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Entry  Thu Jul 5 13:38:05 2012, yuta, Summary, Locking, cavity g-factor from mode scan 
    Reply  Sun Jul 8 00:27:54 2012, yuta, Summary, Locking, calibrating phase tracking mode scan data 
       Reply  Sun Jul 8 00:58:08 2012, Koji, Summary, Locking, calibrating phase tracking mode scan data 
Message ID: 6938     Entry time: Sun Jul 8 00:27:54 2012     In reply to: 6922     Reply to this: 6939
Author: yuta 
Type: Summary 
Category: Locking 
Subject: calibrating phase tracking mode scan data 

FSR for X/Y arm are 3.97 +/- 0.03 MHz and 3.96 +/- 0.02 MHz respectively. This means X/Y arm lengths are 37.6 +/- 0.3 m and 37.9 +/- 0.2 m respectively.
I calibrated the mode scan results using 11MHz sideband as frequency reference.
Calibration factor between the phase of the phase tracker and IR frequency is 9.81 +/- 0.05 kHz/deg for X arm, 9.65 +/- 0.02 kHz/deg for Y arm.

Calculation:
  For the mode scan measurements, we swept the phase of the phase tracker linearly with time. Previous calculation was done without calibrating seconds into actual IR frequency. The first order calibration can be done using modulation frequency as reference. Note that I'm still assuming our sweep was linear here.

  Relation between FSR and modulation frequency can be written in

f_mod = n * nu_FSR + nu_f

  where f_mod is the modulation frequency, n is an integer, nu_f = mod(nu_FSR,f_mod).
  nu_FSR and nu_f are measurable values (in seconds) from the mode scan. We know that f_mod = 11065910 Hz (elog #6027). We also know that nu_FSR is designed to be ~3.7 MHz(=c/2L). So, n = 2.
  We can calculate f_mod in seconds, so we can calibrate seconds into IR frequency.


Calibrating X arm mode scan:
  From the 8FSR mode-scan data (see elog #6859), positions of TEM00 and upper/lower 11 MHz sidebands in seconds are;

TEM00    242.00     214.76     187.22     159.27     131.33     102.96     74.61     46.00     17.51
upper    236.70     209.05     181.36     153.42     125.06      96.86     68.43     40.20
lower    220.35     192.96     165.03     136.98     108.92      80.65     52.25     23.90


  So, FSR and nu_f in seconds are;

FSR    27.24     27.54     27.95     27.94     28.37     28.35     28.61     28.49
nu_f   21.80     21.82     22.14     22.19     22.26     22.28     22.40     22.40


  By using formula above, modulation frequency in seconds are;

f_mod    76.28    76.90    78.04    78.07    79.00    78.98    79.62    79.38

  By taking average, FSR and f_mod in seconds are

FSR    28.1 +/- 0.2
f_mod    78.3 +/- 0.4

  We know that f_mod = 11065910 Hz, so conversion constant from seconds to frequency is

k1 = 0.1413 +/- 0.0007 MHz/sec

  We swept the phase by 3600 deg in 250 sec, so conversion constant from degree to frequency is

k2 = 9.81 +/- 0.05 kHz/deg

  Also, using k1, FSR for X arm is

FSR = 3.97 +/- 0.03 MHz

  This means, X arm length is

L = c/(2*FSR) = 37.6 +/- 0.3 m


Calibrating Y arm mode scan:
  From the 8FSR mode-scan data (see elog #6832), positions of TEM00 and upper/lower 11 MHz sidebands in seconds are;

TEM00    246.70     218.15     190.06     161.87     133.26     104.75     76.01     47.19     18.60
upper    240.86     212.78     184.32     155.73     127.23      98.48     69.78     41.26
lower    224.53     195.73     167.31     139.13     110.81      82.27     53.60     24.50


  So, FSR and nu_f in seconds are;

FSR    28.55     28.09     28.19     28.61     28.51     28.74     28.82     28.59
nu_f   22.44     22.57     22.60     22.61     22.47     22.48     22.50     22.68


  By using formula above, modulation frequency in seconds are;

f_mod    79.54    78.75    78.98    79.825    79.485    79.955    80.14    79.855


  By taking average, FSR and f_mod in seconds are

FSR    28.5 +/- 0.1
f_mod    79.6 +/- 0.2

  We know that f_mod = 11065910 Hz, so conversion constant from seconds to frequency is

k1 = 0.1390 +/- 0.0003 MHz/sec

  We swept the phase by 3600 deg in 250 sec, so conversion constant from degree to frequency is

k2 = 9.65 +/- 0.02 kHz/deg

  (k2 of X arm and Y arm is different because delay-line lengths are different)
  Using k1, FSR for X arm is

FSR = 3.96 +/- 0.02 MHz

  This means, X arm length is

L = c/(2*FSR) = 37.9 +/- 0.2 m


Summary of mode scan results:
X arm
  Mode matching    MMR = 91.2 +/- 0.3 % (elog #6859) Note that we had ~2% of 01/10 mode.
  FSR         FSR = 3.97 +/- 0.03 MHz (this elog)
  finesse    F = 416 +/- 6 (elog #6859)
  g-factor    g1*g2 = 0.3737 +/- 0.002 (elog #6922)

  length        L = 37.6 +/- 0.3 m (this elog)
  ETM RoC  R2 = 60.0 +/- 0.5 m (this elog and #6922; assuming ITM is flat)

Y arm
  Mode matching    MMR = 86.7 +/- 0.3 % (elog #6828) Note that we had ~5% of 01/10 mode.
  FSR         FSR = 3.96 +/- 0.02 MHz (this elog)
  finesse    F = 421 +/- 6 (elog #6832)
  g-factor    g1*g2 = 0.3765 +/- 0.003 (elog #6922)

  length       L = 37.9 +/- 0.2 m (this elog)
  ETM RoC R2 = 60.7 +/- 0.3 m (this elog and #6922; assuming ITM is flat)

  I think these are all the important arm parameters we can derive just from mode scan measurement.

  Every errors shown above are statistical error in 1 sigma. We need linearity check to put systematic error. Also, we need more precise calibration after that, too, if the sweep has considerably large non-linearity. To do the linearity check, I think the most straight forward way is to install frequency divider to monitor actual beat frequency during the sweep.

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