Inhomogeneous birefringence analysis using a tensor-valued backprojection

Abstract

Injection-molded lenses have an inhomogeneous stress-induced birefringence that can degrade optical performance. This paper presents a new approach for measuring and analyzing inhomogeneous anisotropic samples. The birefringence distribution is characterized by 3D index ellipsoids, and a tomographic reconstruction of this 3D distribution is developed from a linear line projection relationship between the spatially varying index ellipsoids and tomographic polarimetry. This forward representation enables a tensor-valued backprojection for reconstructing the birefringence distribution of an inhomogeneous anisotropic sample. In this approach, each index ellipsoid is represented by a Hermitian matrix, and the 3D birefringence distribution is defined as the distribution of these matrices. This paper is centered on the introduction of the fundamental algorithm and the presentation of a general solution by applying the Radon transform and the backprojection to a tensor field, without requiring specific parameters such as stress fields. Consequently, the computational approach presented in this paper demonstrates that, using 60 tomographic views, reconstruction errors for parameters that characterize spatially varying index ellipsoids remain less than 5%. Here, the error is defined as the ratio of reconstruction variation to the respective maximum values of the original distributions.