OSA's Digital Library

Applied Optics

Applied Optics


  • Editor: Joseph N. Mait
  • Vol. 52, Iss. 2 — Jan. 10, 2013
  • pp: 212–218

Transfer matrix for treating stratified media including birefringent crystals

Thomas Essinger-Hileman  »View Author Affiliations

Applied Optics, Vol. 52, Issue 2, pp. 212-218 (2013)

View Full Text Article

Enhanced HTML    Acrobat PDF (271 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



Birefringent crystals are extensively used to manipulate polarized light. The generalized transfer matrix developed allows efficient calculation of the full polarization state of light transmitted through and reflected by a stack of arbitrarily many discrete layers of isotropic and birefringent materials at any frequency and angle of incidence. The matrix of a uniaxial birefringent crystal with arbitrary rotation is calculated, along with its reduction to the matrix of an isotropic medium. This method is of great practical importance where tight control of systematic effects is needed in optical systems employing birefringent crystals, one example being wave plates used by cosmic microwave background polarimetry with wide field-of-view telescopes.

© 2013 Optical Society of America

OCIS Codes
(160.1190) Materials : Anisotropic optical materials
(220.0220) Optical design and fabrication : Optical design and fabrication
(230.0230) Optical devices : Optical devices
(260.0260) Physical optics : Physical optics
(260.1440) Physical optics : Birefringence

ToC Category:
Physical Optics

Original Manuscript: October 11, 2012
Revised Manuscript: November 25, 2012
Manuscript Accepted: November 30, 2012
Published: January 8, 2013

Thomas Essinger-Hileman, "Transfer matrix for treating stratified media including birefringent crystals," Appl. Opt. 52, 212-218 (2013)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. D. H. Goldstein, Polarized Light (CRC, 2011).
  2. J. Tinbergen, Astronomical Polarimetry (Cambridge University, 1996).
  3. D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).
  4. S. Pancharatnam, “Achromatic combinations of birefringent plates,” Proc. Indian Acad. Sci. A 41, 137–144 (1955).
  5. F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés,” Ann. Phys. 5, 596–640 (1950).
  6. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
  7. E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1987).
  8. D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972). [CrossRef]
  9. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979). [CrossRef]
  10. T. M. Essinger-Hileman, “Probing inflationary cosmology: the atacama B-mode search (ABS),” Ph.D. thesis (Princeton University, 2011).
  11. Note that this ordering of the Pauli matrices is different from that often used in the literature: σI=(1001)σQ=(100−1)σU=(0110)σV=(0−ii0).
  12. W. C. Jones, T. E. Montroy, B. P. Crill, C. R. Contaldi, T. S. Kisner, A. E. Lange, C. J. MacTavish, C. B. Netterfield, and J. E. Ruhl, “Instrumental and analytic methods for bolometric polarimetry,” Astron. Astrophys. 470, 771–785 (2007). [CrossRef]
  13. S. A. Bryan, T. E. Montroy, and J. E. Ruhl, “Modeling dielectric half-wave plates for cosmic microwave background polarimetry using a Mueller matrix formalism,” Appl. Opt. 49, 6313–6323 (2010). [CrossRef]
  14. H. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw-Hill, 1983).
  15. G. Szivessy, Handbuch der Physik, Vol. 20 (Julius Springer, 1928), Chap. 11, p. 715.

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.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited