THIS ALOG DISAPPEARED FROM THE ALOG LAST NIGHT, THIS IS A REPOST.

J. Kissel, K. Izumi

I've finished the analysis of the calibration of the H1 SUS ETMY actuators that are involved with global DARM control, where we've used the "free swinging Michelson method" (i.e. using the IR laser's wavelength as a frequency / length reference). The results are as follows:

    'iStage'     '[m/ct] @ DC'    '1-sigma Unc.'
    'ITMX L2'    [ 3.9461e-13]    [    0.013313]
    'ETMX L3'    [ 8.0906e-14]    [    0.026998]
    'ETMY L1'    [  4.851e-11]    [    0.026588]
    'ETMY L2'    [ 3.8468e-13]    [    0.026885]
    'ETMY L3'    [ 1.4543e-15]    [    0.027057]

The messages:
- Kiwamu is still finalizing his analysis, but we can safely see that a similar calibration using the ALS DIFF VCO as a frequency / length reference agrees with the above results.
- Assuming the ETMX ESD driver's DC gain is 40 [V/V] and the new ETMY driver's DC gain is 2 [V/V], then this translates to an ESD force coefficient of
    'optic'      '[N/V^2]'
    'ETMX L3'    [2.21e-10]
    'ETMY L3'    [7.96e-11]
This means that the ETMY ESD drive strength is 2.78 times weaker than ETMX.
- Though we've used the  full complex transfer functions for transfer function ratios and multiplication to get the final answers for each stage, what discrepancies we find in the phase of the final transfer function that determines the magnitude answer are ignored. This is because we determine the phase response / frequency dependence of the DARM actuator and DARM Sensor collectively when comparing the DARM Open Loop Gain Transfer Function against a model of the full loop, e.g. LHO aLOG 18186.
Curiosities that don't affect the answer, but are none-the-less irksome:
- We need to flip the sign of the measurement of the ITMY L2 stage drive to MICH, the ETMX L3 stage drive DARM, and the ETMY L3 stage drive to DARM in order for the overall phase of the final results to make sense.
- The phase of the ETMY L2 stage is offset from the model by ~10 [deg]. 
- We'll now use the above numbers to change the DC calibration of the DARM model parameter file that has been used for creating the DARM loop model, compare against a DARM OLGTF measurement and therefore make a statement about the optical gain.
- This method of determining the actuation coefficient -- especially for the very weak ETMY L3 stage -- is *very* time consuming, cumbersome and only gets quality results in the 4 to 7 [Hz] band, therefore we should do it rarely if at all in the future. PCal should become our new standard technique of determining the actuation coefficients and these kind of measurements should be the checks!

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Details
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MEASUREMENT METHOD
A lot of the methodology for this calibration technique has been outlined before (see most recently LHO aLOG 14135, most clearly (IMHO) P0900120, and originally in T030097), but I'll repeat it briefly here, because one needs to understand how we've augmented the technique further in order to obtain the ETMY L3 actuation strength.

One begins with the measurement from which the technique gets its name:
(A) After locking and aligning the IFO into a dark Michelson configuration (with PRM, SRM, and the ETMs misaligned), break the lock and measure the AS port's demodulated Q-phase signal ("AS_Q") and DC power ("AS_DC") while the Michelson freely swings through fringes. For the aLIGO IFO, that's H1:LSC-MICH_IN1_Q (a scaled version of H1:LSC-ASAIR_A_RF45_Q_ERR_DQ) and H1:LSC-ASAIR_A_LF_OUT, and we grabbed ~300 [sec] worth of data. As the Michelson evolves through fringes the AS_Q error signal proportional to the displacement of the mirrors:

AS_Q = (1/2) * A_{pp} * sin( (4*pi/lambda) * dl )             (1)

where A_{pp} is the peak-to-peak amplitude of the signal, lambda is the laser wavelength, and dl is the Michelson displacement, (lx - ly). We assume that the displacement is small (thanks SEI team!) such that 

AS_Q ~ A_{pp} * (2*pi/lambda) * dl = k * dl                   (2)

leaving the AS_Q signal proportional to the Michelson displacement by a real constant, k, the optical gain of the Michelson. Over the years, we've found that simply taking the peak-to-peak of the fringing time-series is prone systematic errors in determining A_{pp} due to drifts in alignment during the long uncontrolled stretch. As such, we've taken advantage of the sin / cos relationship of AS_Q and AS_DC and plot the ellipse they form as the Michelson fringes. We chuck up the long time series into several smaller time series and plot the ellipses, fit each to determine the semi-major axis of the AS_Q vs. AS_DC ellipse, and take the mean of each fit's semi-major axis to determine A_{pp} (see first page of attached). Because the fit of each chunk is a measurement of the inherent value of A_{pp}, we assign the standard error of the mean (i.e. d(A_{pp}) = std( A_{pp}^{i} ) / sqrt(N) ) as the uncertainty. For this measurement, 

k = 1.112e+08 +/- 1.7181e+05 [(MICH Displacement [m])/ (MICH Sensor [ct])]


(B) We eventually want to drive a given stage of the ITMs to determine the actuation strength of that stage with our newly calibrated MICH sensor, MICH_IN1 (AS_Q). However, we need the Michelson locked on a dark fringe so that the MICH error signal remains linear. So we must measure the MICH loop suppression. Now with the MICH locked, measure the MICH_IN2 / MICH EXC loop suppression transfer function. A little bit of loop math will show that 

MICH_IN2 / MICH EXC = 1 / (1 + G_{M})                          

as desired. Note that one *could* measure the open loop gain transfer function, G_{M}, directly, as MICH_IN1 / MICH_IN2 = - G_{M}, but for ease of uncertainty propagation, we've just measured the suppression directly. The open loop gain and suppression are shown in pgs 2 and 3. The uncertainty for each frequency point is determined from the coherence of the measurement and the number of averages,

                         1 - C  
d|TF| =  |TF| *  sqrt ( ------- )  [ same units as |TF| ]
                         2 C N  
                                                              (3)
                  1 - C  
d <(TF) = sqrt ( ------- )   [ rad ]
                  2 C N   

ref LHO aLOG 10506, or originally Bendat and Piersol, "Random Data" 2nd Ed, p317.

(C) Pick any stage of either ITM and take a driven transfer function of iStage ITM drive to MICH_IN1. With the data from (1) and (2), create an absolute calibration of this [(MICH IN1 [ct])/ (iStage ITM drive [ct])] transfer function in terms of [m] by inverting the optical gain and loop suppression:

  ITM Optic disp. [m]      MICH IN1       1            
--------------------- = ( ---------- ) * --- * (1 + G)                                       (6)
iStage ITM Drive [ct]      ITM EXC        k           

                              MICH IN1 [ct]            MICH [m]        MICH IN2 [ct]  -1          
                      = ( --------------------- ) * ------------  *  ( ------------- )       (5)
                          iStage ITM Drive [ct]      MICH IN1 [ct]     MICH EXC [ct] 

Each frequency point of the loop suppression and itm drive transfer function's uncertainty is determined by Eqs. 2 & 3, and are propagated to the uncertainty in each frequency point of the overall calibration by adding the relative magnitude and phase in quadrature. To compress each frequency point into an assessment of the overall DC actuation strength of that stage, we divide the resultant transfer function by a model of the actuation from that stage to it's optic. This way, if we've modeled the actuation strength as a function of frequency correctly, each frequency point becomes an independent measure of the overall strength of the actuator. As such, the single number and uncertainty for this stage is formed by the weighted mean and chi-squared weighted variance in the weighted mean,

          sum (x_{i} * sigma_{i}^{-2}) 
bar{x} =  ----------------------------                                                       (7)
             sum ( sigma_{i}^{-2} )                                                          Wikipedia
                             1                  1             (x_{i} - 1)^2
sigma_{bar{x}}^2 = ---------------------- *  ------- * sum ( --------------- )               (8)
                   sum ( sigma_{i}^{-2} )    (N - 1)           sigma_{i}^2                   Wikipedia
     

(D) The rest of the game is just using various parts of the interferometer to propagate this known "reference" actuator's absolute calibration to other optics. This is done by locking up a configuration that involves the reference actuator and actuator / optic you want to calibrate, and measuring the response of that IFO to both drives and taking the ratio of transfer functions:

  Optic Disp [m]           MICH [m]           ITM EXC [ct]        Some IFO [ct]
------------------   = ( ----------- ) *  ( -------------- ) * ( ----------------- )          (9)
iStage Drive [ct]        ITM EXC [ct]        Some IFO [ct]       iStage Drive [ct]

and the uncertainty is propagated in the same way as described in step C -- the frequency points of each new transfer function's uncertainty is determined by the coherence and number of averages, the uncertainty in the frequency points of the resulting product are the quadrature sum of the component TFs, the absolute calibration of the stage to optic is divided by a model, and the hopefully unity magnitude residual's weighted mean and weighted uncertainty are taken as the absolution calibration.

We found out Wednesday that the former method used in prior IFOs /science runs of just using a single arm IFO and propagating directly to the ETM doesn't work for the lowest strength actuators (i.e. ETMY L3) because the frequency noise of the Single Arm pollutes the measurement enough that one cannot get any coherence. As such, we've used many measurements to propagate the the absolute calibration to the ETMY L3: using the X single-arm to propagate ITMX L2 to ETMX L3, and then the full IFO in DC readout to propogate ETMX L3 to each stage of ETMY, e.g.

   ETMY L3 [m]         MICH [m]          ITM EXC [ct]        XARM IN1 [ct]         ETMX L3 EXC [ct]     DARM IN1 [ct]
---------------- = ( ----------- ) *  ( -------------- ) * ( -------------- ) * ( -------------- ) * ( --------------- )         (10)
ETMY L3 EXC [ct]     ITM EXC [ct]        XARM IN1 [ct]        ETMX L3 [ct]         DARM IN1 [ct]       ETMY L3 EXC [ct]


So if you're keeping track, that's six transfer functions and and a time series we have to take, and the transfer functions all need to take up a lot of time to get enough coherence that the uncertainty doesn't blow up. It's an all day adventure just to get all of these transfer functions. Thankfully, because of Kiwamu's care in getting the coherence super high, we still manage to get good data with reasonable error bars between 4 and 7 [Hz]. Any higher a frequency than this, and the original ITM to MICH transfer function hits the MICH noise floor. Below 4 [hz] the ITM to ETM transfer functions with the single arm loose coherence from frequency noise. So in addition to the many transfer functions we have to take, we only get a small frequency band where we can really get any sort of precision, and MICH isn't good enough to let us measure into the gravitational wave band.

WHERE EVERY THING EXISTS
The raw .xml files live here:
/ligo/svncommon/CalSVN/aligocalibration/trunk/Runs/PreER7/H1/Measurements/FreeSwingMich/2015-05-28/
2015-05-28_H1DARM_ETMX_L3_Drive.xml
2015-05-28_H1DARM_ETMY_L1_State1_Drive.xml
2015-05-28_H1DARM_ETMY_L2_State2_Drive.xml
2015-05-28_H1DARM_ETMY_L3_LVLP_Drive.xml
2015-05-28_H1MICH_freeswingingdata.xml
2015-05-28_H1MICH_ITMDrives.xml
2015-05-28_H1MICH_OLG.xml
2015-05-28_H1XARM_ETMandITMDrives.xml
which have been exported to .txt files with corresponding _ts, _tf, or _coh tags to indicate the contents.

The analysis is done with 
/ligo/svncommon/CalSVN/aligocalibration/trunk/Runs/PreER7/H1/Scripts/analyze_mich_freeswinging_data_20150528.m


Stay tuned for more actuation coefficient measurement technique comparisons, and an update to the DARMmodel, and therefore the CAL-CS and GDS calibration pipelines, and therefore the DARM spectrum and Inspiral range.