[EricQ, gautam]

Last night, we set about trying to see if we could measure and verify the predictions of the simulations, and if there are indeed HOM sidebands co-resonating with the carrier. Koji pointed out that if we clip the transmitted beam from the arm incident on a PD, then the power of the higher order HG modes no longer integrate to 0 (i.e. the orthogonality is broken), and so if there are indeed some co-resonating modes, we should be able to see the beat between them on a spectrum analyzer. The procedure we followed was:

1. Choose a suitable PD to measure the beat. We chose to use the Thorlabs PDA10CF because it has ~150MHz bandwidth, and also the responsivity is reasonable at 1064nm.
2. We started our measurements at the Y-end. There was a sufficiently fast lens in the beam path between the transmon QPD and the high gain PD at the Y end, so we went ahead and simply switched out the high gain thorlabs PDA520 for the PDA10CF. To power the PDA10CF, we borrowed the power cable from the green REFL PD temporarily.
3. We maximized the DC power of the photodiode signal using an oscilloscope. Then to introduce the above-mentioned clipping and orthogonality-breaking, we misaligned the beam on the PD until the DC power was ~2/3 the maximum value. 
4. We then hooked up the PD output to the Agilent network analizyer (with a DC block).
5. We measured the spectrum of the PD signal around 11.066MHz (with 100kHz span) and higher harmonics up to 55MHz and used a narrow bandwidth (100Hz) and long integration time (64 averages) to see if we could find any peaks. More details in the results section.
6. Having satisfied ourselves with the Y-end measurements, we 

• restored the power cable to the green beat PD
• re-installed the thorlabs PDA520 
• verified that both IR and green could be locked to the arm

We then repeated the above steps at the X-end (but here, an additional lens had to be installed to focus the IR beam onto the PDA10CF - there was, however, sufficient space on the table so we didn't need to remove the PDA520 for this measurement).

Results:

Y-end: DC power on the photodiode at optimal alignment ~ 200mV => spectra taken by deliberately misaligning the beam incident on the PD till the DC power was ~120mV (see remarks about these values).

I converted the peak heights seen on the spectrum analyzer in volts to power by dividing by transimpedance (=5*10^3 V/A into a 50ohm load) * responsivity at 1064nm (~0.6A/W for PDA10CF).

Remarks:

1. This effect flagged by the simulations seems to be real. Unfortunately I can't get a more quantitative picture because we can't quantify the mode-overlap between the carrier 00 mode and any higher order mode on the beat PD (as we know nothing about the profile of these modes), but the simulations did suggets that the 2nd order 22MHz and 4th order 44MHz HOMs are the ones closest to the carrier 00 resonance (see Attachments #2 and #3), which is kind of borne out by these results. 
2. I disbelieve the conversions into power that I have done above, but have just put them in for now, because a DC power of 200mW at the Y-end suggests that there is >160uW of light transmitted from the arm, which is at least twice what we expect from a simple FP cavity calculation with the best-known parameters. If I've missed out something obvious in doing this conversion, please let me know! 
3. For the Y-arm, the region around 55MHz had a peak (presumably from the sideband HOM beating with the carrier) but also a bunch of other weird sub-structures. I'm attaching a photo of the analyzer screen. Not sure what to make of this...