Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)3095310.22363/2658-4670-2022-30-2-149-159Research ArticleInvestigation of adiabatic waveguide modes model for smoothly irregular integrated optical waveguidesSevastyanovAnton L.<p>PhD in Physical and Mathematical Sciences, Deputy head of department: Department of Digitalization of Education</p>alsevastyanov@gmail.comhttps://orcid.org/0000-0002-0280-485XHigher School of Economics0305202230214915903052022Copyright © 2022, Sevastyanov A.L.2022<p style="text-align: justify;">The model of adiabatic waveguide modes (AWMs) in a smoothly irregular integrated optical waveguide is studied. The model explicitly takes into account the dependence on the rapidly varying transverse coordinate and on the slowly varying horizontal coordinates. Equations are formulated for the strengths of the AWM fields in the approximations of zero and first order of smallness. The contributions of the first order of smallness introduce depolarization and complex values characteristic of leaky modes into the expressions of the AWM electromagnetic fields. A stable method is proposed for calculating the vertical distribution of the electromagnetic field of guided modes in regular multilayer waveguides, including those with a variable number of layers. A stable method for solving a nonlinear equation in partial derivatives of the first order (dispersion equation) for the thickness profile of a smoothly irregular integrated optical waveguide in models of adiabatic waveguide modes of zero and first orders of smallness is described. Stable regularized methods for calculating the AWM field strengths depending on vertical and horizontal coordinates are described. Within the framework of the listed matrix models, the same methods and algorithms for the approximate solution of problems arising in these models are used. Verification of approximate solutions of models of adiabatic waveguide modes of the first and zero orders is proposed; we compare them with the results obtained by other authors in the study of more crude models.</p>smoothly irregular thin-film dielectric waveguidesadiabatic waveguide modesregularized methods for calculating field strengthsмодели квантовых измеренийвозмущение дискретного спектракомплексные собственные значенияпучки операторов[B. Z. Katsenelenbaum, Theory of irregular waveguides with slowly varying parameters [Teoriya neregulyarnyh volnovodov s medlenno menyayushchimisya parametrami]. Moscow: Akad. Nauk SSSR, 1961, in Russian.][V. V. Shevchenko, Continuous transitions in open waveguides [Plavnyye perekhody v otkrytykh volnovodakh]. Moscow: Nauka, 1969, in Russian.][M. V. Fedoryuk, “Justification of the method of cross-sections for an acoustic waveguide with inhomogeneous filling”, USSR Computational Mathematics and Mathematical Physics, vol. 13, no. 1, pp. 162-173, 1973. DOI: 10.1016/0041-5553(74)90012-3.][A. A. Egorov and L. A. Sevast’yanov, “Structure of modes of a smoothly irregular integrated optical four-layer three-dimensional waveguide”, Quantum Electronics, vol. 39, no. 6, pp. 566-574, 2009. DOI: 10.1070/QE2009v039n06ABEH013966.][A. A. Egorov et al., “Simulation of guided modes (eigenmodes) and synthesis of a thin-film generalised waveguide Luneburg lens in the zero-order vector approximation”, Quantum Electronics, vol. 40, no. 9, pp. 830-836, 2010. DOI: 10.1070/QE2010v040n09ABEH014332.][A. A. Egorov, L. A. Sevastianov, and A. L. Sevastianov, “Method of adiabatic modes in research of smoothly irregular integrated optical waveguides: zero approximation”, Quantum Electronics, vol. 44, no. 2, pp. 167-173, 2014. DOI: 10.1070/QE2014v044n02ABEH015303.][A. L. Sevastianov, “Asymptotic method for constructing a model of adiabatic guided modes of smoothly irregular integrated optical waveguides”, Discrete and Continuous Models and Applied Computational Science, vol. 20, no. 3, pp. 252-273, 2020. DOI: 10.22363/2658-4670-2020-283-252-273.][A. L. Sevastianov, “Single-mode propagation of adiabatic guided modes in smoothly irregular integral optical waveguides”, Discrete and Continuous Models and Applied Computational Science, vol. 28, no. 4, pp. 361- 377, 2020. DOI: 10.22363/2658-4670-2020-28-4-361-377.][G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch oscillations in an array of curved optical waveguides”, Physical Review Letters, vol. 83, no. 5, pp. 963-966, 1999. DOI: 10.1103/PhysRevLett.83.963.][S. Longhi, D. Janner, M. Marano, and P. Laporta, “Quantum-mechanical analogy of beam propagation in waveguides with a bent axis: dynamic-mode stabilization and radiation-loss suppression”, Physical Review E, vol. 67, no. 3, p. 036601, 2003. DOI: 10.1103/PhysRevE.67.036601.][I. Vorobeichik et al., “Electromagnetic realization of orders-of-magnitude tunneling enhancement in a double well system”, Physical Review Letters, vol. 90, p. 176806, 17 2003. DOI: 10.1103/PhysRevLett.90.176806.][S. Longhi, “Coherent destruction of tunneling in waveguide directional couplers”, Physical Review A, vol. 71, p. 065801, 6 2005. DOI: 10.1103/PhysRevA.71.065801.][R. Khomeriki and S. Ruffo, “Nonadiabatic Landau-Zener tunneling in waveguide arrays with a step in the refractive index”, Physical Review Letters, vol. 94, p. 113904, 11 2005. DOI: 10.1103/PhysRevLett.94.113904.][K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules”, Reviews of Modern Physics, vol. 70, pp. 1003-1025, 3 1998. DOI: 10.1103/RevModPhys.70.1003.][F.T.HioeandJ.H.Eberly,“N-Levelcoherencevectorandhigherconservation laws in quantum optics and quantum mechanics”, Physical Review Letters, vol. 47, pp. 838-841, 12 1981. DOI: 10.1103/PhysRevLett.47.838.][J. Oreg, F. T. Hioe, and J. H. Eberly, “Adiabatic following in multilevel systems”, Physical Review A, vol. 29, pp. 690-697, 2 1984. DOI: 10.1103/PhysRevA.29.690.][C. E. Carroll and F. T. Hioe, “Three-state systems driven by resonant optical pulses of different shapes”, Journal of the Optical Society of America B: Optical Physics, vol. 5, no. 6, pp. 1335-1340, 1988. DOI: 10.1364/JOSAB.5.001335.][J. Oreg, K. Bergmann, B. W. Shore, and S. Rosenwaks, “Population transfer with delayed pulses in four-state systems”, Physical Review A, vol. 45, pp. 4888-4896, 7 1992. DOI: 10.1103/PhysRevA.45.4888.][N. V. Vitanov and S. Stenholm, “Analytic properties and effective twolevel problems in stimulated Raman adiabatic passage”, Physical Review A, vol. 55, pp. 648-660, 1 1997. DOI: 10.1103/PhysRevA.55.648.][V. M. Babich and V. S. Buldyrev, Asymptotic methods in short-wavelength diffraction theory (Alpha Science Series on Wave Phenomena), English. Harrow, UK: Alpha Science International, 2009.][Y. A. Kravtsov and Y. I. Orlov, Geometrical optics of inhomogeneous media. Berlin: Springer-Verlag, 1990.][A. L. Sevastyanov, “Single-mode waveguide spread of light in a smooth irregular integral optical waveguide [Komp’yuternoe modelirovanie polej napravlyaemyh mod tonkoplenochnoj obobshchennoj volnovodnoj linzy Lyuneberga]”, in Russian, Ph.D. dissertation, Peoples’ Friendship University of Russia, Moscow, 2010.][M. D. Malykh, “On integration of the first order differential equations in a finite terms”, Journal of Physics: Conference Series, vol. 788, p. 012026, 2017. DOI: 10.1088/1742-6596/788/1/012026.][A. D. Polyanin and V. E. Nazaikinskii, Handbook of linear partial differential equations for engineers and scientists, 2nd ed. Boca Raton, London: CRC Press, 2016. DOI: 10.1201/b19056.]