Muller Boundary Integral Equations for Solving Generalized Complex-Frequency Eigenvalue Problem
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Muller Boundary Integral Equations for Solving Generalized Complex-Frequency Eigenvalue Problem A. O. Oktyabrskaya1* , A. O. Spiridonov2** , and E. M. Karchevskii3*** (Submitted by E. K. Lipachev) 1
Institute of Computational Mathematics and Information Technologies, Kazan Federal University, Kazan, Tatarstan, 420008 Russia 2 Laboratory of Computational Technologies and Computer Modeling, Kazan Federal University, Kazan, Tatarstan, 420008 Russia 3 Department of Applied Mathematics, Kazan Federal University, Kazan, Tatarstan, 420008 Russia Received March 6, 2020; revised March 14, 2020; accepted March 20, 2020
Abstract—The current paper clarifies the connection between the generalized complex-frequency eigenvalue problem and the eigenvalue problem for the Muller boundary integral equations. It is proved that these problems are spectrally equivalent if a specially tailored eigenvalue problem does not have any solution. DOI: 10.1134/S1995080220070343 Keywords and phrases: eigenvalue problem, Muller boundary integral equations.
1. INTRODUCTION Various two-dimensional (2-D) microcavity lasers have been investigated numerically with the aid of a modified electromagnetic eigenvalue problem, specifically tailored to extract the threshold values of gain in addition to the emission frequencies (see, e.g., [17, 20, 21], and references therein). Such a modified formulation called the Lasing Eigenvalue Problem (LEP) was first introduced in 2004 in [14] and since then has gained credit in the photonics community. More recently, LEP analysis was applied to study stand-alone and periodic nanolasers with localized surface-plasmon modes [2, 3, 11, 13]. The greatest progress may have been achieved for two-dimensional microcavities with uniform gain in [15], where the original problem was reduced equivalently to a nonlinear spectral problem for the system of Muller boundary integral equations (BIEs), which was solved accurately by the Nystrom method. Derived first by Muller [10] this system has become a reliable and efficient tool for analysis of the electromagnetic field in the presence of a 2-D homogeneous dielectric object with an arbitrary smooth boundary. Particularly, Muller BIEs were used for computations of eigenmodes of active [15–17] and passive microcavities [4, 6]. We note that most of the authors used a classical physical model based on the search of complex-valued natural frequencies of passive open (lossless and lossy) cavities—this is usually called the Complex-Frequency Eigenvalue Problem (CFEP). To use one theoretical framework for the both of models, LEP and CFEP, it was introduced a generalized model, which was called the Generalized Complex-Frequency Eigenvalue Problem (GCFEP) [19]. In the current paper, we clarify the connection between GCFEP and Muller BIEs. In Section 2, we formulate GCFEP, following [19]. In Section 4, we reduce GCFEP to the eigenvalue problem for the system of Muller BIEs. To get the system of Muller BIEs, we add term by term the traces of the integral representations for eigenfunctions and their nor
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