Wave Equations for Spontaneous Brillouin Scattering (SBS) in Optical Fibers

This work describes the research of the differential equations governing spontaneous Brillouin scattering (SBS) which is a nonlinear process in which an electromagnetic optical pump wave generates an acoustic wave through the process of electrostriction. Electrostriction is the tendency of materials to become compressed in the presence of an electric field. The acoustic wave modulates the refractive index of the medium. This index grating scatters the optical wave by Bragg diffraction creating a backwards optical Stokes wave. Bragg gratings are ‘written’ on the internal surface of an optical element or fiber, like ‘bumps’ on the internal surface of a tube. The index grating is moving away from the source so the Brillouin frequency, ωS is shifted down in frequency. The moving corrugating surface scatters the incident light with a Doppler effect, giving scattered photons with shifted frequencies. A Stokes[1] wave refers to the relative phase of light reflected at a boundary between the grating and fiber surface of different refractive indices.In this paper, I describe a theoretical model of Brillouin scattering that shows how spontaneous Brillouin scattering is initiated by thermally excited acoustic waves distributed within a Brillouin-active medium (fiber optic ’tube’ or hollow ‘thread’). This model predicts how the sBs Stokes power depend upon the laser light intensity and upon the physical properties of the sBs medium. A brief study of a numerical solution to the Brillouin wave equations is presented.[1] The scattering of light to a wave downshifted in frequency is named after applied mathematician Sir George Stokes