Ray traces in a distributed-index planar microlens

In this paper ray-optical treatment in a planar microlens with 3-D distributed refractive index is presented. An integral form of the ray equation has been derived when the index distribution is expressed by the sum of decomposed functions of radial distance r and axial distance z as in actual fabricated devices. The integral equation is solved independently with respect to r and z, and an analytic form of the ray trace has been obtained in some cases. By applying the result to a planar microlens with index distribution which can be almost approximated by the profile assumed here, its optical properties such as focal length, numerical aperture, and spherical aberration have been estimated.