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

Abstract

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.

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

“…; x ∈ Γ 2 : ð13Þ Furthermore, according to Spiridonov et al [17], we get a system of BIEs (named after Muller [35]). We write the system in the operator form:…”

“…Here, m ¼ 1; 2; 3; 4; y ∈ Γ j ; j ¼ 1; 2; x ∈ Γ 1 for l ¼ 1; 2; x ∈ Γ 2 for l ¼ 3; 4: Function g denotes elements u j and v j ; j ¼ 1; 2, depending on the context. The kernels have the form [17]:…”

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

“…The formulation of GCFEP for 2D microcavity lasers with piercing holes is given in [18]. A generic geometry of the analyzed microcavities is shown in Figure 1.…”

“…Recently, for numerical simulation of more complicated 2D microcavity lasers, namely, active cavities with piercing holes [18], a modified version of the Muller BIEs, together with a trigonometric Galerkin discretization technique, was proposed [19,20]. 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.…”

“…Preliminarily, in Section 2, we follow [18] and briefly recall the main steps of reducing of the original problem to the nonlinear eigenvalue problem for the set of Muller BIEs. Section 3 presents the equations related to the trigonometric Galerkin discretization of the mentioned BIEs.…”