Photoelectron shot noise

4.3 Photoelectron shot noise

For gravitational wave signals to be detected, the output of the interferometer must be held at one of a number of possible points on an interference fringe. An obvious point to choose is halfway up a fringe since the change in photon number produced by a given differential change in arm length is greatest at this point. The interferometer may be stabilised to this point by sensing any changes in intensity at the interferometer output with a photodiode and feeding the resulting signal back, with suitable phase, to a transducer capable of changing the position of one of the interferometer mirrors. Information about changes in the length of the interferometer arms can then be obtained by monitoring the signal fed back to the transducer.

As mentioned earlier it is very important that the system used for sensing the optical fringe movement on the output of the interferometer can resolve strains in space of 2× 10^–23 (Hz)^–1/2 or lower, or differences in the lengths of the two arms of less than 10^–19 m(Hz)^–1/2, a minute displacement compared to the wavelength of light $≃$ 10^–6 m. A limitation to the sensitivity of the optical readout scheme is set by shot noise in the detected photocurrent. From consideration of the number of photoelectrons (assumed to obey Poisson statistics) measured in a time $τ$ it can be shown [42 ] that the detectable strain sensitivity depends on the level of laser power $P$ of wavelength $λ$ used to illuminate the interferometer of arm length $L$ , and on the time $τ$ , such that:

where $c$ is the velocity of light and $h$ is Planck’s constant and we assume that the photodetectors have a quantum efficiency $≃$ 1. Thus achievement of the required strain sensitivity level requires a laser, operating at a wavelength of 10^–6 m, to provide 6 × 10⁶ W power at the input to a simple Michelson interferometer. This is a formidable requirement; however there are a number of techniques which allow a large reduction in this power requirement and these will be discussed in the next section.