Birefringence simulation of annealed ingot of magnesium fluoride single crystal

We developed a method for simulating birefringence of an annealed ingot of MgF₂ single crystal for lithography optics. This method provides the optical path difference caused by crystal symmetry and residual stress existing in the crystal. The method consists of the heat conduction analysis, the residual stress analysis and the birefringence analysis. Because there exists no experimental data on the inelastic behavior of MgF₂ single crystal, the residual stress was estimated with the elastic thermal stress analysis using the finite element method by assuming a stress-free temperature. In this analysis, the temperature dependence of material properties and crystal anisotropy were taken into account. In the birefringence analysis, the distributions of optical path difference were calculated by an approximate method using the result of the residual stress analysis. This approximate method uses the average stress along the wave normal and is equivalent to the exact method in case of low stress dealt with the present study. In this analysis, it is possible to consider both the intrinsic birefringence and the stress birefringence in any crystal orientation. The distribution of the optical path difference in the annealed ingot obtained from the present calculation agrees qualitatively with that of the experiment. Its calculated value also agrees reasonably well with that of the experiment, when a stress-free temperature is adequately selected.