Lasing of circular photonic molecules composed of three and four microdisks

Photonic molecules (PMs) composed of several identical active microdisks placed in a circle are investigated in this paper. We study the lasing eigenvalue problems for such PMs as electromagnetic field boundary value problems formulated in special manner, which enables one to extract not only natural frequencies but also thresholds of optical modes. The problem is reduced to an infinite matrix equation of the Fredholm second kind. Then we use a secant method to compute the roots of truncated determinant equation that converge to the eigenvalues of original problem when the determinant size is taken progressively larger. Here, all modes in a circular PM composed of M microdisks split into 2M independent symmetry classes. Coupling of microcavities leads to the shift in frequencies and modification of thresholds in comparison to the single microdisk. With carefully tuned separation it is possible to achieve a considerable reduction of thresholds in PMs composed of three and four microdisks