Scalar potential reconstruction method of axisymmetric 3D refractive index fields with background-oriented schlieren

Deflection angles of light rays passing through a refractive index field can be measured by the background-oriented schlieren (BOS) technique. Assuming that the deflection angle is sufficiently small and the paraxial approximation can apply to the light rays, a vector consisting of deflection angles in two orthogonal directions is shown to be derived from a gradient of a scalar potential. The scalar potential can be written as an integration of the refractive index field over the light ray path. Thus, a method to reconstruct an axisymmetric 3D refractive index field with the scalar potential is proposed here. An arbitrary measured deflection angle vector, however, is generally written not only with a scalar potential but with a vector potential. Thus, the Poisson's equation is derived to extract a scalar potential from a measured deflection angle vector. The axisymmetric 3D refractive index field is able to be reconstructed using the Abel transformation [1] of the scalar potential derived by applying the 2D Fourier transformation to the Poisson's equation. The scalar potential reconstruction method is validated by reconstructing a spherically symmetric refractive index field where a deflection angle vector field is able to be calculated accurately.