Interpolated spectrum evaluation

Now we look at the faster spectrum evaluation technique outlined in Section 4.3. For the bent-straight waveguide couplers involved in the ringresonator of Figure 4.2, the wavelength dependence of the ``original'' scattering matrix entries ${\sf {S}}_{v,w}$ is depicted in Figure 3.6 (absolute square) and Figure 3.7 (real and imaginary part). By extracting phase gains for propagation along the coupler length, one obtains a ``modified'' scattering matrix ${\sf {S}}'$ (4.10). The wavelength dependence of ${\sf {S}}_{v,w}'$ is shown in Figures 4.3, 4.4.

As the bend modes in the present coupler are not that lossy (see Figure 3.2), the plots of the absolute square of ${\sf {S}}_{v,w}'$ in Figure 4.3 do not differ much from ${\sf {S}}_{v,w}$ in Figure 3.6, although for the TM modes one can observe a slight change. But the plots of the real and imaginary parts of ${\sf {S}}_{v,w}'$ clearly bring forward the essence of the discussion in Section 4.3. While the corresponding plots in Figure 3.7 show considerable oscillations, the curves in Figure 4.4 are almost linear such that ${\sf {S}}_{v,w}'$ can be reliability interpolated.

Figure 4.3: Wavelength dependence of the absolute square of the entries of the scattering matrix $\sf {S}^{'}$. The coupler configuration and the interpretation of the curves are identical to Figure 3.6.

Figure 4.4: Wavelength dependence of the complex valued entries of the scattering matrix $\sf {S}'$. For the coupler configuration and the interpretation of the curves, refer to Figure 3.7.

The left side plots of Figure 4.5 shows the resonator spectrum as obtained by interpolating bend mode propagation constants and CMT scattering matrices for only two (linear interpolation) or three different wavelengths (quadratic interpolation), according to Section 4.3. While small deviations remain for the linear approximation, on the scale of the figure the curves related to quadratic interpolation are hardly distinguishable from the direct CMT results. Thus the interpolation approach provides a very effective means to predict the resonator spectrum, in particular if narrow dips /peaks in the responses of high-quality resonators would have to be resolved.

Figure: Left: CMT results, where the spectrum has been evaluated directly and by interpolation of CMT computations for nodal wavelengths $1.015\,\mu$m and $1.085\,\mu$m (linear), or $1.015\,\mu$m, $1.05\,\mu$m, and $1.085\,\mu$m (quadratic interpolation). Right: Comparison of spectra computed with the approximation of constant coupler scattering matrices as discussed in Section 1.4.3 and computed with direct CMT calculations. The microresonator setting is as in Figure 4.2.

In Section 1.4.3 we discussed an approximate spectrum evaluation method, where one assumes that for a narrow wavelength interval the scattering matrices are approximately constant, and the resonances are ``exclusively'' due to the phase gains experienced by the cavity modes while propagating along the cavity. By using the scattering matrices ${\sf {S}}'$ and ${\sf {\tilde{S}}}'$ (corresponding to a total cavity length of $2 \pi R$), we can verify this approximation. In fact according to the right part of Figure 4.5, this scenario is quite well realized. For the present configuration with a low loss cavity mode, the spectrum evaluated with ${\sf {S}}'$, ${\sf {\tilde{S}}}'$ at $\lambda=1.05\,\mu$m and a wavelength dependent propagation constant $\gamma_{\mbox{\scriptsize b}0}$ agrees quite well with ``direct'' CMT calculations. Note that, far from the reference wavelength $\lambda=1.05\,\mu$m, the approximation of constant matrices ${\sf {S}}'$, ${\sf {\tilde{S}}}'$ slightly deteriorates.

Beyond modal amplitudes and power levels, the CMT solutions permit to access the full optical electromagnetic field. Figure 4.6 collects plots of the principal components for off resonance and resonant configurations for both polarizations. Off resonance, one observes the large Through transmission, small amplitudes of the waves in the Drop-port, and also only minor wave amplitudes in the cavity. At the resonances, the straight transmission is almost completely suppressed; the major part of the input power arrives at the Drop-port. For the present strongly coupled configurations, the power that enters and leaves the cavity at the two couplers leads to considerably different field intensities in the left and right halves of the ring. Here the radiative parts of the bend modes are appreciable outside the cavity, in particular for the more lossy TM waves.

Figure 4.6: CMT results for the microring structure of Figure 4.2, local intensities (first and third row) and snapshots of the physical field (second and forth row) of the principal components of TE $(E_{y})$ (first and second row) and TM $(H_{y})$ (third and forth row) polarized waves, for an off-resonance wavelength (first column) and at a resonance (second column). For visualization purposes the coupler computational window has been extended to $[z_{\mbox{\scriptsize i}}, z_{\mbox{\scriptsize o}}]=[-4, 8]\,\mu\mbox{m}$.