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.
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