Abstract

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 C1 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

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  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]

2005 (2)

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]

S. Baskal and Y. S. Kim, 'The language of Einstein spoken by optical instruments,' Opt. Spectrosc. 99, 443-446 (2005).
[CrossRef]

2004 (5)

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]

M. V. Berry, 'Physics of nonhermitian degeneracies,' Czech. J. Phys. 55, 1039-1046 (2004).
[CrossRef]

W. D. Heiss, 'Exceptional points -- their universal occurrence and their physical significance,' Czech. J. Phys. 55, 1091-1099 (2004).
[CrossRef]

T. Stehmann, W. D. Heiss, and F. G. Scholtz, 'Observation of exceptional points in electronic circuits,' J. Phys. A 37, 7813-7819 (2004).
[CrossRef]

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]

2003 (6)

T. Tudor, 'Generalized observables in polarization optics,' J. Phys. A 36, 9567-9590 (2003).
[CrossRef]

F. Keck, H. J. Korsch, and S. Mossmann, 'Unfolding a diabolic point: a generalized crossing scenario,' J. Phys. A 36, 2125-2137 (2003).
[CrossRef]

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]

M. V. Berry, 'Mode degeneracies and the Petermann excess-noise factor for unstable lasers,' J. Mod. Opt. 50, 63-81 (2003).
[CrossRef]

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]

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]

2000 (4)

S. F. Li, 'Jones-matrix analysis with Pauli matrices: application to ellipsometry,' J. Opt. Soc. Am. A 17, 920-926 (2000).
[CrossRef]

D. Han, Y. S. Kim, and M. E. Noz, 'Interferometers and decoherence matrices,' Phys. Rev. E 61, 5907-5913 (2000).
[CrossRef]

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]

W. D. Heiss, 'Repulsion of resonant states and exceptional points,' Phys. Rev. E 61, 929-932 (2000).
[CrossRef]

1999 (2)

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]

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]

1998 (4)

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]

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]

M. V. Berry and D. H. J. O'Dell, 'Diffraction by volume gratings with imaginary potentials,' J. Phys. A 31, 2093-2101 (1998).
[CrossRef]

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]

1997 (2)

1996 (1)

M. V. Berry and S. Klein, 'Geometric phases from stacks of crystal plates,' J. Mod. Opt. 43, 165-180 (1996).
[CrossRef]

1994 (2)

S.-Y. Lu and R. A. Chipman, 'Homogeneous and inhomogeneous Jones matrices,' J. Opt. Soc. Am. A 11, 766-773 (1994).
[CrossRef]

R. Bhandari and G. D. Love, 'Polarization eigenmodes of a QHQ retarder -- some new features,' Opt. Commun. 110, 479-484 (1994).
[CrossRef]

1993 (1)

R. Bhandari, 'Interferometry without beam splitters -- a sensitive technique for spinor phases,' Phys. Lett. A 180, 21-24 (1993).
[CrossRef]

1992 (1)

R. Bhandari, 'Observation of Dirac singularities with light polarization. II,' Phys. Lett. A 171, 267-270 (1992).
[CrossRef]

1991 (1)

R. Bhandari, 'Evolution of light beams in polarization and direction,' Phys. Lett. B 175, 111-122 (1991).

1989 (1)

R. Simon and N. Mukunda, 'Universal SU(2) gadget for polarization optics,' Phys. Lett. A 138, 474-480 (1989).
[CrossRef]

1988 (1)

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]

1971 (1)

1970 (1)

E. B. Davies and J. T. Lewis, 'An operatorial approach to quantum probability,' Commun. Math. Phys. 17, 239-260 (1970).
[CrossRef]

1965 (1)

1955 (2)

S. Pancharatnam, 'The propagation of light in absorbing biaxial crystals. Part I. Theoretical,' Proc. Indian Acad. Sci., Sect. A 42, 86-109 (1955).

S. Pancharatnam, 'Achromatic combinations of birefringent plates. Part II: An achromatic quarter-wave plate,' Proc. Indian Acad. Sci., Sect. A 41, 137-144 (1955).

1949 (1)

Agarwal, G. S.

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1996).

Barnett, S. M.

S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Clarendon, 2003).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1996).

Baskal, S.

S. Baskal and Y. S. Kim, 'The language of Einstein spoken by optical instruments,' Opt. Spectrosc. 99, 443-446 (2005).
[CrossRef]

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]

Berry, M. V.

M. V. Berry, 'Physics of nonhermitian degeneracies,' Czech. J. Phys. 55, 1039-1046 (2004).
[CrossRef]

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]

M. V. Berry, 'Mode degeneracies and the Petermann excess-noise factor for unstable lasers,' J. Mod. Opt. 50, 63-81 (2003).
[CrossRef]

M. V. Berry and D. H. J. O'Dell, 'Diffraction by volume gratings with imaginary potentials,' J. Phys. A 31, 2093-2101 (1998).
[CrossRef]

M. V. Berry and S. Klein, 'Geometric phases from stacks of crystal plates,' J. Mod. Opt. 43, 165-180 (1996).
[CrossRef]

Bhandari, R.

R. Bhandari, 'Halfwave retarder for all polarization states,' Appl. Opt. 36, 2799-2801 (1997).
[CrossRef] [PubMed]

R. Bhandari and G. D. Love, 'Polarization eigenmodes of a QHQ retarder -- some new features,' Opt. Commun. 110, 479-484 (1994).
[CrossRef]

R. Bhandari, 'Interferometry without beam splitters -- a sensitive technique for spinor phases,' Phys. Lett. A 180, 21-24 (1993).
[CrossRef]

R. Bhandari, 'Observation of Dirac singularities with light polarization. II,' Phys. Lett. A 171, 267-270 (1992).
[CrossRef]

R. Bhandari, 'Evolution of light beams in polarization and direction,' Phys. Lett. B 175, 111-122 (1991).

Brosseau, Ch.

Ch. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).

Busch, P.

P. Busch, P. J. Lathi, and P. Mittelstaedt, The Quantum Theory of Measurement (Springer, 1996).

Chipman, R. A.

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2002).

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, 1997), Vol. 1.

Davies, E. B.

E. B. Davies and J. T. Lewis, 'An operatorial approach to quantum probability,' Commun. Math. Phys. 17, 239-260 (1970).
[CrossRef]

de Muynck, M.

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]

Dennis, M. R.

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]

Diu, B.

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, 1997), Vol. 1.

Feynman, R. P.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics III (Quantum Mechanics) (Addison-Wesley, 1965).

Georgieva, E.

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]

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]

Han, D.

D. Han, Y. S. Kim, and M. E. Noz, 'Interferometers and decoherence matrices,' Phys. Rev. E 61, 5907-5913 (2000).
[CrossRef]

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]

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]

Heiss, W. D.

W. D. Heiss, 'Exceptional points -- their universal occurrence and their physical significance,' Czech. J. Phys. 55, 1091-1099 (2004).
[CrossRef]

T. Stehmann, W. D. Heiss, and F. G. Scholtz, 'Observation of exceptional points in electronic circuits,' J. Phys. A 37, 7813-7819 (2004).
[CrossRef]

W. D. Heiss, 'Repulsion of resonant states and exceptional points,' Phys. Rev. E 61, 929-932 (2000).
[CrossRef]

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]

Hsü, H.-Y.

Keck, F.

F. Keck, H. J. Korsch, and S. Mossmann, 'Unfolding a diabolic point: a generalized crossing scenario,' J. Phys. A 36, 2125-2137 (2003).
[CrossRef]

Kim, Y. S.

S. Baskal and Y. S. Kim, 'The language of Einstein spoken by optical instruments,' Opt. Spectrosc. 99, 443-446 (2005).
[CrossRef]

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]

D. Han, Y. S. Kim, and M. E. Noz, 'Interferometers and decoherence matrices,' Phys. Rev. E 61, 5907-5913 (2000).
[CrossRef]

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]

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]

Kirillov, O. N.

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]

Klein, S.

M. V. Berry and S. Klein, 'Geometric phases from stacks of crystal plates,' J. Mod. Opt. 43, 165-180 (1996).
[CrossRef]

Korsch, H. J.

F. Keck, H. J. Korsch, and S. Mossmann, 'Unfolding a diabolic point: a generalized crossing scenario,' J. Phys. A 36, 2125-2137 (2003).
[CrossRef]

Laloë, F.

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, 1997), Vol. 1.

Lathi, P. J.

P. Busch, P. J. Lathi, and P. Mittelstaedt, The Quantum Theory of Measurement (Springer, 1996).

Leighton, R. B.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics III (Quantum Mechanics) (Addison-Wesley, 1965).

Lewis, J. T.

E. B. Davies and J. T. Lewis, 'An operatorial approach to quantum probability,' Commun. Math. Phys. 17, 239-260 (1970).
[CrossRef]

Li, S. F.

Love, G. D.

R. Bhandari and G. D. Love, 'Polarization eigenmodes of a QHQ retarder -- some new features,' Opt. Commun. 110, 479-484 (1994).
[CrossRef]

Lu, S.-Y.

Mailybaev, A. A.

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]

Marathay, A. S.

Mittelstaedt, P.

P. Busch, P. J. Lathi, and P. Mittelstaedt, The Quantum Theory of Measurement (Springer, 1996).

Monzón, J. J.

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]

Mossmann, S.

F. Keck, H. J. Korsch, and S. Mossmann, 'Unfolding a diabolic point: a generalized crossing scenario,' J. Phys. A 36, 2125-2137 (2003).
[CrossRef]

Mukunda, N.

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]

R. Simon and N. Mukunda, 'Universal SU(2) gadget for polarization optics,' Phys. Lett. A 138, 474-480 (1989).
[CrossRef]

Müller, M.

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]

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2002).

Noz, M. E.

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]

D. Han, Y. S. Kim, and M. E. Noz, 'Interferometers and decoherence matrices,' Phys. Rev. E 61, 5907-5913 (2000).
[CrossRef]

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]

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]

O'Dell, D. H. J.

M. V. Berry and D. H. J. O'Dell, 'Diffraction by volume gratings with imaginary potentials,' J. Phys. A 31, 2093-2101 (1998).
[CrossRef]

Pancharatnam, S.

S. Pancharatnam, 'Achromatic combinations of birefringent plates. Part II: An achromatic quarter-wave plate,' Proc. Indian Acad. Sci., Sect. A 41, 137-144 (1955).

S. Pancharatnam, 'The propagation of light in absorbing biaxial crystals. Part I. Theoretical,' Proc. Indian Acad. Sci., Sect. A 42, 86-109 (1955).

S. Pancharatnam, Collected Works of S. Pancharatnam (Oxford U. Press, 1975).

Pascovici, G.

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]

Philipp, M.

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]

Radmore, P. M.

S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Clarendon, 2003).

Richartz, M.

Richter, A.

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]

Rotter, I.

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]

Sanchez-Soto, L. L.

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]

Sands, M.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics III (Quantum Mechanics) (Addison-Wesley, 1965).

Scholtz, F. G.

T. Stehmann, W. D. Heiss, and F. G. Scholtz, 'Observation of exceptional points in electronic circuits,' J. Phys. A 37, 7813-7819 (2004).
[CrossRef]

Seyranian, A. P.

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]

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U. Press, 1962).

Simon, R.

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]

R. Simon and N. Mukunda, 'Universal SU(2) gadget for polarization optics,' Phys. Lett. A 138, 474-480 (1989).
[CrossRef]

Stehmann, T.

T. Stehmann, W. D. Heiss, and F. G. Scholtz, 'Observation of exceptional points in electronic circuits,' J. Phys. A 37, 7813-7819 (2004).
[CrossRef]

Tudor, T.

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]

T. Tudor, 'Generalized observables in polarization optics,' J. Phys. A 36, 9567-9590 (2003).
[CrossRef]

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]

von Brentano, P.

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]

Whitney, C.

Appl. Opt. (1)

Commun. Math. Phys. (1)

E. B. Davies and J. T. Lewis, 'An operatorial approach to quantum probability,' Commun. Math. Phys. 17, 239-260 (1970).
[CrossRef]

Czech. J. Phys. (2)

M. V. Berry, 'Physics of nonhermitian degeneracies,' Czech. J. Phys. 55, 1039-1046 (2004).
[CrossRef]

W. D. Heiss, 'Exceptional points -- their universal occurrence and their physical significance,' Czech. J. Phys. 55, 1091-1099 (2004).
[CrossRef]

J. Mod. Opt. (2)

M. V. Berry, 'Mode degeneracies and the Petermann excess-noise factor for unstable lasers,' J. Mod. Opt. 50, 63-81 (2003).
[CrossRef]

M. V. Berry and S. Klein, 'Geometric phases from stacks of crystal plates,' J. Mod. Opt. 43, 165-180 (1996).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (1)

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]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

J. Phys. A (6)

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]

T. Tudor, 'Generalized observables in polarization optics,' J. Phys. A 36, 9567-9590 (2003).
[CrossRef]

F. Keck, H. J. Korsch, and S. Mossmann, 'Unfolding a diabolic point: a generalized crossing scenario,' J. Phys. A 36, 2125-2137 (2003).
[CrossRef]

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]

M. V. Berry and D. H. J. O'Dell, 'Diffraction by volume gratings with imaginary potentials,' J. Phys. A 31, 2093-2101 (1998).
[CrossRef]

T. Stehmann, W. D. Heiss, and F. G. Scholtz, 'Observation of exceptional points in electronic circuits,' J. Phys. A 37, 7813-7819 (2004).
[CrossRef]

Opt. Commun. (1)

R. Bhandari and G. D. Love, 'Polarization eigenmodes of a QHQ retarder -- some new features,' Opt. Commun. 110, 479-484 (1994).
[CrossRef]

Opt. Spectrosc. (1)

S. Baskal and Y. S. Kim, 'The language of Einstein spoken by optical instruments,' Opt. Spectrosc. 99, 443-446 (2005).
[CrossRef]

Optik (Stuttgart) (2)

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]

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]

Phys. Lett. A (4)

R. Simon and N. Mukunda, 'Universal SU(2) gadget for polarization optics,' Phys. Lett. A 138, 474-480 (1989).
[CrossRef]

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]

R. Bhandari, 'Interferometry without beam splitters -- a sensitive technique for spinor phases,' Phys. Lett. A 180, 21-24 (1993).
[CrossRef]

R. Bhandari, 'Observation of Dirac singularities with light polarization. II,' Phys. Lett. A 171, 267-270 (1992).
[CrossRef]

Phys. Lett. B (1)

R. Bhandari, 'Evolution of light beams in polarization and direction,' Phys. Lett. B 175, 111-122 (1991).

Phys. Rev. A (1)

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]

Phys. Rev. E (5)

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]

W. D. Heiss, 'Repulsion of resonant states and exceptional points,' Phys. Rev. E 61, 929-932 (2000).
[CrossRef]

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]

D. Han, Y. S. Kim, and M. E. Noz, 'Interferometers and decoherence matrices,' Phys. Rev. E 61, 5907-5913 (2000).
[CrossRef]

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]

Proc. Indian Acad. Sci., Sect. A (2)

S. Pancharatnam, 'Achromatic combinations of birefringent plates. Part II: An achromatic quarter-wave plate,' Proc. Indian Acad. Sci., Sect. A 41, 137-144 (1955).

S. Pancharatnam, 'The propagation of light in absorbing biaxial crystals. Part I. Theoretical,' Proc. Indian Acad. Sci., Sect. A 42, 86-109 (1955).

Proc. R. Soc. London, Ser. A (1)

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]

Other (9)

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, 1997), Vol. 1.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics III (Quantum Mechanics) (Addison-Wesley, 1965).

S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Clarendon, 2003).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1996).

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2002).

S. Pancharatnam, Collected Works of S. Pancharatnam (Oxford U. Press, 1975).

W. A. Shurcliff, Polarized Light (Harvard U. Press, 1962).

P. Busch, P. J. Lathi, and P. Mittelstaedt, The Quantum Theory of Measurement (Springer, 1996).

Ch. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).

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Equations (75)

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P ( n ) = 1 2 ( σ 0 + n σ ) ,
R ( n , δ ) = e i ( δ 2 ) n σ = σ 0 cos δ 2 + i n σ sin δ 2 .
σ 0 = [ 1 0 0 1 ] , σ 1 = [ 0 1 1 0 ] , σ 2 = [ 0 i i 0 ] ,
σ 3 = [ 1 0 0 1 ] .
A = a 0 σ 0 + a σ ,
A = a 0 * σ 0 + a * σ ,
AA = A A .
B = b 0 σ + b σ ,
AB = ( a 0 b 0 + a b ) σ 0 + ( b 0 a + a 0 b ) σ + i ( a × b ) σ .
( a 0 a 0 * + a a * ) σ 0 + ( a 0 * a + a 0 a * ) σ + i ( a × a * ) σ = ( a 0 * a 0 + a * a ) σ 0 + ( a 0 a * + a 0 * a ) σ + i ( a * × a ) σ .
a × a * = 0 .
a * = λ a ,
a = e i α r .
A = e i α 0 a 0 σ 0 + e i α r σ ,
A = e i α 0 a 0 σ 0 + e i α r σ .
AA = A A = ( a 0 2 + r 2 ) σ 0 + 2 a 0 r σ cos ( α α 0 ) ,
UU = I σ 0 ,
a 0 r cos ( α α 0 ) = 0 ,
a 0 2 + r 2 = 1 .
α α 0 = π 2 modulo π ,
a 0 = cos δ 2 , r = n sin δ 2 ,
U = e i α 0 ( σ 0 cos δ 2 + in σ sin δ 2 ) = e i α 0 e i ( δ 2 ) n σ .
H = H ,
α 0 = 0 , π ( modulo 2 π ) ,
α = 0 ( modulo π ) .
H = ± a 0 σ 0 + r σ .
P 2 = P .
( a σ ) ( b σ ) = ( a b ) + i ( a × b ) σ ,
( a 0 2 + r 2 ) σ 0 + 2 a 0 r σ = ± a 0 σ 0 + r σ .
a 0 2 + r 2 = ± a 0 ,
2 a 0 r = r .
a 0 = 0 , r = 0 ,
P = O ,
P = σ 0 1 .
P = 1 2 ( σ 0 + n σ ) ,
S = 2 P l ,
S = n σ ,
P = P P θ P P x .
n x ( 1 , 0 , 0 ) ,
n θ ( cos 2 θ , sin 2 θ , 0 ) ,
P P θ = 1 2 ( σ 0 + σ 1 ) ,
P P θ = 1 2 ( σ 0 + σ 1 cos 2 θ + σ 2 sin 2 θ ) .
P = 1 4 ( σ 0 + σ 1 cos 2 θ + σ 2 sin 2 θ + σ 1 + σ 0 cos 2 θ i σ 3 sin 2 θ ) , = 1 2 cos θ [ ( σ 0 + σ 1 ) cos θ + ( σ 2 i σ 3 ) sin θ ] .
r ( cos θ , sin θ , i sin θ ) .
R P θ 2 ( π ) P P x .
n P = ( 1 , 0 , 0 ) ,
n R = ( cos θ , sin θ , 0 ) ,
P P x = 1 2 ( σ 0 + σ 1 ) ,
R P θ 2 ( π ) = i ( σ 1 cos θ + σ 2 sin θ ) .
i 2 ( σ 1 cos θ + σ 2 sin θ ) ( σ 0 + σ 1 ) , = i 2 [ ( σ 0 + σ 1 ) cos θ + ( σ 2 i σ 3 ) sin θ ] ,
r ( cos θ , sin θ , i sin θ ) ;
C = R P x ( π 2 ) P P 45 ° .
n P ( 0 , 1 , 0 ) ,
n R ( 1 , 0 , 0 ) .
P P 45 ° = 1 2 ( σ 0 + σ 2 ) ,
R P x ( π 2 ) = 1 2 ( σ 0 + i σ 1 ) ,
C = 1 2 2 ( σ 0 + i σ 1 + σ 2 σ 3 ) .
r ( i , 1 , 1 ) .
R T = R P θ ( π ) R P x ( π ) .
n ( 1 , 0 , 0 ) ,
n ( cos 2 θ , sin 2 θ , 0 ) ,
R P x ( π ) = i σ 1 ,
R P θ ( π ) = i ( σ 1 cos 2 θ + σ 2 sin 2 θ ) .
R T = ( σ 1 2 cos 2 θ + σ 2 σ 1 sin 2 θ ) = ( I cos 2 θ i σ 3 sin 2 θ ) .
n ( 0 , 0 , + 1 ) ,
R T = e i π R R ( 4 θ ) .
R P = R P x ( π 2 ) R P θ ( π ) R p x ( π 2 ) .
n ( 1 , 0 , 0 ) ,
R P x ( π ) = 1 2 ( I + i σ 1 ) .
n ( cos 2 θ , sin 2 θ , 0 ) ,
R P θ ( π ) = i ( σ 1 cos 2 θ + σ 2 sin 2 θ ) .
R P = i 2 ( σ 1 cos 2 θ + σ 2 sin 2 θ + i I cos 2 θ + σ 3 sin 2 θ + i I cos 2 θ σ 3 sin 2 θ σ 1 cos 2 θ + σ 2 sin 2 θ ) ,
R P = ( I cos 2 θ i σ 2 sin 2 θ ) .
n ( 0 , 1 , 0 ) ,
R P x ( π 2 ) R P θ ( π ) R P x ( π 2 ) = R P 45 ° ( 4 θ ) R P 45 ° ( 4 θ ) .

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