OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 31, Iss. 6 — Jun. 1, 2014
  • pp: 1141–1157

Classical light beams and geometric phases

N. Mukunda, S. Chaturvedi, and R. Simon  »View Author Affiliations

JOSA A, Vol. 31, Issue 6, pp. 1141-1157 (2014)

View Full Text Article

Enhanced HTML    Acrobat PDF (402 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(260.5430) Physical optics : Polarization
(350.5500) Other areas of optics : Propagation

ToC Category:
Physical Optics

Original Manuscript: December 17, 2013
Revised Manuscript: March 11, 2014
Manuscript Accepted: March 12, 2014
Published: May 2, 2014

N. Mukunda, S. Chaturvedi, and R. Simon, "Classical light beams and geometric phases," J. Opt. Soc. Am. A 31, 1141-1157 (2014)

Sort:  Author  |  Year  |  Journal  |  Reset  


  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]

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.

Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited