Abstract

If the state of polarization of a monochromatic light beam is changed in a cyclical manner, the beam acquires—in addition to the usual dynamic phase—a geometric phase. This geometric or Pancharatnam–Berry phase equals half the solid angle of the contour traced out on the Poincaré sphere. We show that such a geometric interpretation also exists for the Pancharatnam connection, the criterion according to which two beams with different polarization states are said to be in phase. This interpretation offers what is to our knowledge a new and intuitive method to calculate the geometric phase that accompanies non-cyclic polarization changes.

© 2010 Optical Society of America

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References

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  1. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London, Ser. A 392, 45–57 (1984).
    [CrossRef]
  2. M. V. Berry, “Anticipations of the geometric phase,” Phys. Today 43(12), 34–40 (1990).
    [CrossRef]
  3. S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Indian Acad. Sci., Sect. A 44, 247–262 (1956).
  4. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).
  5. M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
    [CrossRef]
  6. R. Bhandari, “Polarization of light and topological phases,” Phys. Rep. 281, 1–64 (1997).
    [CrossRef]
  7. P. Hariharan, “The geometric phase,” in Progress in Optics, E.Wolf, ed., Vol. 48 (Elsevier, 2005), pp. 149–193.
    [CrossRef]
  8. R. Bhandari and J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988).
    [CrossRef] [PubMed]
  9. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam–Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. 27, 1141–1143 (2002).
    [CrossRef]
  10. G. Biener, Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Manipulation of polarization-dependent multivortices with quasi-periodic subwavelength structures,” Opt. Lett. 31, 1594–1596 (2006).
    [CrossRef] [PubMed]
  11. J. Samuel and R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
    [CrossRef] [PubMed]
  12. T. F. Jordan, “Berry phases for partial cycles,” Phys. Rev. A 38, 1590–1592 (1988).
    [CrossRef] [PubMed]
  13. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
  14. F. Eriksson, “On the measure of solid angles,” Math. Mag. 63, 184–187 (1990).
    [CrossRef]
  15. L. E. J. Brouwer, “Über Abbildung von Mannigfaltigkeiten,” Math. Ann. LXXI, 97–115 (1912).
  16. T. H. Chyba, L. J. Wang, L. Mandel, and R. Simon, “Measurement of the Pancharatnam phase for a light beam,” Opt. Lett. 13, 562–564 (1988).
    [CrossRef] [PubMed]
  17. H. Schmitzer, S. Klein, and W. Dultz, “Nonlinearity of Pancharatnam’s topological phase,” Phys. Rev. Lett. 71, 1530–1533 (1993).
    [CrossRef] [PubMed]
  18. R. Bhandari, “Observation of Dirac singularities with light polarization. I,” Phys. Lett. A 171, 262–266 (1992).
    [CrossRef]
  19. R. Bhandari, “Observation of Dirac singularities with light polarization. II,” Phys. Lett. A 171, 267–270 (1992).
    [CrossRef]
  20. T. van Dijk, H. F. Schouten, W. Ubachs, and T. D. Visser, “The Pancharatnam–Berry phase for non-cyclic polarization changes,” Opt. Express 18, 10796–10804 (2010).
    [CrossRef] [PubMed]

2010 (1)

2006 (1)

2002 (1)

1997 (1)

R. Bhandari, “Polarization of light and topological phases,” Phys. Rep. 281, 1–64 (1997).
[CrossRef]

1993 (1)

H. Schmitzer, S. Klein, and W. Dultz, “Nonlinearity of Pancharatnam’s topological phase,” Phys. Rev. Lett. 71, 1530–1533 (1993).
[CrossRef] [PubMed]

1992 (2)

R. Bhandari, “Observation of Dirac singularities with light polarization. I,” Phys. Lett. A 171, 262–266 (1992).
[CrossRef]

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

1990 (2)

F. Eriksson, “On the measure of solid angles,” Math. Mag. 63, 184–187 (1990).
[CrossRef]

M. V. Berry, “Anticipations of the geometric phase,” Phys. Today 43(12), 34–40 (1990).
[CrossRef]

1988 (4)

R. Bhandari and J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988).
[CrossRef] [PubMed]

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

T. F. Jordan, “Berry phases for partial cycles,” Phys. Rev. A 38, 1590–1592 (1988).
[CrossRef] [PubMed]

T. H. Chyba, L. J. Wang, L. Mandel, and R. Simon, “Measurement of the Pancharatnam phase for a light beam,” Opt. Lett. 13, 562–564 (1988).
[CrossRef] [PubMed]

1987 (1)

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

1984 (1)

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

1956 (1)

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

1941 (1)

1912 (1)

L. E. J. Brouwer, “Über Abbildung von Mannigfaltigkeiten,” Math. Ann. LXXI, 97–115 (1912).

Berry, M. V.

M. V. Berry, “Anticipations of the geometric phase,” Phys. Today 43(12), 34–40 (1990).
[CrossRef]

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

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

Bhandari, R.

R. Bhandari, “Polarization of light and topological phases,” Phys. Rep. 281, 1–64 (1997).
[CrossRef]

R. Bhandari, “Observation of Dirac singularities with light polarization. I,” Phys. Lett. A 171, 262–266 (1992).
[CrossRef]

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

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

R. Bhandari and J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988).
[CrossRef] [PubMed]

Biener, G.

Bomzon, Z.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).

Brouwer, L. E. J.

L. E. J. Brouwer, “Über Abbildung von Mannigfaltigkeiten,” Math. Ann. LXXI, 97–115 (1912).

Chyba, T. H.

Dultz, W.

H. Schmitzer, S. Klein, and W. Dultz, “Nonlinearity of Pancharatnam’s topological phase,” Phys. Rev. Lett. 71, 1530–1533 (1993).
[CrossRef] [PubMed]

Eriksson, F.

F. Eriksson, “On the measure of solid angles,” Math. Mag. 63, 184–187 (1990).
[CrossRef]

Gorodetski, Y.

Hariharan, P.

P. Hariharan, “The geometric phase,” in Progress in Optics, E.Wolf, ed., Vol. 48 (Elsevier, 2005), pp. 149–193.
[CrossRef]

Hasman, E.

Jones, R. C.

Jordan, T. F.

T. F. Jordan, “Berry phases for partial cycles,” Phys. Rev. A 38, 1590–1592 (1988).
[CrossRef] [PubMed]

Klein, S.

H. Schmitzer, S. Klein, and W. Dultz, “Nonlinearity of Pancharatnam’s topological phase,” Phys. Rev. Lett. 71, 1530–1533 (1993).
[CrossRef] [PubMed]

Kleiner, V.

Mandel, L.

Niv, A.

Pancharatnam, S.

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

Samuel, J.

R. Bhandari and J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988).
[CrossRef] [PubMed]

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

Schmitzer, H.

H. Schmitzer, S. Klein, and W. Dultz, “Nonlinearity of Pancharatnam’s topological phase,” Phys. Rev. Lett. 71, 1530–1533 (1993).
[CrossRef] [PubMed]

Schouten, H. F.

Simon, R.

Ubachs, W.

van Dijk, T.

Visser, T. D.

Wang, L. J.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).

J. Mod. Opt. (1)

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

Math. Ann. (1)

L. E. J. Brouwer, “Über Abbildung von Mannigfaltigkeiten,” Math. Ann. LXXI, 97–115 (1912).

Math. Mag. (1)

F. Eriksson, “On the measure of solid angles,” Math. Mag. 63, 184–187 (1990).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Lett. A (2)

R. Bhandari, “Observation of Dirac singularities with light polarization. I,” Phys. Lett. A 171, 262–266 (1992).
[CrossRef]

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

Phys. Rep. (1)

R. Bhandari, “Polarization of light and topological phases,” Phys. Rep. 281, 1–64 (1997).
[CrossRef]

Phys. Rev. A (1)

T. F. Jordan, “Berry phases for partial cycles,” Phys. Rev. A 38, 1590–1592 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

R. Bhandari and J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988).
[CrossRef] [PubMed]

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

H. Schmitzer, S. Klein, and W. Dultz, “Nonlinearity of Pancharatnam’s topological phase,” Phys. Rev. Lett. 71, 1530–1533 (1993).
[CrossRef] [PubMed]

Phys. Today (1)

M. V. Berry, “Anticipations of the geometric phase,” Phys. Today 43(12), 34–40 (1990).
[CrossRef]

Proc. Indian Acad. Sci., Sect. A (1)

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

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

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

Other (2)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).

P. Hariharan, “The geometric phase,” in Progress in Optics, E.Wolf, ed., Vol. 48 (Elsevier, 2005), pp. 149–193.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

The great circle through A, B, and basis state X. (a) If state B does not lie on the segment between A and X , then the sum of the three geodesics X A , A B , and B X is zero. (b) If B lies on the segment between A and X , then the sum of the three geodesics equals the great circle.

Fig. 2
Fig. 2

Illustration of the intersection of the plane given by Eq. (10) and the Poincaré sphere. This intersection is a circle (indicated by the dashed curve) that runs through the points A , X , and B. All points on the circle segment that runs from A to B to X constitute the set { B } of states that have the same phase difference γ A B with respect to A as the state B. The great circle through A and X is shown as a solid-dotted curve.

Fig. 3
Fig. 3

Selected contours of the phase γ A B for the case A = ( 0 , 0.8 , 0.6 ) . The basis state X, the equator (Eq.), and the meridian through X are also shown.

Fig. 4
Fig. 4

Contours of equal phase of γ A B for the case that the state A is taken to be (0.6,0,0.8). Two singular points with opposite topological charges can be seen at A and X .

Fig. 5
Fig. 5

Contours of equal phase of γ A B for the case that the state A is taken to be (0,0,1). The singularity at A is seen to have a topological charge of + 1 .

Fig. 6
Fig. 6

Illustration of the generalized solid angle Ω X A B C . In going from state A to state B, the beam acquires a geometric phase equal to half the solid angle Ω X A B , which is positive. In going from B to C the acquired phase equals half the solid angle Ω X B C , which is negative. Since the triangle B K X does not contribute, this is equivalent to the generalized solid angle Ω X A B C , which equals half the solid angle of the triangle A B K (positive), plus half the solid angle of the triangle X K C (negative).

Fig. 7
Fig. 7

Illustration of the equality Ω N A B C + Ω C B A X = Ω N A X C . Such a construction can be made for any choice of states.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

E = cos   α e ̂ 1 + sin   α   exp ( i θ ) e ̂ 2 ,
E A = ( cos   α A , sin   α A e i θ A ) T ,
E B = e i γ A B ( cos   α B , sin   α B e i θ B ) T .
| E A + E B | 2 = | E A | 2 + | E B | 2 + 2   Re ( E A E B )
Im ( E A E B ) = 0 ,
Re ( E A E B ) > 0.
γ A B = Ω X A B / 2.
tan ( Ω A B C 2 ) = A ( B × C ) 1 + B C + A C + A B .
tan   γ A B = A y B z A z B y 1 + B x + A x + A x B x + A y B y + A z B z .
c x B x + c y B y + c z B z + D = 0 ,
c x = tan   γ A B ( 1 + A x ) ,
c y = tan   γ A B A y + A z ,
c z = tan   γ A B A z A y ,
D = c x .
γ A B = Ω X A B 2 = 1 2 π / 2 π 0 ϕ sin   θ d ϕ d θ = 1 2 ϕ .
γ A B C γ A B + γ B C = ( Ω X A B + Ω X B C ) / 2 = Ω X A B C / 2 ,
γ A B C γ A B + γ B C = ( Ω N A B + Ω N B C ) / 2 = Ω N A B C / 2.
Ω N A B C Ω X A B C = Ω N A B C + Ω C B A X = Ω N A X C .
γ A B C γ A B C = Ω N A X C / 2.

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