Abstract

We present a novel setup that allows the observation of the geometric phase that accompanies polarization changes in monochromatic light beams for which the initial and final states are different (so-called non-cyclic changes). This Pancharatnam-Berry phase can depend in a linear or in a nonlinear fashion on the orientation of the optical elements, and sometimes the dependence is singular. Experimental results that confirm these three types of behavior are presented. The observed singular behavior may be applied in the design of optical switches.

© 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. Lond. A Math. Phys. Sci. 392, 45–57 (1984).
    [CrossRef]
  2. M. V. Berry, “Anticipations of the geometric phase,” Phys. Today 43(12), 34–40 (1990).
    [CrossRef]
  3. J. Samuel, and R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
    [CrossRef] [PubMed]
  4. T. F. Jordan, “Berry phases for partial cycles,” Phys. Rev. A 38, 1590–1592 (1988).
    [CrossRef] [PubMed]
  5. A. Shapere, and F. Wilczek, eds., Geometric Phases in Physics (World Scientific, Singapore, 1989).
  6. S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).
  7. S. Pancharatnam, Collected Works (Oxford University Press, Oxford, 1975).
  8. M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
    [CrossRef]
  9. M. Born, and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, seventh (expanded) ed. (Cambridge University Press, Cambridge, 1999).
    [PubMed]
  10. R. Bhandari, “Polarization of light and topological phases,” Phys. Rep. 281, 1–64 (1997).
    [CrossRef]
  11. P. Hariharan, “The geometric phase,” in: Progress in Optics (E. Wolf, ed.) 48, 149–201 (Elsevier, Amsterdam, 2005).
  12. R. Bhandari, and J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988).
    [CrossRef] [PubMed]
  13. 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]
  14. H. Schmitzer, S. Klein, and W. Dultz, “Nonlinearity of Pancharatnam’s topological phase,” Phys. Rev. Lett. 71, 1530–1533 (1993).
    [CrossRef] [PubMed]
  15. R. Bhandari, “Observation of Dirac singularities with light polarization. I,” Phys. Lett. A 171, 262–266 (1992).
    [CrossRef]
  16. R. Bhandari, “Observation of Dirac singularities with light polarization. II,” Phys. Lett. A 171, 267–270 (1992).
    [CrossRef]
  17. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
  18. T. van Dijk, H. F. Schouten, and T. D. Visser, “Geometric interpretation of the Pancharatnam connection and noncyclic polarization changes,” submitted (2010).
  19. C. Brosseau, Fundamentals of Polarized Light (Wiley, New York, 1998).
  20. A. G. Wagh, and V. C. Rakhecha, “On measuring the Pancharatnam phase. I. Interferometry,” Phys. Lett. A 197, 107–111 (1995).
    [CrossRef]
  21. J. F. Nye, Natural Focusing and Fine Structure of Light (IOP Publishing, Bristol, 1999).
  22. G. I. Papadimitriou, C. Papazoglou, and A. S. Pomportsis, Optical Switching (Wiley, Hoboken, 2007).

1997

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

1995

A. G. Wagh, and V. C. Rakhecha, “On measuring the Pancharatnam phase. I. Interferometry,” Phys. Lett. A 197, 107–111 (1995).
[CrossRef]

1993

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

1992

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

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

1988

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]

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

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

1984

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

1956

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

1941

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. Lond. A Math. Phys. Sci. 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]

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]

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]

Mandel, L.

Pancharatnam, S.

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

Rakhecha, V. C.

A. G. Wagh, and V. C. Rakhecha, “On measuring the Pancharatnam phase. I. Interferometry,” Phys. Lett. A 197, 107–111 (1995).
[CrossRef]

Samuel, J.

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]

Schmitzer, H.

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

Simon, R.

Wagh, A. G.

A. G. Wagh, and V. C. Rakhecha, “On measuring the Pancharatnam phase. I. Interferometry,” Phys. Lett. A 197, 107–111 (1995).
[CrossRef]

Wang, L. J.

J. Mod. Opt.

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.

Opt. Lett.

Phys. Lett. A

A. G. Wagh, and V. C. Rakhecha, “On measuring the Pancharatnam phase. I. Interferometry,” Phys. Lett. A 197, 107–111 (1995).
[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]

Phys. Rep.

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

Phys. Rev. A

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

Phys. Rev. Lett.

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

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

Phys. Today

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

Proc. Indian Acad. Sci. A

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

Proc. R. Soc. Lond. A Math. Phys. Sci.

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

Other

A. Shapere, and F. Wilczek, eds., Geometric Phases in Physics (World Scientific, Singapore, 1989).

S. Pancharatnam, Collected Works (Oxford University Press, Oxford, 1975).

M. Born, and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, seventh (expanded) ed. (Cambridge University Press, Cambridge, 1999).
[PubMed]

P. Hariharan, “The geometric phase,” in: Progress in Optics (E. Wolf, ed.) 48, 149–201 (Elsevier, Amsterdam, 2005).

T. van Dijk, H. F. Schouten, and T. D. Visser, “Geometric interpretation of the Pancharatnam connection and noncyclic polarization changes,” submitted (2010).

C. Brosseau, Fundamentals of Polarized Light (Wiley, New York, 1998).

J. F. Nye, Natural Focusing and Fine Structure of Light (IOP Publishing, Bristol, 1999).

G. I. Papadimitriou, C. Papazoglou, and A. S. Pomportsis, Optical Switching (Wiley, Hoboken, 2007).

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Figures (7)

Fig. 1.
Fig. 1.

Non-closed path ABCDE on the Poincaré sphere for a monochromatic light beam that passes through a sequence of polarizers and compensators.

Fig. 2.
Fig. 2.

Sketch of the Mach-Zehnder setup. The light from a He-Ne laser (right-hand top) is split into two beams. All polarizing elements are placed in the upper arm, the lower arm only contains a gray filter. The compensators are depicted with striped holders, the linear polarizers with non-striped holders. The last two pairs of elements are mounted together. The interference pattern of the recombined beams is recorded with either a photo diode or a CCD camera (left-hand bottom).

Fig. 3.
Fig. 3.

Measured intensity as a function of the orientation angle ϕ2. The solid curve is a fit of the measured data to the function C1 + C2 cos(2ϕ2 +C3). The vertical symbols indicate error bars.

Fig. 4.
Fig. 4.

Geometric phase of the final state E when the initial state A coincides with the north pole (blue curve), and when A lies between the equator and the north pole (red curve), both as a function of the orientation angle ϕ1. The solid curves are theoretical predictions [Eq. (14)], the dots and error bars represent measurements. In this example ϕ2 = 0.

Fig. 5.
Fig. 5.

Geometric phase of the final state E when the initial state A coincides with the south pole (blue curve), and when A lies between the equator and the south pole (red curve), both as a function of the orientation angle ϕ1. The solid curves are theoretical predictions [Eq. (14)], the dots and error bars represent measurements. In this example ϕ2 = 0.

Fig. 6.
Fig. 6.

Color-coded plot of the phase of the final state E as a function of the initial state A as described by the two parameters αA and θA [cf. Eq. (1)]. In this example ϕ1 = 3π/4, and ϕ2 = 1.8.

Fig. 7.
Fig. 7.

Singular behavior of the geometric phase of the final state E when the initial state A lies on the equator, as a function of the orientation angle ϕ1. The solid curve is a theoretical prediction [Eq. (14)], the dots and error bars represent measurements. In this example θA = 0.27, αA = 0.0 and ϕ2 = 0.

Equations (16)

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

EA=cosαAsinαAeiθA , (0αAπ/2;πθAπ) ,
EB=eiγcosαBsinαBeiθB , (0αBπ/2;πθBπ) .
EA+EB2=EA2+EB2+2Re(EA·EB*)
Im(EA·EB*)=0,
Re(EA·EB*)>0.
P(ϕ)=(cos2ϕcosϕsinϕcosϕsinϕsin2ϕ),
C(δ,θ)=(cos(δ/2)+isin(δ/2)cos(2θ)isin(δ/2)sin(2θ)isin(δ/2)sin(2θ)cos(δ/2)isin(δ/2)cos(2θ)).
EE=C(π/2,ϕ2π/4)·P(ϕ2)·C(π/2,ϕ1π/4)·P(ϕ1)·EA.
EB=P(ϕ1)·EA=T(A,ϕ1) cosϕ1sinϕ1 ,
EC=C(π/2,ϕ1π/4)·EB=T(A,ϕ1) eiϕ1 1/2i/2 ,
ED=P(ϕ2)·EC=T(A,ϕ1)ei(ϕ2ϕ1) cosϕ2sinϕ2 ,
EE=C(π/2,ϕ2π/4)·ED=T(A,ϕ1) ei(2ϕ2ϕ1) 1/2i/2 ,
T(A,ϕ1)=cosαAcosϕ1+sinαAeiθAsinϕ1cosαAcosϕ1+sinαAeiθAsinϕ1
Ψ=arg[T(A,ϕ1)ei(2ϕ2ϕ1)]
EA2+EE2+2Re(EA·EE*)=1+T(A,ϕ1)2+2H(A,ϕ1)cos(2ϕ2ϕ1+ϕH),
H(A,ϕ1)eiϕH=T*(A,ϕ1)EA·1/2i/2,

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