This article discusses how a trade-off between the high directionality of emissions and low threshold gain can be achieved in active eccentric microring cavities. Our findings are based on the lasing eigenvalue problem formalism, considered using the method of analytical regularisation, and an extremely fast and accurate dedicated Galerkin method, applied to a set of associated Muller boundary integral equations. This method allows us to investigate symmetric and antisymmetric modes separately, on the threshold of nonattenuation in time emission. Numerical results show that the directivities of emission of working modes in a given frequency range, together with their threshold values of gain, are controlled by the size and location of the air hole in the cavity. The high efficiency of the developed code allows us to make an elementary optimisation of the considered cavity; this code is a promising engineering tool in the design of microring lasers.This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

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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.

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