Beat signals

NascentOxygen

Certainly the sidebands themselves aren't of the same frequency, so additively they are never going to cancel. But when each is multiplied (aka mixed) with the carrier frequency, among the products of that process are two low frequency components each of frequency Ω. In this phase modulation, one of those low-frequency components is in anti-phase to the other.

f95toli

It is because the sidebands are the results of a carrier being passed through a phase modulator where is is mixed with the IF, i.e. they are the products of the same process which means that they maintain a phase relationship when the signal is downmixed again in the lock-in.

I use a variation of this technique in my experiment (although not for spectroscopy), and I can assure you that the math not only looks good but also works in practice

NascentOxygen

I'm coming to the conclusion that there is a bit of poetic licence here. When the author says that the upper sideband is out of phase with the lower sideband, I think he probably being conservative with words, and what he really means is that the modulation carried on that upper sideband, when demodulated returns a demodulated signal which is in antiphase with the demodulated signal recovered from the lower sideband.

In which case, any mathematics is moot.

If my hunch is right, this probably arises so often in communications engineering that it is a well-understood shorthand. Let's see if anyone can support me on this.