A Imperfectly Phase-Modulated Light / Residual Amplitude Modulation

[Twigg]

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 $b << a$.

 

[tech99]

If we add two waves in a phasor diagram you will see that the ampitudes add vectorially and also that the resultant is altered in phase. If either wave changes in some way then both amplitude and phase of the resultant alter.

 

[Twigg]

Oh! Thanks for that suggestion @tech99. Thinking of the two modulated exponentials as phasors really helps! A lot easier to draw the resultant than to do the algebra here. And I catch your point about there being some amplitude modulation as the two modulated phasors move out of sync with each other.