Wave propagation in media with focused gain

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Abstract
A solution of the scalar wave equation in a rotationally symmetric focused Gaussian amplifying medium is constructed from coupled Gaussian-Laguerre solutions of the free-space wave equation and is formally exact. The solution allows for the construction of the characteristic eigenmodes of the system and has direct application to Raman lasers, dye lasers, and the free-electron laser. The eigenvalues corresponding to the eigenmodes are directly related to mode growth rates and phase velocities. Numerical results for single- and multiple-pass growth are evaluated and exhibit an unpredicted gain enhancement that can be related to the phenomenon of "gain focusing" first described for uniform parabolic gain distributions. The solution for a focused parabolic gain distribution is also presented. Output intensity distributions are found for initial conditions corresponding to spontaneous emission and to a specified initial field. The same method of solution can be applied to a medium with a focused Gaussian refractive index.