We developed a method for simulating birefringence of an annealed ingot of calcium fluoride single crystal caused by the residual after annealing process. The method comprises the heat conduction analysis that provides the temperature distribution during the annealing, the elastic thermal stress analysis using the assumption of the stress-free temperature that provides the residual stress annealing, and the birefringence analysis of an annealed ingot induced by the residual stress. The finite element method was applied the heat conduction analysis and the elastic thermal stress analysis. In these analyses, the temperature dependence of material and the crystal anisotropy were taken into account. In the birefringence analysis, the photoelastic effect gives the change of indices, from which the optical path difference in the annealed ingot is calculated by the Jones calculus. The relation between the calculus and the approximate method using the stress components averaged along the optical path is discussed theoretically. It is that the result of the approximate method agrees very well with that of the Jones calculus in birefringence analysis. The pattern of the optical path difference in the annealed ingot obtained from the present birefringence calculation methods reasonably well with that of the experiment. The calculated values also agree reasonably well with those of the experiment, when a free temperature is adequately selected. (c) 2007 Elsevier B.V. All rights reserved.