Aplanatic afocal system using two GRIN lenses with spherical surfaces

The correction of aberrations is fundamental in optical systems, and one can use lenses with aspheric surfaces, free form surfaces, diffractive optics, metasurfaces, and gradient-index (GRIN) media. Here we propose to use a GRIN structure for correcting spherical aberration and coma in an afocal system. The complexity of having GRIN lenses in the system is balanced by the fact that the external lens surfaces are spherical, for both GRIN lenses are spherical and concentric to one common point, the intermediate focal point. For collimated light, each lens is free from spherical aberration. The whole system can also be free from coma aberration if the Abbe Sine Condition (ASC) is fulfilled. To create an aplanatic afocal system, it is necessary to have a constant ratio between the heights of the incoming and outgoing rays for all rays in the axial beam, so that the angular magnification of the system is preserved. The afocal system presented in this paper has been designed using geometrical optics. An analytical expression to satisfy the ASC has been derived by matching the central refractive index value of the outer surfaces lenses. The system consists of one positive and one negative GRIN lens with their inner surfaces being concentric to the intermediate focus. The angular magnification m is given by the ratio of the radius of curvature of the front surface of the first lens and that of the back surface of the second lens and in our example has been chosen as m = 2.