1 March 1998 Paraxial ray tracing in inhomogeneous optical media
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Starting from the variational form of Fermat's principle, we identify the optical Lagrangian and obtain the standard form of the ray equations in inhomogeneous optical media. Specializing to cylindrically symmetric media, we define the paraxial ray approximation through the exact first integral of the ray equation. This enables us not only to relate the constants of integration to the boundary conditions at the entry point of the ray into the medium, but more importantly furnishes a systematic way of introducing corrections to the paraxial approximation. We also exhibit power-law refractive index profiles, which result in periodic ray trajectories.
A. A. Rangwala, A. M. Ghodgaonkar, R. D. Tewari, "Paraxial ray tracing in inhomogeneous optical media," Optical Engineering 37(3), (1 March 1998). https://doi.org/10.1117/1.601937

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