OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 27, Iss. 12 — Dec. 1, 2010
  • pp: 2558–2562

Generalization of ray tracing in a linear inhomogeneous anisotropic medium: a coordinate-free approach

Alireza Akbarzadeh and Aaron J. Danner  »View Author Affiliations

JOSA A, Vol. 27, Issue 12, pp. 2558-2562 (2010)

View Full Text Article

Enhanced HTML    Acrobat PDF (285 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



The Hamiltonian of an optical medium is important in both the design and the description of optical devices in the geometrical optics limit. The results calculated in this article show in detail how ray tracing in anisotropic materials in arbitrary coordinate systems and curved spaces can be carried out. Writing Maxwell’s equations in the most general form, we derive a coordinate-free form for the eikonal equation and hence the Hamiltonian of a general purpose medium. The expression works for both orthogonal and non-orthogonal coordinate systems, and we show how it can be simplified for biaxial and uniaxial media in orthogonal coordinate systems. In order to show the utility of the equations in a real case, we study both the isotropic and the uniaxially transmuted birefringent Eaton lens and derive the ray trajectories in spherical coordinates for each case.

© 2010 Optical Society of America

OCIS Codes
(080.2710) Geometric optics : Inhomogeneous optical media
(080.2740) Geometric optics : Geometric optical design
(160.1190) Materials : Anisotropic optical materials
(260.1440) Physical optics : Birefringence
(080.5692) Geometric optics : Ray trajectories in inhomogeneous media
(260.2710) Physical optics : Inhomogeneous optical media

ToC Category:
Geometric Optics

Original Manuscript: August 16, 2010
Manuscript Accepted: September 30, 2010
Published: November 8, 2010

Alireza Akbarzadeh and Aaron J. Danner, "Generalization of ray tracing in a linear inhomogeneous anisotropic medium: a coordinate-free approach," J. Opt. Soc. Am. A 27, 2558-2562 (2010)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef] [PubMed]
  2. U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009). [CrossRef]
  3. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]
  4. U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110–112 (2009). [CrossRef]
  5. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008). [CrossRef]
  6. N. A. Mortensen, “Prospects for poor-man’s cloaking with low-contrast all-dielectric optical elements,” J. Eur. Opt. Soc. Rapid Publ. 4, 09008 (2009). [CrossRef]
  7. J. E. Eaton, “On spherically symmetric lenses,” IRE Trans. Antennas Propag. 4, 66–71 (1952).
  8. T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” New J. Phys. 10, 115038 (2008). [CrossRef]
  9. O. N. Stavroudis, “Ray-tracing formulas for uniaxial crystals,” J. Opt. Soc. Am. 52, 187–191 (1962). [CrossRef]
  10. L. Quan-Ting, “Simple ray tracing formulas for uniaxial optical crystals,” Appl. Opt. 29, 1008–1010 (1990). [CrossRef]
  11. S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. I. Algorithms,” J. Opt. Soc. Am. A 10, 2371–2382 (1993). [CrossRef]
  12. S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A 10, 2383–2393 (1993). [CrossRef]
  13. M. Sluijter, D. K. G. de Boer, and J. J. M. Braat, “General polarized ray-tracing method for inhomogeneous uniaxially anisotropic media,” J. Opt. Soc. Am. A 25, 1260–1273 (2008). [CrossRef]
  14. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006). [CrossRef] [PubMed]
  15. Q. Cheng-Wei, Y. Hai-Ying, L. Le-Wei, S. Zouhdi, and Y. Tat-Soon, “Backward waves in magnetoelectrically chiral media: propagation, impedance, and negative refraction,” Phys. Rev. B 75, 155120 (2007). [CrossRef]
  16. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).

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.


Fig. 1 Fig. 2 Fig. 3

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited