6. LiNbO3 nanostructures and photonic crystals

due to an increased sensitivity of the TM waves to membrane roughness. It is noteworthy that the coupling losses remain the same regardless of the membrane thickness, which confirms the

The EO overlap coefficient is deduced from the Fourier transform of the reflected spectral density (see Figure 8), which is also the autocorrelation of the impulse response. Due to the Fabry-Perot oscillations inside the cavity formed by the waveguide, a peak appears in the Fourier transform, which coincides with a round trip of the light between the two facets of the waveguide. From this peak, we can deduce the global effective group index: ngeff = t2�c0/(2�Ltot), c0 being the speed of light and Ltot denoting the waveguide length. The resulting effective group indexes are neffTE = 2.189�0.005 and neffTM = 2.269�0.005 for TE and TM polarizations, respectively, in an X-cut Y-propagating waveguide with a membrane thickness of 4.5 μm. The effective group index is measured voltage by voltage from Figure 8, for the assessment of the group index variation per voltage: Δng/ΔV. The results are exposed in Table 2. Γ is calculated

from Δng/ΔV by using expressions (6) and (7) for the TE and TM waves, respectively:

<sup>Γ</sup>TE <sup>¼</sup> <sup>Δ</sup>neg

<sup>Γ</sup>TM <sup>¼</sup> <sup>Δ</sup>nog

<sup>Δ</sup><sup>V</sup> <sup>∙</sup> <sup>2</sup>∙<sup>g</sup> n3 <sup>e</sup> ∙r<sup>33</sup>

<sup>Δ</sup><sup>V</sup> <sup>∙</sup> <sup>2</sup>∙<sup>g</sup> n3 <sup>o</sup> ∙r<sup>13</sup>

Table 2 confirms the twofold enhancement of the EO interaction when the membrane is thinned down to 4.5 μm, and it shows that this enhancement is even higher for the TMpropagating wave, although this was not anticipated from the FEM calculations. This latter result can be of great interest to seek for isotropic EO behavior of the guided wave in the

Figure 8. Zoom view of the Fourier transform of the reflected optical density spectrum. These measurements are achieved as a function of the applied voltage through a 12.0 mm long tapered-membrane-based waveguide with a width

of 6 μm and a thickness of 4.5 μm. The total length was Ltot = 1.2 mm.

(6)

(7)

efficiency of the tapers to mode match with the fibers.

166 Emerging Waveguide Technology

presence of an applied voltage.

It is well established that the machining of materials at the wavelength scale can yield a specific control of the light flux, which is of great interest to enhance significantly the electro-optical efficiency. In particular, photonic crystals (PhCs) are periodic structures that are designed to affect the motion of photons in a similar way that periodicity of a semiconductor crystal affects the behavior of electrons. The non-existence of propagating electromagnetic modes inside the structures at certain frequencies introduces unique optical phenomena such as tight light confinement. The part of the spectrum for which wave propagation is not possible is called the photonic bandgap (PBG).

The first reported PhC-based EO LiNbO3 modulator [52] was configured to behave as an electro-absorbent modulator: it was designed to have a photonic bandgap, and the spectral edge was exploited for intensity modulation. Hence, an active length of 11 μm was sufficient to modulate the light with a driving voltage of 13 V [52]. A schematic diagram of such a device is seen in Figure 9. It consists of a square lattice PhC integrated on an annealed proton exchanged optical waveguide and surrounded by capacitive coplanar electrodes. The LiNbO3 PhC was fabricated through focused ion beam (FIB) milling The refractive index was modified by the electric field Ez generated between the electrodes, which moved the spectral position of the PBG and consequently the transmitted intensity. The geometric properties of the PhC were chosen to benefit from slow light effects.

This compact modulator showed an EO interaction 312 times higher than the one predicted from Eq. (2): the extraordinary effect is shown in Figure 10 where the PBG is spectrally shifted

Figure 9. Image of the first reported PhC-based EO modulator [53]. (a) Schematic view of the ultra-compact modulator. (b) SEM image of the photonic crystal surrounded by electrodes.

by 100 nm when only 40 V is applied on the electrodes. This effect is attributed to an enhanced field factor f induced by slow light effect:

$$\mathbf{f} = \sqrt{\frac{\mathbf{v}\_{\mathcal{S}}^{\text{BIL.K}}}{\mathbf{v}\_{\mathcal{S}}^{\text{PC}}}} \tag{8}$$

subject yet, but the inscription of a 1D-PhC cavity as reported by Courjal et al. [51] appears like a good start. The component is made of two 1D PhCs separated by 200 μm in a free-suspended membrane. The 1D PhCs are designed to be reflectors at 1550 nm wavelength and their pair constitutes a Fabry-Perot cavity. One of them can be visualized in Figure 12(b). The free spectral range of 2.7 nm seen in Figure 10(b) is in good agreement with the theoretical

factor is measured to be of 2580, revealing moderated reflectivity of the 1D PhC (R ≈ 50%). The spectral position of a resonance peak shown in Figure 10(b) is shifted by 40.0 pm/V, which is not as high as the shift reported in Figure 10(a), but this is balanced with the transmission

The driving voltage is assessed by measuring the output optical power at 1550 nm. The measured electro-optical response reported in Figure 11 shows a 9.6 V driving voltage for the TE-polarized wave and 25.2 V for the TM-polarized wave. So the figure of merit for each polarization is, respectively, 1.9 and 5 mVm, for TE and TM waves, which is 40-fold higher than the ones classicallymeasured inMach-Zehnder intensity modulators. In the compact modulator of Figure 9, the EO interaction was enhanced by means of slow light effects. Here the remarkable EO sensitivity is rather attributed to a tip effect that enhances the electric field in the vicinity of the Bragg grating.

Figure 11. Experimental EO response of a TE wave (a) and TM wave (b) at the output of a 200 μm-long Fabry-Perot

Figure 12. SEM views of LiNbO3 free-suspended membranes done by optical-grade dicing. The patterns are inscribed by FIB milling. (a) 1D-PhC implemented in a 4.5 μm-thick waveguide. (b) Nanostructure in a 450 nm thick membrane. Such a

thickness is a record for a free-suspended layer fabricated through mechanical approach.

/(2Δn), where Δ is the distance between the two 1D PhCs. The quality

Lithium Niobate Optical Waveguides and Microwaveguides

http://dx.doi.org/10.5772/intechopen.76798

169

prediction: FSR=λ<sup>2</sup>

losses that are diminished by 5 dB.

integrated in a 4.5 μm thick membrane.

where vBULK <sup>g</sup> is the group velocity in the bulk material and vPC <sup>g</sup> is the group velocity in the photonic crystal. The EO enhancement is explained by Roussey et al. [53] by an effective EO coefficient being enhanced by the local field factor as follows:

$$r\_{33}^{PC} = f^3 r\_{33} \tag{9}$$

The local field factor was calculated by using the slope of the band dispersion diagram and was evaluated to be 6.8 which was in good agreement with the measurements. To our knowledge, this huge effect is still a record in terms of EO sensitivity. However, as performing and attractive as this configuration may appear, the transmission losses were of 10 dB (see Figure 10(a)), which is prohibitive for commercial applications. The small extinction ratio (12 dB) was also limiting for many demanding EO applications. These limitations are mainly due to the weak light confinement inside the APE waveguide, combined with the conical shape of photonic crystals that provoked deviation of light inside the substrate [54].

Since then, other approaches have been proposed, based on microring resonators or PhCs in thin films, enabling low propagation losses and high extinction ratio [42]. However, as mentioned in Section 5.1, these techniques are prohibitive in terms of fiber coupling, due to the mode mismatch between the fiber and the guided mode inside the thin film, which also induces insertion losses larger than 10 dB.

In this context, the suspended waveguides mentioned in Section 5.2 appear as good candidates for hosting photonic crystal and nanostructures. There are not many publications on the

Figure 10. Spectral transmission responses through two PhC-based EO LiNbO3 modulators, TE polarization. (a) Normalized transmission response through the modulator seen in Figure 11. (b) Normalized transmission response through a suspended Fabry-Perot [Caspar16]. The two measurements are done by direct butt-coupling with SMFs.

subject yet, but the inscription of a 1D-PhC cavity as reported by Courjal et al. [51] appears like a good start. The component is made of two 1D PhCs separated by 200 μm in a free-suspended membrane. The 1D PhCs are designed to be reflectors at 1550 nm wavelength and their pair constitutes a Fabry-Perot cavity. One of them can be visualized in Figure 12(b). The free spectral range of 2.7 nm seen in Figure 10(b) is in good agreement with the theoretical prediction: FSR=λ<sup>2</sup> /(2Δn), where Δ is the distance between the two 1D PhCs. The quality factor is measured to be of 2580, revealing moderated reflectivity of the 1D PhC (R ≈ 50%). The spectral position of a resonance peak shown in Figure 10(b) is shifted by 40.0 pm/V, which is not as high as the shift reported in Figure 10(a), but this is balanced with the transmission losses that are diminished by 5 dB.

by 100 nm when only 40 V is applied on the electrodes. This effect is attributed to an enhanced

s

photonic crystal. The EO enhancement is explained by Roussey et al. [53] by an effective EO

The local field factor was calculated by using the slope of the band dispersion diagram and was evaluated to be 6.8 which was in good agreement with the measurements. To our knowledge, this huge effect is still a record in terms of EO sensitivity. However, as performing and attractive as this configuration may appear, the transmission losses were of 10 dB (see Figure 10(a)), which is prohibitive for commercial applications. The small extinction ratio (12 dB) was also limiting for many demanding EO applications. These limitations are mainly due to the weak light confinement inside the APE waveguide, combined with the conical shape of photonic

Since then, other approaches have been proposed, based on microring resonators or PhCs in thin films, enabling low propagation losses and high extinction ratio [42]. However, as mentioned in Section 5.1, these techniques are prohibitive in terms of fiber coupling, due to the mode mismatch between the fiber and the guided mode inside the thin film, which also

In this context, the suspended waveguides mentioned in Section 5.2 appear as good candidates for hosting photonic crystal and nanostructures. There are not many publications on the

Figure 10. Spectral transmission responses through two PhC-based EO LiNbO3 modulators, TE polarization. (a) Normalized transmission response through the modulator seen in Figure 11. (b) Normalized transmission response through a

suspended Fabry-Perot [Caspar16]. The two measurements are done by direct butt-coupling with SMFs.

ffiffiffiffiffiffiffiffiffiffiffiffi vBULK g vPC g

(8)

<sup>g</sup> is the group velocity in the

r<sup>33</sup> (9)

f ¼

r PC <sup>33</sup> ¼ f 3

<sup>g</sup> is the group velocity in the bulk material and vPC

coefficient being enhanced by the local field factor as follows:

crystals that provoked deviation of light inside the substrate [54].

induces insertion losses larger than 10 dB.

field factor f induced by slow light effect:

where vBULK

168 Emerging Waveguide Technology

The driving voltage is assessed by measuring the output optical power at 1550 nm. The measured electro-optical response reported in Figure 11 shows a 9.6 V driving voltage for the TE-polarized wave and 25.2 V for the TM-polarized wave. So the figure of merit for each polarization is, respectively, 1.9 and 5 mVm, for TE and TM waves, which is 40-fold higher than the ones classicallymeasured inMach-Zehnder intensity modulators. In the compact modulator of Figure 9, the EO interaction was enhanced by means of slow light effects. Here the remarkable EO sensitivity is rather attributed to a tip effect that enhances the electric field in the vicinity of the Bragg grating.

Figure 11. Experimental EO response of a TE wave (a) and TM wave (b) at the output of a 200 μm-long Fabry-Perot integrated in a 4.5 μm thick membrane.

Figure 12. SEM views of LiNbO3 free-suspended membranes done by optical-grade dicing. The patterns are inscribed by FIB milling. (a) 1D-PhC implemented in a 4.5 μm-thick waveguide. (b) Nanostructure in a 450 nm thick membrane. Such a thickness is a record for a free-suspended layer fabricated through mechanical approach.

So, PhC-based EO modulators are extremely promising for the achievement of low-power consuming and compact modulators either by exploiting slow light effects in the nanostructure or tip effects in structured electrodes. The recent developments reported on free-suspended membranes appear to be also very promising for preserving low losses. The achievement of commercial performances should be achieved by a better control in fabrication of the 1D PhCs.

References

10.1063/1.1937997

10.1364/JOSA.55.000828

com/

2321099

series.pdf

DOI: 10.1364/OE.17.018489

newport.com/n/electro-optic-modulator-faqs

Walter de Gruyter GmbH & Co KG; 2015 978-3-11-030449-7

[1] Turner EH. High frequency electro-optic coefficients of lithium niobate. Applied Physics

Lithium Niobate Optical Waveguides and Microwaveguides

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[2] Wooten EL, Kissa KM, Yi-Yan A, Murphy EJ, Lafaw DA, Hallemeier PF, Maack D, Attanasio DV, Fritz DJ, McBrien GJ, Bossi DE. A review of lithium niobate modulators for fiber-optic communications systems. IEEE Journal on Selected Topics in Quatum

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Many other 2D suspended devices may be envisioned by using the same technique: as a perspective, Figure 12 shows SEM images of diced membranes hosting micro- and nanopatterns that can be used as LiNbO3 MEMs, opto-mechanical, or nonlinear wires.

### 7. Conclusion

In conclusion, lithium niobate is a material of current interest in both industry and research, thanks to its robust and reproducible properties and also thanks to the emergence of new technologies such as ion slicing or optical grade dicing. Hence, and even though the first modulators date from the 1970s, progress is still visible, to move toward ever more compactness with low energy consumption and low losses. In this chapter, we have taken particular interest in three important parameters: the Γ electro-optical overlap coefficient that characterizes EO efficiency, the η overlap integral with optical fibers, which quantifies coupling losses, and the propagation losses, the last two parameters being important contributors to overall insertion losses. We have shown how the use of confined guides (deep ridges or thin films) increases the EO efficiency by a factor of 2 and how tapers allow low integration losses. Finally, we have highlighted the spectacular effects of nanostructuring in confined guides to gain 1–2 orders of magnitude on the EO efficiency. These developments open the way toward compact and low-consuming photonic integrated circuit but also offer the promise of multi-control in optical circuits.

### Acknowledgements

This work was supported by the SATT Grand-Est under project μguide, partially by ANR under project ANR-16-CE24-0024-01, and by the Labex ACTION program (contract ANR-11- LABX-01-01). The work was partly supported by the French RENATECH network and its FEMTO-ST technological facility.

### Author details

Nadège Courjal<sup>1</sup> \*, Maria-Pilar Bernal<sup>1</sup> , Alexis Caspar<sup>1</sup> , Gwenn Ulliac<sup>1</sup> , Florent Bassignot<sup>2</sup> , Ludovic Gauthier-Manuel<sup>1</sup> and Miguel Suarez<sup>1</sup>

\*Address all correspondence to: nadege.courjal@femto-st.fr


### References

So, PhC-based EO modulators are extremely promising for the achievement of low-power consuming and compact modulators either by exploiting slow light effects in the nanostructure or tip effects in structured electrodes. The recent developments reported on free-suspended membranes appear to be also very promising for preserving low losses. The achievement of commercial performances should be achieved by a better control in fabrication of the 1D PhCs. Many other 2D suspended devices may be envisioned by using the same technique: as a perspective, Figure 12 shows SEM images of diced membranes hosting micro- and nano-

In conclusion, lithium niobate is a material of current interest in both industry and research, thanks to its robust and reproducible properties and also thanks to the emergence of new technologies such as ion slicing or optical grade dicing. Hence, and even though the first modulators date from the 1970s, progress is still visible, to move toward ever more compactness with low energy consumption and low losses. In this chapter, we have taken particular interest in three important parameters: the Γ electro-optical overlap coefficient that characterizes EO efficiency, the η overlap integral with optical fibers, which quantifies coupling losses, and the propagation losses, the last two parameters being important contributors to overall insertion losses. We have shown how the use of confined guides (deep ridges or thin films) increases the EO efficiency by a factor of 2 and how tapers allow low integration losses. Finally, we have highlighted the spectacular effects of nanostructuring in confined guides to gain 1–2 orders of magnitude on the EO efficiency. These developments open the way toward compact and low-consuming photonic

This work was supported by the SATT Grand-Est under project μguide, partially by ANR under project ANR-16-CE24-0024-01, and by the Labex ACTION program (contract ANR-11- LABX-01-01). The work was partly supported by the French RENATECH network and its

, Alexis Caspar<sup>1</sup>

, Gwenn Ulliac<sup>1</sup>

, Florent Bassignot<sup>2</sup>

,

patterns that can be used as LiNbO3 MEMs, opto-mechanical, or nonlinear wires.

integrated circuit but also offer the promise of multi-control in optical circuits.

7. Conclusion

170 Emerging Waveguide Technology

Acknowledgements

Author details

Nadège Courjal<sup>1</sup>

FEMTO-ST technological facility.

\*, Maria-Pilar Bernal<sup>1</sup>

\*Address all correspondence to: nadege.courjal@femto-st.fr

Ludovic Gauthier-Manuel<sup>1</sup> and Miguel Suarez<sup>1</sup>

1 FEMTO-ST Institute, Besançon, France 2 FEMTO-Engeeniring, Besançon, France


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**Chapter 9**

Provisional chapter

**Raman Solitons in Nanoscale Optical Waveguides, with**

DOI: 10.5772/intechopen.75121

A mathematical analysis is conducted to illustrate the controllability of the Raman soliton self-frequency shift with polynomial nonlinearity in metamaterials by using collective variable method. The polynomial nonlinearity is due to the expanding nonlinear polarization PNL in a series over the field E up to the seventh order. Gaussian assumption is selected to these pulses on a generalized mode. The numerical simulation of soliton

Much attention has been devoted to the understanding of metamaterials [1–4]. Through its engineered structures, researchers are able to control and manipulate the electromagnetic fields [5]. Using the freedom of design that metamaterials provide, electromagnetic fields can be redirected at will and propose a design strategy [6]. A general recipe for the design of media that create perfect invisibility within the accuracy of geometrical optics is developed. The imperfections of invisibility can be made arbitrarily small to hide objects that are much larger

Especially, mathematical operations can be performed based on suitably designed metamaterials blocks, such as spatial differentiation, integration, or convolution [8]. Soliton pulse can evolve owning to delicate balance between dispersion and nonlinearity. However, it is

> © 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.

parameter variation is given for the Gaussian pulse parameters.

Keywords: Raman solitons, polynomial nonlinearity, collective variables

Raman Solitons in Nanoscale Optical Waveguides, with

**Metamaterials, Having Polynomial Law Nonlinearity**

Metamaterials, Having Polynomial Law Nonlinearity

**Using Collective Variables**

Using Collective Variables

http://dx.doi.org/10.5772/intechopen.75121

Abstract

1. Introduction

than the wavelength [7].

Yanan Xu, Jun Ren and Matthew C. Tanzy

Yanan Xu, Jun Ren and Matthew C. Tanzy

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter


#### **Raman Solitons in Nanoscale Optical Waveguides, with Metamaterials, Having Polynomial Law Nonlinearity Using Collective Variables** Raman Solitons in Nanoscale Optical Waveguides, with Metamaterials, Having Polynomial Law Nonlinearity Using Collective Variables

DOI: 10.5772/intechopen.75121

Yanan Xu, Jun Ren and Matthew C. Tanzy Yanan Xu, Jun Ren and Matthew C. Tanzy

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.75121

#### Abstract

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[42] Poberaj G, Hu H, Sohler W, Günter P. Lithium niobate on insulator (LNOI) for microphotonic devices. Laser & Photonics Reviews. 2012;6:488-503. DOI: 10.1002/lpor.20110

[43] NanoLN, LiNbO3 thin films [Internet]. 2017. Available from http://www.nanoln.com/en/

[44] Partow Technologies LLC. Ino sliced thin films. [Internet]. 2017. Available from: http://

[45] Volk MF, Suntsov S, Rüter CE, Kip D. Low loss ridge waveguides in lithium niobate thin films by optical grade diamond blade dicing. Optics Express. 2016;24:1386. DOI: 10.1364/

[46] Guarino A, Poberaj G, Rezzonico D, Degl'Innocenti R, Günter P. Electro-optically tunable microring resonators in lithium niobate. Nature Photonics. 2007;1:407-410. DOI: 10.1038/

[47] Chen L, Chen J, Nagy J, Reano R. Highly linear ring modulator from hybrid silicon and

[48] Lu H, Sadani B, Courjal N, Ulliac G, Smith N, Stenger V, Collet M, Baida FI, Bernal M-P. Enhanced electro-optical lithium niobate photonic crystal wire waveguide on a smart-cut

[49] Geiss R, Saravi S, Sergeyev A, Diziain S, Setzpfandt F, Schrempel F, Grange R, Kley E, Tünnermann A, Pertsch T. Fabrication of nanoscale lithium niobate waveguides for secondharmonic generation. Optics Letters. 2015;40:2715-2718. DOI: 10.1364/OL.40.002715

[50] Chen L, Xu Q, Wood G, Reano RM. Hybrid silicon and lithium niobate electro-optical ring

[51] Courjal N, Caspar A, Calero V, Ulliac G, Suarez M, Guyot, Bernal MP. Simple production of membrane-based LiNbO3 micro-modulators with integrated tapers. Optics Letters.

[52] Roussey M, Bernal MP, Courjal N, Van Labeke D, Baida FI. Electro-optic effect exaltation on lithium niobate photonic crystals due to slow photons. Applied Physics Letters. 2006;

[53] Roussey M, Baida FI, Bernal M-P. Experimental and theoretical observations of the slowlight effect on a tunable photonic crystal. JOSA B. 2007;24(6):1416-1422. DOI: 10.1364/

[54] Burr GW, Diziain S, Bernal M-P. The impact of finite-depth cylindrical and conical holes in lithium niobate photonic crystals. Optics Express. 2008;16:6302-6316. DOI: 10.1364/OE.16.00

lithium niobate. Optics Express. 2015;23:13255. DOI: 10.1364/OE.23.013255

thin film. Optics Express. 2012;20:2974-2981. DOI: 10.1364/OE.20.002974

modulator. Optica. 2014;1:112-118. DOI: 10.1364/OPTICA.1.000112

2016;41:5110-5113. DOI: 10.1364/OL.41.005110

89:241110-241113. DOI: 10.1063/1.2402946

2293-2296. DOI: 10.1063/1.121801

www.partow-tech.com/thinfilms/

pinfo.asp?ArticleID=13

OE.24.001386

nphoton.2007.93

JOSAB.24.001416

6302

0035

174 Emerging Waveguide Technology

A mathematical analysis is conducted to illustrate the controllability of the Raman soliton self-frequency shift with polynomial nonlinearity in metamaterials by using collective variable method. The polynomial nonlinearity is due to the expanding nonlinear polarization PNL in a series over the field E up to the seventh order. Gaussian assumption is selected to these pulses on a generalized mode. The numerical simulation of soliton parameter variation is given for the Gaussian pulse parameters.

Keywords: Raman solitons, polynomial nonlinearity, collective variables

#### 1. Introduction

Much attention has been devoted to the understanding of metamaterials [1–4]. Through its engineered structures, researchers are able to control and manipulate the electromagnetic fields [5]. Using the freedom of design that metamaterials provide, electromagnetic fields can be redirected at will and propose a design strategy [6]. A general recipe for the design of media that create perfect invisibility within the accuracy of geometrical optics is developed. The imperfections of invisibility can be made arbitrarily small to hide objects that are much larger than the wavelength [7].

Especially, mathematical operations can be performed based on suitably designed metamaterials blocks, such as spatial differentiation, integration, or convolution [8]. Soliton pulse can evolve owning to delicate balance between dispersion and nonlinearity. However, it is

© 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.

always a challenge to compensate for the loss when engineering these types of waveguide using metamaterials. The strong perturbation of a soliton envelope caused by the stimulated Raman scattering confines the energy scalability preventing the so-called dissipative soliton resonance [9]. It is important to know the limit we can reach expanding the nonlinear polarization PNL in a series over the field E [10]. The fourth-order nonlinear susceptibility χð Þ<sup>4</sup> , the fifth-order nonlinearity χð Þ<sup>5</sup> , and the seventh-order nonlinearity χð Þ<sup>7</sup> have been measured [11, 12]. The polynomial mode nonlinearity is due to the nonlinear polarization of metamaterials in the power-series expansion form where terms are kept up to the seventh order in the field E [10, 12–15]. This chapter conducts mathematical analysis to illustrate the controllability of the Raman soliton self-frequency shift with polynomial nonlinearity in metamaterials by using collective variable method.

3. Mathematical formulation

residual free energy (RFE) E, where

with the following simple way:

matter physics [27].

not be exactly zero [28].

E ¼ ð∞ �∞ j j q 2 dt ¼ ð∞ �∞

Cj <sup>¼</sup> <sup>∂</sup><sup>E</sup> ∂Zj

Then, we define a second set of constraints:

Through Eqs. (2)–(6), it leads to the equations of motion:

¼ ∂ ∂Zj

<sup>C</sup>\_ <sup>j</sup> <sup>¼</sup> dCj

ð∞ �∞ j j q 2 dt � � <sup>¼</sup>

The rate of change of Cj with respect to the normalized distance is defined as.

dz <sup>¼</sup> <sup>2</sup><sup>ℜ</sup> <sup>d</sup>

The pulse may not only be able to translate as a whole entity, but it may also execute more or less complex internal vibrations depending on the type of the perturbations in the system. This particle-like behavior has led to the formulation of the collective variable ð Þ CV techniques [25]. The basic idea is that the soliton solution depends on a collective of variables, called CVs, symbolically Zjð Þ j ¼ 1;…N , which represent pulse width, amplitude, chirp, frequency, and so on [25–28]. To this end, the original field is decomposed into two components, say Φð Þ z; t at

Raman Solitons in Nanoscale Optical Waveguides, with Metamaterials, Having Polynomial Law Nonlinearity Using…

where the first component f constitutes soliton solution and the second one q represents the residual radiation that is known as small amplitude dispersive waves. Introduction of these N

In order for the system to remain in the original phase space and best fit for the static solution, the CV method is obtained by configuring the function f Zð Þ <sup>1</sup>; Z2;…;ZN; t and minimizes

The approximation of neglecting the residual field is called "bare approximation" in condensed

Let CVs evolve only in a particular direction to minimize the PFE in the dynamical system

ð∞ �∞

dz <sup>ð</sup><sup>∞</sup> ∞ ∂q<sup>∗</sup> ∂Zj

where ℜ stands for the real part. Here, the weak equality indicates that the constraints Cj need

dCj

∂q ∂Zj <sup>q</sup><sup>∗</sup> <sup>þ</sup> ∂q<sup>∗</sup> ∂Zj q

Φð Þ¼ z; t f Zð <sup>1</sup>; Z2;…; ZN; tÞ þ q zð Þ ; t , (2)

j j <sup>Φ</sup>ð Þ� <sup>z</sup>; <sup>t</sup> <sup>f</sup>ðZ1; <sup>Z</sup>2;…;ZN; <sup>t</sup><sup>Þ</sup> <sup>2</sup>

dt: (3)

� �dt: (4)

http://dx.doi.org/10.5772/intechopen.75121

177

qdt � � � � , (5)

dz <sup>≈</sup> <sup>0</sup>: (6)

position z in the metamaterials and at time t, in the following way:

CVs increases the phase space of the dynamical system.

#### 2. Governing model

The dimensionless form nonlinear Schrödinger's equation (NLSE) that governs the propagation of Raman soliton through optical metamaterials, with polynomial law nonlinearity, is given by [16–24].

$$\begin{split} \mathrm{i}\frac{\partial}{\partial t}\Phi(z,t) + a\frac{\partial^2}{\partial z^2}\Phi(z,t) &+ \left(c\_1|\Phi(z,t)|^2 + c\_2|\Phi(z,t)|^4 + c\_3|\Phi(z,t)|^6\right)\Phi(z,t) \\ &= i\alpha \frac{\partial}{\partial z}\Phi(z,t) + i\lambda \frac{\partial}{\partial z}\left(|\Phi(z,t)|^2\Phi(z,t)\right) + i\nu \frac{\partial}{\partial z}\left(|\Phi(z,t)|^2\right)\Phi(z,t) .\end{split} \tag{1}$$

In this model, Φð Þ z; t represents the complex valued wave function with the independent variables being z and t that represent spatial and temporal variables, respectively. The first term represents the temporal evolution of nonlinear wave, while the coefficient a is the group velocity dispersion (GVD). The coefficients of cj for j ¼ 1; 2; 3 correspond to the nonlinear terms. Together, they form polynomial mode nonlinearity. The polynomial mode nonlinearity is due to the nonlinear polarization of metamaterials in the power-series expansion form where terms are kept up to the seventh order in the field E [10, 12–15]. It must be noted here that when c<sup>2</sup> ¼ c<sup>3</sup> ¼ 0 and c<sup>1</sup> 6¼ 0, the model Eq. (1) collapses to Kerr mode nonlinearity which is due to third-order polarization PNL [15]. However, if <sup>c</sup><sup>3</sup> <sup>¼</sup> 0 and <sup>c</sup><sup>1</sup> 6¼ 0 and <sup>c</sup><sup>2</sup> 6¼ 0, one arrives at parabolic mode nonlinearity, and it is from the fifth-order polarization PNL [15, 30]. Thus, polynomial mode stands as an extension version to Kerr and parabolic modes. Actually, the Raman effect is not influenced by the properties of the metamaterials; however, the Raman coefficient combines with the dispersive magnetic permeability of the metamaterials leading to additional higher-order nonlinear terms [10, 12, 14]. The group velocity and self-phase modulation term produce the delicate balance dispersion and nonlinearity that accounts for the formation of the stable soliton. On the right hand side, α describes intermodel dispersion, λ represents the self-steepening term in order to avoid the formation of shocks, and ν is the complex higher-order nonlinear dispersion coefficient.

### 3. Mathematical formulation

always a challenge to compensate for the loss when engineering these types of waveguide using metamaterials. The strong perturbation of a soliton envelope caused by the stimulated Raman scattering confines the energy scalability preventing the so-called dissipative soliton resonance [9]. It is important to know the limit we can reach expanding the nonlinear polarization PNL in a series over the field E [10]. The fourth-order nonlinear susceptibility χð Þ<sup>4</sup> , the fifth-order nonlinearity χð Þ<sup>5</sup> , and the seventh-order nonlinearity χð Þ<sup>7</sup> have been measured [11, 12]. The polynomial mode nonlinearity is due to the nonlinear polarization of metamaterials in the power-series expansion form where terms are kept up to the seventh order in the field E [10, 12–15]. This chapter conducts mathematical analysis to illustrate the controllability of the Raman soliton self-frequency shift with polynomial nonlinearity in

The dimensionless form nonlinear Schrödinger's equation (NLSE) that governs the propagation of Raman soliton through optical metamaterials, with polynomial law nonlinearity, is

<sup>∂</sup><sup>z</sup> j j <sup>Φ</sup>ð Þ <sup>z</sup>; <sup>t</sup> <sup>2</sup>

In this model, Φð Þ z; t represents the complex valued wave function with the independent variables being z and t that represent spatial and temporal variables, respectively. The first term represents the temporal evolution of nonlinear wave, while the coefficient a is the group velocity dispersion (GVD). The coefficients of cj for j ¼ 1; 2; 3 correspond to the nonlinear terms. Together, they form polynomial mode nonlinearity. The polynomial mode nonlinearity is due to the nonlinear polarization of metamaterials in the power-series expansion form where terms are kept up to the seventh order in the field E [10, 12–15]. It must be noted here that when c<sup>2</sup> ¼ c<sup>3</sup> ¼ 0 and c<sup>1</sup> 6¼ 0, the model Eq. (1) collapses to Kerr mode nonlinearity which is due to third-order polarization PNL [15]. However, if <sup>c</sup><sup>3</sup> <sup>¼</sup> 0 and <sup>c</sup><sup>1</sup> 6¼ 0 and <sup>c</sup><sup>2</sup> 6¼ 0, one arrives at parabolic mode nonlinearity, and it is from the fifth-order polarization PNL [15, 30]. Thus, polynomial mode stands as an extension version to Kerr and parabolic modes. Actually, the Raman effect is not influenced by the properties of the metamaterials; however, the Raman coefficient combines with the dispersive magnetic permeability of the metamaterials leading to additional higher-order nonlinear terms [10, 12, 14]. The group velocity and self-phase modulation term produce the delicate balance dispersion and nonlinearity that accounts for the formation of the stable soliton. On the right hand side, α describes intermodel dispersion, λ represents the self-steepening term in order to avoid the formation of shocks, and ν is the

<sup>∂</sup>z<sup>2</sup> <sup>Φ</sup>ð Þþ <sup>z</sup>; <sup>t</sup> <sup>c</sup>1j j <sup>Φ</sup>ð Þ <sup>z</sup>; <sup>t</sup> <sup>2</sup> <sup>þ</sup> <sup>c</sup>2j j <sup>Φ</sup>ð Þ <sup>z</sup>; <sup>t</sup> <sup>4</sup> <sup>þ</sup> <sup>c</sup>3j j <sup>Φ</sup>ð Þ <sup>z</sup>; <sup>t</sup> <sup>6</sup>

Φð Þ z; t  þ iν ∂

<sup>∂</sup><sup>z</sup> j j <sup>Φ</sup>ð Þ <sup>z</sup>; <sup>t</sup> <sup>2</sup>

Φð Þ z; t

(1)

Φð Þ z; t :

metamaterials by using collective variable method.

∂2

complex higher-order nonlinear dispersion coefficient.

<sup>Φ</sup>ð Þþ <sup>z</sup>; <sup>t</sup> <sup>i</sup><sup>λ</sup> <sup>∂</sup>

2. Governing model

176 Emerging Waveguide Technology

given by [16–24].

i ∂ ∂t

Φð Þþ z; t a

¼ iα ∂ ∂z The pulse may not only be able to translate as a whole entity, but it may also execute more or less complex internal vibrations depending on the type of the perturbations in the system. This particle-like behavior has led to the formulation of the collective variable ð Þ CV techniques [25]. The basic idea is that the soliton solution depends on a collective of variables, called CVs, symbolically Zjð Þ j ¼ 1;…N , which represent pulse width, amplitude, chirp, frequency, and so on [25–28]. To this end, the original field is decomposed into two components, say Φð Þ z; t at position z in the metamaterials and at time t, in the following way:

$$\Phi(z,t) = f(Z\_1, Z\_2, \dots, Z\_N, t) + q(z,t),\tag{2}$$

where the first component f constitutes soliton solution and the second one q represents the residual radiation that is known as small amplitude dispersive waves. Introduction of these N CVs increases the phase space of the dynamical system.

In order for the system to remain in the original phase space and best fit for the static solution, the CV method is obtained by configuring the function f Zð Þ <sup>1</sup>; Z2;…;ZN; t and minimizes residual free energy (RFE) E, where

$$E = \int\_{-\infty}^{\infty} |q|^2 dt = \int\_{-\infty}^{\infty} |\Phi(z, t) - f(Z\_1, Z\_2, \dots, Z\_N, t)|^2 dt. \tag{3}$$

The approximation of neglecting the residual field is called "bare approximation" in condensed matter physics [27].

Let CVs evolve only in a particular direction to minimize the PFE in the dynamical system with the following simple way:

$$\mathbf{C}\_{\dot{\jmath}} = \frac{\partial \mathbf{E}}{\partial Z\_{\dot{\jmath}}} = \frac{\partial}{\partial Z\_{\dot{\jmath}}} \left( \int\_{-\infty}^{\infty} |q|^2 dt \right) = \int\_{-\infty}^{\infty} \left( \frac{\partial q}{\partial Z\_{\dot{\jmath}}} q^\* + \frac{\partial q^\*}{\partial Z\_{\dot{\jmath}}} q \right) dt. \tag{4}$$

The rate of change of Cj with respect to the normalized distance is defined as.

$$\dot{\mathbf{C}}\_{\rangle} = \frac{d\mathbf{C}\_{\rangle}}{dz} = 2\Re\left(\frac{d}{dz}\left(\int\_{\circ}^{\circ} \frac{\partial q^{\*}}{\partial Z\_{\circ}} q dt\right)\right),\tag{5}$$

where ℜ stands for the real part. Here, the weak equality indicates that the constraints Cj need not be exactly zero [28].

Then, we define a second set of constraints:

$$\frac{d\mathbb{C}\_j}{dz} \approx 0.\tag{6}$$

Through Eqs. (2)–(6), it leads to the equations of motion:

$$\dot{\mathbf{C}}\_{\dot{f}} = -2\Re \sum\_{k=1}^{N} \left( \int\_{-\infty}^{\infty} \frac{\partial f^\*}{\partial \mathbf{Z}\_{\dot{f}}} \frac{\partial f}{\partial \mathbf{Z}\_k} dt - \int\_{-\infty}^{\infty} \frac{\partial^2 f^\*}{\partial \mathbf{Z}\_{\dot{f}} \partial \mathbf{Z}\_k} q dt \right) \frac{d\mathbf{Z}\_k}{dt} + R\_{\dot{\mathbf{P}}} \tag{7}$$

where

$$R\_{\dot{\gamma}} = 2\mathcal{R} \int\_{-\infty}^{\infty} \frac{\partial f^\*}{\partial Z\_{\dot{\gamma}}} \frac{d\Phi}{dz} dt,\tag{8}$$

4. Computational results

In this case, with N ¼ 6:

where

f Zð <sup>1</sup>;Z2;Z3;Z4; Z5;Z6; tÞ ¼ Z1exp �ð Þ t � Z<sup>2</sup>

phase. Also, m is the Gaussian parameter, where m > 0.

∂C<sup>1</sup> ∂Z<sup>1</sup>

0

∂C<sup>2</sup> ∂Z<sup>1</sup>

∂C<sup>3</sup> ∂Z<sup>1</sup>

∂C<sup>4</sup> ∂Z<sup>1</sup>

BBBBBBBBBBBBBBBBBBBBBB@

∂C<sup>5</sup> ∂Z<sup>1</sup>

∂C<sup>6</sup> ∂Z<sup>1</sup> ∂C<sup>1</sup> ∂Z<sup>2</sup>

∂C<sup>2</sup> ∂Z<sup>2</sup>

∂C<sup>3</sup> ∂Z<sup>2</sup>

∂C<sup>4</sup> ∂Z<sup>2</sup>

∂C<sup>5</sup> ∂Z<sup>2</sup>

∂C<sup>6</sup> ∂Z<sup>2</sup>

z ¼

R ¼

∂C ∂z ¼

In this part the adiabatic parameter dynamics of solitons in optical metamaterials with poly-

Raman Solitons in Nanoscale Optical Waveguides, with Metamaterials, Having Polynomial Law Nonlinearity Using…

2m=X<sup>2</sup> <sup>3</sup> þ i Z4

where Z<sup>1</sup> is the soliton amplitude, Z<sup>2</sup> is the center position of the soliton, Z<sup>3</sup> is the inverse width of the pulse, Z<sup>4</sup> is the soliton chirp, Z<sup>5</sup> is the soliton frequency, and Z<sup>6</sup> is the soliton

> ∂C<sup>1</sup> ∂Z<sup>3</sup>

> ∂C<sup>2</sup> ∂Z<sup>3</sup>

> ∂C<sup>3</sup> ∂Z<sup>3</sup>

> ∂C<sup>4</sup> ∂Z<sup>3</sup>

> ∂C<sup>5</sup> ∂Z<sup>3</sup>

> ∂C<sup>6</sup> ∂Z<sup>3</sup>

0

BBBBBBBBB@

Z1 Z2 Z3 Z4 Z5 Z6 ∂C<sup>1</sup> ∂Z<sup>5</sup>

∂C<sup>2</sup> ∂Z<sup>5</sup>

∂C<sup>3</sup> ∂Z<sup>5</sup>

∂C<sup>4</sup> ∂Z<sup>5</sup>

∂C<sup>5</sup> ∂Z<sup>5</sup>

∂C<sup>6</sup> ∂Z<sup>5</sup>

1

CCCCCCCCCA

R1 R2 R3 R4 R5 R6 1

CCCCCCCCCA

0

BBBBBBBBB@

∂C<sup>1</sup> ∂Z<sup>5</sup>

∂C<sup>2</sup> ∂Z<sup>5</sup>

∂C<sup>3</sup> ∂Z<sup>5</sup>

∂C<sup>4</sup> ∂Z<sup>5</sup>

∂C<sup>5</sup> ∂Z<sup>5</sup>

∂C<sup>6</sup> ∂Z<sup>5</sup> ∂C<sup>1</sup> ∂Z<sup>6</sup> 1

∂C<sup>2</sup> ∂Z<sup>6</sup>

∂C<sup>3</sup> ∂Z<sup>6</sup>

∂C<sup>4</sup> ∂Z<sup>6</sup> CCCCCCCCCCCCCCCCCCCCCCA

, (17)

, (18)

, (16)

∂C<sup>5</sup> ∂Z<sup>6</sup>

∂C<sup>6</sup> ∂Z<sup>6</sup>

<sup>2</sup> ð Þ <sup>t</sup> � <sup>Z</sup><sup>2</sup>

� �, (15)

<sup>2</sup> <sup>þ</sup> iZ5ð Þþ <sup>t</sup> � <sup>Z</sup><sup>2</sup> iZ<sup>6</sup>

http://dx.doi.org/10.5772/intechopen.75121

179

nomial nonlinearity will be obtained by CV method. A Gaussian is given by

for 1 ≤ j ≤ N.

The set of Eqs. (5)–(8) is equivalent to the matrix equation:

$$
\dot{\mathbf{C}} = \frac{\partial \mathbf{C}}{\partial \mathbf{z}} \dot{\mathbf{z}} + \mathbf{R},
\tag{9}
$$

where

$$z = \begin{pmatrix} Z\_1 \\ Z\_2 \\ \dots \\ Z\_N \end{pmatrix},\tag{10}$$

$$R = \begin{pmatrix} R\_1 \\ R\_2 \\ \dots \\ R\_N \end{pmatrix}'\tag{11}$$

while the N � N Jacobian is given by

$$\frac{\partial \mathcal{C}}{\partial \mathbf{z}} = \frac{\partial (\mathcal{C}\_1, \mathcal{C}\_2, \dots, \mathcal{C}\_N)}{\partial (Z\_1, Z\_2, \dots, Z\_N)} = \left(\frac{\partial \mathcal{C}\_j}{\partial Z\_k}\right)\_{N \times N} \tag{12}$$

with

$$\frac{\partial \mathbb{C}\_{j}}{\partial \mathcal{Z}\_{k}} = -2\Re \left( \int\_{-\infty}^{\infty} \frac{\partial f^{\*}}{\partial \mathcal{Z}\_{j}} \frac{\partial f}{\partial \mathcal{Z}\_{k}} dt - \int\_{-\infty}^{\infty} \frac{\partial^{2} f^{\*}}{\partial \mathcal{Z}\_{j} \partial \mathcal{Z}\_{k}} q dt \right), \tag{13}$$

for 1 ≤ j, k ≤ N.

At this stage, through Eq. (6) we can solve Eq. (9) by the following CV equations of motion:

$$
\dot{X} = \left(\frac{\partial \mathcal{C}}{\partial \mathbf{z}}\right)^{-1} \mathcal{R}.\tag{14}
$$

The set of Eqs. (4)–(14) represents the complete CV treatment for the generalized NLSE Eq. (1).

### 4. Computational results

<sup>C</sup>\_ <sup>j</sup> ¼ �2<sup>ℜ</sup> <sup>X</sup>

where

178 Emerging Waveguide Technology

for 1 ≤ j ≤ N.

where

with

for 1 ≤ j, k ≤ N.

N

ð∞ �∞ ∂f ∗ ∂Zj

Rj ¼ 2ℜ

∂f ∂Zk dt � ð∞ �∞

> ð∞ �∞

<sup>C</sup>\_ <sup>¼</sup> <sup>∂</sup><sup>C</sup> ∂z

z ¼

R ¼

<sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>∂</sup>ð Þ <sup>C</sup>1;C2;…;CN ∂ð Þ Z1;Z2;…;ZN

> ð∞ �∞

∂f ∗ ∂Zj

∂f ∂Zk dt � ð∞ �∞

At this stage, through Eq. (6) we can solve Eq. (9) by the following CV equations of motion:

The set of Eqs. (4)–(14) represents the complete CV treatment for the generalized NLSE Eq. (1).

<sup>X</sup>\_ <sup>¼</sup> <sup>∂</sup><sup>C</sup> ∂z � ��<sup>1</sup>

∂f ∗ ∂Zj dΦ

Z1 Z2 … ZN

R1 R2 … RN

1

CCCCCA

1

CCCCA

<sup>¼</sup> <sup>∂</sup>Cj ∂Zk � �

� �

N�N

qdt

∂2 f ∗ ∂Zj∂Zk

0

BBBBB@

0

BBBB@

∂2 f ∗ ∂Zj∂Zk

� � dZk

qdt

dt <sup>þ</sup> Rj, (7)

dz dt, (8)

z\_ þ R, (9)

, (10)

, (11)

R: (14)

, (12)

, (13)

k¼1

The set of Eqs. (5)–(8) is equivalent to the matrix equation:

while the N � N Jacobian is given by

∂C

¼ �2ℜ

∂Cj ∂Zk In this part the adiabatic parameter dynamics of solitons in optical metamaterials with polynomial nonlinearity will be obtained by CV method. A Gaussian is given by

$$f(\mathbf{Z}\_1, \mathbf{Z}\_2, \mathbf{Z}\_3, \mathbf{Z}\_4, \mathbf{Z}\_5, \mathbf{Z}\_6; \mathbf{t}) = \mathbf{Z}\_1 \exp\left[ -(\mathbf{t} - \mathbf{Z}\_2)^{2\mathbf{n}/\lambda\_3^2} + \mathbf{i}\frac{\mathbf{Z}\_4}{2}(\mathbf{t} - \mathbf{Z}\_2)^2 + \mathbf{i}\mathbf{Z}\_5(\mathbf{t} - \mathbf{Z}\_2) + \mathbf{i}\mathbf{Z}\_6 \right],\tag{15}$$

where Z<sup>1</sup> is the soliton amplitude, Z<sup>2</sup> is the center position of the soliton, Z<sup>3</sup> is the inverse width of the pulse, Z<sup>4</sup> is the soliton chirp, Z<sup>5</sup> is the soliton frequency, and Z<sup>6</sup> is the soliton phase. Also, m is the Gaussian parameter, where m > 0.

In this case, with N ¼ 6:

∂C ∂z ¼ ∂C<sup>1</sup> ∂Z<sup>1</sup> ∂C<sup>1</sup> ∂Z<sup>2</sup> ∂C<sup>1</sup> ∂Z<sup>3</sup> ∂C<sup>1</sup> ∂Z<sup>5</sup> ∂C<sup>1</sup> ∂Z<sup>5</sup> ∂C<sup>1</sup> ∂Z<sup>6</sup> ∂C<sup>2</sup> ∂Z<sup>1</sup> ∂C<sup>2</sup> ∂Z<sup>2</sup> ∂C<sup>2</sup> ∂Z<sup>3</sup> ∂C<sup>2</sup> ∂Z<sup>5</sup> ∂C<sup>2</sup> ∂Z<sup>5</sup> ∂C<sup>2</sup> ∂Z<sup>6</sup> ∂C<sup>3</sup> ∂Z<sup>1</sup> ∂C<sup>3</sup> ∂Z<sup>2</sup> ∂C<sup>3</sup> ∂Z<sup>3</sup> ∂C<sup>3</sup> ∂Z<sup>5</sup> ∂C<sup>3</sup> ∂Z<sup>5</sup> ∂C<sup>3</sup> ∂Z<sup>6</sup> ∂C<sup>4</sup> ∂Z<sup>1</sup> ∂C<sup>4</sup> ∂Z<sup>2</sup> ∂C<sup>4</sup> ∂Z<sup>3</sup> ∂C<sup>4</sup> ∂Z<sup>5</sup> ∂C<sup>4</sup> ∂Z<sup>5</sup> ∂C<sup>4</sup> ∂Z<sup>6</sup> ∂C<sup>5</sup> ∂Z<sup>1</sup> ∂C<sup>5</sup> ∂Z<sup>2</sup> ∂C<sup>5</sup> ∂Z<sup>3</sup> ∂C<sup>5</sup> ∂Z<sup>5</sup> ∂C<sup>5</sup> ∂Z<sup>5</sup> ∂C<sup>5</sup> ∂Z<sup>6</sup> ∂C<sup>6</sup> ∂Z<sup>1</sup> ∂C<sup>6</sup> ∂Z<sup>2</sup> ∂C<sup>6</sup> ∂Z<sup>3</sup> ∂C<sup>6</sup> ∂Z<sup>5</sup> ∂C<sup>6</sup> ∂Z<sup>5</sup> ∂C<sup>6</sup> ∂Z<sup>6</sup> 0 BBBBBBBBBBBBBBBBBBBBBB@ 1 CCCCCCCCCCCCCCCCCCCCCCA , (16)

$$z = \begin{pmatrix} Z\_1 \\ Z\_2 \\ Z\_3 \\ Z\_4 \\ Z\_5 \\ Z\_6 \end{pmatrix},\tag{17}$$
 
$$\mathcal{R} = \begin{pmatrix} R\_1 \\ R\_2 \\ R\_3 \\ R\_4 \\ R\_5 \\ R\_6 \end{pmatrix},\tag{18}$$

where

$$\mathcal{R}\_1 = -a\mathcal{Z}\_1(\mathcal{Z}\_4 + \mathcal{X}\_1\mathcal{Z}\_5) \frac{\Gamma\left(\frac{1}{2m}\right)}{m\left(\frac{2}{Z\_3^2}\right)^{\frac{1}{2m}}} - \left(\frac{2a\mathcal{Z}\_1^2\mathcal{X}\_4}{m} + \frac{12\lambda\mathcal{Z}\_1^3}{Z\_3^2}\right) \frac{\Gamma\left(\frac{1}{m}\right)}{\left(\frac{4}{Z\_3^2}\right)^{\frac{1}{m}}} - 2a\mathcal{Z}\_1 + 4\mathcal{Z}\_1^3\nu\_\prime \tag{19}$$

$$\begin{split} R\_{2} = 2aZ\_{1}^{2}((m-1)Z\_{4} + Z\_{1}Z\_{5}) + (1-2m) \left( \frac{aZ\_{1}^{2}Z\_{5}}{2m} \left(\frac{2}{Z\_{3}^{3}}\right)^{\frac{1}{m}} + \left(\frac{aZ\_{1}^{2}}{16} - \frac{\nu Z\_{5}^{2}}{8m}\right) \left(\frac{4}{Z\_{3}^{2}}\right)^{\frac{1}{m}} \right) \Gamma\left(-\frac{1}{2m}\right) \\ = -\left(\frac{Z\_{1}^{2}Z\_{4}}{m} \left(c\_{1}\left(\frac{Z\_{3}^{2}}{4}\right)^{\frac{1}{m}} + c\_{2}\left(\frac{Z\_{3}^{2}}{6}\right)^{\frac{1}{m}} + C\_{3}\left(\frac{Z\_{3}^{2}}{8}\right)^{\frac{1}{m}}\right) - \frac{4\lambda Z\_{1}^{4}Z\_{4}Z\_{5}}{\left(\frac{6}{Z\_{3}^{3}}\right)^{\frac{1}{m}}} \Gamma\left(\frac{1}{m}\right) \\ + \left(\frac{aZ\_{1}^{2}Z\_{5}}{m\left(\frac{2}{Z\_{3}^{2}}\right)^{\frac{1}{m}} - m\left(\frac{6}{Z\_{3}^{2}}\right)^{\frac{1}{m}}}\right) \Gamma\left(\frac{1}{2m}\right) + \frac{(m-1)\lambda Z\_{1}^{4}Z\_{3}^{2}}{3m} \left(\frac{6}{Z\_{3}^{2}}\right)^{\left(\frac{1}{m}\right)} \Gamma\left(-\frac{1}{m}\right). \end{split} \tag{20}$$

<sup>R</sup><sup>3</sup> <sup>¼</sup> <sup>ν</sup>Z<sup>5</sup> 1 Z3 � <sup>3</sup>aZ<sup>3</sup> 1Z 2 <sup>m</sup> � <sup>1</sup> � � <sup>3</sup> Z<sup>4</sup> m <sup>Γ</sup> <sup>1</sup> m � � � <sup>2</sup>αZ<sup>2</sup> 1 X3 � <sup>3</sup>λZ<sup>4</sup> 1 2Z<sup>3</sup> þ c1Z<sup>4</sup> 1 2Z<sup>3</sup> 4 Z2 3 � � <sup>1</sup> ð Þ <sup>2</sup><sup>m</sup> � aZ<sup>2</sup> <sup>1</sup>ð Þ Z<sup>4</sup> þ Z1Z<sup>5</sup> 2 Z2 3 � � <sup>1</sup> ð Þ<sup>m</sup> þ c2Z<sup>6</sup> 1 3Z<sup>3</sup> 6 Z2 3 � � <sup>1</sup> ð Þ <sup>2</sup><sup>m</sup> þ c3Z<sup>8</sup> 1 4Z<sup>3</sup> 8 Z2 3 � � <sup>1</sup> ð Þ <sup>2</sup><sup>m</sup> 0 BB@ 1 CCA <sup>Γ</sup> <sup>1</sup> 2m � � <sup>2</sup>m<sup>2</sup> , (21) <sup>R</sup><sup>4</sup> <sup>¼</sup> aZ<sup>2</sup> <sup>1</sup>ð Þ 1 � 2m 2 Z2 3 � � <sup>1</sup> ð Þ <sup>2</sup><sup>m</sup> 0 BB@ 1 CCA <sup>Γ</sup> <sup>1</sup> 2m � � <sup>4</sup><sup>m</sup> � aZ<sup>3</sup> 1 2 Z2 3 � � <sup>1</sup> ð Þ<sup>m</sup> � <sup>λ</sup>Z<sup>4</sup> <sup>1</sup>Z<sup>5</sup> 4 Z2 3 � � <sup>1</sup> ð Þ<sup>m</sup> 0 BB@ 1 CCA <sup>Γ</sup> <sup>1</sup> m � � m þ αZ<sup>2</sup> <sup>1</sup>Z<sup>4</sup> 2 <sup>2</sup> Z2 3 � � <sup>2</sup> ð Þ<sup>m</sup> þ 2λZ<sup>5</sup> 1 4 Z2 3 � � <sup>2</sup> ð Þ<sup>m</sup> 0 BB@ 1 CCA <sup>Γ</sup> <sup>2</sup> m � � m �Z<sup>2</sup> 1 c1Z<sup>2</sup> 1 4 Z2 3 � � <sup>3</sup> ð Þ <sup>2</sup><sup>m</sup> þ c2Z<sup>4</sup> 1 6 Z2 3 � � <sup>3</sup> ð Þ <sup>2</sup><sup>m</sup> þ c3Z<sup>6</sup> 1 8 Z2 3 � � <sup>3</sup> ð Þ <sup>2</sup><sup>m</sup> � <sup>α</sup>Z<sup>5</sup> 2 Z2 3 � � <sup>3</sup> ð Þ <sup>2</sup><sup>m</sup> 0 BB@ 1 CCA <sup>Γ</sup> <sup>3</sup> 2m � � <sup>m</sup> , (22)

$$\begin{split} \mathcal{R}\_5 = 2a(1 - 2m)Z\_1^2 - \left(\frac{aZ\_1^3}{\left(\frac{2}{Z\_3^2}\right)^{\left(\frac{1}{2m}\right)}}\right)\frac{\Gamma\left(\frac{1}{2m}\right)}{m} - 2\left(\frac{aZ\_1^2Z\_4}{\left(\frac{2}{X\_3^2}\right)^{\left(\frac{1}{2m}\right)}} - \frac{\lambda Z\_1^4Z\_4}{\left(\frac{4}{X\_3^2}\right)^{\left(\frac{1}{2m}\right)}}\right)\frac{\Gamma\left(\frac{3}{2m}\right)}{m} \\ - 2\left(\frac{aZ\_1^2X\_5}{\left(\frac{2}{X\_3^2}\right)^{\left(\frac{1}{2}\right)}} - \frac{\lambda Z\_1^4Z\_5}{\left(\frac{4}{X\_3^2}\right)^{\left(\frac{1}{2}\right)}}\right)\frac{\Gamma\left(\frac{1}{m}\right)}{m} - \frac{aZ\_1^3\Gamma\left(\frac{1}{2m}\right)}{m\left(\frac{2}{X\_3^2}\right)^{\left(\frac{1}{2m}\right)}} + 2a(1 - 2m)aZ\_1^2. \end{split} \tag{23}$$

<sup>R</sup><sup>6</sup> <sup>¼</sup> <sup>2</sup> <sup>α</sup>Z1Z<sup>4</sup> 2 Z2 3 � � <sup>1</sup> ð Þ<sup>m</sup>

BB@

0

soliton phase.

þ λZ<sup>4</sup> <sup>1</sup>Z<sup>4</sup>

4 Z2 3 � � <sup>1</sup> ð Þ<sup>m</sup>

1

<sup>Γ</sup> <sup>1</sup> m � �

Figure 1. (a) Surface plot of soliton amplitude, (b) surface plot of soliton center position, (c) surface plot of soliton inverse width of the pulse, (d) surface plot of soliton chirp variation, (e) surface plot of soliton frequency, and (f) surface plot of

Raman Solitons in Nanoscale Optical Waveguides, with Metamaterials, Having Polynomial Law Nonlinearity Using…

m þ

aZ1ð Þ 2m � 1 m

αZ1Z<sup>5</sup> 2 Z2 3 � � <sup>1</sup> ð Þ <sup>2</sup><sup>m</sup>

> Z2 3 2 � � <sup>1</sup> ð Þ <sup>2</sup><sup>m</sup>

0

BB@

þ λZ<sup>4</sup> <sup>1</sup>Z<sup>5</sup>

4 Z2 3 � � <sup>1</sup> ð Þ <sup>2</sup><sup>m</sup>

<sup>Γ</sup> � <sup>1</sup> 2m � �

1

<sup>Γ</sup> <sup>1</sup> 2m � �

http://dx.doi.org/10.5772/intechopen.75121

181

m

(24)

� <sup>2</sup>aZ<sup>2</sup> 1:

CCA

CCA

þ

Raman Solitons in Nanoscale Optical Waveguides, with Metamaterials, Having Polynomial Law Nonlinearity Using… http://dx.doi.org/10.5772/intechopen.75121 181

R<sup>1</sup> ¼ �aZ1ð Þ Z<sup>4</sup> þ X1Z<sup>5</sup>

180 Emerging Waveguide Technology

<sup>R</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup>aZ<sup>2</sup>

þ

<sup>R</sup><sup>4</sup> <sup>¼</sup> aZ<sup>2</sup>

0

BB@

0

BB@

2 Z2 3 � � <sup>1</sup> ð Þ <sup>2</sup><sup>m</sup> <sup>Γ</sup> <sup>1</sup> 2m � �

� <sup>2</sup>aZ<sup>2</sup>

0 @

aZ<sup>2</sup> <sup>1</sup>Z<sup>5</sup> 2m

<sup>1</sup>X<sup>4</sup> m

1

CCA <sup>Γ</sup> <sup>1</sup> 2m � �

> 2 <sup>m</sup> � <sup>1</sup> � �

c2Z<sup>6</sup> 1

1

<sup>Γ</sup> <sup>1</sup> m � �

m þ

c2Z<sup>4</sup> 1 6 Z2 3 � � <sup>3</sup> ð Þ <sup>2</sup><sup>m</sup>

CCA

þ

<sup>m</sup> � <sup>2</sup> <sup>α</sup>Z<sup>2</sup>

m �

0

BB@

<sup>3</sup> Z<sup>4</sup> m

þ

4Z<sup>3</sup> 8 Z2 3 � � <sup>1</sup> ð Þ <sup>2</sup><sup>m</sup>

þ

<sup>1</sup>Z<sup>4</sup>

2 X2 3 � � <sup>3</sup> ð Þ <sup>2</sup><sup>m</sup>

aZ<sup>3</sup> <sup>1</sup><sup>Γ</sup> <sup>1</sup> 2m � �

m <sup>2</sup> Z2 3 � � <sup>1</sup> ð Þ <sup>2</sup><sup>m</sup>

þ

2 Z3 3

 ! <sup>1</sup> 2m þ

> Z2 3 6 � �<sup>1</sup> m þ C<sup>3</sup>

!

! <sup>Γ</sup> <sup>1</sup>

12λZ<sup>3</sup> 1 Z2 3

m � �

� <sup>2</sup>αZ<sup>1</sup> <sup>þ</sup> <sup>4</sup>Z<sup>3</sup>

Z2 3

� <sup>4</sup>λZ<sup>4</sup>

6 Z2 3

! <sup>1</sup> ð Þ <sup>2</sup><sup>m</sup>

� <sup>3</sup>λZ<sup>4</sup> 1 2Z<sup>3</sup>

<sup>2</sup>m<sup>2</sup> ,

2λZ<sup>5</sup> 1 1

<sup>Γ</sup> <sup>2</sup> m � �

m

<sup>m</sup> ,

(22)

(23)

CCA

<sup>Γ</sup> <sup>3</sup> 2m � �

1

CCA

4 Z2 3 � � <sup>2</sup> ð Þ<sup>m</sup>

1Z<sup>2</sup> 3

� <sup>2</sup>αZ<sup>2</sup> 1 X3

1

<sup>Γ</sup> <sup>1</sup> 2m � �

þ

� <sup>α</sup>Z<sup>5</sup> 2 Z2 3 � � <sup>3</sup> ð Þ <sup>2</sup><sup>m</sup>

1

<sup>Γ</sup> <sup>3</sup> 2m � �

m

1,

CCA

<sup>þ</sup> <sup>2</sup>að Þ <sup>1</sup> � <sup>2</sup><sup>m</sup> aZ<sup>2</sup>

CCA

 ! <sup>1</sup> 2m

1 <sup>A</sup><sup>Γ</sup> � <sup>1</sup> 2m � �

<sup>1</sup>Z4Z<sup>5</sup> 6 Z2 3 � �<sup>1</sup> m

1

CCA <sup>Γ</sup> <sup>1</sup> m � �

<sup>Γ</sup> � <sup>1</sup> m � � ,

(20)

(21)

<sup>1</sup>ν, (19)

4 Z2 3 � �<sup>1</sup> m

αZ<sup>2</sup> 1 <sup>16</sup> � <sup>ν</sup>Z<sup>5</sup> 1 8m � � 4

> Z2 3 8 � �<sup>1</sup> m

<sup>þ</sup> ð Þ <sup>m</sup> � <sup>1</sup> <sup>λ</sup>Z<sup>4</sup>

<sup>Γ</sup> <sup>1</sup> m � �

> c3Z<sup>8</sup> 1

0

BB@

αZ<sup>2</sup> <sup>1</sup>Z<sup>4</sup>

2 <sup>2</sup> Z2 3 � � <sup>2</sup> ð Þ<sup>m</sup>

c3Z<sup>6</sup> 1 8 Z2 3 � � <sup>3</sup> ð Þ <sup>2</sup><sup>m</sup>

> � <sup>λ</sup>Z<sup>4</sup> <sup>1</sup>Z<sup>4</sup>

4 X2 3 � � <sup>3</sup> ð Þ <sup>2</sup><sup>m</sup>

3m

m <sup>2</sup> Z2 3 � � <sup>1</sup> 2m

> <sup>1</sup>Z<sup>4</sup> m

c1 Z2 3 4 � �<sup>1</sup> m þ c<sup>2</sup>

� <sup>λ</sup>Z<sup>4</sup> 1Z<sup>2</sup> 5

m <sup>6</sup> Z2 3 � � <sup>1</sup> ð Þ <sup>2</sup><sup>m</sup>

> � <sup>3</sup>aZ<sup>3</sup> 1Z

> > þ

� <sup>λ</sup>Z<sup>4</sup> <sup>1</sup>Z<sup>5</sup>

c1Z<sup>2</sup> 1 4 Z2 3 � � <sup>3</sup> ð Þ <sup>2</sup><sup>m</sup>

<sup>Γ</sup> <sup>1</sup> 2m � �

1

<sup>Γ</sup> <sup>1</sup> m � �

CCA

4 Z2 3 � � <sup>1</sup> ð Þ<sup>m</sup>

3Z<sup>3</sup> 6 Z2 3 � � <sup>1</sup> ð Þ <sup>2</sup><sup>m</sup>

<sup>R</sup><sup>3</sup> <sup>¼</sup> <sup>ν</sup>Z<sup>5</sup> 1 Z3

<sup>1</sup>ð Þ Z<sup>4</sup> þ Z1Z<sup>5</sup> 2 Z2 3 � � <sup>1</sup> ð Þ<sup>m</sup>

> 1 2 Z2 3 � � <sup>1</sup> ð Þ<sup>m</sup>

> > 0

BB@

1

CCA

�Z<sup>2</sup> 1

1 2 Z2 3 � � <sup>1</sup> ð Þ <sup>2</sup><sup>m</sup>

> � <sup>λ</sup>Z<sup>4</sup> <sup>1</sup>Z<sup>5</sup>

4 X2 3 � � <sup>1</sup> ð Þ<sup>m</sup>

<sup>1</sup>ðð Þ m � 1 Z<sup>4</sup> þ Z1Z5Þ þ ð Þ 1 � 2m

þ

c1Z<sup>4</sup> 1

1

<sup>Γ</sup> <sup>1</sup> 2m � �

CCA

2Z<sup>3</sup> 4 Z2 3 � � <sup>1</sup> ð Þ <sup>2</sup><sup>m</sup>

<sup>1</sup>ð Þ 1 � 2m

<sup>R</sup><sup>5</sup> <sup>¼</sup> <sup>2</sup>að Þ <sup>1</sup> � <sup>2</sup><sup>m</sup> <sup>Z</sup><sup>2</sup>

0

BB@

¼ � <sup>Z</sup><sup>2</sup>

αZ<sup>2</sup> <sup>1</sup>Z<sup>5</sup>

m <sup>2</sup> Z2 3 � � <sup>1</sup> ð Þ <sup>2</sup><sup>m</sup>

� aZ<sup>2</sup>

<sup>4</sup><sup>m</sup> � aZ<sup>3</sup>

<sup>1</sup> � aZ<sup>3</sup>

0

BB@

<sup>1</sup>X<sup>5</sup>

�<sup>2</sup> <sup>α</sup>Z<sup>2</sup>

BB@

0

2 Z2 3 � � <sup>1</sup> ð Þ<sup>m</sup>

0

BB@

0

BB@

Figure 1. (a) Surface plot of soliton amplitude, (b) surface plot of soliton center position, (c) surface plot of soliton inverse width of the pulse, (d) surface plot of soliton chirp variation, (e) surface plot of soliton frequency, and (f) surface plot of soliton phase.

$$\begin{split} \mathcal{R}\_{6} = 2 \left( \frac{a \mathcal{Z}\_{1} \mathcal{Z}\_{4}}{\left( \frac{2}{Z\_{3}^{2}} \right)^{(\frac{1}{2})} + \frac{\lambda \mathcal{Z}\_{1}^{4} \mathcal{Z}\_{4}}{\left( \frac{4}{Z\_{3}^{2}} \right)^{(\frac{1}{2})}} \right) \frac{\Gamma \left( \frac{1}{m} \right)}{m} + \left( \frac{a \mathcal{Z}\_{1} \mathcal{Z}\_{5}}{\left( \frac{2}{Z\_{3}^{2}} \right)^{(\frac{1}{2m})} + \frac{\lambda \mathcal{Z}\_{1}^{4} \mathcal{Z}\_{5}}{\left( \frac{4}{Z\_{3}^{2}} \right)}} \right) \frac{\Gamma \left( \frac{1}{2m} \right)}{m} \\ &+ \frac{a \mathcal{Z}\_{1} (2m - 1)}{m} \left( \frac{Z\_{3}^{2}}{2} \right)^{(\frac{1}{2m})} \Gamma \left( - \frac{1}{2m} \right) - 2a \mathcal{Z}\_{1}^{2} . \end{split} \tag{24}$$

### 5. Numerical simulation

The nonlinear dynamical system discussed in the previous section is plotted to illustrate the collective variables numerically; see Figure 1. The parameter values are as follows: m ¼ 1, <sup>a</sup> <sup>¼</sup> <sup>9</sup>:<sup>9</sup> � <sup>10</sup>�<sup>2</sup> , <sup>λ</sup> <sup>¼</sup> <sup>9</sup>:<sup>8</sup> � <sup>10</sup>�<sup>2</sup> , <sup>α</sup> <sup>¼</sup> <sup>1</sup> � <sup>10</sup>�<sup>1</sup> , <sup>c</sup><sup>1</sup> <sup>¼</sup> <sup>9</sup>:<sup>9</sup> � <sup>10</sup>�<sup>1</sup> , <sup>c</sup><sup>2</sup> ¼ �<sup>8</sup> � <sup>10</sup>�<sup>2</sup> , <sup>c</sup><sup>3</sup> ¼ �<sup>8</sup> � <sup>10</sup>�<sup>3</sup> , and <sup>ν</sup> <sup>¼</sup> <sup>1</sup>:<sup>01</sup> � <sup>10</sup>�<sup>1</sup> [10, 12, 15, 30].

Author details

\*, Jun Ren<sup>2</sup> and Matthew C. Tanzy<sup>1</sup>

1 Department of Mathematical Sciences, Delaware State University, Dover, DE, USA 2 Department of Physics and Engineering, Delaware State University, Dover, DE, USA

[1] Liu Y, Zhang X. Recent advances in transformation optics. Nanoscale. 2012;4:5277-5292.

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[2] Schurig D, Mock JJ, Justice BJ, Cummer SA, Pendry JB, Starr AF, Smith DR. Metamaterials

[3] Pendry JB. Negative refraction makes a perfect lens. Physics Review Letter. 2000;85(18):

[4] Grzegorczyk TM, Kong JA. Review of left-handed metamaterials: Evolution from theoretical and numerical studies to potential applications. Journal of Electromagnetic Waves

[5] Li J. A literature survey of mathematical study of Metamaterials. International Journal of

[6] Pendry JB, Schurig D, Smith DR. Controlling electromagnetic fields. Science. 2006;

[8] Silva A, Monticone F, Castaldi G, Galdi V, Alu A, Engheta N. Performing mathematical

[9] Kalashnikov VL, Sorokin E. Dissipative Raman solitons. Optics Express. 2014;22(24):

[10] Mitev VM, Pavlov LI, Stamenov KV. Seventh and ninth order nonlinear susceptibility measurement in alkali metal vapour. Optical and Quantum Electrics. 1979;11:229-236

[11] Ekvall K, Lundevall C, van der Meulen P. Studies of the fifth-order nonlinear susceptibil-

[7] Leonhardt U. Optical conformal mapping. Science. 2006;312(23):1777-1780

ity of ultraviolet-grade fused silica. Optics Letters. 2001;26(12):896-898

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\*Address all correspondence to: yxu@desu.edu

DOI: 10.1039/C2NR31140B

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Numerical Analysis. 2016;13(2):230-243

Yanan Xu1

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This continuous surface plot shows the dynamical relationship between the time and collective variables, in Figure 1. It shows soliton amplitude, center position, inverse width of the pulse, chirp, and phase keeping the original shape as time goes by. This is because group velocity and self-phase modulation term produce the delicate balance dispersion and nonlinearity that accounts for the formation of the stable soliton. It also describes Stokes Raman scattering that is due to transmitted wave at higher frequency and anti-Stokes Raman scattering where transmitted wave is at lower frequency by Figure 1(e) [29]. These results are consistent with Raman soliton scattering effect.

#### 6. Conclusion

This chapter gives Raman soliton solutions in optical metamaterials that is studied with polynomial nonlinearity. The polynomial mode nonlinearity is due to expanding the nonlinear polarization PNL in a series over the field E up to the seventh order [13–15]. The polynomial mode nonlinearity is an extension of the Kerr and parabolic mode nonlinearity, which are from third- and fifth-order polarization PNL [15, 30], respectively. The analytical results are supplemented with numerical simulation by collective variables. The continuous surface plot shows the dynamical relationship between the time and collective variables. It shows soliton amplitude, center position, inverse width of the pulse, chirp, and phase keeping the original shape as time goes by since group velocity and self-phase modulation term produce the delicate balance dispersion and nonlinearity. It also describes Stokes Raman scattering that is due to transmitted wave at higher frequency and anti-Stokes Raman scattering where transmitted wave is at lower frequency by Figure 1(e) [8, 29].

In the future, the set of plot with m 6¼ 1 will be plotted, and third-order dispersion (TOD) and fourth-order dispersion (FOD) will be included [15]. Nonlinear polarization of medium in the form of a power-series expansion, keeping the terms up to the ninth order, will be explored [10].

#### Acknowledgements

This work was supported by NSF EAGER grants: 1649173.

### Author details

5. Numerical simulation

182 Emerging Waveguide Technology

, <sup>λ</sup> <sup>¼</sup> <sup>9</sup>:<sup>8</sup> � <sup>10</sup>�<sup>2</sup>

mitted wave is at lower frequency by Figure 1(e) [8, 29].

This work was supported by NSF EAGER grants: 1649173.

and <sup>ν</sup> <sup>¼</sup> <sup>1</sup>:<sup>01</sup> � <sup>10</sup>�<sup>1</sup> [10, 12, 15, 30].

<sup>a</sup> <sup>¼</sup> <sup>9</sup>:<sup>9</sup> � <sup>10</sup>�<sup>2</sup>

6. Conclusion

explored [10].

Acknowledgements

The nonlinear dynamical system discussed in the previous section is plotted to illustrate the collective variables numerically; see Figure 1. The parameter values are as follows: m ¼ 1,

This continuous surface plot shows the dynamical relationship between the time and collective variables, in Figure 1. It shows soliton amplitude, center position, inverse width of the pulse, chirp, and phase keeping the original shape as time goes by. This is because group velocity and self-phase modulation term produce the delicate balance dispersion and nonlinearity that accounts for the formation of the stable soliton. It also describes Stokes Raman scattering that is due to transmitted wave at higher frequency and anti-Stokes Raman scattering where transmitted wave is at lower frequency by Figure 1(e) [29]. These results are consistent with Raman soliton scattering effect.

This chapter gives Raman soliton solutions in optical metamaterials that is studied with polynomial nonlinearity. The polynomial mode nonlinearity is due to expanding the nonlinear polarization PNL in a series over the field E up to the seventh order [13–15]. The polynomial mode nonlinearity is an extension of the Kerr and parabolic mode nonlinearity, which are from third- and fifth-order polarization PNL [15, 30], respectively. The analytical results are supplemented with numerical simulation by collective variables. The continuous surface plot shows the dynamical relationship between the time and collective variables. It shows soliton amplitude, center position, inverse width of the pulse, chirp, and phase keeping the original shape as time goes by since group velocity and self-phase modulation term produce the delicate balance dispersion and nonlinearity. It also describes Stokes Raman scattering that is due to transmitted wave at higher frequency and anti-Stokes Raman scattering where trans-

In the future, the set of plot with m 6¼ 1 will be plotted, and third-order dispersion (TOD) and fourth-order dispersion (FOD) will be included [15]. Nonlinear polarization of medium in the form of a power-series expansion, keeping the terms up to the ninth order, will be

, <sup>c</sup><sup>1</sup> <sup>¼</sup> <sup>9</sup>:<sup>9</sup> � <sup>10</sup>�<sup>1</sup>

, <sup>c</sup><sup>2</sup> ¼ �<sup>8</sup> � <sup>10</sup>�<sup>2</sup>

, <sup>c</sup><sup>3</sup> ¼ �<sup>8</sup> � <sup>10</sup>�<sup>3</sup>

,

, <sup>α</sup> <sup>¼</sup> <sup>1</sup> � <sup>10</sup>�<sup>1</sup>

Yanan Xu1 \*, Jun Ren<sup>2</sup> and Matthew C. Tanzy<sup>1</sup>


#### References


[12] Yang K, Kumar J. Susceptibilities of a poly(4BCMU) film through electroabsorption spectroscopy. Optics Letters. 2000;25(16):1186-1188

[26] Tchofo-Dinda P, Moubissi AB, Nakkeeran K. Collective variable theory for optical soli-

Raman Solitons in Nanoscale Optical Waveguides, with Metamaterials, Having Polynomial Law Nonlinearity Using…

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185

[27] Boesch R, Stancioff P, Willis CR. Hamiltonian equations for multiple-collective-variable theories of nonlinear Klein-Gordon equations: A projection-operator approach. Physics

[28] Tchofo Dinda P, Moubissi AB, Nakkeeran K. A collective variable approach for dispersion-managed solitons. Journal of Physics A: Mathematical and General. 2001;34:

[29] WIKIPEDIA. (2018, January 23). Retrieved from: https://en.wikipedia.org/w/index.php?

[30] Min X, Yang R, Tian J, Xue W, Christian JM. Exact dipole solitary wave solution in metamaterials with higher-order dispersion. Journal of Modern Optics. 2016;63(S3):S44-S50

tons in fibers. Physical Review E. 2011;64:016608(15)

Review B. 1988;38(10):6713-6735

title=Ramanscattering&action=history.

103-110


[26] Tchofo-Dinda P, Moubissi AB, Nakkeeran K. Collective variable theory for optical solitons in fibers. Physical Review E. 2011;64:016608(15)

[12] Yang K, Kumar J. Susceptibilities of a poly(4BCMU) film through electroabsorption

[13] Scalora M, Sychin MS, Akozbek N, Poliakov EY, D'Aguanno G, Mattiucci N, Bloemer MJ, Zheltikove AM. Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: Application to negative index materials. Physical Review Letters. 2005;

[14] Powers PE. Fundamentals of Nonlinear Optics. 1st ed. Boca Raton, FL: CRC Press; 2011 [15] Syrchin MS, Zheltikov AM, Scalora M. Analytical treatment of self-phase-modulation beyond the slowly varying envelope approximation. Physical Review A. 2004;69:053803

[16] Biswas A, Khan KR, Mahmood MF, Belic M. Bright and dark solitons in optical

[17] Biswas A, Mirzazadeh M, Savescu M, Milovic D, Khan KR, Mahmood MF, Belic M. Singular solitons in optical metamaterials by ansatz method and simplest equation

[18] Ebadi G, Mohavir A, Guzman JV, Khan KR, Mahmood MF, Moraru L, Biswas A, Belic M. Solitons in optical metamaterials by F-expansion scheme. Optoelectronics and Advanced

[19] Xiang Y, Dai X, Wen S, Guo J, Fan D. Controllable Raman soliton self-frequency shift in

[20] Krishnan EV, Al Gabshi M, Zhou Q, Khan KR, Mahmood MF, Xu Y, Biswas A, Belic M. Solitons in optical metamaterials by mapping method. Journal of Optoelectronics and

[21] Biswas A, Mirzazadeh M, Eslami M, Milovic D, Belic M. Solitons in optical metamaterials by functional variable method and first integral approach. Frequenz. 2014;68(11–12):525-530 [22] Bhrawy AH, Alshaery AA, Hilal EM, Milovic D, Moraru L, Savescu M, Biswas A. Optical solitons with polynomial and triple power law nonlinearities and spatio-temporal disper-

[23] Biswas A, Milovic D. Traveling wave solutions of the nonlinear Schrodinger's equation in non-Kerr law media. Communications in Nonlinear Science and Numerical Simulation.

[24] Xu Y, Savescu M, Khan KR, Mahmood MF, Biswas A, Belic M. Soliton propagation through nanoscale waveguides in optical metamaterials. Optics and Laser Technology.

[25] Asseu O, Diby A, Yoboue P, Kamagete A. Spatio-temporal pulsating dissipative Solitons through collective variable methods. Journal of Applied Mathematics and Physics. 2016:

sion. Proceedings of the Romanian Academy, Series A. 2014;15(3):235-240

spectroscopy. Optics Letters. 2000;25(16):1186-1188

metamaterials. Optik. 2014;125(13):3299-3302

Advanced Materials. 2015;17(3–4):511-516

2009;14(5):1993-1998

2016;7:177-186

1032-1041

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Materials: Rapid Communications. 2014;8(9–10):828-834

nonlinear metamaterials. Physical Review A. 2011;84:033815

95:013902

184 Emerging Waveguide Technology


**Chapter 10**

Provisional chapter

**Silicon-on-Insulator Slot Waveguides: Theory and**

DOI: 10.5772/intechopen.75539

Silicon-on-Insulator Slot Waveguides: Theory and

**Applications in Electro-Optics and Optical Sensing**

This chapter deals with the basic concept of silicon-on-insulator (SOI) slot waveguides, including slot waveguide theory, fabrication steps, and applications. First, in the theory section, a modal field expression and the characteristic equation is derived, which is also valid for higher-order modes. SOI slot waveguide structures are simulated and characteristic values like the effective refractive indices and the field confinement factors are determined. The fabrication section describes typical SOI fabrication steps and the limits of current fabrication techniques. Additionally, developments regarding loss reduction in SOI slot waveguides are given from the fabrication point of view. This is followed by the theory and practice of slot waveguide based electro-optical modulators. Here, the SOI slot waveguide is embedded in an organic nonlinear optical material in order to achieve record-low voltage-length products. In the field of optical sensors, it is shown that slot waveguides enable remarkable waveguide sensitivity for both refractive index sensing

Keywords: silicon-on-insulator (SOI), slot waveguide, polymer-based optical waveguides,

Recent developments in cloud computing, social media, and the Internet of things have heightened the need for integrated photonic communication systems. To compensate for the emerged data traffic, electro-optical (EO) modulators with large bandwidth and high-speed operation are necessary. This in turn needs novel material systems and waveguide structures since the established depletion-type modulators are speed limited due to carrier injection and

> © 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.

silicon-organic hybrid waveguide, optical waveguides technology, electro-optic

Applications in Electro-Optics and Optical Sensing

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.75539

Patrick Steglich

Abstract

and surface sensing.

1. Introduction

waveguides, waveguide sensing

Patrick Steglich

#### **Silicon-on-Insulator Slot Waveguides: Theory and Applications in Electro-Optics and Optical Sensing** Silicon-on-Insulator Slot Waveguides: Theory and Applications in Electro-Optics and Optical Sensing

DOI: 10.5772/intechopen.75539

Patrick Steglich Patrick Steglich

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.75539

#### Abstract

This chapter deals with the basic concept of silicon-on-insulator (SOI) slot waveguides, including slot waveguide theory, fabrication steps, and applications. First, in the theory section, a modal field expression and the characteristic equation is derived, which is also valid for higher-order modes. SOI slot waveguide structures are simulated and characteristic values like the effective refractive indices and the field confinement factors are determined. The fabrication section describes typical SOI fabrication steps and the limits of current fabrication techniques. Additionally, developments regarding loss reduction in SOI slot waveguides are given from the fabrication point of view. This is followed by the theory and practice of slot waveguide based electro-optical modulators. Here, the SOI slot waveguide is embedded in an organic nonlinear optical material in order to achieve record-low voltage-length products. In the field of optical sensors, it is shown that slot waveguides enable remarkable waveguide sensitivity for both refractive index sensing and surface sensing.

Keywords: silicon-on-insulator (SOI), slot waveguide, polymer-based optical waveguides, silicon-organic hybrid waveguide, optical waveguides technology, electro-optic waveguides, waveguide sensing

#### 1. Introduction

Recent developments in cloud computing, social media, and the Internet of things have heightened the need for integrated photonic communication systems. To compensate for the emerged data traffic, electro-optical (EO) modulators with large bandwidth and high-speed operation are necessary. This in turn needs novel material systems and waveguide structures since the established depletion-type modulators are speed limited due to carrier injection and

© 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.

removal. In contrast, polymer-based EO modulators exhibit high-speed operation but suffer from process compatibility with the well-established silicon-on-insulator (SOI) technology, which makes high volume and cheap production challenging.

shown in Figure 1. During the last decade, a new waveguide approach based on vertical silicon slot waveguides has been proposed [1] and experimentally demonstrated to be suitable as an optical phase shifter [2]. An SOI slot waveguide consists typically of two silicon rails with a height of h ¼ 220 nm. This thickness of 220 nm has become a standard used in particular by most multi-project wafer foundries [5]. As illustrated in Figure 1, both silicon rails are located on top of a buried oxide (BOX) substrate and are separated from each other by a slot width s. The width of the silicon rails is denoted as wg. Slot waveguides enable a high field confinement in a narrow low-index region. Infiltration of the interior of the slot waveguide with an EO polymer allows the use of the Pockels effect. Because of this effect, slot wave guides have high potential in the field of optical switching and high-speed modulation even at frequencies of 100 GHz [3]. As a consequence, various devices like Mach-Zehnder interferometers and ring resonators have been recently developed using slot waveguide phase shifters [6]. In fact, slot waveguides have become the key element in order to implement organic materials into silicon photonics. Beside the strip and slot waveguide, the strip-loaded slot waveguide is utilized in

Silicon-on-Insulator Slot Waveguides: Theory and Applications in Electro-Optics and Optical Sensing

http://dx.doi.org/10.5772/intechopen.75539

189

The overwhelming advantage of SOI slot waveguides lies in its compatibility with CMOS fabrication processes. This compatibility ensures a cost efficient mass production environment for such integrated photonic devices. Figure 2 shows, as an example, a scanning electron microscopy (SEM) picture of three slot waveguides with different slot widths from the top view and one slot waveguide in the cross-sectional view recorded with a focused ion beam (FIB). They were fabricated in an SiGe BiCMOS pilot line at the Institute of High-Performance Microelectronics (IHP) in Frankfurt (Oder) using 200 mm SOI wafers and 248 nm lithography.

One major advantage of slot waveguides compared to strip waveguides is the fact that the guided light is partially confined in between two silicon rails. Consequently, the light is forced

Figure 1. Typical silicon-on-insulator waveguide types. The most common waveguide type is the strip waveguide, where the light is highly confined inside the silicon core. In case of slot waveguides, the light is confined near two silicon rails. To apply a voltage to the silicon rails, strip-loads serve as electrical connections. The corresponding waveguide is called a

this work. The cross section of each waveguide type is shown in Figure 1.

2.2. Silicon-on-insulator slot waveguide theory

strip-loaded slot waveguide.

After the invention by Almeida et al. in 2004 [1], slot waveguides became one key element to combine the well-established SOI technology with nonlinear optical polymers [2]. In recent years, there has been an increasing interest in slot waveguide structures. An SOI slot waveguide consists of a small gap in between two silicon rails, where a nonlinear optical polymer can be deposited. This narrow gap has two effects. First, the guided light is partly confined inside the gap. Second, the gap leads to an extremely large electric field, while voltages as low as 1 V are applied. Consequently, a record-high operation speed and large bandwidth with low energy consumption has been demonstrated using the silicon-organic hybrid (SOH) photonics [3].

Researchers have also discovered the advantages of slot waveguides for optical sensing. In this case, the slot waveguide takes advantage from the fact that more than 70% of the guided light can be confined near the silicon rails. Therefore, the light has a stronger interaction with the analyte compared to common strip waveguides, where only a fraction, 20%, of the light can be interacting. This strong light-analyte interaction leads to a large waveguide sensitivity, which has pushed the development of recent integrated optical sensors.

This work examines the theory and applications of SOI slot waveguides for EO modulators and optical sensors. We provide a theoretical guideline for a deeper understanding of SOI slot waveguides. The present work is structured as follows. The following section focuses on a detailed theoretical description and analysis of slot waveguide structures. This is followed by a section on the fabrication scheme using standard SOI technology. Finally, application notes in electro-optics and optical sensing are given in Sections 4 and 5, respectively.

### 2. Silicon-on-insulator slot waveguide

This section deals with the basic concepts of SOI slot waveguides. For a deeper understanding of slot waveguide structure, the breakthrough paper of Almeida et al. [1] is recommended. However, the modal field expression given in [1] has some transcription errors and the given characteristic equation is not explicit for solving higher-order modes. Therefore, the correct modal field expression and the characteristic equation, which is also valid for higher-order modes is provided in this chapter, following the comprehensive work of Liu et al. [4].

#### 2.1. Introduction to silicon-on-insulator waveguides

In general, silicon waveguides can be readily fabricated from SOI wafers using standard CMOS (complementary metal-oxide-semiconductor) processes. A typical SOI wafer consists of a buried oxide (BOX) layer between the silicon wafer and a thin silicon layer. Optical lithography and etching techniques are used to form the silicon waveguide. The most common silicon waveguide is the strip waveguide. This waveguide has a rectangular geometry as shown in Figure 1. During the last decade, a new waveguide approach based on vertical silicon slot waveguides has been proposed [1] and experimentally demonstrated to be suitable as an optical phase shifter [2]. An SOI slot waveguide consists typically of two silicon rails with a height of h ¼ 220 nm. This thickness of 220 nm has become a standard used in particular by most multi-project wafer foundries [5]. As illustrated in Figure 1, both silicon rails are located on top of a buried oxide (BOX) substrate and are separated from each other by a slot width s. The width of the silicon rails is denoted as wg. Slot waveguides enable a high field confinement in a narrow low-index region. Infiltration of the interior of the slot waveguide with an EO polymer allows the use of the Pockels effect. Because of this effect, slot wave guides have high potential in the field of optical switching and high-speed modulation even at frequencies of 100 GHz [3]. As a consequence, various devices like Mach-Zehnder interferometers and ring resonators have been recently developed using slot waveguide phase shifters [6]. In fact, slot waveguides have become the key element in order to implement organic materials into silicon photonics. Beside the strip and slot waveguide, the strip-loaded slot waveguide is utilized in this work. The cross section of each waveguide type is shown in Figure 1.

The overwhelming advantage of SOI slot waveguides lies in its compatibility with CMOS fabrication processes. This compatibility ensures a cost efficient mass production environment for such integrated photonic devices. Figure 2 shows, as an example, a scanning electron microscopy (SEM) picture of three slot waveguides with different slot widths from the top view and one slot waveguide in the cross-sectional view recorded with a focused ion beam (FIB). They were fabricated in an SiGe BiCMOS pilot line at the Institute of High-Performance Microelectronics (IHP) in Frankfurt (Oder) using 200 mm SOI wafers and 248 nm lithography.

#### 2.2. Silicon-on-insulator slot waveguide theory

removal. In contrast, polymer-based EO modulators exhibit high-speed operation but suffer from process compatibility with the well-established silicon-on-insulator (SOI) technology,

After the invention by Almeida et al. in 2004 [1], slot waveguides became one key element to combine the well-established SOI technology with nonlinear optical polymers [2]. In recent years, there has been an increasing interest in slot waveguide structures. An SOI slot waveguide consists of a small gap in between two silicon rails, where a nonlinear optical polymer can be deposited. This narrow gap has two effects. First, the guided light is partly confined inside the gap. Second, the gap leads to an extremely large electric field, while voltages as low as 1 V are applied. Consequently, a record-high operation speed and large bandwidth with low energy consumption has been demonstrated using the silicon-organic hybrid (SOH) pho-

Researchers have also discovered the advantages of slot waveguides for optical sensing. In this case, the slot waveguide takes advantage from the fact that more than 70% of the guided light can be confined near the silicon rails. Therefore, the light has a stronger interaction with the analyte compared to common strip waveguides, where only a fraction, 20%, of the light can be interacting. This strong light-analyte interaction leads to a large waveguide sensitivity, which

This work examines the theory and applications of SOI slot waveguides for EO modulators and optical sensors. We provide a theoretical guideline for a deeper understanding of SOI slot waveguides. The present work is structured as follows. The following section focuses on a detailed theoretical description and analysis of slot waveguide structures. This is followed by a section on the fabrication scheme using standard SOI technology. Finally, application notes in

This section deals with the basic concepts of SOI slot waveguides. For a deeper understanding of slot waveguide structure, the breakthrough paper of Almeida et al. [1] is recommended. However, the modal field expression given in [1] has some transcription errors and the given characteristic equation is not explicit for solving higher-order modes. Therefore, the correct modal field expression and the characteristic equation, which is also valid for higher-order

In general, silicon waveguides can be readily fabricated from SOI wafers using standard CMOS (complementary metal-oxide-semiconductor) processes. A typical SOI wafer consists of a buried oxide (BOX) layer between the silicon wafer and a thin silicon layer. Optical lithography and etching techniques are used to form the silicon waveguide. The most common silicon waveguide is the strip waveguide. This waveguide has a rectangular geometry as

modes is provided in this chapter, following the comprehensive work of Liu et al. [4].

which makes high volume and cheap production challenging.

has pushed the development of recent integrated optical sensors.

2. Silicon-on-insulator slot waveguide

2.1. Introduction to silicon-on-insulator waveguides

electro-optics and optical sensing are given in Sections 4 and 5, respectively.

tonics [3].

188 Emerging Waveguide Technology

One major advantage of slot waveguides compared to strip waveguides is the fact that the guided light is partially confined in between two silicon rails. Consequently, the light is forced

Figure 1. Typical silicon-on-insulator waveguide types. The most common waveguide type is the strip waveguide, where the light is highly confined inside the silicon core. In case of slot waveguides, the light is confined near two silicon rails. To apply a voltage to the silicon rails, strip-loads serve as electrical connections. The corresponding waveguide is called a strip-loaded slot waveguide.

Figure 2. SEM image (left) of slot waveguides with different slot widths fabricated using 248 nm lithography. FIB image (right) of a slot waveguide cross section [13].

to interact directly with the cladding material. The reason for the high confinement is the large index contrast of the high-index silicon nsi and the low-index cladding material nclad. At the interface the normal electric field, which is according to Figure 1, the Ex field component, undergoes a large discontinuity. This results in a field enhancement in the low-index region, which is proportional to the ratio of the refractive indices of the cladding material to that of silicon,

$$E\_{\rm x,slot} = \frac{n\_{si}^2}{n\_{clad}^2} E\_{\rm x,si} \tag{1}$$

n xð Þ¼ f xð Þ¼

Letting

index of the slot is denoted as nslot.

8 >><

>>:

γ2

γ2

d2 Ex dx<sup>2</sup> � <sup>γ</sup><sup>2</sup>

8

>>>>>>>><

>>>>>>>>:

d2 Ex dx<sup>2</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup>

d2 Ex dx<sup>2</sup> � <sup>γ</sup><sup>2</sup>

the Helmholtz equation can be expressed as

γ2 si <sup>¼</sup> <sup>k</sup><sup>2</sup> 0n2

and β ¼ k0neff denotes the mode propagation constant with the effective refractive index neff .

Figure 3. Schematic of the slot waveguide structure with infinite height (slab waveguide approximation). The refractive

slot <sup>¼</sup> <sup>β</sup><sup>2</sup> � <sup>k</sup><sup>2</sup>

clad <sup>¼</sup> <sup>β</sup><sup>2</sup> � <sup>k</sup><sup>2</sup>

nslot, if xj j < a nsi, if a < j j x < b

Silicon-on-Insulator Slot Waveguides: Theory and Applications in Electro-Optics and Optical Sensing

(3)

191

(7)

nclad, if xj j > b

0n2

0n2

slotEx ¼ 0, if xj j < a

cladEx ¼ 0, if xj j > b

The effective refractive index is obtained by solving the Eigen equation of the waveguide structure. In case of the TM fundamental mode, the following condition is assumed: nsi > nclad ≥ nslot. The analysis of other cases like nsi > nslot > nclad is out of the scope of this thesis. For a more detailed analysis the work of Liu et al. [4] is recommended. However, by choosing reasonable modal field functions in each layer of the slab-based slot waveguide and

siEx ¼ 0, if a < j j x < b

slot (4)

http://dx.doi.org/10.5772/intechopen.75539

clad (6)

si � <sup>β</sup><sup>2</sup> (5)

Here, Ex,slot and Ex,si represent the electric field inside the slot and inside the silicon, respectively. From Eq. (1), it is apparent that the modal field distribution in the slot depends on the quotient of the refractive indices of silicon and the slot material. The larger the nsi to nclad ratio, the stronger is the normal electric field component in the slot. Considering silicon with a refractive index of 3.48 and air as cladding material with a refractive index of 1.0, the resulting field amplitude is more than 12 times higher in the slot region according to Eq. (1). The high confinement inside the slot is of special benefit for EO and biosensing applications.

As shown in Figure 2, a conventional slot waveguide structure with finite height consists of two rectangle silicon rails. In the following, we will transform the three-dimensional (3D) rectangular silicon rails of the slot waveguide into a two-dimensional (2D) slab waveguide, where the height of the slot waveguide becomes infinite as illustrated in Figure 3. The coordinate system is set in the center of the slab-based slot waveguide. This slab waveguide approximation makes it easier to find an analytical solution and is simpler and more intuitive than numerical methods like the finite element method (FEM).

For the Ex component of the fundamental TM (transverse magnetic) mode, the Helmholtz equation for each layer becomes

$$\frac{d^2 E\_x}{d\mathbf{x}^2} + \left[k\_0^2 n^2(\mathbf{x}) - \beta^2\right] E\_x = 0,\tag{2}$$

where

Silicon-on-Insulator Slot Waveguides: Theory and Applications in Electro-Optics and Optical Sensing http://dx.doi.org/10.5772/intechopen.75539 191

Figure 3. Schematic of the slot waveguide structure with infinite height (slab waveguide approximation). The refractive index of the slot is denoted as nslot.

$$m(\mathbf{x}) = f(\mathbf{x}) = \begin{cases} n\_{\text{slot}} & \text{if } |\mathbf{x}| < a \\\\ n\_{\text{si}} & \text{if } a < |\mathbf{x}| < b \\\\ n\_{\text{clad}} & \text{if } |\mathbf{x}| > b \end{cases} \tag{3}$$

and β ¼ k0neff denotes the mode propagation constant with the effective refractive index neff . Letting

$$
\gamma\_{\text{slot}}^2 = \beta^2 - k\_0^2 n\_{\text{slot}}^2 \tag{4}
$$

$$
\gamma\_{si}^2 = k\_0^2 n\_{si}^2 - \beta^2 \tag{5}
$$

$$
\gamma\_{clad}^2 = \beta^2 - k\_0^2 n\_{clad}^2 \tag{6}
$$

the Helmholtz equation can be expressed as

to interact directly with the cladding material. The reason for the high confinement is the large index contrast of the high-index silicon nsi and the low-index cladding material nclad. At the interface the normal electric field, which is according to Figure 1, the Ex field component, undergoes a large discontinuity. This results in a field enhancement in the low-index region, which is proportional to the ratio of the refractive indices of the cladding material to that of

Figure 2. SEM image (left) of slot waveguides with different slot widths fabricated using 248 nm lithography. FIB image

Ex,slot <sup>¼</sup> <sup>n</sup><sup>2</sup>

confinement inside the slot is of special benefit for EO and biosensing applications.

numerical methods like the finite element method (FEM).

d2 Ex dx<sup>2</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup>

equation for each layer becomes

where

si n2 clad

Here, Ex,slot and Ex,si represent the electric field inside the slot and inside the silicon, respectively. From Eq. (1), it is apparent that the modal field distribution in the slot depends on the quotient of the refractive indices of silicon and the slot material. The larger the nsi to nclad ratio, the stronger is the normal electric field component in the slot. Considering silicon with a refractive index of 3.48 and air as cladding material with a refractive index of 1.0, the resulting field amplitude is more than 12 times higher in the slot region according to Eq. (1). The high

As shown in Figure 2, a conventional slot waveguide structure with finite height consists of two rectangle silicon rails. In the following, we will transform the three-dimensional (3D) rectangular silicon rails of the slot waveguide into a two-dimensional (2D) slab waveguide, where the height of the slot waveguide becomes infinite as illustrated in Figure 3. The coordinate system is set in the center of the slab-based slot waveguide. This slab waveguide approximation makes it easier to find an analytical solution and is simpler and more intuitive than

For the Ex component of the fundamental TM (transverse magnetic) mode, the Helmholtz

0n2

Ex,si (1)

ð Þ� <sup>x</sup> <sup>β</sup><sup>2</sup> Ex <sup>¼</sup> <sup>0</sup>, (2)

silicon,

(right) of a slot waveguide cross section [13].

190 Emerging Waveguide Technology

$$\begin{cases} \frac{d^2 E\_x}{dx^2} - \gamma\_{\text{slot}}^2 E\_x = 0, & \text{if } |\mathbf{x}| < a \\\\ \frac{d^2 E\_x}{dx^2} + \gamma\_{\text{sil}}^2 E\_x = 0, & \text{if } a < |\mathbf{x}| < b \\\\ \frac{d^2 E\_x}{dx^2} - \gamma\_{\text{clad}}^2 E\_x = 0, & \text{if } |\mathbf{x}| > b \end{cases} \tag{7}$$

The effective refractive index is obtained by solving the Eigen equation of the waveguide structure. In case of the TM fundamental mode, the following condition is assumed: nsi > nclad ≥ nslot. The analysis of other cases like nsi > nslot > nclad is out of the scope of this thesis. For a more detailed analysis the work of Liu et al. [4] is recommended. However, by choosing reasonable modal field functions in each layer of the slab-based slot waveguide and employing the electromagnetic field boundary conditions, the modal field solution and the characteristic equation can be obtained. In consideration of the natural boundary condition and waveguide symmetry the general solution for the transverse E-field profile Ex of the fundamental TM mode is

$$E\_x(\mathbf{x}) = \begin{cases} A\_1 \cosh\left(\mathcal{V}\_{\text{slot}} \mathbf{x}\right), & \text{if } |\mathbf{x}| < a \\ A\_2 \cos\left(\mathcal{V}\_{\text{sil}} |\mathbf{x}|\right) + B\_2 \sin\left(\mathcal{V}\_{\text{sil}} |\mathbf{x}|\right), & \text{if } a < |\mathbf{x}| < b \\ A\_3 e^{-\mathcal{V}\_{\text{slot}}|\mathbf{x}|}, & \text{if } |\mathbf{x}| > b \end{cases} \tag{8}$$

Here, A1, A2, A<sup>3</sup> and B<sup>2</sup> are constants. Assuming that there is no free charge at the boundary, the boundary continuous conditions for the TM mode are the continuity of the normal component of the electric displacement field n<sup>2</sup> cladEx, clad <sup>¼</sup> <sup>n</sup><sup>2</sup> siEx,si � � and <sup>n</sup><sup>2</sup> slotEx,slot <sup>¼</sup> <sup>n</sup><sup>2</sup> siEx,si � � and the continuity of dEx/ dx. At boundary surfaces j j x ¼ a and j j x ¼ b, the boundary continuous conditions are used to derive the coefficients and the eigenvalue equation for β. The eigenvalue equation is a transcendental equation and is given by

$$\tan^{-1}\left(\frac{n\_{si}^2}{n\_{\rm clad}^2}\frac{\mathcal{V}\_{\rm clad}}{\mathcal{V}\_{si}}\right) + \tan^{-1}\left(\frac{n\_{si}^2}{n\_{\rm dot}^2}\frac{\mathcal{V}\_{\rm slot}}{\mathcal{V}\_{si}}\tanh(\mathcal{V}\_{\rm slot}a)\right) + m \cdot \pi = \mathcal{V}\_{si}(b-a) \tag{9}$$

where m is an integer. The analytical solution of the fundamental TM mode is obtained by substituting the coefficients into the general solution for Ex (Eq. (8)), which leads to the modal field expression

$$E\_x(\mathbf{x}) = A \begin{cases} \frac{1}{n\_{\rm{slot}}^2} \cosh\left(\gamma\_{\rm{slot}} x\right), & \text{if } |\mathbf{x}| < a \\\\ \frac{1}{n\_{\rm{sl}}^2} \cosh\left(\gamma\_{\rm{slot}} a\right) \cos\left(\gamma\_{\rm{sl}} (|\mathbf{x}| - a)\right) \\\\ + \frac{1}{n\_{\rm{slot}}^2} \sum\_{\rm{sif}} \sinh\left(\gamma\_{\rm{sl}ot} a\right) \sin\left(\gamma\_{\rm{si}} (|\mathbf{x}| - a)\right), & \text{if } a < |\mathbf{x}| < b \\\\ \frac{1}{n\_{\rm{cl}}^2} \left(\cosh\left(\gamma\_{\rm{sl}ot} a\right) \cos\left(\gamma\_{\rm{sl}} (b - a)\right)\right) \\\\ + \frac{n\_{\rm{sl}}^2}{n\_{\rm{slot}}^2} \sum\_{\rm{sif}} \sinh\left(\gamma\_{\rm{sl}ot} a\right) \sin\left(\gamma\_{\rm{sl}} (b - a)\right)\right) e^{-\gamma\_{\rm{sl}}|\mathbf{x}|}, & \text{if } |\mathbf{x}| > b \end{cases} \tag{10}$$

Here, the quasi-TE eigenmode presents the major E-field component along the x-direction [1]. Consequently, the quasi-TE eigenmode in the 3D slot waveguide structure is analogous to the TM eigenmode in the slab-based slot waveguide structure, which was studied in the previous section. In the following simulation study, the waveguide parameters, i.e., the silicon rail width wg and the slot width s, are variable whereas the height h is fixed to 220 nm (see Figure 5).

Figure 4. Calculated Ex field (normalized) for a slot waveguide structure with infinite height. Parameters: λ ¼ 1550 nm,

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a ¼ 110 nm, b ¼ 220 nm, nslot ¼ nclad ¼ 1:44, nsi ¼ 3:48.

silicon and buried oxide substrate, respectively.

The wavelength is assumed to be λ ¼ 1550 nm because it is a typical telecommunication wavelength in silicon photonics. Triangular vector elements with a maximum and minimum

Figure 5. Cross-sectional view of an SOI strip waveguide (a) and an SOI slot waveguide (b). The cladding oxide is removed to hybridize the silicon waveguide with an EO polymer (EO modulator) or to bring the waveguide in contact with an analyte (biochemical sensor). The refractive indices are denoted as nclad, nsi and nbox for the caldding material,

where A is an arbitrary constant. As an example, Figure 4 shows the normalized Ex field distribution of a typical silicon slot waveguide. This figure gives an evidence of the large discontinuity and the high E-field confinement inside the slot region.

#### 2.3. Simulation of slot waveguides

In this section, a commercial full-vectorial finite element method (FEM) based mode solver from COMSOL Multiphysics is employed to simulate the slot waveguide structure with finite height.

employing the electromagnetic field boundary conditions, the modal field solution and the characteristic equation can be obtained. In consideration of the natural boundary condition and waveguide symmetry the general solution for the transverse E-field profile Ex of the

Here, A1, A2, A<sup>3</sup> and B<sup>2</sup> are constants. Assuming that there is no free charge at the boundary, the boundary continuous conditions for the TM mode are the continuity of the normal compo-

cladEx, clad <sup>¼</sup> <sup>n</sup><sup>2</sup>

continuity of dEx/ dx. At boundary surfaces j j x ¼ a and j j x ¼ b, the boundary continuous conditions are used to derive the coefficients and the eigenvalue equation for β. The eigenvalue

> γslot γsi

where m is an integer. The analytical solution of the fundamental TM mode is obtained by substituting the coefficients into the general solution for Ex (Eq. (8)), which leads to the modal

tanh <sup>γ</sup>slot<sup>a</sup> � � !

<sup>A</sup>1cosh <sup>γ</sup>slot<sup>x</sup> � �, if xj j <sup>&</sup>lt; <sup>a</sup> <sup>A</sup><sup>2</sup> cos <sup>γ</sup>sij j <sup>x</sup> � � <sup>þ</sup> <sup>B</sup><sup>2</sup> sin <sup>γ</sup>sij j <sup>x</sup> � �, if a <sup>&</sup>lt; j j <sup>x</sup> <sup>&</sup>lt; <sup>b</sup>

<sup>A</sup>3e�γcladj j <sup>x</sup> , if xj j <sup>&</sup>gt; <sup>b</sup>

siEx,si � � and n<sup>2</sup>

sinh <sup>γ</sup>slot<sup>a</sup> � � sin <sup>γ</sup>sið Þ j j <sup>x</sup> � <sup>a</sup> � �, if a <sup>&</sup>lt; j j <sup>x</sup> <sup>&</sup>lt; <sup>b</sup>

slotEx,slot <sup>¼</sup> <sup>n</sup><sup>2</sup>

�γcladj j <sup>x</sup> , if xj j <sup>&</sup>gt; <sup>b</sup>

siEx,si � � and the

þ m∙π ¼ γsið Þ b � a (9)

(8)

(10)

fundamental TM mode is

192 Emerging Waveguide Technology

Exð Þ¼ x

nent of the electric displacement field n<sup>2</sup>

tan�<sup>1</sup> <sup>n</sup><sup>2</sup> si n2 clad

Exð Þ¼ x A

2.3. Simulation of slot waveguides

field expression

8 ><

>:

equation is a transcendental equation and is given by

γclad γsi

1 n2 slot

8

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

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

1 n2 si

þ 1 n2 slot

þ n2 si n2 slot

1 n2 clad

γslot γsi

γslot γsi

discontinuity and the high E-field confinement inside the slot region.

<sup>þ</sup> tan�<sup>1</sup> <sup>n</sup><sup>2</sup>

si n2 slot

cosh <sup>γ</sup>slot<sup>x</sup> � �, if xj j <sup>&</sup>lt; <sup>a</sup>

cosh <sup>γ</sup>slot<sup>a</sup> � � cos <sup>γ</sup>sið Þ j j <sup>x</sup> � <sup>a</sup> � �

cosh <sup>γ</sup>slot<sup>a</sup> � � cos <sup>γ</sup>sið Þ <sup>b</sup> � <sup>a</sup> � � �

sinh <sup>γ</sup>slot<sup>a</sup> � � sin <sup>γ</sup>sið Þ <sup>b</sup> � <sup>a</sup><sup>Þ</sup> � �<sup>e</sup>

where A is an arbitrary constant. As an example, Figure 4 shows the normalized Ex field distribution of a typical silicon slot waveguide. This figure gives an evidence of the large

In this section, a commercial full-vectorial finite element method (FEM) based mode solver from COMSOL Multiphysics is employed to simulate the slot waveguide structure with finite height.

!

Figure 4. Calculated Ex field (normalized) for a slot waveguide structure with infinite height. Parameters: λ ¼ 1550 nm, a ¼ 110 nm, b ¼ 220 nm, nslot ¼ nclad ¼ 1:44, nsi ¼ 3:48.

Here, the quasi-TE eigenmode presents the major E-field component along the x-direction [1]. Consequently, the quasi-TE eigenmode in the 3D slot waveguide structure is analogous to the TM eigenmode in the slab-based slot waveguide structure, which was studied in the previous section. In the following simulation study, the waveguide parameters, i.e., the silicon rail width wg and the slot width s, are variable whereas the height h is fixed to 220 nm (see Figure 5).

The wavelength is assumed to be λ ¼ 1550 nm because it is a typical telecommunication wavelength in silicon photonics. Triangular vector elements with a maximum and minimum

Figure 5. Cross-sectional view of an SOI strip waveguide (a) and an SOI slot waveguide (b). The cladding oxide is removed to hybridize the silicon waveguide with an EO polymer (EO modulator) or to bring the waveguide in contact with an analyte (biochemical sensor). The refractive indices are denoted as nclad, nsi and nbox for the caldding material, silicon and buried oxide substrate, respectively.

element size of 6 and 2 nm, respectively have been adopted for meshing the profile with over <sup>1</sup>:<sup>2</sup> � 104 mesh elements. The overall simulation area was 3 � <sup>3</sup> <sup>μ</sup>m. In order to yield the mode field distribution and effective refractive index, the refractive index distribution n xð Þ ; y need to be declared to calculate eigenvalues and eigenfunctions of the wave equation. Finally, we get the E-field intensity distribution for the quasi-TE and quasi-TM mode as shown in Figure 6. In the following, we will neglect the quasi-TM mode because it is over two orders of magnitude smaller than the quasi-TE mode.

The material properties for our simulations were taken from a Sellmeier fit of the optical data from Malitson ð Þ SiO<sup>2</sup> [16] and Salzberg and Villa ð Þ Si [17], which corresponds to having nSiO<sup>2</sup> ¼ nbox ¼ 1:444 and nsi ¼ 3:48 at λ ¼ 1550 nm. The refractive index of the surrounding material is in general variable because it can be gas, fluid or solid, depending on the application. For exemple, we chose nclad ¼ 1:7, which corresponds to a commercially available and reliable organic material named M3 (commercialized by GigOptix Inc.). M3 is successfully used for several slot waveguide based EO modulators like in [2, 18–20].

As result, the effective refractive index can be obtained from this simulation. Figure 7 shows the calculated effective refractive indices neff as a function of the slot width s and the rail width wg. From this figure, it can be seen that the effective refractive index becomes higher by increasing the rail width wg and by decreasing the slot width s. Therefore, such parameters have to be taken into account in order to design slot waveguide based mode coupler, ring resonators, or similar photonic components.

#### 2.4. SOI slot waveguide optimization

In order to design and improve the waveguide geometry for applications in the field of EO modulators and optical sensors, it is necessary to calculate characteristic values, which describe the optical field confinement and therefore the interaction of light with the surrounding material. One key figure of merit is the field confinement factor. In particular, it describes how well the guided modal field is confined in a certain region and is defined as the ratio of the time averaged energy flow through the domain of interest (Dint) to the time averaged energy flow through the

Figure 7. Calculated effective refractive indices neff of SOI slot waveguides as a function of the slot width s and the rail

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Here, E and H are the electric and magnetic field vectors, respectively, and ez is the unit vector in the z direction. There are four different cases in order to choose the domain of interest, as illustrated in Figure 8. In case of common strip waveguides for electro-optics, the domain of interest is usually equal to the core region (Dint ¼ Dcore) because high confinement inside the core leads to lower optical losses. In contrast to that, for bio-sensing applications, the region of the cladding region is considered to be the domain of interest (Dint ¼ Dclad), which is valid for strip and slot waveguides as well. The reason for this is that the main goal is to have a high light interaction with the surrounding material. Considering slot waveguides for electrooptics, the domain of interest is equal to the slot region (Dint ¼ Dslot). There is also a change in refractive index outside the slot since the electric field is also located outside. This contribution is, however, very small compared to that one inside the slot if we assume the linear EO effect (Pockels effect). The Pockels effect requires indeed a non-centro-symmetrical orientation of the EO polymers or non-centro-symmetric organic crystals. Because of that, the main part of the electric field, which gives a contribution to the refractive index change, is the x-component.

DintRe <sup>E</sup> � <sup>H</sup><sup>∗</sup> f gezdxdy

DtotRe <sup>E</sup> � <sup>H</sup><sup>∗</sup> f gezdxdy (11)

Γ ¼ ÐÐ

ÐÐ

total domain (Dtot)

width wg.

Figure 6. FEM simulation of the normalized E-field intensity j j Ex <sup>2</sup> <sup>þ</sup> Ey <sup>2</sup> <sup>þ</sup> j j Ez <sup>2</sup> for the first TE and first TM mode of SOI strip and slot waveguides. Parameters: λ ¼ 1550 nm, nbox ¼ 1:444, nsi ¼ 3:48.

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element size of 6 and 2 nm, respectively have been adopted for meshing the profile with over <sup>1</sup>:<sup>2</sup> � 104 mesh elements. The overall simulation area was 3 � <sup>3</sup> <sup>μ</sup>m. In order to yield the mode field distribution and effective refractive index, the refractive index distribution n xð Þ ; y need to be declared to calculate eigenvalues and eigenfunctions of the wave equation. Finally, we get the E-field intensity distribution for the quasi-TE and quasi-TM mode as shown in Figure 6. In the following, we will neglect the quasi-TM mode because it is over two orders of magnitude

The material properties for our simulations were taken from a Sellmeier fit of the optical data from Malitson ð Þ SiO<sup>2</sup> [16] and Salzberg and Villa ð Þ Si [17], which corresponds to having nSiO<sup>2</sup> ¼ nbox ¼ 1:444 and nsi ¼ 3:48 at λ ¼ 1550 nm. The refractive index of the surrounding material is in general variable because it can be gas, fluid or solid, depending on the application. For exemple, we chose nclad ¼ 1:7, which corresponds to a commercially available and reliable organic material named M3 (commercialized by GigOptix Inc.). M3 is successfully

As result, the effective refractive index can be obtained from this simulation. Figure 7 shows the calculated effective refractive indices neff as a function of the slot width s and the rail width wg. From this figure, it can be seen that the effective refractive index becomes higher by increasing the rail width wg and by decreasing the slot width s. Therefore, such parameters have to be taken into account in order to design slot waveguide based mode coupler, ring

In order to design and improve the waveguide geometry for applications in the field of EO modulators and optical sensors, it is necessary to calculate characteristic values, which describe the optical field confinement and therefore the interaction of light with the surrounding material.

> 

<sup>2</sup> <sup>þ</sup> j j Ez <sup>2</sup> for the first TE and first TM mode of SOI

used for several slot waveguide based EO modulators like in [2, 18–20].

smaller than the quasi-TE mode.

194 Emerging Waveguide Technology

resonators, or similar photonic components.

Figure 6. FEM simulation of the normalized E-field intensity j j Ex <sup>2</sup> <sup>þ</sup> Ey

strip and slot waveguides. Parameters: λ ¼ 1550 nm, nbox ¼ 1:444, nsi ¼ 3:48.

2.4. SOI slot waveguide optimization

Figure 7. Calculated effective refractive indices neff of SOI slot waveguides as a function of the slot width s and the rail width wg.

One key figure of merit is the field confinement factor. In particular, it describes how well the guided modal field is confined in a certain region and is defined as the ratio of the time averaged energy flow through the domain of interest (Dint) to the time averaged energy flow through the total domain (Dtot)

$$\Gamma = \frac{\iint\_{D\_{\rm int}} \text{Re}\{E \times H^\*\} e\_z d\mathbf{x} dy}{\iint\_{D\_{\rm int}} \text{Re}\{E \times H^\*\} e\_z d\mathbf{x} dy} \tag{11}$$

Here, E and H are the electric and magnetic field vectors, respectively, and ez is the unit vector in the z direction. There are four different cases in order to choose the domain of interest, as illustrated in Figure 8. In case of common strip waveguides for electro-optics, the domain of interest is usually equal to the core region (Dint ¼ Dcore) because high confinement inside the core leads to lower optical losses. In contrast to that, for bio-sensing applications, the region of the cladding region is considered to be the domain of interest (Dint ¼ Dclad), which is valid for strip and slot waveguides as well. The reason for this is that the main goal is to have a high light interaction with the surrounding material. Considering slot waveguides for electrooptics, the domain of interest is equal to the slot region (Dint ¼ Dslot). There is also a change in refractive index outside the slot since the electric field is also located outside. This contribution is, however, very small compared to that one inside the slot if we assume the linear EO effect (Pockels effect). The Pockels effect requires indeed a non-centro-symmetrical orientation of the EO polymers or non-centro-symmetric organic crystals. Because of that, the main part of the electric field, which gives a contribution to the refractive index change, is the x-component.

To validate our simulation approach and compare it with literature data, the field confinement factors of standard SOI strip waveguides are investigated additionally and depicted in Figure 10. The domains of interest are the silicon core and the cladding in this case. Note that the substrate is not included in the calculated domains and, therefore, the sum of the core and cladding field confinement factor is not equal to unity. As can be seen from Figure 10, there is a large confinement of about 0:76 in the silicon core region of a strip waveguide with a typical waveguide width of w ¼ 500 nm. These results are in good agreement with the literature [7]. However, to maximize the sensitivity of SOI slot waveguide based biochemical sensors, it is necessary to maximize the field confinement factor of the cladding (which will be referred to as Γclad) in order to increase the light-analyte interaction. From Figure 11, it can be seen that the confinement in the cladding region of a slot waveguide is increased by decreasing the rail width wg and increasing the slot width s. We chose this parameter range to be compatible with typical semiconductor production platforms. For wg ¼ 180 nm, the highest confinement in the cladding region is obtained in the parameter range of our simulation. For a slot waveguide with wg ¼ 180 nm and s ¼ 180 nm we obtain a field confinement factor of Γclad ¼ 0:69. This is an enhancement of about five times compared to a conventional strip waveguide with a typical waveguide width of w ¼ 500 nm. With that result, the high sensitivity of slot waveguide based label-free sensors as stated by Claes et al. can be explained [8]. However, due to the difficulty in functionalizing the interior of the slot, the sensitivity could be smaller than expected.

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Figure 10. Calculated field confinement factor Γstrip of conventional SOI strip waveguides for the core and cladding

region as a function of the waveguide width wg.

Figure 8. Domains of interest: core Dcore, cladding Dclad and slot Dslot regions are highlighted in green. Please note that the substrate is not included in the cladding region because it does not contribute to the electro-optical or sensing effect. Only the amount of light, which is interacting with the cladding material, i.e. in the cladding or in the slot region, is of interest.

The x-component is homogeneous inside the slot. Outside there is just a little projection of the x-component which contributes as shown in Figure 9. This figure shows a simulation of the optical and electrical field in case of a strip-loaded slot waveguide. The strip-load serves as the electrical contact. In the following, however, the field confinement factor will be determined for slot waveguides without strip-load; the results are also valid for strip-loaded slot waveguides since the difference is negligible.

Figure 9. (a) The normalized optical field distribution for the quasi-TE eigenmode of a strip-loaded slot waveguide structure and (b) normalized x-component of the electric field (Ex). The largest overlap between optical and electrical field is achieved inside the slot. Because of that, Pockels effect outside the slot region is negligible and therefore, the field confinement factor in the slot region Γslot should be taken into account in order to avoid an underestimation of the refractive index change Δn.

To validate our simulation approach and compare it with literature data, the field confinement factors of standard SOI strip waveguides are investigated additionally and depicted in Figure 10. The domains of interest are the silicon core and the cladding in this case. Note that the substrate is not included in the calculated domains and, therefore, the sum of the core and cladding field confinement factor is not equal to unity. As can be seen from Figure 10, there is a large confinement of about 0:76 in the silicon core region of a strip waveguide with a typical waveguide width of w ¼ 500 nm. These results are in good agreement with the literature [7].

However, to maximize the sensitivity of SOI slot waveguide based biochemical sensors, it is necessary to maximize the field confinement factor of the cladding (which will be referred to as Γclad) in order to increase the light-analyte interaction. From Figure 11, it can be seen that the confinement in the cladding region of a slot waveguide is increased by decreasing the rail width wg and increasing the slot width s. We chose this parameter range to be compatible with typical semiconductor production platforms. For wg ¼ 180 nm, the highest confinement in the cladding region is obtained in the parameter range of our simulation. For a slot waveguide with wg ¼ 180 nm and s ¼ 180 nm we obtain a field confinement factor of Γclad ¼ 0:69. This is an enhancement of about five times compared to a conventional strip waveguide with a typical waveguide width of w ¼ 500 nm. With that result, the high sensitivity of slot waveguide based label-free sensors as stated by Claes et al. can be explained [8]. However, due to the difficulty in functionalizing the interior of the slot, the sensitivity could be smaller than expected.

The x-component is homogeneous inside the slot. Outside there is just a little projection of the x-component which contributes as shown in Figure 9. This figure shows a simulation of the optical and electrical field in case of a strip-loaded slot waveguide. The strip-load serves as the electrical contact. In the following, however, the field confinement factor will be determined for slot waveguides without strip-load; the results are also valid for strip-loaded slot wave-

Figure 9. (a) The normalized optical field distribution for the quasi-TE eigenmode of a strip-loaded slot waveguide structure and (b) normalized x-component of the electric field (Ex). The largest overlap between optical and electrical field is achieved inside the slot. Because of that, Pockels effect outside the slot region is negligible and therefore, the field confinement factor in the slot region Γslot should be taken into account in order to avoid an underestimation of the

Figure 8. Domains of interest: core Dcore, cladding Dclad and slot Dslot regions are highlighted in green. Please note that the substrate is not included in the cladding region because it does not contribute to the electro-optical or sensing effect. Only the amount of light, which is interacting with the cladding material, i.e. in the cladding or in the slot region, is of interest.

guides since the difference is negligible.

196 Emerging Waveguide Technology

refractive index change Δn.

Figure 10. Calculated field confinement factor Γstrip of conventional SOI strip waveguides for the core and cladding region as a function of the waveguide width wg.

Figure 11. Calculated field confinement factor Γclad of an SOI slot waveguide for the cladding region in dependence on the slot width s and the rail width wg.

3. Fabrication of silicon-on-insulator slot waveguides

width s and the rail width wg.

reduce the slot width it is possible to use electron beam lithography.

hence, the resulting conductivity is about 11:3 Ω<sup>1</sup>

After the physical layout is developed and simulated, the optimized design need to be converted to a chip layout. Then the fabrication can be realized in an SOI pilot line. This section describes a typical process flow, which can be used with a 200 mm SOI wafer and 248 nm optical lithography or electron beam lithography. Commonly, SOI slot waveguides are fabricated by optical lithography, which enables slot width in the range of 120 nm to 200 nm. To

Figure 12. Calculated field confinement factor Γslot of an SOI slot waveguide for the slot region as a function of the slot

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The complete fabrication scheme is illustrated in Figure 13. The basis is an SOI wafer with a 220 nm silicon layer on top of a 2 μm buried oxide layer. In fact, the silicon layer is the actual waveguide layer. It can be doped in a certain area (Figure 13a–c) with boron in order to reduce electric conductivity. In this case, the boron doping concentration is about 1017 cm<sup>3</sup> and,

create thin slabs, which will be used as the electric connection from the electrodes to the slot waveguide (Figure 13d–e). This is followed by etching the slot down until the substrate (Figure 13f–h). It is essential to etch it completely because a connection between both silicon rails would lead to a current flow, which in turn induces heat. To further reduce the electric conductivity, another doping step with higher boron doping concentration needs to be performed, as illustrated in Figure 13i–k. This higher doping area is placed a few hundreds

cm1. Then, the waveguide is reduced to

In order to improve SOI slot waveguides for EO applications, it is necessary to find the highest confinement in the slot region. Figure 12 shows the calculated field confinement factors for the slot region Γslot as a function of the slot width s and the rail width wg as parameter. As can be seen from this figure, there is one maximum of the highest field confinement of Γslot ¼ 0:216 at a slot width of s ¼ 116 nm and a rail width of wg ¼ 200 nm. This is about three times smaller compared to an SOI strip waveguide (see Figure 10). However, in this case, it is more convenient to relate the field confinement factor Γslot to the area where the light is confined as figure of merit, ξslot ¼ Γslot=Aslot and ξclad ¼ Γclad=Aclad. In our case, Aslot is equal to the slot domain of the slot waveguide and Astrip is equal to the silicon core domain of the strip waveguide. For the optimized slot waveguide structure (s ¼ 116 nm, wg ¼ 200 nm), this figure of merit is about <sup>ξ</sup>slot <sup>¼</sup> <sup>7</sup>:<sup>83</sup> � 1012 <sup>m</sup>�<sup>1</sup> and for a typical strip waveguide (<sup>w</sup> <sup>¼</sup> 500 nm), it is about <sup>ξ</sup>slot <sup>¼</sup> <sup>6</sup>:<sup>82</sup> � 1012 <sup>m</sup>�1. Even <sup>ξ</sup>slot is 15% higher than <sup>ξ</sup>strip; one should keep in mind that a modulator with extremely high field confinement usually suffers very high optical loss. This off-loss can make any modulator useless. With the calculated field confinement factor, an underestimation of the required interaction length for an EO phase modulator is ensured, but such estimation might lead to a field confinement factor that looks prohibitive for using such modulators. The key benefit of slot waveguides is nevertheless the use of organic materials with EO coefficients that are more than one order of magnitude higher compared to semiconductors like GaAs or strained silicon.

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Figure 12. Calculated field confinement factor Γslot of an SOI slot waveguide for the slot region as a function of the slot width s and the rail width wg.

#### 3. Fabrication of silicon-on-insulator slot waveguides

In order to improve SOI slot waveguides for EO applications, it is necessary to find the highest confinement in the slot region. Figure 12 shows the calculated field confinement factors for the slot region Γslot as a function of the slot width s and the rail width wg as parameter. As can be seen from this figure, there is one maximum of the highest field confinement of Γslot ¼ 0:216 at a slot width of s ¼ 116 nm and a rail width of wg ¼ 200 nm. This is about three times smaller compared to an SOI strip waveguide (see Figure 10). However, in this case, it is more convenient to relate the field confinement factor Γslot to the area where the light is confined as figure of merit, ξslot ¼ Γslot=Aslot and ξclad ¼ Γclad=Aclad. In our case, Aslot is equal to the slot domain of the slot waveguide and Astrip is equal to the silicon core domain of the strip waveguide. For the optimized slot waveguide structure (s ¼ 116 nm, wg ¼ 200 nm), this figure of merit is about <sup>ξ</sup>slot <sup>¼</sup> <sup>7</sup>:<sup>83</sup> � 1012 <sup>m</sup>�<sup>1</sup> and for a typical strip waveguide (<sup>w</sup> <sup>¼</sup> 500 nm), it is about <sup>ξ</sup>slot <sup>¼</sup> <sup>6</sup>:<sup>82</sup> � 1012 <sup>m</sup>�1. Even <sup>ξ</sup>slot is 15% higher than <sup>ξ</sup>strip; one should keep in mind that a modulator with extremely high field confinement usually suffers very high optical loss. This off-loss can make any modulator useless. With the calculated field confinement factor, an underestimation of the required interaction length for an EO phase modulator is ensured, but such estimation might lead to a field confinement factor that looks prohibitive for using such modulators. The key benefit of slot waveguides is nevertheless the use of organic materials with EO coefficients that are more than one order of magnitude higher compared to semi-

Figure 11. Calculated field confinement factor Γclad of an SOI slot waveguide for the cladding region in dependence on the

conductors like GaAs or strained silicon.

slot width s and the rail width wg.

198 Emerging Waveguide Technology

After the physical layout is developed and simulated, the optimized design need to be converted to a chip layout. Then the fabrication can be realized in an SOI pilot line. This section describes a typical process flow, which can be used with a 200 mm SOI wafer and 248 nm optical lithography or electron beam lithography. Commonly, SOI slot waveguides are fabricated by optical lithography, which enables slot width in the range of 120 nm to 200 nm. To reduce the slot width it is possible to use electron beam lithography.

The complete fabrication scheme is illustrated in Figure 13. The basis is an SOI wafer with a 220 nm silicon layer on top of a 2 μm buried oxide layer. In fact, the silicon layer is the actual waveguide layer. It can be doped in a certain area (Figure 13a–c) with boron in order to reduce electric conductivity. In this case, the boron doping concentration is about 1017 cm<sup>3</sup> and, hence, the resulting conductivity is about 11:3 Ω<sup>1</sup> cm1. Then, the waveguide is reduced to create thin slabs, which will be used as the electric connection from the electrodes to the slot waveguide (Figure 13d–e). This is followed by etching the slot down until the substrate (Figure 13f–h). It is essential to etch it completely because a connection between both silicon rails would lead to a current flow, which in turn induces heat. To further reduce the electric conductivity, another doping step with higher boron doping concentration needs to be performed, as illustrated in Figure 13i–k. This higher doping area is placed a few hundreds

4. Electro-optic modulation in slot waveguides

Electro-optic modulators based on slot waveguides typically examine the linear EO effect, also known as Pockels effect, in organic materials such as organic crystals or polymers. In general, an EO polymer consists of a passive matrix of polymer chains that gives the material its mechanical, thermal, and chemical stability. This matrix is doped with active molecules that have a strong Pockels effect. Most EO devices based on organic EO materials use dipolar molecules embedded in or covalently attached to polymeric material lattices. A device-relevant EO activity requires a macroscopic non-centrosymmetric orientation, which is usually achieved by electric field poling of molecules endowed with their own lack of symmetry. This

Silicon-on-Insulator Slot Waveguides: Theory and Applications in Electro-Optics and Optical Sensing

leads to a linear EO effect, which is appropriated for high-frequency applications.

<sup>Δ</sup><sup>n</sup> ¼ � <sup>1</sup> 2 n3

represented by [9].

The linear EO effect in terms of the refractive index change Δn can be determined by

eoprE <sup>¼</sup> <sup>1</sup> neop

where neop represents the refractive index of the EO polymer, r is the linear EO coefficient, E is the electric field, and χð Þ<sup>2</sup> denotes the second-order susceptibility. In SOI slot waveguides, the electric field can be determined by E ¼ U=s, where U is the applied voltage. Comparing the coefficients in Eq. (12) gives the relation between linear EO coefficient and second-order susceptibility

> <sup>r</sup> ¼ � <sup>2</sup>χð Þ<sup>2</sup> n4 eop

The linear EO coefficient r describes the EO activity of a centrosymmetric material and can be

r∝ Nβ cos <sup>3</sup>

Here, N is the chromophore number density or the number of nonlinear molecules in the material contributing to the polarization, β represents the molecular first hyperpolarizability,

θ denotes the angle between molecular dipole axis and the electric field vector. Eq. (14)

3. Maximizing the average non-centrosymmetric order parameter cos <sup>3</sup>θ by inducing a

Such requirements are best met by EO polymers which are dipolar and exhibit a highly polarizable donor-π-acceptor (D-π-A) system. This D-π-A system can support a charge transfer between electron donating and electron withdrawing groups [10]. For electron donating

and cos <sup>3</sup>θ denotes the average non-centrosymmetric order parameter.

suggests strategies to increase the linear EO effect: 1. Increasing the chromophore number density N.

high molecular orientation.

2. Using chromophores with high first hyperpolarizability β.

χð Þ<sup>2</sup> E, (12)

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201

: (13)

θ (14)

Figure 13. Schematic representation of the fabrication flow of a silicon-on-insulator slot waveguide.

of nanometers away from the slot waveguide (for example, 800 nmÞ to avoid optical losses due to two photon absorption and free carrier absorption. The boron doping concentration is 1020 cm�<sup>3</sup> and, hence, the resulting conductivity is about 1177:86 Ω�<sup>1</sup> cm�1. Finally, the electric contacts are formed by depositing aluminum on the doped silicon (Figure 13i–n). A silicide intermediate layer can be used between the aluminum and doped silicon in order to reduce the resistivity.

However, only a few SOI slot waveguide-based ring resonators are published so far. The main reason for this fact is that slot-waveguide structures suffer from relatively high losses, mainly caused by sidewall roughness [24]. As a consequence, slot waveguide resonators like microring resonators have typically small optical quality factors (Q-factors). One possible approach to tackle this problem is to reduce propagation losses in the slot waveguides by atomic layer deposition of thin dielectric films, for example, amorphous titanium dioxide (TiO2) [25, 26]. This reduces, on the one hand, propagation losses but on the other hand, it is associated with additional and not economical production steps, which makes it rather unattractive from the commercial point of view. Another way to tackle this problem is to improve the coupling efficiency. For example, an improvement of 10 times mechanical, thermal the optical Q-factor was demonstrated recently [27].

### 4. Electro-optic modulation in slot waveguides

Electro-optic modulators based on slot waveguides typically examine the linear EO effect, also known as Pockels effect, in organic materials such as organic crystals or polymers. In general, an EO polymer consists of a passive matrix of polymer chains that gives the material its mechanical, thermal, and chemical stability. This matrix is doped with active molecules that have a strong Pockels effect. Most EO devices based on organic EO materials use dipolar molecules embedded in or covalently attached to polymeric material lattices. A device-relevant EO activity requires a macroscopic non-centrosymmetric orientation, which is usually achieved by electric field poling of molecules endowed with their own lack of symmetry. This leads to a linear EO effect, which is appropriated for high-frequency applications.

The linear EO effect in terms of the refractive index change Δn can be determined by

$$
\Delta n = -\frac{1}{2} n\_{\text{evp}}^3 r E = \frac{1}{n\_{\text{evp}}} \chi^{(2)} E,\tag{12}
$$

where neop represents the refractive index of the EO polymer, r is the linear EO coefficient, E is the electric field, and χð Þ<sup>2</sup> denotes the second-order susceptibility. In SOI slot waveguides, the electric field can be determined by E ¼ U=s, where U is the applied voltage. Comparing the coefficients in Eq. (12) gives the relation between linear EO coefficient and second-order susceptibility

$$r = -\frac{2\chi^{(2)}}{n\_{\rm opt}^4}.\tag{13}$$

The linear EO coefficient r describes the EO activity of a centrosymmetric material and can be represented by [9].

$$r \propto N\beta \langle \cos^3 \theta \rangle \tag{14}$$

Here, N is the chromophore number density or the number of nonlinear molecules in the material contributing to the polarization, β represents the molecular first hyperpolarizability, and cos <sup>3</sup>θ denotes the average non-centrosymmetric order parameter.

θ denotes the angle between molecular dipole axis and the electric field vector. Eq. (14) suggests strategies to increase the linear EO effect:

1. Increasing the chromophore number density N.

of nanometers away from the slot waveguide (for example, 800 nmÞ to avoid optical losses due to two photon absorption and free carrier absorption. The boron doping concentration is 1020 cm�<sup>3</sup>

Figure 13. Schematic representation of the fabrication flow of a silicon-on-insulator slot waveguide.

formed by depositing aluminum on the doped silicon (Figure 13i–n). A silicide intermediate layer can be used between the aluminum and doped silicon in order to reduce the resistivity.

However, only a few SOI slot waveguide-based ring resonators are published so far. The main reason for this fact is that slot-waveguide structures suffer from relatively high losses, mainly caused by sidewall roughness [24]. As a consequence, slot waveguide resonators like microring resonators have typically small optical quality factors (Q-factors). One possible approach to tackle this problem is to reduce propagation losses in the slot waveguides by atomic layer deposition of thin dielectric films, for example, amorphous titanium dioxide (TiO2) [25, 26]. This reduces, on the one hand, propagation losses but on the other hand, it is associated with additional and not economical production steps, which makes it rather unattractive from the commercial point of view. Another way to tackle this problem is to improve the coupling efficiency. For example, an improvement of 10 times mechanical, thermal the optical Q-factor

cm�1. Finally, the electric contacts are

and, hence, the resulting conductivity is about 1177:86 Ω�<sup>1</sup>

was demonstrated recently [27].

200 Emerging Waveguide Technology


Such requirements are best met by EO polymers which are dipolar and exhibit a highly polarizable donor-π-acceptor (D-π-A) system. This D-π-A system can support a charge transfer between electron donating and electron withdrawing groups [10]. For electron donating

Figure 14. D-π-a system. EO polymers are typically composed of electron donating (D) and electron accepting (A) side groups.

usually the following side groups are used: N CH ð Þ<sup>3</sup> <sup>2</sup>, OCH3, OH, while for electron accepting groups the following ones: NO, O2N, CHO, CN. The π-electron conjugated segment serves to transmit the charge, as illustrated in Figure 14.

In this work, we demonstrate EO modulation in an SOI slot waveguide using the azobenzene dye Disperse Red 1 doped in poly(methyl methacrylate). This guest-host polymer system is infiltrated into the slot waveguide by spin-coating. A ring resonator is employed to demonstrate intensity modulation. Here, the slot waveguide is partially introduced in the ring. The geometry values are the same as in our previous work in Ref. [11]. This ring resonator has been recently shown to operate efficiently as EO switch [12] and is fabricated in a SiGe BiCMOS pilot line on a 200 nm SOI wafer at the Institute of Innovative Microelectronics IHP in Frankfurt (Oder), Germany. This work, however, demonstrates intensity modulation by applying a function generator to the ground-signal-ground electrodes instead of a DC source. For this experiment, an external cavity laser operating the optical C-band is used as light source, which is polarized to obtain a quasi TE mode in the slot waveguide. A schematic of the complete experimental set-up is shown in Figure 15. For our demonstration, we applied a sine signal with a peak-to-peak voltage of 1 V at a frequency of 25 kHz. The oscilloscope traces in Figure 16 show the applied electrical

signal (blue) and the observed optical signal (green). The present experiment demonstrates the feasibility of a polymer filled SOI slot waveguide for EO modulation at low voltages. It is worthwhile to note that the applied voltage of 1 V is compatible with common CMOS-based

Figure 16. Oscilloscope traces of the electrical sine signal (blue) and the modulated optical signal (green). The peak-to-

Silicon-on-Insulator Slot Waveguides: Theory and Applications in Electro-Optics and Optical Sensing

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203

The working principle of photonic ring resonators for biochemical sensing is based on refractive index sensing, i.e., the measurement of the resonance wavelength shift Δλ due to a refractive index change Δns of the solution containing the analyte in specific concentration. This refractive index change is typically subjected to a change in concentration. In order to measure the resonance wavelengths, the light of a tunable laser is in-coupled to the photonic chip out-coupled via an optical fiber, and detected by a photodiode. According to the resonance conditions, only selected wavelengths can propagate in the ring and distinct resonance

In principle, the resonant wavelength λres of the device depends on the effective refractive index neff of the optical waveguide that, in turn, is determined by the refractive index of the analyte. Therefore, by detecting changes in the resonant wavelength Δλ of the ring, small changes in refractive index Δns of the solution can be determined. The wavelength shift can

drivers; making the present approach attractive from the commercial point of view.

peak voltage of the electrical signal is 1 V and the modulation frequency is 25 kHz.

5. Optical sensing in slot waveguides

peaks appear in the output spectrum.

be calculated by [13].

Figure 15. Schematic of the experimental set-up for EO modulation. An external cavity laser (ECL) is polarized and coupled to the SOI chip. An erbium doped fiber amplifier (EDFA) is used to amplify the outcoupled light. A 2 nm optical filter is applied to avoid signal noise from the EDFA and a variable attenuator is used to avoid damaging the InGaAs photodetector. Finally, the modulated signal is recorded with a digital sampling oscilloscope (DSO). The digital multimeter was used for wavelength adjustment.

Silicon-on-Insulator Slot Waveguides: Theory and Applications in Electro-Optics and Optical Sensing http://dx.doi.org/10.5772/intechopen.75539 203

Figure 16. Oscilloscope traces of the electrical sine signal (blue) and the modulated optical signal (green). The peak-topeak voltage of the electrical signal is 1 V and the modulation frequency is 25 kHz.

signal (blue) and the observed optical signal (green). The present experiment demonstrates the feasibility of a polymer filled SOI slot waveguide for EO modulation at low voltages. It is worthwhile to note that the applied voltage of 1 V is compatible with common CMOS-based drivers; making the present approach attractive from the commercial point of view.

#### 5. Optical sensing in slot waveguides

usually the following side groups are used: N CH ð Þ<sup>3</sup> <sup>2</sup>, OCH3, OH, while for electron accepting groups the following ones: NO, O2N, CHO, CN. The π-electron conjugated segment serves to

Figure 14. D-π-a system. EO polymers are typically composed of electron donating (D) and electron accepting (A) side

In this work, we demonstrate EO modulation in an SOI slot waveguide using the azobenzene dye Disperse Red 1 doped in poly(methyl methacrylate). This guest-host polymer system is infiltrated into the slot waveguide by spin-coating. A ring resonator is employed to demonstrate intensity modulation. Here, the slot waveguide is partially introduced in the ring. The geometry values are the same as in our previous work in Ref. [11]. This ring resonator has been recently shown to operate efficiently as EO switch [12] and is fabricated in a SiGe BiCMOS pilot line on a 200 nm SOI wafer at the Institute of Innovative Microelectronics IHP in Frankfurt (Oder), Germany. This work, however, demonstrates intensity modulation by applying a function generator to the ground-signal-ground electrodes instead of a DC source. For this experiment, an external cavity laser operating the optical C-band is used as light source, which is polarized to obtain a quasi TE mode in the slot waveguide. A schematic of the complete experimental set-up is shown in Figure 15. For our demonstration, we applied a sine signal with a peak-to-peak voltage of 1 V at a frequency of 25 kHz. The oscilloscope traces in Figure 16 show the applied electrical

Figure 15. Schematic of the experimental set-up for EO modulation. An external cavity laser (ECL) is polarized and coupled to the SOI chip. An erbium doped fiber amplifier (EDFA) is used to amplify the outcoupled light. A 2 nm optical filter is applied to avoid signal noise from the EDFA and a variable attenuator is used to avoid damaging the InGaAs photodetector. Finally, the modulated signal is recorded with a digital sampling oscilloscope (DSO). The digital

transmit the charge, as illustrated in Figure 14.

groups.

202 Emerging Waveguide Technology

multimeter was used for wavelength adjustment.

The working principle of photonic ring resonators for biochemical sensing is based on refractive index sensing, i.e., the measurement of the resonance wavelength shift Δλ due to a refractive index change Δns of the solution containing the analyte in specific concentration. This refractive index change is typically subjected to a change in concentration. In order to measure the resonance wavelengths, the light of a tunable laser is in-coupled to the photonic chip out-coupled via an optical fiber, and detected by a photodiode. According to the resonance conditions, only selected wavelengths can propagate in the ring and distinct resonance peaks appear in the output spectrum.

In principle, the resonant wavelength λres of the device depends on the effective refractive index neff of the optical waveguide that, in turn, is determined by the refractive index of the analyte. Therefore, by detecting changes in the resonant wavelength Δλ of the ring, small changes in refractive index Δns of the solution can be determined. The wavelength shift can be calculated by [13].

$$
\Delta\lambda = \frac{\lambda\_{\rm res}}{n\_{\rm g}} \Delta n\_{\rm s} \Gamma\_{\rm clad} \tag{15}
$$

chloride, and ethylene glycol, has been demonstrated using homogenous sensing [14]. How-

Silicon-on-Insulator Slot Waveguides: Theory and Applications in Electro-Optics and Optical Sensing

In contrast, surface sensing signals originate from analytes in the close vicinity of the ring resonator surface [14]. In this case, the surface of the silicon waveguide is modified with an antibody. Only specific analytes can be attached to the antibody, i.e., this analyte-antibody binding leads to a specific measurement. Once the analyte-antibody binding takes place, the residuals can be removed by drying or flushing to enhance the specific measurement. When the antigen, for example, proteins or DNA, interacts with the antibody, the optical wave is influenced in its propagation, the resonance condition changes, and the resonance peak shifts by Δλ. The magnitude of the wavelength shift Δλ provides information on the quantity of the adsorbed analyte, where the detection limit of the sensor is defined by the minimum resolvable wavelength shift. Moreover, by measuring the peak wavelength shift during the binding

However, the main advantage of SOI slot waveguides is the fact that more light can interact with the analyte. A possible explanation for this is the large field confinement factor in the

> Swg <sup>¼</sup> <sup>δ</sup>ng δns

The waveguide sensitivity, however, does not take into account the resonant structure of widely used ring resonators. Therefore, a second sensitivity is defined as the ring resonator sensitivity

> Srr <sup>¼</sup> δλres δns

SOI slot waveguide based ring resonators have been demonstrated to achieve more than three times larger ring resonator sensitivities compared to conventional strip waveguide based ring resonators [8]. Notwithstanding this, slot waveguide based ring resonators have not been yet translated into a large overall sensitivity since the optical losses increase the detection limit described by the full width at half maximum (FWHM). At large FWHM the resonance peak shift may not be resolved. To take both into account (ring resonator sensitivity and the

FOM <sup>¼</sup> Srr

However, the experimental demonstration of surface sensing is beyond the scope of this work, as it needs sophisticated surface modification techniques. Instead, this work focuses on homogenous sensing and demonstrates an exemplary detection of 2-propanol to prove the feasibility of the novel ring resonator concept. This demonstration has the aim of verifying that

We will now turn to the experimental demonstration of homogeneous sensing. To validate our theoretical investigations we fabricated a hybrid-waveguide ring resonator with an SiGe

: (16)

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205

: (17)

FWHM : (18)

cladding region [15]. This fact leads to an enhanced waveguide sensitivity defined as

ever, homogeneous sensing usually lacks detection specificity.

process, the binding kinetic can be observed.

FWHM), we can define a figure of merit FOM given by

the concept has practical potential.

given by

where Γclad is the field confinement factor in the cladding which has to satisfy the condition Δneff ¼ Γcladns.

Generally, there are two types of sensing mechanisms, namely homogenous sensing and surface sensing, that a ring resonator can perform. The difference is the origin of sensing signal as shown in Figure 17. Homogeneous sensing signals results from the refractive index change induced by the presence of the analytes in the whole region of the evanescent field, which leads to a non-specific measurement. Detection of chemicals, for example, 2-propanol, sodium

Figure 17. Schematic of the surface sensing principle. The cross section shows that the receptor molecules (antibody) are located on top of slot waveguide surface. The resonance peak is shifted due to a specific binding event. From this data the quantity of the adsorbed analyte is obtained.

chloride, and ethylene glycol, has been demonstrated using homogenous sensing [14]. However, homogeneous sensing usually lacks detection specificity.

<sup>Δ</sup><sup>λ</sup> <sup>¼</sup> <sup>λ</sup>res ng

Δneff ¼ Γcladns.

204 Emerging Waveguide Technology

where Γclad is the field confinement factor in the cladding which has to satisfy the condition

Generally, there are two types of sensing mechanisms, namely homogenous sensing and surface sensing, that a ring resonator can perform. The difference is the origin of sensing signal as shown in Figure 17. Homogeneous sensing signals results from the refractive index change induced by the presence of the analytes in the whole region of the evanescent field, which leads to a non-specific measurement. Detection of chemicals, for example, 2-propanol, sodium

Figure 17. Schematic of the surface sensing principle. The cross section shows that the receptor molecules (antibody) are located on top of slot waveguide surface. The resonance peak is shifted due to a specific binding event. From this data the

quantity of the adsorbed analyte is obtained.

ΔnsΓclad, (15)

In contrast, surface sensing signals originate from analytes in the close vicinity of the ring resonator surface [14]. In this case, the surface of the silicon waveguide is modified with an antibody. Only specific analytes can be attached to the antibody, i.e., this analyte-antibody binding leads to a specific measurement. Once the analyte-antibody binding takes place, the residuals can be removed by drying or flushing to enhance the specific measurement. When the antigen, for example, proteins or DNA, interacts with the antibody, the optical wave is influenced in its propagation, the resonance condition changes, and the resonance peak shifts by Δλ. The magnitude of the wavelength shift Δλ provides information on the quantity of the adsorbed analyte, where the detection limit of the sensor is defined by the minimum resolvable wavelength shift. Moreover, by measuring the peak wavelength shift during the binding process, the binding kinetic can be observed.

However, the main advantage of SOI slot waveguides is the fact that more light can interact with the analyte. A possible explanation for this is the large field confinement factor in the cladding region [15]. This fact leads to an enhanced waveguide sensitivity defined as

$$\mathcal{S}\_{w\emptyset} = \frac{\delta n\_{\emptyset}}{\delta n\_{s}}.\tag{16}$$

The waveguide sensitivity, however, does not take into account the resonant structure of widely used ring resonators. Therefore, a second sensitivity is defined as the ring resonator sensitivity given by

$$S\_{rr} = \frac{\delta \lambda\_{\rm res}}{\delta n\_s}.\tag{17}$$

SOI slot waveguide based ring resonators have been demonstrated to achieve more than three times larger ring resonator sensitivities compared to conventional strip waveguide based ring resonators [8]. Notwithstanding this, slot waveguide based ring resonators have not been yet translated into a large overall sensitivity since the optical losses increase the detection limit described by the full width at half maximum (FWHM). At large FWHM the resonance peak shift may not be resolved. To take both into account (ring resonator sensitivity and the FWHM), we can define a figure of merit FOM given by

$$FOM = \frac{S\_{rr}}{FWHM}.\tag{18}$$

However, the experimental demonstration of surface sensing is beyond the scope of this work, as it needs sophisticated surface modification techniques. Instead, this work focuses on homogenous sensing and demonstrates an exemplary detection of 2-propanol to prove the feasibility of the novel ring resonator concept. This demonstration has the aim of verifying that the concept has practical potential.

We will now turn to the experimental demonstration of homogeneous sensing. To validate our theoretical investigations we fabricated a hybrid-waveguide ring resonator with an SiGe

finite element method based mode solver. We have determined field confinement factors for the slot and cladding region for applications as EO modulator and optical sensor, respectively. According to the present simulation study, SOI slot waveguides provide about five times higher field confinement in the cladding region with respect to conventional SOI strip waveguides. These results can be used for design optimizations in order to achieve optimum SOI

Silicon-on-Insulator Slot Waveguides: Theory and Applications in Electro-Optics and Optical Sensing

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The present study was designed to determine the optical and electro-optical characteristics of SOI slot waveguide-based devices. The presented hybrid-waveguide ring resonator consisting of a slot waveguide implemented in a strip waveguide-based ring resonator has a set of characteristics, which makes it an excellent and unique candidate for low power switches and modulators in the field of optical communication systems. The device was fabricated using CMOS fabrication processes, which enables high integration density and scalable, large-volume manufacturing. One of the more significant findings to emerge from this study is the significantly improved EO response compared to common ring-based modulators making use of the plasma dispersion effect, giving rise to the perspective of sub-femtojoule EO switching. In addition, we have performed intensity modulation at 25 kHz. So far, these characteristics are not combined in any other ring resonator to our knowledge. Moreover, there is still significant room for enhancing the device performance in terms of EO response, for example, by taking advantage of the continuously improving EO polymers. Taken together, the presented results suggest that the slot waveguide-based ring resonators are promising candidates for future EO switches, modulators and tunable filters. Future work should investigate the poling procedure to achieve a larger linear EO effect and therefore, higher modulation frequencies using the same EO polymer. Another limitation is the utilized EO polymer. Using advanced EO polymers indicate that the current results could be further improved. Notwithstanding these limitations, the device performance observed in this study indicate the great promises of using EO polymers which are essential for various hybrid photonic devices particularly for low-power applications. The demonstrated ring resonator opens a new route toward promising applications in the realm

A further aspect investigated in this work was the use of SOI slot waveguides for optical sensing. As proof of principle, homogeneous sensing of different concentrations of 2-propanol in de-ionized water was performed. This study set out to determine the overall sensitivity, taking into account the optical losses and the ring resonator sensitivity. It turns out that the slot waveguide-based ring resonator has a higher overall sensitivity than most traditional ring resonator sensors based on strip waveguides. Thus, it appears to be promising for a wide

slot waveguide dimensions for sensing applications.

of polymer-based non-linear photonics.

Author details

Patrick Steglich

range of sensing applications, including biochemical sensors.

Address all correspondence to: patrick.steglich@th-wildau.de Technical University of Applied Sciences Wildau, Germany

Figure 18. Measured resonance wavelength shift versus refractive index of the test liquid (100, 75, and 50% 2-propanol in deionized water). The slope of the linear fit gives the resonator sensitivity of Srr ¼ 106:29 nm=RIU [21].

BiCMOS pilot line. The slot width of the fabricated device is s ¼ 150 nm and the rail width is wg ¼ 180 nm. A comprehensive study and all geometric parameters of the hybrid-waveguide ring resonator can be found in Ref. [21].

As preliminary test, we measured 100, 75 and 50% concentrations of 2-propanol in de-ionized water. The test liquids are directly dropped onto the surface of the sensor. The refractive indices of the liquids were independently measured by means of a refractometer (Kern ORT 1RS) obtaining 1.3766, 1.3656, and 1.3546 for 100, 75, and 50%, respectively, in good agreement with literature values [22, 23]. Figure 18 shows the resonance wavelength shift Δλ versus refractive index ns of the liquid solutions [21]. Each measurement is carried out five times to prove reproducibility. Regression analysis was used to predict the ring resonator sensitivity. A ring resonator sensitivity of Srr ¼ 106:29 nm=RIU and a FOM of 1337 were observed. For comparison, a strip waveguidebased ring resonator has typically a ring resonator sensitivity of about 70 nm=RIU and a FOM of 903. This result demonstrates the feasibility of SOI slot waveguides for biochemical sensing and the advantage in terms of sensitivity with respect to SOI strip waveguides.

#### 6. Summary

In conclusion, this work has theoretically and experimentally studied aspects of SOI slot waveguides. Simulation and analysis of SOI slot waveguides have been carried out using a finite element method based mode solver. We have determined field confinement factors for the slot and cladding region for applications as EO modulator and optical sensor, respectively. According to the present simulation study, SOI slot waveguides provide about five times higher field confinement in the cladding region with respect to conventional SOI strip waveguides. These results can be used for design optimizations in order to achieve optimum SOI slot waveguide dimensions for sensing applications.

The present study was designed to determine the optical and electro-optical characteristics of SOI slot waveguide-based devices. The presented hybrid-waveguide ring resonator consisting of a slot waveguide implemented in a strip waveguide-based ring resonator has a set of characteristics, which makes it an excellent and unique candidate for low power switches and modulators in the field of optical communication systems. The device was fabricated using CMOS fabrication processes, which enables high integration density and scalable, large-volume manufacturing. One of the more significant findings to emerge from this study is the significantly improved EO response compared to common ring-based modulators making use of the plasma dispersion effect, giving rise to the perspective of sub-femtojoule EO switching. In addition, we have performed intensity modulation at 25 kHz. So far, these characteristics are not combined in any other ring resonator to our knowledge. Moreover, there is still significant room for enhancing the device performance in terms of EO response, for example, by taking advantage of the continuously improving EO polymers. Taken together, the presented results suggest that the slot waveguide-based ring resonators are promising candidates for future EO switches, modulators and tunable filters. Future work should investigate the poling procedure to achieve a larger linear EO effect and therefore, higher modulation frequencies using the same EO polymer. Another limitation is the utilized EO polymer. Using advanced EO polymers indicate that the current results could be further improved. Notwithstanding these limitations, the device performance observed in this study indicate the great promises of using EO polymers which are essential for various hybrid photonic devices particularly for low-power applications. The demonstrated ring resonator opens a new route toward promising applications in the realm of polymer-based non-linear photonics.

A further aspect investigated in this work was the use of SOI slot waveguides for optical sensing. As proof of principle, homogeneous sensing of different concentrations of 2-propanol in de-ionized water was performed. This study set out to determine the overall sensitivity, taking into account the optical losses and the ring resonator sensitivity. It turns out that the slot waveguide-based ring resonator has a higher overall sensitivity than most traditional ring resonator sensors based on strip waveguides. Thus, it appears to be promising for a wide range of sensing applications, including biochemical sensors.

### Author details

BiCMOS pilot line. The slot width of the fabricated device is s ¼ 150 nm and the rail width is wg ¼ 180 nm. A comprehensive study and all geometric parameters of the hybrid-waveguide

Figure 18. Measured resonance wavelength shift versus refractive index of the test liquid (100, 75, and 50% 2-propanol in

deionized water). The slope of the linear fit gives the resonator sensitivity of Srr ¼ 106:29 nm=RIU [21].

As preliminary test, we measured 100, 75 and 50% concentrations of 2-propanol in de-ionized water. The test liquids are directly dropped onto the surface of the sensor. The refractive indices of the liquids were independently measured by means of a refractometer (Kern ORT 1RS) obtaining 1.3766, 1.3656, and 1.3546 for 100, 75, and 50%, respectively, in good agreement with literature values [22, 23]. Figure 18 shows the resonance wavelength shift Δλ versus refractive index ns of the liquid solutions [21]. Each measurement is carried out five times to prove reproducibility. Regression analysis was used to predict the ring resonator sensitivity. A ring resonator sensitivity of Srr ¼ 106:29 nm=RIU and a FOM of 1337 were observed. For comparison, a strip waveguidebased ring resonator has typically a ring resonator sensitivity of about 70 nm=RIU and a FOM of 903. This result demonstrates the feasibility of SOI slot waveguides for biochemical sensing and

In conclusion, this work has theoretically and experimentally studied aspects of SOI slot waveguides. Simulation and analysis of SOI slot waveguides have been carried out using a

the advantage in terms of sensitivity with respect to SOI strip waveguides.

ring resonator can be found in Ref. [21].

206 Emerging Waveguide Technology

6. Summary

Patrick Steglich

Address all correspondence to: patrick.steglich@th-wildau.de

Technical University of Applied Sciences Wildau, Germany
