Reports until 14:23, Tuesday 23 December 2014
H1 ISC
stefan.ballmer@LIGO.ORG - posted 14:23, Tuesday 23 December 2014 - last comment - 16:34, Tuesday 23 December 2014(15807)
Converting contrast defect to ITM RoC change (diopter change) 
The following elog is in support of Evan's upcoming elog on last nights TCS transient (turning off TCS)

Bottom line:

The contrast defect (CD) can be modeled as

CD = CD_0 + pi^2/8 * D^2*w^4/lambda^2 

where:
CD    : contrast defect
CD_0  : residual contrast defect, not due to ITM RoC mismatch
D     : Diopter change in (one) ITM, double pass, i.e. D=2*( 1/R_new - 1/R_old )
w     : beam spot radius on the ITM, nominally 53mm
lambda: wave length = 1.064e-6m 

Derivation: 

- incident beam:            |Psi> =  N exp(-r^2/w^2)
- ITMX reflection operator: exp(i*k*D*r^2/2)
- ITMY reflection operator: 1
- reflected field:          |r> = |Psi> * exp(i*k*D*r^2/4) *   cos(k*D*r^2/4)
- dark port field:          |t> = |Psi> * exp(i*k*D*r^2/4) * i sin(k*D*r^2/4)

- Power in dark port due to RoC mismatch:
    = k^2*D^2*w^4/32 = pi^2/8 * D^2*w^4/lambda^2 
  The bottom line formula for contrast defect follows from this.

Some other useful expressions:
- Power reflected from the Michelson:
    = 1 -  = 1 - pi^2/8 * D^2*w^4/lambda^2 
  Note that this power will be in a different mode, so the mode matching into the recycling cavity is expected to change.

And for reference, some Gaussian integrals:
   h_n := 
   h_n = n/4 w^2 h_(n-2)
   h_0 = 1
   h_2 = 1/2 w^2   
   h_4 = 1/2 w^4   
   h_6 = 3/4 w^6   
   h_8 = 3/2 w^8
Comments related to this report
evan.hall@LIGO.ORG - 16:34, Tuesday 23 December 2014 (15815)
Link

Stefan, Evan

Since we turned off the TCS last night and left PRMI locked on carrier, we have roughly 8 hours of good data that tells us the (1) the contrast defect and (2) the behavior of the ITM thermal lens as a function of time. To extract these quantities, we did the following:

  • We extrated minute trends of LSC-POP_A_LF_OUT, LSC-REFL_A_LF_OUT, and LSC-ASAIR_A_LF_OUT from last night.
  • We calibrated POP_A power to PRC power as follows:
    • FM10 already calibrates the POP_A counts into microwatts of power incident on the PD.
    • In Kiwamu's recycling gain measurement (LHO#15793), he misaligned PRM and locked MICH on a dark fringe. There was 7.2 μW on POP_A. Assuming a PSL power of 10.7 W (measured today), an IMC transmission of 83 % (LHO#9954), and a PRM transmission of 3.0 % (galaxy), this means there was 260 mW of power in the PRC.
    • The calibration PRC / POP_A is therefore 3.7×104 W/W.
  • We calibrated ASAIR_A power to SRC power as follows. Note that what I call "SRC power" is power leaving the BS going toward SRM (this is to distinguish it from "AS power" seen after the SRM).
    • ASAIR_A is already calibrated into microwatts of power incident on the PD.
    • In Stefan's contrast defect measurement (LHO#15332), he found the maximum SRC power from the free-swinging michelson was 1890 ct on ASAIR_B (not A). Today I measured the calibration to A is ASAIR_A / ASAIR_B = 3.52 ct/ct. The maximum SRC power from the free-swinging Michelson I take to be 10.7 W × 0.83 × 0.03 × 0.986 (this last number is the ITM reflectivity).
    • The calibration SRC / ASAIR_A is therefore 39 W/W.
  • We calibrated REFL_A to REFL (i.e., the power heading from the PRM to IM4) as follows:
    • FM10 already calibrates the REFL_A counts into milliwatts of power incident on the PD.
    • In yesterday's RFAM measurement (LHO#15781), we measured 81 mW on REFL_A with the PRM aligned and the ITMs misaligned. The REFL power I take to be 10.7 W × 0.83 × 0.97.
    • The calibration REFL / REFL_A is therefore 106 W/W.
  • It can be shown that the ratio PSRC / PPRC = TM RI, where TM is the carrier power transmissivity of the Michelson, and RI is the power reflectivity of each ITM. I assume RI = 0.986. Therefore, we can solve for TM as a function of time given the calibrated power measurements. This is given in the upper plot of the attachment. From last night's data alone, the starting value of TM is 1.115×10−3. However, Daniel has previously corrected the contrast defect for the presence of the 45 MHz sidebands (LHO#15432), finding a defect of 0.99×10−3. In the plot, I have rescaled TM by 0.99 / 1.115 to reflect this.
  • To convert ths into a change into diopters according to Stefan's formula, we need to know the "best" contrast defect achievable (i.e., the residual defect assuming no RoC mismatch). Stefan's measurement (LHO#15332) showed a minimum in ASAIR_B of (−0.2 + 1.78) ct with the Michelson locked on a dark fringe. According to Daniel's analysis, 0.17 ct of this is due to the 45 MHz sideband. So the net minimum is 1.41 ct. With a bright fringe power of 1890 ct, I take the "best" contrast defect to be 8×10−4.
  • By using this "best" defect and Stefan's formula above, we compute change in the ITM thermal lens (in diopters) as a function of time. This is given in the lower plot of the attachment.

In principle, we can also use the above data to extract the mode-matching into the PRMI as a function of time. Perhaps we will pursue this later.

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