We develop a steady-state analytical and numerical model of the optical
response of power-recycled Fabry-Perot Michelson laser gravitational-wave
detectors to thermal focusing in optical substrates. We assume that the thermal
distortions are small enough that we can represent the unperturbed intracavity
field anywhere in the detector as a linear combination of basis functions
related to the eigenmodes of one of the Fabry-Perot arm cavities, and we take
great care to preserve numerically the nearly ideal longitudinal phase
resonance conditions that would otherwise be provided by an external
servo-locking control system. We have included the effects of nonlinear thermal
focusing due to power absorption in both the substrates and coatings of the
mirrors and beamsplitter, the effects of a finite mismatch between the
curvatures of the laser wavefront and the mirror surface, and the diffraction
by the mirror aperture at each instance of reflection and transmission. We
demonstrate a detailed numerical example of this model using the MATLAB program
Melody for the initial LIGO detector in the Hermite-Gauss basis, and compare
the resulting computations of intracavity fields in two special cases with
those of a fast Fourier transform field propagation model. Additional
systematic perturbations (e.g., mirror tilt, thermoelastic surface
deformations, and other optical imperfections) can be included easily by
incorporating the appropriate operators into the transfer matrices describing
reflection and transmission for the mirrors and beamsplitter.Comment: 24 pages, 22 figures. Submitted to JOSA
