I've been developing an idea for making a direct measurement of the SRC Gouy phase at RF. It's a very different approach from what has been tried before. Prior to attempting this at the sites, I'm interested in making a proof-of-concept measurement demonstrating the technique on the 40m. The finesse of the 40m SRC will be slightly higher than at the sites due to its lower-transmission SRM. Thus if this technique does not work at the 40m, it almost certainly will not work at the sites.
The idea is, with the IFO locked in a signal-recycled Michelson configuration (PRM and both ETMs misaligned), to inject an auxiliary laser from the AS port and measure its reflection from the SRC using one of the pre-OMC pickoff RFPDs. At the sites, this auxiliary beam is provided by the newly-installed squeezer laser. Prior to injection, an AM sideband is imprinted on the auxiliary beam using an AOM and polarizer. The sinusoidal AOM drive signal is provided by a network analyzer, which sweeps in frequency across the MHz band and demodulates the PD signal in-phase to make an RF transfer function measurement. At the FSR, there will be a AM transmission resonance (reflection minimum). If HOMs are also present (created by either partially occluding or misaligning the injection beam), they too will generate transmission resonances, but at a frequency shift proportional to the Gouy phase. For the theoretical 19 deg one-way Gouy phase at the sites, this mode spacing is approximately 300 kHz. If the transmission resonances of two or more modes can be simultaneously measured, their frequency separation will provide a direct measurement of the SRC Gouy phase.
The above figure illustrates this measurement configuration. An attached PDF gives more detail and the expected response based on Finesse modeling of this IFO configuration.