Rigorous Formulation of the Lasing Eigenvalue Problem as a Spectral Problem for a Fredholm Operator Function

Spiridonov, Alexander O.; Karchevskii, Evgenii M.; Nosich, Alexander I.

doi:10.1134/s1995080218080127

Cited by 19 publications

(13 citation statements)

References 14 publications

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“…As proved in [27], some of the kernels K q,s j have the logarithmic singularities and all the others are continuous. Therefore, for each k ∈ L and γ ∈ R the operator B(k, γ) is compact [27]. A(k, γ) has a bounded inverse operator for each γ ∈ R and k ∈ Im + .…”

Section: Analytical Regularization Of the Generalized Complex-frequenmentioning

confidence: 86%

“…If γ > 0, then the cavity is active, and some of the eigenvalues k can have Imk < K, where K > 0, i.e., the imaginary part of some k ∈ L 0 can be equal to or greater than zero. If there exits γ > 0, such that the eigenvalue k of GCFEP is positive, then the pair (k, γ) and the corresponding eigenfunction u satisfy all the conditions of LEP [11,16,27]. Such values of γ are different for different eigenvalues k. They are the values of the threshold material gain that is needed to compensate for the radiation losses and provide not attenuating in the time field function for the corresponding k. Theorem 1.…”

Section: Analytical Regularization Of the Generalized Complex-frequenmentioning

confidence: 99%

“…and let C j = C(Γ j ) be the Banach space of continuous functions on Γ j , j = 1, 2, supplied with the maximum norm, C = C 1 × C 2 , and W = C × C. Denote by I the identical operator in the space W. Then GCFEP (1)-(5) is equivalent to the following nonlinear eigenvalue problem [27] (see also [35]):…”

Section: Analytical Regularization Of the Generalized Complex-frequenmentioning

confidence: 99%

“…Although the usefulness of LEP is quite obvious, its comprehensive mathematical theory has not been provided so far, however, certain efforts were presented in [27]. Development of such theory is actually the aim of our work.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Mathematical and Numerical Modeling of On-Threshold Modes of 2-D Microcavity Lasers with Piercing Holes

Spiridonov

Karchevskii

Nosich³

2019

Axioms

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This study considers the mathematical analysis framework aimed at the adequate description of the modes of lasers on the threshold of non-attenuated in time light emission. The lasers are viewed as open dielectric resonators equipped with active regions, filled in with gain material. We introduce a generalized complex-frequency eigenvalue problem for such cavities and prove important properties of the spectrum of its eigensolutions. This involves reduction of the problem to the set of the Muller boundary integral equations and their discretization with the Nystrom technique. Embedded into this general framework is the application-oriented lasing eigenvalue problem, where the real emission frequencies and the threshold gain values together form two-component eigenvalues. As an example of on-threshold mode study, we present numerical results related to the two-dimensional laser shaped as an active equilateral triangle with a round piercing hole. It is demonstrated that the threshold of lasing and the directivity of light emission, for each mode, can be efficiently manipulated with the aid of the size and, especially, the placement of the piercing hole, while the frequency of emission remains largely intact.

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Section: Analytical Regularization Of the Generalized Complex-frequenmentioning

confidence: 86%

Section: Analytical Regularization Of the Generalized Complex-frequenmentioning

confidence: 99%

Section: Analytical Regularization Of the Generalized Complex-frequenmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Mathematical and Numerical Modeling of On-Threshold Modes of 2-D Microcavity Lasers with Piercing Holes

Spiridonov

Karchevskii

Nosich³

2019

Axioms

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“…Developing the ideas of the articles cited in the previous paragraph, we have proposed a convenient (for numerical analysis operator) formulation of LEP [Spiridonov et al, 2018] and have constructed a modification of the Nyström method taking the symmetry of the problem [Spiridonov, Karchevskii, 2016] into account. What is important is that all the authors proceeded from the assumption that the boundary of the cavity was smooth.…”

Section: Introductionmentioning

confidence: 99%

Mathematical and numerical modeling of a drop-shaped microcavity laser

Spiridonov¹,

Karchevskii²

2019

CRM

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This paper studies electromagnetic fields, frequencies of lasing, and emission thresholds of a drop-shaped microcavity laser. From the mathematical point of view, the original problem is a nonstandard two-parametric eigenvalue problem for the Helmholtz equation on the whole plane. The desired positive parameters are the lasing frequency and the threshold gain, the corresponding eigenfunctions are the amplitudes of the lasing modes. This problem is usually referred to as the lasing eigenvalue problem. In this study, spectral characteristics are calculated numerically, by solving the lasing eigenvalue problem on the basis of the set of Muller boundary integral equations, which is approximated by the Nyström method. The Muller equations have weakly singular kernels, hence the corresponding operator is Fredholm with zero index. The Nyström method is a special modification of the polynomial quadrature method for boundary integral equations with weakly singular kernels. This algorithm is accurate for functions that are well approximated by trigonometric polynomials, for example, for eigenmodes of resonators with smooth boundaries. This approach leads to a characteristic equation for mode frequencies and lasing thresholds. It is a nonlinear algebraic eigenvalue problem, which is solved numerically by the residual inverse iteration method. In this paper, this technique is extended to the numerical modeling of microcavity lasers having a more complicated form. In contrast to the microcavity lasers with smooth contours, which were previously investigated by the Nyström method, the drop has a corner. We propose a special modification of the Nyström method for contours with corners, which takes also the symmetry of the resonator into account. The results of numerical experiments presented in the paper demonstrate the practical effectiveness of the proposed algorithm.

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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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Rigorous Formulation of the Lasing Eigenvalue Problem as a Spectral Problem for a Fredholm Operator Function

Cited by 19 publications

References 14 publications

Mathematical and Numerical Modeling of On-Threshold Modes of 2-D Microcavity Lasers with Piercing Holes

Mathematical and Numerical Modeling of On-Threshold Modes of 2-D Microcavity Lasers with Piercing Holes

Mathematical and numerical modeling of a drop-shaped microcavity laser

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

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