Scattering matrix analysis of the full resonator

Treating the resonator shown in Figure 4.1 as a black box with four external ports A, B, $\tilde{\mbox{A}}$, $\tilde{\mbox{B}}$, let's assume that the response of the resonator is characterized by an abstract bidirectional resonator scattering matrix $\boldsymbol {\cal{S}}$. Let $\boldsymbol {A}_{\mbox{\scriptsize i}}$, $\boldsymbol {B}_{\mbox{\scriptsize i}}$, $\boldsymbol {\tilde{A}}_{\mbox{\scriptsize i}}$, $\boldsymbol {\tilde{B}}_{\mbox{\scriptsize i}}$ be the amplitudes of incoming fields, and $\boldsymbol {A}_{\mbox{\scriptsize o}}$, $\boldsymbol {B}_{\mbox{\scriptsize o}}$, $\boldsymbol {\tilde{A}}_{\mbox{\scriptsize o}}$, $\boldsymbol {\tilde{B}}_{\mbox{\scriptsize o}}$ be the outgoing field amplitudes at the respective ports. Then one can write

$$\begin{pmatrix}\boldsymbol {A}_{\mbox{\scriptsize o}} \\ \boldsym... ...scriptsize i}} \\ \boldsymbol {\tilde{B}}_{\mbox{\scriptsize i}} \end{pmatrix},$$ (4.5)

where the zeros represent negligible backreflections. The interpretation of the scattering matrix elements is as for the bent-straight waveguide coupler (see Section 3.3).

Again following the reciprocity arguments for linear circuits made of nonmagnetic materials (see Section 3.3), the above scattering matrix is symmetric, i.e.

$$\boldsymbol {\cal{S}}_{\mbox{\scriptsize BA}} = (\boldsymbol {\ca... ...de{\mbox{\scriptsize B}} \tilde{\mbox{\scriptsize A}}})^{\mbox{\scriptsize T}},$$ (4.6)

where the superscript T represents the transpose.

If the resonator shown in Figure 4.1 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 the transmission $\boldsymbol {A} \rightarrow \boldsymbol {\tilde{B}}$ to be equal to the transmission $\boldsymbol {B} \rightarrow \boldsymbol {\tilde{A}}$. Similarly, one expects equal transmissions $\boldsymbol {\tilde{A}} \rightarrow \boldsymbol {B}$ and $\boldsymbol {\tilde{B}} \rightarrow \boldsymbol {A}$. Therefore one has

$$\boldsymbol {\cal{S}}_{\tilde{\mbox{\scriptsize B}} \mbox{\script... ...}} = \boldsymbol {\cal{S}}_{\mbox{\scriptsize A} \tilde{\mbox{\scriptsize B}}}.$$ (4.7)

From Eq. (4.6), (4.7), one obtains

$$\boldsymbol {\cal{S}}_{\tilde{\mbox{\scriptsize B}} \mbox{\scriptsize A}} = (\boldsymbol {\cal{S}}_{\mbox{\scriptsize B} \tilde{\mbox{\scriptsize A}}})^{\mbox{\scriptsize T}}.$$ (4.8)

In case of monomodal port waveguides, this simplifies to

$${\cal{S}}_{\tilde{\mbox{\scriptsize B}} \mbox{\scriptsize A}} = {\cal{S}}_{\mbox{\scriptsize B} \tilde{\mbox{\scriptsize A}}},$$ (4.9)

which means that, irrespective of different separation distances, as long as there is a symmetry with respect to the $z=0$ plane, the output power at port $\tilde{\mbox{B}}$ for unit power input at port A and no input at port $\tilde{\mbox{A}}$ is exactly the same as the power observed at port B for unit power input at port $\tilde{\mbox{A}}$ and no input at port A.

In Section 4.5.3 we show that the numerical implementation respects these abstract constraints.