I have done some thinking about the mGyro locking setup, and I think I have found a way to make it shot noise limited.

• Photodetection

The REFL diodes for the gyro should be set up in the aLIGO configuration (i.e. resonant notch readout). We will use an LMH6624 as the MAX4107 is obsolete. Since we are operating at 33 MHz here, the GBW of this amplifier is not as much of an issue as it is for the transmission setup, and we can use it in a higher-gain setting. I have chosen 4.5k/50 -> G = 91 for its gain at the moment, though this can be adjusted slightly without much issue. The notch is a 31-pF cap in series with a 750-nH inductor, where the non-inverting input of the LMH6624 is between them. This gives a transimpedance gain of ~75 dB V/A @ 33 MHz. Here is a transfer function and a closeup

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The responsivity of the 2-mm diodes (Perkin-Elmer C30642) we will be using is ~0.8 A/W @ 1064nm, and the Cougar adds an additional 10 dB of gain to give a total optical response of 1.4 x 10⁴ V/W.

The aLIGO design is such that the resonant transimpedance is first-order insensitive to fluctuations in the diode capacitance from things like bias voltage fluctuations.

The maximum current rating of the C30642 is 100 mA, meaning that we can now use the full available optical power of ~50 mW per direction. This is a huge improvement from the < 1 mW we were able to use with the broadband PDs.

Below is the input-referred noise spectrum of this photodetector. At 50 mW incident power, given optimum modulation---which should be in reach now, given the 20-dB increase in available modulation depth from the EOM resonant circuit---the expected shot noise current from the RF sidebands alone (i.e. zero contrast defect) is ~4.6 x 10^-11 A/rHz. The PD beats that at and around 33 MHz.

• Servo

When designing the frequency response of the new servo, an obvious starting place was to ensure that the overall OLTF had a gain of unity at our current UGF of ~15 kHz. This is because we know the bandwidth is currently limited by PZT resonances which we will have to tailor notch filters for in order to progress further. From here, the idea is to increase the low-frequency gain as much as possible without eating up phase margin at the UGF.

In order to estimate what the gain of the new servo has to be at 15 kHz, we can multiply together all the individual improvements in the optical response:

We have eta: 0.7 -> 0.8, Z_PD: 10000 V/A -> 18000 V/A, P: ~0.5 mW -> 50 mW, and G_SB is the improvement in optical gain from the larger modulation depth (~ a few). This isn't exact, but a good estimate is about 43-46 dB.

Below is the magnitude response of PDH box #1437 in the configuration when our UGF was 15 kHz, showing a servo gain of about unity. This means that we can't go too wrong by aiming for a gain of ~-40 dB @ 15 kHz with the new servo.

Below is the current "PDH2" board schematic, which now has the correct filter stage components filled in. It has 4 poles at 50 Hz (switchable to DC), two zeros at 960 Hz, one zero at 9.6 kHz, and one zero at 96 kHz. This last one hands the 1/f off to the cavity pole. I am now remembering that the cavity pole is supposed to be at 98 kHz, not 96 kHz, so we should change this. We should get a more precise measurement of it before we do, though.

Here is the transfer function in the "0" state with no boosts, showing a gain of ~-40 dB @ 15 kHz. The circuit in this configuration has less than 9 degrees of phase lag @ 300 kHz; at 100 kHz, it is less than 3 degrees.

Here is the noise curve of the PDH2 (I have included only op amp noise, as the resistor noise is drowned by it). I pinpointed the major low-frequency contribution to "U7", the second, variable-gain stage (second plot).

Multiplying the expected shot noise current from above by the PD transimpedance ( 85 dB V/A ), we get a shot noise voltage of 5.8 x 10^-7 V/rHz. The PDH2 beats this from a few kHz down to 100 mHz.

So, the above outline gives a way to minimize electronics noise in the locking scheme over a reasonable band. This shot noise level corresponds to ~1.8 x 10^-11 (rad/s)/rHz, well below the requirement curve at the three frequencies for which it is defined. To improve further this we could:

1. Use quieter op amps
2. Increase the input power
3. Increase the finesse (right now it is about 450)

For completeness, here is the PDH2 TF in mode "4" with all four boosts engaged: