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. |