# Numerical and Analytical Modeling of Guided Modes of a Planar Gradient Waveguide

• Edik Ayrjan
• Migran Gevorkyan
• Dmitry Kulyabov
• Konstantin Lovetskiy
• Nikolai Nikolaev
• Anton Sevastianov
• Leonid Sevastianov
• Eugeny Laneev
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 678)

## Abstract

The mathematical model of light propagation in a planar gradient optical waveguide consists of the Maxwell’s equations supplemented by the matter equations and boundary conditions. In the coordinates adapted to the waveguide geometry, the Maxwell’s equations are separated into two independent sets for the TE and TM polarizations. For each there are three types of waveguide modes in a regular planar optical waveguide: guided modes, substrate radiation modes, and cover radiation modes. We implemented in our work the numerical-analytical calculation of typical representatives of all the classes of waveguide modes.

In this paper we consider the case of a linear profile of planar gradient waveguide, which allows for the most complete analytical description of the solution for the electromagnetic field of the waveguide modes. Namely, in each layer we are looking for a solution by expansion in the fundamental system of solutions of the reduced equations for the particular polarizations and subsequent matching them at the boundaries of the waveguide layer.

The problem on eigenvalues (discrete spectrum) and eigenvectors is solved in the way that first we numerically calculate (approximately, with double precision) eigenvalues, then numerically and analytically—eigenvectors. Our modelling method for the radiation modes consists in reducing the initial potential scattering problem (in the case of the continuous spectrum) to the equivalent ones for the Jost functions: the Jost solution from the left for the substrate radiation modes and the Jost solution from the right for the cover radiation modes.

## Keywords

Waveguide propagation of electromagnetic radiation Equations of waveguide modes of regular waveguide Numerical-analytical modelling

## References

1. 1.
2. 2.
Ayrjan, E.A., Egorov, A.A., Michuk, E.N., Sevastyanov, A.L., Sevastianov, L.A., Stavtsev, A.V.: Representations of guided modes of integrated-optical multilayer thin-film waveguides, p. 52. Dubna, preprint JINR E11-2011-31 (2011)Google Scholar
3. 3.
Ayryan, E.A., Egorov, A.A., Sevastyanov, L.A., Lovetskiy, K.P., Sevastyanov, A.L.: Mathematical modeling of irregular integrated optical waveguides. In: Adam, G., Buša, J., Hnatič, M. (eds.) MMCP 2011. LNCS, vol. 7125, pp. 136–147. Springer, Heidelberg (2012)
4. 4.
Barnoski, M.: Introduction to Integrated Optics. Plenunm, New York (1974)
5. 5.
Conwell, E.: Modes in optical waveguides formed by diffusion. Appl. Phys. Lett. 23, 328–329 (1973)
6. 6.
Conwell, E.: WKB approximation for optical guide modes in a medium with exponentially varying index. J. Appl. Phys. 47, 1407 (1975)
7. 7.
Divakov, D.V., Sevastianov, L.A.: Application of incomplete Galerkin method to irregular transition in open planar waveguides. Matematicheskoe Modelirovanie 27(7), 44–50 (2015)
8. 8.
Egorov, A.A., Sevastyanov, A.L., Airyan, E.A., Lovetskiy, K.P., Sevastianov, L.A.: Adiabatic modes of smoothly irregular optical wavegide: zero-order vector theory. Matematicheskoe Modelirovanie 22(8), 42–54 (2010)
9. 9.
Egorov, A.A., Lovetskii, K.P., Sevastianov, A.L., Sevastianov, L.A.: Integrated Optics: Theory and Computer Modelling. RUDN Publisher, Moscow (2015)Google Scholar
10. 10.
Egorov, A.A., Lovetskiy, K.P., Sevastianov, A.L., Sevastianov, L.A.: Simulation of guided modes (eigenmodes) and synthesis of a thin-film generalised waveguide luneburg lens in the zero-order vector approximation. Quantum Electron. 40(9), 830–836 (2010)
11. 11.
Egorov, A.A., Sevastyanov, L.A.: Structure of modes of a smoothly irregular integrated-optical four-layer three-dimensional waveguide. Quantum Electron. 39(6), 566–574 (2009)
12. 12.
Fitio, V.M., Romakh, V.V., Bobitski, Y.V.: Numerical method for analysis of waveguide modes in planar gradient waveguides. Mater. Sci. 20(3), 256–261 (2014)Google Scholar
13. 13.
Fitio, V.M., Romakh, V.V., Bobitski, Y.V.: Search of mode wavelengths in planar waveguides by using Fourier transform of wave equation. Semicond. Phys. Quantum Electron. Optoelectron. 19(1), 28–33 (2016)
14. 14.
Gevorkyan, M.N., Kulyabov, D.S., Lovetskiy, K.P., Sevastyanov, A.L., Sevastyanov, L.A.: Waveguide modes of a planar optical waveguide. Math. Modell. Geom. 3(1), 43–63 (2015)Google Scholar
15. 15.
Hunsperger, R.G.: Integrated Optics: Theory and Technology. Springer, Heidelberg (1995)
16. 16.
Marcuse, D.: Light Transmission Optics. Van Nostrand Reinhold Company, New York (1972)Google Scholar
17. 17.
Nikolaev, N., Shevchenko, V.V.: Inverse method for the reconstruction of refractive index profile and power management in gradient index optical waveguides. Opt. Quantum Electron. 39(10), 891–902 (2007)
18. 18.
Rganov, A.G., Grigas, S.E.: Defining the parameters of multilayer waveguide modes of dielectric waveguides. Numer. Methods Program. 10, 258–262 (2009)Google Scholar
19. 19.
Rganov, A.G., Grigas, S.E.: Numerical algorithm for waveguide and leaky modes determination in multilayer optical waveguides. Tech. Phys. 55(11), 1614–1618 (2010)
20. 20.
Sevastianov, L., Divakov, D., Nikolaev, N.: Modelling of an open transition of the “horn” type between open planar waveguides. In: EPJ Web of Conferences, vol. 108, p. 02020 (2016)Google Scholar
21. 21.
Sevastianov, L.A., Egorov, A.A.: The theoretical analysis of waveguide propagation of electromagnetic waves in dielectric smoothly-irregular integrated structures. Math. Modell. Geom. 105(4), 576–584 (2008)Google Scholar
22. 22.
Sevastianov, L.A., Egorov, A.A., Sevastyanov, A.L.: Method of adiabatic modes in studying problems of smoothly irregular open waveguide structures. Phys. At. Nucl. 76(2), 224–239 (2013)
23. 23.
Sevastyanov, L.A.: The complete system of modes of open planar waveguide. In: Proceedings of the VI International Scientific Conference Lasers in Science, Technology, and Medicine, pp. 72–76. Publishing House of IRE, Suzdal (1995)Google Scholar
24. 24.
Shevchenko, V.V.: On the spectral expansion in eigenfunctions and associated functions of a non self-adjoint problem of sturm-liouville type on the entire axis. Differ. Equ. 15, 2004–2020 (1979)Google Scholar
25. 25.
Snyder, A.W., Love, J.D.: Optical Waveguide Theory. Chapman and Hall, New York (1983)Google Scholar
26. 26.
Tamir, T.: Integrated Optics. Springer-Verlag, Berlin (1979)Google Scholar
27. 27.
Unger, H.G.: Planar Optical Waveguides and Fibres. Clarendon Press, Oxford (1977)Google Scholar

© Springer International Publishing AG 2016

## Authors and Affiliations

• Edik Ayrjan
• 1
• 4
• Migran Gevorkyan
• 2
• Dmitry Kulyabov
• 1
• 2
• Konstantin Lovetskiy
• 2
• Nikolai Nikolaev
• 2
• Anton Sevastianov
• 2
• Leonid Sevastianov
• 2
• 3
• Eugeny Laneev
• 2
1. 1.Laboratory of Information TechnologiesJoint Institute for Nuclear ResearchDubna, Moscow RegionRussia
2. 2.RUDN University (Peoples’ Friendship University of Russia)MoscowRussia
3. 3.Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear ResearchDubna, Moscow RegionRussia
4. 4.Yerevan Physics InstituteYerevanArmenia