Birefringence Simulation of Annealed Ingot of Magnesium Fluoride Single Crystal

Abstract

We developed a method for simulating birefringence of an annealed ingot of MgF <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> 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 residual stress analysis and the birefringence analysis. The residual stres analysis comprises the heat conduction analysis and the elastic thermal stress analysis. The heat conduction analysis provides the temperature distribution during ingot annealing and the elastic thermal stress analysis provides the residual stress after annealing by using the result of the heat conduction analysis. Because there exists no experimental data on the inelastic behavior of MgF <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> single crystal, the residual stress was estimated with the elastic thermal stress analysis by assuming a stress-free temperature, by which the inelastic behavior could be treated approximately. The finite element method was applied to both the heat conduction analysis and the elastic thermal stress analysis. In these analyses, the temperature dependence of material properties and the 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.