# H-polarized natural modes of microsize graphene strip grating in gain-material layer: regularizing Galerkin technique and lasing threshold conditions

## TL;DR

This paper investigates the lasing threshold conditions for a graphene-based micro-laser using a specialized mathematical technique.

## Contribution

The study introduces a regularizing Galerkin technique to solve lasing eigenvalue problems with guaranteed convergence.

## Key findings

- Threshold conditions for various lasing modes were identified and analyzed.
- The method enables accurate computation of eigenpairs with controlled accuracy.
- The lasing behavior changes predictably with structural parameter variations.

## Abstract

We study the threshold conditions for the natural modes of the microsize plasmonic laser shaped as an infinite flat graphene strip grating symmetrically embedded into the gain-material layer, in the H-polarization case. For this purpose, we solve the lasing eigenvalue problem (LEP), which is a classical source-free electromagnetic field boundary-value problem, adapted to the presence of the active region by the corresponding sign of the imaginary part of the refractive index. In such a way we look for the eigenpairs, i.e. the stimulated emission real-valued frequency and the threshold gain index, specific to each mode. We transform LEP to a hypersingular integral equation for the on-strip current density and discretize it by the regularizing Galerkin technique. This procedure leads to a determinantal equation with guaranteed convergence to the exact LEP eigenpairs and controlled accuracy of their computation. The numerical analysis allows us to study the threshold conditions for various lasing modes of the microsize laser, identify them and trace their change when varying the parameters of the lasing structure.

This article is part of the theme issue ‘Analytically grounded full-wave methods for advances in computational electromagnetics’.