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  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.
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