From the theory of aplanatic points of a sphere it follows that there exist two points such that a spherical wave from first point is transformed into a spherical wave diverging apparently from second.
1. APLANATIC POINTS OF A SPHERE
Let us take a ball lens with radius R and index of refraction n0 immersed in the medium of refractive index n1. The center of the sphere is take at z=0. From the theory of aplanatic points of a sphere (see Luneburg's Mathematical theory of Optics) it follows that there exist two points z0 and z1 on the z-axis such that a spherical wave from z0 is transformed into spherical wave diverging apparently from z1.
It follows that aplanatic ball lens can change the beam divergence without introduction of aberrations. This principle has been used by many optical designers. We can apply the same reasoning to rods and transformation of cylindrical diverging wavefront using aplanatic points of the rods.
Aplanatic ball lenses have been traditionally used to make high power microscope objectives and endoscope objectives. More recently aplanatic ball lenses have been used to reduce beam divergence from optical fibers. Similarly, aplanatic rod lenses could be used to reduce beam divergence from different planar light sources like LEDs, laser diodes or planar waveguides.
1. R. Y. Luneberg, "Mathematical Theory of Optics", University of California Press, Berkeley, 1964.