Semi-analytical finite ray-tracing through the quadratic symmetric GRIN lens

The propagation of light within a gradient index (GRIN) media can be analyzed with the use of differential equations for a given non-homogenous refractive index profile. Numerical methods are often necessary to perform ray-tracing in GRIN media; however, analytical solutions exist for several types of GRIN lenses. In this paper, paraxial and non-paraxial differential equations are derived to calculate the ray path in a GRIN lens. It is shown that the paraxial equation has an analytical solution for a GRIN media with a quadratic profile within the paraxial region. The analytical solution can be obtained by using Legendre polynomials or by the Frobenius method involving a power series. Using the Legendre or Frobenius solution, one can calculate the refractive indices along the ray path. A new recursive relationship is proposed to map the trajectory of light at finite heights. To illustrate the finite ray-tracing method utilizing a non-paraxial differential equation, two lenses (with spherical and elliptical iso-indicial contours) are considered. The lenses' back focal distances, for rays entering the lenses at varying finite heights, are calculated. For each lens, its spherical aberration is estimated. The effective focal length and the shape of the principal surface are also obtained. The accuracy of the results is then compared to the numerical ray-tracing using an optical design software, Zemax OpticStudio. The predicted spherical aberration for the spherical lens differs from numerical ray-tracing by less than λ14 at the marginal zone, while the error for the effective focal length is less than λ100.