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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: 6922     Entry time: Thu Jul 5 13:38:05 2012     Reply to this: 6938
Author: yuta 
Type: Summary 
Category: Locking 
Subject: cavity g-factor from mode scan 

Cavity g-factor for X arm is 0.3737 +/- 0.002, Y arm is 0.3765 +/- 0.003.
If ITMs are flat and arm length L = 39 +/- 1 m, this means RoC of ETMX and ETMY is 62 +/- 2 m and 63 +/- 2 m respectively.

Calculation:
  Transverse mode spacing is expressed by

nu_TMS / nu_FSR = arccos(sqrt(g1*g2)) / pi

  where g1 and g2 is g-factor

gi = 1 - L/Ri

 of ITM/ETM.

  For mode-scan, we swept laser frequency nu. Let's assume this sweep was linear and we can replace laser frequency with time. From the mode-scan result, TMS can be derived by

  t_TMS = sum((n_i-n)*(t_i-t)) / sum((n_i-n)^2)

  where n_i is the order of transverse mode, n is average of n_i's, t_i is the time i-th order mode appeared and t is average of t_i's.
  Since I could only recognize up to 3rd order mode, this can be rewritten as

  t_TMS = 1.5/5 * t_0 + 0.5/5 * t_1 - 0.5/5 * t_2 - 1.5/5 * t_3

  FSR is time between TEM00s. So, g1*g2 can be calculated by

g1*g2 = (cos(pi*t_TMS/t_FSR))^2


X arm result:

  From the 8FSR mode-scan data (see elog #6859), X arm HOM positions in sec are;

HOM 0    242.00     214.76     187.22     159.27     131.33    102.96     74.61     46.00     17.51
HOM 1    234.29     206.78     179.20     150.96     122.90     94.58     66.27     38.10
HOM 2    226.36     198.91     170.80     142.92     114.62     86.51     58.05     29.65
HOM 3    218.14     190.65     162.71     134.78     106.68     78.27     49.95     21.25


  Calculated FSR and TMS in sec are;

FSR    27.24     27.54     27.95     27.94     28.37     28.35     28.61     28.49
TMS     7.951     8.020     8.193     8.151     8.223     8.214     8.220     8.270

  Calculated cavity g-factor are;

g1*g2    0.3699     0.3720     0.3662     0.3704     0.3761     0.3765     0.3839     0.3748

  By taking average,

g1*g2 = 0.3737 +/- 0.002  (error in 1 sigma)


Y arm result:
  From 8FSR mode-scan data (see elog #6832), Y arm HOM positions in sec are;

HOM 0    246.70     218.15     190.06     161.87     133.26    104.75     76.01     47.19     18.60
HOM 1    238.83     210.55     181.88     153.47     124.93     96.08     67.51     39.01
HOM 2    230.48     202.21     173.64     144.80     116.43     86.17     59.84     31.43
HOM 3    222.15     193.47     165.33     137.13     108.60     80.04     51.17     22.25


  Calculated FSR and TMS in sec are;

FSR    28.55     28.09     28.19     28.61     28.51     28.74     28.82     28.59
TMS     8.200     8.238     8.243     8.289     8.248     8.404     8.219     8.240


  Calculated cavity g-factor are;

g1*g2    0.3841     0.3657     0.3683     0.3765     0.3778     0.3683     0.3904     0.3811

  By taking average,

g1*g2 = 0.3765 +/- 0.003  (error in 1 sigma)


Conclusion:
  If ITMs are flat and arm length L = 39 +/- 1 m, this means RoC of ETMX and ETMY is 62 +/- 2 m and 63 +/- 2 m respectively. Designed RoC is 57.35 m.
  Error of RoC is dominated by arm length error. So, we need more precise measurement of the length. This can be done when scan is calibrated and we can measure FSR in frequency.
  Also, we need evaluation of linearity of the sweep. This also can be done by calibration.

ELOG V3.1.3-