Changing both separation distances identically

Figure 4.13 shows the effect of symmetrical changes of both separation distances on the spectral response. As evident from Figures 3.4, 3.11, as the distances $g$, $\tilde{g}$ are reduced, the cross coupling between the straight waveguide and the cavity increases, and the self coupling decreases. Due to the reduced self coupling of the upper straight waveguide, the nonresonant level of throughput power decreases; whereas as more power is coupled from the upper waveguide to the cavity, and then from the cavity to the lower waveguide, the nonresonant level of drop power increases.

Figure 4.13: Spectral response of a ring resonator for different gap widths. The resonator configuration is as in Figure 4.5.

Due to the increased cavity self coupling ( $\vert$S$_{\mbox{\scriptsize bb}}\vert$), as seen from Eq. (1.15), the FWHM decreases. This is clearly revealed in plots of Figure 4.13. As shown in Table 4.5, there is also a change in the resonance positions, here defined as the wavelengths corresponding to the extrema in the transmitted and the dropped power.

Table: Resonance positions of the ring resonator of Figure 4.5, for different separation distances $g=\tilde{g}$. For large separations, the resonance positions computed by the CMT based simulations tend to the eigenfrequencies of the ``isolated'' ring cavity [39, private communication].

$g=\tilde{g}~[\,\mu$m$]$	TE$_{0}$ resonance wavelengths [ $\,\mu$m]
$0.1$	$1.01700$	$1.03950$	$1.06350$
$0.2$	$1.01836$	$1.04138$	$1.06540$
$0.3$	$1.01890$	$1.04200$	$1.06606$
$0.4$	$1.01913$	$1.04227$	$1.06647$
$0.5$	$1.01924$	$1.04239$	$1.06661$
$0.6$	$1.01928$	$1.04245$	$1.06668$
eigenfrequencies	$1.01928$	$1.04248$	$1.06678$

For larger separation distances, the influence of neighboring straight waveguides on the cavity field propagation is marginal, and the resulting resonances of the microresonator (cavity coupled to two straight waveguides) tend to resonances of the ``isolated'' cavity. The comparison with resonance wavelengths of the isolated cavity obtained by the complex eigenfrequency model [39], shown in Table 4.5, confirms this fact.

The same trend is observed in case of a resonator with a disk cavity, see Figure 4.14 and Table 4.6. The disk resonance wavelengths will converge to the wavelength corresponding to the eigenfrequencies of the isolated disk (note that corresponding data for eigenfrequencies of the isolated disk was not available).

Figure 4.14: Spectral response of disk resonators as in Figure 4.8 with different coupler gaps. The plots show the resonances of the TE$_{0}$ mode (the pronounced extrema) and of the TE$_{1}$ mode (the secondary extrema). The CMT simulations involve the first three WGMs.

Table: Resonance positions of the disk resonator of Figure 4.8, for different separations $g=\tilde{g}$. Note that for $g=\tilde{g}=0.1 \,\mu$m, the broad peaks related to the TE$_{0}$ and TE$_{1}$ resonances interfere strongly.

	Resonance wavelengths [ $\,\mu$m]
$g=\tilde{g}$ [ $\,\mu$m]	TE$_{0}$			TE$_{1}$
0.1	1.01952	1.04311	1.06787	-	1.04559	1.07161
0.2	1.01937	1.04307	1.06790	1.02324	1.04813	1.07430
0.3	1.01953	1.04324	1.06809	1.02394	1.04893	1.07520
0.4	1.01962	1.04334	1.06820	1.02425	1.04928	1.07562
0.5	1.01966	1.04340	1.06826	1.02440	1.04946	1.07584
0.6	1.01969	1.04342	1.06829	1.02449	1.04957	1.07596