The Threshold Mode Structure Analysis of the Two-Dimensional Photonic Crystal Lasers

Abstract

Abstract-In this work, threshold mode structures of twodimensional (2D) photonic crystal (PC) lasers are presented. The subjects of this paper are finite photonic crystal structures with circular holes arranged in square and triangular lattices. In each case, both transverse magnetic (TM) and transverse electric (TE) polarization are studied. The analysis is based on the coupled-wave equations and analyzes modes' behavior for the wide range of coupling coefficient values. The laser mode is characterized by threshold gain and frequency deviation, and these quantities depend on coupling constants, which means that the threshold gain of the mode and the mode's frequency deviation depend on the coupling constants. Presented analysis gives an interesting insight into behavior of the modes in photonic crystal lasers.

Threshold Mode Structure of Square and Triangular Lattice Gain and Index Coupled Photonic Crystal Lasers

“…These characteristic quantities were calculated by the numerical solution of Equations (23)- (26) for TM polarization and Equations (30)-(33) for TE polarization. The similar description of modes, shown in Figure 8, where no gain coupling is considered has already been presented in [46] or [49].…”

“…They are based on a coupled-wave theory (CWT) [15], [41] and focused on square and triangular lattice photonic crystals e.g., [32], [33], [42]- [48]. Most of the works e.g., [32], [42]- [46] contain a mathematical description and numerical results of the threshold analysis of two-dimensional (2-D) square and triangular lattice photonic crystal laser with TM and TE polarization. They introduce general coupled mode relations for a threshold gain, a Bragg frequency deviation and field distributions, and give calculation results for some specific values of coupling coefficients.…”

“…Further, in [42] the effect of boundary reflections has been investigated, and it has been shown that the mode properties can be adjusted by changing refractive index or boundary conditions. In, [46], the achievements of these works were summarized and supplemented with the analysis for the wide range of coupling coefficient. These studies concerned structures which were infinite in the direction normal to the 2D PC plane.…”

“…In Equations (30)- (33), the spatial dependence of four magnetic field components H i s , i=2, 4,6,8 was neglected, and it was assumed that α ≪ δ. These steps let to formulate not eight but four partial differential equations (for details see [32] or [46]). The Bragg frequency deviation δ is given by (27).…”

Spatial Intensity Distribution in Plasmonic Particle Array Lasers

“…This section summarizes those aspects of CWT required to describe our experiments. CWT was developed by Kogelnik and Shank in 1972 to describe one-dimensionally periodic DFB lasers [43] and was later further extended to different one-dimensional (1D) and two-dimensional (2D) DFB laser geometries [44][45][46][47]. In CWT, a DFB laser is described as a waveguide with a weakly scattering periodic perturbation in its optical constant, which leads to coupling between forward-and backward-propagating inplane waves [43,44] by Bragg diffraction.…”