Cancellation of laser noise in an unequal-arm interferometer detector of gravitational radiation

Equal-arm interferometric detectors of gravitational radiation allow phase measurements many orders of magnitude below the intrinsic phase stability of the laser injecting light into their arms. This is because the noise in the laser light is common to both arms, experiencing exactly the same delay, and thus cancels when it is differenced at the photo detector. In this situation, much lower level secondary noises then set overall performance. If, however, the two arms have different lengths (as will necessarily be the case with space-borne interferometers), the laser noise experiences different delays in the two arms and will hence not directly cancel at the detector. In this paper we present a method for exactly canceling the laser noise in a one-bounce unequal-arm Michelson interferometer. The method requires separate measurements of the phase difference in each arm, made by interfering the returning laser light in each arm with the outgoing light. Let these two time series of phase difference be $z_i,$ $i=1,2.\mathrm{}$ By forming the quantity $\left[z_1\right(t-{2L}_2/c\left)-z_1\mathrm{}\right(t\left)\mathrm{}\right]-\left[z_2\right(t-{2L}_1/c\left)-z_2\mathrm{}\right(t\left)\mathrm{}\right],$ where $L_i$ are the arm lengths, gravitational wave signals remain while the laser noise is canceled. Unlike other proposed methods, this procedure accurately cancels the laser noise if the arm lengths are known. This method is direct in time and allows for time-varying arm-lengths. In this paper we demonstrate that this method precisely cancels the laser noise, present the transfer function of gravitational waves after forming this linear combination, and discuss system requirements (such as required knowledge of the arm lengths). We verify the technique with numerical simulation of periodic gravitational wave signals embedded in laser and shot noise having spectra expected for a space-borne interferometer, and compare our time-domain approach with approximate correction methods based on Fourier transforms of the $z_i$ processes.