OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Stephen A. Burns
  • Vol. 23, Iss. 12 — Dec. 1, 2006
  • pp: 3133–3138

Two solvable problems of planar geometrical optics

Francesco Borghero and George Bozis  »View Author Affiliations

JOSA A, Vol. 23, Issue 12, pp. 3133-3138 (2006)

View Full Text Article

Enhanced HTML    Acrobat PDF (86 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



In the framework of geometrical optics we consider a two-dimensional transparent inhomogeneous isotropic medium (dispersive or not). We show that (i) for any family belonging to a certain class of planar monoparametric families of monochromatic light rays given in the form f ( x , y ) = c of any definite color and satisfying a differential condition, all the refractive index profiles n = n ( x , y ) allowing for the creation of the given family can be found analytically (inverse problem) and that (ii) for any member of a class of two-dimensional refractive index profiles n = n ( x , y ) satisfying a differential condition, all the compatible families of light rays can be found analytically (direct problem). We present appropriate examples.

© 2006 Optical Society of America

OCIS Codes
(080.0080) Geometric optics : Geometric optics
(080.2710) Geometric optics : Inhomogeneous optical media
(080.2720) Geometric optics : Mathematical methods (general)

Original Manuscript: February 21, 2006
Revised Manuscript: June 14, 2006
Manuscript Accepted: June 16, 2006

Francesco Borghero and George Bozis, "Two solvable problems of planar geometrical optics," J. Opt. Soc. Am. A 23, 3133-3138 (2006)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. J. A. Arnaud, "Application of the mechanical theory of light to fiber optics," J. Opt. Soc. Am. 65, 174-181 (1975). [CrossRef]
  2. M. A. Hindy, "Refractive-index profile in fiber optics," Microwave Opt. Technol. Lett. 29, 252-256 (2001). [CrossRef]
  3. M. S. Dineen, "Extended summaries of several articles from the March Optics and Photonics News special issue on fiber optics," J. Opt. Netw. 1, 143-148 (2002).
  4. P. J. Sands, "Inhomogeneous lenses. VI. Derivative of paraxial coefficients," J. Opt. Soc. Am. 63, 1210-1216 (1973). [CrossRef]
  5. K. Maeda and J. Hamasaki, "A method of determining the refractive-index profile of a lenslike medium," J. Opt. Soc. Am. 67, 1672 (1977). [CrossRef]
  6. Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media, Springer Series on Wave Phenomena (Springer-Verlag, 1990), Vol. 6. [CrossRef]
  7. J. A. Grzesik, "Focusing properties of a three-parameter class of oblate, Luneburg-like inhomogeneous lenses," J. Electromagn. Waves Appl. 19, 1005-1019 (2005). [CrossRef]
  8. B. Wang, P. J. Bos, and C. D. Hoke, "Light propagation in variable-refractive-index materials with liquid-crystal-infiltrated microcavities," J. Opt. Soc. Am. A 20, 2123-2130 (2003). [CrossRef]
  9. M.-H. Wei and W.-C. Chen, "Theoretical analysis on the refractive-index distribution and bandwidth of gradient-index polymer optical fibers from a centrifugal field," Appl. Opt. 42, 2174-2180 (2003). [CrossRef] [PubMed]
  10. E. Acosta, D. Vazquez, L. Garner, and G. Smith, "Tomographic method for measurement of the gradient refractive index of the crystalline lens. I. The spherical fish lens," J. Opt. Soc. Am. A 22, 424-433 (2005). [CrossRef]
  11. K. S. Kunz, "Propagation of microwaves between a parallel pair of doubly curved conducting surfaces," J. Appl. Phys. 25, 642-653 (1954). [CrossRef]
  12. J. C. Minãno, "Refractive-index distribution in two-dimensional geometry for a given one-parameter manifold of rays," J. Opt. Soc. Am. A 2, 1821-1825 (1985). [CrossRef]
  13. G. Beliakov and D. Y. C. Chan, "Analysis of inhomogeneous optical systems by the use of ray tracing. I. Planar systems," Appl. Opt. 36, 5303-5309 (1997). [CrossRef] [PubMed]
  14. R. F. Rinehart, "A solution of the problem of rapid scanning for radar antennae," J. Appl. Phys. 19, 860-862 (1948). [CrossRef]
  15. M. Born and E. Wolf, Principles of Optics, 7th ed. revised (Cambridge U. Press, 2002).
  16. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).
  17. S. Nemoto and T. Makimoto, "Refractive-index distribution for a prescribed ray path," J. Opt. Soc. Am. 69, 450-459 (1979). [CrossRef]
  18. F. Borghero and G. Bozis, "A two-dimensional inverse problem of geometrical optics," J. Phys. A 38, 175-184 (2005). [CrossRef]
  19. A. Fletcher, T. Murphy, and A. Young, "Solutions of two optical problems," Proc. R. Soc. London, Ser. A 223, 216-225 (1954). [CrossRef]
  20. G. Toraldo di Francia, "A family of perfect configuration lenses of revolution," Opt. Acta 1, 157-163 (1955). [CrossRef]
  21. S. Grigoriadou, G. Bozis, and B. Elmabsout, "Solvable cases of Szebehely's equation," Celest. Mech. Dyn. Astron. 74, 211-221 (1999). [CrossRef]
  22. J. Sochacki, "Exact analytical solution of the generalized Luneburg lens problem," J. Opt. Soc. Am. 73, 789-795 (1983). [CrossRef]
  23. J. C. Maxwell, The Scientific Papers of James Clerk Maxwell, W.D.Niven, ed. (Cambridge U. Press, 1890), Vol. I, Solutions of Problems no. 2, pp. 76-79.
  24. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972).
  25. J. Hietarinta, "Direct methods for the search of the second invariant," Phys. Rep. 147, 87-154 (1987). [CrossRef]
  26. G. Contopoulos and G. Bozis, "Complex force fields and complex orbits," Acta Geogr. Sin. 8, 1-14 (2000).

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.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited