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

Axioms ◽  
2019 ◽  
Vol 8 (3) ◽  
pp. 101 ◽  
Author(s):  
Spiridonov ◽  
Karchevskii ◽  
Nosich
Keyword(s):  
Eigenvalue Problem ◽  
Light Emission ◽  
Boundary Integral Equations ◽  
Active Regions ◽  
Boundary Integral ◽  
Complex Frequency ◽  
Microcavity Lasers ◽  
Gain Material ◽  
Real Emission ◽  
Mathematical And Numerical Modeling

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.

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

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 184
Author(s):  
Alexander O. Spiridonov ◽  
Anna I. Repina ◽  
Ilya V. Ketov ◽  
Sergey I. Solov’ev ◽  
Evgenii M. Karchevskii
Keyword(s):  
Eigenvalue Problem ◽  
Galerkin Method ◽  
Numerical Solution ◽  
Original Problem ◽  
Boundary Integral Equations ◽  
Nonlinear Eigenvalue Problem ◽  
Boundary Integral ◽  
Generalized Eigenfunctions ◽  
Eigenvalue And Eigenfunction ◽  
Lasing Modes

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.

scholarly journals The hot spots conjecture can be false: some numerical examples

2021 ◽  
Vol 47 (6) ◽  
Author(s):  
Andreas Kleefeld
Keyword(s):  
Eigenvalue Problem ◽  
Hot Spots ◽  
Neumann Boundary Condition ◽  
Boundary Integral Equations ◽  
Neumann Boundary ◽  
Boundary Integral ◽  
Zero Eigenvalue ◽  
Numerical Examples ◽  
Linear Eigenvalue Problem ◽  
Hot Spots Conjecture

AbstractThe hot spots conjecture is only known to be true for special geometries. This paper shows numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole. The underlying eigenvalue problem for the Laplace equation with Neumann boundary condition is solved with boundary integral equations yielding a non-linear eigenvalue problem. Its discretization via the boundary element collocation method in combination with the algorithm by Beyn yields highly accurate results both for the first non-zero eigenvalue and its corresponding eigenfunction which is due to superconvergence. Additionally, it can be shown numerically that the ratio between the maximal/minimal value inside the domain and its maximal/minimal value on the boundary can be larger than 1 + 10− 3. Finally, numerical examples for easy to construct domains with up to five holes are provided which fail the hot spots conjecture as well.

Integral Representations for the Solution of Dynamic Bending of a Plate with Displacement-Traction Boundary Data

2003 ◽  
Vol 10 (3) ◽  
pp. 467-480
Author(s):  
Igor Chudinovich ◽  
Christian Constanda
Keyword(s):  
Existence And Uniqueness ◽  
Boundary Data ◽  
Boundary Integral Equations ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Boundary Integral ◽  
Integral Representations ◽  
Transverse Shear Deformation ◽  
Uniqueness Of Solutions ◽  
Nonstationary Boundary

Abstract The existence of distributional solutions is investigated for the time-dependent bending of a plate with transverse shear deformation under mixed boundary conditions. The problem is then reduced to nonstationary boundary integral equations and the existence and uniqueness of solutions to the latter are studied in appropriate Sobolev spaces.

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