Spatiotemporal model of the LIGO interferometer

Abstract

We develop a detailed spatiotemporal model in the adiabatic approximation of the optical response of the Laser Interferometer Gravitational-Wave Observatory (LIGO) optical system to environmental perturbations. We begin by deriving a first-order linear time-dependent evolution equation that models the electromagnetic field as a function of position within a Fabry–Perot interferometer. This model allows both the length of the resonator and the misalignment angles of the end mirrors to vary in time and describes both resonant and nonresonant phenomena. After defining a biorthogonality relation that must be satisfied by general unperturbed spatial eigenfunctions of the Fabry–Perot interferometer, we expand the intracavity field as a linear combination of these functions and convert the spatiotemporal evolution equation into a linear system of coupled time-dependent iteration and/or differential equations. We then calculate the adiabatic connection equations that link the two LIGO Fabry–Perot interferometers through the power recycling cavity, which comprises two mirrors (the power recycling input mirror and the beam splitter) that have the same mechanical degrees of freedom as those in the Fabry–Perot arm cavities. We develop a detailed instance of this model for the evolution of the intracavity field within a resonator with sufficiently small misalignment angles that a Hermite–Gauss basis set can be used. We develop a detailed general approach to signal demodulation for simulation of servo-control systems and describe its implementation in the Hermite–Gauss approximation for Cartesian split-plane detectors. Finally, we demonstrate the use of this small-angle model to simulate the effect of angular misalignment on the longitudinal response function.