[Sugioka, Sugimoto, Komori, Fujimoto]

Abstract:

For the performance evaluation of the current PRFPMI, we measured the relationship between the DARM offset power: $P_\mathrm{offset}$  [mW] and the DARM offset length: $x_\mathrm{offset}$ [pm].
The estimated equation is as follows:

$P_\mathrm{offset}=ax_\mathrm{offset}^2+b$
$a$ = 0.42 mW/pm^2
$b$ = 0.13 mW (Contrast defect, from 31330).

Based on the result above, the DARM offset power of ~15 mW corresponds to the DARM offset length of ~5.9 pm in the current PRFPMI.

Details:

Firstly, to measure the optical gain in [mW/pm] at DARM offset power of ~15 mW, we measured the transfer function from DARM displacement (K1: CAL-CS_PROC_DARM_DISPLACEMENT_DQ) to OMC transmission in [mW] (K1:LSC-OMC_DC_IN1).
Fig. 1 shows the observed spectra of DARM displacement (red) and OMC transmission (blue), and Fig. 2 shows the transfer function between them.
Since the OMC transmission is suppressed by DARM control, the transfer function from DARM displacement to OMC transmission can be expressed as:

$\frac{S_\mathrm{OG}}{1+G_\mathrm{DARM}} ,$

where $S_\mathrm{OG}$ is the optical gain in [mW/pm] including the DARM cavity pole and $G_\mathrm{DARM}$ is the DARM OLTF.
The gain value of -20.69 dB at 1 kHz is out of the control bandwidth (DARM UGF~100 Hz) and can be regarded as the optical gain at 1 kHz.
Considering the DARM cavity pole of 18.5 Hz and the observed optical gain of -20.69 dB @ 1kHz, the optical gain at DC: $S_\mathrm{OG,DC}$ can be calculated as:

$S_\mathrm{OG,DC}$ = 5.0 mW/pm @ DARM offset power of 15 mW.

Then, we assued the relationship between the DARM offset power: $P_\mathrm{offset}$  [mW] and the DARM offset length: $x_\mathrm{offset}$ [pm] as follows:

$P_\mathrm{offset}=ax_\mathrm{offset}^2+b.$

Also, the optical gain corresponds to the derivative:

$S_\mathrm{OG,DC}=\frac{dP_\mathrm{offset}}{dx_\mathrm{offset}}=2ax_\mathrm{offset}.$

Using the measured optical gain and the contrast defect (0.13 mW from 31330), the equations above can be solved for the coefficient $a$ [mW/pm^2] and the result is 

$a$  = 0.42 mW/pm^2.

Based on the obtained relationship above, in the current PRFPMI setup, the DARM offset length at ~15 mW DARM offset power is estimated as:

$x_\mathrm{offset}$ = 5.9 pm @ DARM offset power of 15 mW.