  We grow accustomed to the notion that optical susceptibilities can be treated
as a local property of a medium. In the context of nonlinear optics, both Kerr
and Raman processes are considered local, meaning that optical fields at one
location do not produce a nonlinear response at distinct locations in space.
This is because the electronic and phononic disturbances produced within the
material are confined to a region that is smaller than an optical wavelength.
By comparison, Brillouin interactions can result in a highly nonlocal nonlinear
response, as the elastic waves generated through the Brillouin process can
occupy a region in space much larger than an optical wavelength. The
nonlocality of these interactions can be exploited to engineer new types of
processes, where highly delocalized phonon modes serve as an engineerable
channel that mediates scattering processes between light waves propagating in
distinct optical waveguides. These types of nonlocal optomechanical responses
have been recently demonstrated as the basis for information transduction,
however the nontrivial dynamics of such systems has yet to be explored. In this
work, we show that the third-order nonlinear process resulting from spatially
extended Brillouin-active phonon modes involves mixing products from spatially
separated, optically decoupled waveguides, yielding a nonlocal
'joint-susceptibility'. We further explore the coupling of multiple acoustic
modes and show that multi-mode acoustic interference enables a tailorable
nonlocal-nonlinear susceptibility, exhibiting a multi-pole frequency response.
