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.

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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

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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 D0D_0D0​ and total destructive interference at D1D_1D1​.

Choose some vertical slice along the interferometer, let α\alphaα be the probability amplitude of finding the photon in the top path, and β\betaβ 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.