Numerical evaluation of the Luneburg integral and ray tracing

The Luneburg integral has many applications in optics and optoelectronics, among which is determination of the refractive-index profile of a Luneburg lens with a full or nonfull aperture. Consequently, computationally efficient and accurate methods for evaluating this integral represent an important challenge. An alternative approach to numerical evaluation of the Luneburg integral that is five times faster than existing methods is described. Several improvements in the ray-tracing procedure in gradient-index media are also presented. A combination of these methods increases the speed of ray tracing through the generalized Luneburg lens by as many as 2 orders of magnitude compared with earlier algorithms. The precision of our method can be easily controlled. © 1996 Optical Society of America.