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. |