  Inelastic scattering processes such as Brillouin scattering can often
function in cascaded regimes and this is likely to occur in certain integrated
opto-acoustic devices. We develop a Hamiltonian formalism for cascaded
Brillouin scattering valid for both quantum and classical regimes. By regarding
Brillouin scattering as the interaction of a single acoustic envelope and a
single optical envelope that covers all Stokes and anti-Stokes orders, we
obtain a compact model that is well suited for numerical implementation,
extension to include other optical nonlinearities or short pulses, and
application in the quantum-optics domain. We then theoretically analyze
intra-mode forward Brillouin scattering (FBS) for arbitrary waveguides with and
without optical dispersion. In the absence of optical dispersion, we find an
exact analytical solution. With a perturbative approach, we furthermore solve
the case of weak optical dispersion. Our work leads to several key results on
intra-mode FBS. For negligible dispersion, we show that cascaded intra-mode FBS
results in a pure phase modulation and discuss how this necessitates specific
experimental methods for the observation of fibre-based and integrated FBS.
Further, we discuss how the descriptions that have been established in these
two classes of waveguides connect to each other and to the broader context of
cavity opto-mechanics and Raman scattering. Finally, we draw an unexpected
striking similarity between FBS and discrete diffraction phenomena in waveguide
arrays, which makes FBS an interesting candidate for future research in
quantum-optics.
