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.

Figures (5)

• Figure 1: Convergence of the slab-product ABCD matrix (slab slicing) to the closed-form solution in Eqs. \ref{['eq:A_closedform']}--\ref{['eq:D_closedform']} for a $L=1$ cm crystal with axially varying GRIN strength due to absorption. Symbols correspond to the slab method with $N$ uniform segments, while the red dashed lines show the closed-form values of $A$, $B$, $C$, and $D$ evaluated at $z=L$.
• Figure 2: Closed-form ABCD elements of an exponential-decay GRIN medium compared with the standard constant-GRIN model. Panels (a)--(d) show the longitudinal evolution of the matrix elements $A(z)$, $B(z)$, $C(z)$, and $D(z)$ obtained from Eqs. \ref{['eq:A_closedform']}--\ref{['eq:D_closedform']} for a fixed GRIN parameter $g=4~\mathrm{cm^{-2}}$ (i.e., $\gamma=\sqrt{g}=2~\mathrm{cm^{-1}}$). Solid curves correspond to absorption coefficients $\alpha=0.1$, $1$, and $10~\mathrm{cm^{-1}}$, while circle markers denote the standard ABCD elements of a constant GRIN medium with the same $\gamma$.
• Figure 3: Propagation of a zero-order Bessel--Gaussian beam in an exponential-decay GRIN medium for different absorption coefficients, $\alpha$. The intensity distribution in the $y$--$z$ plane is calculated from \ref{['eq:bg_field_closedform']} using the closed-form ABCD elements. All panels use the same GRIN parameter and crystal length, $g=4~\mathrm{cm^{-2}}$ and $L=1~\mathrm{cm}$, with an input Gaussian envelope radius $\omega_0=330~\mu\mathrm{m}$ and cone angle $\theta=0.003$ rad ($m=0$). The color scale shows $\log_{10}(I/I_{\max})$, where $I_{\max}$ is the maximum intensity within each panel.
• Figure 4: Transverse intensity patterns of the zero-order Bessel--Gaussian beam in an exponential-decay GRIN medium at four representative propagation planes for different absorption coefficients. All panels use the same GRIN parameter and crystal length, $g=4~\mathrm{cm^{-2}}$ and $L=1~\mathrm{cm}$, with an input Gaussian envelope radius $\omega_0=330~\mu\mathrm{m}$ and cone angle $\theta=0.003$ rad ($m=0$).
• Figure 5: Representative transverse intensity distributions of the zero-order Bessel--Gaussian beam in an exponential-decay GRIN medium at four propagation planes for different absorption coefficients. All panels use $g=4~\mathrm{cm^{-2}}$ and $L=1~\mathrm{cm}$, with $\omega_0=330~\mu\mathrm{m}$ and cone angle $\theta=0.003$ rad ($m=0$).