Changing only one of the separation distances

Figure 4.15 shows CMT simulation results for microring resonators for varying one of the gaps $g$ or $\tilde{g}$, while keeping the other constant. Keeping the separation $\tilde{g}$ constant, if $g$ is reduced, then the coupling between the upper bus waveguide and the cavity increases. Due to reduced self coupling of the input waveguide, the nonresonant level of the through power decreases, as more power is coupled to the cavity, the nonresonant drop power increases. On the other hand, keeping the separation $g$ constant, if $\tilde{g}$ is reduced, then the coupling between the lower bus waveguide and the cavity increases. As more power is coupled from the cavity to the Drop-port waveguide, the nonresonant level of the drop power increases, whereas the nonresonant through power decreases.

Figure 4.15: Effect of changing one of the separation distances on the spectrum response of the ring resonator. The first two plots show the response for different gaps $g$ for a constant separation distance $\tilde{g}$; for the two lower plots, $g$ is kept fixed, while $\tilde{g}$ varies (note the different values chosen in the upper and lower graphs).

For the simulations shown in Figure 4.15, changing the separation distance also affects the position of the resonances. In both cases, enlarging one of the separation distances, increases the resonance wavelength.

The simulation results of Figure 4.15 show a very peculiar behaviour for $g=\tilde{g}$, for which the resonant through power and drop power attain extrema. Figure 4.16 shows the variation of the resonant power transmission for different settings of the separation distances for the ring resonator. As evident from these plots, when both coupler gaps are identical, the drop power is maximum ($\approx 1$), and the through power is minimum ($\approx 0$). This can be explained as following.

Figure: Resonant power transmission of a ring resonator for different coupler gaps. The plots show the through power and the drop power at the respective resonance wavelengths around $\lambda=1.043 \,\mu$m for $\tilde{g}$ = constant (left plot), $g$ = constant (middle plot) and $g + \tilde{g}$ = constant (right plot).

In Section 1.4.3, we derived an expression for the drop power at resonance for a microresonator configuration with identical couplers ( $g=\tilde{g}$). Generalizing that expression to non-identical couplers ( $g \neq \tilde{g}$) gives

$$P_{\mbox{\scriptsize D}}\vert _{\mbox{\scriptsize res}} = P_{\mbo... ...x{\scriptsize bb}}\vert \mbox{e}^{\displaystyle - \alpha (L + \tilde{L})})^2 }.$$ (4.11)

If the cavity mode under consideration has negligible attenuation ( $\alpha \approx 0$), and if the coupling is assumed to be lossless (power is conserved, i.e. $\vert$S$_{\mbox{\scriptsize bs}}\vert^2 = 1 - \vert\mbox{\sf {S}}_{\mbox{\scriptsize bb}}\vert^2$, $\vert\tilde{\mbox{\sf {S}}}_{\mbox{\scriptsize sb}}\vert^2 = 1 - \vert\tilde{\mbox{\sf {S}}}_{\mbox{\scriptsize bb}}\vert^2$), then

$$P_{\mbox{\scriptsize D}}\vert _{\mbox{\scriptsize res}} = P_{\mbox{\scriptsize I}} \frac{ (1 - \vert\tilde{\mbox{\sf {S}}}_... ...iptsize bb}}\vert\vert \tilde{\mbox{\sf {S}}}_{\mbox{\scriptsize bb}}\vert)^2}.$$ (4.12)

For $\vert$S$_{\mbox{\scriptsize bb}}\vert=\vert\tilde{\mbox{\sf {S}}}_{\mbox{\scriptsize bb}}\vert$, this leads to $P_{\mbox{\scriptsize D}}=P_{\mbox{\scriptsize I}}$, i.e. complete transfer of the input power to the Drop-port. This implies $\vert$S$_{\mbox{\scriptsize ss}}\vert=\vert\tilde{\mbox{\sf {S}}}_{\mbox{\scriptsize ss}}\vert$, $\vert$S$_{\mbox{\scriptsize bs}}\vert=\vert\tilde{\mbox{\sf {S}}}_{\mbox{\scriptsize bs}}\vert$, $\vert$S$_{\mbox{\scriptsize sb}}\vert=\vert\tilde{\mbox{\sf {S}}}_{\mbox{\scriptsize sb}}\vert$, which is realized for identical couplers ( $g=\tilde{g}$), as illustrated in Figure 4.16.

If the attenuation losses are not negligible, then the ideal complete power transfer is not achieved. This becomes apparent for the simulation results of the disk resonator, shown in Figure 4.17. For the low loss TE$_{0}$ modes, the power drop is almost 1. But as the TE$_{1}$ mode has substantially higher attenuation (see Figure 3.8), the TE$_{1}$ resonance power drop for the symmetrical resonator is far from complete power transfer (also see Fig. 4.7).

Figure 4.17: Effect of changing one of the gaps on the spectrum response of the disk resonator. The interpretation of the curves is the same as for Figure 4.15.

Comparison of the first two plots of Figure 4.16 shows that for low loss case, $g$ and $\tilde{g}$ have almost identical effects on $P_{\mbox{\scriptsize D}}$, $P_{\mbox{\scriptsize T}}$ at resonance. Moreover, the plots in Figure 4.15 and 4.17 reveal that as one of the gaps increases, the width of the resonance reduces.