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

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at the beamsplitter remains unchanged, and all of the light still goes back towards
the laser. If the mirrors are moved differentially, the interference at the output is no
longer perfectly destructive, and some light leaks out. For very small phase shifts,
the amount of light returning to the laser is effectively unchanged. This means that
the dark fringe condition effectively diagonalizes the response of the detector into
common and differential modes. No information about differential motion of the end
mirrors goes back towards the laser, and likewise, no common mode information exits
the output of the interferometer.
When a mirror is added at the output, otherwise known as the “dark port,” dif-
ferential signals are the only ones which sense its presence due to the diagonalization
by the dark fringe. If the arms of the interferometer are equal, then the two equal but
oppositely signed signals in the arms experience the same arm cavity, and after being
summed at the beam splitter, the same signal cavity. For the purposes of analyzing
the interferometer’s response to differential arm signals, the rather complicated con-
figuration of Figure 2.1 can be reduced to the 3 mirror coupled cavity of Figure 2.2.
The transmissivity of this configuration is given by Eq. (2.1).
MET M MIT M MSEM
Signal input
φarm
Figure 2.2: Equivalent three mirror coupled cavity for differentially modulated signals
generated in the arms. The effect of losses in the beamsplitter, as well as in the ITM
substrates, can be included in an effective signal mirror reflectivity and transmissivity,
r ′ sem and t ′ sem.
tcc(f) =
BS
φsec
titmt ′ seme −i(2πf(τa+τs)+φdt)/2
¡
¡
PD3
1 − ritmretme −i2πfτa − ritmr ′ seme −i(2πfτs+φdt) + retmr ′ semAitme −i(2πf(τa+τs)+φdt)
(2.1)
The characteristic times τa = 2larm/c and τs = 2lsec/c are the round trip travel times
in the arm and signal cavities. The quantity Aitm represents 1 minus the losses of the
ITM coating. The frequency f is relative to a carrier frequency which is resonant in
the arms, and off-resonant in the signal cavity by the round-trip phase φdt, otherwise