Fluctuations in radiation pressure

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 $P ∕2$ in each arm, the power spectral density of the fluctuating motion of the test mass $m$ induced by the fluctuations in the radiation pressure at angular frequency $ω$ is given by

where $h$ is Planck’s constant, $c$ is the speed of light and $λ$ 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 × 10³ 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.