eSummary:

I don't think the proposed scheme for sensing and controlling the homodyne phase will work without some re-thinking of the scheme. I'll try and explain my thinking here and someone can correct me if I've made a fatal flaw in the reasoning somewhere.

Field spectrum cartoon:

Attachment #1 shows a cartoon of the various field components.

• The input field is assumed to be purely phase modulated (at 11 MHz and 55 MHz) creating pairs of sidebands that are in quadrature to the main carrier field.
• The sideband fields are drawn with positive and negative imaginary parts to indicate the relative negative sign between these terms in the Jacobi-Anger expansion.
• For our air BHD setup, the spectrum of the LO beam will also be the same.
• At the antisymmetric (= dark) port of the beamsplitter, the differential mode signal field will always be in the phase quadrature.
• I'm using the simple Michelson as the test setup:
  • The ITMs have real and (nearly) identical reflectivities for all frequency components incident on it.
  • The sideband fields are rotated by 90 degrees due to the i in the Michelson transmission equation.
  • The Schnupp asymmetry preferentially transmits the 55 MHz sideband to the AS port compared to the 11 MHz sideband - note that in the simple Michelson config, I calculate T(11 MHz) = 0.02%, T(55 MHz) = 0.6% (both numbers not accounting for the PRM attenuation).
• I think the cartoon Hang drew up is for the DRFPMI configuration, with the SRC operated in RSE.
  • The main difference relative to the simple Michelson is that the signal field picks up an additional 90 degrees of phase propagating through the SRC.
• For completeness, I also draw the case of the DRFPMI where the SRC is operated at nearly the orthogonal tuning.
  • I think the situation is similar to the simple Michelson

So is there a 90 degree relative shift between the signal quadrature in the simple Michelson vs the DRFPMI? But wait, there are more problems...

Closing a feedback loop using the 44 MHz signal:

We still need to sense the 44 MHz signal with a photodiode, acquire the signal into our CDS system, and close a feedback loop.

• The 44 MHz signal is itself supposed to be generated by the interference between the TEM00 55 MHz sideband from the IFO output with the TEM00 11 MHz sideband from the LO field (let's neglect any mode mismatch, HOMs etc for the moment).
• By splitting this beat signal photocurrent in two, mixing each part with an electrical 44 MHz signal, and digitizing the IF output of said mixers, we should in principle be able to reconstruct the magnitude and phase of the signal.
• The problem is that we know from other measurements that this signal is going to go through multiple fringes, and hence, we don't have a signal that is linear in the quantity we would like to control, namely the homodyne phase (either quadrature signal can be a candidate linear signal around a zero crossing, but when the signals are going through multiple fringes, neither signal stays linear).
• One possible way to get around this problem is to use a phase tracker servo - basically, close a purely digital feedback loop, using one of the demodulated quadratures as an error signal, and changing the demodulation phase digitally such that the signal stays entirely in the orthogonal quadrature. However, such a scheme relies on the signal magnitude remaining constant. If the "error signal" goes to zero for multiple reasons (rotation out of the quadrature being considered, or just that the signal itself goes to zero), then this technique won't work. Of course, the phase tracker doesn't know what the "phase" of the signal is, when it's magnitude is (nearly) zero.
• It is true that we always expect a "background" level of 44 MHz signal, from the 11 MHz and 55 MHz sidebands in the LO beam directly interfering, but this doesn't contain any useful information, and in fact, it'd only contaminate the phase tracker error signal I think.
• So we can't rely on the error staying in one quadrature (like we do for the regular IFO PDH signals, where there is no relative phase propagation between the LO and RF sideband optical fields and so once we set the demodulation phase, we can assume the signal will always stay in that quadrature, and hence we can close a feedback loop), and we can't track the quadrature. What to do? I tried to dynamically change the phase tracker servo gain based on the signal magnitude (calculated in the RTCDS code using sqrt(I**2 + Q^2), but this did not yield good results...

Next steps:

I don't have any bright ideas at the moment - anyone has any suggestions?🤔

Aside:

I wanted to check what kind of signal the photodiode sees when only the LO field is incident on the photodiode. So with the IFO field blocked, I connected the PDA10CF to the Agilent analyzer in "Spectrum" mode, through a DC block. The result is shown in Attachment #2. To calculate the PM/AM ratio, I assumed a modulation depth of 0.2. The RIN was calculated by dividing the spectrum by the DC value of the PDA10CF output, which was ~1V DC. The frequencies are a little bit off from the true modulation frequencies because (i) I didn't sync the AG4395 to a Rb 10 MHz signal, and (ii) the span/BW ratio was set rather coarsely at 3kHz.

I would expect only 44 MHz and 66 MHz peaks, from the interference between the 11 MHz and 55 MHz sideband fields, all other field products are supposed to cancel out (or are in orthogonal quadratures). This is most definitely not what I see - is this level of RIN normal and consistent with past characterization? I've got no history in this particular measurement.