mireabulletinRussian Technological JournalRussian Technological Journal2782-32102500-316XRTU MIREA10.32362/2500-316X-2023-11-4-84-93mireabulletin-737Research ArticleМАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕMATHEMATICAL MODELINGМодели волноводов, сочетающих градиентные и нелинейно-оптические слоиModels of waveguides combining gradient and nonlinear optical layershttps://orcid.org/0000-0002-7158-9145СавотченкоС. Е.SavotchenkoS. E.

Савотченко Сергей Евгеньевич, д.ф.-м.н, доцент, профессор кафедры высшей математики

308012, Белгород, ул. Костюкова, д. 46

Scopus Author ID 6603577988

Sergey E. Savotchenko, Dr. Sci. (Phys.-Math.), Associate Professor, Professor, High Mathematics Department

46, Kostyukova ul., Belgorod, 308012

Scopus Author ID 6603577988

savotchenkose@mail.ruБелгородский государственный технологический университет им. В.Г. ШуховаРоссияV.G. Shukhov Belgorod State Technological UniversityRussian Federation2023010820231148493Copyright © Савотченко С.Е., 20232023Савотченко С.Е.Savotchenko S.E.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.rtj-mirea.ru/jour/article/view/737Цели

Цели. Теоретические исследования волноводных свойств границ раздела нелинейно-оптических и градиентных сред являются важными для использования в оптоэлектронике. Комбинированные волноводы, сочетающие слои с различными оптическими свойствами, представляются наиболее перспективными, поскольку для них можно подобрать оптимальные характеристики с помощью широкого ряда управляющих параметров. Цель работы – разработка теории композитных оптически-нелинейных градиентных волноводов с произвольным профилем, в рамках которой возможно получение точных аналитических выражений для поверхностных волн и волноводных мод в явном виде. Основной особенностью предлагаемой в данной работе теории является то, что она применима для описания поверхностных волн и волноводных мод, поле в которых сосредоточено внутри градиентного слоя и не выходит за его границу, не контактирующую с нелинейным слоем.

Методы

Методы. Использованы аналитические методы теории оптических волноводов, нелинейной оптики.

Результаты

Результаты. Проведено теоретическое описание волноводных свойств границы раздела двух сред с принципиально различными оптическими характеристиками. Сформулированная модель плоского волновода применима для сред с произвольным распределением пространственного профиля диэлектрической проницаемости. Получено аналитическое выражение, описывающее поверхностную волну, распространяющуюся вдоль границы раздела среды со ступенчатой нелинейностью и градиентного слоя с произвольным профилем диэлектрической проницаемости. Также получены аналитические выражения для поверхностных волн, распространяющихся вдоль границы раздела среды с керровской нелинейностью (как самофокусирующей, так и дефокусирующей) с градиентными средами, характеризующимися экспоненциальным и линейным профилями диэлектрической проницаемости.

Выводы

Выводы. Предложенная теория позволяет наглядно описать в явном аналитическом виде узко локализованные световые потоки в таких волноводах. Показано, что сочетание различных полупроводниковых кристаллов в композитном волноводе позволяет получить с одной стороны от волноведущего интерфейса нелинейнооптический слой, а с другой – слой с градиентным профилем диэлектрической проницаемости.

Objectives

Objectives. Theoretical studies of the waveguide properties of interfaces between nonlinear optical and graded-index media are important for application in optoelectronics. Waveguides combining layers with different optical properties seem to be the most promising, since they can be matched to optimal characteristics using a wide range of control parameters. The paper aims to develop a theory of composite optically nonlinear gradedindex waveguides with an arbitrary profile, within which it is possible to obtain exact analytical expressions for surface waves and waveguide modes in an explicit form. The main feature of the theory proposed in this paper is its applicability for describing surface waves and waveguide modes, in which the field is concentrated inside the gradient layer and does not exceed its boundary, avoiding contact with the nonlinear layer.

Methods

Methods. Analytical methods of the theory of optical waveguides and nonlinear optics are used.

Results

Results. A theoretical description of the waveguide properties of the interface between two media having significantly different optical characteristics is carried out. The formulated model of a plane waveguide is applicable to media having an arbitrary spatial permittivity profile. An analytical expression describing a surface wave propagating along the interface between a medium having stepwise nonlinearity and a gradient layer with an arbitrary permittivity profile is obtained. Additionally, analytical expressions for surface waves propagating along the interface between a medium with Kerr nonlinearity (both self-focusing and defocusing), as well as graded-index media characterized by exponential and linear permittivity profiles, are obtained.

Conclusions

Conclusions. The proposed theory supports a visual description in an explicit analytical form of a narrowly localized light beam within such waveguides. It is shown that by combining different semiconductor crystals in a composite waveguide, it is possible to obtain a nonlinear optical layer on one side of the waveguide interface and a layer with a graded-index dielectric permittivity profile on the other.

нелинейная оптиканелинейные волныоптическая нелинейностькерровская нелинейностьоптический волноводградиентный волноводnonlinear opticsnonlinear wavesoptical nonlinearityKerr nonlinearityoptical waveguidegraded-index waveguideРабота выполнена с использованием оборудования центра высоких технологий БГТУ им. В.Г. Шухова.ReferencesAdams M.J. An Introduction to Optical Waveguides. Chichester: Wiley; 1981. 401 p.Adams M.J. An Introduction to Optical Waveguides. Chichester: Wiley; 1981. 401 p.Chen C.-L. Foundations for Guided-Wave Optics. New York: John Wiley & Sons Inc.; 2005. 462 p. https://doi.org/10.1002/0470042222Chen C.-L. Foundations for Guided-Wave Optics. New York: John Wiley & Sons Inc.; 2005. 462 p. https://doi.org/10.1002/0470042222Dragoman D., Dragoman M. Advanced Optoelectronic devices. Berlin: Springer; 1999. 424 р.Dragoman D., Dragoman M. Advanced Optoelectronic devices. Berlin: Springer; 1999. 424 р.Bednarik M., Cervenka M. Electromagnetic waves in graded-index planar waveguides. J. Opt. Soc. Am. B. 2020;37(12):3631–3643. https://doi.org/10.1364/JOSAB.408679Bednarik M., Cervenka M. Electromagnetic waves in graded-index planar waveguides. J. Opt. Soc. Am. B. 2020;37(12):3631–3643. https://doi.org/10.1364/JOSAB.408679Čada M., Qasymeh M., Pištora J. Optical Wave Propagation in Kerr Media. In: Wave Propagation. Theories and Applications. London: IntechOpen; 2013. P. 175–192. http://doi.org/10.5772/51293Čada M., Qasymeh M., Pištora J. Optical Wave Propagation in Kerr Media. In: Wave Propagation. Theories and Applications. London: IntechOpen; 2013. P. 175–192. http://doi.org/10.5772/51293Shvartsburg A.B., Maradudin A. Waves in Gradient Metamaterials. Singapore: World Scientific; 2013. 339 p. https://doi.org/10.1142/8649Shvartsburg A.B., Maradudin A. Waves in Gradient Metamaterials. Singapore: World Scientific; 2013. 339 p. https://doi.org/10.1142/8649Al-Bader S.J., Jamid H.A. Graded-index optical waveguides with nonlinear cladding. J. Opt. Soc. Am. A. 1988;5(3):374–379. https://doi.org/10.1364/JOSAA.5.000374Al-Bader S.J., Jamid H.A. Graded-index optical waveguides with nonlinear cladding. J. Opt. Soc. Am. A. 1988;5(3):374–379. https://doi.org/10.1364/JOSAA.5.000374Taya S.A., Kullab H.M., Qadoura I.M. Dispersion properties of slab waveguides with double negative material guiding layer and nonlinear substrate. J. Opt. Soc. Am. B. 2013;30(7):2008–2013. https://doi.org/10.1364/JOSAB.30.002008Taya S.A., Kullab H.M., Qadoura I.M. Dispersion properties of slab waveguides with double negative material guiding layer and nonlinear substrate. J. Opt. Soc. Am. B. 2013;30(7):2008–2013. https://doi.org/10.1364/JOSAB.30.002008Almawgani A.H.M., Taya S.A., Hussein A.J., Colak I. Dispersion properties of a slab waveguide with a gradedindex core layer and a nonlinear cladding using the WKB approximation method. J. Opt. Soc. Am. B. 2022;39(6): 1606–1613. https://doi.org/10.1364/JOSAB.458569Almawgani A.H.M., Taya S.A., Hussein A.J., Colak I. Dispersion properties of a slab waveguide with a gradedindex core layer and a nonlinear cladding using the WKB approximation method. J. Opt. Soc. Am. B. 2022;39(6): 1606–1613. https://doi.org/10.1364/JOSAB.458569Hussein A.J., Nassar Z.M., Taya S.A. Dispersion properties of slab waveguides with a linear gradedindex film and a nonlinear substrate. Microsyst. Technol. 2021;27(7):2589–2594. https://doi.org/10.1007/s00542-020-05016-zHussein A.J., Nassar Z.M., Taya S.A. Dispersion properties of slab waveguides with a linear gradedindex film and a nonlinear substrate. Microsyst. Technol. 2021;27(7):2589–2594. https://doi.org/10.1007/s00542-020-05016-zTaya S.A., Hussein A.J., Colak I. An exact solution of a slab waveguide dispersion relation with a linear gradedindex guiding layer (TM case). Microsyst Technol. 2022;28(22):1213–1219. https://doi.org/10.1007/s00542-022-05281-0Taya S.A., Hussein A.J., Colak I. An exact solution of a slab waveguide dispersion relation with a linear gradedindex guiding layer (TM case). Microsyst Technol. 2022;28(22):1213–1219. https://doi.org/10.1007/s00542-022-05281-0Taya S.A., Hussein A.J., Ramahi O.M., Colak I., Chaouche Y.B. Dispersion curves of a slab waveguide with a nonlinear covering medium and an exponential graded-index thin film (transverse magnetic case). J. Opt. Soc. Am. B. 2021;38(11):3237–3243. https://doi.org/10.1364/JOSAB.439034Taya S.A., Hussein A.J., Ramahi O.M., Colak I., Chaouche Y.B. Dispersion curves of a slab waveguide with a nonlinear covering medium and an exponential graded-index thin film (transverse magnetic case). J. Opt. Soc. Am. B. 2021;38(11):3237–3243. https://doi.org/10.1364/JOSAB.439034Hussein A.J., Taya S.A., Vigneswaran D., Udiayakumar R., Upadhyay A., Anwar T., Amiri I.S. Universal dispersion curves of a planar waveguide with an exponential gradedindex guiding layer and a nonlinear cladding. Results in Physics. 2021;20:103734. https://doi.org/10.1016/j.rinp.2020.103734Hussein A.J., Taya S.A., Vigneswaran D., Udiayakumar R., Upadhyay A., Anwar T., Amiri I.S. Universal dispersion curves of a planar waveguide with an exponential gradedindex guiding layer and a nonlinear cladding. Results in Physics. 2021;20:103734. https://doi.org/10.1016/j.rinp.2020.103734Savotchenko S.E. The surface waves propagating along the contact between the layer with the constant gradient of refractive index and photorefractive crystal. J. Opt. 2022;24(4):045501. https://doi.org/10.1088/2040-8986/ac51e9Savotchenko S.E. The surface waves propagating along the contact between the layer with the constant gradient of refractive index and photorefractive crystal. J. Opt. 2022;24(4):045501. https://doi.org/10.1088/2040-8986/ac51e9Savotchenko S.E. The composite planar waveguide structure consisting of the linearly graded-index layer and the nonlinear layer formed with an increasing the electric field. Optik. 2022;252:168542. https://doi.org/10.1016/j.ijleo.2021.168542Savotchenko S.E. The composite planar waveguide structure consisting of the linearly graded-index layer and the nonlinear layer formed with an increasing the electric field. Optik. 2022;252:168542. https://doi.org/10.1016/j.ijleo.2021.168542Savotchenko S.E. Light localization in a linearly graded-index substrate covered by intensity dependent nonlinear self-focusing cladding. J. Opt. 2022;24(6): 065503. https://doi.org/10.1088/2040-8986/ac6babSavotchenko S.E. Light localization in a linearly graded-index substrate covered by intensity dependent nonlinear self-focusing cladding. J. Opt. 2022;24(6): 065503. https://doi.org/10.1088/2040-8986/ac6babSavotchenko S.E. Discrete spectrum of waveguide modes of a linearly graded-index film introduced into a medium with a stepwise nonlinearity. Optik. 2023;281(6):170835. https://doi.org/10.1016/j.ijleo.2023.170835Savotchenko S.E. Discrete spectrum of waveguide modes of a linearly graded-index film introduced into a medium with a stepwise nonlinearity. Optik. 2023;281(6):170835. https://doi.org/10.1016/j.ijleo.2023.170835Savotchenko S.E. Guided waves in a graded-index substrate covered by an intensity-dependent defocusing nonlinear medium. Appl. Phys. B: Lasers and Optics. 2022;128(8):153. https://doi.org/10.1007/s00340-022-07872-1Savotchenko S.E. Guided waves in a graded-index substrate covered by an intensity-dependent defocusing nonlinear medium. Appl. Phys. B: Lasers and Optics. 2022;128(8):153. https://doi.org/10.1007/s00340-022-07872-1Savotchenko S.E. Nonlinear surface waves propagating along the contact between the graded-index layer and the medium with near surface layer where Kerr nonlinearity disappears with increasing light intensity. Optik. 2023;272:170373. https://doi.org/10.1016/j.ijleo.2022.170373Savotchenko S.E. Nonlinear surface waves propagating along the contact between the graded-index layer and the medium with near surface layer where Kerr nonlinearity disappears with increasing light intensity. Optik. 2023;272:170373. https://doi.org/10.1016/j.ijleo.2022.170373Savotchenko S.E. Surface waves propagating along the interface between a parabolic graded-index medium and a self-focusing nonlinear medium: exact analytical solution. J. Opt. 2022;24(10):105501. https://doi.org/10.1088/2040-8986/ac8e80Savotchenko S.E. Surface waves propagating along the interface between a parabolic graded-index medium and a self-focusing nonlinear medium: exact analytical solution. J. Opt. 2022;24(10):105501. https://doi.org/10.1088/2040-8986/ac8e80Savotchenko S.E. Surface waves propagating along the interface between parabolic graded-index medium and photorefractive crystal with diffusion nonlinearity. Phys. B: Condensed Matter. 2023;648(2):414434. https://doi.org/10.1016/j.physb.2022.414434Savotchenko S.E. Surface waves propagating along the interface between parabolic graded-index medium and photorefractive crystal with diffusion nonlinearity. Phys. B: Condensed Matter. 2023;648(2):414434. https://doi.org/10.1016/j.physb.2022.414434Savotchenko S.E. Surface waves propagating along the interface separating an exponential graded-index medium and the medium with a step change in the dielectric constant. Optik. 2022;271(12):170092. https://doi.org/10.1016/j.ijleo.2022.170092Savotchenko S.E. Surface waves propagating along the interface separating an exponential graded-index medium and the medium with a step change in the dielectric constant. Optik. 2022;271(12):170092. https://doi.org/10.1016/j.ijleo.2022.170092Savotchenko S.E. Waveguide properties of interface separating a photorefractive crystal with diffusion nonlinearity and an exponential graded-index medium. Phys. Lett. A. 2022;455(12):128516. https://doi.org/10.1016/j.physleta.2022.128516Savotchenko S.E. Waveguide properties of interface separating a photorefractive crystal with diffusion nonlinearity and an exponential graded-index medium. Phys. Lett. A. 2022;455(12):128516. https://doi.org/10.1016/j.physleta.2022.128516Savotchenko S.E. New types of transverse electric nonlinear waves propagating along a linearly graded-index layer in a medium with Kerr nonlinearity. Opt. Quant. Electron. 2023;55(1):74. https://doi.org/10.1007/s11082-022-04323-1Savotchenko S.E. New types of transverse electric nonlinear waves propagating along a linearly graded-index layer in a medium with Kerr nonlinearity. Opt. Quant. Electron. 2023;55(1):74. https://doi.org/10.1007/s11082-022-04323-1Savotchenko S.E. Temperature controlled waveguide properties of the linearly graded-index film in semiconductor crystal with the photorefractive nonlinearity. Appl. Phys. B: Lasers and Optics. 2023;129(1):7. https://doi.org/10.1007/s00340-022-07950-4Savotchenko S.E. Temperature controlled waveguide properties of the linearly graded-index film in semiconductor crystal with the photorefractive nonlinearity. Appl. Phys. B: Lasers and Optics. 2023;129(1):7. https://doi.org/10.1007/s00340-022-07950-4Khadzhi P.I., Fedorov L.V., Torstveit S. Nonlinear surface waves for the simplest model of nonlinear medium. Phys. Tech. Lett. 1991;61:110–113.Khadzhi P.I., Fedorov L.V., Torstveit S. Nonlinear surface waves for the simplest model of nonlinear medium. Phys. Tech. Lett. 1991;61:110–113.Kaplan А.E. Multistable self-trapping of light and multistable soliton pulse propagation. IEEE J. Quant. Electron. 1985;21(9):1538–1543. https://doi.org/10.1109/JQE.1985.1072828Kaplan А.E. Multistable self-trapping of light and multistable soliton pulse propagation. IEEE J. Quant. Electron. 1985;21(9):1538–1543. https://doi.org/10.1109/JQE.1985.1072828Kartashov Y.V., Malomed B.A., Torner L. Solitons in nonlinear lattices. Rev. Mod. Phys. 2011;83(1):247–305. http://doi.org/10.1103/RevModPhys.83.247Kartashov Y.V., Malomed B.A., Torner L. Solitons in nonlinear lattices. Rev. Mod. Phys. 2011;83(1):247–305. http://doi.org/10.1103/RevModPhys.83.247Laudyn U.A., Rutkowska K.A., Rutkowski R.T., Karpierz M.A., Woliński T.R., Wójcik J. Nonlinear effects in photonic crystal fibers filled with nematic liquid crystals. Cent. Eur. J. Phys. 2008;6(3):612–618. https://doi.org/10.2478/s11534-008-0096-zLaudyn U.A., Rutkowska K.A., Rutkowski R.T., Karpierz M.A., Woliński T.R., Wójcik J. Nonlinear effects in photonic crystal fibers filled with nematic liquid crystals. Cent. Eur. J. Phys. 2008;6(3):612–618. https://doi.org/10.2478/s11534-008-0096-zPolyakov V.V., Polyakova K.P., Seredkin V.A., Patrin G.S. The enhanced magneto-optical Kerr effect in Co/TiO2 multilayer films. Tech. Phys. Lett. 2012;38(10):921–923. https://doi.org/10.1134/S1063785012100227Polyakov V.V., Polyakova K.P., Seredkin V.A., Patrin G.S. The enhanced magneto-optical Kerr effect in Co/TiO2 multilayer films. Tech. Phys. Lett. 2012;38(10):921–923. https://doi.org/10.1134/S1063785012100227Jarque E.C., Malyshev V.A. Nonlinear reflection from a dense saturable absorber: from stability to chaos. Opt. Commun. 1997;14291(3):66–70. https://doi.org/10.1016/S0030-4018(97)00275-7Jarque E.C., Malyshev V.A. Nonlinear reflection from a dense saturable absorber: from stability to chaos. Opt. Commun. 1997;14291(3):66–70. https://doi.org/10.1016/S0030-4018(97)00275-7Schuzgen A., Peyghambarian N., Hughes S. Doppler Shifted Self Reflection from a Semiconductor. Phys. Stat. Sol. (B). 1999;206(1):125–130. https://doi.org/10.1002/(SICI)1521-3951(199803)206:1<125::AID-PSSB125>3.0.CO;2-8Schuzgen A., Peyghambarian N., Hughes S. Doppler Shifted Self Reflection from a Semiconductor. Phys. Stat. Sol. (B). 1999;206(1):125–130. https://doi.org/10.1002/(SICI)1521-3951(199803)206:1<125::AID-PSSB125>3.0.CO;2-8Ляхомская К.Д., Хаджи П.И. Эффект самоотражения и простейшие модели нелинейной среды. Журн. техн. физики. 2000;70(11):86–90. Lyakhomskaya K.D., Khadzhi P.I. Self-reflection effect in naïve model of nonlinear media. Tech. Phys. 2000;45(11):1457–1461. https://doi.org/10.1134/1.1325030 [Original Russian Text: Lyakhomskaya K.D., Khadzhi P.I. Self-reflection effect in naïve model of nonlinear media. Zhurnal Tekhnicheskoi Fiziki. 2000;70(11):86–90 (in Russ.).]Lyakhomskaya K.D., Khadzhi P.I. Self-reflection effect in naïve model of nonlinear media. Tech. Phys. 2000;45(11):1457–1461. https://doi.org/10.1134/1.1325030 [Original Russian Text: Lyakhomskaya K.D., Khadzhi P.I. Self-reflection effect in naïve model of nonlinear media. Zhurnal Tekhnicheskoi Fiziki. 2000;70(11):86–90 (in Russ.).]Christian J.M., McDonald G.S., Chamorro-Posada P. Bistable Helmholtz bright solitons in saturable materials. J. Opt. Soc. Am. B. 2009;26(12):2323–2330. https://doi.org/10.1364/JOSAB.26.002323Christian J.M., McDonald G.S., Chamorro-Posada P. Bistable Helmholtz bright solitons in saturable materials. J. Opt. Soc. Am. B. 2009;26(12):2323–2330. https://doi.org/10.1364/JOSAB.26.002323Korovai O.V. Nonlinear s-polarized quasi-surface waves in the symmetric structure with a metamaterial core. Phys. Solid State. 2015;57(7):1456–1462. https://doi.org/10.1134/S1063783415070197Korovai O.V. Nonlinear s-polarized quasi-surface waves in the symmetric structure with a metamaterial core. Phys. Solid State. 2015;57(7):1456–1462. https://doi.org/10.1134/S1063783415070197Enns R.H., Rangnekar S.S., Kaplan A.E. Bistablesoliton pulse propagation: Stability aspect. Phys. Rev. A. 1987;36(3):1270–1279. https://doi.org/10.1103/PhysRevA.36.1270Enns R.H., Rangnekar S.S., Kaplan A.E. Bistablesoliton pulse propagation: Stability aspect. Phys. Rev. A. 1987;36(3):1270–1279. https://doi.org/10.1103/PhysRevA.36.1270Khadzhi P.I., Rusanov A.M., Gaivan S.L. Cavity-free optical bistability of a thin semiconductor film in the exciton region of the spectrum. 1999;29(6):539–541. https://doi.org/10.1070/QE1999v029n06ABEH001526Khadzhi P.I., Rusanov A.M., Gaivan S.L. Cavity-free optical bistability of a thin semiconductor film in the exciton region of the spectrum. 1999;29(6):539–541. https://doi.org/10.1070/QE1999v029n06ABEH001526Khadzhi P.I., Gaivan S.L. Nonlinear interaction of an ultrashort light pulse with a thin semiconductor film under conditions of two-photon excitation of biexcitons. Quantum Electron. 1995;25(9):897–900. https://doi.org/10.1070/QE1995v025n09ABEH000497Khadzhi P.I., Gaivan S.L. Nonlinear interaction of an ultrashort light pulse with a thin semiconductor film under conditions of two-photon excitation of biexcitons. Quantum Electron. 1995;25(9):897–900. https://doi.org/10.1070/QE1995v025n09ABEH000497Corovai A.V., Khadzhi P.I. Optical properties of a semiconductor upon two-photon excitation of biexcitons by a powerful pump pulse and one-photon probing in the M band. Quantum Electron. 2001;31(10):937–939. https://doi.org/10.1070/QE2001v031n10ABEH002080Corovai A.V., Khadzhi P.I. Optical properties of a semiconductor upon two-photon excitation of biexcitons by a powerful pump pulse and one-photon probing in the M band. Quantum Electron. 2001;31(10):937–939. https://doi.org/10.1070/QE2001v031n10ABEH002080Roussignol P., Ricard D., Flytzanis C. Nonlinear optical properties of commercial semiconductor-doped glasses. Appl. Phys. A. 1987;44:285–292. https://doi.org/10.1007/BF00624594Roussignol P., Ricard D., Flytzanis C. Nonlinear optical properties of commercial semiconductor-doped glasses. Appl. Phys. A. 1987;44:285–292. https://doi.org/10.1007/BF00624594Vanhaudenarde A., Trespidi M., Frey R. Refractive-index changes during photodarkening in semiconductor-doped glasses. J. Opt. Soc. Am. B. 1994;11(8):1474–1479. https://doi.org/10.1364/JOSAB.11.001474Vanhaudenarde A., Trespidi M., Frey R. Refractive-index changes during photodarkening in semiconductor-doped glasses. J. Opt. Soc. Am. B. 1994;11(8):1474–1479. https://doi.org/10.1364/JOSAB.11.001474Catunda T., Cury L.A. Transverse self-phase modulation in ruby and GdAlO3:Cr+3 crystals. J. Opt. Soc. Am. B. 1990;7(8): 1445–1455. https://doi.org/10.1364/JOSAB.7.001445Catunda T., Cury L.A. Transverse self-phase modulation in ruby and GdAlO3:Cr+3 crystals. J. Opt. Soc. Am. B. 1990;7(8): 1445–1455. https://doi.org/10.1364/JOSAB.7.001445Wang S.Q., Wang X., Birge R., Downie J.D., Timucin D., Gary C. Propagation of a Gaussian beam in a bacteriorhodopsin film. J. Opt. Soc. Am. B. 1998;15(5): 1602–1609. https://doi.org/10.1364/JOSAB.15.001602Wang S.Q., Wang X., Birge R., Downie J.D., Timucin D., Gary C. Propagation of a Gaussian beam in a bacteriorhodopsin film. J. Opt. Soc. Am. B. 1998;15(5): 1602–1609. https://doi.org/10.1364/JOSAB.15.001602Mendoza-Alvarez J.G., Nunes F.D., Patel N.B. Refractive index dependence on free carriers for GaAs. J. Appl. Phys. 1980;51(8):4365-4367. https://doi.org/10.1063/1.328298Mendoza-Alvarez J.G., Nunes F.D., Patel N.B. Refractive index dependence on free carriers for GaAs. J. Appl. Phys. 1980;51(8):4365-4367. https://doi.org/10.1063/1.328298Ravindran S., Datta A., Alameh K., Lee Y.T. GaAs based long-wavelength microring resonator optical switches utilising bias assisted carrier-injection induced refractive index change. Opt. Express. 2012;20(14):15610–15627. https://doi.org/10.1364/OE.20.015610Ravindran S., Datta A., Alameh K., Lee Y.T. GaAs based long-wavelength microring resonator optical switches utilising bias assisted carrier-injection induced refractive index change. Opt. Express. 2012;20(14):15610–15627. https://doi.org/10.1364/OE.20.015610Zucker J.E., Chang T.Y., Wegener M., Sauer N.J., Jones K.L., Chemla D.S. Large refractive index changes in tunable-electron-density InGaAs/InAlAs quantum wells. IEEE Photon. Technol. Lett. 1990;2(1):29–31. https://doi.org/10.1109/68.47032Zucker J.E., Chang T.Y., Wegener M., Sauer N.J., Jones K.L., Chemla D.S. Large refractive index changes in tunable-electron-density InGaAs/InAlAs quantum wells. IEEE Photon. Technol. Lett. 1990;2(1):29–31. https://doi.org/10.1109/68.47032Ishida K., Nakamura H., Matsumura H. InGaAsP/InP optical switches using carrier induced refractive index change. Appl. Phys. Lett. 1987;50(3):141–142. https://doi.org/10.1063/1.97695Ishida K., Nakamura H., Matsumura H. InGaAsP/InP optical switches using carrier induced refractive index change. Appl. Phys. Lett. 1987;50(3):141–142. https://doi.org/10.1063/1.97695Вигдорович Е.Н. Радиационная стойкость эпитаксиальных структур на основе GaAs. Russ. Technol. J. 2019;7(3):41–49. https://doi.org/10.32362/2500-316X-2019-7-3-41-49 [Vigdorovich E.N. Radiation resistance of epitaxial structures based on GaAs. Russ. Technol. J. 2019;7(3):41–49 (in Russ.). https://doi.org/10.32362/2500-316X-2019-7-3-41-49 ]Вигдорович Е.Н. Радиационная стойкость эпитаксиальных структур на основе GaAs. Russ. Technol. J. 2019;7(3):41–49. https://doi.org/10.32362/2500-316X-2019-7-3-41-49 [Vigdorovich E.N. Radiation resistance of epitaxial structures based on GaAs. Russ. Technol. J. 2019;7(3):41–49 (in Russ.). https://doi.org/10.32362/2500-316X-2019-7-3-41-49 ]Karasiński P., Rogoziński R. Influence of refractive profile shape on the distribution of modal attenuation in planar structures with absorption cover. Opt. Commun. 2007;269(1):76–88. https://doi.org/10.1016/j.optcom.2006.07.067Karasiński P., Rogoziński R. Influence of refractive profile shape on the distribution of modal attenuation in planar structures with absorption cover. Opt. Commun. 2007;269(1):76–88. https://doi.org/10.1016/j.optcom.2006.07.067Shutyi A., Sementsov D., Kazakevich A.V., Sannikov D. Waveguide regimes of a graded-index planar waveguide with cladding. Tech. Phys. 1999;44(1):1329–1333. https://doi.org/10.1134/1.1259518Shutyi A., Sementsov D., Kazakevich A.V., Sannikov D. Waveguide regimes of a graded-index planar waveguide with cladding. Tech. Phys. 1999;44(1):1329–1333. https://doi.org/10.1134/1.1259518

The authors declare that there are no conflicts of interest present.