  The dynamics of cascaded-order Brillouin lasers make them ideal for
applications such as rotation sensing, highly coherent optical communications,
and low-noise microwave signal synthesis. Remark- ably, when implemented at the
chip-scale, recent experimental studies have revealed that Brillouin lasers can
operate in the fundamental linewidth regime where optomechanical and quantum
noise sources dominate. To explore new opportunities for enhanced performance,
we formulate a simple model to describe the physics of cascaded Brillouin
lasers based on the coupled mode dynamics governed by electrostriction and the
fluctuation-dissipation theorem. From this model, we obtain analytical formulas
describing the steady state power evolution and accompanying noise properties,
including expressions for phase noise, relative intensity noise and power
spectra for beat notes of cascaded laser orders. Our analysis reveals that
cascading modifies the dynamics of intermediate laser orders, yielding noise
properties that differ from single-mode Brillouin lasers. These modifications
lead to a Stokes order linewidth dependency on the coupled order dynamics and a
broader linewidth than that predicted with previous single order theories. We
also derive a simple analytical expression for the higher order beat notes that
enables calculation of the Stokes linewidth based on only the relative measured
powers between orders instead of absolute parameters, yielding a method to
measure cascaded order linewidth as well as a prediction for sub-Hz operation.
We validate our results using stochastic numerical simulations of the cascaded
laser dynamics.
