On the equivalence of differential and integral formulations of the problem for eigenwaves of weakly guiding dielectric waveguides
https://doi.org/10.26907/2541-7746.2025.3.566-587
Abstract
The use of the method of Muller boundary integral equations for solving the problem of eigenwaves of weakly guiding dielectric waveguides was justified. A theorem was proved about the spectral equivalence between the original differential problem and the problem for the system of Muller boundary integral equations on the physical sheet of the Riemann surface, where the eigenvalues, the propagation constants of the eigenwaves, are sought. With this aim, the localization regions of the spectra of the original problem and a so-called “turned inside out” problem generating spurious eigenvalues were analyzed. A sufficient condition of equivalence was obtained: the problems are equivalent if the problem turned inside out has only a trivial solution. Consequently, as confirmed by the results of the numerical experiments, only true eigenvalues on the physical sheet of the Riemann surface can be found by using the method of Muller boundary integral equations.
About the Authors
E. E. Shabardin
Kazan Federal University
Russian Federation
Russian Federation
Egor E. Shabardin, Student
Kazan
E. M. Karchevskii
Kazan Federal University
Russian Federation
Russian Federation
Evgenii M. Karchevskii, Dr. Sci. (Physics and Mathematics), Full Professor
Kazan
References
1. Snyder A.W., Love J.D. Optical Waveguide Theory. New York, NY, Springer, 1983. viii, 738 p. https://doi.org/10.1007/978-1-4613-2813-1.
2. Kunz K.S., Luebbers R.J. The Finite Difference Time Domain Method for Electromagnetics. Boca Raton, FL, CRC Press, 1993. 464 p. https://doi.org/10.1201/9780203736708.
3. Taflove A., Hagness S.C. Computational Electrodynamics: The Finite-Difference Time-Domain Method. Boston, MA, London, Artech House, 2005. 1038 p.
4. Elsherbeni A.Z., Demir V. The Finite-Difference Time-Domain Method for Electromagnetics with MATLAB Simulations. Ser.: The ACES Series on Computational Electromagnetics and Engineering. SciTech Publ., 2015. 560 p. https://doi.org/10.1049/SBEW514E.
5. Obayya S.Computational Photonics. Chichester, Hoboken, NJ, John Wiley & Sons, 2010. xv, 306 p. https://doi.org/10.1002/9780470667064.
6. Shao Y., Peng Z., Lim K.H., Lee J.-F. Non-conformal domain decomposition methods for timeharmonic Maxwell equations. Proc. R. Soc. A, 2012, vol. 468, no. 2145, pp. 2433–2460. https://doi.org/10.1098/rspa.2012.0028.
7. Koczka G., Bauernfeind T., Preis K., B´ır´o O. An iterative domain decomposition method for solving wave propagation problems. Electromagnetics, 2014, vol. 34, nos. 3–4, pp. 210–221. https://doi.org/10.1080/02726343.2014.877751.
8. Beilina L., Lindstr¨om E. An adaptive finite element/finite difference domain decomposition method for applications in microwave imaging. Electronics, 2022, vol. 11, no. 9, art. 1359. https://doi.org/10.3390/electronics11091359.
9. Wartak M.S. Computational Photonics: An Introduction with MATLAB. Cambridge, Cambridge Univ. Press, 2013. xiv, 452 p. https://doi.org/10.1017/CBO9780511794247.
10. Lavrinenko A.V., Lægsgaard J., Gregersen N., Schmidt F., Søndergaard T. Numerical Methods in Photonics. Boca Raton, FL, CRC Press, 2015. 364 p. https://doi.org/10.1201/b17408.
11. Dautov R.Z., Karchevskii E.M. Numerical modeling of optical fibers using the finite element method and an exact non-reflecting boundary condition. Comput. Methods Appl. Math., 2018, vol. 18, no. 4, pp. 581–601. https://doi.org/10.1515/cmam-2017-0049.
12. Colton D., Kress R. Integral Equation Methods in Scattering Theory. Ser.: Classics in Applied Mathematics. Vol. 72. Philadelphia, PA, SIAM, 2013. 287 p.
13. M¨uller C. Foundations of the Mathematical Theory of Electromagnetic Waves. Ser.: Grundlehren der mathematischen Wissenschaften. Vol. 155. Berlin, Heidelberg, Springer, 1969. viii, 356 p. https://doi.org/10.1007/978-3-662-11773-6.
14. Karchevskii E.M. Analysis of the eigenmode spectra of dielectric waveguides. Comput. Math. Math. Phys., 1999, vol. 39, no. 9, pp. 1493–1498.
15. Karchevskii E.M. The fundamental wave problem for cylindrical dielectric waveguides. Differ. Equations, 2000, vol. 36, no. 7, pp. 1109–1111. https://doi.org/10.1007/BF02754515.
16. Frolov A., Kartchevskiy E. Integral equation methods in optical waveguide theory. In: Beilina L., Shestopalov Y.V. (Eds.) Inverse Problems and Large-Scale Computations. Ser.: Springer Proceedings in Mathematics & Statistics. Vol. 52. Cham, Springer, 2013, pp. 119–133. https://doi.org/10.1007/978-3-319-00660-4_9.
17. Spiridonov A.O., Karchevskiy E.M. Projection methods for computation of spectral characteristics of weakly guiding optical waveguides. Proc. Int. Conf. Days on Diffraction 2013. St. Petersburg, IEEE, 2013, pp. 131–135. http://dx.doi.org/10.1109/DD.2013.6712817.
18. Boriskina S.V., Sewell P., Benson T.M., Nosich A.I. Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization. J. Opt. Soc. Am. A, 2004, vol. 21, no. 3, pp. 393–402. http://dx.doi.org/10.1364/JOSAA.21.000393.
19. Nosich A.I., Smotrova E.I., Boriskina S.V., Benson T.M., Sewell P. Trends in microdisk laser research and linear optical modelling. Opt. Quantum Electron., 2007, vol. 39, no. 15, pp. 1253– 1272. http://dx.doi.org/10.1007/s11082-008-9203-z.
20. Smotrova E.I., Byelobrov V.O., Benson T.M., Ctyrok´y J., Sauleau R., Nosich A.I. Optical theorem ˇ helps understand thresholds of lasing in microcavities with active regions. IEEE J. Quantum Electron., 2011, vol. 47, no. 1, pp. 20–30. http://dx.doi.org/10.1109/JQE.2010.2055836.
21. Smotrova E.I., Tsvirkun V., Gozhyk I., Lafargue C., Ulysse C., Lebental M., Nosich A.I. Spectra, thresholds, and modal fields of a kite-shaped microcavity laser. J. Opt. Soc. Am. B, 2013, vol. 30, no. 6, pp. 1732–1742. http://dx.doi.org/10.1364/JOSAB.30.001732.
22. Spiridonov A.O., Karchevskii E.M., Nosich A.I. Spectra, thresholds, and modal fields of a circular microcavity laser transforming into a square. Proc. 2015 17th Int. Conf. on Transparent Optical Networks (ICTON). Budapest, IEEE, 2015, pp. 1–4. http://dx.doi.org/10.1109/ICTON.2015.7193654.
23. Spiridonov A.O., Karchevskii E.M. Mathematical and numerical analysis of the spectral characteristics of dielectric microcavities with active regions. Proc. 2016 Days on Diffraction (DD). St. Petersburg, IEEE, 2016, pp. 390–395. http://dx.doi.org/10.1109/DD.2016.7756880.
24. Spiridonov A.O., Karchevskii E.M., Nosich A.I. Mathematical and numerical modeling of onthreshold modes of 2-D microcavity lasers with piercing holes. Axioms, 2019, vol. 8, no. 3, art. 101. http://dx.doi.org/10.3390/axioms8030101.
25. Oktyabrskaya A.O., Repina A.I., Spiridonov A.O., Karchevskii E.M., Nosich A.I. Numerical modeling of on-threshold modes of eccentric-ring microcavity lasers using the Muller integral equations and the trigonometric Galerkin method. Opt. Commun., 2020, vol. 476, art. 126311. http://dx.doi.org/10.1016/j.optcom.2020.126311.
26. Spiridonov A.O., Oktyabrskaya A.O., Karchevskii E.M., Nosich A.I. Mathematical and numerical analysis of the generalized complex-frequency eigenvalue problem for two-dimensional optical microcavities. SIAM J. Appl. Math., 2020, vol. 80, no. 4, pp. 1977–1998. http://dx.doi.org/10.1137/19M1261882.
27. Yi C.-H., Dietz B., Han J.-H., Ryu J.-W. Decay rates of optical modes unveiling the island structures in mixed phase space. Phys. Rev. A, 2025, vol. 111, no. 3, art. 033509. https://doi.org/10.1103/PhysRevA.111.033509.
28. Du W., Li C., Sun J., Xu H., Yu P., Ren A., Wu J., Wang Z. Nanolasers based on 2D materials. Laser Photonics Rev., 2020, vol. 14, no. 12, art. 2000271. http://dx.doi.org/10.1002/lpor.202000271.
29. Smotrova E.I., Nosich A.I., Benson T.M., Sewell P. Cold-cavity thresholds of microdisks with uniform and nonuniform gain: Quasi-3-D modeling with accurate 2-D analysis. IEEE J. Sel. Top. Quantum Electron., 2005, vol. 11, no. 5, pp. 1135–1142. http://dx.doi.org/10.1109/JSTQE.2005.853848.
30. Smotrova E.I., Nosich A.I. Mathematical study of the two-dimensional lasing problem for the whispering-gallery modes in a circular dielectric microcavity. Opt. Quantum Electron., 2004, vol. 36, no. 1, pp. 213–221. http://dx.doi.org/10.1023/B:OQEL.0000015641.19947.9c.
31. Herasymova D.O., Dukhopelnykov S.V., Natarov D.M., Zinenko T.L., Lucido M., Nosich A.I. Threshold conditions for transversal modes of tunable plasmonic nanolasers shaped as single and twin graphene-covered circular quantum wires. Nanotechnology, 2022, vol. 33, no. 49, art. 495001. http://dx.doi.org/10.1088/1361-6528/ac8e0c.
32. Kaliberda M.E., Pogarsky S.A., Kostenko O.V., Nosych O.I., Zinenko T.L. Circular quantum wire symmetrically loaded with a graphene strip as the plasmonic micro/nano laser: Threshold conditions analysis. Opt. Express, 2024, vol. 32, no. 7, pp. 12213–12227. https://doi.org/10.1364/OE.514643.
33. Yevtushenko F.O., Dukhopelnykov S.V., Zinenko T.L., Nosych O.I. Microsize graphene strip grating on gain substrate laser: Comparison of the H- and E-polarized mode threshold conditions. Opt. Express, 2025, vol. 33, no. 6, pp. 14266–14287. https://doi.org/10.1364/OE.554771.
34. Oktyabrskaya A.O., Spiridonov A.O., Karchevskii E.M. Muller boundary integral equations for solving generalized complex-frequency eigenvalue problem. Lobachevskii J. Math., 2020, vol. 41, no. 7, pp. 1377–1384. https://doi.org/10.1134/s1995080220070343.
35. Ketov I.V., Karchevskii E.M. Characteristic equations for true and fictitious eigenvalues of Muller boundary integral equations. Lobachevskii J. Math., 2023, vol. 44, no. 9, pp. 4016–4027. http://dx.doi.org/10.1134/S1995080223090147.
36. Misawa R., Niino K., Nishimura N. Boundary integral equations for calculating complex eigenvalues of transmission problems. SIAM J. Appl. Math., 2017, vol. 77, no. 2, pp. 770–788. https://doi.org/10.1137/16m1087436.
37. Steinbach O., Unger G. Combined boundary integral equations for acoustic scattering-resonance problems. Math. Methods Appl. Sci., 2016, vol. 40, no. 5, pp. 1516–1530. https://doi.org/10.1002/mma.4075.
38. Shabardin E.E., Spiridonov A.O., Repina A.I., Karchevskii E.M. Numerical modeling of photonic crystal fibers using Muller boundary integral equations. Lobachevskii J. Math., 2024, vol. 45, no. 10, pp. 4787–4797. https://doi.org/10.1134/S1995080224605861.
39. Abramowitz M., Stegun I.A. (Eds.) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards. Ser.: Applied Mathematics Series. Vol. 55. Washington, DC, US Gov. Print. Office, 1972. xiv, 1046 p.
40. Spiridonov A.O., Karchevskii E.M., Nosich A.I. Rigorous formulation of the lasing eigenvalue problem as a spectral problem for a Fredholm operator function. Lobachevskii J. Math., 2018, vol. 39, no. 8, pp. 1148–1157. https://doi.org/10.1134/s1995080218080127.
41. Ketov I.V., Oktyabrskaya A.O., Spiridonov A.O., Karchevskii E.M. True and fictitious eigenvalues of the set of Muller boundary integral equations. Software Syst., 2022, vol. 35, no. 3, pp. 316–328. https://doi.org/10.15827/0236-235X.139.316-328. (In Russian)
Review
For citations:
Shabardin E.E., Karchevskii E.M. On the equivalence of differential and integral formulations of the problem for eigenwaves of weakly guiding dielectric waveguides. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki. 2025;167(3):566-587. (In Russ.) https://doi.org/10.26907/2541-7746.2025.3.566-587
Views:
61
Email this article 