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Analytical predictions for nonlinear optical processes in silicon slot waveguides

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Abstract

Analytical expressions derived for dispersion relations and fields are used to obtain analytical predictions for the performance of modulators fabricated in second-order \((\upchi ^{2})\) and third-order \((\upchi ^{3})\) nonlinear materials filled into waveguide slot of silicon-on-insulator waveguides. Analytical expressions are used to model the electron-optical modulation and Kerr effect on slot waveguides for the theoretical modeling. The results obtained for slot waveguides with second-order nonlinearities are compared with those obtained under the planar-waveguide approximations reported as in earlier work. Also, exhaustive solutions are obtained for third-order nonlinear slot waveguides. This work employs long standing and reliable mathematical methods to develop the computational modal of photonics integrated devices. Moreover, the present work allows one a new perspective on design of various optical devices based on low-index sub-wavelength nonlinear slot waveguides.

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Acknowledgements

Prof. Vishnu Priye, who was exchange scholar at University of Colorado, Boulder, USA (June 1, 2013, to August 31, 2013) acknowledges financial support from Indian Institute of Technology (Indian School Of Mines), Dhanbad, University of Colorado, Boulder, and Lightwave Logic, USA.

Author information

Authors and Affiliations

  1. Photonics R&D Laboratory, Department of Electronics Engineering, Indian Institute of Technology (Indian School of Mines), Dhanbad, India

    Vishnu Priye & Nishit Malviya

  2. Guided Wave Optics Lab (GWOL), University of Colorado, Boulder, USA

    Alan Mickelson

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  1. Vishnu Priye
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  2. Nishit Malviya
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  3. Alan Mickelson
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Corresponding author

Correspondence to Nishit Malviya.

Appendices

Appendix I

1.1 Terms of second-order nonlinear propagation constant

We provide here the expressions for the constants that appear in Eq. (16):

$$\begin{aligned} R_1= & {} \left[ {-\sin \left( {\kappa _f \left( {b-a} \right) } \right) +\left( {\frac{n_f^2 }{n_c^2 }\frac{\gamma _c }{\kappa _f }} \right) \cos \left( {\kappa _f \left( {b-a} \right) } \right) } \right] ,\\ R_2= & {} \left( {\frac{n_f^2 }{\kappa _f }} \right) \left[ -\cos \left( {\kappa _f \left( {b-a} \right) } \right) \right. \\&\left. +\left( {\frac{n_f^2 }{n_c^2 }\frac{\gamma _c }{\kappa _f }} \right) \sin \left( {\kappa _f \left( {b-a} \right) } \right) \right] ,\\ R_3= & {} \left( {\frac{n_f^2 }{n_c^2 }\frac{\gamma _c }{\kappa _f }} \right) ; \quad R_4 =\left( {\frac{n_f^2 }{\kappa _f }} \right) ;\\ \alpha _1= & {} -\,a\sinh \left( {\gamma _s a} \right) \left( {\frac{1}{2\kappa _f }\left( {\frac{n_f^2 }{n_c^2 }\frac{\gamma _s }{\kappa _f }} \right) +\frac{1}{2\gamma _s }} \right) ,\\ \alpha _2= & {} \left( {\frac{a}{2n_f^2 }\cosh \left( {\gamma _s a} \right) } \right) -\left( {\frac{1}{2n_f^2 \kappa _f }\left( {\frac{n_f^2 }{n_c^2 }\frac{\gamma _c }{\kappa _f }} \right) \sinh \left( {\gamma _s a} \right) } \right) \\&-\left( {\frac{1}{2n_{\mathrm{s}0}^2 \gamma _s }\left( {\sinh \left( {\gamma _s a} \right) +\gamma _s a\cosh \left( {\gamma _s a} \right) } \right) } \right) ,\\ \alpha _3= & {} \left( {\left( {\frac{-\,b}{2\kappa _f }} \right) \sin \left( {\kappa _f \left( {b-a} \right) } \right) \cosh \left( {\gamma _s a} \right) } \right) \\&+\left( {\left( {\frac{b}{2\kappa _f }} \right) \left( {\frac{n_f^2 }{n_c^2 }\frac{\gamma _c }{\kappa _f }} \right) \cos \left( {\kappa _f \left( {b-a} \right) } \right) \sinh \left( {\gamma _s a} \right) }\right) \\&+\left( {\frac{-1}{2\gamma _c }} \right) \left( {\cosh \left( {\gamma _s a} \right) \cos \left( {\kappa _f \left( {b-a} \right) } \right) } \right. \\&+\left. {\left( {\frac{n_f^2 }{n_c^2 }\frac{\gamma _c }{\kappa _f }} \right) \sinh \left( {\gamma _s a} \right) \sin \left( {\kappa _f \left( {b-a} \right) } \right) } \right) , \end{aligned}$$
$$\begin{aligned} \alpha _4= & {} \left( {\frac{-1}{2n_f^2 \kappa _f }} \right) \left[ \sin \left( {\kappa _f \left( {b-a} \right) } \right) \right. \\&\left. +\,\kappa _f b\cos \left( {\kappa _f \left( {b-a} \right) } \right) \right] \cosh \left( {\gamma _s a} \right) \\&+\left( {\frac{1}{2n_f^2 \kappa _f }} \right) \left[ {\cos \left( {\kappa _f \left( {b-a} \right) } \right) -\kappa _f b\sin \left( {\kappa _f \left( {b-a} \right) } \right) } \right] \\&\times \,\left( {\frac{n_f^2 }{n_{\mathrm{s}0}^2 }\frac{\gamma _c }{\kappa _f }} \right) \sinh \left( {\gamma _s a} \right) \\&+\left( {\frac{\gamma _c b-1}{2n_c^2 \gamma _c }} \right) \left[ \cosh \left( {\gamma _s a} \right) \cos \kappa _f \left( {b-a} \right) \right. \\&\left. +\left( {\frac{n_f^2 }{n_{\mathrm{s}0}^2 }\frac{\gamma _s }{\kappa _f }} \right) \sinh \left( {\gamma _s a} \right) \sin \left( {\kappa _f \left( {b-a} \right) } \right) \right] ,\\ \theta _1= & {} \frac{a\sinh \left( {\gamma _s a} \right) }{2\gamma { }_s}k_0^2 \left( {2n_{\mathrm{s}0} n_1 } \right) ,\\ \theta _2= & {} \left( {\frac{1}{2n_{\mathrm{s}0}^2 \gamma _s }\left( {\sinh \left( {\gamma _s a} \right) +\gamma _s a\cosh \left( {\gamma _s a} \right) } \right) } \right) k_0^2 \left( {2n_{\mathrm{s}0} n_1 } \right) \\&+\left( {\frac{1}{n_{\mathrm{s}0}^4 }\left( {2n_{\mathrm{s}0} n_1 } \right) \cosh \left( {\gamma _s a} \right) } \right) . \end{aligned}$$

Appendix II

1.1 Third-order nonlinear propagation constant terms

We provide here the expressions for the constants that appear in Eq. (23):

$$\begin{aligned} T_1= & {} -2\beta _0 a\sinh \left( {\gamma _s a} \right) \left[ {\frac{1}{2\gamma _s }+\frac{1}{2\kappa _f }\left( {\frac{n_f^2 }{n_{\mathrm{s}0}^2 }\frac{\gamma _s }{\kappa _f }} \right) } \right] ,\\ T_2= & {} \left( {-\frac{2\beta _0 }{2\gamma _s n_{\mathrm{s}0}^2 }} \right) \left( {\sinh \left( {\gamma _s a} \right) +\gamma _s a\cosh \left( {\gamma _s a} \right) } \right) \\&-\left( {\frac{2\beta _0 }{2\kappa _f n_f^2 }} \right) \left( {\frac{n_f^2 }{n_{\mathrm{s}0}^2 }\frac{\gamma _s }{\kappa _f }} \right) \sinh \left( {\gamma _s a} \right) \\&+\left( {\frac{\beta _0 a}{n_f^2 }} \right) \cosh \left( {\gamma _s a} \right) ,\\ T_3= & {} \left( {-\frac{2\beta _0 b}{2\kappa _f }} \right) \sin \left( {\kappa _f \left( {b-a} \right) } \right) \cosh \left( {\gamma _s a} \right) \\&+\left( {\frac{2\beta _0 b}{2\kappa _f }} \right) \cos \left( {\kappa _f \left( {b-a} \right) } \right) \sinh \left( {\gamma _s a} \right) \left( {\frac{n_f^2 }{n_{\mathrm{s}0}^2 }\frac{\gamma _s }{\kappa _f }} \right) \\&-\left( {\frac{2\beta _0 b}{2\gamma _c }} \right) \left[ \cos \left( {\gamma _s } \right) \cos \left( {\kappa _f \left( {b-a} \right) } \right) \right. \\&\left. {+\frac{n_f^2 }{n_f^2 }\frac{\gamma _s }{\kappa _f }\sinh \left( {\gamma _s a} \right) \sin \left( {\kappa _f \left( {b-a} \right) } \right) } \right] ,\\ T_4= & {} \left( {\frac{\beta _0 }{\kappa _f n_f^2 }} \right) \left[ \left( \cos \left( {\kappa _f \left( {b-a} \right) } \right) \right. \right. \\&\left. \left. -\,\kappa _f b\sin \left( {\kappa _f \left( {b-a} \right) } \right) \right) \left( {\frac{n_f^2 }{n_{\mathrm{s}0}^2 }\frac{\gamma _s }{\kappa _f }} \right) \sinh \left( {\gamma _s a} \right) \right] \\&-\left( {\frac{\beta _0 }{\kappa _f n_f^2 }} \right) \left( \sin \left( {\kappa _f \left( {b-a} \right) } \right) \right. \\&\left. +\,\kappa _f b\cos \left( {\kappa _f \left( {b-a} \right) } \right) \right) \cosh \left( {\gamma _s a} \right) \\&+\left( {\frac{\beta _0 }{\gamma _c n_c^2 }} \right) \left( {\gamma _c b-1}\right) \left[ \cos \left( {\gamma _s } \right) \cos \left( {\kappa _f \left( {b-a} \right) } \right) \right. \\&\left. {+\,\frac{n_f^2 }{n_f^2 }\frac{\gamma _s }{\kappa _f }\sinh \left( {\gamma _s a} \right) \sin \left( {\kappa _f \left( {b-a} \right) } \right) } \right] , \end{aligned}$$
$$\begin{aligned} Q_1= & {} \left( {\frac{a}{8\gamma _s}} \right) \frac{k_0 \alpha \left| {A_0 } \right| ^{2}}{\left( {\omega \varepsilon _0 n_{\mathrm{s}0}^2 } \right) ^{2}}\left( {3\beta _0^2 +\gamma _s } \right) \sinh \left( {\gamma _s a} \right) \\&+\left( {\frac{1}{32\gamma _s }} \right) \frac{k_0 \alpha \left| {A_0 } \right| ^{2}}{\left( {\omega \varepsilon _0 n_{\mathrm{s}0}^2 } \right) ^{2}}\left( {\beta _0^2 +\gamma _s^2 } \right) \cosh \left( {3\gamma _s a} \right) ,\\ Q_2= & {} \left( {\frac{1}{8\gamma _s n_{\mathrm{s}0}^2 }} \right) \frac{k_0 \alpha \left| {A_0 } \right| ^{2}}{\left( {\omega \varepsilon _0 n_{\mathrm{s}0}^2 } \right) ^{2}}\left( {3\beta _0^2 +\gamma _s^2 }\right) \\&\times \left( {\sinh \left( {\gamma _s a} \right) +\gamma _s a\cosh \left( {\gamma _s a} \right) } \right) \\&+\left( {\frac{\gamma _s }{k_0 n_{\mathrm{s}0}^4 }} \right) \frac{k_0 \alpha \left| {A_0 } \right| ^{2}}{\left( {\omega \varepsilon _0 n_{\mathrm{s}0}^2 } \right) ^{2}}\\&\times \left( {\beta _0^2 \cosh ^{2}\left( {\gamma _s a} \right) \sinh \left( {\gamma _s a} \right) +\gamma _s^2 \sinh ^{3}\left( {\gamma _s a} \right) } \right) \\&+\left( {\frac{3}{32\gamma _s n_{\mathrm{s}0}^2 }} \right) \frac{k_0 \alpha \left| {A_0 } \right| ^{2}}{\left( {\omega \varepsilon _0 n_{\mathrm{s}0}^2} \right) ^{2}}\left( {\beta _0^2 +\gamma _s^2 } \right) \sinh \left( {3\gamma _s a} \right) ,\\ P_1= & {} -\frac{\kappa _f }{n_f^2 }\sin \left( {\kappa _f \left( {b-a} \right) } \right) +\frac{\gamma _c }{n_c^2 }\cos \left( {\kappa _f \left( {b-a} \right) } \right) ,\\ P_2= & {} -\frac{\kappa _f }{n_f^2 }\cos \left( {\kappa _f \left( {b-a} \right) } \right) +\frac{\gamma _c }{n_c^2 }\sin \left( {\kappa _f \left( {b-a} \right) } \right) . \end{aligned}$$

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Priye, V., Malviya, N. & Mickelson, A. Analytical predictions for nonlinear optical processes in silicon slot waveguides. J Comput Electron 17, 857–865 (2018). https://doi.org/10.1007/s10825-018-1150-8

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