Single-mode propagation of adiabatic guided modes in smoothly irregular integral optical waveguides

Cover Page

Cite item

Full Text

Abstract

This paper investigates the waveguide propagation of polarized electromagnetic radiation in a thin-film integral optical waveguide. To describe this propagation, the adiabatic approximation of solutions of Maxwell’s equations is used. The construction of a reduced model for adiabatic waveguide modes that retains all the properties of the corresponding approximate solutions of the Maxwell system of equations was carried out by the author in a previous publication in DCM & ACS, 2020, No 3. In this work, for a special case when the geometry of the waveguide and the electromagnetic field are invariant in the transverse direction. In this case, there are separate nontrivial TEand TM-polarized solutions of this reduced model. The paper describes the parametrically dependent on longitudinal coordinates solutions of problems for eigenvalues and eigenfunctions - adiabatic waveguide TE and TM polarizations. In this work, we present a statement of the problem of finding solutions to the model of adiabatic waveguide modes that describe the stationary propagation of electromagnetic radiation. The paper presents solutions for the single-mode propagation of TE and TM polarized adiabatic waveguide waves.

Full Text

1. Introduction In works [1]-[5] a cycle of studies of the propagation of polarized light in integrated-optical smoothly irregular thin-film waveguides was carried out within the framework of the model of adiabatic waveguide waves. They showed the advantages of the model and its advantages over other models in the description of open dielectric waveguides [6]-[8]. At the same time, until recently, the question of substantiating this model remained open. In work [9] the substantiation of the model was carried out, which is a reduction of a more complex in use general model based on Maxwell’s equations. In the present work, within the framework of the model of adiabatic waveguide waves, the problem of stationary propagation of polarized light in a smoothly irregular © Sevastianov A. L., 2020 This work is licensed under a Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/ integral-optical waveguide is posed, an auxiliary problem for eigenvalues and eigenfunctions (adiabatic waveguide modes) is formulated and solved. The solution of the stationary problem by the generalized Kantorovich method is proposed, its solution is obtained in the single-mode propagation mode. 2. Basic concepts and notation Waveguide propagation of monochromatic polarized electromagnetic radiation in integrated optical waveguides is described by Maxwell’s equations. The electromagnetic field is described using complex amplitudes. The material environment is considered, consisting of dielectric subdomains that fill the entire three-dimensional space. The latter means that the dielectric constants of the subdomains are different and real, and the magnetic permeability is everywhere equal to the magnetic permeability of the vacuum. It follows from the foregoing that in the absence of external currents and charges, the induced currents and charges are equal to zero. In the absence of external charges and currents, the scalar Maxwell equations follow from the vector ones, and the boundary conditions for the normal components follow from the boundary conditions for the tangential components. The constitutive equations of connection in the case under consideration are assumed to be linear. Thus, the electromagnetic field in a space filled with dielectrics in the Gaussian system of units is described by the equations [10]: rotE = - 1

×

About the authors

Anton L. Sevastianov

Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.
Email: sevastianov-al@rudn.ru

Candidate of Physical and Mathematical Sciences, assistant professor of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

References

  1. L. A. Sevastianov and A. A. Egorov, “Theoretical analysis of the waveguide propagation of electromagnetic waves in dielectric smoothlyirregular integrated structures,” Optics and Spectroscopy, vol. 105, no. 4, pp. 576-584, 2008. doi: 10.1134/S0030400X08100123.
  2. A. A. Egorov and L. A. Sevastianov, “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.
  3. A. A. Egorov, K. P. Lovetskiy, A. L. Sevastianov, and L. A. Sevastianov, “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.
  4. A. A. Egorov et al., “Adiabatic modes of smoothly irregular optical waveguide: zero approximation of vector theory [Adiabaticheskie mody plavno-neregulyarnogo opticheskogo volnovoda: nulevoe priblizhenie vektornoj teorii],” Russian, Matem. modelirovaniye, vol. 22, no. 8, pp. 42- 54, 2010, [in Russian].
  5. A. A. Egorov, A. L. Sevast’yanov, and L. A. Sevast’yanov, “Stable computer modeling of thin-film generalized waveguide Luneburg lens,” Quantum Electronics, vol. 44, no. 2, pp. 167-173, 2014. DOI: 10.1070/ QE2014v044n02ABEH015303.
  6. A. A. Egorov, K. P. Lovetsky, A. L. Sevastyanov, and L. A. Sevastyanov, “Luneberg Thin-Film Waveguide Lens: From Problem Statement to Solution. Theory and mathematical modeling of adiabatic modes [Tonkoplenochnaya volnovodnaya linza Lyuneberga: ot postanovki problemy do ee resheniya. Teoriya i matematicheskoe modelirovanie adiabaticheskih mod],” Russian, in Trudy RNTORES im. A.S. Popova. Vyp. 5. The 5th International Conference “Acousto-Optical and Radar Measurement and Information Processing Methods” (ARMIMP-2012), Moscow-Suzdal, [in Russian], 2012, pp. 186-190.
  7. A. A. Egorov, A. L. Sevastyanov, E. A. Ayryan, and L. A. Sevastyanov, “Stable computer modeling of thin-film generalized waveguide Luneburg lens [Ustojchivoe komp’yuternoe modelirovanie tonkoplenochnoj obobshchennoj volnovodnoj linzy Lyuneberga],” Russian, Matem. modelirovaniye, vol. 26, no. 11, pp. 37-44, 2014, [in Russian].
  8. E. Ayryan, G. Dashitsyrenov, E. Laneev, K. Lovetskiy, L. Sevastianov, and A. Sevastianov, “Mathematical synthesis of the thickness profile of the waveguide Lüneburg lens using the adiabatic waveguide modes method,” in Saratov Fall Meeting 2016: Laser Physics and Photonics XVII; and Computational Biophysics and Analysis of Biomedical Data III, V. L. Derbov, D. E. Postnov, V. L. Derbov, and D. E. Postnov, Eds., International Society for Optics and Photonics, vol. 10337, SPIE, 2017, pp. 134-145. doi: 10.1117/12.2267920.
  9. A. L. Sevastyanov, “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. 28, no. 3, pp. 252-273, 2020. doi: 10.22363/2658-4670- 2020-28-3-252-273.
  10. A. S. Il’inskii, V. V. Kravtsov, and A. G. Sveshnikov, Mathematical Models of Electrodynamics [Matematicheskie modeli elektrodinamiki], Russian. Moscow: Vyssh. Shkola, 1991, [in Russian].
  11. L. A. Sevastyanov, A. A. Egorov, and A. L. Sevastyanov, “Method of adiabatic modes in studying problems of smoothly irregular open waveguide structures,” Physics of Atomic Nuclei, vol. 776, no. 2, pp. 224- 239, 2013. doi: 10.1134/S1063778813010134.
  12. L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Analysis. New York: Wiley, 1964.
  13. M. V. Fedoryuk, “A justification of the method of transverse sections for an acoustic wave guide with nonhomogeneous content,” Mathematical Physics, vol. 13, no. 1, pp. 162-173, 1973. doi: 10.1016/0041-5553(74) 90012-3.
  14. B. Z. Katsenelenbaum, Theory of Irregular Waveguides with Slowly Varying Parameters [Teoriya neregulyarnyh volnovodov s medlenno menyayushchimisya parametrami], Russian. Moscow: Akad. Nauk SSSR, 1961, [in Russian].
  15. V. V. Shevchenko, Continuous Transitions in Open Waveguides [Plavnye perekhody v otkrytyh volnovodah], Russian. Moscow: Nauka, 1969, [in Russian].
  16. M. J. Adams, An Introduction to Optical Waveguides. New York: Wiley, 1981.
  17. T. Tamir, “Guided-wave optoelectronics,” in Integrated Optics, T. Tamir, Ed., Berlin: Springer-Verlag, 1990.
  18. A. W. Snyder and J. D. Love, Optical Waveguide Theory. New York: Chapman and Hall, 1983.
  19. D. V. Divakov and A. L. Sevastianov, “The Implementation of the Symbolic-Numerical Method for Finding the Adiabatic Waveguide Modes of Integrated Optical Waveguides in CAS Maple,” in Computer Algebra in Scientific Computing, M. England, W. Koepf, T. M. Sadykov, W. M. Seiler, and E. V. Vorozhtsov, Eds., Cham: Springer International Publishing, 2019, pp. 107-121. doi: 10.1007/978-3-030-26831-2_8.
  20. D. V. Divakov, A. A. Tiutiunnik, and A. L. Sevastianov, “SymbolicNumeric Study of Geometric Properties of Adiabatic Waveguide Modes,” in Computer Algebra in Scientific Computing, F. Boulier, M. England, T. M. Sadykov, and E. V. Vorozhtsov, Eds., Cham: Springer International Publishing, 2020, pp. 228-244. doi: 10.1007/978-3-030-600266_13.
  21. D. V. Divakov and A. A. Tiutiunnik, “Symbolic study of eigenvectors for constructing a general solution to a system of ODEs with a symbolic matrix of coefficients,” Programmirovaniye, no. 2, pp. 3-16, 2020.
  22. L. A. Sevastyanov, A. L. Sevastyanov, and A. A. Tyutyunnik, “Analytical Calculations in Maple to Implement the Method of Adiabatic Modes for Modelling Smoothly Irregular Integrated Optical Waveguide Structures,” in Computer Algebra in Scientific Computing, V. P. Gerdt, W. Koepf, W. M. Seiler, and E. V. Vorozhtsov, Eds., Cham: Springer International Publishing, 2014, pp. 419-431. doi: 10.1007/978-3-319-10515-4_30.
  23. A. A. Egorov, K. P. Lovetskii, A. L. Sevastianov, and L. A. Sevastianov, Integrated optics: theory and computer modeling [Integral’naya optika: teoriya i komp’yuternoe modelirovanie], Russian. Moscow: PFUR Publishing house, 2015, [in Russian].
  24. A. A. Egorov, K. P. Lovetskii, A. L. Sevastyanov, and L. A. Sevastyanov, “Model of a smoothly irregular multilayer integrated-optical waveguide in the zero vector approximation: theory and numerical analysis [Model’ mnogoslojnogo plavno-neregulyarnogo integral’no-opticheskogo volnovoda v nulevom vektornom priblizhenii: teoriya i chislennyj analiz],” Russian, Zhurnal radioelektroniki, no. 3, 2019, [in Russian]. DOI: 10. 30898/1684-1719.2019.3.11.
  25. M. D. Malykh, “On integration of the first order differential equations in a finite terms,” Journal of Physics: Conference Series, vol. 788, p. 012 026, Jan. 2017. doi: 10.1088/1742-6596/788/1/012026.
  26. A. D. Polyanin and V. E. Nazaikinskii, Handbook of linear partial differentialequations for engineers and scientists, 2nd Edition. BocaRaton, London: CRC Press, 2016.
  27. A. A. Samarskiy and A. N. Tikhonov, “Excitation of radio waveguides. I [O vozbuzhdenii radiovolnovodov. I ],” Russian, Zhurnal tekhnicheskoy fiziki, vol. 17, no. 11, pp. 1283-1296, 1947, [in Russian].
  28. A. A. Samarskiy and A. N. Tikhonov, “Excitation of radio waveguides. II [O vozbuzhdenii radiovolnovodov. II ],” Russian, Zhurnal tekhnicheskoy fiziki, vol. 17, no. 12, pp. 1431-1440, 1947, [in Russian].
  29. A. A. Samarskiy and A. N. Tikhonov, “Excitation of radio waveguides. III [O vozbuzhdenii radiovolnovodov. III ],” Russian, Zhurnal tekhnicheskoy fiziki, vol. 18, no. 7, pp. 971-983, 1948, [in Russian].
  30. A. A. Samarskii and A. N. Tikhonov, “Representation of the field in a waveguide as the sum of the TE and TM fields [O predstavlenii polya v volnovode v vide summy polej TE i TM],” Russian, Zhurnal tekhnicheskoy fiziki, vol. 18, no. 7, pp. 959-970, 1948, [in Russian].
  31. V. V. Shevchenko, “Spectral decomposition in eigenand associated functions of a nonselfadjoint problem of Sturm-Liouville type on the entire axis [O spektral’nom razlozhenii po sobstvennym i prisoedinennym funkciyam odnoj nesamosopryazhennoj zadachi tipa Shturma-Liuvillya na vsej osi],” Russian, Differ. Uravn., vol. 15, no. 11, pp. 2004-2020, 1979, [in Russian].
  32. E. M. Karchevskii, “Determination of the propagation constants of dielectric-waveguide eigenmodes by methods of potential theory,” Computational Mathematics and Mathematical Physics, vol. 38, no. 1, pp. 132- 136, 1998.
  33. R. Z. Dautov and E. M. Karchevskii, “On a spectral problem of the theory of dielectric waveguides,” Computational Mathematics and Mathematical Physics, vol. 39, no. 8, pp. 1293-1299, 1999.
  34. E. M. Karchevskii, “Analysis of the eigenmode spectra of dielectric waveguides,” Computational Mathematics and Mathematical Physics, vol. 39, no. 9, pp. 1493-1498, 1999.
  35. E. M. Karchevskii, “Investigation of a numerical method for solving a spectral problem in the theory of dielectric waveguides,” Russian Mathematics (Izvestiya VUZ. Matematika), vol. 43, no. 1, pp. 8-15, 1999.
  36. R. Z. Dautov and E. M. Karchevskii, “Existence and properties of solutions to the spectral problem of the dielectric waveguide theory,” Computational Mathematics and Mathematical Physics, vol. 40, no. 8, pp. 1200-1213, 2000.
  37. R. Z. Dautov and E. M. Karchevskii, “Solution of the vector problem of the natural waves of cylindrical dielectric waveguides based on a nonlocal boundary condition,” Computational Mathematics and Mathematical Physics, vol. 42, no. 7, pp. 1012-1027, 2002.
  38. E. M. Karchevskii and S. I. Solov’ev, “Existence of eigenvalues of a spectral problem in the theory of dielectric waveguides,” Russian Mathematics (Izvestiya VUZ. Matematika), vol. 47, no. 3, pp. 75-77, 2003.
  39. E. M. Karchevskii, A. I. Nosich, and G. W. Hanson, “Mathematical analysis of the generalized natural modes of an inhomogeneous optical fiber,” SIAM Journal on Applied Mathematics, vol. 65, no. 6, pp. 2033- 2048, 2005. doi: 10.1137/040604376.
  40. A. F. Stevenson, “General Theory of Electromagnetic Horns,” Journal of Applied Physics, vol. 22, no. 12, p. 1447, 1951. doi: 10.1063/1.1699891.
  41. S. A. Schelkunoff, “Conversion of Maxwell’s equations into generalized Telegraphist’s equations,” The Bell System Technical Journal, vol. 34, no. 5, pp. 995-1043, 1955. doi: 10.1002/j.1538-7305.1955.tb03787. x.
  42. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE Journal of Quantum Electronics, vol. 9, pp. 919-933, 1973. doi: 10.1109/JQE. 1973.1077767.
  43. L. V. Kantorovich, “A direct method for the approximate solution of the problem of the minimum of a double integral [Odin pryamoj metod priblizhennogo resheniya zadachi o minimume dvojnogo integrala],” Russian, Izvestiya Akademii nauk SSSR. VII seriya. Otdeleniye matematicheskikh i yestestvennykh nauk, no. 5, pp. 647-652, 1933, [in Russian].
  44. B. Z. Katsenelenbaum, “Irregular waveguides with slowly varying parameters [Neregulyarnye volnovody s medlenno menyayushchimisya parametrami],” Russian, Doklady Akademii Nauk SSSR, vol. 102, no. 4, p. 711, 1955, [in Russian].
  45. B. Z. Katsenelenbaum, “On the general theory of irregular waveguides [K obshchej teorii neregulyarnyh volnovodov],” Russian, Doklady Akademii Nauk SSSR, vol. 116, no. 2, pp. 203-206, 1957, [in Russian].
  46. A. G. Sveshnikov, “An approximate method for calculating a weakly irregular waveguide [Priblizhennyj metod rascheta slabo neregulyarnogo volnovoda],” Russian, Doklady Akademii Nauk SSSR, vol. 110, no. 2, pp. 197-199, 1956, [in Russian].
  47. G. Y. Lyubarsky and A. Y. Povzner, “On the theory of wave propagation in irregular waveguides [K teorii rasprostraneniya voln v neregulyarnyh volnovodah],” Russian, Zhurnal tekhnicheskoy fiziki, no. 29, pp. 170-179, 1959, [in Russian].
  48. N. E. Maltsev, “Some modifications of the method of cross sections [Nekotorye modifikacii metoda poperechnyh sechenij],” Russian, Akusticheskii zhurnal, no. 16, pp. 102-109, 1970, [in Russian].
  49. B. Z. Katsenelenbaum, “Curved waveguides of constant cross-section [Izognutye volnovody postoyannogo secheniya],” Russian, Radiotekhnika i elektronika, no. 2, pp. 171-185, 1956, [in Russian].
  50. B. Z. Katsenelenbaum, “Symmetric dielectric transition in a circular waveguide for the H01 wave [Simmetrichnyj dielektricheskij perekhod v volnovode kruglogo secheniya dlya volny H01],” Russian, Radiotekhnika i elektronika, no. 3, p. 339, 1956, [in Russian].
  51. B. Z. Katsenelenbaum, “Long symmetric waveguide transition for the H01 wave [Dlinnyj simmetrichnyj volnovodnyj perekhod dlya volny H01],” Russian, Radiotekhnika i elektronika, no. 5, pp. 531-546, 1957, [in Russian].
  52. A. G. Sveshnikov, “On the propagation of radio waves in weakly curved waveguides [O rasprostranenii radiovoln v slaboizognutyh volnovodah],” Russian, Radiotekhnika i elektronika, vol. 1, no. 9, p. 1222, 1956, [in Russian].
  53. A. G. Sveshnikov, “Waves in curved pipes [Volny v izognutyh trubah],” Russian, Radiotekhnika i elektronika, vol. 3, no. 5, p. 641, 1958, [in Russian].
  54. A. G. Sveshnikov, “Irregular waveguides [Neregulyarnye volnovody],” Russian, Izv. Vuzov. Radiofizika, vol. 2, no. 5, p. 720, 1959, [in Russian].
  55. A. G. Sveshnikov, “An approximate method for calculating a weakly irregular waveguide [Priblizhennyj metod rascheta slabo neregulyarnogo volnovoda],” Russian, Doklady Akademii Nauk SSSR, vol. 80, no. 3, pp. 345-347, 1956, [in Russian].
  56. A. G. Sveshnikov, “On the proof of a method of calculation for irregular waveguides,” USSR Computational Mathematics and Mathematical Physics, vol. 3, no. 1, pp. 219-232, 1963.
  57. A. G. Sveshnikov, “A substantiation of a method for computing the propagation of electromagnetic oscillations in irregular waveguides,” USSR Computational Mathematics and Mathematical Physics, vol. 3, no. 2, pp. 413-429, 1963. doi: 10.1016/0041-5553(63)90027-2.
  58. A. G. Sveshnikov, “On the bending of waveguides,” USSR Computational Mathematics and Mathematical Physics, vol. 1, no. 3, pp. 882-888, 1962.
  59. A. G. Sveshnikov and A. S. Il’inskii, “Calculation of waveguide transition of composite form,” USSR Computational Mathematics and Mathematical Physics, vol. 3, no. 3, pp. 635-649, 1963.
  60. A. S. Il’inskii and A. G. Sveshnikov, “Methods for investigating irregular waveguides,” USSR Computational Mathematics and Mathematical Physics, vol. 8, no. 2, pp. 167-180, 1968.
  61. A. G. Sveshnikov, “The incomplete Galerkin method [Nepolnyj metod Galerkina],” Russian, Dokl. Akad. Nauk SSSR, vol. 236, no. 5, pp. 1076- 1079, 1977, [in Russian].
  62. A. N. Bogolyubov and A. G. Sveshnikov, “Application of an iteration method to the investigation of plane waveguides with inhomogeneous filling,” USSR Computational Mathematics and Mathematical Physics, vol. 14, no. 4, pp. 125-133, 1974.
  63. A. N. Bogolyubov and A. G. Sveshnikov, “Justification of a finite-difference method for analyzing optical waveguides,” USSR Computational Mathematics and Mathematical Physics, vol. 19, no. 6, pp. 139-150, 1979.
  64. A. N. Bogolyubov, A. L. Delitsyn, and A. G. Sveshnikov, “On the problem of excitation of a waveguide filled with an inhomogeneous medium,” Computations mathematics and mathematical physics, vol. 39, no. 11, pp. 1794-1813, 1999.
  65. A. N. Bogolyubov and M. D. Malykh, “Remark on the Radiation Conditions for an Irregular Waveguide,” Computations mathematics and mathematical physics, vol. 43, no. 4, pp. 560-563, 2003.
  66. A. N. Bogolyubov and M. D. Malykh, “Theory of Perturbations of Spectral Characteristics of Waveguide Systems,” Computations mathematics and mathematical physics, vol. 43, no. 7, pp. 1049-1061, 2003.

Copyright (c) 2020 Sevastianov A.L.

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies