I have the following question regarding directional couplers. Below is the question.
My solution was to multiply the scattering matrices of both couplers as in the following
\begin{pmatrix}0&-\frac{i}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&0\\ \:\:-\frac{i}{\sqrt{2}}&0&0&-\frac{1}{\sqrt{2}}\\ \:\:-\frac{1}{\sqrt{2}}&0&0&-\frac{i}{\sqrt{2}}\\ \:\:\:\:0&-\frac{1}{\sqrt{2}}&-\frac{i}{\sqrt{2}}&0\end{pmatrix}\begin{pmatrix}0&-\frac{i}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&0\\ \:\:\:-\frac{i}{\sqrt{2}}&0&0&-\frac{1}{\sqrt{2}}\\ \:\:\:-\frac{1}{\sqrt{2}}&0&0&-\frac{i}{\sqrt{2}}\\ \:\:\:\:\:0&-\frac{1}{\sqrt{2}}&-\frac{i}{\sqrt{2}}&0\end{pmatrix}
Then I get the following matrix
\begin{pmatrix}0&0&0&i\\ \:0&0&i&0\\ \:0&i&0&0\\ \:i&0&0&0\end{pmatrix}
So does that mean that the resulting phase and amplitude for port 2' and 3' relative to port 1 are zero! Please help me with this as am really doubting that this is correct.
