Methods of analytical regularization in the spectral theory of open waveguides

Karchevskii, Evgenii M.; Nosich, Alexander I.

doi:10.1109/mmet.2014.6928740

Cited by 12 publications

(4 citation statements)

References 19 publications

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“…, 2n − 1. Applying approximations (33) and equating both sides of the functional equality following from (14) on the mesh points Ξ h , we reduce (14) to the following finite-dimensional nonlinear eigenvalue problem:…”

Section: Nyström Methodsmentioning

confidence: 99%

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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: Nyström Methodsmentioning

confidence: 99%

“…To apply the method of analytical regularization [33,34] for GCFEP, we use the following integral representations of the eigenfunctions u:…”

Section: Analytical Regularization Of the Generalized Complex-frequenmentioning

confidence: 99%

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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“…To solve LEP numerically, nonlinear spectral problems for systems of boundary integral equations were proposed in [Karchevskii, Nosich, 2014], [Nosich, 2016]. The kernels of the systems are weakly singular, therefore the corresponding operators are Fredholm with zero index.…”

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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“…Its potential has been recently demonstrated in the LEP analysis of the emission spectra, thresholds and modal fields of a kite-shaped 2-D microcavity laser [3]. In the present work we will at first consider the LEP for arbitrary 2-D smooth-contour dielectric microcavity from the mathematical point of view using the methods of analytical regularization [4]. We intend to prove that the original boundary-value problem for the Helmholtz equation on the plane with Reichardt radiation condition is equivalent to a nonlinear eigenvalue problem for the Muller BIE with a compact operator.…”

Section: Introductionmentioning

confidence: 99%

Spectra, thresholds, and modal fields of a circular microcavity laser transforming into a square

Spiridonov

Karchevskii

Nosich³

2015

2015 17th International Conference on Transparent Optical Networks (ICTON)

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We investigate the lasing spectra, threshold gain values, and modal fields for a two-dimensional microcavity laser with a square contour defined by the super-circle equation. The cavity modes are considered accurately using the linear electromagnetic formalism of the Lasing Eigenvalue Problem (LEP) with exact boundary and radiation conditions. We reduce the original problem to a nonlinear eigenvalue problem for the Muller boundary integral equation and build a numerical algorithm of its solution. Computations demonstrate that under the deformation of microcavity from the circle to a square there exist modes that preserve low thresholds.

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Methods of analytical regularization in the spectral theory of open waveguides

Cited by 12 publications

References 19 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

Spectra, thresholds, and modal fields of a circular microcavity laser transforming into a square

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