experimental physics - Effect of Lock-in Amplifier Phase Noise

Suppose that an oscillator with phase noise is used to drive a modulating process and that the same oscillator is used (with an appropriate phase shift) is used to synchronously demodulate the signal from a detector monitoring the process. I will assume that the oscillator frequency is sufficiently low that the system contributes no relative phase noise between the signal and the reference (i.e. the system is perfectly coherent)

Since the reference modulating waveforms are identical (albeit noisy), will the oscillator phase noise have any influence on the demodulated signal? Does the linewidth and pedestal of the oscillator used affect the SNR of the lock-in detector?

Intuitively I feel that there is no effect, however, most lock-in equipment proclaim exceptionally low phase-noise characteristics.

Answer

Let your oscillator signal be $r(t)=A(1+m(t))cos(\omega t +\phi(t))$; here $m(t)$ and $\phi(t)$ are the AM (amplitude modulation) and PM (phase modulation) noise processes. If you use this $r(t)$ to detect synchronously another signal, say, $x(t)=B(t)cos(\omega t +\theta(t))$ that is derived from the oscillator then aside from other noise sources you have $$x(t)=B(t)cos(\omega t +\theta(t)) = A(1+m(t-\tau))cos(\omega (t-\tau) +\phi(t-\tau) + \theta_0),$$ where $\tau$ is a time delay and $\theta_0$ is a phase delay, both independent of all the other quantities. Synchronous detection really just what nowadays is more frequently called IQ detection, you multiply the reference and its $\pi/2$ shifted version with the incoming signal and low pass filter the baseband part, then th result is further processed, mostly digitally. The result has two terms $$c(t) \propto cos(\mu(t)+\theta_1) = cos(\phi(t)-\phi(t-\tau)+\theta_1)$$ and $$s(t) \propto sin(\mu(t)+\theta_1))=sin(\phi(t)-\phi(t-\tau)+\theta_1) $$ where $\theta_1=\omega \tau +\theta_0$ is a fixed phase shift and AM noise is ignored.

Notice the phase noise shows up in the expression as $\mu(t)=\phi(t)-\phi(t-\tau)$. For small delays, $\mu(t) \approx \dot\phi(t)\tau$, and here $\dot\phi(t)$ is what conventionally called the FM noise of the oscillator. In short, smaller the delay $\tau$ is the less the effect of the phase noise will be on the measurement. I have ignored the AM noise but usually in a mixer (the synchronous detector - multiplier) there can be AM to PM conversion effects, and these can be quite painful to eliminate if very low noise measurements are needed. Usually, the easy thing to do is to follow the oscillator with a very good amplitude limiter to avoid such noise conversion.