OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 23 — Nov. 12, 2007
  • pp: 15175–15186

Anisotropy without tensors: a novel approach using geometric algebra

Sérgio A. Matos, Marco A. Ribeiro, and Carlos R. Paiva  »View Author Affiliations

Optics Express, Vol. 15, Issue 23, pp. 15175-15186 (2007)

View Full Text Article

Enhanced HTML    Acrobat PDF (651 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



The most widespread approach to anisotropic media is dyadic analysis. However, to get a geometrical picture of a dielectric tensor, one has to resort to a coordinate system for a matrix form in order to obtain, for example, the index-ellipsoid, thereby obnubilating the deeper coordinate-free meaning of anisotropy itself. To overcome these shortcomings we present a novel approach to anisotropy: using geometric algebra we introduce a direct geometrical interpretation without the intervention of any coordinate system. By applying this new approach to biaxial crystals we show the effectiveness and insight that geometric algebra can bring to the optics of anisotropic media.

© 2007 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(160.1190) Materials : Anisotropic optical materials
(260.1180) Physical optics : Crystal optics
(260.1440) Physical optics : Birefringence
(260.2110) Physical optics : Electromagnetic optics

ToC Category:
Physical Optics

Original Manuscript: August 22, 2007
Revised Manuscript: October 8, 2007
Manuscript Accepted: October 26, 2007
Published: November 1, 2007

Sérgio A. Matos, Marco A. Ribeiro, and Carlos R. Paiva, "Anisotropy without tensors: a novel approach using geometric algebra," Opt. Express 15, 15175-15186 (2007)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. J. C. Maxwell, A Treatise on Electricity and Magnetism (Dover, New York, 1954) Vol.  2, p. 443.
  2. M. Born and E. Wolf, Principles of Optics, 7th expanded ed., (Cambridge University Press, Cambridge, 1999) pp. 790-852.
  3. A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley Classics Library, Hoboken, 2003).
  4. I. Richter, P. C. Sun, F. Xu, and Y. Fainman, "Design considerations of form birefringence microstructures," Appl. Opt. 34, 2421-2429 (1995). http://www.opticsinfobase.org/abstract.cfm?URI=ao--34-14-2421 [CrossRef] [PubMed]
  5. U. Levy, C. H. Tsai, L. Pang, and Y. Fainman, "Engineering space-time variant inhomogeneous media for polarization control," Opt. Lett. 29, 1718-1720 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=ol-29-15-1718 [CrossRef] [PubMed]
  6. 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]
  7. I. V. Lindell, Differential Forms in Electromagnetics, (IEEE Press, Piscataway, 2004) pp. 123-161.
  8. I. V. Lindell, Methods for Electromagnetic Field Analysis, (IEEE Press, Piscataway, 2nd ed., 1995) pp. 17-52.
  9. D. Hestenes, New Foundations for Classical Mechanics (Kluwer Academic Publishers, Dordrecht, 2nd ed., 1999).
  10. P. Lounesto, Clifford Algebras and Spinors (Cambridge University Press, Cambridge, 2nd ed., 2001). [CrossRef]
  11. C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University Press, Cambridge, 2003).
  12. L. Dorst, D. Fontijne, and S. Mann, Geometric Algebra for Computer Science - An Object-oriented Approach to Geometry (Elsevier - Morgan Kaufmann Publishers, San Francisco, 2007).
  13. D. Hestenes, "Oersted Medal Lecture 2002: Reforming the mathematical language of physics," Am. J. Phys. 71, 104-121 (2003). http://modelingnts.la.asu.edu/pdf/OerstedMedalLecture.pdf. [CrossRef]
  14. P. Puska, "Covariant isotropic constitutive relations in Clifford’s geometric algebra," Progress in Electromagnetics Research - PIER 32, 413-428 (2001). http://ceta.mit.edu/PIER/pier32/16.00080116.puska.pdf. [CrossRef]
  15. C. R. Paiva and M. A. Ribeiro, "Doppler shift from a composition of boosts with Thomas rotation: A spacetime algebra approach," J. Electromagn. Waves Appl. 20, 941-953 (2006). http://dx.doi.org/10.1163/156939306776149806. [CrossRef]
  16. D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, (Kluwer Academic Publishers, Dordrecht, 1984) pp. 63-136.
  17. H. C. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach, (McGraw-Hill, Singapore, 1985) pp. 215-216.
  18. M. A. Ribeiro, S. A. Matos, and C. R. Paiva, "A geometric algebra approach to anisotropic media," in Proc. 2007 IEEE Antennas and Propagation Society International Symposium, Honolulu, Hawaii, USA, (2007), pp. 4032-4035.
  19. J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, (Oxford University Press, Oxford, 1985) pp. 24-25.

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.

Supplementary Material

» Media 1: MOV (495 KB)     
» Media 2: MOV (77 KB)     

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited