Analytical solutions for film stress and bending deformation of coated optical lenses

Abstract

Axisymmetric film/substrate systems are irreplaceable for many applications, owing to special structural requirements. Stoney's formula is widely used to analyze the thin plate film/substrate system. Many researches have been presented to relax limitations of Stoney's formula, which considered the substrate as a thin plate without any curvature. For the optical lens, the axisymmetric thin shell substrate with different curvatures on its upper and lower surfaces is common due to the need of optical focusing. The film stress of the optical lens needs to be optimized and measured due to its effects on system deformations and optical performances. However, the optical lens cannot be solved by Stoney's formula and its extensions due to the curvatures. In this paper, analytical solutions for the thin shell optical lens with different curvatures on upper and lower substrate surfaces are developed by the thin shell theory with the axisymmetric distribution film stress resultant (FSR) in consideration. The FSR is modeled by thermal stress equivalent method, which allows the calculation for the axisymmetric distribution FSR induced by many factors. The forward and inverse solutions between the FSR and the bending deformation are provided in the form of simple formulas with correction coefficient vectors depending on the global substrate thickness, which enables its feasibility in practical applications. Moreover, the two solutions are verified by the finite element method considering the deformation normal to the substrate surface measured by the interferometer in inverse solution, and small errors are found when the optical lens conforms to the assumptions of the method in this paper. The proposed method is efficient to predict the optical lens’ bending deformations, and extends the usage of curvature-based techniques from plates to shells with different curvatures on upper and lower surfaces in the film stress measurement, which makes great significance to the thin film/substrate application.