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The green laser is phase locked to the main laser (PSL) with a PLL. The relative frequency between the PSL and green will be determined by this PLL. The noise of the PLL local oscillator will be directly introduced as the differential frequency fluctuation of PSL and green. The required level of this differential frequency noise can be roughly derived as follows.
The FSR of the arm cavity is about 50kHz. Since the finesse of the arm cavity is about 1500, the FWHM of the carrier resonance is about 33Hz. When the arm is locked to the green laser, the PSL carrier has to be sitting at somewhere in the FSR, ideally without moving. In reality the carrier moves around in the FSR due to the differential frequency noise. Since we will slowly bring the PSL carrier to the resonance, we want keep the wondering of the PSL carrier in the FSR much less than the FWHM of the resonance. Let's say we want keep it smaller than 1/1000 of the resonance width. Then the required differential frequency noise is &(df_{\mathrm{RMS}); = 33/1000=0.033Hz.
