EDITS:

We slightly modify the way how we normalize the sensing matrix, but the conclusions remain the same. 

Previously, for a given dof (i.e., SRM or BS), we normalize its response in the sensors by [ Norm(phi_ASA) = sqrt(A_I**2 + A_Q**2 + B_I**2 + B_Q**2)(phi_ASA) ], where A/B for gouy phase and I/Q for demod phase, and the normalization factor is thus a function of sensor A's gouy phase at the AS port. 

Now we instead take the max of Norm over all the ASA gouy phase space, which slightly increase the conditional number. 

For completeness we also computed the conditional number for AS36 as a comparison. Here because the SRM shows up in the I phase (degenerate with spot centering signal), we relax the requirement of using only Q-phase signals which we place on the AS72 case. Yet we don't want to invert a overdetermined 4x2 matrix. Therefore we choose one of (AS36A_I, AS36B_I) to max SRM response and choose one of (AS36A_Q, AS36B_Q) to max BS response. Even under this relaxed condition, AS36 still has much larger (i.e. worse) conditional number than AS72 when 02/20 modes are present.  

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We studied the AS72.8 ASC scheme (beating RF 45.5MHz and RF 118.3MHz sidebands at the AS port) for the new monolithic SRMs. In the simulation we adopt RoC_srm = -5.675m and T_srm = 32.35%. The results are attached.  

CONCLUSIONS:

1. If the differential ITM thermal  is small (1/f < 1/300km), the new scheme is robust against small variations of SRC one-way gouy phase. 

2. If the SRC one-way gouy phase is at the nominal value of 17.5 deg (0.5 deg smaller than the original one due to the change of SRM RoC; other SRC parameters fixed), the 02/20 modes of RF +118.3MHz is close to resonance and thus differential thermal lens changes the error signal noticeably. In the worst-case scenario the sensing matrix has a conditional number of 11 18. I.e., the sensing matrix is as degenerate as [1, 1; 0.7, 1] [1, 1; 0.8, 1]. As a comparison, with same amount of diff lens and relax the requirement of decoupling WFSs from QPDs, the AS36 sensing matrix has a conditional number of 36.

3. If we can tune the SRC one-way gouy phase (e.g. with a ring heater to tune SR3 RoC), then increasing the SRC one-way gouy phase will make the ASC signal more robust. For SRC one-way gouy phase of 22 deg, the largest conditional number is no more than 2 3 which can be easily inverted. 

DETAILS:

1. Most of our parameters are based on the Finesse input file (T1300904) for H1. We changed SRM parameters as described above; we also added differential ITM thermal lens of 1/f <= 1/100 km at both ITMX and ITMY, and allowed the SRC gouy phase to vary to account for its uncertainty as well as actions may happen (e.g. adding SR3 ring heater or moving SR2-SR3 distance) after O2. The modulation depth of the RF 118.3MHz sideband is assumed to be 1/1000 of the mod depth of RF 9.1MHz (37042), so the overall optical gain of AS73 is roughly 1/100 relative to AS36 (10 times higher transmissivity x 1000 times lower mod depth). 

2. All the error signals are plotted as functions of AS A sensor's gouy phase to account for our lack of knowledge of it absolute location. AS B is assumed to be 90 degree gouy phase apart from A. 

3. We phase the AS72.8 signal s.t. all the SUM goes into the I-phase (in the plots, resp phase of 0 or +-180). The spot centering motion will show up in the phase.

4. To decouple the wave-front distortion from the centering loops, we form the sensing matrix for SRM/BS using ASA-Q and ASB-Q. 

5. To quantify how degenerate the 2x2 sensing matrix is, we utilize the normalized conditional number. I.e., we first normalize each dof's total response (quadratic sum of I/Q phases and A/B sensors) to 1, and then do a singular value decomposition of the 2x2 sensing matrix with ASA-Q and ASB-Q. The normalized conditional number is then the ratio between the largest and the smallest singular values, and the larger this conditional number, the more degenerate a matrix is. E.g., [1, 0; 0, 1] has conditional number of 1, whereas [1, 1; 1, 1] has conditional number of infinity. The matrix [1, 1; 0.3, 1] has conditional number of ~4.