Mach-Zehnder interferometer

A Mach-Zehnder interferometer consists of two beam splitters (thick grey lines) and two mirrors (thin lines). A light beam splits at the first beam splitter, travels some distance in top top and bottom paths, and then recombines at the second beam splitter.

Depending on the path length distance, the top and bottom beam will have some different phase. By varying this distance it can be arranged to have e.g. total constructive interference at $D_0$ and total destructive interference at $D_1$.

Choose some vertical slice along the interferometer, let $\alpha$ be the probability amplitude of finding the photon in the top path, and $\beta$ in the bottom path. We can represent the behavior of the interferometer as a series of matrix multiplications, each giving the probability amplitudes at one part of the interferometer.

Say the beam splitters are symmetric, so they reflect and transmit 1/2 the intensity. Then the matrix for the beam splitter can be given by

The matrix acts on the probability amplitude, not the intensity, so we get a $\sqrt 2$