Muller Boundary Integral Equations for Solving Generalized Complex-Frequency Eigenvalue Problem

Oktyabrskaya, Alina O.; Spiridonov, Alexander O.; Karchevskii, Evgenii M.

doi:10.1134/s1995080220070343

Cited by 11 publications

(9 citation statements)

References 20 publications

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“…Consequently, for each pair of eigenvalues k > 0 and γ > 0, operator Aðk; γÞ is an index-zero Fredholm operator [36]. According to Oktyabrskaya et al [37], we see that if u ∈ U is an eigenfunction of the Problems (1)-( 5) with corresponding eigenvalues k > 0 and γ > 0, functions u j and v j defined in Equations ( 11)-( 13) belong to spaces C j ; j ¼ 1; 2; and form a nontrivial solution of the system in Equation ( 14) with the same k > 0 and γ > 0.…”

Section: Analytical Regularization Of the Lasing Eigenvalue Problemmentioning

confidence: 99%

Trade‐off between threshold gain and directionality of emission for modes of two‐dimensional eccentric microring lasers analysed using lasing eigenvalue problem

Repina

Oktyabrskaya

Spiridonov

et al. 2021

IET Microwaves Antenna & Prop

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This article discusses how a trade-off between the high directionality of emissions and low threshold gain can be achieved in active eccentric microring cavities. Our findings are based on the lasing eigenvalue problem formalism, considered using the method of analytical regularisation, and an extremely fast and accurate dedicated Galerkin method, applied to a set of associated Muller boundary integral equations. This method allows us to investigate symmetric and antisymmetric modes separately, on the threshold of nonattenuation in time emission. Numerical results show that the directivities of emission of working modes in a given frequency range, together with their threshold values of gain, are controlled by the size and location of the air hole in the cavity. The high efficiency of the developed code allows us to make an elementary optimisation of the considered cavity; this code is a promising engineering tool in the design of microring lasers.This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

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Section: Analytical Regularization Of the Lasing Eigenvalue Problemmentioning

confidence: 99%

Trade‐off between threshold gain and directionality of emission for modes of two‐dimensional eccentric microring lasers analysed using lasing eigenvalue problem

Repina

Oktyabrskaya

Spiridonov

et al. 2021

IET Microwaves Antenna & Prop

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“…However, there is no full equivalence between GCFEP and the eigenvalue problem for the system of Muller BIEs [14]. Namely, it was proven in [15] that for each eigenfunction of GCFEP there is a corresponding eigenvector of the system of Muller BIEs. Still, the assertion in the opposite direction is not true: there is one more problem that is reduced to the Muller BIEs, called "turned inside out GCFEP" [15].…”

Section: Introductionmentioning

confidence: 99%

“…Namely, it was proven in [15] that for each eigenfunction of GCFEP there is a corresponding eigenvector of the system of Muller BIEs. Still, the assertion in the opposite direction is not true: there is one more problem that is reduced to the Muller BIEs, called "turned inside out GCFEP" [15]. If GCFEP and the turned inside out GCFEP together have only the trivial solutions, then the system of Muller BIEs has only the trivial solution [15], and the resolvent set of the corresponding operator-valued function is not empty.…”

Section: Introductionmentioning

confidence: 99%

“…Still, the assertion in the opposite direction is not true: there is one more problem that is reduced to the Muller BIEs, called "turned inside out GCFEP" [15]. If GCFEP and the turned inside out GCFEP together have only the trivial solutions, then the system of Muller BIEs has only the trivial solution [15], and the resolvent set of the corresponding operator-valued function is not empty. This result is important for the theoretical investigation of the spectrum of the eigenvalue problem.…”

Section: Introductionmentioning

confidence: 99%

“…Mathematically, this means that there is an additional region (the hole) inside the cavity domain, and hence, an additional boundary in the integral-equation formulation. This makes the theoretical analysis more difficult compared with [8,14], as well as [15], where the problems with one boundary were investigated, as it was done originally by Muller [9]. In [21], the authors generalized results of [15] and clarified the connection between GCFEP and the eigenvalue problem for the system of Muller BIEs in this more complicated situation.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Spiridonov

Repina

Ketov

et al. 2021

Axioms

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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.

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Accurate Simulation of On-Threshold Modes of Microcavity Lasers with Active Regions Using Galerkin Method

Repina

Oktyabrskaya

Ketov

et al. 2021

Lecture Notes in Computational Science and Engineering

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Muller Boundary Integral Equations for Solving Generalized Complex-Frequency Eigenvalue Problem

Cited by 11 publications

References 20 publications

Trade‐off between threshold gain and directionality of emission for modes of two‐dimensional eccentric microring lasers analysed using lasing eigenvalue problem

Trade‐off between threshold gain and directionality of emission for modes of two‐dimensional eccentric microring lasers analysed using lasing eigenvalue problem

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

Accurate Simulation of On-Threshold Modes of Microcavity Lasers with Active Regions Using Galerkin Method

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