P010010-00-R - LIGO - California Institute of Technology

167
The broadband RF sideband phases are then (modulo 2π)
φ ± + = ±π (5.44)
φ ± s = ±2π (5.45)
φ ± − = ±α = ± Ωbbδ
c
where δ is the macroscopic asymmetry l1 − l2.
(5.46)
The power and signal cavities are both relatively short (of order 10 m), which
corresponds to a free spectral range of roughly 10 MHz. Even with an unlikely finesse
of 1000, the bandwidth of these cavities wouldn’t be much less than 10 kHz (for typical
numbers, the bandwidth of the power/signal coupled cavity tends to be roughly 100
kHz). This allows the assumption that the transmission for the DC RF sidebands and
their noise sidebands is constant over the bandwidth of interest, so the dependence
on the frequency of the noise sideband ±ω can be ignored.
The transmissivity for the RF sidebands is derived in Appendix B.3.
t± = ±tprmAbstsem sin(α ′ )g ±
ifo
ifo = e−i(φdt+η ± s +η ±
+ )/2
g ±
tD±
tD± = 1 − rprmAbs cos(α ′ )e −iη±
+ − rsemAbs cos(α ′ )e −i(η± s +φdt) +
rprmrsemA 2 bse −i(η±
+ +η± s +φdt)
where the primed α is defined to include the η phase as
α ′ = α ∓ η ± −
(5.47)
(5.48)
(5.49)
(5.50)
The approach used in this thesis is to modify the length of the signal cavity to