Developing a reconstruction algorithm for 3D birefringence from tomographic polarimetry

Abstract

Stress-induced birefringence is created by the manufacturing process of injection-molded lenses. Characterizing the degradation in optical performance is important information to guide manufacturing improvements. The 3D birefringence distribution is proportional to the inhomogenous stress field. Birefringence is an anistropic property therefore retardance measurements change if the optical component is rotated. Reconstruction of a 3D birefringence distribution from a series of retardance measurements poses a challenge due to the extension of tomographic algorithms to tensor-valued quantities. Our approach is a closed-form forward model to linearly relate the polarimetric measurements to a line projection through an discrete arrangement of index ellipsoids. The rank of this linear system is investigated for varying arrangements of index ellipsoids. For example, given an homogenous birefringence distribution, only three non-parallel line projections are required for a full-rank operator that reconstructs the index ellipsoid of a planar slice. These three line projections are full-rank for some arrangements of three unique index ellipsoids and rank deficient for others. This mathematical framework is developed to design a tomographic polarimeter for inspecting the stress-induced 3D birefringence distribution of injection-molded optics. INTRODUCTIONInjection-molded lenses have an inhomogeneous stress-induced birefringence that can degrade optical performance. Many studies have been conducted to investigate residual stress and resulting birefringence caused by molding conditions. 1, 2 These studies have contributed to improved performance and higher yield of injection molded optical elements. Numerical modeling software such as Moldflow 3 can be used to establish the relationship between molding conditions and birefringence distribution and thereby identify process improvements for reducing birefringence. 4 The magnitude of stress-induced birefringence is also dependent on the mechanical properties of the material. 5 Commercially available analyzers like Photonic Lattice's 2D birefringence analyzer 6 are widely used to analyze birefringence distributions. However, such instruments are limited to measuring 2D birefringence. This paper presents a new approach for measuring and analyzing anisotropic inhomogeneous samples. A tomographic reconstruction of the 3D birefringence distribution is developed from a linear line projection relationship between the spatially varying index ellipsoids and tomographic polarimetric observations. To simplify the geometry of the line projection, the refractive index gradients are assumed to be small enough that light propagates in a straight line within the optic. This forward representation allows tomographic inverse solutions to be applied to quantify the 3D birefringence distribution of an inhomogeneous and anisotropic sample. TOMOGRAPHIC POLARIZATION Index EllipsoidAn index ellipsoid defines the polarization-dependent index of refraction (e.g. birefringence) of a material for all dire...