Exponentially Convergent Galerkin Method for Numerical Modeling of Lasing in Microcavities with Piercing Holes
The paper investigates an algorithm for the numerical solution of a parametric eigenvalue problem for the Helmholtz equation on the plane specially tailored for the accurate mathematical modeling of lasing modes of microring lasers. The original problem is reduced to a nonlinear eigenvalue problem f...
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MDPI AG
2021-08-01
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| Online Access: | https://www.mdpi.com/2075-1680/10/3/184 |
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| author | Alexander O. Spiridonov Anna I. Repina Ilya V. Ketov Sergey I. Solov’ev Evgenii M. Karchevskii |
| author_facet | Alexander O. Spiridonov Anna I. Repina Ilya V. Ketov Sergey I. Solov’ev Evgenii M. Karchevskii |
| author_sort | Alexander O. Spiridonov |
| collection | DOAJ |
| description | The paper investigates an algorithm for the numerical solution of a parametric eigenvalue problem for the Helmholtz equation on the plane specially tailored for the accurate mathematical modeling of lasing modes of microring lasers. The original problem is reduced to a nonlinear eigenvalue problem for a system of Muller boundary integral equations. For the numerical solution of the obtained problem, we use a trigonometric Galerkin method, prove its convergence, and derive error estimates in the eigenvalue and eigenfunction approximation. Previous numerical experiments have shown that the method converges exponentially. In the current paper, we prove that if the generalized eigenfunctions are analytic, then the approximate eigenvalues and eigenfunctions exponentially converge to the exact ones as the number of basis functions increases. To demonstrate the practical effectiveness of the algorithm, we find geometrical characteristics of microring lasers that provide a significant increase in the directivity of lasing emission, while maintaining low lasing thresholds. |
| first_indexed | 2024-03-10T07:54:32Z |
| format | Article |
| id | doaj.art-c5312a272fdb438fb2216e3bc6e2e9b1 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T07:54:32Z |
| publishDate | 2021-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-c5312a272fdb438fb2216e3bc6e2e9b12023-11-22T12:02:38ZengMDPI AGAxioms2075-16802021-08-0110318410.3390/axioms10030184Exponentially Convergent Galerkin Method for Numerical Modeling of Lasing in Microcavities with Piercing HolesAlexander O. Spiridonov0Anna I. Repina1Ilya V. Ketov2Sergey I. Solov’ev3Evgenii M. Karchevskii4Laboratory of Computational Technologies and Computer Modeling, Kazan Federal University, 18 Kremlevskaya St., 420008 Kazan, RussiaDepartment of System Analysis and Information Technologies, Kazan Federal University, 18 Kremlevskaya St., 420008 Kazan, RussiaDepartment of Applied Mathematics, Kazan Federal University, 18 Kremlevskaya St., 420008 Kazan, RussiaDepartment of Numerical Mathematics, Kazan Federal University, 18 Kremlevskaya St., 420008 Kazan, RussiaDepartment of Applied Mathematics, Kazan Federal University, 18 Kremlevskaya St., 420008 Kazan, RussiaThe paper investigates an algorithm for the numerical solution of a parametric eigenvalue problem for the Helmholtz equation on the plane specially tailored for the accurate mathematical modeling of lasing modes of microring lasers. The original problem is reduced to a nonlinear eigenvalue problem for a system of Muller boundary integral equations. For the numerical solution of the obtained problem, we use a trigonometric Galerkin method, prove its convergence, and derive error estimates in the eigenvalue and eigenfunction approximation. Previous numerical experiments have shown that the method converges exponentially. In the current paper, we prove that if the generalized eigenfunctions are analytic, then the approximate eigenvalues and eigenfunctions exponentially converge to the exact ones as the number of basis functions increases. To demonstrate the practical effectiveness of the algorithm, we find geometrical characteristics of microring lasers that provide a significant increase in the directivity of lasing emission, while maintaining low lasing thresholds.https://www.mdpi.com/2075-1680/10/3/184nonlinear eigenvalue problemboundary integral equationtrigonometric Galerkin methodaccuracy estimatemicroring laser |
| spellingShingle | Alexander O. Spiridonov Anna I. Repina Ilya V. Ketov Sergey I. Solov’ev Evgenii M. Karchevskii Exponentially Convergent Galerkin Method for Numerical Modeling of Lasing in Microcavities with Piercing Holes Axioms nonlinear eigenvalue problem boundary integral equation trigonometric Galerkin method accuracy estimate microring laser |
| title | Exponentially Convergent Galerkin Method for Numerical Modeling of Lasing in Microcavities with Piercing Holes |
| title_full | Exponentially Convergent Galerkin Method for Numerical Modeling of Lasing in Microcavities with Piercing Holes |
| title_fullStr | Exponentially Convergent Galerkin Method for Numerical Modeling of Lasing in Microcavities with Piercing Holes |
| title_full_unstemmed | Exponentially Convergent Galerkin Method for Numerical Modeling of Lasing in Microcavities with Piercing Holes |
| title_short | Exponentially Convergent Galerkin Method for Numerical Modeling of Lasing in Microcavities with Piercing Holes |
| title_sort | exponentially convergent galerkin method for numerical modeling of lasing in microcavities with piercing holes |
| topic | nonlinear eigenvalue problem boundary integral equation trigonometric Galerkin method accuracy estimate microring laser |
| url | https://www.mdpi.com/2075-1680/10/3/184 |
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