Precision dynamics of resonantly enhanced optical parametric amplifiers

Optical parametric amplification is a crucial technology for the production of continuous-wave squeezed vacuum, which is now applied to gravitational-wave interferometry and other highly sensitive measurements of optical phase. The amplifiers employed in gravitational-wave detection are resonantly enhanced with optical cavities, which introduces nontrivial dynamics to their operation and requires a frequency-domain model of the relations between optical fields. In this work, input-output relations between optical fields entering, circulating in, and exiting a model amplifier are solved directly to produce frequency-domain relations that fully incorporate the cavity dynamics. We also provide low-order zero-pole-gain expansions for these relations to facilitate their analysis from a feedback control perspective. In the limit of small cavity decay rates, our relations are shown to reduce to relations previously derived under a Hamiltonian formalism. Expressions for amplifier figures of merit and for squeezed quadrature variances are also analyzed. Along the way, we examine how information about the amplifier performance can be extracted from the frequency dependence of the input-output relations, providing characterization methods that are complementary to methods that examine the static amplifier behavior.