Simulation results

Now we assess the validity of the perturbation expression (2.31). For the moderately lossy bent waveguide configuration considered in Figure 2.14, the estimation of the change in the phase propagation constants by the perturbation expression agrees very well with the directly computed values.

Figure: Phase propagation constants estimated by the perturbational expression, for a bent waveguide configuration with $n_{\mbox{\scriptsize s}} = n_{\mbox{\scriptsize c}} = 1.0$, $d=0.5\,\mu$m, $R=5 \,\mu$m. Dashed lines denote $\beta /k$ obtained by direct calculations, dots are reference points $n_{\mbox{\scriptsize f}}=1.5$ and $n_{\mbox{\scriptsize f}} = 1.75$, and the slope of the solid line segments is given by expression (2.31).

As an another example, for Figure 2.15, the perturbational expression (2.31) is assessed for the WGMs. For the moderately lossy fundamental and first order WGMs, the agreement is excellent, but for the second order WGMs with considerable losses (e.g. $n_{\mbox{\scriptsize f}}=1.5$, $\gamma / k=1.0422 -$   i$\ 5.7410\cdot 10^{-3}$ (TE$_{2}$), $1.0339 -$   i$\ 1.21610\cdot 10^{-2}$ (TM$_{2}$)) there are major deviations. Apparently, here the changes in the mode profiles and the attenuation constants due to the core refractive index perturbation are not negligible, such that the ansatz (2.26) is not appropriate for these fields.

Figure: Phase propagation constants of WGMs evaluated by the perturbational expression  (2.31), for a bent waveguide configuration with $n_{\mbox{\scriptsize c}}=1.0$, $d=R=5\,\mu$m. Interpretation of the curves is as for Figure 2.14.