Coupler with monomodal bent waveguide

As the first example, we consider bent-straight waveguide couplers according to Figure 3.1, formed by straight and circularly bent cores of widths $w_{\mbox{\scriptsize s}} = 0.4\,\mu\mbox{m}$ and $w_{\mbox{\scriptsize c}}=0.5\,\mu\mbox{m}$ with refractive index $n_{\mbox{\scriptsize c}}= n_{\mbox{\scriptsize s}}=1.5$, embedded in a background with refractive index $n_{\mbox{\scriptsize b}}=1$. The configurations differ with respect to the radius $R$ of the outer bend interface, and with respect to the distance $g$ between the cores. The interaction of waves with vacuum wavelength $\lambda=1.05\,\mu$m is studied, $k=2 \pi / \lambda$ is the associated vacuum wavenumber. Both constituent waveguides are single-mode at the target wavelength, with mode profiles that are well confined to the respective cores. Figure 3.2 illustrates an example for the two basis fields. The longer outer tail of the bend profile is accompanied by a slight shift of the profile maximum towards the exterior of the bend.

Figure: Normalized fundamental TE mode profiles $\vert\tilde{E}_y\vert$ (top) and snapshots of the propagating physical fields $E_y$ (bottom) of the constituent bent (left) and straight waveguides (right) related to the coupler configurations of Section 3.4.1, for $R=5 \,\mu$m. The effective mode indices of the basis fields are $\gamma/k = 1.29297 -$   i$\, 7.5205 \cdot 10^{-6}$ (bend mode), and $\beta /k=1.3137$ (straight waveguide).

The CMT simulations of the couplers are carried out on computational windows of $[x_{\mbox{\scriptsize l}}, x_{\mbox{\scriptsize r}}] \times [z_{\mbox{\scripts... ...size o}}] = [0, R+10\,\mu\mbox{m}] \times [-R+1\,\mu\mbox{m}, R-1\,\mu\mbox{m}]$, if $R \le 5\,\mu$m, otherwise on a window $[x_{\mbox{\scriptsize l}}, x_{\mbox{\scriptsize r}}] \times [z_{\mbox{\scripts... ...ptsize o}}] = [R-5\,\mu\mbox{m}, R+10\,\mu\mbox{m}] \times [-8, 8]\,\mu\mbox{m}$, discretized with stepsizes of $h_x = 0.005\,\mu$m and $h_z=0.1\,\mu$m. For the two basis fields the CMT analysis generates $2\times 2$ transfer matrices T and scattering matrices S that can be viewed as being $z$-dependent in the sense as discussed in Section 3.4. Figure 3.3 shows the evolution of the matrix elements with the position $z= z_{\mbox{\scriptsize o}}$ of the coupler output plane.

Figure: Elements of the transfer matrix T and scattering matrix S for TE (first row) and TM (second row) polarized light, versus the output plane position $z_{\mbox{\scriptsize o}}$, for couplers as introduced in Section 3.4.1 with $R=5 \,\mu$m and $g=0.2\,\mu$m.

The matrix elements ${\sf {T}}_{o,i}$ and ${\sf {S}}_{o,i}$ relate the amplitudes of an input mode $i$ to an output mode $o$; for the present normalized modes the absolute squares can thus be viewed as the relative fractions of optical power transferred from mode $i$ at the input plane $z=z_{\mbox{\scriptsize i}}$ to mode $o$ at the output plane $z= z_{\mbox{\scriptsize o}}$ of the coupler. After an initial interval, where these quantities remain stationary, one observes variations around the central plane $z=0$ of the coupler, which correspond to the interaction of the waves. Here the nonorthogonal basis fields are strongly overlapping; it is therefore not surprising that the levels of specific components of $\vert{\sf {T}}_{o,i}\vert^2$ and $\vert{\sf {S}}_{o,i}\vert^2$ exceed $1$ in this interval.

After the region of strongest interaction, near the end of the $z$-computational interval, one finds that the elements $\vert{\sf {T}}_{\mbox{\scriptsize b0},i}\vert^2$ that map to the bend mode amplitude become stationary again, while the elements $\vert{\sf {T}}_{\mbox{\scriptsize s0},i}\vert^2$ related to the output to the straight mode still show an oscillatory behaviour. This is due to the interference effects as explained in Section 3.2.3. The proper amplitudes of the modes of the bus channel can be extracted by applying the projection operation (3.19); the corresponding matrix elements $\vert{\sf {S}}_{\mbox{\scriptsize s0},i}\vert^2$ attain stationary values, such that the ``coupling strength'' predicted for the involved modes does not depend on the (to a certain degree arbitrary) position of the coupler output plane.

Anyway, the scattering matrix S, that enters the relations (1.4), (1.5) for the transmission properties of the resonator device, should be considered a static quantity, computed for the fixed computational interval $[z_{\mbox{\scriptsize i}}, z_{\mbox{\scriptsize o}}]$. From the design point of view, one is interested in the elements of this matrix (the ``coupling coefficients'') as a function of the resonator / coupler design parameters. Figure 3.4 summarizes the variation of S with the width of the coupler gap, for a series of different bend radii.

Figure: Scattering matrix elements $\vert{\sf {S}}_{o,i}\vert^2$ versus the gap width $g$, for couplers as considered in Section 3.4.1 with cavity radii $R= 3, 5, 10, 15 \,\mu$m, for TE (first row) and TM (second row) polarized waves.

Uniformly for all radii and for both polarizations one observes the following trends. For large gap widths, the non-interacting fields lead to curves that are constant, at levels of unity ( $\vert{\sf {S}}_{\mbox{\scriptsize s0}, \mbox{\scriptsize s0}}\vert^2$, full transmission along the straight waveguide), moderately below unity ( $\vert{\sf {S}}_{\mbox{\scriptsize b0}, \mbox{\scriptsize b0}}\vert^2$, attenuation of the isolated bend mode, stronger for the TM field), or zero ( $\vert{\sf {S}}_{\mbox{\scriptsize b0}, \mbox{\scriptsize s0}}\vert^2$ and $\vert{\sf {S}}_{\mbox{\scriptsize s0}, \mbox{\scriptsize b0}}\vert^2$, decoupled fields). As the gap width decreases, the growing interaction strength between the modes in the two cores causes increasing cross coupling $\vert{\sf {S}}_{\mbox{\scriptsize b0}, \mbox{\scriptsize s0}}\vert^2$, $\vert{\sf {S}}_{\mbox{\scriptsize s0}, \mbox{\scriptsize b0}}\vert^2$ and decreasing self coupling $\vert{\sf {S}}_{\mbox{\scriptsize s0}, \mbox{\scriptsize s0}}\vert^2$, $\vert{\sf {S}}_{\mbox{\scriptsize b0}, \mbox{\scriptsize b0}}\vert^2$. This continues until a maximum level of power transfer is attained (where the level should depend on the ``phase mismatch'' between the basis fields, though a highly questionable notion in case of the bend modes [114]).

If the gap is further reduced, the cross coupling coefficients decrease, even if a growing strength of the interaction can be expected; the decrease can be attributed to a process of ``forth and back coupling'', as shown in Figure 3.5, where along the propagation axis a major part of the optical power changes first from the input channel to the second waveguide, then back to the input core. One should therefore distinguish clearly between the magnitude of the coefficients (3.8) in the differential equations that govern the coupling process, and the solution of these equations for a finite interval, the net effect of the coupler, represented by the scattering matrix S.

Figure: Forth and back coupling as the separation distance $g$ is reduced. The plots show the real physical $E_{y}$ field for the coupler configuration as in Section 3.4.1, with $R=15\,\mu$m.

For the symmetric computational windows used for the present simulations, the abstract reasoning of Section 3.3 predicts symmetric coupler scattering matrices. According to Figures 3.3 and 3.4, this constraint is respected remarkably well by the CMT simulations. In Figure 3.3, the curves related to $\vert{\sf {S}}_{\mbox{\scriptsize s0}, \mbox{\scriptsize b0}}\vert^2$ and $\vert{\sf {S}}_{\mbox{\scriptsize b0}, \mbox{\scriptsize s0}}\vert^2$ end in nearly the same level at $z= z_{\mbox{\scriptsize o}}$. Figure 3.4 shows pairs of close curves for the cross coupling coefficients, where larger deviations occur only for rather extreme configurations with small bend radii and gaps close to zero; the deviations are more pronounced for the TM case. Here one might question the validity of the assumptions underlying the CMT ansatz (3.3). Otherwise the symmetry of the scattering matrices provides a useful means to assess the accuracy of the CMT simulations, beyond merely the power balance constraint.

In the ``standard resonator model'', as seen from Eqs. (1.4), (1.5), the spectral response of the resonators depends on the scattering matrix entries. Therefore it is useful to look at the wavelength dependence of the scattering matrix, which is shown in the plots of Figure 3.6 and 3.7.

Figure: Wavelength dependence of the entries (absolute square) of the scattering matrix $\sf {S}$, $\vert\sf{S}_{\mbox{\scriptsize b}0,\mbox{\scriptsize b}0}\vert^2$ (solid line), $\vert\sf{S}_{\mbox{\scriptsize b}0,\mbox{\scriptsize s}0}\vert^2$ (dashed line), $\vert\sf{S}_{\mbox{\scriptsize s}0,\mbox{\scriptsize b}0}\vert^2$ (circles), and $\vert\sf{S}_{\mbox{\scriptsize s}0,\mbox{\scriptsize s}0}\vert^2$ (squares). The coupler configuration is as in Section 3.4.1, with $R=5 \,\mu$m and gap widths $g=0.2\,\mu$m$, 0.3 \,\mu$m$, 0.4 \,\mu$m.

Note that the scattering matrix entries are complex numbers. While the absolute square of these entries, as shown in Figure 3.6, shows a monotonic behaviour, the corresponding real and imaginary parts, shown in Figure 3.7, oscillate. These oscillations are due to phase changes experienced by the modal fields while propagating along the coupler. We will elaborate this point further in Section 4.3, with a numerical example in Section 4.4.1.

Figure 3.7: Wavelength dependence of complex valued entries of the scattering matrix $\sf {S}$. For each separation distance, the plots in the first row show $\Re({\sf{S}}_{\mbox{\scriptsize b}0,\mbox{\scriptsize b}0})$ (line with circles), $\Im({\sf{S}}_{\mbox{\scriptsize b}0,\mbox{\scriptsize b}0})$ (line with squares), $\Re({\sf{S}}_{\mbox{\scriptsize s}0,\mbox{\scriptsize s}0})$ (solid line), $\Im({\sf{S}}_{\mbox{\scriptsize s}0,\mbox{\scriptsize s}0})$ (dashed line), and the plots in the second row show $\Re({\sf{S}}_{\mbox{\scriptsize b}0,\mbox{\scriptsize s}0})$ (circles), $\Im({\sf{S}}_{\mbox{\scriptsize b}0,\mbox{\scriptsize s}0})$ (squares), $\Re({\sf{S}}_{\mbox{\scriptsize s}0,\mbox{\scriptsize b}0})$ (solid line), $\Im({\sf{S}}_{\mbox{\scriptsize s}0,\mbox{\scriptsize b}0})$ (dashed line). $\Re$ and $\Im$ denote the real and imaginary parts of the complex numbers. Coupler configurations are as in Figure 3.6.

As the wavelength increases, the bent waveguide and straight waveguide modes become less confined, and the interaction between these modes increases. This results in a steady increase of the cross coupling coefficients $\vert{\sf {S}}_{\mbox{\scriptsize b0}, \mbox{\scriptsize s0}}\vert^2, \vert{\sf {S}}_{\mbox{\scriptsize s0}, \mbox{\scriptsize b0}}\vert^2$, and a decrease of the self coupling coefficients $\vert{\sf {S}}_{\mbox{\scriptsize b0}, \mbox{\scriptsize b0}}\vert^2, \vert{\sf {S}}_{\mbox{\scriptsize s0}, \mbox{\scriptsize s0}}\vert^2$. For varying wavelengths and for both polarizations, the simulation results in Figures 3.6 and 3.7 show that reciprocity is maintained very well, i.e. the cross coupling coefficients ${\sf {S}}_{\mbox{\scriptsize b}0, \mbox{\scriptsize s}0}$ and ${\sf {S}}_{\mbox{\scriptsize s}0, \mbox{\scriptsize b}0}$ coincide as complex numbers (see the plots in the second row of Figure 3.7). Slight deviations can be observed for the configurations with $g=0.2\,\mu$m and TM polarization, for the structure with the strongest interaction and lossy, less regular fields.