Technique to transform optical deflection equations into linear difference ones for tomography

Abstract

The random-direction partial derivatives in optical deflection equations were transformed into numerical differences in reference frames for tomography. The nonlinear deflection equations were transformed into linear tomography ones. A detecting ray will turn its propagating direction when it runs through a heterogeneous refractive index field. Its deflecting angle a is the function of refractive index n by n's first order partial derivative. So, the optical deflection equation involves nonlinear first order partial derivative. This kind of detecting ray equations can't be resolved by tomography algorithm directly. At first, the nonlinear partial derivative should be transformed into numerical difference. Here, a practical transforming algorithm was put forward. The diagnosed field was divided into tiny foursquare grids. Each grid and its refractive index were approximated to a correct cone with an irregular bottom. With the approximation, the space partial increment calculation was much simplified at any grid, in any direction and to any detecting ray. It was assumed that the refractive index distribution should be coplanar in the area between three grid centers of the three close-adjacent grids. With the assumption, the refractive index partial increment could be calculated with a numerical difference function of close-adjacent grid refractive indexes. With the approximation and assumption, the partial derivative was transformed into numerical difference. As the result, partial derivative related to any detecting ray could be transformed into numerical difference. Nonlinear deflection equations could be transformed into linear difference ones. So, the deflected angles can directly be applied to reconstruction as projections.