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. |