Reciprocity of scattering matrix

For the analysis of the bent-straight waveguide couplers discussed in Section 3.2, we have considered only forward propagating modes. By considering backward propagating modes also, the corresponding full scattering matrix that relates the bidirectional amplitudes of the outgoing waves to the amplitudes of the corresponding incoming modes is given as:

$$\begin{pmatrix}\boldsymbol {a}^{-}\\ \boldsymbol {A}^{-}\\ \bolds... ...dsymbol {A}^{\ \ }\\ \boldsymbol {b}^{-} \\ \boldsymbol {B}^{-}\end{pmatrix}\,.$$ (3.21)

Here the superscripts $^{-}$ indicate the amplitudes of backwards (anticlockwise) propagating waves, where the zeroes implement the assumption of negligible backreflections. The entries of the submatrices S$_{vw}$ with $v, w =$   b, s represent the ``coupling'' from the modes of waveguide $w$ to the fields supported by waveguide $v$.

A fundamental property of any linear optical circuit made of nonmagnetic materials is that the transmission between any two ``ports'' does not depend upon the propagation direction. The proof can be based e.g. on the integration of a reciprocity identity over the spatial domain covered by that circuit [43]. More specifically, the full scattering matrix of the reciprocal circuit has to be symmetric. The argument holds for circuits with potentially attenuating materials, in the presence of radiative losses, and irrespectively of the particular shape of the connecting cores. It relies crucially on the precise definition of the ``ports'' of the circuit, where independent ports can be realized either by mode orthogonality or by spatially well separated outlets.

Assuming that the requirements of that argument can be applied at least approximately to our present bent-straight waveguide couplers, one expects that the coupler scattering matrix is symmetric. For the submatrices this implies the following equalities (T denotes the transpose):

$$_{\mbox{\scriptsize bb}} = (\mbox{\sf {S}}_{\mbox{\scriptsize bb}... ...ize ss}} = (\mbox{\sf {S}}_{\mbox{\scriptsize ss}}^{-})^{\mbox{\scriptsize T}}.$$ (3.22)

If the coupler shown in Figure 3.1 is defined symmetrical with respect to the central plane $z=0$ and if identical mode profiles are used for the incoming and outgoing fields, then one can further expect (see [43]) the transmission $\boldsymbol {A} \rightarrow \boldsymbol {b}$ to be equal to the transmission $\boldsymbol {B}^{-} \rightarrow \boldsymbol {a}^{-}$. Similarly, one expects equal transmissions $\boldsymbol {a} \rightarrow \boldsymbol {B}$ and $\boldsymbol {b}^{-} \rightarrow \boldsymbol {A}^{-}$, or

$$_{\mbox{\scriptsize bs}} = \mbox{\sf {S}}_{\mbox{\scriptsize bs}}... ...{\sf {S}}_{\mbox{\scriptsize sb}} = \mbox{\sf {S}}_{\mbox{\scriptsize sb}}^{-}.$$ (3.23)

Consequently, according to equations (3.22) and (3.23), also the unidirectional scattering matrix

$$= \begin{pmatrix}\mbox{\sf {S}}_{\mbox{\scriptsize bb}} & \mbox{\... ..._{\mbox{\scriptsize sb}} & \mbox{\sf {S}}_{\mbox{\scriptsize ss}} \end{pmatrix}$$ (3.24)

associated with the forward, clockwise propagation through the coupler, i.e. the lower left quarter block of the full matrix in equation (3.21) can be expected to be symmetric:

$$_{\mbox{\scriptsize bs}} = (\mbox{\sf {S}}_{\mbox{\scriptsize sb}... ...bs}}^{-} = (\mbox{\sf {S}}_{\mbox{\scriptsize sb}}^{-})^{\mbox{\scriptsize T}}.$$ (3.25)

Here translated to the multimode coupler setting, this means that ``the coupling from the straight waveguide to the cavity bend is equal to the coupling from the cavity bend to the bus waveguide''.