Geometrical-Optics Inspired Solution
Analytical expressions that describe the behavior of radial, axial, and spherical GRIN profiles have been extensively studied [4, 3, 50]. These expressions use paraxial approximations but provide insight into the physics of GRIN lenses. This section will describe the theory behind color correction for simple index profiles. In addition, the geometrical parameters to provide color correction for a given material will be provided and trade-offs between performance and complexity will be discussed.
The simplest GRIN profile is one which possess a quadratic variation in index along the radial direction. Consider such a radial GRIN whose index distribution is given by
where ar controls concavity and strength of the GRIN profile and D is the diameter of the lens. Since this index distribution has no axial dependence, a single value nS can be used to describe the effective index on the front and back lens surfaces. The mean index can be used for simplicity and occurs at a radial position -Jl/ 3 D/2. The mean index is then assigned to nS. It follows then that the surfaces have an Abbe number defined in the usual way
while the Abbe number of the GRIN is defined as 
where the subscripts 1, 2, 3 refer to the index values at the short, center, and long wavelengths, respectively. By combining the powers and dispersions of the surfaces and GRIN, one can form a new aggregate lens system and can prescribe it to possess a desired effective power and dispersion. The achromatic system of equations for this lens is given as
which are similar to those of the traditional achromatic doublet (Eq. 6.41). In this convention, fT and SfT are the target (i.e., total) system power and short-long wavelength focal drift, respectively, while fS and fG are the surface and GRIN powers at the central wavelength, respectively. Recall that the focal drift is inversely proportional to the target Abbe number SfT = fT/vT . Solving this linear system of equations in the achromatic limit (i.e., SfT ^0) yields solutions for the surface and GRIN powers:
The requisite surface power can then be found by
Next, by using the lens maker’s equation, expressions for the individual surface radii of curvature can be found:
Likewise, by substituting in an expression for the GRIN power , the lens thickness can be determined.
which results in a thickness T given by
Note that while the thickness is explicitly defined, there is some flexibility in values for the surface radii since one can be specified and the other solved for. Geometries such as meniscus, biconvex, plano-convex, and convex-plano are all accessible in an achromatic radial GRIN. An axial GRIN, however, can offer a different range of solutions.