40m Upgrade parameters
Preliminary, 5/16/2002
Alan Weinstein, ajw@ligo.caltech.edu
______________________________________

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 ~2m PRC: 36 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
        = 38.55 m   for f1 = 33.207e6 Hz, n1 = 8.040
        (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
        = 38.55 m   for f2 = 166.033e6 Hz, n11 = 42.20,    not resonant.

  L_MC = n2 * c / 2 f1
       = 13.542 m   for n2 = 3
         This is the shortest mode cleaner that will pass both f1 and f2

  f2 = n3 * f1
     = 166.033 MHz for n3 = 5
       so that both f1 and f2 are resonant in both the MC and the PRC

  L_PRC = (n4 + 0.5) * c / 2 f1
        = 2.257 m    for f1 = 33.207 e6 Hz, n4 = 0.

  L_M = Michelson asymmetry = c / 4 f2
        = 0.451  m  for f2 = 166.033 MHz.

  L_SRC = (n3 - dt / 2) * c / 2 f2 - L_PRC
        = 2.151   for n3 = 5  and dt = 0.235
        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 4000 Hz,
        maximizing the S/N for a binary inspiral in the presence of 
        thermal test mass noise.

  L(PRM to BS)  = 0.300 m
  L(SRM to BS)  = L_SRC + L(PRM to BS) - L_PRC = 0.200
  L(BS to ITMx) = L_PRC - L(PRM to BS) + L_M/2 = 2.183
  L(BS to ITMy) = L_PRC - L(PRM to BS) - L_M/2 = 1.731

  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.

__________________________________________
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

40m uses SAME transmissivities, to acheive SAME cavity finesses
(different light storage times, cavity poles).

- T_ITM = 0.005
- T_SRM = 0.070
- T_PRM = 0.060, 0.065, 0.070
- T_ETM = 15 ppm
- Average power loss per mirror = 37.5 ppm
- laser power = 1 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 = 1578 Hz
- arm finesse = 1235
- Arm power gain = 775
- PRC finesse = 47
- PRC power gain = 16.5 (carrier)

DC fields @ 1   Watt  input power,
Assuming modulation depth of 0.1 for both pairs of RF sidebands:

           Input  SymPort AsymPort      PRC      SRC     Armx     Army
SB -2     0.0025   0.0025   0.0000   0.0000   0.0001   0.0000   0.0000
SB -1     0.0025   0.0023   0.0001   0.0577   0.0017   0.0012   0.0018
Carrier   0.9900   0.0062   0.0000   14.067   0.0000   5414.8   5414.8
SB +1     0.0025   0.0021   0.0003   0.1482   0.0039   0.0043   0.0035
SB +2     0.0025   0.0000   0.0024   0.0345   0.0321   0.0001   0.0001

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 half-symmetric arm cavity with g factor = 1/3:

R_ETM =   57.375 m
R_ITM =   flat (> 5625 m)
waist =   3.027 mm at ITM (1/e^2 intensity)
zr =      27.05 m
w_ITM =   3.027 mm
w_ETM =   5.242 mm
d1ppm_ETM =  28 mm
GuoyArm =  0.955 rad
R =       21192 m  (back of ITM)
w =       3.027 mm (back of ITM)
R_BS =      412 m
w_BS =    3.033 mm
R_RM =      348 m
w_RM =    3.036 mm
R_RM2       238 m  (back of RM)
w_RM2 =   3.036 mm (back of RM)
R_SM =      365 m
w_SM =    3.035 mm


OK, this was the pre-10/31/01 spec, which was used
to order the mirrors. On 10/31/01, some lengths
were changed (eg, Larm = 38.25 -> 38.55 m)
and in 6/02, the polished mirrors were received.

Here's a revised (8/14/02) spec:
optic   spec    toler   delivered     new-spec
ETM     57.375  0.600   57.57,57.68   57.625
PRM     348        20   355,356,343   328+-20
SRM     365        25   367,377,386   343+-25

The ROC as delivered lead to <1% mode mismatch.

See also 
http://www.ligo.caltech.edu/~ajw/40m_optspecs.txt
for info about optics as delivered.

__________________________________________

Mirror dimensions:

ITM (FSi, Heraeus low absorption) and
ETM (Fsi, Corning):
          125 mm diameter
           50 mm thick
           30 mm clear aperture
          1.3 kg mass

BS  (FSi, Heraeus low absorption),
PRM (Fsi, Corning), and
SRM (Fsi, Corning):
           75 mm diameter
           25 mm thickness
           30 mm clear aperture
         0.23 kg mass
