### 5.3 Fluctuations in radiation pressure

A phenomenon which becomes increasingly important as the effective laser power in the arms is
increased is the effect on the test masses of fluctuations in the radiation pressure of the light in the arms.
From [26], for a simple Michelson interferometer with power in each arm, the power spectral density
of the fluctuating motion of the test mass induced by the fluctuations in the radiation pressure at
angular frequency is given by
where is Planck’s constant, is the speed of light and the wavelength of the laser light. In the
case of systems with cavities in the arms where the number of effective reflections is approximately 50,
fluctuations in the amplitude of the light arise immediately after the beamsplitter where vacuum
fluctuations enter into the system. These are then enhanced by the power build up effects of the optical
cavities. In this case
Evaluating this at 10 Hz, for powers of around 5 × 10^{3} W after the beamsplitter and masses of 30 kg,
indicates an amplitude spectral density for each mass of 9 × 10^{–20} m(Hz)^{–1/2}. Given the target sensitivity
of the detector at 10 Hz in Figure 3 of approximately 7 × 10^{–23} (Hz)^{–1/2} which translates to a motion of
each test mass of close to 10^{–19} m(Hz)^{–1/2} it is clear that radiation pressure may be a significant limitation
at low frequency. Of course the effects of the radiation pressure fluctuations can be reduced by increasing
the test masses, or by decreasing the laser power at the expense of deproving sensitivity at higher
frequencies.
It should be noted that as discussed in [26, 15, 16] and [60] for a simple Michelson system, the
optimisation of laser power to minimise the combined effect photon shot noise and radiation pressure
fluctuations allows one to reach exactly the sensitivity limit predicted by the Heisenberg Uncertainty
Principle, in its position and momentum formulation. An extension of this analysis to a system with cavities
in the arms has been carried out by one of the authors [43] with the same result and it seems
likely to be true for the more complicated optical systems using power and signal recycling
also.