Analytical solutions for anisotropic time-dependent heat equations with Robin boundary condition for cubic-shaped solid-state laser crystals

Abstract

The problem of finding analytical solutions for time-dependent or time-independent heat equations, especially for solid-state laser media, has required a lot of work in the field of thermal effects. However, to calculate the temperature distributions analytically, researchers often have to make some approximations or employ complex methods. In this work, we present full analytical solutions for anisotropic time-dependent heat equations in the Cartesian coordinates with various source terms corresponding to various pumping schemes. Moreover, the most general boundary condition of Robin (or impedance boundary condition), corresponding to the convection cooling mechanism, was applied. This general condition can be easily switched to constant temperature and thermal insulation as special cases. To this end, we first proposed a general approach to solving time-dependent heat equations with an arbitrary heat source. We then applied our approach to explore the temperature distribution for three cases: steady-state pumping or long transient, single-shot pumping or short transient, and repetitively pulsed pumping. Our results show the possibility of an easier and more accurate approach to analytical calculations of the thermal dispersion, thermal stresses (strains), thermal bending, thermal phase shift, and other thermal effects.

Pulsed Bessel–Gauss beams: a depleted wave model for type II second-harmonic generation

“…where we have shifted the time origin to 2t p [22] so that ψ 1 t 0; r; z ψ 2 t 0; r; z ≈ 0. We choose t p 50 μs [22] as the duration time of pulses and α 51532 m −1 as the transverse wave vector.…”

“…We choose t p 50 μs [22] as the duration time of pulses and α 51532 m −1 as the transverse wave vector. α is calculated according to the equation α 2π∕λ 1 sinβ with β 1° [23] as the cone angle of the axicon used in the Bessel-Gauss setups.…”

Numerical modelling of self-heating effects on guiding modes of high-power photonic crystal fibre lasers

“…The most general boundary condition for temperature at the fibre surface is [22] ( 2) where h is the heat convection coefficient having a typical value for cooling with air as h = 17 w/m 2 /K, T ∞ = 298 K is ambient temperature, ε = 0.95 [8] Table 1. Figure 2 shows the temperature distribution across the PCFL for P h = 10 W/m.…”

Analytical solution of transient temperature in continuous wave end-pumped laser slab: Reduction of temperature distribution and time of thermal response

“…Few literature dealt with analytical solution of transient temperature in laser slab. Some of them are that presented by Sabaeian [8] who presented analytical solution for anisotropic transient heat equation for solid-state laser crystal with cubic geometry with robin boundary condition using separation of variable technique. Another work presented by Zhang et al [9] solved the heat equation in 3-D model for anisotropic crystal with special boundary condition.…”