The Response of Laser Interferometric Gravitational Wave Detectors Beyond the Eikonal Equation

Abstract

The response of Michelson interferometers to weak plane gravitational waves is computed at one order of accuracy beyond the eikonal equation. The modulation of the electromagnetic field amplitude and polarisation are taken into account by solving the transport equations of geometrical optics with boundary conditions adapted to laser interferometry. An explicit formula for the interferometer output power is derived, comprising perturbations of the optical path length, frequency, amplitude and polarisation.

The Electromagnetic Field in Gravitational Wave Interferometers

“…Analyticity in ω g /ω implies that our result coincides with the one obtained by a suitably-truncated (cf. [21]) eikonal approximation of our model. In other words, we have provided a justification that eikonal expansions give the correct result at the expected level of accuracy.…”

“…To express the returning field in a concise notation and to compare with the final results of Ref. [21] (see Section 8), we define, for any wave-vector k…”

“…to denote the values of the GW phase at the (retarded) times of light reflection and emission, respectively. In order to make contact with the notation of [21], given a vector k we set…”

“…To relate the light polarisation vectors sent into the two interferometer arms (which are both derived from the incoming laser beam) and to compute the output laser field from the returning light rays, one must model the polarisation transfer at the beam splitter. Following the model put forward in [21], we assume the polarisation of the field emitted by the laser to have the form…”

“…Similarly to the first ray, its amplitude is reduced by 1/ √ 2, but additionally there is the standard phase shift of π. For a more detailed description of the beam splitter transformations, see [21,Sect. 6].…”