Convergence of the Galerkin method for solving a nonlinear problem of the eigenmodes of microdisk lasers

A.I. Repina

Kazan Federal University, Kazan, 420008 Russia

E-mail: airepinas@gmail.com

Received January 15, 2021


ORIGINAL ARTICLE

Full text PDF

DOI: 10.26907/2541-7746.2021.1.5-20

For citation: Repina A.I. Convergence of the Galerkin method for solving a nonlinear problem of the eigenmodes of microdisk lasers. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2021, vol. 163, no. 1, pp. 5–20. doi: 10.26907/2541-7746.2021.1.5-20. (In Russian)

Abstract

This paper investigates an eigenvalue problem for the Helmholtz equation on the plane modeling the laser radiation of two-dimensional microdisk resonators. It was reduced to an eigenvalue problem for a holomorphic Fredholm operator-valued function A(k). For its numerical solution, the Galerkin method was proposed, and its convergence was proved. Namely, a sequence of the finite-dimensional holomorphic operator functions An(k) that converges regularly to A(k) was constructed. Further, it was established that there is a sequence of eigenvalues kn of the operator-valued functions An(k) converging to k0 for each eigenvalue k0 of the operator-valued function A(k). If {kn}n∈N is a sequence of eigenvalues of the operator-valued functions An(k) converging to a number of k0 , then k0 is an eigenvalue of A(k). The estimates for the rate of convergence of {kn}n∈N to k0 depend either on the order of the pole k0 of the operator-valued function A−1(k), or on the algebraic multiplicities of all eigenvalues of An(k) in a neighborhood of k0 , or on the number of different eigenvalues of An(k) in this neighborhood. The reasoning is based on the fundamental results of the theory of holomorphic operator-valued functions and is important for the theory of microdisk lasers, because it significantly expands the class of devices interesting for applications that allow mathematical modeling based on numerical methods that are strictly justified.

Keywords: microdisk laser, nonlinear eigenvalue problem, system of Muller boundary integral equations, Galerkin method

Acknowledgments. The study was supported by the Kazan Federal University Strategic Academic Leadership Program.

References

  1. 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, pp. 1–16. doi: 10.1002/lpor.202070066.
  2. 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. doi: 10.1007/s11082-008-9203-z.
  3. 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. doi: 10.1364/josaa.21.000393.
  4. 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. 3, pp. 213–221. doi: 10.1023/B:OQEL.0000015641.19947.9c.
  5. Spiridonov A.O., Karchevskii E.M., Benson T.M., Nosich A.I. Why elliptic microcavity lasers emit light on bow-tie-like modes instead of whispering-gallery-like modes. Opt. Commun., 2019, vol. 439, pp. 112–117. doi: 10.1016/j.optcom.2019.01.056.
  6. Zolotukhina A.S., Spiridonov A.O., Karchevskii E.M., Nosich A.I. Lasing modes of a microdisk with a ring gain area and of an active microring. Opt. Quantum Electron., 2015, vol. 47, no. 12, pp. 3883–3891. doi: 10.1007/s11082-015-0240-0.
  7. Zolotukhina A.S., Spiridonov A.O., Karchevskii E.M., Nosich A.I. Electromagnetic analysis of optimal pumping of a microdisk laser with a ring electrode. Appl. Phys. B: Lasers Opt., 2017, vol. 123, no. 1, art. 32, pp. 1–6. doi: 10.1007/s00340-016-6625-3.
  8. Spiridonov A.O., Karchevskii E.M. Mathematical and numerical analysis of the spectral characteristics of dielectric microcavities with active regions. Proc. Int. Conf. Days in Diffraction (DD-2016). IEEE, 2016, pp. 390–395. doi: 10.1109/DD.2016.7756880.
  9. 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. doi: 10.1364/JOSAB.30.001732.
  10. Mu¨ller C. Foundations of the Mathematical Theory of Electromagnetic Waves. Berlin, Heidelberg, Springer, 1969. 356 p.
  11. 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. doi: 10.1137/19M1261882.
  12. Kozlov V., Maz'ya V. Differential Equations with Operator Coefficients with Applications to Boundary Value Problems for Partial Differential Equations. Heidelberg, Springer, 1999. 444 p.
  13. Karma O. Approximation in eigenvalue problems for holomorphic Fredholm operator functions I. Numer. Funct. Anal. Optim., 1996, vol. 17, nos. 3–4, pp. 365–387. doi: 10.1080/01630569608816699.
  14. Karma O. Approximation in eigenvalue problems for holomorphic Fredholm operator functions II (convergence rate). Numer. Funct. Anal. Optim., 1996, vol. 17, nos. 3–4, pp. 389–408. doi: 10.1080/01630569608816700.
  15. 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, pp. 1–9. doi: 10.1016/j.optcom.2020.126311.
  16. Repina A.I., Oktyabrskaya A.O., Ketov I.V., Karchevskii E.M. Laser modes of active eccentric microring cavities. Proc. Int. Conf. on Transparent Optical Networks (ICTON-2020). IEEE, 2020, pp. 1–4. doi: 10.1109/ICTON51198.2020.9203752.
  17. Repina A.I., Oktyabrskaya A.O. Mathematical modeling of photonic crystal resonators based on the Lasing Eigenvalue Problem. Proc. Int. Conf. on Control Systems, Mathematical Modelling, Automation and Energy Efficiency (SUMMA-2019), 2019, pp. 472–477. doi: 10.1109/SUMMA48161.2019.8947540.
  18. Repina A.I., Oktyabrskaya A.O., Karchevskii E.M. Unidirectional emission of active eccentric microring cavities. Proc. 2020 IEEE East-West Design & Test Symp. (EWDTS-2020), 2020, pp. 274–278. doi: 10.1109/EWDTS50664.2020.9225153.
  19. Oktyabrskaya A.O., Repina A.I., Karchevskii E.M. Laser modes of active circular microcavity with circular piercing hole. Proc. Int. Conf. on Electronics and Nanotechnology (ELNANO-2020), 2020, pp. 207–210. doi: 10.1109/ELNANO50318.2020.9088894.
  20. Spiridonov A.O., Karchevskii E.M., Nosich A.I. Mathematical and numerical modeling of on-threshold modes of 2-D microcavity lasers with piercing holes. Axioms, 2019, vol. 8, no. 3, art. 101, pp. 1–16. doi: 10.3390/axioms8030101.
  21. Reichardt H. Ausstrahlungsbedingungen fur die Wellengleihung. Abh. Math. Semin. Hamburg, 1960, vol. 24, pp. 41–53. (In German)
  22. Kartchevski E.M., Nosich A.I., Hanson G.W. Mathematical analysis of the generalized natural modes of an inhomogeneous optical fiber. SIAM J. Appl. Math., 2005, vol. 65, no. 6, pp. 2033–2048.
  23. Colton D., Kress R. Integral Equation Methods in Scattering Theory. Philadelphia, SIAM, 2013. xvi, 271 p.
  24. Frolov A., Kartchevskiy E. Integral equation methods in optical waveguide theory. Springer Proc. Math. Stat., 2013, vol. 52, pp. 119–133.
  25. Karchevskii E.M. The fundamental wave problem for cylindrical dielectric waveguides. Diff. Equat., 2000, vol. 36, no. 7, pp. 1109–1111. doi: 10.1007/BF02754515.
  26. Spiridonov A.O., Karchevskii E.M. Projection methods for computation of spectral characteristics of weakly guiding optical waveguides. Proc. Int. Conf. Days in Diffraction (DD-2013), 2013, pp. 131–135. doi: 10.1109/DD.2013.6712817.
  27. 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. doi: 10.1134/S1995080218080127.
  28. Kress R. Linear integral equations. In: Springer Monographs in Mathematics. Vol. 82. New York, Springer, 1999. xvi, 412 p. doi: 10.1007/978-1-4614-9593-2.
  29. 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. doi: 10.1134/S1995080220070343.


The content is available under the license Creative Commons Attribution 4.0 License.