Stress and temperature analysis for surface cooling or heating of laser window materials

Abstract

The maximum reduction in the average temperature 〈T〉z=b−1 ∫0b d z T of a slab of thickness b that can be obtained in both a short period of time, typically [inverted lazy s]1 sec, and a long period of time (steady-state approached) without causing material failure is calculated. The slab is originally at a constant temperature, and 〈T〉z is the temperature that determines the optical properties of a normally illuminated slab. The results indicate that Ge28Sb12Se60 glass can be cooled rapidly enough for use in laser systems at relatively high power levels (several hundred W/cm2) in the pulse mode of operation discussed previously, in contrast to previous beliefs. There is a time constant τs for the nonexponential approach of the surface temperature Ts of a semi-infinite medium to the temperature Tc of the coolant. Consider the thermally thin-disk case of τb≪τs, where τb is time required for heat to diffuse across the thickness b, roughly speaking. The value of 〈T〉z then approaches Tc exponentially with a time constant τc, which is much smaller than τs. For the thermally thick-disk case of τs≪τb, 〈T〉z[inverted lazy s](t/τc)Tc for t≪τs; 〈T〉z≃(t/τb)1/2 Tc for τs≪t≪τb; and 〈T〉z≃Tc for t≫τb. The solution to the heat-flow equation for T is known in simple closed form for the case of a semi-infinite medium. The result is quite accurate for finite disks, both thermally thick and thin, when t≪τb is satisfied. For this case of t≪τb, the maximum value of the magnitude of the stress component |σxx|(=|σyy|) is determined by the surface temperature Ts only. For a given temperature difference across the disk, the maximum value of |σxx| is less for the case of t≥τb than for the case of t≪τb. Two examples, GaAs and Ge28Sb12Se60 glass, are considered.