# Imperfectly Phase-Modulated Light / Residual Amplitude Modulation

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I'm trying to understand this paper and others on the same topic. I struggle conceptually with their first equation, which is an expression for an imperfectly phase modulated optical field from an electro-optic phase modulator (EOM) that is contaminated with a little bit of amplitude modulation (residual amplitude modulation, or RAM for short): $$E^{PM,RAM}_{inc}(x,y,t) = E(x,y) e^{i\omega t} [ae^{i(\delta_o sin \Omega t + \phi_o)} + be^{i(\delta_e sin \Omega t + \phi_e)} ]$$
where ##E(x,y)## is the TEM profile of the beam, ##\omega## is the beam's carrier frequency, ##a## and ##b## are alignment factors determined by the polarization angles (I think they're the amount of the beam that's aligned with the ordinary or extraordinary axes?), ##\delta_{o,e}## are the modulation indexes in the ordinary and extraordinary axes, ##\Omega## is the phase modulation frequency, and ##\phi_{o,e}## are phase offsets in the ordinary and extraordinary axes. The ordinary and extraordinary axes here are determined by the crystallographic alignment of the EOM crystal. I don't really have a clear picture in my head.

The big question for me is: how does this represent amplitude modulation? I see what looks like phase modulation on the fast and slow axes of the electro-optic crystal, OK. How does the linear combination of two phase modulated signals with different modulation depths turn into an amplitude modulation?

Any input here at all is appreciated. Sorry I couldn't be more use framing my question. Thanks!

Edit: It can be assumed that ##a \approx 1## and ##b << a##.

• ergospherical

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