It is notoriously difficult to perform exact ray tracing through a GRIN lens, which is done numerically using optical design software. Even exact paraxial ray tracing equations are not available for GRIN lenses. One could also use an approximate method, where the ray path within the GRIN lens is assumed to be parabolic.17 However, it would be desirable to have an exact method for paraxial ray tracing so that all optical characteristics of the lens can be found in closed form. Due to the linear dependence of the iso-indicial contours radius on the normalized axial distance, , we are able to derive a closed-form solution for paraxial ray tracing in the geometry-invariant GRIN lens. Paraxial ray tracing is based on two main equations.25 According to the first one we have Display Formula
(17)where and are respectively the refractive indices before and after the interface surface, and are the angles of the incident and refracted rays, is the height of the ray at the surface, and is the radius of the surface. For the next surface located at the axial distance from the first one, the height of the incident ray, , is obtained by Display Formula
(18)Following the same approach used to derive Eq. (11), we rewrite the axial thickness of the infinitely thin shells as , then Eq. (18) becomes Display Formula
(19)Using Eq. (3) and substituting the definition of the derivative from Eq. (19) into Eq. (17) results in Display Formula
(20)Finally considering and as continuous functions of , we expand Eq. (20) around the origin for and keep only the first order terms, which gives us Display Formula
(21)Solving Eq. (21) for the anterior and posterior hemispheres (where corresponds to and , respectively) leads to a general ray equation: Display Formula
(22)where is Gaussian (ordinary) hypergeometric function and Display FormulaDisplay FormulaDisplay FormulaDisplay Formulawhere and are respectively the angle and the height of the incident ray after refraction by the anterior surface of the lens and the expressions for and are given in the appendix. Using Eqs. (19) and (22), the angle of the ray can be found as Display Formula
(23)Both the height and the angle of the ray are necessary to describe the optical properties of the GRIN lens, which is the main goal of Secs. 6 and 7.