Rat Race (Hybrid 180°) Scattering Matrix
Just trace the signal from port 1 to the other ports, regarding it as a wave with delays causing the given phase shifts. The zeros of transmission are caused by the two paths from the input to the output arriving at the output just out of phase, so they cancel each other out. It is easy to see by adding up the phase shifts that s12 and s14 must be the opposite of each other. Similarly, the paths from 2 to 3 and 3 to 4 are the same length, so they must be equal. Then, if you assume the losses are negligible, the overall s matrix must be unitary, so the magnitudes of the non zero entries is 1/sqrt(2).