Spectrum evaluation

A quantitative evaluation of the present microresonator model requires the propagation constants of the cavity modes $\gamma_{\mbox{\scriptsize b}p}$, hidden in G, $\tilde{\mbox{\sf {G}}}$, and the scattering matrices S, $\tilde{\mbox{\sf {S}}}$ of couplers (I) and (II). Once these quantities are available, the optical transmission through the resonator is given by equations (4.3).

In principle the spectral response of the device can be obtained by repeating the entire solution procedure for different wavelengths in an interesting range. That direct approach requires repeated computations of the bend propagation constants and the scattering matrices. A large part of the numerical effort can be avoided, if one calculates the relevant quantities merely for a few distant wavelengths, and then uses complex interpolations of these values for the actual spectrum evaluation. The interpolation procedure, however, should be applied to quantities that vary but slowly with the wavelength.

In line with the reasoning concerning the resonances in Section 1.4.3, one can expect that any rapid wavelength dependence of the transmission is determined mainly by the phase gain of the waves circulating in the cavity. Rapid changes in these phase relations are due to a comparably slow wavelength dependence of the bend propagation constants $\gamma_{\mbox{\scriptsize b}p}$, that is multiplied by the lengths $L$, $\tilde{L}$ of the external cavity segments. If a substantial part of the cavity is already contained in the coupler regions, then the elements of the scattering matrices S exhibit also fast phase oscillations with the wavelength, as depicted in Figure 3.7, such that S directly is not suitable for the interpolation. Apart from these rapid changes, which can be attributed to the unperturbed propagation of the basis modes along the bent and straight waveguides, the interaction between the waves in the two coupled cores introduces an additional wavelength dependence, which in turn can be expected to be slow.

To separate the two scales of wavelength dependence in S, one divides by the exponentials that correspond to the undisturbed wave propagation of the bend and straight modes towards and from the symmetry plane $z=0$:

$${\mbox{\sf {S}}}' = \mbox{\sf {Q}}^{0}\,\mbox{\sf {S}}\,{(\mbox{\sf {P}}^{0})}^{-1}$$ (4.10)

Here P$^{0}$ and Q$^{0}$ are diagonal matrices with entries ${\sf {P}}^0_{j,j}$ and ${\sf {Q}}^0_{j,j}$ as defined for P and Q in equation (3.20). Formally, one can view ${\mbox{\sf {S}}}'$ as the scattering matrix of a coupler with zero length, where the interaction takes place instantaneously at $z=0$. This modification of S, applied analogously to $\tilde{\mbox{\sf {S}}}$, is compensated by redefining the lengths of the external cavity segments as $L' = \tilde{L'} = \pi R$, by changing the matrices G and $\tilde{\mbox{\sf {G}}}$ accordingly, and, where necessary, by taking into account the altered phase relations on the external straight segments.

After these modifications, the new matrices G$'$ and $\tilde{\mbox{\sf {G}}'}$ capture the phase gain of the cavity field along the full circumference. The modified scattering matrices S$'$ and $\tilde{\mbox{\sf {S}}'}$ show only a slow wavelength dependence (see Figure 4.4), such that the interpolation can be successfully applied to these matrices and to the bend propagation constants in G$'$ and $\tilde{\mbox{\sf {G}}'}$. The resonant features of the device are now entirely effected by the analytical relations (4.3), such that one obtains an excellent agreement between the transmission spectra computed with the interpolated quantities and the direct calculation, while the computational effort is significantly reduced.