Pre-Phase A Report - Lisa - Nasa

102 Chapter 4 Measurement Sensitivity
N ≈ n1(f)L2 − n2(f)L1
L1
(4.52)
H ≈ h(f) L1 + L2
. (4.53)
L1
From the above expressions for P (f), it is clear that the laser phase noise can be corrected
for to the measurement noise level N(f) if the arm lengths are known accurately enough,
and if the measurement system has sufficient dynamic range.
In general, we assume that the difference in length of the two arms is known to 200 m, and
the mean length to 20 km. The error in the arm length difference can then cause an error
of 4×10 −9 of the laser phase noise at f =1mHz,and4×10 −10 at f =0.1 mHz. If the
fractional difference in arm lengths is 1%, the errors in the laser phase noise corrections
due to the uncertainty in the mean arm length are the same magnitude as the values
given above.
Our model for the laser phase noise before correction at frequencies of 0.1 mHz to 1 Hz
is based on the thermal stability of the reference cavity to which the laser in spacecraft
1 is locked for frequencies below about 3 mHz, plus the noise in locking to the
cavity at higher frequencies. Our estimate for the locking noise comes from the results
of Salomon et al. [109]. As a typical case, we take the fractional frequency noise in the
laser to be 2×10−11 / √ Hz at 0.1 mHz, 1×10−13 / √ Hz at 1 mHz, and 2×10−15 / √ Hz at
10 mHz. As an example, the frequency noise at 1 mHz corresponds to a phase noise level
of 5×10−3 m/ √ Hz . To correct for this to a measurement noise level of 4 pm/ √ Hz requires
a phase noise reduction of a factor 8×10−10 .
The measurement of the difference in arm length for arms 1 and 2 is then corrected for the
laser phase noise, using the approximately known lengths of the arms. The requirement on
knowing the difference in arm lengths is about 200 m, as assumed earlier. The arm lengths
will be determined by combining ground tracking of the spacecraft with the observed arm
length changes from the laser phase measurements, or, alternatively, by measuring a group
delay with a modulation tone on the laser beam. If the arm lengths are very close to equal,
the noise at the harmonics of the round-trip travel frequency will be less well determined,
but the accuracy will still be sufficient for correcting the measured arm length difference.
A similar process is used to correct the time series of the length of arm 3 minus the average
for arms 1 and 2.
The advantage of on-board correction for the laser phase noise is that the data can be
compressed by a factor of perhaps 5 before they are transmitted to the ground. This is
because the arm length changes will be very smooth and the gravitational wave signals
relatively small. The one disadvantage of having to correct for the laser phase noise is
that a small fractional error will be made in the amplitude and phase of the gravitational
wave signals in some cases. However, this error appears to be correctable for sources
where the source direction and polarization can be determined.
4.3.3 Clock noise
An ultra-stable oscillator (USO) is required onboard for the phase measurements, for
compensation of the orbital Doppler shifts, and for providing offset frequencies for laser
3-3-1999 9:33 Corrected version 2.08