Coupler with multimodal bent waveguide

If the core width of a bent waveguide is increased beyond a certain limit, then as discussed in Section 2.4.5, the whispering gallery regime is reached, where the modes are guided by just the outer dielectric interface. Figure 3.8 illustrates the first four lowest order whispering gallery modes that are supported by a structure with the parameters of the previous ring segments, where the interior has been filled with the core material. If the resulting disk is employed as the cavity in a resonator structure, all bend modes with reasonably low losses must be suspected to be relevant for the functioning of the device. Therefore we now consider bent-straight coupler configurations, where the bend supports multiple whispering gallery modes.

Figure: TE polarized whispering gallery modes; basis fields for the CMT analysis of the multimode couplers of Section 3.4.2. The plots show the absolute value $\vert\tilde{E}_{y}\vert$ of the radial mode profile (left) and snapshots of the propagating physical field $E_y$ (right). The effective mode indices $\gamma _j/k$ related to the bend radius $R=5 \,\mu$m are $1.32793 -$   i$\,9.531\cdot 10^{-7}$ (TE$_{0}$), $1.16931 -$   i$\,4.032\cdot 10^{-4}$ (TE$_{1}$), $1.04222 -$   i$\,5.741\cdot 10^{-3}$ (TE$_{2}$), and $0.92474 -$   i$\,1.313 \cdot 10^{-2}$ (TE$_{3}$), for $\lambda=1.05\,\mu$m. All modes are power normalized.

A parameter set similar to Section 3.4.1 is adopted, with $n_{\mbox{\scriptsize c}}= n_{\mbox{\scriptsize s}}=1.5$, $n_{\mbox{\scriptsize b}}=1.0$, $R=5 \,\mu$m, $w_{\mbox{\scriptsize c}}=R$, $w_{\mbox{\scriptsize s}} = 0.4\,\mu\mbox{m}$, $g=0.2\,\mu$m, for the target wavelength $\lambda=1.05\,\mu$m. The CMT analysis of the coupler structures is carried out on a computational window $[x_{\mbox{\scriptsize l}}, x_{\mbox{\scriptsize r}}]=[0, 15]\,\mu\mbox{m}$, $[z_{\mbox{\scriptsize i}}, z_{\mbox{\scriptsize o}}]= [-4, 4]\,\mu\mbox{m}$ with large extent in the (radial) $x$-direction, in order to capture the radiative parts of the lossy higher order bend fields. Stepsizes for the numerical integrations are $h_x = 0.005\,\mu$m, $h_z=0.1\,\mu$m, as before.

It is not a priori evident, how many basis fields are relevant for a particular simulation. Figure 3.9 shows the effect of the inclusion of the higher order bend modes on the evolution of the primary coefficients of the scattering matrix S.

Figure 3.9: CMT analysis of the multimode coupler of Section 3.4.2, effect of the inclusion of higher order cavity modes on the evolution of the scattering matrix. Results for TE waves with one (dashed line), two (dash-dotted line), three (solid line), and four cavity modes (dotted line) taken into account. Note the different vertical axes of the plots.

The self coupling coefficient $\vert{\sf {S}}_{\mbox{\scriptsize b0}, \mbox{\scriptsize b0}}\vert^2$ of the fundamental bend field is hardly influenced at all, and there is only a minor effect on the cross coupling coefficients $\vert{\sf {S}}_{\mbox{\scriptsize s0}, \mbox{\scriptsize b0}}\vert^2$ and $\vert{\sf {S}}_{\mbox{\scriptsize b0}, \mbox{\scriptsize s0}}\vert^2$. Inclusion of the first order bend field reduces merely the self coupling coefficient $\vert{\sf {S}}_{\mbox{\scriptsize s0}, \mbox{\scriptsize s0}}\vert^2$ of the straight mode by a substantial amount, due to the additional coupling to that basis field. Apparently, for the present structure it is sufficient to take just the two or three lowest order bend modes into account. This hints at one of the advantages of CMT approach, where one can precisely analyze the significance of the individual basis modes. We will resume this issue in Section 4.4.2.

With three cavity fields and the mode of the straight waveguide, the CMT simulations lead to coupler scattering matrices of dimension $4\times 4$. Curves for the evolution of the $16$ elements of the propagation and scattering matrices T, S are collected in Figure 3.10. Just as in Section 3.4.1, the application of the projection procedure to extract the stationary levels of $\vert{\sf {S}}_{\mbox{\scriptsize s0},j}\vert^2$, $\vert{\sf {S}}_{j,\mbox{\scriptsize s0}}\vert^2$ from the nonstationary quantities $\vert{\sf {T}}_{\mbox{\scriptsize s0},j}\vert^2$, $\vert{\sf {T}}_{j,\mbox{\scriptsize s0}}\vert^2$ at the exit port of the coupler is essential.

Figure: Evolution of the propagation matrix T and scattering matrix S for the coupler configuration with multimode bend as specified in Section 3.4.2; CMT results with four basis fields.

Again, the agreement of the exit levels of all cross coupling coefficients indicates that reciprocity is satisfied. In contrast to Figure 3.3, the noticeable decay of the self coupling coefficients $\vert{\sf {S}}_{\mbox{\scriptsize b1},\mbox{\scriptsize b1}}\vert^2$, $\vert{\sf {S}}_{\mbox{\scriptsize b2},\mbox{\scriptsize b2}}\vert^2$ is due to the strong attenuation of the basis fields, as directly introduced into S via equation (3.20).

According to Figure 3.11, the elements of the scattering matrix exhibit a similar variation with the gap width as found for the former monomode bent-straight waveguide coupler (cf. Figure 3.4).

Figure 3.11: Scattering matrix elements $\vert{\sf {S}}_{o,i}\vert^2$ versus the gap width $g$ for the coupler structures of Section 3.4.2 for TE (top) and TM polarization (bottom). The CMT simulations take three whispering gallery modes and the field of the straight waveguide into account.

With growing separation distance the cross coupling coefficients tend to zero. The constant levels attained by the self coupling coefficients of the bent modes are determined by the power the respective mode loses in traversing the computational window. Also here, with the exception of configurations with almost closed gap, we find that cross coupling coefficients with reversed indices coincide, i.e. that the simulations obey reciprocity.