Strict mathematical model of mechanical stress birefringence for optical plates

The mechanical stress birefringence (SBR) has received attention due to its effect on polarization in immersion lithography. In this paper, we present a strict mathematical model to obtain the correct SBR and slow-axis distributions of optical plates. First, the linear conditions of the model are solved to ensure the reasonable assembly of optical plates. Then we strictly define the plane principal stresses and slow-axis angle, and we give the correct expressions. Utilizing this model, we simulate a transmissivity variable plate, which is divided into 11 layers to obtain the effective SBR experienced by incident light crossing the plate. In this case, the simulation results achieve convergence. The validity of the model is verified by comparing the SBR and slow-axis distributions obtained by different expressions of the plate. This model is of great significance for polarization analysis in lithography systems and the reasonable assembly of optical elements.