Diffraction limit calculationcalculator

I am trying to understand why light undergoes a phase shift when reflecting off one side of a half-silvered mirror, but not the other side.

Diffraction limit calculationpdf

However, this explanation seems to completely neglect the silver coating. It seems to me that there is no air-glass boundary at all, but instead an air-silver boundary and a silver-glass boundary.

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Diffraction limitcalculator

Notice that the light travels through the glass twice, once from air to glass to mirror, and once from mirror to glass to air; one direction has increasing index and the other doesn't.

Assuming a lossless beamsplitter the energy conservation condition $\left | E_0 \right |^2 = \left | E_t \right |^2 + \left | E_r \right |^2$ gives $\phi^{'}_t - \phi^{''}_t = \phi^{'}_r - \phi^{'}_r + \pi$, thus:

Diffraction limit calculationexample

With coherent light from a laser one can assume that the path lengths of a Michelson interferometer are equal (for the purpose of your question) and due to the narrow bandwidth of the laser light one can also assume low dispersion (equal path lengths through the glass).

"Phase shift between the transmitted and the reflected optical fields of a semireflecting lossless mirror is π/2" (1980), by Vittorio Degiorgio, in the American Journal of Physics 48, 81 (1980); https://doi.org/10.1119/1.12238

There is a phase change for a reflection when a wave propagating in a lower-refractive index medium reflects from a higher-refractive index medium, but not in the opposite case.