Whispering gallery modes

If the core width of a bent waveguide is increased, then as in the case of straight waveguides, the mode profile changes, and eventually the bent waveguide becomes multimodal. But at the same time another interesting phenomenon occurs, which can not happen with straight waveguides. If the core width of a bent waveguide is increased beyond a certain limit, a regime is reached where the bend modes are guided by just the outer dielectric interface and the precise location of the inner dielectric interface becomes irrelevant. These modes are known as ``whispering gallery modes'' (WGMs).

The model of Section 2.2 covers those configurations with the formal choice $n_{\mbox{\scriptsize s}} = n_{\mbox{\scriptsize f}}$ in Figure 2.1, where $d$ becomes irrelevant. The above mentioned transition of a bend mode towards a WGM is shown in Figure 2.10. For illustration purposes, we adopt a set of parameters from Ref. [57], that specifies a high-contrast curved interface with a rather small radius, i.e. a parameter regime that differs considerably from the previous bent slabs. For the present configuration, the field profiles of the TE$_{0}$ mode for $d=1.5 \,\mu$m$, 2.0 \,\mu$m, and $4 \,\mu$m are almost identical.

Figure: Transition of a bend mode towards a whispering gallery mode. The core width $d$ of a bent waveguide with the parameters $n_{\mbox{\scriptsize f}}=1.5$, $n_{\mbox{\scriptsize c}}=1.0$, $R=4.0\,\mu$m, and $\lambda=1.0\,\mu$m is increased, until the whispering gallery mode regime is reached.

In fact, plots of the propagation constants for these modes in Figure 2.11 show that for core widths larger than $1.0 \,\mu$m, $n_{\mbox{\scriptsize eff}}$ becomes almost independent from the core width. For increasing core width, a larger part of the mode profile is trapped inside the core. This results in the increase of the phase constant $\beta$ and the decrease of the attenuation constant $\alpha$.

Figure 2.11: Effect of increasing the core width on the TE$_{0}$ bent mode propagation constant (given in terms of effective refractive index $n_{\mbox{\scriptsize eff}} = \gamma / k$). The waveguide configuration is as for Figure 2.10.

A comparison of the present analytical simulation results for the propagation constants of the four lowest order WGMs with FDTD results from Ref. [57] is shown in Table 2.4. Again we find a very good agreement.

Table: Propagation constants $\gamma$ for the whispering gallery modes of a curved dielectric interface with the parameters $n_{\mbox{\scriptsize f}}=1.5$, $n_{\mbox{\scriptsize c}}=1.0$, $R=4.0\,\mu$m, and $\lambda=1.0\,\mu$m. Third column: Results from Ref. [57].

	$\gamma /k$,  present	$\gamma /k$,  Ref. [57]
TE$_{0}$	$1.3106 -$i$\,1.1294\cdot 10^{-5}$	$1.310-$i$\,1.133\cdot 10^{-5}$
TE$_{1}$	$1.1348 -$i$\,1.8862\cdot 10^{-3}$	$1.134-$i$\,1.888\cdot 10^{-3}$
TE$_{2}$	$0.9902 -$i$\,1.1676\cdot 10^{-2}$	$-$
TE$_{3}$	$0.8558 -$i$\,1.8832\cdot 10^{-2}$	$-$

As shown in Figure 2.12, for growing mode order, qualitatively one finds the increase of the attenuation, the outwards shift of the outermost profile intensity maxima, the raise of the exterior field levels, and the wider radial extent of the profiles, just as for the modes of the bent cores in Figs. 2.7 and 2.9. In contrast to the impression given e.g. in Refs. [96,98], the complex mode profiles exhibit minima in the absolute value of the principal field component, not nodal points.

Figure 2.12: Profiles (top) and physical field evolution (bottom) of the three lowest order whispering gallery modes according to the specification of Table 2.4.

Despite the substantial differences in the attenuation levels of these modes, the higher order fields may well play a role for the representation of resonances of the corresponding disc-shaped microresonator cavity, due to the rather short circumference. Therefore we conclude this section with two examples of interferences of whispering gallery modes in Figure 2.13. As for the bend slabs in Figure 2.8 one observes an interior beating pattern and ray-like bundles of waves in the exterior, here on much shorter ranges in terms of the local wavelength.

Figure 2.13: Interference patterns of the modes of Figure 2.12 and Table 2.4; the plots show the absolute value $\vert E_y\vert$ (left) and snapshots of the time harmonic physical field $E_y$ (right). Superpositions of the two (top) and three (bottom) lowest order fields are considered, initialized with unit amplitudes of the normalized profiles (with positive $\tilde{E}_y(R)$) at $z=0$.