# Phase shift for reflection and transmission on absorptive medium with complex refractive index

I am trying to come with a hand-waving explanation as to why the reflection and transmission peaks of a metal-coated Fabry-Perot cavity are shifted in frequency (see the paper by Monzon & al).

For this, as they put it in the paper, the explanation is to be found in the absorption (hence complex refractive index) of the first mirror. I am struggling to find a hand-waving explanation as to why this is possible indeed.

This being said, I might be able to understand it through some phase jump upon transmission from an absorptive medium to air, but phase jumps (discontinuities) should be unphysical, right? Though, referring myself to other posts, like this one, the reflectivity is taken from the addition of the complex indexes, and not from the addition of their modules; which makes me believe the reflection index for the amplitudes of the field is complex $$R = |r|^2 \ \ \ \ \ \ \ \ \ \ \ r \in \mathbb{C}$$

if $$r$$ is defined as $$r_{1\rightarrow2} = \frac{n_1 - n_2}{n_1 + n_2}$$ I can only assume $$t$$ will be similarly defined as $$t_{1\rightarrow2} = \frac{2 n_1}{n_1+n_2}$$

Hence my question at hand: the reflection does take a phase shift different than the usual pi upon reflection, just because the coefficient r is complex. If this is so, the transmission coefficient should also induce a phase shift, should it not? I did find several courses and persons talking about the phase shift in r, but never do I see the phase shift in transmission mentioned, although i don't see any mathematical reason why this should not exist. As a matter of fact, if it does, I would be able to explain the frequency shift between the reflection and transmission peaks (which we do observe, by the way, in our lab).

I'm not particularly sure what you're taking reflection and transmission coefficients to be, but the correct route is through the Fresnel amplitude coefficients like $$r_{1\rightarrow2} = \frac{n_1 - n_2}{n_1 + n_2},$$ which you multiply with the incident amplitude to give you the reflected amplitude. If medium 2 is absorptive, then $$n_2$$ will have a nonzero imaginary part, which means that $$r_{1\rightarrow2}$$ will have a nontrivial phase, which will also be imprinted onto the reflected-light amplitude. This phase shift is absolutely real (it's what enables things like the extreme-UV reflection waveplate in this paper to work) and it is exactly equivalent to adding a little bit of optical path to the inside of your cavity. It is entirely plausible that such an effect will contribute to a shift in the resonant frequency of a cavity that includes such a mirror.