Mathematical and Numerical Analysis of the Generalized Complex-Frequency Eigenvalue Problem for Two-Dimensional Optical Microcavities

Exponentially Convergent Galerkin Method for Numerical Modeling of Lasing in Microcavities with Piercing Holes

“…Various 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].…”

“…This is because Muller BIEs are the Fredholm second-kind equations, which guarantee the convergence of their numerical solutions. By the same reasons, 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).…”

“…It is based on the search for complexvalued natural frequencies of open passive resonators. To be able to build a general theory for both LEP and CFEP models, a generalized model was proposed in [8]. It obtained the following name: generalized complex-frequency eigenvalue problem (GCFEP) [8].…”

“…To be able to build a general theory for both LEP and CFEP models, a generalized model was proposed in [8]. It obtained the following name: generalized complex-frequency eigenvalue problem (GCFEP) [8]. The reason for reducing GCFEP to the Muller BIEs was to get a system of weakly singular integral equations [13] on the boundary of the microcavity laser.…”

“…This result is important for the theoretical investigation of the spectrum of the eigenvalue problem. Using it and the fundamental results of the theory of projection methods for holomorphic Fredholm operator-valued functions [16,17], the convergence of a Nystrom method was proven in [8].…”