Reciprocity and the scattering matrix of waveguide modes

The implications of the Lorentz reciprocity theorem for a scatterer connected to waveguides with arbitrary modes, including degenerate, evanescent, and complex modes, are discussed. In general it turns out that a matrix $\mathit{CS}$ is symmetric, where $C$ is the matrix of generalized orthogonality coefficients and $S$ is the scattering matrix. Examples are given, including a scatterer surrounded by waveguides or free space, and discontinuities of waveguides.

Figure 1

The “black box”, or scatterer, is contained in volume $v$ with boundary $s$. (a) The general case with several waveguides connected to the surface $s$. (b) The special case with two waveguides, one on each side of $v$. The surface $s$ is regarded as two infinite planes.

Figure 2

The discontinuity between parallel plate, metallic waveguides. On the left-hand side (1) there is a waveguide of thickness $d^1$ inserted into another waveguide of thickness $d^2$. On the right-hand side (2), only the waveguide of thickness $d^2$ is present.

Figure 3

The matrix $|c_k{r}_{kj}^1|$. The numbers used are $d^1=d$, $d^2=2d$, and $\omega d/c=20$. In the plot, modes $1,2,...,15$ correspond to even TE modes in the central waveguide on the left-hand side, while modes $16,...,30$ correspond to even TE modes in the upper and lower waveguides. Modes 1, 2, 3, 16, 17, and 18 are propagating, while the remaining modes are evanescent. The plot was generated using 100 modes on each side in the calculations, which leads to an asymmetry of ${10}^{-3}$ for the matrix. This error is reduced to ${10}^{-5}$ by using 1000 modes.

Figure 4

The matrix $|c_k{r}_{kj}^1|$. The refractive index of the dielectric waveguide core is set to 1.5, and $\omega d/c=1$. The modes are computed numerically using periodic boundary conditions with period $L=100d$. To reduce the effect of the artificial boundary conditions, we introduce a small loss everywhere, such that $\epsilon \to \epsilon +i0.05$. The coefficients for the 50 lowest-order TE waveguide modes are displayed in the figure. Only the first waveguide mode is bound, modes $2,...,31$ are unbound and propagating (i.e., with small imaginary parts of the propagation constants), and the remaining modes are unbound and evanescent (i.e., with large imaginary parts of the propagation constants). The plot was generated using 100 modes on the waveguide side. On the free space side we used complex exponentials as the modal fields for simplicity. Since these fields are neither even nor odd, we included twice as many modes there, corresponding to positive and negative transverse wavenumbers. When the matrix Fresnel equation (33) is overdetermined, the matrix inverse should be replaced by the Moore-Penrose pseudoinverse. Using 100 waveguide modes leads to a matrix asymmetry of ${10}^{-3}$. Again, the error is reduced to ${10}^{-5}$ when using 1000 waveguide modes.

Figure 5

(a) Phase of $r_{kj}^1$ and (b) phase of $c_k{r}_{kj}^1$ for the dielectric waveguide in Fig. 4. The phase plots clearly show that multiplication by the $c_k$ coefficients is necessary to obtain a symmetric matrix.