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. 24, Iss. 1 — Jan. 1, 2007
  • pp: 204–210

Pauli algebraic forms of normal and nonnormal operators

Tiberiu Tudor and Aurelian Gheondea  »View Author Affiliations

JOSA A, Vol. 24, Issue 1, pp. 204-210 (2007)

View Full Text Article

Enhanced HTML    Acrobat PDF (96 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



A unified treatment of the Pauli algebraic forms of the linear operators defined on a unitary linear space of two dimensions over the field of complex numbers C 1 is given. The Pauli expansions of the normal and nonnormal operators, unitary and Hermitian operators, orthogonal projectors, and symmetries are deduced in this frame. A geometrical interpretation of these Pauli algebraical results is given. With each operator, one can associate a generally complex vector, its Pauli axis. This is a natural generalization of the well-known Poincaré axis of some normal operators. A geometric criterion of distinction between the normal and nonnormal operators by means of this vector is established. The results are exemplified by the Pauli representations of the normal and nonnormal operators corresponding to some widespread composite polarization devices.

© 2006 Optical Society of America

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

ToC Category:
Physical Optics

Original Manuscript: March 17, 2006
Revised Manuscript: June 18, 2006
Manuscript Accepted: July 3, 2006

Tiberiu Tudor and Aurelian Gheondea, "Pauli algebraic forms of normal and nonnormal operators," J. Opt. Soc. Am. A 24, 204-210 (2007)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. M. V. Berry and S. Klein, 'Geometric phases from stacks of crystal plates,' J. Mod. Opt. 43, 165-180 (1996). [CrossRef]
  2. S. F. Li, 'Jones-matrix analysis with Pauli matrices: application to ellipsometry,' J. Opt. Soc. Am. A 17, 920-926 (2000). [CrossRef]
  3. C. Whitney, 'Pauli-algebraic operators in polarization optics,' J. Opt. Soc. Am. 61, 1207-1213 (1971). [CrossRef]
  4. R. Bhandari, 'Halfwave retarder for all polarization states,' Appl. Opt. 36, 2799-2801 (1997). [CrossRef] [PubMed]
  5. R. Bhandari and G. D. Love, 'Polarization eigenmodes of a QHQ retarder -- some new features,' Opt. Commun. 110, 479-484 (1994). [CrossRef]
  6. R. Bhandari, 'Interferometry without beam splitters -- a sensitive technique for spinor phases,' Phys. Lett. A 180, 21-24 (1993). [CrossRef]
  7. R. Bhandari, 'Observation of Dirac singularities with light polarization. II,' Phys. Lett. A 171, 267-270 (1992). [CrossRef]
  8. R. Bhandari, 'Evolution of light beams in polarization and direction,' Phys. Lett. B 175, 111-122 (1991).
  9. C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, 1997), Vol. 1.
  10. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics III (Quantum Mechanics) (Addison-Wesley, 1965).
  11. S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Clarendon, 2003).
  12. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1996).
  13. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2002).
  14. M. V. Berry, 'Physics of nonhermitian degeneracies,' Czech. J. Phys. 55, 1039-1046 (2004). [CrossRef]
  15. W. D. Heiss, 'Exceptional points -- their universal occurrence and their physical significance,' Czech. J. Phys. 55, 1091-1099 (2004). [CrossRef]
  16. 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]
  17. M. V. Berry, 'Mode degeneracies and the Petermann excess-noise factor for unstable lasers,' J. Mod. Opt. 50, 63-81 (2003). [CrossRef]
  18. M. V. Berry and M. R. Dennis, 'The optical singularities of birefringent dichroic chiral crystals,' Proc. R. Soc. London, Ser. A 459, 1261-1292 (2003). [CrossRef]
  19. M. V. Berry and D. H. J. O'Dell, 'Diffraction by volume gratings with imaginary potentials,' J. Phys. A 31, 2093-2101 (1998). [CrossRef]
  20. 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]
  21. T. Stehmann, W. D. Heiss, and F. G. Scholtz, 'Observation of exceptional points in electronic circuits,' J. Phys. A 37, 7813-7819 (2004). [CrossRef]
  22. 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]
  23. W. D. Heiss, 'Repulsion of resonant states and exceptional points,' Phys. Rev. E 61, 929-932 (2000). [CrossRef]
  24. F. Keck, H. J. Korsch, and S. Mossmann, 'Unfolding a diabolic point: a generalized crossing scenario,' J. Phys. A 36, 2125-2137 (2003). [CrossRef]
  25. T. Tudor, 'Operatorial form of the theory of polarization optical devices: I. Spectral theory of the basic devices,' Optik (Stuttgart) 114, 539-547 (2003). [CrossRef]
  26. T. Tudor, 'Operatorial form of the theory of polarization optical devices: II. Spectral theory of the composite devices,' Optik (Stuttgart) 115, 173-180 (2004). [CrossRef]
  27. S.-Y. Lu and R. A. Chipman, 'Homogeneous and inhomogeneous Jones matrices,' J. Opt. Soc. Am. A 11, 766-773 (1994). [CrossRef]
  28. T. Tudor, 'Generalized observables in polarization optics,' J. Phys. A 36, 9567-9590 (2003). [CrossRef]
  29. E. B. Davies and J. T. Lewis, 'An operatorial approach to quantum probability,' Commun. Math. Phys. 17, 239-260 (1970). [CrossRef]
  30. P. Busch, P. J. Lathi, and P. Mittelstaedt, The Quantum Theory of Measurement (Springer, 1996).
  31. M. de Muynck, 'An alternative to the Lüders generalization of the von Neumann projection and its interpretation,' J. Phys. A 31, 431-444 (1998). [CrossRef]
  32. Ch. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).
  33. R. Simon and N. Mukunda, 'Universal SU(2) gadget for polarization optics,' Phys. Lett. A 138, 474-480 (1989). [CrossRef]
  34. A. S. Marathay, 'Operator formalism in the theory of partial polarization,' J. Opt. Soc. Am. 55, 969-980 (1965).
  35. S. Pancharatnam, 'The propagation of light in absorbing biaxial crystals. Part I. Theoretical,' Proc. Indian Acad. Sci., Sect. A 42, 86-109 (1955).
  36. S. Pancharatnam, Collected Works of S. Pancharatnam (Oxford U. Press, 1975).
  37. W. A. Shurcliff, Polarized Light (Harvard U. Press, 1962).
  38. M. Richartz, H.-Y. Hsü, 'Analysis of elliptical polarization,' J. Opt. Soc. Am. 39, 136-157 (1949). [CrossRef]
  39. S. Pancharatnam, 'Achromatic combinations of birefringent plates. Part II: An achromatic quarter-wave plate,' Proc. Indian Acad. Sci., Sect. A 41, 137-144 (1955).
  40. S. Baskal, E. Georgieva, Y. S. Kim, and M. E. Noz, 'Lorentz group in classical ray optics,' J. Opt. B: Quantum Semiclassical Opt. 6, S455-S472 (2004). [CrossRef]
  41. R. Simon and N. Mukunda, 'Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,' J. Opt. Soc. Am. A 15, 2147-2155 (1998). [CrossRef]
  42. G. S. Agarwal, 'SU(2) structure of the Poincaré sphere for light beams with orbital angular momentum,' J. Opt. Soc. Am. A 16, 2914-2916 (1999). [CrossRef]
  43. D. Han, Y. S. Kim, and M. E. Noz, 'Jones-matrix formalism as a representation of the Lorentz group,' J. Opt. Soc. Am. A 14, 2290-2298 (1997). [CrossRef]
  44. E. Georgieva, 'Slide-rule-like property of Wigner's little groups and cyclic S matrices for multilayer optics,' Phys. Rev. E 68, 026606 (2003). [CrossRef]
  45. J. J. Monzón and L. L. Sanchez-Soto, 'Multilayer optics as an analog computer for testing special relativity,' Phys. Lett. A 262, 18-26 (1999). [CrossRef]
  46. D. Han, Y. S. Kim, and M. E. Noz, 'Interferometers and decoherence matrices,' Phys. Rev. E 61, 5907-5913 (2000). [CrossRef]
  47. D. Han, Y. S. Kim, and M. E. Noz, 'Linear canonical transformation of coherent and squeezed states in the Wigner phase space,' Phys. Rev. A 37, 807-814 (1988). [CrossRef] [PubMed]
  48. S. Baskal and Y. S. Kim, 'The language of Einstein spoken by optical instruments,' Opt. Spectrosc. 99, 443-446 (2005). [CrossRef]

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