OSA's Digital Library

Applied Optics

Applied Optics


  • Vol. 35, Iss. 7 — Mar. 1, 1996
  • pp: 1011–1014

Numerical evaluation of the Luneburg integral and ray tracing

Gleb Beliakov  »View Author Affiliations

Applied Optics, Vol. 35, Issue 7, pp. 1011-1014 (1996)

View Full Text Article

Enhanced HTML    Acrobat PDF (167 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

Original Manuscript: August 28, 1995
Published: March 1, 1996

Gleb Beliakov, "Numerical evaluation of the Luneburg integral and ray tracing," Appl. Opt. 35, 1011-1014 (1996)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. S. Doric, E. Munro, “General solution of the nonfullaperture Luneburg lens problem,” J. Opt. Soc. Am. 73, 1083–1086 (1983).
  2. J. Sochacki, C. Gomez-Reino, “Nonfull-aperture Luneburg lenses: a novel solution,” Appl. Opt. 24, 1371–1373 (1985).
  3. W. H. Southwell, “Ray tracing in gradient-index media,” J. Opt. Soc. Am. 72, 908–911 (1982).
  4. G. Beliakov, “Reconstruction of optical characteristics of waveguide lenses by the use of ray tracing,” Appl. Opt. 33, 3401–3404 (1994).
  5. G. Beliakov, “Study of mathematical model of the refractive index reconstruction of a smooth thin film waveguide by the use of ray tracing,” Ph.D. dissertation (Russian Peoples’ Friendship University, Moscow, 1992), in Russian.
  6. L. Montagnino, “Ray tracing in inhomogeneous media,” J. Opt. Soc. Am. 58, 1667–1668 (1968).
  7. A. Sharma, D. V. Kumar, A. K. Ghatak, “Tracing rays through graded-index media: a new method,” Appl. Opt. 21, 984–987 (1982).
  8. S. Doric, E. Munro, “Improvements of the ray trace through the generalized Luneburg lens,” Appl. Opt. 22, 443–445 (1983).
  9. J. Sochacki, D. Rogus, C. Gomez-Reino, “Paraxial designing of planar waveguide. Variable-index focusing elements: Part 1. Lenses of circular symmetry,” Fiber Integr. Opt. 8, 121–127 (1989).
  10. J. Sochacki, “Functional approach to the Luneburg's integral for the planar Luneburg lenses design,” IEEE J. Lightwave Technol. LT-3, 684–687 (1985).
  11. J. Sochacki, J. R. Flores, R. Staronski, C. Gomez-Reino, “Improvements in the computation of refractive-index profiles for the generalized Luneburg lens,” J. Opt. Soc. Am. A 8, 1248–1255 (1991).
  12. E. Colombini, “Index-profile computation for the generalized Luneburg lens,” J. Opt. Soc. Am. 71, 1403–1405 (1981).
  13. W. H. Southwell, “Index profiles for generalized Luneburg lenses and their use in planar optical waveguides,” J. Opt. Soc. Am. 67, 1010–1014 (1977).
  14. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1989), p. 122.
  15. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Vol. 55 of the Appl. Math. Ser. (Dover, New York, 1972), p. 916.

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited