Bessel Gaussian Beam Propagation in a Thermally Induced Axially Varying GRIN Medium

Abstract: High power end pumped solid state lasers often operate in regimes where pump induced heating creates a strong refractive index gradient (thermal lensing) that governs resonator stability and mode quality. When the pump is absorbed according to the Beer Lambert law, the thermal load, and hence the GRIN strength, vary along the crystal length, so the standard ABCD matrix of a constant-gradient GRIN element is no longer directly applicable. Here, we derive a closed-form ABCD transmission matrix for a thermally loaded laser crystal pumped by atop-hat beam while explicitly accounting for axial absorption. Starting from the steady-state heat equation, we obtain the temperature field and the associated thermo-optic index profile. We then solve the paraxial eikonal ray equation analytically and express the transfer-matrix elements in terms of Bessel and Neumann functions. The resulting matrix is validated against the conventional slab product method and shown to recover the uniform-medium and constant gradient GRIN limits. Finally, we illustrate its utility by model ing Bessel Gaussian beam propagation through the axially varying thermally induced GRIN medium.