Laser heating of a slab having temperature-dependent surface absorptance

The exact solution obtained here for the temperature in a laser-heated slab with surface absorptance A=Ai+A1 T (0,t) proportional to temperature and with negligible radial heat flow, shows that the temperature is considerably greater in general than for A1=0. In the limit of long time, for negligible cooling, the surface temperature increases exponentially as TA{ exp[(I/Il)(t/τl)]−1} for I≲Il and as TA{2 exp[(I/Il)2(t/τl)]−1} for I≳Il, where TA=Ai/A1, Il=K/lA1, τl=Cl2/K, K is the thermal conductivity, C is the heat capacity per unit volume, and l is the slab thickness. The limiting temperature derived here for t≳τl/5 and that derived by Sparks and Loh for t≲τl/5 afford simple closed-form expressions for T (z,t) for all z and t. With cooling, the temperature increases without bound (thermal runaway) for I≳Ihl, where Ihl=Il[Hs+H/(1+H)] with Hs and H normalized heat-transfer coefficients at the front and rear surfaces of the slab. The intrinsic cw damage threshold, which is slightly less than Ihl, can be quite low—from 100 W/cm2 to 150 kW/cm2 for the examples considered. The ability to withstand a given irradiance in a small beam does not imply the ability to withstand the same irradiance in a large beam.