Phase shift in Mach-Zehnder interferometer

[Sonderval]

TL;DR

Is the phase shift 90 or 180 degree on reflection?

I'm confused by the phase shifts in a Mach-Zehnder interferometer because I keep finding two different explanations.

One explanation (for example, given on Wikipedia, but also elsewhere) states that on each reflection, the phase shift is 180 degrees, but only, if light is reflected from the front of a beam-splitter but not from the back. In this explanation, the second beam splitter is reversed (as shown on the Wiki page).

However, there are other explanations like this given here:
https://chem.libretexts.org/Bookshe...m_Computer_with_a_Mach-Zehnder_Interferometerwhere it is stated that each reflection has a 90-degree phase shift.

Both explanations work (in the sense that they both give the correct result for constructive/destructive interference), but which one is correct? Or are they both correct because it is also possible to construct mirrors with 90-degree phase shifts (for example, by choosing the correct path length in a dielectric and putting the mirror surface always at the back)?

Thanks for any help.

 

[Aidyan]

This is an article that might help:
https://www.cs.princeton.edu/courses/archive/fall06/cos576/papers/zetie_et_al_mach_zehnder00.pdf

 

[Cthugha]

The correct answer is: It depends. For any symmetric and lossless beam splitter, you can easily show that the phase difference between the transmitted and reflected beam is necessarily π/2. However, not all beam splitters are symmetric. For example, the dielectric ones also mentioned in the wikipedia article are obviously not symmetric. If these are lossless, you instead arrive at a phase relation that ensures that ensures that \phi_{r,1}-\phi_{t,1}+\phi_{r,2}-\phi_{t,2}=\pi, where 1 and 2 would correspond to the red and blue beams in the wiki article, respectively.

A reasonable overview is given in this article and references therein:
https://arxiv.org/ftp/arxiv/papers/1509/1509.00393.pdf
as well as in Agnesi and Degiorgio, "Beam splitter phase shifts: wave optics approach", Optics & Laser Technology 95, 72 (2017), which is unfortunately behind a paywall.

 

[Sonderval]

@Aidyan and Cthugha
Funnily, the references you both provided on first sight again seemed to contradict each other - Zetie talking about 180° phase shift on reflection, Cthugha explaining that the shift is 90°
But thanks to the reference by Henault, I finally understand it: There are symmetric and asymmetric beam splitters, and in a symmetric beam splitters, the phase shift between transmitted and reflected is 90°.
Thanks for your help, that was what I needed.

 

[vanhees71]

Indeed, of course you have to analyze the specific experimental setup. The rules for phase shifts are explained already in classical electrodynamics. Just look up "Fresnel's formulae" in a good textbook on E&M or optics (e.g., Sommerfeld, Lectures vol. 4).

 

[Sonderval]

@vanhees71
Thanks. I was just confused because my sources never mentioned that there would be alternative ways of doing it - some used the asymmetric beam splitters, some used symmetric ones, but none mentioned that both exist.