Lasing spectra and thresholds of supermodes in an active microdisk assisted with a passive microring in view of the mode overlap coefficients

We consider lasing modes in the microcavity shaped as a circular active disk placed in the centre of a passive dielectric ring as solutions to 2-D linear eigenproblem for the Maxwell equations with exact boundary and radiation conditions. The gain is introduced into the cavity material within the active region via the ldquoactiverdquo imaginary part of the refractive index, and the modal frequencies and threshold values of gain are sought as the eigenvalues. It is found that the modes in such a cavity are supermodes, i.e. optically coupled modes of the active and passive regions. The modal thresholds of lasing demonstrate complicated behaviour, which can be explained by introducing the mode overlap coefficients for each of the partial regions of the cavity. It is shown that the reduction of threshold always accompanies the pulling of the mode E-field into the active region while the threshold growth corresponds to the situation when the E-field is pushed from the active region into the passive one, which is either the ring or the air-gap.