  Understanding how laser light scatters from realistic mirror surfaces is
crucial for the design, com- missioning and operation of precision
interferometers, such as the current and next generation of gravitational-wave
detectors. Numerical simulations are indispensable tools for this task but
their utility can in practice be limited by the computational cost of
describing the scattering process. In this paper we present an efficient method
to significantly reduce the computational cost of optical simulations that
incorporate scattering. This is accomplished by constructing a near optimal
representation of the complex, multi-parameter 2D overlap integrals that
describe the scattering process (referred to as a reduced order quadrature). We
demonstrate our technique by simulating a near-unstable Fabry-Perot cavity and
its control signals using similar optics to those installed in one of the LIGO
gravitational-wave detectors. We show that using reduced order quadrature
reduces the computational time of the numerical simulation from days to minutes
(a speed-up of $\approx 2750 \times$) whilst incurring negligible errors. This
significantly increases the feasibility of modelling interferometers with
realistic imperfections to overcome current limits in state-of-the-art optical
systems. Whilst we focus on the Hermite-Gaussian basis for describing the
scattering of the optical fields, our method is generic and could be applied
with any suitable basis. An implementation of this reduced order quadrature
method is provided in the open source interferometer simulation software
Finesse.
