The normal lock acquisition of the power recycling cavity uses the B2_3f_ACp signal, which extracts the beating of the first and second modulation harmonics in reflection of the interferometer. The reason is that the gain of normal demodulated signal B2_ACp changes sign when the power recycling cavity passes from over-coupled to under-coupled, or in other words when the reflectivity of the Michelson (as seen from the symmetric port) becomes smaller than the PR reflectivity. In few words the reason is that B2_ACp is extracted from the beating of carrier and first order sidebands which are both resonant inside PRC.

Instead the second order harmonic is not and therefore a robust error signal can be obtained from its beating with the first order sideband, which is resonant.

This suggested the possibility of getting completely rid of the 3f signal by using the 8MHz modulation. Indeed the 8MHz sidebands are not resonant in PRC, while carrier and 6MHz sidebands are. Therefore the beating of carrier with 8MHz should give an error signal with properties similar to B2_3f.

A simple Finesse simulation has been set up. Maintaining the cavities on resonance, both PRCL and MICH are swept. MICH detuning resembles the different dark fringe offsets during lock acquisition, while the PRCL one is used to study the behavior of the error signals. The fringe offset is reconstructed as the normalized ratio of B1_DC over B5_DC. Demodulation phases have ot been tuned, therefore both phase an quadrature signals are shown. No high order mode has been used.

The following two plots show the behavior of B2_ACp. Clearly it can not give a working error signal to control PRCL for all values of fringe offsets.

The following two plots show instead B2_3f_ACp. A good error signal with zero close to the operating point (PRCL phi = 0) is always present.

Finally, these two plots shows the 8MHz demodulated signal. The behavior is very similar, meaning that B2_8MHz_ACp could in principle be used in place of B2_3f_ACp from the beginning of the lock acquisition sequence.

Finesse input file can be downloaded here.