup> � b, d<sup>2</sup> = b<sup>2</sup> � b1, and d<sup>3</sup> = a � b2. The refractive index in the particular case with the three dielectric layers and the metallic layer in the cross section of the straight hollow waveguide

Applications and Solving Techniques of Propagated Wave in Waveguides Filled with Inhomogeneous Dielectric…

n<sup>0</sup> 0 ≤ r < b � ε1=2

" #

" #

" #

" #

2

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup> <sup>þ</sup> <sup>ε</sup>1=<sup>2</sup> <sup>2</sup>

n<sup>1</sup> b þ ε1=2 ≤ r < b<sup>1</sup> � ε2=2

2

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup><sup>1</sup> <sup>þ</sup> <sup>ε</sup>2=<sup>2</sup> <sup>2</sup>

2

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup><sup>2</sup> <sup>þ</sup> <sup>ε</sup>3=<sup>2</sup> <sup>2</sup>

2

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>a</sup> <sup>þ</sup> <sup>ε</sup>4=<sup>2</sup> <sup>2</sup>

where the parameters ε1, ε2, ε3, and ε<sup>4</sup> are very small [e.g., ε<sup>1</sup> = ε<sup>2</sup> = ε<sup>3</sup> = ½ � a � b =50, ε<sup>4</sup> = δm/50]. The refractive indices of the air, dielectric and metallic layers are denoted as n0, n1, n2, n3, and

0 0 ≤ r < b � ε1=2

ε1

0 b þ ε1=2 ≤ r < b<sup>1</sup> � ε2=2

ε1

0 b<sup>1</sup> þ ε2=2 ≤ r < b<sup>2</sup> � ε3=2

ε1

0 b<sup>2</sup> þ ε3=2 ≤ r < a � ε4=2

ε4

½ � r � ð Þ b þ ε1=2 ε<sup>1</sup>

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup> <sup>þ</sup> <sup>ε</sup>1=<sup>2</sup> <sup>2</sup> h i<sup>2</sup>

½ � r � ð Þ b<sup>1</sup> þ ε2=2 ε<sup>2</sup>

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup><sup>1</sup> <sup>þ</sup> <sup>ε</sup>2=<sup>2</sup> <sup>2</sup> h i<sup>2</sup>

½ � r � ð Þ b<sup>2</sup> þ ε3=2 ε<sup>3</sup>

½ � r � ð Þ a þ ε4=2 ε<sup>4</sup>

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>a</sup> <sup>þ</sup> <sup>ε</sup>4=<sup>2</sup> <sup>2</sup> h i<sup>2</sup>

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup><sup>1</sup> <sup>þ</sup> <sup>ε</sup>2=<sup>2</sup> <sup>2</sup> h i<sup>2</sup>

2

2

2

2

n<sup>3</sup> b<sup>2</sup> þ ε3=2 ≤ r < a � ε3=2

n<sup>2</sup> b<sup>1</sup> þ ε2=2 ≤ r < b<sup>2</sup> � ε2=2

b � ε1=2 ≤ r < b þ ε1=2

http://dx.doi.org/10.5772/intechopen.76793

b<sup>1</sup> � ε2=2 ≤ r < b<sup>1</sup> þ ε2=2

,

b � ε1=2 ≤ r < b þ ε1=2

b<sup>1</sup> � ε2=2 ≤ r < b<sup>1</sup> þ ε2=2

:

(13)

b<sup>2</sup> � ε3=2 ≤ r < b<sup>2</sup> þ ε3=2

a � ε4=2 ≤ r < a þ ε4=2

(12)

305

b<sup>2</sup> � ε2=2 ≤ r < b<sup>2</sup> þ ε3=2

a � ε3=2 ≤ r < a þ ε4=2

(Figure 1(c)) is calculated as follows:

8

>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>:

<sup>n</sup><sup>0</sup> <sup>þ</sup> ð Þ <sup>n</sup><sup>1</sup> � <sup>n</sup><sup>0</sup> exp 1 � <sup>ε</sup><sup>1</sup>

<sup>n</sup><sup>1</sup> <sup>þ</sup> ð Þ <sup>n</sup><sup>2</sup> � <sup>n</sup><sup>1</sup> exp 1 � <sup>ε</sup><sup>2</sup>

<sup>n</sup><sup>2</sup> <sup>þ</sup> ð Þ <sup>n</sup><sup>3</sup> � <sup>n</sup><sup>2</sup> exp 1 � <sup>ε</sup><sup>3</sup>

<sup>n</sup><sup>3</sup> <sup>þ</sup> ð Þ nm � <sup>n</sup><sup>3</sup> exp 1 � <sup>ε</sup><sup>4</sup>

nm, respectively. In this study, we suppose that n<sup>3</sup> > n<sup>2</sup> > n1.

ε1

ε2

ε3

ε4

ε3

ε4

ε1

ε2

�<sup>4</sup> <sup>n</sup><sup>1</sup> � <sup>n</sup><sup>0</sup> ð Þ exp 1 � <sup>ε</sup><sup>1</sup>

�<sup>4</sup> <sup>n</sup><sup>2</sup> � <sup>n</sup><sup>1</sup> ð Þ exp 1 � <sup>ε</sup><sup>2</sup>

�<sup>4</sup> <sup>n</sup><sup>3</sup> � <sup>n</sup><sup>2</sup> ð Þ exp 1 � <sup>ε</sup><sup>3</sup>

�<sup>4</sup> nm � <sup>n</sup><sup>3</sup> ð Þ exp 1 � <sup>ε</sup><sup>4</sup>

<sup>n</sup><sup>0</sup> <sup>þ</sup> <sup>n</sup><sup>1</sup> � <sup>n</sup><sup>0</sup> ð Þ exp 1 � <sup>ε</sup><sup>1</sup>

<sup>n</sup><sup>2</sup> <sup>þ</sup> <sup>n</sup><sup>3</sup> � <sup>n</sup><sup>2</sup> ð Þ exp 1 � <sup>ε</sup><sup>2</sup>

<sup>n</sup><sup>3</sup> <sup>þ</sup> nm � <sup>n</sup><sup>3</sup> ð Þ exp 1 � <sup>ε</sup><sup>3</sup>

<sup>n</sup><sup>3</sup> <sup>þ</sup> nm � <sup>n</sup><sup>3</sup> ð Þ exp 1 � <sup>ε</sup><sup>4</sup>

ε1

ε2

ε3

ε4

The transverse derivative of the dielectric profile is calculated as follows:

2

" #

2

" #

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup> <sup>þ</sup> <sup>ε</sup>1=<sup>2</sup> <sup>2</sup>

" #

2

" #

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup><sup>1</sup> <sup>þ</sup> <sup>ε</sup>2=<sup>2</sup> <sup>2</sup>

" #

2

" #

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup><sup>2</sup> <sup>þ</sup> <sup>ε</sup>3=<sup>2</sup> <sup>2</sup>

" #

2

" #

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>a</sup> <sup>þ</sup> <sup>ε</sup>4=<sup>2</sup> <sup>2</sup>

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup> <sup>þ</sup> <sup>ε</sup>1=<sup>2</sup> <sup>2</sup>

2

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup><sup>1</sup> <sup>þ</sup> <sup>ε</sup>2=<sup>2</sup> <sup>2</sup>

2

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup><sup>2</sup> <sup>þ</sup> <sup>ε</sup>3=<sup>2</sup> <sup>2</sup>

2

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>a</sup> <sup>þ</sup> <sup>ε</sup>4=<sup>2</sup> <sup>2</sup>

0 else

nm else

n rð Þ¼

8

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

grð Þ¼ r

where r ¼ ð Þ <sup>x</sup> � <sup>a</sup>=<sup>2</sup> <sup>2</sup> <sup>þ</sup> ð Þ <sup>y</sup> � <sup>b</sup>=<sup>2</sup> <sup>2</sup> .

The proposed techniques in this subsection and the subsection 3.1 relate to the method for the propagation along the straight rectangular metallic waveguide [32]. The techniques and the particular applications to solve the rectangular and circular profiles in the cross section of the straight rectangular waveguide are important in order to improve the mode model [32]. The method is based on the Laplace and Fourier transform and their inverse transforms. Laplace transform is necessary to obtain the comfortable and simple input-output connections of the fields. The output transverse field profiles are computed by the inverse Laplace and Fourier transforms.

The matrix G is given by the form.

$$\mathbf{G} = \begin{bmatrix} \mathcal{S}\_{00} & \mathcal{S}\_{-10} & \mathcal{S}\_{-20} & \cdots & \mathcal{S}\_{-nm} & \cdots & \mathcal{S}\_{-NM} \\\\ \mathcal{S}\_{10} & \mathcal{S}\_{00} & \mathcal{S}\_{-10} & \cdots & \mathcal{S}\_{-(n-1)m} & \cdots & \mathcal{S}\_{-(N-1)M} \\\\ \mathcal{S}\_{20} & \mathcal{S}\_{10} & \ddots & \ddots & \ddots & & \\ \vdots & \mathcal{S}\_{20} & \ddots & \ddots & \ddots & & \\ \mathcal{S}\_{mn} & \ddots & \ddots & \ddots & \mathcal{S}\_{00} & \vdots \\\\ \vdots & & & & & \\ \mathcal{S}\_{NM} & \cdots & \cdots & \cdots & \cdots & \cdots & \mathcal{S}\_{00} \end{bmatrix}. \tag{11}$$

Similarly, the G<sup>x</sup> and G<sup>y</sup> matrices are obtained by the derivatives of the dielectric profile. These matrices relate to the method that is based on the Laplace and Fourier transforms and their inverse [32].

Several examples will demonstrate in the next section in order to explore the effects of the rectangular and circular materials in the cross section (Figure 1(a) and 1(b)) along the straight waveguide on the output field. All the graphical results will be demonstrated as a response to a half-sine (TE10) input-wave profile and the rectangular and circular materials in the cross section of the straight rectangular waveguide.

#### 3.3. The technique based on ωε function for the circular profile in the cross section of the straight circular waveguide

The cross section of hollow waveguide (Figure 1(c)) is made of a tube of various types of three dielectric layers and a metallic layer. The internal and external diameters are denoted as 2b, 2b1, 2b2, 2a, and 2(a+δm), respectively, where δ<sup>m</sup> is the thickness of the metallic layer. In addition, we denote the thickness of the dielectric layers as d1, d2, and d3, respectively, where d<sup>1</sup> = b<sup>1</sup> � b, d<sup>2</sup> = b<sup>2</sup> � b1, and d<sup>3</sup> = a � b2. The refractive index in the particular case with the three dielectric layers and the metallic layer in the cross section of the straight hollow waveguide (Figure 1(c)) is calculated as follows:

gyð Þ¼� n; m

304 Emerging Waveguide Technology

q

Fourier transforms.

inverse [32].

The matrix G is given by the form.

G ¼

section of the straight rectangular waveguide.

straight circular waveguide

⋮

where r ¼

2g<sup>0</sup> ab

cos nπ a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>x</sup> � <sup>a</sup>=<sup>2</sup> <sup>2</sup> <sup>þ</sup> ð Þ <sup>y</sup> � <sup>b</sup>=<sup>2</sup> <sup>2</sup>

ð<sup>2</sup><sup>π</sup> 0

8 ><

>:

ð<sup>r</sup>1þε1=<sup>2</sup> r1�ε1=2

r cos θ þ

.

� � � �

ε1

cos

<sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>r</sup><sup>1</sup> � <sup>ε</sup>1=<sup>2</sup> <sup>2</sup> h i<sup>2</sup>

<sup>b</sup> <sup>r</sup> sin <sup>θ</sup> <sup>þ</sup>

� � ���

mπ

The proposed techniques in this subsection and the subsection 3.1 relate to the method for the propagation along the straight rectangular metallic waveguide [32]. The techniques and the particular applications to solve the rectangular and circular profiles in the cross section of the straight rectangular waveguide are important in order to improve the mode model [32]. The method is based on the Laplace and Fourier transform and their inverse transforms. Laplace transform is necessary to obtain the comfortable and simple input-output connections of the fields. The output transverse field profiles are computed by the inverse Laplace and

> <sup>g</sup><sup>00</sup> <sup>g</sup>�<sup>10</sup> <sup>g</sup>�<sup>20</sup> … <sup>g</sup>�nm … <sup>g</sup>�NM <sup>g</sup><sup>10</sup> <sup>g</sup><sup>00</sup> <sup>g</sup>�<sup>10</sup> … <sup>g</sup>�ð Þ <sup>n</sup>�<sup>1</sup> <sup>m</sup> … <sup>g</sup>�ð Þ <sup>N</sup>�<sup>1</sup> <sup>M</sup>

g<sup>20</sup> g<sup>10</sup> ⋱⋱ ⋱ ⋮ g<sup>20</sup> ⋱⋱ ⋱

gnm ⋱ ⋱⋱ g<sup>00</sup> ⋮

gNM … …… … … g<sup>00</sup>

Similarly, the G<sup>x</sup> and G<sup>y</sup> matrices are obtained by the derivatives of the dielectric profile. These matrices relate to the method that is based on the Laplace and Fourier transforms and their

Several examples will demonstrate in the next section in order to explore the effects of the rectangular and circular materials in the cross section (Figure 1(a) and 1(b)) along the straight waveguide on the output field. All the graphical results will be demonstrated as a response to a half-sine (TE10) input-wave profile and the rectangular and circular materials in the cross

3.3. The technique based on ωε function for the circular profile in the cross section of the

The cross section of hollow waveguide (Figure 1(c)) is made of a tube of various types of three dielectric layers and a metallic layer. The internal and external diameters are denoted as 2b,

ε1

a 2 <sup>2</sup>½ � <sup>r</sup> � ð Þ <sup>r</sup><sup>1</sup> � <sup>ε</sup>1=<sup>2</sup> exp 1 � <sup>q</sup>εð Þ<sup>r</sup> � � sin <sup>θ</sup>

b 2 <sup>1</sup> <sup>þ</sup> <sup>g</sup><sup>0</sup> exp 1 � <sup>q</sup>εð Þ<sup>r</sup> � � � �

: (11)

(10)

rdrdθ,

$$n(r) = \begin{cases} n\_0 & 0 \le r < b - \varepsilon\_1/2\\ n\_0 + (n\_1 - n\_0) \exp\left[1 - \frac{\varepsilon\_1^2}{\varepsilon\_1^2 - \left[r - (b + \varepsilon\_1/2)\right]^2}\right] & b - \varepsilon\_1/2 \le r < b + \varepsilon\_1/2\\ n\_1 & b - \varepsilon\_1/2 \le r < b - \varepsilon\_2/2\\ n\_1 + (n\_2 - n\_1) \exp\left[1 - \frac{\varepsilon\_2^2}{\varepsilon\_2^2 - \left[r - (b\_1 + \varepsilon\_2/2)\right]^2}\right] & b\_1 - \varepsilon\_2/2 \le r < b\_1 + \varepsilon\_2/2\\ n\_2 & \varepsilon\_2 + (n\_3 - n\_2) \exp\left[1 - \frac{\varepsilon\_3^2}{\varepsilon\_3^2 - \left[r - (b\_2 + \varepsilon\_3/2)\right]^2}\right] & b\_2 - \varepsilon\_2/2 \le r < b\_2 + \varepsilon\_3/2\\ n\_3 & \varepsilon\_3 + (n\_m - n\_3) \exp\left[1 - \frac{\varepsilon\_4^2}{\varepsilon\_4^2 - \left[r - (a + \varepsilon\_4/2)\right]^2}\right] & b\_2 + \varepsilon\_3/2 \le r < a - \varepsilon\_3/2\\ n\_m & \end{cases} \tag{12}$$

where the parameters ε1, ε2, ε3, and ε<sup>4</sup> are very small [e.g., ε<sup>1</sup> = ε<sup>2</sup> = ε<sup>3</sup> = ½ � a � b =50, ε<sup>4</sup> = δm/50]. The refractive indices of the air, dielectric and metallic layers are denoted as n0, n1, n2, n3, and nm, respectively. In this study, we suppose that n<sup>3</sup> > n<sup>2</sup> > n1.

The transverse derivative of the dielectric profile is calculated as follows:

grð Þ¼ r 0 0 ≤ r < b � ε1=2 �<sup>4</sup> <sup>n</sup><sup>1</sup> � <sup>n</sup><sup>0</sup> ð Þ exp 1 � <sup>ε</sup><sup>1</sup> 2 ε1 <sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup> <sup>þ</sup> <sup>ε</sup>1=<sup>2</sup> <sup>2</sup> " #½ � r � ð Þ b þ ε1=2 ε<sup>1</sup> 2 <sup>n</sup><sup>0</sup> <sup>þ</sup> <sup>n</sup><sup>1</sup> � <sup>n</sup><sup>0</sup> ð Þ exp 1 � <sup>ε</sup><sup>1</sup> 2 ε1 <sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup> <sup>þ</sup> <sup>ε</sup>1=<sup>2</sup> <sup>2</sup> " # ε1 <sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup> <sup>þ</sup> <sup>ε</sup>1=<sup>2</sup> <sup>2</sup> h i<sup>2</sup> b � ε1=2 ≤ r < b þ ε1=2 0 b þ ε1=2 ≤ r < b<sup>1</sup> � ε2=2 �<sup>4</sup> <sup>n</sup><sup>2</sup> � <sup>n</sup><sup>1</sup> ð Þ exp 1 � <sup>ε</sup><sup>2</sup> 2 ε2 <sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup><sup>1</sup> <sup>þ</sup> <sup>ε</sup>2=<sup>2</sup> <sup>2</sup> " #½ � r � ð Þ b<sup>1</sup> þ ε2=2 ε<sup>2</sup> 2 <sup>n</sup><sup>2</sup> <sup>þ</sup> <sup>n</sup><sup>3</sup> � <sup>n</sup><sup>2</sup> ð Þ exp 1 � <sup>ε</sup><sup>2</sup> 2 ε2 <sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup><sup>1</sup> <sup>þ</sup> <sup>ε</sup>2=<sup>2</sup> <sup>2</sup> " # ε1 <sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup><sup>1</sup> <sup>þ</sup> <sup>ε</sup>2=<sup>2</sup> <sup>2</sup> h i<sup>2</sup> b<sup>1</sup> � ε2=2 ≤ r < b<sup>1</sup> þ ε2=2 0 b<sup>1</sup> þ ε2=2 ≤ r < b<sup>2</sup> � ε3=2 �<sup>4</sup> <sup>n</sup><sup>3</sup> � <sup>n</sup><sup>2</sup> ð Þ exp 1 � <sup>ε</sup><sup>3</sup> 2 ε3 <sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup><sup>2</sup> <sup>þ</sup> <sup>ε</sup>3=<sup>2</sup> <sup>2</sup> " #½ � r � ð Þ b<sup>2</sup> þ ε3=2 ε<sup>3</sup> 2 <sup>n</sup><sup>3</sup> <sup>þ</sup> nm � <sup>n</sup><sup>3</sup> ð Þ exp 1 � <sup>ε</sup><sup>3</sup> 2 ε3 <sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup><sup>2</sup> <sup>þ</sup> <sup>ε</sup>3=<sup>2</sup> <sup>2</sup> " # ε1 <sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>b</sup><sup>1</sup> <sup>þ</sup> <sup>ε</sup>2=<sup>2</sup> <sup>2</sup> h i<sup>2</sup> b<sup>2</sup> � ε3=2 ≤ r < b<sup>2</sup> þ ε3=2 0 b<sup>2</sup> þ ε3=2 ≤ r < a � ε4=2 �<sup>4</sup> nm � <sup>n</sup><sup>3</sup> ð Þ exp 1 � <sup>ε</sup><sup>4</sup> 2 ε4 <sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>a</sup> <sup>þ</sup> <sup>ε</sup>4=<sup>2</sup> <sup>2</sup> " #½ � r � ð Þ a þ ε4=2 ε<sup>4</sup> 2 <sup>n</sup><sup>3</sup> <sup>þ</sup> nm � <sup>n</sup><sup>3</sup> ð Þ exp 1 � <sup>ε</sup><sup>4</sup> 2 ε4 <sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>a</sup> <sup>þ</sup> <sup>ε</sup>4=<sup>2</sup> <sup>2</sup> " # ε4 <sup>2</sup> � ½ � <sup>r</sup> � ð Þ <sup>a</sup> <sup>þ</sup> <sup>ε</sup>4=<sup>2</sup> <sup>2</sup> h i<sup>2</sup> a � ε4=2 ≤ r < a þ ε4=2 0 else : 8 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

(13)

The proposed technique in this subsection relates to the theoretical model that based on Maxwell's equations, Fourier-Bessel series, Laplace transform, and the inverse Laplace transform [33]. In this theoretical model, the longitudinal components of the fields are developed into the Fourier-Bessel series. The transverse components of the fields are expressed as functions of the longitudinal components in the Laplace plane and are obtained by using the inverse Laplace transform by the residue method. The separation of the variables is obtained by using the orthogonal relations.

C Nð Þ� log

approaches Ey.

with the analytical solution [35].

shown good agreement.

profile.

max <sup>j</sup>ENþ<sup>2</sup>

Applications and Solving Techniques of Propagated Wave in Waveguides Filled with Inhomogeneous Dielectric…

∣max E<sup>N</sup>þ<sup>2</sup> y

for <sup>N</sup> <sup>≥</sup> 1, where 2ð Þ <sup>N</sup> <sup>þ</sup> <sup>1</sup> <sup>2</sup> is the number of the modes. By increasing the order <sup>N</sup>, then Eyð Þ <sup>N</sup>

Figure 4(a) shows the geometry of the slab profile for practical case of the slab dielectric

Figure 4(b) demonstrates the output result of the comparison between the theoretical model

Figure 4(c) shows the criterion of the convergence in Eq. (18). Between N = 7 and N = 9 the value of the criterion is equal to �2. According to Figure 4(b) and 4(c), the comparison has

All the next graphical results are demonstrated as a response to a half-sine (TE10) input-wave

Figure 5(a)–(f)relates to discontinuous problem according to Figure 1(a). Figure 5(a)–(d) shows the results of the output field as a response to a half-sine (TE10) input-wave profile. In this case, c = d = 3.3 mm and the center of the rectangle is located at the point (0.5 a, 0.5 b), as shown in Figure 1 (a) for e<sup>r</sup> = 3, 5, 7, and 10, respectively. The other parameters are a = b = 2 cm , z = 15 cm, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and β = 58 1=m. The output fields are strongly affected by the input wave profile (TE<sup>10</sup>

Figure 5(e) shows the output amplitude and the Gaussian shape of the central peak in the same cross section of Figure 5(a)–(d), where a = 2 cm, b = 2 cm, y = b/2 = 10 mm, c = 3.3 mm,

Figure 4. (a). The cross section of the straight rectangular waveguide. (b). The results between our model and the

mode), the rectangular profile, and the location of the center of the rectangle (0.5 a, 0.5 b).

d = 3.3 mm, z = 15 cm, k<sup>0</sup> = 167 1=m, and for e<sup>r</sup> =3, 5, 7, and 10, respectively.

analytical method. (c). The criterion of the convergence according to Eq. (18).

material, where a = 20 mm, b = 10 mm, d = 3.3 mm, t = 8.35 mm, λ ¼ 6:9 cm, and e<sup>r</sup> = 9.

8 < :

<sup>y</sup> � EN y j

> y � �<sup>∣</sup>

9 =

;, (18)

307

http://dx.doi.org/10.5772/intechopen.76793

� �

� � � min <sup>E</sup><sup>N</sup>

The output transverse components of the fields of the straight hollow waveguide are finally expressed in a form of transfer matrix function, and the derivation has been already explained in detail in Ref. [33]. The contribution of the proposed technique to calculate the refractive indices (n rð Þ) and the transverse derivative of the dielectric profile (gr) for three dielectric layers is important to improve the method that is based on the mode model [33]. This improved method is important to reduce the transmission losses of the dielectric coated metal waveguides.

#### 4. Numerical results

Several examples for all geometry of the rectangular and circular waveguides and the dielectric profile are demonstrated in this section for three cases, as shown in Figure 1(a)–(c).

#### 4.1. Numerical results for the rectangular dielectric material in the cross section of the straight rectangular waveguide

The analytical method for the dielectric slab [35] is shown in Figure 4(a). The slab profile in the cross section is based on transcendental equation, as follows:

$$E\_{y1} = j \frac{k\_z}{\varepsilon\_0} \sin \left( \nu \mathbf{x} \right) \qquad \quad 0 < \mathbf{x} < t \tag{14}$$

$$E\_{y2} = j \frac{k\_z}{\varepsilon\_0} \frac{\sin\left(\nu t\right)}{\cos\left(\mu(t - a/2)\right)} \cos\left[\mu(\mathbf{x} - a/2)\right] \qquad \qquad t < \mathbf{x} < t + d \tag{15}$$

$$E\_{y3} = j \frac{k\_z}{\varepsilon\_0} \sin \left[ \nu (a - \mathbf{x}) \right] \qquad \qquad t + d < \mathbf{x} < a,\tag{16}$$

where ν � ffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 <sup>o</sup> � k 2 z q and μ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εrk<sup>2</sup> <sup>o</sup> � <sup>k</sup><sup>2</sup> z q result from the transcendental equation.

$$\left(\frac{a-d}{d}\right)\frac{d\mu}{2}\tan\left(\frac{d\mu}{2}\right) - \left(t\nu\right)\cot\left(t\nu\right) = 0.\tag{17}$$

The criterion for the convergence of the solution is given by

Applications and Solving Techniques of Propagated Wave in Waveguides Filled with Inhomogeneous Dielectric… http://dx.doi.org/10.5772/intechopen.76793 307

$$C(N) \equiv \log \left\{ \frac{\max\left( |E\_y^{N+2} - E\_y^N| \right)}{|\max\left( E\_y^{N+2} \right) - \min\left( E\_y^N \right)|} \right\},\tag{18}$$

for <sup>N</sup> <sup>≥</sup> 1, where 2ð Þ <sup>N</sup> <sup>þ</sup> <sup>1</sup> <sup>2</sup> is the number of the modes. By increasing the order <sup>N</sup>, then Eyð Þ <sup>N</sup> approaches Ey.

The proposed technique in this subsection relates to the theoretical model that based on Maxwell's equations, Fourier-Bessel series, Laplace transform, and the inverse Laplace transform [33]. In this theoretical model, the longitudinal components of the fields are developed into the Fourier-Bessel series. The transverse components of the fields are expressed as functions of the longitudinal components in the Laplace plane and are obtained by using the inverse Laplace transform by the residue method. The separation of the variables is obtained

The output transverse components of the fields of the straight hollow waveguide are finally expressed in a form of transfer matrix function, and the derivation has been already explained in detail in Ref. [33]. The contribution of the proposed technique to calculate the refractive indices (n rð Þ) and the transverse derivative of the dielectric profile (gr) for three dielectric layers is important to improve the method that is based on the mode model [33]. This improved method

Several examples for all geometry of the rectangular and circular waveguides and the dielectric

The analytical method for the dielectric slab [35] is shown in Figure 4(a). The slab profile in the

sin ð Þ νx 0 < x < t (14)

sin ½ � νð Þ a � x t þ d < x < a, (16)

� ð Þ tν cotð Þ¼ tν 0: (17)

cos <sup>μ</sup>ð Þ <sup>t</sup> � <sup>a</sup>=<sup>2</sup> � � cos <sup>μ</sup>ð Þ <sup>x</sup> � <sup>a</sup>=<sup>2</sup> � � <sup>t</sup> <sup>&</sup>lt; <sup>x</sup> <sup>&</sup>lt; <sup>t</sup> <sup>þ</sup> <sup>d</sup> (15)

result from the transcendental equation.

is important to reduce the transmission losses of the dielectric coated metal waveguides.

profile are demonstrated in this section for three cases, as shown in Figure 1(a)–(c).

cross section is based on transcendental equation, as follows:

Ey<sup>3</sup> ¼ j

q

a � d d � � dμ

The criterion for the convergence of the solution is given by

Ey<sup>1</sup> ¼ j

sin ð Þ νt

kz ε0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εrk<sup>2</sup> <sup>o</sup> � <sup>k</sup><sup>2</sup> z

<sup>2</sup> tan

dμ 2 � �

kz ε0

4.1. Numerical results for the rectangular dielectric material in the cross section of the

by using the orthogonal relations.

306 Emerging Waveguide Technology

4. Numerical results

straight rectangular waveguide

Ey<sup>2</sup> ¼ j

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 <sup>o</sup> � k 2 z

q

where ν �

kz ε0

and μ �

Figure 4(a) shows the geometry of the slab profile for practical case of the slab dielectric material, where a = 20 mm, b = 10 mm, d = 3.3 mm, t = 8.35 mm, λ ¼ 6:9 cm, and e<sup>r</sup> = 9.

Figure 4(b) demonstrates the output result of the comparison between the theoretical model with the analytical solution [35].

Figure 4(c) shows the criterion of the convergence in Eq. (18). Between N = 7 and N = 9 the value of the criterion is equal to �2. According to Figure 4(b) and 4(c), the comparison has shown good agreement.

All the next graphical results are demonstrated as a response to a half-sine (TE10) input-wave profile.

Figure 5(a)–(f)relates to discontinuous problem according to Figure 1(a). Figure 5(a)–(d) shows the results of the output field as a response to a half-sine (TE10) input-wave profile. In this case, c = d = 3.3 mm and the center of the rectangle is located at the point (0.5 a, 0.5 b), as shown in Figure 1 (a) for e<sup>r</sup> = 3, 5, 7, and 10, respectively. The other parameters are a = b = 2 cm , z = 15 cm, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and β = 58 1=m. The output fields are strongly affected by the input wave profile (TE<sup>10</sup> mode), the rectangular profile, and the location of the center of the rectangle (0.5 a, 0.5 b).

Figure 5(e) shows the output amplitude and the Gaussian shape of the central peak in the same cross section of Figure 5(a)–(d), where a = 2 cm, b = 2 cm, y = b/2 = 10 mm, c = 3.3 mm, d = 3.3 mm, z = 15 cm, k<sup>0</sup> = 167 1=m, and for e<sup>r</sup> =3, 5, 7, and 10, respectively.

Figure 4. (a). The cross section of the straight rectangular waveguide. (b). The results between our model and the analytical method. (c). The criterion of the convergence according to Eq. (18).

Figure 5. The output field as a response to a half-sine (TE10) input-wave profile where c = d = 3.3 mm and the center of the rectangle is located at the point (0.5 a, 0.5 b) according to Figure 1(a), where: (a). e<sup>r</sup> = 3; (b). e<sup>r</sup> = 5; (c). e<sup>r</sup> = 7; (d). e<sup>r</sup> = 10. The other parameters are a = b = 2 cm, z = 15 cm, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and β = 58 1=m . (e). The output field for e<sup>r</sup> =3, 5, 7, and 10, respectively, where a = b = 2 cm, y = b/2 = 10 mm. (f). The output profiles for N = 1, 3, 5, and 7, where e<sup>r</sup> = 10.

By increasing only the value of e<sup>r</sup> of the rectangular dielectric profile, the TE<sup>10</sup> wave profile decreased, the Gaussian shape of the output field increased, and the relative amplitude decreased. In addition, by increasing only the value of er, the width of the Gaussian shape decreased.

The output profiles are shown in Figure 6(f) for N = 1, 3, 5, and 7, where e<sup>r</sup> = 10. The other

Figure 6. The output field as a response to a half-sine (TE10) input-wave profile where the radius of all circle is equal to 2.5 mm and the center of the circle is located at the point (0.5 a, 0.5 b) according to Figure 1(b), where: (a). e<sup>r</sup> = 3; (b). e<sup>r</sup> = 5; (c). e<sup>r</sup> = 7; (d). e<sup>r</sup> = 10. The other parameters are a = b = 2 cm, z = 15 cm, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and β = 58 1=m . (e). The output field for e<sup>r</sup> =3, 5, 7, and 10, respectively, where a = b = 20 mm, y = b/2 = 10 mm. The radius of all circles is equal to 2.5 mm. (f). The output

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309

By increasing only the parameter e<sup>r</sup> of the circular dielectric profile from 3 to 10, the Gaussian shape of the output transverse profile of the field increased, the TE<sup>10</sup> wave profile decreased,

The output fields of Figure 6(a)–(f) are strongly affected by the input wave profile (TE<sup>10</sup> mode),

It is interesting to see a similar behavior of the output results in the cases of rectangular profiles (Figure 5(a)–(f)) that relate to Figure 1(a) and in the cases of circular profiles (Figure 6(a)–(f)) that relate to Figure 1(b), for every value of er, respectively. According to these output results, we see the similar behavior for every value of er, but the amplitudes of the output fields are

The comparison between our theoretical result of the output power density with the laboratory result in the case of the straight hollow waveguide with one dielectric layer and a metallic layer

parameters are a = b = 2 cm, z = 15 cm, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and β = 58 1=m.

the circular profile, and the location of the center of the circle (0.5 a, 0.5 b).

4.3. Numerical results for the three dielectric layers and a metallic layer

and the relative amplitude of the output field decreased.

profiles for N = 1, 3, 5, and 7, where e<sup>r</sup> = 10.

in the cross section of the straight hollow waveguide

different.

The output profiles and the amplitudes for N = 1, 3, 5, and 7 are shown in Figure 5(f), for e<sup>r</sup> = 10. By increasing only the parameter of the order N, the output field approaches to the final output field.

According to the results, we see that the output fields are strongly affected by the input wave profile (TE<sup>10</sup> mode), the rectangular profile, and the location of the center of the rectangle (0.5 a, 0.5 b).

#### 4.2. Numerical results for the circular dielectric material in the cross section of the straight rectangular waveguide

Figure 6(a)–(f) relates to discontinuous problem according to Figure 1(b). Figure 6(a)–(d) demonstrates the results of the output field as a response to TE<sup>10</sup> input-wave profile, for e<sup>r</sup> = 3, 5, 7, and 10, respectively. The radius of the circle in this case is equal to 2.5 mm and the center of the circle is located at the point (0.5 a, 0.5 b), as shown in Figure 1(b).

The other parameters are a = b = 2 cm , z = 15 cm, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and β = 58 1=m.

Figure 6(e) shows the output amplitude and the Gaussian shape of the central peak in the same cross section of Figure 6(a)–(d), where e<sup>r</sup> =3, 5, 7, and 10, respectively.

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Figure 6. The output field as a response to a half-sine (TE10) input-wave profile where the radius of all circle is equal to 2.5 mm and the center of the circle is located at the point (0.5 a, 0.5 b) according to Figure 1(b), where: (a). e<sup>r</sup> = 3; (b). e<sup>r</sup> = 5; (c). e<sup>r</sup> = 7; (d). e<sup>r</sup> = 10. The other parameters are a = b = 2 cm, z = 15 cm, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and β = 58 1=m . (e). The output field for e<sup>r</sup> =3, 5, 7, and 10, respectively, where a = b = 20 mm, y = b/2 = 10 mm. The radius of all circles is equal to 2.5 mm. (f). The output profiles for N = 1, 3, 5, and 7, where e<sup>r</sup> = 10.

The output profiles are shown in Figure 6(f) for N = 1, 3, 5, and 7, where e<sup>r</sup> = 10. The other parameters are a = b = 2 cm, z = 15 cm, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and β = 58 1=m.

By increasing only the parameter e<sup>r</sup> of the circular dielectric profile from 3 to 10, the Gaussian shape of the output transverse profile of the field increased, the TE<sup>10</sup> wave profile decreased, and the relative amplitude of the output field decreased.

The output fields of Figure 6(a)–(f) are strongly affected by the input wave profile (TE<sup>10</sup> mode), the circular profile, and the location of the center of the circle (0.5 a, 0.5 b).

It is interesting to see a similar behavior of the output results in the cases of rectangular profiles (Figure 5(a)–(f)) that relate to Figure 1(a) and in the cases of circular profiles (Figure 6(a)–(f)) that relate to Figure 1(b), for every value of er, respectively. According to these output results, we see the similar behavior for every value of er, but the amplitudes of the output fields are different.

#### 4.3. Numerical results for the three dielectric layers and a metallic layer in the cross section of the straight hollow waveguide

By increasing only the value of e<sup>r</sup> of the rectangular dielectric profile, the TE<sup>10</sup> wave profile decreased, the Gaussian shape of the output field increased, and the relative amplitude decreased.

Figure 5. The output field as a response to a half-sine (TE10) input-wave profile where c = d = 3.3 mm and the center of the rectangle is located at the point (0.5 a, 0.5 b) according to Figure 1(a), where: (a). e<sup>r</sup> = 3; (b). e<sup>r</sup> = 5; (c). e<sup>r</sup> = 7; (d). e<sup>r</sup> = 10. The other parameters are a = b = 2 cm, z = 15 cm, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and β = 58 1=m . (e). The output field for e<sup>r</sup> =3, 5, 7, and 10, respectively, where a = b = 2 cm, y = b/2 = 10 mm. (f). The output profiles for N = 1, 3, 5, and 7, where e<sup>r</sup> = 10.

The output profiles and the amplitudes for N = 1, 3, 5, and 7 are shown in Figure 5(f), for e<sup>r</sup> = 10. By increasing only the parameter of the order N, the output field approaches to the final

According to the results, we see that the output fields are strongly affected by the input wave profile (TE<sup>10</sup> mode), the rectangular profile, and the location of the center of the rectangle (0.5 a, 0.5 b).

4.2. Numerical results for the circular dielectric material in the cross section of the straight

Figure 6(a)–(f) relates to discontinuous problem according to Figure 1(b). Figure 6(a)–(d) demonstrates the results of the output field as a response to TE<sup>10</sup> input-wave profile, for e<sup>r</sup> = 3, 5, 7, and 10, respectively. The radius of the circle in this case is equal to 2.5 mm and the center

The other parameters are a = b = 2 cm , z = 15 cm, k<sup>0</sup> = 167 1=m, λ = 3.75 cm, and β = 58 1=m.

Figure 6(e) shows the output amplitude and the Gaussian shape of the central peak in the

of the circle is located at the point (0.5 a, 0.5 b), as shown in Figure 1(b).

same cross section of Figure 6(a)–(d), where e<sup>r</sup> =3, 5, 7, and 10, respectively.

In addition, by increasing only the value of er, the width of the Gaussian shape decreased.

output field.

rectangular waveguide

308 Emerging Waveguide Technology

The comparison between our theoretical result of the output power density with the laboratory result in the case of the straight hollow waveguide with one dielectric layer and a metallic layer is demonstrated in Figure 7(a) and 7(b). In this example, the diameter (2a) of the waveguide is 2 mm, the thickness of the dielectric layer [dð Þ AgI ] is 0:75 μm, and the minimum spot-size (w0) is 0.3 mm. The length of the straight waveguide is 1 m. The refractive indices of the air, the dielectric layer (AgI) and the metallic layer (Ag) are nð Þ<sup>0</sup> ¼ 1, nð Þ AgI ¼ 2:2, and nð Þ Ag ¼ 13:5 � j75:3, respectively. The value of the refractive index of the material at a wavelength of λ=10.6 μm is taken from the table compiled by Miyagi et al. [36].

The results of the output power density ð Þ jSavj (e.g., Figure 7(a)) show the behavior of the solutions for the TEM<sup>00</sup> mode in excitation. The comparison between our theoretical result (Figure 7(a)) and the published experimental data [37], as shown also in Figure 7(b) shows good agreement of a Gaussian shape as expected, except for the secondary small propagation mode. The experimental result is taken into account the roughness of the internal wall of the waveguide, but our theoretical model is not taken the roughness.

Miyagi and Kawakami showed that transmission losses of the dielectric-coated metal waveguides are drastically reduced when a multiple dielectric layer is formed instead of a single dielectric layer [31]. The simplest and most efficient multilayer structure is three dielectric layers that deposited on a metal layer.

By changing only the spot size from w<sup>0</sup> = 0.1 to 0.3 mm, with the same other parameters, the output power density is changed, as shown in Figure 8(a) and 8(b). The results of the output power density ð Þ jSavj show the behavior of the solutions for the TEM<sup>00</sup> mode in excitation. By changing only the spot size from w<sup>0</sup> = 0.3 to 0.1 mm, the width of the output Gaussian becomes more narrow. The output field results are strongly affected by the spot size and the structure of the three layers and the metallic layer in the cross section of the straight hollow waveguide.

Figure 8. The output power density for the straight hollow waveguide with three dielectric layers, where (a). a = 0.8 mm and w<sup>0</sup> = 0.1 mm. (b). a = 0.8 mm and w<sup>0</sup> = 0.3 mm. (c). a = 0.6 mm and w<sup>0</sup> = 0.3 mm. The other parameters are: b = 0.5 mm, <sup>λ</sup> <sup>¼</sup> <sup>10</sup>:<sup>6</sup> <sup>μ</sup><sup>m</sup> , <sup>n</sup>ð Þ <sup>0</sup> = 1, <sup>n</sup><sup>1</sup> = 2.22�j 10�6, <sup>n</sup><sup>2</sup> = 4�j 10�6, <sup>n</sup><sup>3</sup> = 6�j 10�6, nm <sup>=</sup> <sup>n</sup>ð Þ Ag = 13.5�j 75.3, and the length of the straight

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311

This mode model can be a useful tool to predict the relevant parameters in the case of the hollow waveguide with three dielectric layers and a metallic layer (Ag) for practical applications (output fields, output power density and output power transmission), before carrying

The main objective of this chapter was to develop the techniques that enable us to solve problems with inhomogeneous dielectric materials in the cross section of the straight rectangular and circular waveguides. The proposed techniques are very effective in relation to the conventional methods because they allow the development of expressions in the cross section only according to the specific discontinuous problem. In this way, the mode model methods become an improved methods to solve discontinuous problems in the cross section (and not

Three examples of inhomogeneous dielectric profiles in the cross section of the straight waveguides were shown in Figure 1(a)–(c). Figure 1(a) and 1(b) show the rectangular and circular profiles in the cross section of the straight rectangular waveguide, respectively. Figure 1(c) shows three dielectric layers and a metallic layer in the cross section of the straight hollow waveguide. The second objective is to understand the influence of the inhomogeneous dielec-

The proposed techniques are important to improve the methods that are based on Laplace and Fourier transforms and their inverse transforms also for the discontinuous rectangular and

out experiments in the laboratory.

only for continuous problems).

tric materials on the output fields.

5. Conclusions

waveguide is 1 m.

Thus, in this subsection, we present the output results that relate to the proposed technique and the particular application for the cross section of the straight hollow waveguide with three dielectric layers (and not only with one dielectric layer) and a metallic layer. In this case, we can improve the method [33], in the case of the three dielectric layers and a metallic layer in the cross section of the straight hollow waveguide.

Figure 8(a)–(c) relates to discontinuous problem according to Figure 1(c). The output power density for the straight hollow waveguide with three dielectric layers is shown in Figure 8(a)– (c). Figure 8(a) is shown for a = 0.8 mm and w<sup>0</sup> = 0.1 mm. Figure 8(b) is shown for a = 0.8 mm and w<sup>0</sup> = 0.3 mm. Figure 8(c) is shown for a = 0.6 mm and w<sup>0</sup> = 0.3 mm. The other parameters are b = 0.5 mm, <sup>λ</sup> <sup>¼</sup> <sup>10</sup>:<sup>6</sup> <sup>μ</sup><sup>m</sup> , <sup>n</sup>ð Þ<sup>0</sup> = 1, <sup>n</sup><sup>1</sup> = 2.22�j 10�<sup>6</sup> , <sup>n</sup><sup>2</sup> = 4�j 10�<sup>6</sup> , <sup>n</sup><sup>3</sup> = 6�j 10�<sup>6</sup> , nm = nð Þ Ag = 13.5�j 75.3, and the length of the straight waveguide is 1 m.

Figure 7. The comparison between our theoretical results of the output power density with the laboratory results in the case of the straight hollow waveguide with one dielectric layer. The parameters are a = 1 mm, w<sup>0</sup> = 0.3 mm, dð Þ AgI = 0.75 μm, λ = 10.6 μm, nð Þ<sup>0</sup> = 1, nð Þ AgI = 2.2, nð Þ Ag = 13.5�j 75, and the length of the straight waveguide is 1 m. (a). theoretical result. (b). laboratory result.

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Figure 8. The output power density for the straight hollow waveguide with three dielectric layers, where (a). a = 0.8 mm and w<sup>0</sup> = 0.1 mm. (b). a = 0.8 mm and w<sup>0</sup> = 0.3 mm. (c). a = 0.6 mm and w<sup>0</sup> = 0.3 mm. The other parameters are: b = 0.5 mm, <sup>λ</sup> <sup>¼</sup> <sup>10</sup>:<sup>6</sup> <sup>μ</sup><sup>m</sup> , <sup>n</sup>ð Þ <sup>0</sup> = 1, <sup>n</sup><sup>1</sup> = 2.22�j 10�6, <sup>n</sup><sup>2</sup> = 4�j 10�6, <sup>n</sup><sup>3</sup> = 6�j 10�6, nm <sup>=</sup> <sup>n</sup>ð Þ Ag = 13.5�j 75.3, and the length of the straight waveguide is 1 m.

By changing only the spot size from w<sup>0</sup> = 0.1 to 0.3 mm, with the same other parameters, the output power density is changed, as shown in Figure 8(a) and 8(b). The results of the output power density ð Þ jSavj show the behavior of the solutions for the TEM<sup>00</sup> mode in excitation. By changing only the spot size from w<sup>0</sup> = 0.3 to 0.1 mm, the width of the output Gaussian becomes more narrow. The output field results are strongly affected by the spot size and the structure of the three layers and the metallic layer in the cross section of the straight hollow waveguide.

This mode model can be a useful tool to predict the relevant parameters in the case of the hollow waveguide with three dielectric layers and a metallic layer (Ag) for practical applications (output fields, output power density and output power transmission), before carrying out experiments in the laboratory.

#### 5. Conclusions

is demonstrated in Figure 7(a) and 7(b). In this example, the diameter (2a) of the waveguide is 2 mm, the thickness of the dielectric layer [dð Þ AgI ] is 0:75 μm, and the minimum spot-size (w0) is 0.3 mm. The length of the straight waveguide is 1 m. The refractive indices of the air, the dielectric layer (AgI) and the metallic layer (Ag) are nð Þ<sup>0</sup> ¼ 1, nð Þ AgI ¼ 2:2, and nð Þ Ag ¼ 13:5 � j75:3, respectively. The value of the refractive index of the material at a wavelength of

The results of the output power density ð Þ jSavj (e.g., Figure 7(a)) show the behavior of the solutions for the TEM<sup>00</sup> mode in excitation. The comparison between our theoretical result (Figure 7(a)) and the published experimental data [37], as shown also in Figure 7(b) shows good agreement of a Gaussian shape as expected, except for the secondary small propagation mode. The experimental result is taken into account the roughness of the internal wall of the

Miyagi and Kawakami showed that transmission losses of the dielectric-coated metal waveguides are drastically reduced when a multiple dielectric layer is formed instead of a single dielectric layer [31]. The simplest and most efficient multilayer structure is three dielectric

Thus, in this subsection, we present the output results that relate to the proposed technique and the particular application for the cross section of the straight hollow waveguide with three dielectric layers (and not only with one dielectric layer) and a metallic layer. In this case, we can improve the method [33], in the case of the three dielectric layers and a metallic layer in the

Figure 8(a)–(c) relates to discontinuous problem according to Figure 1(c). The output power density for the straight hollow waveguide with three dielectric layers is shown in Figure 8(a)– (c). Figure 8(a) is shown for a = 0.8 mm and w<sup>0</sup> = 0.1 mm. Figure 8(b) is shown for a = 0.8 mm and w<sup>0</sup> = 0.3 mm. Figure 8(c) is shown for a = 0.6 mm and w<sup>0</sup> = 0.3 mm. The other parameters

Figure 7. The comparison between our theoretical results of the output power density with the laboratory results in the case of the straight hollow waveguide with one dielectric layer. The parameters are a = 1 mm, w<sup>0</sup> = 0.3 mm, dð Þ AgI = 0.75 μm, λ = 10.6 μm, nð Þ<sup>0</sup> = 1, nð Þ AgI = 2.2, nð Þ Ag = 13.5�j 75, and the length of the straight waveguide is 1 m. (a). theoretical

, <sup>n</sup><sup>2</sup> = 4�j 10�<sup>6</sup>

, <sup>n</sup><sup>3</sup> = 6�j 10�<sup>6</sup>

, nm = nð Þ Ag =

λ=10.6 μm is taken from the table compiled by Miyagi et al. [36].

waveguide, but our theoretical model is not taken the roughness.

layers that deposited on a metal layer.

310 Emerging Waveguide Technology

result. (b). laboratory result.

cross section of the straight hollow waveguide.

are b = 0.5 mm, <sup>λ</sup> <sup>¼</sup> <sup>10</sup>:<sup>6</sup> <sup>μ</sup><sup>m</sup> , <sup>n</sup>ð Þ<sup>0</sup> = 1, <sup>n</sup><sup>1</sup> = 2.22�j 10�<sup>6</sup>

13.5�j 75.3, and the length of the straight waveguide is 1 m.

The main objective of this chapter was to develop the techniques that enable us to solve problems with inhomogeneous dielectric materials in the cross section of the straight rectangular and circular waveguides. The proposed techniques are very effective in relation to the conventional methods because they allow the development of expressions in the cross section only according to the specific discontinuous problem. In this way, the mode model methods become an improved methods to solve discontinuous problems in the cross section (and not only for continuous problems).

Three examples of inhomogeneous dielectric profiles in the cross section of the straight waveguides were shown in Figure 1(a)–(c). Figure 1(a) and 1(b) show the rectangular and circular profiles in the cross section of the straight rectangular waveguide, respectively. Figure 1(c) shows three dielectric layers and a metallic layer in the cross section of the straight hollow waveguide. The second objective is to understand the influence of the inhomogeneous dielectric materials on the output fields.

The proposed techniques are important to improve the methods that are based on Laplace and Fourier transforms and their inverse transforms also for the discontinuous rectangular and circular profiles in the cross section. The technique based on ωε function was explained in detail in this chapter.

the cases of infrared regime. These models can be a useful tool to predict the relevant parameters

Applications and Solving Techniques of Propagated Wave in Waveguides Filled with Inhomogeneous Dielectric…

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313

Department of Electrical Engineering, Sami Shamoon College of Engineering, Beer Sheva,

[1] Chiang KS. Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides. Optical and Quantum Electronics. 1993;26:S113-

[2] Zhao S. Full-vectorial matched interface and boundary (MIB) method for the modal analysis of dielectric waveguides. Journal of Lightwave Technology. 2008;26:2251-2259

[3] Harrington JA, Matsuura Y. Review of hollow waveguide technology. SPIE. 1995;2396:

[4] Harrington JA, Harris DM, Katzir A, editors. Biomedical Optoelectronic Instrumentation.

[5] Harrington JA. A review of IR transmitting, hollow waveguides. Fiber and Integrated

[6] Marhic ME. Mode-coupling analysis of bending losses in IR metallic waveguides. Applied

[7] Croitoru N, Goldenberg E, Mendlovic D, Ruschin S, Shamir N. Infrared chalcogenide tube

[8] Jiao CQ. Selective suppression of electromagnetic modes in a rectangular waveguide by using distributed wall losses. Progress in Electromagnetics Research Letters. 2011;22:119-

[9] Mineo M, Di Carlo A, Paoloni C. Analytical design method for corrugated rectangular waveguide SWS THZ vacuum tubes. Journal of Electromagnetic Waves and Applications.

[10] Smartt CJ, Benson TM, Kendall PC. Exact transcendental equation for scalar modes of rectangular dielectric waveguides. Optical and Quantum Electronics. 1994;26:641-644

for practical applications, before carrying out experiments in the laboratory.

Address all correspondence to: zionm@post.tau.ac.il

Author details

Zion Menachem

References

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Israel

The result of the comparison between the theoretical models with the analytical solution [35] is shown in Figure 4(b) and the convergence is shown in Figure 4(c). The comparison has shown good agreement.

The results for the rectangular profile in the cross section of the straight rectangular waveguide (Figure 1(a)) were shown in Figure 5(a)–(f). Figure 5(a)–(d) shows the results of the output field as a response to a half-sine (TE10) input-wave profile. The output fields are strongly affected by the input wave profile (TE<sup>10</sup> mode), the rectangular profile, and the location of the center of the rectangle (0.5 a, 0.5 b).

By increasing only the value of e<sup>r</sup> of the rectangular dielectric material, the Gaussian shape of the output field increased, the TE<sup>10</sup> wave profile decreased, the relative amplitude decreased, and the width of the Gaussian shape decreased. The output field approaches to the final output field, by increasing only the parameter of the order N, as shown in Figure 5(f).

The results for the circular profile in the cross section of the straight rectangular waveguide (Figure 1(b)) were shown in Figure 6(a)–(f). Figure 6(a)–(d) shows the results of the output field as a response to a half-sine (TE10) input-wave profile.

By increasing only the value of the parameter e<sup>r</sup> of the circular dielectric material (Figure 1(b)) in the rectangular cross section from 3 to 10, the Gaussian shape of the output transverse profile of the field increased, the TE<sup>10</sup> wave profile decreased, and the relative amplitude of the output field decreased.

The output fields of Figure 5(a)–(f) and Figure 6(a)–(f) are strongly affected by the input wave profile (TE<sup>10</sup> mode), the rectangular profile or circular profile, and the location of the center of the rectangle or the circle (0.5 a, 0.5 b).

It is interesting to see a similar behavior of the output results in the cases of rectangular profiles (Figure 5(a)–(f)) that relate to Figure 1(a) and in the cases of circular profiles (Figure 6(a)–(f)) that relate to Figure 1(b), for every value of er, respectively. According to these output results, we see the similar behavior for every value of er, but the amplitudes of the output fields are different.

The comparison between our theoretical result (Figure 7(a)) and the published experimental data [37], as shown also in Figure 7(b) shows good agreement of a Gaussian shape as expected, except for the secondary small propagation mode. The experimental result is taken into account the roughness of the internal wall of the waveguide, but our theoretical model is not taken the roughness.

The output power density for the straight hollow waveguide with three dielectric layers (Figure 1(c)) is shown in Figure 8(a)–(c). The output field results are strongly affected by the spot size and the structure of the three layers and the metallic layer in the cross section of the straight hollow waveguide.

These models are useful to predict the structure of the output fields for rectangular and circular profiles in straight waveguides in the cases of microwave and millimeter-wave regimes and in the cases of infrared regime. These models can be a useful tool to predict the relevant parameters for practical applications, before carrying out experiments in the laboratory.

### Author details

circular profiles in the cross section. The technique based on ωε function was explained in

The result of the comparison between the theoretical models with the analytical solution [35] is shown in Figure 4(b) and the convergence is shown in Figure 4(c). The comparison has shown

The results for the rectangular profile in the cross section of the straight rectangular waveguide (Figure 1(a)) were shown in Figure 5(a)–(f). Figure 5(a)–(d) shows the results of the output field as a response to a half-sine (TE10) input-wave profile. The output fields are strongly affected by the input wave profile (TE<sup>10</sup> mode), the rectangular profile, and the location of the

By increasing only the value of e<sup>r</sup> of the rectangular dielectric material, the Gaussian shape of the output field increased, the TE<sup>10</sup> wave profile decreased, the relative amplitude decreased, and the width of the Gaussian shape decreased. The output field approaches to the final

The results for the circular profile in the cross section of the straight rectangular waveguide (Figure 1(b)) were shown in Figure 6(a)–(f). Figure 6(a)–(d) shows the results of the output

By increasing only the value of the parameter e<sup>r</sup> of the circular dielectric material (Figure 1(b)) in the rectangular cross section from 3 to 10, the Gaussian shape of the output transverse profile of the field increased, the TE<sup>10</sup> wave profile decreased, and the relative amplitude of

The output fields of Figure 5(a)–(f) and Figure 6(a)–(f) are strongly affected by the input wave profile (TE<sup>10</sup> mode), the rectangular profile or circular profile, and the location of the center of

It is interesting to see a similar behavior of the output results in the cases of rectangular profiles (Figure 5(a)–(f)) that relate to Figure 1(a) and in the cases of circular profiles (Figure 6(a)–(f)) that relate to Figure 1(b), for every value of er, respectively. According to these output results, we see the similar behavior for every value of er, but the amplitudes of the output fields are different. The comparison between our theoretical result (Figure 7(a)) and the published experimental data [37], as shown also in Figure 7(b) shows good agreement of a Gaussian shape as expected, except for the secondary small propagation mode. The experimental result is taken into account the roughness of the internal wall of the waveguide, but our theoretical model is not

The output power density for the straight hollow waveguide with three dielectric layers (Figure 1(c)) is shown in Figure 8(a)–(c). The output field results are strongly affected by the spot size and the structure of the three layers and the metallic layer in the cross section of the

These models are useful to predict the structure of the output fields for rectangular and circular profiles in straight waveguides in the cases of microwave and millimeter-wave regimes and in

output field, by increasing only the parameter of the order N, as shown in Figure 5(f).

field as a response to a half-sine (TE10) input-wave profile.

detail in this chapter.

312 Emerging Waveguide Technology

good agreement.

center of the rectangle (0.5 a, 0.5 b).

the output field decreased.

taken the roughness.

straight hollow waveguide.

the rectangle or the circle (0.5 a, 0.5 b).

Zion Menachem

Address all correspondence to: zionm@post.tau.ac.il

Department of Electrical Engineering, Sami Shamoon College of Engineering, Beer Sheva, Israel

### References


[11] Chen TT. Wave propagation in an inhomogeneous transversely magnetized rectangular waveguide. Applied Scientific Research. 1960;8:141-148

[26] Yener N. Advancement of algebraic function approximation in eigenvalue problems of lossless metallic waveguides to infinite dimensions, part I: Properties of the operator in infinite dimensions. Journal of Electromagnetic Waves and Applications. 2006;20:1611-

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[27] Yener N. Algebraic function approximation in eigenvalue problems of lossless metallic waveguides: Examples. Journal of Electromagnetic Waves and Applications. 2006;20:731-

[28] Khalaj-Amirhosseini M. Analysis of longitudinally inhomogeneous waveguides using Taylor's series expansion. Journal of Electromagnetic Waves and Applications. 2006;20:

[29] Khalaj-Amirhosseini M. Analysis of longitudinally inhomogeneous waveguides using the Fourier series expansion. Journal of Electromagnetic Waves and Applications. 2006;20:

[30] Reutskiy SY. The methods of external excitation for analysis of arbitrarily-shaped hollow conducting waveguides. Progress in Electromagnetics Research. 2008;82:203-226

[31] Miyagi M, Kawakami S. Design theory of dielectric-coated circular metallic waveguides for infrared transmission. Journal of Lightwave Technology. 1984;LT-2:116-126

[32] Menachem Z, Jerby E. Transfer matrix function (TMF) for propagation in dielectric waveguides with arbitrary transverse profiles. IEEE Transactions on Microwave Theory and

[33] Menachem Z, Tapuchi S. Influence of the spot-size and cross-section on the output fields and power density along the straight hollow waveguide. Progress in Electromagnetics

[34] Vladimirov V. Equations of Mathematical Physics. New York: Marcel Dekker, Inc.; 1971

[36] Miyagi M, Harada K, Kawakami S. Wave propagation and attenuation in the general class of circular hollow waveguides with uniform curvature. IEEE Transactions on Microwave

[37] Croitoru N, Inberg A, Oksman M, Ben-David M. Hollow silica, metal and plastic wave-

[35] Collin RE. Foundation for Microwave engineering. New York: McGraw-Hill; 1996

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[11] Chen TT. Wave propagation in an inhomogeneous transversely magnetized rectangular

[12] Yeap KH, Tham CY, Yassin G, Yeong KC. Attenuation in rectangular waveguides with

[13] Sharma J. Full-wave analysis of dielectric rectangular waveguides. Progress In Electro-

[14] Sumathy M, Vinoy KJ, Datta SK. Analysis of rectangular folded-waveguide millimeterwave slow-wave structures using conformal transformations. Journal of Infrared, Milli-

[15] Rothwell EJ, Temme A, Crowgey B. Pulse reflection from a dielectric discontinuity in a rectangular waveguide. Progress In Electromagnetics Research. 2009;97:11-25

[16] Liu S, Li LW, Leong MS, Yeo TS. Rectangular conducting waveguide filled with uniaxial anisotropic media: a modal analysis and dyadic Green's function. Progress In Electromag-

[17] Han SH, Wang XL, Fan Y. Improved generalized admittance matrix technique and its applications to rigorous analysis of millimeter-wave devices in rectangular waveguide.

[18] Lu W, Lu YY. Waveguide mode solver based on Neumann-to-Dirichlet operators and boundary integral equations. Journal of Computational Physics. 2012;231:1360-1371 [19] Eyges L, Gianino P. Modes of dielectric waveguides of arbitrary cross sectional shape.

[20] Abbas Z, Pollard RD, Kelsall W. A rectangular dielectric waveguide technique for determination of permittivity of materials at W-band. IEEE Transactions on Microwave Theory

[21] Hewlett SJ, Ladouceur F. Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff. Journal of

[22] Hernandez-Lopez MA, Quintillan M. Propagation characteristics of modes in some rectangular waveguides using the finite-difference time-domain method. Journal of Electro-

[23] Vaish A, Parthasarathy H. Analysis of rectangular waveguide using finite element

[24] Baganas K. Inhomogeneous dielectric media: Wave propagation and dielectric permittivity reconstruction in the case of a rectangular waveguide. Journal of Electromagnetic

[25] Novotny L, Hafner C. Light propagation in a cylindrical waveguide with a complex,

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**Chapter 17**

Provisional chapter

**Periodic Rectangular and Circular Profiles in the Cross**

DOI: 10.5772/intechopen.76794

Periodic Rectangular and Circular Profiles in the Cross

**Section of the Straight Waveguide Based on Laplace**

Section of the Straight Waveguide Based on Laplace

**and Fourier Transforms and Their Inverse Transforms**

This chapter presents propagation along the straight rectangular waveguide with periodic rectangular and circular profiles in the cross section. The objectives in this study are to explore the effect of the periodic rectangular and circular profiles in the cross section of the straight waveguide on the output field and to develop the technique to calculate two kinds of the periodic profiles. The method is based on Laplace and Fourier transforms and the inverse Laplace and Fourier transforms. The contribution of the proposed technique is important to improve the method that is based on Laplace and Fourier transforms and their inverse transforms also for the discontinuous periodic rectangular and circular profiles in the cross section (and not only for the continuous profiles). The proposed technique is very effective to solve complex problems, in relation to the conventional methods, especially when we have a large numbers of dielectric profiles. The application is useful for straight waveguides in the microwave and the millimeter wave regimes, with periodic rectangular and circular

and Fourier Transforms and Their Inverse Transforms

**and Applications**

and Applications

Additional information is available at the end of the chapter

profiles in the cross section of the straight waveguide.

Keywords: wave propagation, helical waveguide, dielectric waveguide, power

© 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

http://dx.doi.org/10.5772/intechopen.76794

Zion Menachem

Abstract

transmission

Zion Menachem

**Periodic Rectangular and Circular Profiles in the Cross Section of the Straight Waveguide Based on Laplace and Fourier Transforms and Their Inverse Transforms and Applications** Periodic Rectangular and Circular Profiles in the Cross Section of the Straight Waveguide Based on Laplace and Fourier Transforms and Their Inverse Transforms and Applications

DOI: 10.5772/intechopen.76794

#### Zion Menachem Zion Menachem

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.76794

#### Abstract

This chapter presents propagation along the straight rectangular waveguide with periodic rectangular and circular profiles in the cross section. The objectives in this study are to explore the effect of the periodic rectangular and circular profiles in the cross section of the straight waveguide on the output field and to develop the technique to calculate two kinds of the periodic profiles. The method is based on Laplace and Fourier transforms and the inverse Laplace and Fourier transforms. The contribution of the proposed technique is important to improve the method that is based on Laplace and Fourier transforms and their inverse transforms also for the discontinuous periodic rectangular and circular profiles in the cross section (and not only for the continuous profiles). The proposed technique is very effective to solve complex problems, in relation to the conventional methods, especially when we have a large numbers of dielectric profiles. The application is useful for straight waveguides in the microwave and the millimeter wave regimes, with periodic rectangular and circular profiles in the cross section of the straight waveguide.

Keywords: wave propagation, helical waveguide, dielectric waveguide, power transmission

© 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.

## 1. Introduction

Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides with emphasis on recent developments has been published [1]. Examples of interesting methods, such as the finite difference method and the finite element method have been reviewed. The method for the eigenmode analysis of two-dimensional step-index waveguides has been proposed [2]. The method distinguishes itself other existing interface methods by avoiding the use of the Taylor series expansion and by introducing the concept of the iterative use of low-order jump conditions.

been described [18], and this method is reliable down to modal cutoff. Analysis for new types

Periodic Rectangular and Circular Profiles in the Cross Section of the Straight Waveguide Based on Laplace…

http://dx.doi.org/10.5772/intechopen.76794

319

Propagation characteristics of modes in some rectangular waveguides using the finitedifference time-domain method have been proposed [20]. Analysis of rectangular waveguide using finite element method has been presented for arbitrarily shaped waveguide [21]. Wave propagation and dielectric permittivity reconstruction in the case of a rectangular waveguide

Important method for the analysis of electromagnetic wave propagation along the straight dielectric waveguide with arbitrary profiles has been proposed [23]. The mode model method for wave propagation in the straight waveguide with a circular cross section has been proposed [24]. This method in Refs. [23, 24] related to the methods based on Laplace and Fourier

The objectives in this chapter are to explore the effect of the periodic rectangular and circular profiles in the cross section of the straight waveguide on the output field and to develop the technique to calculate the dielectric profile, the elements of the matrix, and its derivatives of the dielectric profile. The proposed technique is important to improve the mode model also for the

periodic rectangular and circular profiles and not only for the continuous profiles.

2. Periodic rectangular and circular profiles in the cross section of the

In this chapter we introduce two different techniques, and the particular applications allow us to improve the mode model so that we can solve inhomogeneous problems also for periodic profiles in the cross section of the straight waveguide. Thus, in this chapter we introduce two techniques to calculate the dielectric profile, the elements of the matrix, and its derivatives of dielectric profile in the cases of periodic rectangular and circular profiles in the cross section of

The proposed techniques are very effective in relation to the conventional methods because they allow the development of expressions in the cross section only according to the specific discontinuous problem. In this way, the mode model method becomes an improved method to solve discontinuous problems in the cross section (and not only for continuous problems).

Three examples of periodic rectangular profiles are shown in Figure 1(a–c), and three examples of periodic circular profiles are shown in Figure 1(d–f) in the cross section of the straight

An example of periodic structure with two rectangular profiles along x-axis is shown in Figure 1(a), where the centers of the left rectangle and right rectangle are located at the points (0.25 a, 0.5 b) and (0.75 a, 0.5 b), respectively. An example of periodic structure with two rectangular profiles along y-axis is shown in Figure 1(b), where the centers of the upper

An example of periodic structure with four rectangular profiles along x-axis and y-axis is shown in Figure 1(c). The center of the first rectangle is located at the point (0.25 a, 0.25 b),

rectangle and lower rectangle are located at the points (0.5 a, 0.75 b) and (0.5 a, 0.25 b).

of waveguide with Fourier's expansion differential method has been proposed [19].

transforms and the inverse Laplace and Fourier transforms.

have been studied [22].

straight waveguide

rectangular waveguide.

the straight rectangular waveguide.

The method of selective suppression of electromagnetic modes in a rectangular waveguides by using distributed wall losses has been proposed [3]. Analytical design method for corrugated rectangular waveguide has been proposed [4].

A Fourier operator method has been used to derive for the first time an exact closed-form eigenvalue equation for the scalar mode propagation constants of a buried rectangular dielectric waveguide [5]. Wave propagation in an inhomogeneous transversely magnetized rectangular waveguide has been studied with the aid of a modified Sturm-Liouville differential equation [6]. A fundamental and accurate technique to compute the propagation constant of waves in a lossy rectangular waveguide has been proposed [7]. This method is based on matching the electric and magnetic fields at the boundary and allowing the wave numbers to take complex values.

A great amount of numerical results for cylindrical dielectric waveguide array have been presented [8]. Dielectric cylinders have been arrayed by a rectangular mode. When the area of dielectric cylinder in a unit cell varied from a small number to a big one and even maximum, interactions between space harmonics firstly got stronger but finally got weaker. Full-wave analysis of dielectric rectangular waveguides has been presented [9]. The waveguide properties of permeable one-dimensional periodic acoustic structures have been studied [10]. Analysis of rectangular folded-waveguide millimeter wave slow-wave structure using conformal transformation has been developed [11].

A simple closed-form expression to compute the time-domain reflection coefficient for a transient TE<sup>10</sup> mode wave incident on a dielectric step discontinuity in a rectangular waveguide has been presented [12]. In this paper, an exponential series approximation was provided for efficient computation of the reflected and transmitted field waveforms.

A waveguide with layered-periodic walls for different relations between the dielectric permittivities of the central layer and the superlattice layers has been proposed [13]. A full-vectorial boundary integral equation method for computing guided modes of optical waveguides has been presented [14]. A method for the propagation constants of arbitrary cross-sectional shapes has been described [15]. Experiment and simulation of TE<sup>10</sup> cut-off reflection phase in gentle rectangular downtapers has been studied [16].

The rectangular dielectric waveguide technique for the determination of complex permittivity of a wide class of dielectric materials of various thicknesses and cross sections has been described [17]. In this paper, the technique has been presented to determine the dielectric constant of materials. Fourier decomposition method applied to mapped infinite domains has been described [18], and this method is reliable down to modal cutoff. Analysis for new types of waveguide with Fourier's expansion differential method has been proposed [19].

1. Introduction

318 Emerging Waveguide Technology

take complex values.

the iterative use of low-order jump conditions.

rectangular waveguide has been proposed [4].

transformation has been developed [11].

gentle rectangular downtapers has been studied [16].

Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides with emphasis on recent developments has been published [1]. Examples of interesting methods, such as the finite difference method and the finite element method have been reviewed. The method for the eigenmode analysis of two-dimensional step-index waveguides has been proposed [2]. The method distinguishes itself other existing interface methods by avoiding the use of the Taylor series expansion and by introducing the concept of

The method of selective suppression of electromagnetic modes in a rectangular waveguides by using distributed wall losses has been proposed [3]. Analytical design method for corrugated

A Fourier operator method has been used to derive for the first time an exact closed-form eigenvalue equation for the scalar mode propagation constants of a buried rectangular dielectric waveguide [5]. Wave propagation in an inhomogeneous transversely magnetized rectangular waveguide has been studied with the aid of a modified Sturm-Liouville differential equation [6]. A fundamental and accurate technique to compute the propagation constant of waves in a lossy rectangular waveguide has been proposed [7]. This method is based on matching the electric and magnetic fields at the boundary and allowing the wave numbers to

A great amount of numerical results for cylindrical dielectric waveguide array have been presented [8]. Dielectric cylinders have been arrayed by a rectangular mode. When the area of dielectric cylinder in a unit cell varied from a small number to a big one and even maximum, interactions between space harmonics firstly got stronger but finally got weaker. Full-wave analysis of dielectric rectangular waveguides has been presented [9]. The waveguide properties of permeable one-dimensional periodic acoustic structures have been studied [10]. Analysis of rectangular folded-waveguide millimeter wave slow-wave structure using conformal

A simple closed-form expression to compute the time-domain reflection coefficient for a transient TE<sup>10</sup> mode wave incident on a dielectric step discontinuity in a rectangular waveguide has been presented [12]. In this paper, an exponential series approximation was pro-

A waveguide with layered-periodic walls for different relations between the dielectric permittivities of the central layer and the superlattice layers has been proposed [13]. A full-vectorial boundary integral equation method for computing guided modes of optical waveguides has been presented [14]. A method for the propagation constants of arbitrary cross-sectional shapes has been described [15]. Experiment and simulation of TE<sup>10</sup> cut-off reflection phase in

The rectangular dielectric waveguide technique for the determination of complex permittivity of a wide class of dielectric materials of various thicknesses and cross sections has been described [17]. In this paper, the technique has been presented to determine the dielectric constant of materials. Fourier decomposition method applied to mapped infinite domains has

vided for efficient computation of the reflected and transmitted field waveforms.

Propagation characteristics of modes in some rectangular waveguides using the finitedifference time-domain method have been proposed [20]. Analysis of rectangular waveguide using finite element method has been presented for arbitrarily shaped waveguide [21]. Wave propagation and dielectric permittivity reconstruction in the case of a rectangular waveguide have been studied [22].

Important method for the analysis of electromagnetic wave propagation along the straight dielectric waveguide with arbitrary profiles has been proposed [23]. The mode model method for wave propagation in the straight waveguide with a circular cross section has been proposed [24]. This method in Refs. [23, 24] related to the methods based on Laplace and Fourier transforms and the inverse Laplace and Fourier transforms.

The objectives in this chapter are to explore the effect of the periodic rectangular and circular profiles in the cross section of the straight waveguide on the output field and to develop the technique to calculate the dielectric profile, the elements of the matrix, and its derivatives of the dielectric profile. The proposed technique is important to improve the mode model also for the periodic rectangular and circular profiles and not only for the continuous profiles.
