Analytical stress modeling of high-energy laser windows: Application to fusion-cast calcium fluoride windows

The performance of a laser-window material must be assessed not only in terms of its ability to transmit high-power beams without generating undue optical distortion but also in terms of the constraints imposed by stress-related failure modes. In operational use, the stress field images the superposition of stresses originating from the mechanical load created by the pressure differential and the thermal load created by the laser beam. Here, we provide the tools to carry out an analysis of both pressure- and beam-induced stresses, and illustrate the procedure in the context of assessing the performance of a “model” window made of fusion-cast $CaF2$⁠. The analysis assumes (a) operation on a time scale such that lateral heat diffusion can be ignored, and (b) cylindrically symmetric Gaussian beam shapes, which permit straightforward calculations of stress distributions that should be representative of worst case situations. Pressure-induced stresses strongly depend on the window’s aspect ratio, which suggests increasing the thickness to minimize the stress, but considerations relating to the optical performance require minimum allowable thicknesses based on a Weibull statistical analysis of the fracture probability. Beam-induced stresses are best evaluated in terms of (a) thickness-averaged radial and azimuthal stresses, which increase linearly with exposure time and depend on radial distances through the truncation parameter, and (b) across-the-thickness stress deviations relative to the average stress, which are caused by surface absorption and reach steady-state configurations on a time scale much shorter than the characteristic time for lateral heat transport. The average stress is always compressive and equibiaxial in the central region of the window, but its azimuthal component turns tensile in the rim region, thus threatening the structural integrity through brittle fracture. In addition, the coating-induced stress results in on-axis surface compressions that may exceed the yield strength of the windowpane material. In this light we formulate a figure of merit for stress, which demonstrates that promising laser-window materials must combine a small stress factor (expansion coefficient times elastic modulus) with superior thermal properties in terms of the product of heat capacity and thermal conductivity; $CaF2$ and $SrF2$ are the only known candidates that exhibit outstanding optical features at chemical laser wavelengths together with acceptable thermomechanical properties.