Abstract:

To roughly estimate the non-linear effect of DARM signals, Icalculated the second order effect of OMC trans power flutuation.
Though calculation is not entirely correct, non-linear effect seems not negligible.

Detail:

To estimate non-linear effect of DARM, I performed some calculations.

If DARM offset is small enough, OMC transmission power can be written as P = ax**2+b (1) (klog31350).
So, OMC trans power fluctuation can be written as dP = 2ax_0dx + a(dx)**2, where x0 is a DARM offset value and dx is a residual of DARM error signals.
When we evaluated the calibration factor, we assumed the second term can be negligible, so dx = dP/(2ax_0).
If non-linear effect can be negligible, a(dx)**2 = (dP)**2/(4a(x_0)**2) should be small enough compared with the first term, 2ax_0dx, so I calculated it (fig1).
Figure 2 shows the ratio of two values.

According to the calculation, ratio is more than 0.5, so it doesn't seem negligible.

Calculation detail:

Previously, a and b values in the equation (1) are calculated as 0.42 mW/pm^2 and 0.13 mW, respectively (klog31350).
In addition, x_0 when IMC output was calculated as 5.9 pm when OMC transmission was 15mW.
Since IMC output power was increased from 1.26W to 10.5W, the values of a, b, and x_0 can be calculated as follows.
a = 0.42 * (10.5/1.26) = 3.5 mW/pm^2.
b = 0.13 * (10.5/1.26) = 1.0 mW.
x_0 = 5.9*sqrt(1.26/10.5) = 2.04 pm.

I also calculated the a and x_0 by using calibration factor measurement in klog34466, which are 2.85 mW/pm^2 and 2.21 pm, respectively.
Note tat I used the equation of 15mW = ax_0**2 + b and 2ax_0 = optical gain (=2.126e12/0.1688) for the caluculation.
Since the values are slightly different, I used the average of two values for a (3.175 mW/pm^2) and x_0 (2.125 pm) during the above estimation.

Note:

I used the data around July 9 21:15 UTC for the calculation.
Since OMC transmission data includes non-linear effect, above calculation is not correct but it would b some indicator for estimating non-linear effects.