Abstract
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.
Highlights
IntroductionPublished: 11 August 2021Various two-dimensional (2D) models of microdisk and microring lasers (see, e.g., [1,2])can be investigated with the aid of a specific electromagnetic eigenvalue problem adapted to calculate the threshold values of gain, in addition to the emission frequencies, which is called the lasing eigenvalue problem (LEP) [3,4,5,6,7]
Published: 11 August 2021Various two-dimensional (2D) models of microdisk and microring laserscan be investigated with the aid of a specific electromagnetic eigenvalue problem adapted to calculate the threshold values of gain, in addition to the emission frequencies, which is called the lasing eigenvalue problem (LEP) [3,4,5,6,7]
For 2021Various two-dimensional (2D) microcavity lasers with uniform gain, LEP was reduced in [8] to a nonlinear eigenvalue problem for the system of the Muller boundary integral equations (BIEs). This system, obtained by Muller in [9], is widely used in the analysis of electromagnetic-wave scattering from 2D and 3D homogeneous dielectric objects with smooth boundaries [10,11]. This is because Muller BIEs are the Fredholm second-kind equations, which guarantee the convergence of their numerical solutions
Summary
Published: 11 August 2021Various two-dimensional (2D) models of microdisk and microring lasers (see, e.g., [1,2])can be investigated with the aid of a specific electromagnetic eigenvalue problem adapted to calculate the threshold values of gain, in addition to the emission frequencies, which is called the lasing eigenvalue problem (LEP) [3,4,5,6,7]. For 2D microcavity lasers with uniform gain, LEP was reduced in [8] to a nonlinear eigenvalue problem for the system of the Muller boundary integral equations (BIEs) This system, obtained by Muller in [9], is widely used in the analysis of electromagnetic-wave scattering from 2D and 3D homogeneous dielectric objects with smooth boundaries [10,11]. The eigenmodes of fully active [6,8] and passive [12] microcavities can be calculated using Muller BIEs. Many authors, as in [12], have used a physical model called the complex-frequency eigenvalue problem (CFEP). The reason for reducing GCFEP to the Muller BIEs was to get a system of weakly singular
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