Spatiotemporal model of the LIGO interferometer

We develop a detailed spatiotemporal model in the adiabatic approximation o f the optical response of the Laser Interferometer Gravitational-Wave Obser vatory (LIGO) optical system to environmental perturbations. We begin by de riving a first-order linear time-dependent evolution equation that models t he electromagnetic field as a function of position within a Fabry-Perot int erferometer. This model allows both the length of the resonator and the mis alignment angles of the end mirrors to vary in time and describes both reso nant and nonresonant phenomena. After defining a biorthogonality relation t hat must he satisfied by general unperturbed spatial eigenfunctions of the Fabry-Perot interferometer, we expand the intracavity field as a linear com bination of these functions and convert the spatiotemporal evolution equati on into a linear system of coupled time-dependent iteration and/or differen tial equations. We then calculate the adiabatic connection equations that l ink the two LIGO Fabry-Perot interferometers through the power recycling ca vity, which comprises two mirrors (the power recycling input mirror and the beam splitter) that have the same mechanical degrees of freedom as those i n the Fabry-Perot arm cavities. We develop a detailed instance of this mode l for the evolution of the intracavity field within a resonator with suffic iently small misalignment angles that a Hermite-Gauss basis set can be used . We develop a detailed general approach to signal demodulation for simulat ion of servo-control systems and describe its implementation in the Hermite -Gauss approximation for Cartesian split-plane detectors. Finally, we demon strate the use of this small-angle model to simulate the effect of angular misalignment on the longitudinal response function. (C) 1999 Optical Societ y of America [S0740-3232(99)01112-6].