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40m dual recycling upgrade cavity lengths
PRELIMINARY              AJW, 9/15/00

Set the mode cleaner length:
L_MC = 12.69 m
fsr_MC = 11.82 MHz

Set the first PM sideband to resonate in MC and PRC,
and be as low as possible:
f_1 = 3* fsr_MC = 35.46 MHz 

Set the PRC length:
L_PRC = c/(4*f_1) = 2.115

Set the second PM sideband to resonate in MC and PRC,
and be as high as possible:
f_2 = 4*f_1 = 141.84 MHz
(do we dare go to 5*f_1? it might make some things easier;
but would maybe require some demodulating at 6*f_1 = 213 MHz).

Set the asymmetry to put f_2 out the dark port:
delta = c/(4*f_2) = 0.528

Set the SRC length:
Can choose SRC to be the same length as PRC,
except that the detuning means that f_2 plus carrier 
must be resonant there, so we need an offset.
choose dt = 0.5 to get the shot noise curve 
to be minimum at ~ 1500 Hz.

L_SRC = L_PRC + c/(4*f_2) * dt = 2.379

SO, this can be realized at the 40m as follows:
Optical path length of both arms = 38.25 m
Optical path lengths:
 BS - ITMX (inline): 2.134 m
 BS - ITMY (perpin): 1.606 m
 BS - PRM          : 0.245 m
 BS - SRM          : 0.509 m

L_PRC = 0.245+(2.134+1.606)/2 = 2.115   CHECK!
delta = 2.134-1.606 = 0.528             CHECK!
L_SRC = 0.509+(2.134+1.606)/2 = 2.379   CHECK!

I believe that these will fit in our chambers.
Don't forget that physical path lengths are a bit smaller,
due to light travelling through the glass.

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First stab at control signals:

Start with LIGO I, add one extra suspended optic (signal mirror), and
maybe also a suspended monolithic output mode cleaner (one
suspension).

This means one more pair of phase-modulated RF sidebands, one more
length control signal, one more wavefront sensor, and one more
dark-port beam (after the output mode cleaner) for DC-sensing of the
GW signal (independent of L- control).

Ports: MC reflected (MCRP), MC transmitted (MCTP), symmetric port
(SP), asymmetric port 1 (before output mode cleaner) (AP1), asymmetric
port 2 (after output mode cleaner) (AP2), power recycling cavity
pickoff (off of BS AR side) (PRP), X-arm transmitted (XTP), Y-arm
transmitted (YTP).

Each port needs a shutter, quad-photodiode monitor, and camera.

LSC RF photodiodes on SP, AP1, PRP, MCR, as follows 
(with CA = carrier, SB1 roughly 35 MHz, SB2 roughly 142 MHz) 
(TENTATIVE!):

PORT   RF                                   DOF
----   --                                   ---
AP1, CA-SB2,                                L-
SP,  CA-SB1,                                L+ 
SP,  SB2-SB1,                               l+ 
AP1, CA-SB2 and SB1 (double-demod),         l- 
PRP, CA-SB2 and SB1 (double-demod),         l_s 
MCR, at MC control frequency (11.82 MHz?)   MCL
GW (slow) photodiode on AP2                 GW (L-)

WFS telescopes and quad RF photodiodes as follows
(with CA = carrier, SB1 roughly 35 MHz, SB2 roughly 142 MHz) 
(TENTATIVE!):

AP1, CA-SB1,                                D(ETM) 
PRP, SB1-SB2,                               D(ITM) 
SP,  CA-SB2,                                C(ETM) 
PRP, CA-SB2,                                C(ITM) 
PRP, CA-SB2 (different RF phase),           RM 
AP1, SB1-SB2,                               SM 
MCR, at MC control frequency.               MC

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I welcome any and all comments!