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The interaction between a beam of light and an acoustic wave is commonly modelized as a photon-phonon scattering phenomenon. Applying the conservation of energy we have that

$$ \hbar\nu_1=\hbar\nu_0\pm\hbar\nu_p$$

Simplyfing $\hbar$ is easy to see that the scattered beam is shifted in frequency of $\pm\nu_p$.

So, in my understanding the presence itself of a diffraction order is due to the scattering with a phonon. So I would say that doesn't matter if I have a travelling or a stading wave: if I have scattering I also have a Doppler shift, but on Wikipedia (https://en.wikipedia.org/wiki/Acousto-optic_modulator#Frequency) is written that diffraction from a standing wave does not shift the frequency of the diffracted light.

Is this true? If so, why?

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