I am not sure what you say is possible for AOMs. Light needs to travel through acousto-optic devices in order to be diffracted. The Klein-Cook parameter (which identifies the properties of the acousto-optic interaction) even contains the length of the device. In reflection this parameter would go to zero, thus describing no interaction at all (unless the reflection happens after the device).

@AlessandroZunino Sure. Edited.

Note that the Doppler shift from a moving mirror is $\Delta f = 2v/c$, and the frequency shift you describe here ($\text{few}\,\mathrm{Ghz}/100\,\mathrm{THz} \sim 10^{-5}$) corresponds to a mirror speed $v$ like the speed of sound in a solid. The "moving mirror" approach (or "moving scatterer" in the transmission geometry) is baked pretty deep into this setup, though here the "mirrors" ("scatterers") are pressure variations in whatever material comprises the active part of the modulator.

@rob I agree! The moving-mirror / moving-scatterer picture is essential for this approach to work. But OP didn't require a static reflection surface -- the question asks for a static device, and this is perfectly applicable for the devices I describe ;-).

@EmilioPisanty I'm less familiar with EOMs than with EOMs, but I'm not sure for this answer their difference is significant. In the end, in both cases, there is a traveling refractive index wave $\Delta n \sin(\Omega t)$. How this wave is created should not matter. If this is true, then the interaction is described by exactly the same math of the acousto-optic effect. Thus, in both cases, there is no hope to make those devices work in reflection mode.

"it should also be possible to build equivalent devices there" - are there acoustoacoustic modulators?

@AlessandroZunino It's indeed plausible that the idea I presented might not work -- hence the initial qualifier that you can (only) 'probably' make it work. But transmission frequency shifters do exist, and the backup option of putting a mirror behind those is clearly feasible.

@user253751 I have no idea whether such devices have been put into practice. But nonlinear acoustics is certainly a thing, and there isn't any real missing physics link that would prevent you from designing such a device, at least in principle. I imagine there aren't any available because there isn't a big need for them. (Optical frequency shifters are of everyday use in precision spectroscopy: you have a stabilized laser, and then you use an EOM to shift it and scan over the frequency range of interest.)

Is this basically just modulating the light frequency with the oscillation period? (And if yes, why would this shift the frequency in one direction)?

@NorbertSchuch I'm not sure what you mean by "modulating the frequency with the oscillation period", but at least in AOMs you can sort of think of it as photons absorbing a transverse phoNon from the acoustic standing wave in the AOM, which then deflects the beam depending on whether the photon gained or lost energy, so you get distinct beams. You can then retroreflect one of the beams back through the AOM and do some polarisation stuff to pick it off from everything else

see fig 2 here, it's a very common setup researchgate.net/publication/…

@llama I would like to understand where the change in frequency comes from. It cannot just come from a (quasi-static) change in the refractive index: After leaving the cell (i.e. coming from vacuum, going into vacuum), the frequency & wavelength of the beam would be exactly the same as before, it would just leave in a certain direction. -- Differently asking: If I would send a continuous classical light wave $cos(kx-\omega t)$, what would I get out?

So my understanding would be that you get sth. like $\cos(\delta t)\cos(kx-\omega t)$, with $\delta$ the modulation frequency. But this should have components at $\omega$ as well as $\omega\pm\delta$, so (i) it is not clear to me how one would single out the $\omega-\delta$ beam (unless one knows $\omega$), and (ii) only a fraction of the light would be red-shifted.

@NorbertSchuch the proper way to analyse an AOM is that the acoustic waves form a periodic refractive index variation, which acts like a moving grating. The beams you get out are of the form $cos(kx-(\omega+m\delta)t)$, where $m = ...,-1,0,1,2,...$. You're right that only a fraction is diffracted, but in can be up to 80% intensity in the $m=\pm1$ order

OP asks "during the reflection by a different type of mirror, could the light photons lose some energy ". This proposal does not frequency shift an image during reflection.

@llama Thanks for the explanation. But there is the limitation that you cannot exceed 50% redshifted ... and it is not obvious that the $m=-1$ order goes in a certain direction (so you can filter it) -- does it?

@NorbertSchuch in reality the limit is about a GHz due to the fact that you have to use a piezoelectric crystal. The diffracted beam does come out at an angle defined by $m$ and the wavelength of the acoustic wave (and the incident light).

@llama Thanks. So it is not clear that you could build a device which only returns the redshifted beam, independent of the frequency of the ingoing beam.

@NorbertSchuch It's unclear to me whether one can build a device whose only action is to return a red-shifted beam, with the only energetic loss being the redshift.

Thanks for clarifying, Emilio! Is there something which would rule this out (within a certain framework)?

If this device is similar to a moving mirror whose time-averaged position doesn't change, then the time-averaged frequency-shift factor has to be 1 (unless you've invented time travel). What is the frequency shift as a function of time? When does it redshift the reflected light and when does it blueshift it?

@benrg I think it is more a way to (amplitude-)modulate the original signal with a slower frequency, which upon Fourier decomposition yields signals at different frequencies (but without further time dependence). Of course, what you call time-varying and what you attribute to the presence of different frequencies is probably rather a matter of taste.