Simulation results

Now we assess the validity of expressions (4.15), (4.16), and the above spectrum evaluation method for the perturbed resonators. For the assessment, we consider again the configurations of Sections 4.4.1 and 4.4.2. A cavity of radius $R=5 \,\mu$m is coupled to straight waveguides of width $w_{\mbox{\scriptsize s}} = 0.3 \,\mu\mbox{m}$ and core refractive index $n_{\mbox{\scriptsize s}} = 1.5$. The background refractive index is $n_{\mbox{\scriptsize b}}=1.0$, and the gap widths $g=\tilde{g}= 0.2\,\mu$m. The cavity has the form of a ring, with a core width $w_{\mbox{\scriptsize c}} = 0.3 \,\mu\mbox{m}$, and of a disk, with $w_{\mbox{\scriptsize c}}=R$. For the unperturbed setting, the cavity core refractive index is $n_{\mbox{\scriptsize c}} = 1.5$. For the perturbed structure, this value is changed to $n_{\mbox{\scriptsize c} p} = 1.504$. The spectra are evaluated by quadratic interpolation with nodal wavelengths $\lambda=1.015 \,\mu$m$, 1.05\,\mu$m$, 1.085\,\mu$m.

With the help of Eq. (4.16), the shifts in the (real part of) the cavity mode propagation constants at the resonance wavelengths of the unperturbed resonators are calculated; Eq.(4.15) then gives the shifts in the resonance wavelengths. Adding these differences to the unperturbed resonance wavelengths, determines the resonance positions for the perturbed configuration, which are compared with the resonance positions obtained by direct CMT simulations for the perturbed ( $n_{\mbox{\scriptsize c} p} = 1.504$) resonator. Figure 4.21 depicts the spectral responses for the perturbed and the unperturbed ring resonators. The spectral response computed with the method outlined in this section agrees with the direct CMT calculation. Thus for a small perturbation of the cavity core refractive index, using the scattering matrices and the cavity propagation constants of the unperturbed structure, and the shifts in the cavity mode propagation constants, one can quite reliably predict the spectral response for the perturbed resonator.

Figure: Spectrum shift due to tuning of the cavity core refractive index. The microring resonator configuration is as in Section 4.6. The curves of the normalized transmitted power are calculated by the spectrum evaluation method of Section 4.3 for the unperturbed resonator with $n_{\mbox{\scriptsize c}} = 1.5$ (dash-dotted line) and for the perturbed resonator with $n_{\mbox{\scriptsize c} p} = 1.504$ (circles), and by the approximation of Section 4.6 for the perturbed resonator (solid line).

A similar comparison of the transmitted power for the unperturbed and the perturbed disk resonator is shown in Figure 4.22. Just as in the case of the monomodal ring, for the present multimodal disk we find that the perturbed resonator spectrum computed by the approximation method (solid line) discussed in this section agrees well with the direct CMT simulations (circles), what concerns the resonance positions.

Minor deviations are observed in the depths of the resonance dips, in particular for the TE$_{1}$ resonances, where apparently the change in modal attenuation due to the core refractive index change is slightly larger than for the TE$_{0}$ mode. This alteration of the cavity mode losses is not taken into account by the present approximation procedure.

Figure 4.22: Spectral shift due to the cavity core refractive index perturbation, for the microdisk resonator as specified in Section 4.6. The interpretation of the curves is as for Figure 4.21.