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Message ID: 16125     Entry time: Thu May 6 16:13:39 2021
 Author: Anchal Type: Summary Category: IMC Subject: Angular actuation calibration for IMC mirrors

Here's my first attempt at doing angular actuation calibration for IMC mirrors using the method descibed in /users/OLD/kakeru/oplev_calibration/oplev.pdf by Kakeru Takahashi. The key is to see how much is the cavity mode misaligned from the input mode of beam as the mirrors are moved along PIT or YAW.

There two possible kinds of mismatch:

• Parallel displacement of cavity mode axis:
• In this kind of mismatch, the cavity mode is simply away from input mode by some distance $\dpi{150} \large \beta$.
• This results in transmitted power reduction by the gaussian factor of $\dpi{150} \large e^{-\frac{\beta^2}{w_0^2}}$ where $\dpi{150} \large w_0$ is the beam waist of input mode (or nominal waist of cavity).
• For some mismatch, we can approximate this to
$\dpi{150} \large 1 - \frac{\beta^2}{w_0^2}$
• Angular mismatch of cavity mode axis:
• The cavity mode axis could be tilted with respect to input mode by some angle $\dpi{150} \large \alpha$.
• This results in transmitted power reduction by the gaussian factor of $\dpi{150} \large e^{- \frac{\alpha^2}{\alpha_0^2}}$  where $\dpi{150} \large \alpha_0$ is the beam divergence angle of input mode (or nominal waist of cavity) given by $\dpi{150} \large \frac{\lambda}{\pi w_0}$.
• or some mismatch, we can approximate this to
$\dpi{150} \large 1 - \frac{\alpha^2}{\alpha_0^2}$

Kakeru's document goes through cases for linear cavities. For IMC, the mode mismatches are bit different. Here's my take on them:

### MC2:

• MC2 is the easiest case in IMC as it is similar to the end mirror for linear cavity with plane input mirror (the case of which is already studies in sec 0.3.2 in Kaker's document).
• PIT:
• When MC2 PIT is changed, the cavity mode simple shifts upwards (or downwards) to the point where the normal from MC2 is horizontal.
• Since, MC1 and MC3 are plane mirrors, they support this mode just with a different beam spot position, shifted up by $\dpi{150} \large (R-L)\theta$.
• So the mismatch is simple of the first kind. In my calculations however, I counted the two beams on MC1 and MC3 separately, so the factor is twice as much.
• Calling the coefficient to square of angular change $\dpi{150} \large \eta$, we get:
$\dpi{150} \large \eta_{._{2P}} = \frac{2 (R-L)^2}{w_0^2}$
• Here, R is radius of curvature of MC1/3 taken as 21.21m and L is the cavity half-length of IMC taken as 13.545417m.
• YAW:
• For YAW, the case is bit more complicated. Similar to PIT, there will be a horizontal shift of the cavity mode by $\dpi{150} \large (R-L)\theta$.
• But since the MC1 and MC3 mirrors will be fixed, the angle of the two beams from MC1 and MC3 to MC2 will have to shift by $\dpi{150} \large \theta/2$.
• So the overall coefficient would be:
$\dpi{150} \large \eta_{._{2Y}} = \frac{2 (R-L)^2}{w_0^2} + \frac{2}{4\alpha_0^2}$
• The factor of 4 in denominator of seconf term on RHS above comes because only half og angular actuation is felt per arm. The factor of 2 in numerator for for the 2 arms.

### MC1/3:

• First, let's establish that the case of MC1 and MC3 is same as the cavity mode must change identically when the two mirrors are moved similarly.
• YAW:
• By tilting MC1 by $\dpi{150} \large \theta$, we increase the YAW angle between MC1 and MC3 by $\dpi{150} \large \theta$.
• Beam spot on both MC1 and MC3 moves by $\dpi{150} \large (R-L)\theta$.
• The beam angles on both arms get shifted by $\dpi{150} \large \theta/2$.
• So the overall coefficient would be:
$\dpi{150} \large \eta_{._{13Y}} = \frac{2 (R-L)^2}{w_0^2} + \frac{2}{4\alpha_0^2}$
• Note, this coefficient is same as MC2, so it si equivalent to moving teh MC2 by same angle in YAW.
• PIT:
• I'm not very sure of my caluculation here (hence presented last).
• Changing PIT on MC1, should change the beam spot on MC2 but not on MC3. Only the angle of MC3-MC2 arm should deflect by $\dpi{150} \large \theta/2$.
• While on MC1, the beam spot must change by $\dpi{150} \large (R-L)\theta/2$ and the MC1-MC2 arm should deflect by $\dpi{150} \large \theta/2$.
• So the overall coefficient would be:
$\dpi{150} \large \eta_{._{13P}} = \frac{(R-L)^2}{4 w_0^2} + \frac{2}{4\alpha_0^2}$

### Test procedure:

• We first clicked on MC WFS Relief (on C1:IOO-WFS_MASTER) to reduce the large offsets accumulated on WFS outputs. This script took 10 minutes and reduced the offsets to single digits and IMC remained locked throughout the process.
• Then we switched off the WFS to freeze the outputs.
• We moved the MC#_PIT/YAW_OFFSET up and down and measured the C1:IOO-MC_TRANS_SUMFILT_OUT channel as an indicater of IMC mode matching.
• Attachement 1 are the 6 measurements and there fits to a parabola. Fitting code and plots are thanks to Paco.
• We got the curvature of parabolas $\dpi{150} \large \gamma$from these fits in units of 1/cts^2.
• The $\dpi{150} \large \eta$ coefficients calculated above are in units of 1/rad^2.
• We got the angular actuation calibration from these offsets to physical angular dispalcement in units of rad/cts by $\dpi{150} \large \sqrt{\gamma / \eta}$.
• AC calibration:
• I parked the offset to some value to get to the side of parabola. I was trying to reduce transmission from about 14000 cts to 10000-12000 cts in each case.
• Sent excitation using MC#_ASCPIT/YAW_EXC using awg at 77 Hz and 10000 cts.
• Measured the cts on transmission channel at 77 Hz. Divided it by 2 and by the dc offset provided. And divided by the amplitude of cts set in excitation. This gives $\dpi{150} \large \eta_{ac}$ analogous to above DC case.
• Then angular actuation calibration at 77 Hz from these offsets to physical angular dispalcement in units of rad/cts by $\dpi{150} \large \sqrt{\gamma/\eta_{ac}}$.
• Following are the results:
Optic Act
Calibration factor at DC [µrad/cts]
Calibration factor at 77 Hz [prad/cts]
MC1 PIT 7.931+/-0.029 906.99
MC1 YAW 5.22+/-0.04 382.42
MC2 PIT 13.53+/-0.08 869.01
MC2 YAW 14.41+/-0.21 206.67
MC3 PIT 10.088+/-0.026 331.83
MC3 YAW 9.75+/-0.05 838.44

• Note these values are measured with the new settings in effect from 16120. If these are changed, this measurement will not be valid anymore.
• I believe the small values for MC1 actuation have to do with the fact that coil output gains for MC1 are very weird and small, which limit the actuation strength.
• TAbove the resonance frequencies, they will fall off by 1/f^2 from the DC value. I've confirmed that the above numbers are of correct order of magnitude atleast.
• Please let me know if you can point out any mistakes in the calculations above.
 Attachment 1: IMC_Ang_Act_Cal_Kakeru_Tests.pdf  114 kB  Uploaded Thu May 6 17:57:32 2021
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