  Owing to their highly coherent emission and compact form factor, Brillouin
lasers have been identified as a valuable asset for applications including
portable atomic clocks, precision sensors, coherent microwave synthesis and
energy-efficient approaches to coherent communications. While the fundamental
emission linewidth of these lasers can be very narrow, noise within dielectric
materials leads to drift in the carrier frequency, posing vexing challenges for
applications requiring ultra-stable emission. A unified understanding of
Brillouin laser performance may provide critical insights to reach new levels
of frequency stability, however existing noise models focus on only one or a
few key noise sources, and do not capture the thermo-optic drift in the laser
frequency produced by thermal fluctuations or absorbed power. Here, we develop
a coupled mode theory of Brillouin laser dynamics that accounts for dominant
forms of noise in non-crystalline systems, capturing the salient features of
the frequency and intensity noise for a variety of systems. As a result, theory
and experiment can be directly compared to identify key sources of noise and
the frequency bands they impact, revealing strategies to improve the
performance of Brillouin lasers and pave the way for highly-coherent sources of
light on a chip.
