I fitted intra-cavity power dependance on mirror misalignment plot with parabola to get the g-factor.

  Y arm (tangential) g = 0.44 +0.01 -0.01  (measured value before was 0.3765 +/- 0.003 elog #6938)
  PRC (sagittal)       g = 0.97 +0.01 -0.04 (expected value is 0.939 elog #8068)
  PRC (tangential)   g = 0.96 +0.02 -0.05 (expected value is 0.966 elog #8068)

Error bars are just statistical errors from the fitting. Estimated systematic error is ~0.04 (or more).
Here, I assumed PR2/PR3 to be flat to make the calculation simple. I assumed PRC to be curved PRM - flat ITM cavity, and Y arm to be curved ETMY - flat ITMY cavity.

g-factor calculation:
  Intra-cavity power decrease can be written as

dP/P = (dx/w0)**2 + (dt/a0)**2

where dx and dt are translation and tilt of the beam axis introduced by mirror misalignment. w0 is waist size and a0 is divergence angle (= lamb/(pi*w0)).

  When considering a flat-curved cavity with cavity length L, dx and dt can be expressed as;

(dx)    1  ( L*g     L ) (a2)
(  ) = --- (           )*(  )
(dt)   1-g ( -(1-g)  0 ) (a1)

using misalignments of mirrors(a1,a2). Here, mirror1 is curved, and mirror2 is flat. See Kakeru document /users/OLD/kakeru/oplev_calibration/oplev.pdf for derivation.

  So, power decrease by flat mirror misalignment can be expressed as

dP/P = pi*L/lamb * g/(1-g)/sqrt(g*(1-g)) * a2**2

  For curved mirror is

dP/P = pi*L/lamb * 1/(1-g)/sqrt(g*(1-g)) * a1**2

  We can derive g-factor by measuring dP dependance on a1/a2.


Script:
  My script lives in /opt/rtcds/caltech/c1/scripts/dither/gfactormeasurement/plotgfactor.py.
  It least fitts data with parabola (scipy.optimize.leastsq) and gets g-factor value from bisection (scipy.optimize.bisect).


Result:
  Below are the plots of fitted curves.




Systematic effect:
  [oplev calibration] We noticed QPD rotation when calibration oplevs (elog #8232). ~5 deg of rotation makes 10% of systematic error to the oplev calibration and this introduces ~0.04 of error to g-factor values. This

  [oplev linear range] Oplev linear range is ~100 urad, so this is OK.

  [assumption of flat PR2/PR3] Result here doesn't tell you g-factor of PRM itself, but some "effective g-factor" of PRM/PR2/PR3 combination. We can compare with FINESSE result.

  [intra-cavity power drift] If there's significant intra-cavity power drift during the measurement, if effects parabola fitting. We can make this affect small by sweeping the mirror alignment in both direction and take average.


By the way:
  I kept getting PRC g-factor of something like 0.999999 because I had power normalization mistake in my calculation. My script worked for Yarm because TRY is already normalized.
  Also, I was multiplying the oplev calibration factor wrong last night (see elog #8230).

Next:
  - Compare with FINESSE result.
  - Is this g-factor enough? Is this presicion enough? Calculate from mirror angluar motion.
  - More stable lock of PRMI.
  - Try dithering method to measure g-factor to check consistency and also to study systematic effect.