Thin lens is the index of refraction of the lens material, and "R" and "R" are the radii of curvature of the two surfaces. For a thin lens, "d" is much smaller than one of the radii of curvature (either "R" or "R"). In these conditions, the last term of the Lensmaker's equation becomes negligible, and the focal length of a thin lens in air can be approximated by Here "R" is taken to be positive if the first surface is convex, and negative if the surface is concave. The signs are reversed for the back surface of the lens: "R" is
Luneburg lens one side, and formula_14 only on the other: The parameter formula_19 is a constant for any given ray, but differs between rays passing at different distances from the centre of the lens. For rays passing through the centre, it is zero. In some special cases, such as for Maxwell's fish-eye, this first order equation can be further integrated to give a formula for formula_14 as a function or formula_13. In general it provides the relative rates of change of formula_14 and formula_13, which may be integrated numerically to follow the path of the ray through the lens. Luneburg lens A