Advanced LIGO, 4K parameters
Preliminary, 5/15/2001
Alan Weinstein, ajw@ligo.caltech.edu
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Readout scheme:

- Two pairs of phase-modulated sidebands,
  applied in series.
- Both are resonant in the power recycling cavity (PRC).
  Neither are resonant in the F-P arms.
- Chose one to be as high as possible,
  limited by practical high-power modulators and photodetectors: 180 MHz.
  Chose the other to be as low as possible, while still resonant
  in an ~8m PRC: 9 MHz.
  The higher sideband must be an integral multiple of the lower;
  both must pass through the mode cleaner.
- The Schnupp asymmetry is chosen to make the 180 MHz bright
  at the asymmetric port of the BS; the 9 MHz is near the dark fringe,
  and the carrier is at the dark fringe.
- THUS, the 180 MHz mainly senses the signal recycling mirror (SRM),
  leaving the 9 MHz primarily sensitive to the power recycling mirror (PRM).
- Carrier light, beating against the 9 MHz at the symmetric port, senses L+;
  Carrier light, beating against the 180 MHz at the asymmetric port, senses L-;
  and the l_PRC, l_SRC, and l_M (Michelson) are sensed by beating
  the 180 against the 9 MHz at the various ports; carrier light,
  sensing the large arm phase shifts, is not used for sensing the shorter
  degrees of freedom.

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Cavity lengths, asymmetry, RF frequencies:

  L_arm = (n1 + 0.5) * c / 2 f1
        = 4000. m   for f1 = 9e6 Hz, n1 = 239.666.
        (n1 should not be an integer+0.5, so that f1 is not resonant in the arms.
         it need not be exactly an integer; f1 need not be exactly antiresonant.)
  L_arm = (n11 + 0.5) * c / 2 f2
        = 4000. m   for f2 = 180e6 Hz, n11 = 4802.823, not resonant.

  L_MC = n2 * c / 2 f1
       = 16.6551 m   for n2 = 1
         This is the shortest mode cleaner that will pass both f1 and f2

  f2 = n3 * f1
     = 180 MHz for n3 = 20
       so that both f1 and f2 are resonant in both the MC and the PRC

  L_PRC = (n4 + 0.5) * c / 2 f1
        = 8.3276 m   for f1 = 9e6 Hz, n4 = 0.

  L_M = Michelson asymmetry = c / 4 f2
        = 0.4164 m  for f2 = 180e6 Hz.

  L_SRC = (n3 - dt / 2) * c / 2 f2 - L_PRC
        = 9.1444  for n3 = 20 and dt = 0.0382
        where dt is the round-trip carrier tune in the SRC, in units of pi radians.
        This value of dt places the peak response of the IFO at 250 Hz,
        maximizing the S/N for a binary inspiral in the presence of 
        thermal test mass noise.

  L_AVG         = 4.7000 m  ; center of BS chamber to center of ITM chambers
  L(PRM to BS)  = L_PRC - L_AVG = 3.6276 m
  L(SRM to BS)  = L_SRC - L_AVG = 4.4444 m 
  L(BS to ITMx) = L_AVG - L_M/2 = 4.4918 m
  L(BS to ITMy) = L_AVG + L_M/2 = 4.9082 m

  These lengths are OPTICAL path lengths; 
  physical path lengths are a bit shorter 
  for L(BS to ITMx) by approximately (n-1)*(sqrt2*Thickness_BS+Thickness_ITM)
  for L(BS to ITMy) by approximately (n-1)*(Thickness_ITM)
  for L(BS to SRM)  by approximately (n-1)*(sqrt2*Thickness_BS)
  where n = 1.4496 for FSi.

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Mirror transmissivities, DC fields:

AdvLIGO was optimized for best S/N for binary inspiral
in the presence of test mass thermal noise, by
P Fritchel, K Strain, etal, 8/2000, using bench.m.
See: http://www.ligo.caltech.edu/~ligo2/scripts/l2refdes.htm

For sapphire test masses:
- T_ITM = 0.005
- T_SRM = 0.070
- T_PRM = 0.075
- T_ETM = 15 ppm
- Average power loss per mirror = 37.5 ppm
- laser power = 125 W

Due to the SRC detuning, the response of the IFO 
to the +ve and -ve RF sidebands are different (unbalanced).

For these numbers (somewhat unrealistic for sapphire loss),
- arm pole frequency = 15.2 Hz
- arm finesse = 1231
- Arm power gain = 770
- PRC finesse = 41.7
- PRC power gain = 14.3 (carrier)

DC fields @ 125 Watts input power,
Assuming modulation depth of 0.1 for both pairs of RF sidebands:

           Input  SymPort AsymPort      PRC      SRC     Armx     Army
SB -2     0.3117   0.2839   0.0273   0.3904   0.0273   0.0005   0.0008
SB -1     0.3117   0.2995   0.0018   15.515   0.0017   0.0389   0.0395
Carrier   123.75   0.1630   0.0000   1767.1   0.0000   680205   680205
SB +1     0.3117   0.2994   0.0018   15.542   0.0018   0.0386   0.0400
SB +2     0.3117   0.0008   0.3053   4.3635   0.3052   0.0074   0.0075

It all depends on what and where you put the losses;
your mileage may vary!

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Radii of curvature, beam sizes:

Choose a symmetric arm cavity with g factor to make beam spots 
w (1/e^2 power radius) = 6 cm at the two test masses,
to reduce thermoelastic noise.

AdvLIGO, 4KM, g = 0.858, symmetric:

R_ETM =   54261 m
R_ITM =   54261 m
waist = 0.05884 m
zr =      10223 m
w_ITM = 0.05996 m
w_ETM = 0.05996 m
d1ppm_ETM = 0.315 m
GuoyArm =  0.386 rad
R =       37430 m (back of ITM)
w =     0.0599  m (back of ITM)
R_BS =    37376 m
w_BS =  0.05997 m
R_RM =    37335 m
w_RM =  0.05997 m
R_RM2     25754 m (back of RM)
w_RM2 = 0.05997 m (back of RM)
R_SM =    37325 m
w_SM =  0.05997 m

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Mirror dimensions:

ITM/ETM:
Sapphire: 0.314 m diameter
          0.300 m optical aperture
          0.130 m thickness
          40 kg   mass

BS:       0.350 m diameter
          0.060 m thickness

PRM, SRM: 0.260 m diameter
          0.100 m thickness

to be fine-tuned with feedback from FFT results!