OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Stephen A. Burns
  • Vol. 23, Iss. 6 — Jun. 1, 2006
  • pp: 1513–1522

Non-Hermitian polarizers: a biorthogonal analysis

Tiberiu Tudor  »View Author Affiliations


JOSA A, Vol. 23, Issue 6, pp. 1513-1522 (2006)
http://dx.doi.org/10.1364/JOSAA.23.001513


View Full Text Article

Enhanced HTML    Acrobat PDF (130 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

The non-Hermitian operators of the ideal nonorthogonal multilayer optical polarizers are spectrally analyzed in the framework of skew-angular biorthonormal vector bases. It is shown that these polarizers correspond to skew projectors and their operators are generated by skew projectors, exactly as the canonical ideal polarizers correspond to Hermitian projectors. Thus the common feature of all the polarizers (Hermitian and non-Hermitian) is that their “nuclei” are (orthogonal or skew) projectors—the generating projectors. It is shown that if these nonorthogonal polarizers are looked upon as variable devices, two kinds of degeneracy may occur for suitable values of the inner parameter of the device: The corresponding operators may become normal (more precisely, Hermitian) or, on the contrary, very pathological—defective and singular. In the first case their eigenvectors and biorthogonal conjugate eigenvectors collapse into a unique pair of eigenvectors; in the second case their eigenvectors (as well as their biorthogonal conjugates) collapse into a single vector.

© 2006 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(260.5430) Physical optics : Polarization

ToC Category:
Physical Optics

History
Original Manuscript: July 18, 2005
Revised Manuscript: October 18, 2005
Manuscript Accepted: November 14, 2005

Citation
Tiberiu Tudor, "Non-Hermitian polarizers: a biorthogonal analysis," J. Opt. Soc. Am. A 23, 1513-1522 (2006)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-23-6-1513


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. M. V. Berry and S. Klein, "Geometric phases from stacks of crystal plates," J. Mod. Opt. 43, 165-180 (1996). [CrossRef]
  2. W. A. Shurcliff, Polarized Light (Harvard U. Press, 1962).
  3. B. Higman, Applied Group-Theoretic and Matrix Methods (Clarendon, 1955).
  4. C. Lanczos, Applied Analysis (Pitman, 1957).
  5. P. Lancaster and M. Tismenetsky, The Theory of Matrices (Academic, 1985).
  6. M. C. Pease, Methods of Matrix Algebra (Academic, 1965).
  7. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, "Atom waves in crystals of light," Phys. Rev. Lett. 77, 4980-4983 (1996). [CrossRef] [PubMed]
  8. M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystal," Proc. R. Soc. London, Ser. A 459, 1261-1292 (2003). [CrossRef]
  9. M. V. Berry and D. H. J. O'Dell, "Diffraction by volume gratings with imaginary potentials," J. Phys. A 31, 2093-2101 (1998). [CrossRef]
  10. W. D. Heiss, M. Müller, and I. Rotter, "Collectivity, phase transitions, and exceptional points in open quantum systems," Phys. Rev. E 58, 2894-2901 (1998). [CrossRef]
  11. T. Stehmann, W. D. Heiss, and F. G. Scholtz, "Observation of exceptional points in electronic circuits," J. Phys. A 37, 7813-7819 (2004). [CrossRef]
  12. M. Philipp, P. von Brentano, G. Pascovici, and A. Richter, "Frequency and width crossing of two interacting resonances in a microwave cavity," Phys. Rev. E 62, 1922-1926 (2000). [CrossRef]
  13. W. D. Heiss, "Repulsion of resonant states and exceptional points," Phys. Rev. E 61, 929-932 (2000). [CrossRef]
  14. F. Keck, H. J. Korsch, and S. Mossmann, "Unfolding a diabolic point: a generalized crossing scenario," J. Phys. A 36, 2125-2137 (2003). [CrossRef]
  15. M. V. Berry, "Physics of nonhermitian degeneracies," Czech. J. Phys. 55, 1039-1046 (2004).
  16. W. D. Heiss, "Exceptional points—their universal occurrence and their physical significance," Czech. J. Phys. 55, 1091-1099 (2004).
  17. A. P. Seyranian, O. N. Kirillov, and A. A. Mailybaev, "Coupling of eigenvalues of complex matrices at diabolic and exceptional points," J. Phys. A 38, 1723-1740 (2005). [CrossRef]
  18. T. Kato, Perturbation Theory for Linear Operators (Springer, 1966).
  19. A. Messiah, Mécanique Quantique (Dunod, 1964).
  20. T. Tudor, "Operational form of the theory of polarization optical devices: I. Spectral theory of the basic devices," Optik (Stuttgart) 114, 539-547 (2003).
  21. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1996).
  22. T. Tudor, "Generalized observables in polarization optics," J. Phys. A 36, 9567-9590 (2003). [CrossRef]
  23. M. V. Berry, "Mode degeneracies and the Petermann excess-noise factor for unstable lasers," J. Mod. Opt. 50, 63-81 (2003).
  24. E. B. Davies and J. T. Lewis, "An operatorial approach to quantum probability," Commun. Math. Phys. 17, 239-260 (1970). [CrossRef]
  25. M. V. Berry and M. R. Dennis, "Black polarization sandwiches as square roots of zero," J. Opt. A, Pure Appl. Opt. 6, S24-S25 (2004). [CrossRef]
  26. C. Whitney, "Pauli-algebraic operators in polarization optics," J. Opt. Soc. Am. 61, 1207-1213 (1971). [CrossRef]
  27. R. Bhandari, "Halfwave retarder for all polarization states," Appl. Opt. 36, 2799-2801 (1997). [CrossRef] [PubMed]

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.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited