Simulation results

The coupled mode equations (3.9), (3.10) are treated by numerical means on a rectangular computational window $[x_{\mbox{\scriptsize l}}, x_{\mbox{\scriptsize r}}] \times [ z_{\mbox{\scriptsize i}}, z_{\mbox{\scriptsize o}}]$ as introduced in Figure 3.1. The solution involves the numerical quadrature of the integrals (3.7), (3.8) in the $z$-dependent matrices M and F, where a simple trapezoidal rule [113] is applied, using an equidistant discretization of $[x_{\mbox{\scriptsize l}}, x_{\mbox{\scriptsize r}}]$ into intervals of length $h_x$.

Subsequently, a standard fourth order Runge-Kutta scheme [113] serves to generate a numerical solution of the coupled mode equations over the computational domain $[z_{\mbox{\scriptsize i}}, z_{\mbox{\scriptsize o}}]$, which is split into intervals of equal length $h_z$. Exploiting the linearity of equation (3.10), the procedure is formulated directly for the transfer matrix T, i.e. applied to the matrix equation

$$\frac{\mbox{d} \mbox{\sf {T}}(z)}{\mbox{d}z} = \mbox{\sf {M}}(z)^{-1}\,\mbox{\sf {F}}(z)\,\mbox{\sf {T}}(z)$$ (3.26)

with initial condition T$(z_{\mbox{\scriptsize i}}) = \mbox{\sf {I}}$ (the identity matrix), such that $\boldsymbol {C}(z) =$   T$(z)\,\boldsymbol {C}(z_{\mbox{\scriptsize i}})$. While the evaluation of the resonator properties via equations (3.20) and (1.4), (1.5) requires only the solution T$=$   T$(z_{\mbox{\scriptsize o}})$ at the coupler output plane $z= z_{\mbox{\scriptsize o}}$, also the examination of the evolutions of T$(z)$, or S$(z)$, respectively, turns out be instructive.

Having explained how to compute the scattering matrices for the bent-straight waveguide couplers, in Section 3.4.1 and 3.4.2 we summarize a series of numerical results for the theory outlined in Section 3.2 and 3.3. Note that couplers with quite small radius with substantial refractive index contrast are considered as test structures. For the CMT approach, these represent rather extreme configurations, partially with strongly leaky fields, thus with relatively large field strengths in the regions where the CMT ansatz-field clearly violates the Maxwell equations. One expects that for couplers that consist of bent waveguides with large radii, i.e. with better confined bend modes, and more adiabatic interaction in the coupler regions, the CMT approach comes even closer to reality.