Abstract

We present a study of geometric phases in classical wave and polarization optics using the basic mathematical framework of quantum mechanics. Important physical situations taken from scalar wave optics, pure polarization optics, and the behavior of polarization in the eikonal or ray limit of Maxwell’s equations in a transparent medium are considered. The case of a beam of light whose propagation direction and polarization state are both subject to change is dealt with, attention being paid to the validity of Maxwell’s equations at all stages. Global topological aspects of the space of all propagation directions are discussed using elementary group theoretical ideas, and the effects on geometric phases are elucidated.

© 2014 Optical Society of America

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  1. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. A 392, 45–57 (1984).
    [CrossRef]
  2. B. Simon, “Holonomy, the quantum adiabatic theorem, and Berry’s phase,” Phys. Rev. Lett. 51, 2167–2170 (1983).
    [CrossRef]
  3. Y. Aharonov and J. Anandan, “Phase change during a cyclic quantum evolution,” Phys. Rev. Lett. 58, 1593–1596 (1987).
    [CrossRef]
  4. J. Samuel and R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
    [CrossRef]
  5. N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. I. General formalism,” Ann. Phys. 228, 205–268 (1993).
    [CrossRef]
  6. N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. II. The case of unitary group representations,” Ann. Phys. 228, 269–340 (1993).
    [CrossRef]
  7. V. Bargmann, “Note on Wigner’s theorem on symmetry operations,” J. Math. Phys. 5, 862–868 (1964).
    [CrossRef]
  8. L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C.R. Acad. Sci. Paris 110, 1251–1253 (1890).
  9. L. G. Gouy, “Sur la propagation anomale des ondes,” Ann. Chim. Phys. 24, 145–213 (1891).
  10. R. Simon and N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
    [CrossRef]
  11. S. M. Rytov, “Transition from wave to geometrical optics,” Dokl. Akad. Nauk. 18, 238–242 (1938).
  12. V. V. Vladimirskii, “The rotation of polarization plane for curved light ray,” Dokl. Akad. Nauk. 21, 222–227 (1941).
  13. S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Indian Acad. Sci. 44, 247–262 (1956).
  14. Inner products in Hilbert spaces will be generally written as (ψ,ϕ) rather than as 〈ψ|ϕ〉 in Dirac notation. For two-component complex column vectors z, z′ we sometimes write z′†, z′ in place of (z′, z). For three-component complex column vectors, described by Cartesian components in physical space for instance, we write (ψ,ϕ) or ψ†ϕ or ψ*·ϕ as convenient.
  15. R. Simon, N. Mukunda, S. Chaturvedi, and V. Srinivasan, “Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics,” Phys. Lett. A 372, 6847–6852 (2008).
    [CrossRef]
  16. E. M. Rabei, N. Mukunda, and R. Simon, “Bargmann invariants and geometric phases: a generalized connection,” Phys. Rev. A 60, 3397–3409 (1999).
    [CrossRef]
  17. N. Mukunda, S. Chaturvedi, and R. Simon, “Bargmann invariants and off-diagonal geometric phases for multilevel quantum systems: a unitary-group approach,” Phys. Rev. A 65, 012102 (2002).
    [CrossRef]
  18. N. Mukunda, Arvind, E. Ercolessi, G. Marmo, G. Morandi, and R. Simon, “Bargmann invariants, null phase curves, and a theory of the geometric phase,” Phys. Rev. A 67, 042114 (2003).
    [CrossRef]
  19. S. Chaturvedi, E. Ercolessi, G. Morandi, A. Ibort, G. Marmo, N. Mukunda, and R. Simon, “Null phase curves and manifolds in geometric phase theory,” J. Math. Phys. 54, 062106 (2013).
    [CrossRef]
  20. M. Santarsiero, J. C. G. de Sande, G. Piquero, and F. Gori, “Coherence–polarization properties of fields radiated from transversely periodic electromagnetic sources,” J. Opt. 15, 055701 (2013).
    [CrossRef]
  21. B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
    [CrossRef]
  22. B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
    [CrossRef]
  23. N. Mukunda, R. Simon, and E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
    [CrossRef]
  24. N. Mukunda, R. Simon, and E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and applications,” J. Opt. Soc. Am. A 2, 416–426 (1985).
    [CrossRef]
  25. R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Maxwell beams,” J. Opt. Soc. Am. A 3, 536–540 (1986).
    [CrossRef]
  26. R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Cross polarization in laser beams,” Appl. Opt. 26, 1589–1593 (1987).
    [CrossRef]
  27. H. Bacry, “Group theory and paraxial optics,” in Group Theoretical Methods in Physics, W. W. Zachary, ed. (World Scientific, 1985), pp. 215–224.
  28. R. Nityananda and S. Sridhar, “Light beams with general direction and polarization: global description and geometric phase,” arXiv:1212.0943 (2012).
  29. R. Bhandari, “Geometric phases in an arbitrary evolution of a light beam,” Phys. Lett. A 135, 240–244 (1989).
    [CrossRef]
  30. J. H. Hannay, “The Majorana representation of polarization, and the Bery phase of light,” J. Mod. Opt. 45, 1001–1008 (1998).
    [CrossRef]
  31. A. V. Tavrov, Y. Miyamoto, T. Kwabata, and M. Takeda, “Generalized algorithm for the unified analysis and simultaneous evaluation of geometrical spin-redirection phase and Pancharatnam phase in complex interferometric system,” J. Opt. Soc. Am. A 17, 154–161 (2000).
    [CrossRef]
  32. R. Simon and N. Mukunda, “Optical phase space, Wigner representation, and invariant quality parameters,” J. Opt. Soc. Am. A 17, 2440–2463 (2000).
    [CrossRef]
  33. H. Kogelnik, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
    [CrossRef]
  34. R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
    [CrossRef]
  35. R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322–328 (1988).
    [CrossRef]
  36. R. Borghi, M. Santarsiero, and R. Simon, “Shape invariance and a universal form for Gouy phase,” J. Opt. Soc. Am. A 21, 572–579 (2004).
    [CrossRef]
  37. See, for instance, Ref. [16].
  38. The definition of this group is given later, in Eq. (4.9).
  39. The group SO(3) is defined in the axis-angle description later in Eq. (4.2).
  40. The relationship between SU(2) and SO(3) is described, using axis-angle variables, later in Eq. (4.11) below.
  41. It may be useful to recall that for a spherical triangle on S2, the corresponding solid angle (subtended at the center of the sphere) is the “spherical excess,” i.e., the amount by which the sum of the three internal angles exceeds π. This excess occurs because S2 possesses positive curvature.
  42. See, for instance, M. Born and E. Wolf, eds., Principles of Optics, 6th ed. (Pergamon, 1987), Chap. 3.
  43. See, for instance, H. Stephani, ed., General Relativity—An introduction to the Theory of Gravitational Field (Cambridge University, 1985), p. 45.
  44. M. Eisenberg and R. Guy, “A proof of the hairy ball theorem,” Am. Math. Mon. 86, 572–574 (1979).
    [CrossRef]

2013 (2)

S. Chaturvedi, E. Ercolessi, G. Morandi, A. Ibort, G. Marmo, N. Mukunda, and R. Simon, “Null phase curves and manifolds in geometric phase theory,” J. Math. Phys. 54, 062106 (2013).
[CrossRef]

M. Santarsiero, J. C. G. de Sande, G. Piquero, and F. Gori, “Coherence–polarization properties of fields radiated from transversely periodic electromagnetic sources,” J. Opt. 15, 055701 (2013).
[CrossRef]

2010 (2)

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[CrossRef]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[CrossRef]

2008 (1)

R. Simon, N. Mukunda, S. Chaturvedi, and V. Srinivasan, “Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics,” Phys. Lett. A 372, 6847–6852 (2008).
[CrossRef]

2004 (1)

2003 (1)

N. Mukunda, Arvind, E. Ercolessi, G. Marmo, G. Morandi, and R. Simon, “Bargmann invariants, null phase curves, and a theory of the geometric phase,” Phys. Rev. A 67, 042114 (2003).
[CrossRef]

2002 (1)

N. Mukunda, S. Chaturvedi, and R. Simon, “Bargmann invariants and off-diagonal geometric phases for multilevel quantum systems: a unitary-group approach,” Phys. Rev. A 65, 012102 (2002).
[CrossRef]

2000 (2)

1999 (1)

E. M. Rabei, N. Mukunda, and R. Simon, “Bargmann invariants and geometric phases: a generalized connection,” Phys. Rev. A 60, 3397–3409 (1999).
[CrossRef]

1998 (1)

J. H. Hannay, “The Majorana representation of polarization, and the Bery phase of light,” J. Mod. Opt. 45, 1001–1008 (1998).
[CrossRef]

1993 (3)

R. Simon and N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef]

N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. I. General formalism,” Ann. Phys. 228, 205–268 (1993).
[CrossRef]

N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. II. The case of unitary group representations,” Ann. Phys. 228, 269–340 (1993).
[CrossRef]

1989 (1)

R. Bhandari, “Geometric phases in an arbitrary evolution of a light beam,” Phys. Lett. A 135, 240–244 (1989).
[CrossRef]

1988 (2)

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

J. Samuel and R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[CrossRef]

1987 (2)

Y. Aharonov and J. Anandan, “Phase change during a cyclic quantum evolution,” Phys. Rev. Lett. 58, 1593–1596 (1987).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Cross polarization in laser beams,” Appl. Opt. 26, 1589–1593 (1987).
[CrossRef]

1986 (1)

1985 (1)

1984 (2)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. A 392, 45–57 (1984).
[CrossRef]

1983 (2)

B. Simon, “Holonomy, the quantum adiabatic theorem, and Berry’s phase,” Phys. Rev. Lett. 51, 2167–2170 (1983).
[CrossRef]

N. Mukunda, R. Simon, and E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
[CrossRef]

1979 (1)

M. Eisenberg and R. Guy, “A proof of the hairy ball theorem,” Am. Math. Mon. 86, 572–574 (1979).
[CrossRef]

1965 (1)

H. Kogelnik, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

1964 (1)

V. Bargmann, “Note on Wigner’s theorem on symmetry operations,” J. Math. Phys. 5, 862–868 (1964).
[CrossRef]

1956 (1)

S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Indian Acad. Sci. 44, 247–262 (1956).

1941 (1)

V. V. Vladimirskii, “The rotation of polarization plane for curved light ray,” Dokl. Akad. Nauk. 21, 222–227 (1941).

1938 (1)

S. M. Rytov, “Transition from wave to geometrical optics,” Dokl. Akad. Nauk. 18, 238–242 (1938).

1891 (1)

L. G. Gouy, “Sur la propagation anomale des ondes,” Ann. Chim. Phys. 24, 145–213 (1891).

1890 (1)

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C.R. Acad. Sci. Paris 110, 1251–1253 (1890).

Aharonov, Y.

Y. Aharonov and J. Anandan, “Phase change during a cyclic quantum evolution,” Phys. Rev. Lett. 58, 1593–1596 (1987).
[CrossRef]

Anandan, J.

Y. Aharonov and J. Anandan, “Phase change during a cyclic quantum evolution,” Phys. Rev. Lett. 58, 1593–1596 (1987).
[CrossRef]

Arvind,

N. Mukunda, Arvind, E. Ercolessi, G. Marmo, G. Morandi, and R. Simon, “Bargmann invariants, null phase curves, and a theory of the geometric phase,” Phys. Rev. A 67, 042114 (2003).
[CrossRef]

Bacry, H.

H. Bacry, “Group theory and paraxial optics,” in Group Theoretical Methods in Physics, W. W. Zachary, ed. (World Scientific, 1985), pp. 215–224.

Bargmann, V.

V. Bargmann, “Note on Wigner’s theorem on symmetry operations,” J. Math. Phys. 5, 862–868 (1964).
[CrossRef]

Berry, M. V.

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. A 392, 45–57 (1984).
[CrossRef]

Bhandari, R.

R. Bhandari, “Geometric phases in an arbitrary evolution of a light beam,” Phys. Lett. A 135, 240–244 (1989).
[CrossRef]

J. Samuel and R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[CrossRef]

Borghi, R.

Chaturvedi, S.

S. Chaturvedi, E. Ercolessi, G. Morandi, A. Ibort, G. Marmo, N. Mukunda, and R. Simon, “Null phase curves and manifolds in geometric phase theory,” J. Math. Phys. 54, 062106 (2013).
[CrossRef]

R. Simon, N. Mukunda, S. Chaturvedi, and V. Srinivasan, “Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics,” Phys. Lett. A 372, 6847–6852 (2008).
[CrossRef]

N. Mukunda, S. Chaturvedi, and R. Simon, “Bargmann invariants and off-diagonal geometric phases for multilevel quantum systems: a unitary-group approach,” Phys. Rev. A 65, 012102 (2002).
[CrossRef]

de Sande, J. C. G.

M. Santarsiero, J. C. G. de Sande, G. Piquero, and F. Gori, “Coherence–polarization properties of fields radiated from transversely periodic electromagnetic sources,” J. Opt. 15, 055701 (2013).
[CrossRef]

Eisenberg, M.

M. Eisenberg and R. Guy, “A proof of the hairy ball theorem,” Am. Math. Mon. 86, 572–574 (1979).
[CrossRef]

Ercolessi, E.

S. Chaturvedi, E. Ercolessi, G. Morandi, A. Ibort, G. Marmo, N. Mukunda, and R. Simon, “Null phase curves and manifolds in geometric phase theory,” J. Math. Phys. 54, 062106 (2013).
[CrossRef]

N. Mukunda, Arvind, E. Ercolessi, G. Marmo, G. Morandi, and R. Simon, “Bargmann invariants, null phase curves, and a theory of the geometric phase,” Phys. Rev. A 67, 042114 (2003).
[CrossRef]

Gori, F.

M. Santarsiero, J. C. G. de Sande, G. Piquero, and F. Gori, “Coherence–polarization properties of fields radiated from transversely periodic electromagnetic sources,” J. Opt. 15, 055701 (2013).
[CrossRef]

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[CrossRef]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[CrossRef]

Gouy, L. G.

L. G. Gouy, “Sur la propagation anomale des ondes,” Ann. Chim. Phys. 24, 145–213 (1891).

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C.R. Acad. Sci. Paris 110, 1251–1253 (1890).

Guy, R.

M. Eisenberg and R. Guy, “A proof of the hairy ball theorem,” Am. Math. Mon. 86, 572–574 (1979).
[CrossRef]

Hannay, J. H.

J. H. Hannay, “The Majorana representation of polarization, and the Bery phase of light,” J. Mod. Opt. 45, 1001–1008 (1998).
[CrossRef]

Ibort, A.

S. Chaturvedi, E. Ercolessi, G. Morandi, A. Ibort, G. Marmo, N. Mukunda, and R. Simon, “Null phase curves and manifolds in geometric phase theory,” J. Math. Phys. 54, 062106 (2013).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

Kwabata, T.

Marmo, G.

S. Chaturvedi, E. Ercolessi, G. Morandi, A. Ibort, G. Marmo, N. Mukunda, and R. Simon, “Null phase curves and manifolds in geometric phase theory,” J. Math. Phys. 54, 062106 (2013).
[CrossRef]

N. Mukunda, Arvind, E. Ercolessi, G. Marmo, G. Morandi, and R. Simon, “Bargmann invariants, null phase curves, and a theory of the geometric phase,” Phys. Rev. A 67, 042114 (2003).
[CrossRef]

Miyamoto, Y.

Morandi, G.

S. Chaturvedi, E. Ercolessi, G. Morandi, A. Ibort, G. Marmo, N. Mukunda, and R. Simon, “Null phase curves and manifolds in geometric phase theory,” J. Math. Phys. 54, 062106 (2013).
[CrossRef]

N. Mukunda, Arvind, E. Ercolessi, G. Marmo, G. Morandi, and R. Simon, “Bargmann invariants, null phase curves, and a theory of the geometric phase,” Phys. Rev. A 67, 042114 (2003).
[CrossRef]

Mukunda, N.

S. Chaturvedi, E. Ercolessi, G. Morandi, A. Ibort, G. Marmo, N. Mukunda, and R. Simon, “Null phase curves and manifolds in geometric phase theory,” J. Math. Phys. 54, 062106 (2013).
[CrossRef]

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[CrossRef]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[CrossRef]

R. Simon, N. Mukunda, S. Chaturvedi, and V. Srinivasan, “Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics,” Phys. Lett. A 372, 6847–6852 (2008).
[CrossRef]

N. Mukunda, Arvind, E. Ercolessi, G. Marmo, G. Morandi, and R. Simon, “Bargmann invariants, null phase curves, and a theory of the geometric phase,” Phys. Rev. A 67, 042114 (2003).
[CrossRef]

N. Mukunda, S. Chaturvedi, and R. Simon, “Bargmann invariants and off-diagonal geometric phases for multilevel quantum systems: a unitary-group approach,” Phys. Rev. A 65, 012102 (2002).
[CrossRef]

R. Simon and N. Mukunda, “Optical phase space, Wigner representation, and invariant quality parameters,” J. Opt. Soc. Am. A 17, 2440–2463 (2000).
[CrossRef]

E. M. Rabei, N. Mukunda, and R. Simon, “Bargmann invariants and geometric phases: a generalized connection,” Phys. Rev. A 60, 3397–3409 (1999).
[CrossRef]

R. Simon and N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef]

N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. I. General formalism,” Ann. Phys. 228, 205–268 (1993).
[CrossRef]

N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. II. The case of unitary group representations,” Ann. Phys. 228, 269–340 (1993).
[CrossRef]

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Cross polarization in laser beams,” Appl. Opt. 26, 1589–1593 (1987).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Maxwell beams,” J. Opt. Soc. Am. A 3, 536–540 (1986).
[CrossRef]

N. Mukunda, R. Simon, and E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and applications,” J. Opt. Soc. Am. A 2, 416–426 (1985).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

N. Mukunda, R. Simon, and E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
[CrossRef]

Nityananda, R.

R. Nityananda and S. Sridhar, “Light beams with general direction and polarization: global description and geometric phase,” arXiv:1212.0943 (2012).

Pancharatnam, S.

S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Indian Acad. Sci. 44, 247–262 (1956).

Piquero, G.

M. Santarsiero, J. C. G. de Sande, G. Piquero, and F. Gori, “Coherence–polarization properties of fields radiated from transversely periodic electromagnetic sources,” J. Opt. 15, 055701 (2013).
[CrossRef]

Rabei, E. M.

E. M. Rabei, N. Mukunda, and R. Simon, “Bargmann invariants and geometric phases: a generalized connection,” Phys. Rev. A 60, 3397–3409 (1999).
[CrossRef]

Rytov, S. M.

S. M. Rytov, “Transition from wave to geometrical optics,” Dokl. Akad. Nauk. 18, 238–242 (1938).

Samuel, J.

J. Samuel and R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[CrossRef]

Santarsiero, M.

M. Santarsiero, J. C. G. de Sande, G. Piquero, and F. Gori, “Coherence–polarization properties of fields radiated from transversely periodic electromagnetic sources,” J. Opt. 15, 055701 (2013).
[CrossRef]

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[CrossRef]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[CrossRef]

R. Borghi, M. Santarsiero, and R. Simon, “Shape invariance and a universal form for Gouy phase,” J. Opt. Soc. Am. A 21, 572–579 (2004).
[CrossRef]

Simon, B.

B. Simon, “Holonomy, the quantum adiabatic theorem, and Berry’s phase,” Phys. Rev. Lett. 51, 2167–2170 (1983).
[CrossRef]

Simon, B. N.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[CrossRef]

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[CrossRef]

Simon, R.

S. Chaturvedi, E. Ercolessi, G. Morandi, A. Ibort, G. Marmo, N. Mukunda, and R. Simon, “Null phase curves and manifolds in geometric phase theory,” J. Math. Phys. 54, 062106 (2013).
[CrossRef]

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[CrossRef]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[CrossRef]

R. Simon, N. Mukunda, S. Chaturvedi, and V. Srinivasan, “Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics,” Phys. Lett. A 372, 6847–6852 (2008).
[CrossRef]

R. Borghi, M. Santarsiero, and R. Simon, “Shape invariance and a universal form for Gouy phase,” J. Opt. Soc. Am. A 21, 572–579 (2004).
[CrossRef]

N. Mukunda, Arvind, E. Ercolessi, G. Marmo, G. Morandi, and R. Simon, “Bargmann invariants, null phase curves, and a theory of the geometric phase,” Phys. Rev. A 67, 042114 (2003).
[CrossRef]

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[CrossRef]

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[CrossRef]

E. M. Rabei, N. Mukunda, and R. Simon, “Bargmann invariants and geometric phases: a generalized connection,” Phys. Rev. A 60, 3397–3409 (1999).
[CrossRef]

R. Simon and N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef]

N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. II. The case of unitary group representations,” Ann. Phys. 228, 269–340 (1993).
[CrossRef]

N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. I. General formalism,” Ann. Phys. 228, 205–268 (1993).
[CrossRef]

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Cross polarization in laser beams,” Appl. Opt. 26, 1589–1593 (1987).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Maxwell beams,” J. Opt. Soc. Am. A 3, 536–540 (1986).
[CrossRef]

N. Mukunda, R. Simon, and E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and applications,” J. Opt. Soc. Am. A 2, 416–426 (1985).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

N. Mukunda, R. Simon, and E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
[CrossRef]

Simon, S.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[CrossRef]

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[CrossRef]

Sridhar, S.

R. Nityananda and S. Sridhar, “Light beams with general direction and polarization: global description and geometric phase,” arXiv:1212.0943 (2012).

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R. Simon, N. Mukunda, S. Chaturvedi, and V. Srinivasan, “Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics,” Phys. Lett. A 372, 6847–6852 (2008).
[CrossRef]

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[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Cross polarization in laser beams,” Appl. Opt. 26, 1589–1593 (1987).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Maxwell beams,” J. Opt. Soc. Am. A 3, 536–540 (1986).
[CrossRef]

N. Mukunda, R. Simon, and E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and applications,” J. Opt. Soc. Am. A 2, 416–426 (1985).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

N. Mukunda, R. Simon, and E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
[CrossRef]

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N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. I. General formalism,” Ann. Phys. 228, 205–268 (1993).
[CrossRef]

N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. II. The case of unitary group representations,” Ann. Phys. 228, 269–340 (1993).
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[CrossRef]

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[CrossRef]

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E. M. Rabei, N. Mukunda, and R. Simon, “Bargmann invariants and geometric phases: a generalized connection,” Phys. Rev. A 60, 3397–3409 (1999).
[CrossRef]

N. Mukunda, S. Chaturvedi, and R. Simon, “Bargmann invariants and off-diagonal geometric phases for multilevel quantum systems: a unitary-group approach,” Phys. Rev. A 65, 012102 (2002).
[CrossRef]

N. Mukunda, Arvind, E. Ercolessi, G. Marmo, G. Morandi, and R. Simon, “Bargmann invariants, null phase curves, and a theory of the geometric phase,” Phys. Rev. A 67, 042114 (2003).
[CrossRef]

N. Mukunda, R. Simon, and E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

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[CrossRef]

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Other (10)

Inner products in Hilbert spaces will be generally written as (ψ,ϕ) rather than as 〈ψ|ϕ〉 in Dirac notation. For two-component complex column vectors z, z′ we sometimes write z′†, z′ in place of (z′, z). For three-component complex column vectors, described by Cartesian components in physical space for instance, we write (ψ,ϕ) or ψ†ϕ or ψ*·ϕ as convenient.

H. Bacry, “Group theory and paraxial optics,” in Group Theoretical Methods in Physics, W. W. Zachary, ed. (World Scientific, 1985), pp. 215–224.

R. Nityananda and S. Sridhar, “Light beams with general direction and polarization: global description and geometric phase,” arXiv:1212.0943 (2012).

See, for instance, Ref. [16].

The definition of this group is given later, in Eq. (4.9).

The group SO(3) is defined in the axis-angle description later in Eq. (4.2).

The relationship between SU(2) and SO(3) is described, using axis-angle variables, later in Eq. (4.11) below.

It may be useful to recall that for a spherical triangle on S2, the corresponding solid angle (subtended at the center of the sphere) is the “spherical excess,” i.e., the amount by which the sum of the three internal angles exceeds π. This excess occurs because S2 possesses positive curvature.

See, for instance, M. Born and E. Wolf, eds., Principles of Optics, 6th ed. (Pergamon, 1987), Chap. 3.

See, for instance, H. Stephani, ed., General Relativity—An introduction to the Theory of Gravitational Field (Cambridge University, 1985), p. 45.

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Figures (1)

Fig. 1.
Fig. 1.

Illustrating the hyperbolic geometry of the lower-half complex q-plane underlying Gaussian beams and the abcd-law. Free propagation corresponds to the horizontal line passing through q=izR. The two circular geodesics are centered at O1, O2. For the geodesic quadrilateral ψ1ψ2ψ3ψ4ψ1 the angles at ψ1, ψ3 vanish, while the angles at ψ2, ψ4 become π/2 at z1=zR and z2=zR, respectively. Thus, 50% of the total Gouy phase “jump” occurs within a propagation distance 2zR around the waist, zR decreasing quadratically with decreasing waist size w.

Equations (121)

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B={ψH|ψ2=(ψ,ψ)=1}H.
ψBψ=eiαψB,0α<2π.
R=B/U(1)={ρ(ψ)=ψψ|ψB}.
π:BR: ψBπ(ψ)=ρ(ψ)R.
C={ψ(s)B|s1ss2}B,
C=π[C]={ρ(s)=ρ(ψ(s))=ψ(s)ψ(s)|s1ss2}R,
φg[C]=φtot[C]φdyn[C],φtot[C]=arg(ψ(s1),ψ(s2)),φdyn[C]=Ims1s2ds(ψ(s),dψ(s)ds).
Δ3(ψ1,ψ2,ψ3)=(ψ1,ψ2)(ψ2,ψ3)(ψ3,ψ1)
(ψ,ϕ)=cosθ,0<θ<π/2:ψ(s)=ψcoss+ϕsinssinθ,0sθ,ϕ=ϕψcosθ.
(ψ(s),ψ(s))=cos(ss),0s,sθ.
arg(Δ3(ψ1,ψ2,ψ3))=φg[C],C=triangle inRwith verticesρ(ψ1),ρ(ψ2),ρ(ψ3)and connecting geodesics as sides.
ψout(x1,x2)=exp(ix12+x222ƛf)ψin(x1,x2).
Λ(x1,x2)=(E(x1,x2)B(x1,x2))
Λout(x1,x2)=exp(ix12+x222ƛf)Λin(x1,x2),
T:Λin(x1,x2)Λout(x1,x2)=T(Q1,Q2)Λin(x1,x2),Q1=x116×6+ƛG1,Q2=x216×6+ƛG2,
iƛzψ(x,y;z)=ƛ22(2x2+2y2)ψ(x,y;z),
H=12(px2+py2),px=iƛx,py=iƛy.
iƛzψ(x;z)=Hψ(x;z),H=12px2=ƛ222x2.
Imq<0:ψ0(x;q)=(Imqπƛ|q|2)1/4exp(ix22ƛq),dx|ψ0(x;q)|2=1.
ψ(x;0)=ψ0(x;iw22ƛ)=(2πw2)1/4exp(x2w2)ψ(x;z)=eiφG(z)ψ0(x;q(z));φG(z)=12tan1(zzR),q(z)=zizR,zR=Rayleigh range=w2/2ƛ=πw2/λ.
φG(z)=argψ(0;z),φG()φG()=π/2.
qinqout=aqin+bcqin+d,
Δ4(ψ1,ψ2,ψ3,ψ4)=(ψ1,ψ2)(ψ2,ψ3)(ψ3,ψ4)(ψ4,ψ1),argΔ4(ψ1,ψ2,ψ3,ψ4)=φg[C],C=quadrilateral inRwith verticesρ(ψ1),,ρ(ψ4)and geodesics connectingρ(ψ1)toρ(ψ2),,ρ(ψ4)toρ(ψ1)as sides.
iddsψ0(s)=H0ψ0(s),
ψ0(s2)=ei(s2s1)H0ψ0(s1).
φ(s)=arg(ψR,ψ0(s)).
φ(s2)φ(s1)=φg[quadrilateral inRwith verticesρ(ψR),ρ(ψ0(s1)),ρ(ψE),ρ(ψ0(s2)),and geodesic sides].
ψR(z)=limq1=0,q20ψ0(x;q1+iq2)=limq201(πƛq2)1/4exp(x22ƛq2)δ(x);ψE(x)=limq10,q2ψ0(x;q1+iq2)=limq21(πƛq2)1/4exp(x22ƛq2)constant inx.
φ(s)=arg(ψR,ψ0(s))=arg(dxψR(x)*ψ(x;z))=argψ(0;z)=φG(z).
φG(z2)φG(z1)=φg[quadrilateral inR,verticesρ(ψR(x)δ(x)),ρ(ψ(x;z1)),ρ(ψE(x)constant),ρ(ψ(x;z2))and geodesic sides].
Cq=0q=izR+z1q=iq=izR+z2q=0
φG(z2)φG(z1)=φdyn[C]=ImC{(ψ0(x;q),q1ψ0(x;q))dq1+(ψ0(x;q),q2ψ0(x;q))dq2}.
d2=(dq12+dq22)/q22,
φG(z2)φG(z1)=12(arctan(z2/zR)arctan(z1/zR)),
H=C2B3=S3πR2=B3/U(1)=Spol2
En^=(EE)1EτESpol2,τ=(τ1,τ2,τ3)=(σ3,σ1,σ2),
E(z)C2n^(z)=(E(z)E(z))1E(z)τE(z)Spol2.
idE(z)dz=H(z)E(z),H(z)=12τ·a(z),
dn^(z)dz=a(z)n^(z).
E(z2)=U(z2,z1)E(z1),U(z2,z1)SU(2);n^(z2)=R(z2,z1)n^(z1),R(z2,z1)SO(3).
U(z2,z1)=ei(z2z1)H,R(z2,z1)=R(a^,(z2z1)|a|),a^=a/|a|,
E(z)HE(z)=12E(z)E(z)a·n^(z)=constant,
φg[Cpol]=φtot[C]φdyn[C]=argE(1)E(2)Imz1z2dzE(z)dE(z)dz=θ+z1z2dzE(z)H(z)E(z)=12Ω[Cpol],
Cpol=spherical triangle onSpol2:φg[Cpol]=12Ω[Cpol]=arg(E(1)E(2)E(2)E(3)E(3)E(1)).
dds(n(x)dxds)=n(x).
v(s)=x˙(s)=unit tangent;n(s)=v˙(s)/|v˙(s)|=unit principal normal;b(s)=v(s)n(s)=unit binormal;κ(s)=|v˙(s)|=curvature,τ(s)=b(s)·n˙(s)=torsion.
v˙(s)=κ(s)b(s)v(s)=(v(s)v˙(s))v(s),n˙(s)=(κ(s)b(s)+τ(s)v(s))n(s),b˙(s)=τ(s)v(s)b(s).
Ψ(s)=E(s)/E(s)E(s):Ψ˙(s)=κ(s)b(s)Ψ(s),v(s)·Ψ(s)=0.
ea(s)·eb(s)=δab,e1(s)e2(s)=v(s),ea(s)·v(s)=0;e˙a(s)=κ(s)b(s)ea(s),a=1,2.
e1(s1)=n(s1),e2(s1)=b(s1)
dds(ea(s)·n(s)ea(s)·b(s))=(0τ(s)τ(s)0)(ea(s)·n(s)ea(s)·b(s)),a=1,2;(e1(s)e2(s))=(cosχ(s)sinχ(s)sinχ(s)cosχ(s))(n(s)b(s)),χ(s)=s1sdsτ(s).
Ψ(s)=zaea(s),za=ea(s)·Ψ(s)=constant,zz=(z1*z2*)(z1z2)=1.
Ψ(s)n^(z)=zτzSpol2.
cyclic case: x˙(s2)=x˙(s1),x¨(s2)=x¨(s1).
Linear polarizationΨ(s1)=cosθe1(s1)+sinθe2(s1)Ψ(s2)=cosθe1(s2)+sinθe2(s2)=cos(θχ(s2))e1(s1)+sin(θχ(s2))e2(s1),χ(s2)=s1s2dsτ(s).
Cdir={v(s)Sdir2|s1ss2}Sdir2,v(s2)=v(s1).
χ(s2)=s1s2dsτ(s)=Ω[Cdir].
RCP/LCP:Ψ(s1)=12(e1(s1)±ie2(s1))Ψ(s)=12(e1(s)±ie2(s))=12e±iχ(s)(n(s)±ib(s)),Ψ(s2)=e±iχ(s2)Ψ(s1)=eiΩ[Cdir]Ψ(s1).
iddsΨ(s)=H(s)Ψ(s),H(s)=iκ(s)(n(s)v(s)Tv(s)n(s)T).
C={Ψ(s)C3|s1ss2}B5,π[C]=CR4:φg[C]=φtot[C]φdyn[C],φtot[C]=arg(Ψ(s1)Ψ(s2)),φdyn[C]=Ims1s2dsΨ(s)dΨ(s)ds=Im(is1s2dsΨ(s)H(s)Ψ(s))=0,i.e.,φg[C]=φtot[C]=arg(Ψ(s1),Ψ(s2))=Ω[C],
Choices ofPolarizationBehavior ofφg[C]s1,s2Ψ(s)FreeLinearReal0orπΓcyclicRCP/LCPΨ(s2)=Ω[Cdir]Eq.(2.39)eiΩ[Cdir]Ψ(s1)
Ψ(s)=za(s)ea(s):iddsea(s)=H(dir)(s)ea(s),H(dir)(s)=iκ(s)(n(s)v(s)Tv(s)n(s)T);iddsz(s)=H(pol)(s)z(s),H(pol)(s)=12τ·a(s),a(s)real.
iddsΨ(s)=(H(dir)(s)+H(pol)(s))Ψ(s),H(dir)(s)=iκ(s)(n(s)v(s)Tv(s)n(s)T),H(pol)(s)=12aj(s)(τj)abea(s)eb(s)T.
n^(z(s))n^(s):dn^(s)ds=a(s)n^(s).
Im(Ψ(s),dΨ(s)ds)=Im(i(Ψ(s),(H(dir)(s)+H(pol)(s))Ψ(s)))=12a(s)·n^(s),
C={Ψ(s)H|s1ss2}B5,π[C]=CR4.
φg[C]=φtot[C]φdyn[C],φtot[C]=arg(Ψ(s1)Ψ(s2)),φdyn[C]=Ims1s2ds(Ψ(s),ddsΨ(s))=12s1s2dsa(s)·n^(s).
x˙(s2)=x˙(s1),x¨(s2)=x¨(s1);z(s2)=eiθz(s1),n^(s2)=n^(s1);Cpol={n^(s)Spol2|s1ss2}Spol2,closed.
(e1(s2)e2(s2))=(cosΩ[Cdir]sinΩ[Cdir]sinΩ[Cdir]cosΩ[Cdir])(e1(s1)e2(s1)),
φg[C]=arg(z(s1)(cosΩ[Cdir]sinΩ[Cdir]sinΩ[Cdir]cosΩ[Cdir])eiθz(s1))+12s1s2dsa(s)·n^(s)=θ+12s1s2dsa(s)·n^(s)+arg(cosΩ[Cdir]+2isinΩ[Cdir]Imz1(s1)z2(s1)*).
θ+12s1s2dsa(s)·n^(s)=12Ω[Cpol].
RCP/LCP:z1(s1)=12,z2(s1)=±i2;arg(cosΩ[Cdir]+2isinΩ[Cdir]Imz1(s1)z2(s1)*)=Ω[Cdir],
φg[C]=12Ω[Cpol]Ω[Cdir].
U:x·τUx·τU=x·τ=(R(U)x)·τ.
SN2={k^(θ,ϕ)Sdir2|0θ<π,0ϕ<2π},SS2={k^(θ,ϕ)Sdir2|0<θπ,0ϕ<2π},SN2SS2=Sdir2,SN2SS2={k^(θ,ϕ)Sdir2|0<θ<π,0ϕ<2π}.
Rjk(a^,α)=δjkcosα+ajak(1cosα)εjklalsinα,0α2π.
A(k^)=R(e^3,ϕ)R(e^2,θ)R(e^3,ϕ)1=R(e^2cosϕe^1sinϕ,θ),0θ<π,0ϕ<2πA(k^)e^3=k^.
e1(k^)=A(k^)e^1=(sin2ϕ+cosθcos2ϕ(cosθ1)sinϕcosϕsinθcosϕ),e2(k^)=A(k^)e^2=((cosθ1)sinϕcosϕcos2ϕ+cosθsin2ϕsinθsinϕ),k^SN2.
A(k^)=R(e^3,ϕ)R(e^2,θ)R(e^3,ϕ),0<θπ,0ϕ<2π:A(k^)e^3=k^.
e1(k^)=A(k^)e^1=(cosθcos2ϕsin2ϕ(1+cosθ)sinϕcosϕsinθcosϕ),e2(k^)=A(k^)e^2=((1+cosθ)sinϕcosϕcos2ϕcosθsin2ϕsinθsinϕ),k^SS2.
k^SN2SS2:A(k^)=A(k^)R(e^3,2ϕ)=R(k^,2ϕ)A(k^);ea(k^)=R(k^,2ϕ)ea(k^),a=1,2.
R(e^3,2ϕ)=R(e^3,χ(k^))R(e^3,χ(k^))1,k^SN2SS2.
SU(2)={U=2×2complex matrices|UU=12×2,detU=1},
U(a^,α)=ei2αa^·σ=cosα212×2ia^·σsinα2,a^S2,0α2π.
U(a^,α)SU(2)R(a^,α)SO(3).
U(k^)=ei2ϕσ3ei2θσ2ei2ϕσ3,k^SN2;U(k^)=ei2ϕσ3ei2θσ2ei2ϕσ3,k^SS2.
R(e^3,2ϕ)=(U(e^2,4ϕ)00001),U(e^2,4ϕ)=e2iϕσ2=(cos2ϕsin2ϕsin2ϕcos2ϕ).
U(e^2,4ϕ)=V(k^)1V(k^),V(k^)=eiϕσ2eiθ2σ1eiϕσ2,k^SN2;V(k^)=eiϕσ2eiθ2σ1eiϕσ2,k^SS2.
A(k^)=A(k^)(V(k^)1V(k^)00001),i.e.,A(k^)=A(k^)(V(k^)100001)=A(k^)(V(k^)100001).
A(k^)e^3=k^,k^Sdir2.
ASU(3):Ae^3=k^A=A(U00001),ASO(3),USU(2),Ae^3=k^.
A(k^)=R(e^3,ϕ)R(e^2,θ)(ei2θσ1eiϕσ200001).
g1(k^)=A(k^)e^1=cosθ/2e1(k^)+isinθ/2e2(k^),g2(k^)=A(k^)e^2=cosθ/2e2(k^)+isinθ/2e1(k^);k^·ga(k^)=0,ga(k^)*·gb(k^)=δab.
ψ(k^)=zaga(k^),za=ga(k^)*·ψ(k^),z=(z1z2)C2.
ga(k^)*·ψ(k^)=ga(k^)*·ψ(k^)=za.
TSdir2=k^Sdir2Tk^Sdir2Sdir2×R2,(TSdir2)ck^Sdir2(Tk^Sdir2)cSdir2×C2.
H={ψ(k^)C3|k^Sdir2,k^·ψ(k^)=0,ψ2=dΩ(k^)ψ(k^)*·ψ(k^)<},
ψ(k^)=za(k^)ga(k^),za(k^)=ga(k^)*·ψ(k^),ψ2=dΩ(k^)z(k^)z(k^).
H=L2(Sdir2)H(2),
k^·E(k^)=0:E(k^)=zaga(k^),za=ga(k^)*·E(k^)n^(z)=(zz)1zτzSpol2.
(e1(k^)e2(k^))=U0(θ,ϕ)(g1(k^)g2(k^)),U0(θ,ϕ)=(cosθ2+isinθ2sin2ϕisinθ2cos2ϕisinθ2cos2ϕcosθ2isinθ2sin2ϕ);12(e1(k^)+ie2(k^))=12(cosθ2+e2iϕsinθ2)g1(k^)+i2(cosθ2e2iϕsinθ2)g2(k^);12(e1(k^)ie2(k^))=12(cosθ2e2iϕsinθ2)g1(k^)i2(cosθ2+e2iϕsinθ2)g2(k^).
Ψ(s)C3,Ψ(s)*·Ψ(s)=1,k^(s)·Ψ(s)=0.
C={Ψ(s)H|Ψ(s)Ψ(s)=1,k^·Ψ(s)=0,s1ss2}B5,π[C]=CR4.
Ψ(s)=za(s)ga(k^(s)),za(s)=ga(k^(s))*·Ψ(s),z(s)=(z1(s)z2(s)),z(s)z(s)=1.
Ψ˙(s)=ddsΨ(s)=z˙a(s)ga(k^(s))+za(s)g˙a(k^(s)),(Ψ(s),Ψ˙(s))=z(s)z˙(s)+za(s)*ga(k^(s))*·g˙b(k^(s))zb(s).
hab(s)=hba(s)*=iga(k^(s))*·g˙b(k^(s))
φg[C]=φtot[C]φdyn[C],φtot[C]=arg(Ψ(s1)*·Ψ(s2)),φdyn[C]=Ims1s2ds(Ψ(s),Ψ˙(s))=Im{s1s2dsz(s)z˙(s)+is1s2dsz(s)h(s)z(s)}.
h11(s)=h22(s)=12[θ˙(s)sin2ϕ(s)+ϕ˙(s)cos2ϕ(s)sin2θ(s)],h12(s)=h21(s)*=12[θ˙(s)cos2ϕ(s)(sin2θ(s)sin2ϕ(s)+i(1cos2θ(s))ϕ˙(s)].
ga(k^(s2))=ga(k^(s1));Ψ(s2)=eiθΨ(s1)z(s2)=eiθz(s1),n^(s2)=n^(s1).
φg[C]=θIms1s2dsz(s)z˙(s)s1s2dsz(a)h(s)z(s).
φg[C]=12Ω[Cpol]s1s2dsz(s)h(s)z(s).
k^Sdir2:Hk^={EH|k^·E=0}C2;Bk^=B5Hk^={EH|EE=1,k^·E=0};Rk^=Bk^/U(1)={ρ(E)=EER4|EBk^};Bk^S3,Rk^S2,for eachk^Sdir2.
ρRk^ρ=3×3complex matrix,ρ=ρ2=ρ0,Trρ=1,ρk^=0.
k^k^0: Bk^Bk^={E=qk^k^|k^k^|,|q|=1},
T=k^Sdir2Bk^;(a)L=k^Sdir2Rk^·(b)
T=Sdir2×S3,L=Sdir2×Spol2.
pTp=(k^,E),k^Sdir2,EBk^.
p=(k^,E)TEB5.
ifEE*0: k^is fixed upto a sign,resulting in a twofold ambiguity;if EE*=0: k^is fixed upto anSO(2)rotation,more precisely anO(2)rotation,resulting in a continuous ambiguityinvolving linear polarization states.
C0={p(s)=(k^(s),E(s))T|s1ss2}T,
C={E(s)Bk^(s)|s1ss2}B5,

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