Spectrum evaluation for perturbed resonators

By the interpolation method outlined in Section 4.3, in principle one can compute the resonator spectra for the unperturbed and the perturbed configurations separately. Instead of the scattering matrices $\sf {S}$, $\tilde{\sf {S}}$ as given by Eq. (4.1), which are associated with the couplers defined over a larger $z$ interval, one uses the scattering matrices $\sf {S}'$, $\sf {\tilde{S}}'$ as given by Eq. (4.10), which are associated with couplers of a zero length.

Similar to the arguments presented in Section 1.4.3, let's assume that for a slight change of the cavity core refractive index the scattering matrices $\sf {S}'$, $\sf {\tilde{S}}'$ do not change much, and the shifts of the resonances are entirely due to the changes in the cavity mode propagation constants. Then using $\sf {S}'$, $\sf {\tilde{S}}'$ of the unperturbed resonator, and adding the phase propagation constants shifts $\delta \beta_{\mbox{\scriptsize b}p}$ to the propagation constants $\gamma_{\mbox{\scriptsize b}p}$ of the unperturbed cavity segments, one can again follow the interpolation method described in Section 4.3, without recalculating the scattering matrices for the perturbed resonator. In this way, a significant amount of computational work can be avoided.