Material selection for GRIN-based achromatic doublets

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Abstract
We use first-order optical principles to examine the ability of gradient index (GRIN) lenses to correct chromatic aberrations. We consider radial GRIN lenses with flat surfaces, with a flat diffractive surface, and with curved surfaces. We model the GRIN material system as a locally varying, subwavelength blend of three materials. In this model, we demonstrate that the color-correcting properties of each lens type can be expressed solely in terms of the dispersion properties of the base materials. We find, at this level of approximation, that the material condition for a two-material GRIN achromat with curved surfaces is identical to that for a homogeneous doublet achromat comprised of the same two materials. For the more general case of three-material, ternary GRIN elements, we use the theory to develop a figure-of-merit-based optimization approach. This allows us to identify promising material combinations without first fabricating a GRIN element. The optimization approach can be applied to alternate GRIN geometries and arbitrary glass catalogs. We use our model to search a large, commercial glass catalog to identify the best achromatic glass combinations for the three different GRIN lenses described above. Significant numerical effort was required to identify which glass combinations performed best. Ternary glass combinations are necessary to achieve good achromatic performance for flat geometries. Diffraction combined with a graded-index enables improved color correction for the same optical power or nearly a factor of two increase in power for the same level of color correction. Glass pairs that perform well as an achromatic doublet also perform well chromatically when blended in a GRIN singlet.

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