Yuhang and Michael

This measurement is more for the purpose of the KAGRA filter cavity project. We would like to optimise the design of the length control for the filter cavity. This involves setting the parameters of the control loop (bandwidth, filters, RMS noise etc) as well as choosing a control scheme of one of the three available for frequency dependent squeezing (A+ Resonant Locking Field, AdV+ Subcarrier, TAMA CCFC).

The performance of the filter cavity length control is in principle limited by two noise types:

• backscattering - the length noise of the filter cavity causes fluctutations of incident coherent fields, most notably the residual interferometer dark port power that leaks through the squeezing path Faraday isolators. This light has length noise imprinted on it by the filter cavity which degrades squeezing at frequencies where mirror motion is dominant (~ < 10 Hz). Backscatter can be decreased by increasing the gain or bandwidth of the control loop. However, increasing the gain at low frequency causes issues with stability, and increasing the bandwidth causes issues with...
• sensing noise - primarily caused by phase (frequency) noise of the voltage controlled oscillator that generates the sensing field for filter cavity locking. The phase noise of the sensing light causes locking fluctuations of the filter cavity length, which in the case of sensing noise can't be suppressed by increasing the gain of the control loop. In the A+ case, the phase noise of the AOMs that generate the resonant locking field is quite severe and the reinjection of sensing noise becomes prominent at and above about 10 Hz.

Changing the control bandwidth will raise one of these noises and lower the other, and the "bucket" is of the order 10-30 Hz for ~ 100m filter cavities with km scale GW detectors. Thus, we would like to measure the frequency or phase noise of the TAMA PLL at low frequency, as a placeholder for the KAGRA filter cavity sensing control noise. The PLL generates a sine wave that is the frequency difference of the main laser and CC/p-pol. Obviously we would like to keep this difference as stable as possible. We can beat this sine wave with a different one in order to check the stability. When the frequency of the signal and LO is the same, provided that the LO is stable enough, the voltage coming out of the mixing between the local oscillator and PLL is proportional to the phase noise. The relevant equations for this principle of measurement are given in Yuhang's thesis, 4.25 to 4.31. By default, we are using 190 MHz for the CC PLL and 250 MHz for p-pol, but the digital system in TAMA can only generate sufficiently clean signals up to 100 MHz. So we lowered them to 50 MHz. 

The spectrum of the phase noise is related to the spectrum of the mixed voltage by:

PSD (rad/rtHz) = PSD (V/rtHz) / Apk^2

Where Apk^2 is the amplitude of the oscillation that occurs when we change the LO by a small amount.

For the actual measurement, I used the P-POL PLL MON channel as the signal and the CC PLL LO (DDS3 board, Channel 0) as the local oscillator. The level of the DDS3 Ch0 output at 50 MHz was found to be 7.61 dBm after removing the attenuator that is normally attached to it. The signal and LO were send to a small mixer. By setting DDS3 Ch0 to 50.0001 MHz, the Apk is found to be 0.083 V. Then, resetting it back to 50 MHz, the signal and LO are combined at the mixer to give the phase noise spectrum. Of course, you should make sure that the PLL has remained locked throughout the measurement :).

Figure 1 shows the spectrum of PLL phase noise. Figure 2 shows a previous measurement from Yuhang's thesis. In general seems mostly consistent but I think we have more noise in the tens of Hz band right now.