Analytical Solution of Brillouin Amplifier Equations for lossless medium

In order to explain pump depletion in Stimulated Brillouin scattering (SBS), coupled intensity equations describing the interaction of pump and stokes waves in Brillouin medium, must be solved simultaneously. Since this problem has well-defined boundary conditions, such a mathematical problem is known as the two-point boundary value problem. Conventional solution techniques leads transcendental equation which results implicit solution. In this paper, we accurately define Pump and Stokes evolution in lossless medium in terms of conserved quantity and proposed the solution of this conserved quantity using the asymptotic theory. Regarding with the saturation region, the gain approximation of Brillouin Fiber Amplifier (BFA) for the lossless medium, is introduced for the first time to our best of knowledge.