Graded refractive index media appear in numerous industrial applications such as non-isothermal flows, optics material processing, biological imaging. Refractive index gradient has been an early help for combusting flow visualisation. The numerical treatment of radiative transport is difficult in such media due to the curvature of rays, especially when the media are not optically thick. Computer-aided remote probing (inversion) is done today with the help of the diffusion approximation adapted to varying refractive index media but is unsuitable for thin media. Therefore, it is important to develop an approach allowing the use of the radiative transport equation which is the most complete formalism for radiative transfer to date and to couple it to reconstruction schemes. The aim of this study is to demonstrate the reconstruction of an arbitrary refractive index distribution from a least-squares gradient-based iterative inversion algorithm taking advantage of the full transient Radiative Transfer Equation (tRTE). The finite-difference discrete-ordinates method for the tRTE and its adjoint has been implemented, accounting for spatial changes in the distribution of the refractive index in a semi-transparent medium. A least-squares gradient-based iterative algorithm has been designed and elementary tests have been carried to demonstrate reconstruction possibilities.

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