I was reading a paper introducing the Delayed Self-Heterodyne Interferometry technique by Okoshi in 1980. It was aimed to measure the linewidth of lasers and the main set-up was something like a Mach-Zehnder interferometer, but one of the arms was delayed by a fiber path much longer than the laser's coherence length, and there was an AOM in the other arm [I don't know if it's important]. The interference of these two beams was claimed to indicate the linewidth of the laser. What I do not understand is that how these uncorrelated beams do interfere?
For the coherence time the laser’s frequency and phase are stable(ish). Interfering two snippets of the laser from different times, they may have different frequency and phase, but for a coherence time the frequency and phase of each is well defined and they will interfere producing a beat frequency. After a coherence time their phases or frequencies will have drifted enough that the beat frequency and phase drift. However, if the coherence time is longer than your detector bandwidth, you have already recorded the beats that happened during that coherence time. The beat frequency and phase will drift through all possible relations presented by the bandwidth of the laser, but it will all be recorded in the time record.
The coherence length or time is no hard limit for two laser to be able to interfere. In the applicable math nothing changes when if both lasers are coherent or independent. One can write each laser field as E=A*exp(i*omega_0*t+phi(t) ) with the center frequency omega_0 and an added phase noise. If you interfere 2 such laser fields the resulting light intensity is I= |E1+E2|² = (E1+E2)*(E1+E2)' which in the end leads to an expression I(t) = I_0+2A1*A2*cos(phi1(t)-phi2(t))
where the phase difference creates a modulation of Intensity. With less correlation the phase difference rises and it needs more bandwidth to track the phase change.
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