Abstract

The method of obtaining the dynamic and geometric part of the phase introduced by a birefringent medium in two kinds of interferometric experiments is presented. The mathematical formulas for the phases obtained using Jones formalism are visualized with the specific triangles on the Poincaré sphere. Generally, these triangles are similar to those used in the Pancharatnam’s original construction developed by Courtial to calculate the Pancharatnam geometric phase for the light passing through a single birefringent plate. In these graphical constructions following Courtial’s idea, we used the points representing both of the birefringent medium’s eigenvectors. This allowed the most intuitive explanation of the mechanism of dividing the whole phase shift introduced by the birefringent plate into two different parts: dynamical and geometrical. The considered constructions were used as a description of two simple experiments with a birefringent medium in a Mach–Zehnder interferometer and a polariscopic setup. The experimental verifications of our theoretical predictions should convince the reader of the correctness of the assumed model.

© 2011 Optical Society of America

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  1. P. Hariharan, “The geometric phase,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2005), Vol. 48, pp. 149-201.
    [CrossRef]
  2. S. Pancharatnam, “Generalized theory of interference and its applications. Part I. Coherent pencils,” Proc. Indian Acad. Sci. A 44, 247-262 (1956).
  3. S. Pancharatnam, Collected Works (Oxford University, 1975).
  4. M. V. Berry, “The adiabatic phase and Pancharatnam's phase for polarized light,” J. Mod. Opt. 34, 1401-1407 (1987).
    [CrossRef]
  5. M. V. Berry, “Anticipation of the geometric phase,” Phys. Today 43, 34-40 (1990).
    [CrossRef]
  6. G. D. Love, “The unbounded nature of geometrical and dynamical phases in polarization optics,” Opt. Commun. 131, 236-240(1996).
    [CrossRef]
  7. A. G. Wagh, V. C. Rakhecha, J. Summhammer, G. Badurek, H. Weinfurter, B. E. Allman, H. Kaiser, K. Hamacher, D. L. Jacobson, and S. A. Werner, “Experimental separation of geometric and dynamical phases using neutron interferometry,” Phys. Rev. Lett. 78, 755-759 (1997).
    [CrossRef]
  8. R. Bhandari, “SU(2) phase jump and geometrical phases,” Phys. Lett. A 157, 221-225 (1991).
    [CrossRef]
  9. R. Bhandari and J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211-1213(1988).
    [CrossRef] [PubMed]
  10. K. Y. Bliokh, “Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium,” J. Opt. A Pure Appl. Opt. 11, 094009 (2009).
    [CrossRef]
  11. 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]
  12. S. Ramashesan, “The Poincaré sphere and the Pancharatnam phase--some historical remarks,” Curr. Sci. 59, 1154-1158 (1990).
  13. A. G. Wagh and V. C. Rakhecha, “On measuring the Pancharatnam phase. I. Interferometry,” Phys. Lett. A 197, 107-111 (1995).
    [CrossRef]
  14. M. V. Berry and S. Klein, “Geometric phases from stacks of crystal plates,” J. Mod. Opt. 43, 165-180 (1996).
    [CrossRef]
  15. P. Hariharan, S. Mujumdar, and H. Ramachandran,” A simple demonstration of Pancharatnam phase as a geometric phase,” J. Mod. Opt. 46, 1443-1446 (1999).
    [CrossRef]
  16. E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 1-4 (2003).
    [CrossRef]
  17. Y. Ben-Aryeh, “Berry and Pancharatnam topological phases of atomic and optical systems,” J. Opt. B: Quantum Semiclass. Opt. 6, R1-R18 (2004).
    [CrossRef]
  18. N. Murakami, Y. Kato, N. Baba, and T. Ishigaki, “Geometric phase modulation for the separate arms in nulling interferometer,” Opt. Commun. 237, 9-15 (2004).
    [CrossRef]
  19. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Manipulation of the Pancharatnam phase in vectorial vortices,” Opt. Express 14, 4208-4220 (2006).
    [CrossRef] [PubMed]
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  21. J. Courtial, “Wave plates and the Pancharatnam phase,” Opt. Commun. 171, 179-183 (1999).
    [CrossRef]
  22. P. Yeh and C. Gu, “Jones matrix formulation,” in Optics of Liquid Crystal Displays (Wiley, 2010), Chap. 4.1, pp. 173-289.
  23. H. Schmitzer, S. Klein, and W. Dultz, “An experimental test of the path dependency of Pancharatnam's geometric phase,” J. Mod. Opt. 45, 1039-1047 (1998).
    [CrossRef]
  24. R. Bhandari, “Polarization of light and topological phases,” Phys. Rep. 281, 1-64 (1997).
    [CrossRef]
  25. M. Martinelli and P. Vavassori, “A geometric (Pancharatnam) phase approach to the polarization and phase control in the coherent optics circuits,” Opt. Commun. 80, 166-176(1990).
    [CrossRef]
  26. W. A. Woźniak and P. Kurzynowski, “Compact spatial polariscope for light polarization state analysis,” Opt. Express 16, 10471-10479 (2008).
    [CrossRef] [PubMed]
  27. W. A. Woźniak and M. Banach, “Measurements of elliptically birefringent media parameters in optical vortex birefringence compensator,” Appl. Opt. 47, 3390-3396 (2008).
    [CrossRef] [PubMed]

2010 (1)

2009 (1)

K. Y. Bliokh, “Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium,” J. Opt. A Pure Appl. Opt. 11, 094009 (2009).
[CrossRef]

2008 (2)

2006 (1)

2004 (2)

Y. Ben-Aryeh, “Berry and Pancharatnam topological phases of atomic and optical systems,” J. Opt. B: Quantum Semiclass. Opt. 6, R1-R18 (2004).
[CrossRef]

N. Murakami, Y. Kato, N. Baba, and T. Ishigaki, “Geometric phase modulation for the separate arms in nulling interferometer,” Opt. Commun. 237, 9-15 (2004).
[CrossRef]

2003 (1)

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 1-4 (2003).
[CrossRef]

1999 (2)

P. Hariharan, S. Mujumdar, and H. Ramachandran,” A simple demonstration of Pancharatnam phase as a geometric phase,” J. Mod. Opt. 46, 1443-1446 (1999).
[CrossRef]

J. Courtial, “Wave plates and the Pancharatnam phase,” Opt. Commun. 171, 179-183 (1999).
[CrossRef]

1998 (1)

H. Schmitzer, S. Klein, and W. Dultz, “An experimental test of the path dependency of Pancharatnam's geometric phase,” J. Mod. Opt. 45, 1039-1047 (1998).
[CrossRef]

1997 (2)

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

A. G. Wagh, V. C. Rakhecha, J. Summhammer, G. Badurek, H. Weinfurter, B. E. Allman, H. Kaiser, K. Hamacher, D. L. Jacobson, and S. A. Werner, “Experimental separation of geometric and dynamical phases using neutron interferometry,” Phys. Rev. Lett. 78, 755-759 (1997).
[CrossRef]

1996 (2)

G. D. Love, “The unbounded nature of geometrical and dynamical phases in polarization optics,” Opt. Commun. 131, 236-240(1996).
[CrossRef]

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

1995 (1)

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

1991 (1)

R. Bhandari, “SU(2) phase jump and geometrical phases,” Phys. Lett. A 157, 221-225 (1991).
[CrossRef]

1990 (3)

S. Ramashesan, “The Poincaré sphere and the Pancharatnam phase--some historical remarks,” Curr. Sci. 59, 1154-1158 (1990).

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

M. Martinelli and P. Vavassori, “A geometric (Pancharatnam) phase approach to the polarization and phase control in the coherent optics circuits,” Opt. Commun. 80, 166-176(1990).
[CrossRef]

1988 (2)

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]

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

1956 (1)

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

Allman, B. E.

A. G. Wagh, V. C. Rakhecha, J. Summhammer, G. Badurek, H. Weinfurter, B. E. Allman, H. Kaiser, K. Hamacher, D. L. Jacobson, and S. A. Werner, “Experimental separation of geometric and dynamical phases using neutron interferometry,” Phys. Rev. Lett. 78, 755-759 (1997).
[CrossRef]

Baba, N.

N. Murakami, Y. Kato, N. Baba, and T. Ishigaki, “Geometric phase modulation for the separate arms in nulling interferometer,” Opt. Commun. 237, 9-15 (2004).
[CrossRef]

Badurek, G.

A. G. Wagh, V. C. Rakhecha, J. Summhammer, G. Badurek, H. Weinfurter, B. E. Allman, H. Kaiser, K. Hamacher, D. L. Jacobson, and S. A. Werner, “Experimental separation of geometric and dynamical phases using neutron interferometry,” Phys. Rev. Lett. 78, 755-759 (1997).
[CrossRef]

Banach, M.

Ben-Aryeh, Y.

Y. Ben-Aryeh, “Berry and Pancharatnam topological phases of atomic and optical systems,” J. Opt. B: Quantum Semiclass. Opt. 6, R1-R18 (2004).
[CrossRef]

Berry, M. V.

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

M. V. Berry, “Anticipation of the geometric phase,” Phys. Today 43, 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]

Bhandari, R.

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

R. Bhandari, “SU(2) phase jump and geometrical phases,” Phys. Lett. A 157, 221-225 (1991).
[CrossRef]

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.

Bliokh, K. Y.

K. Y. Bliokh, “Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium,” J. Opt. A Pure Appl. Opt. 11, 094009 (2009).
[CrossRef]

Chyba, T. H.

Courtial, J.

J. Courtial, “Wave plates and the Pancharatnam phase,” Opt. Commun. 171, 179-183 (1999).
[CrossRef]

Crawford, P. R.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 1-4 (2003).
[CrossRef]

Dultz, W.

H. Schmitzer, S. Klein, and W. Dultz, “An experimental test of the path dependency of Pancharatnam's geometric phase,” J. Mod. Opt. 45, 1039-1047 (1998).
[CrossRef]

Galvez, E. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 1-4 (2003).
[CrossRef]

Gu, C.

P. Yeh and C. Gu, “Jones matrix formulation,” in Optics of Liquid Crystal Displays (Wiley, 2010), Chap. 4.1, pp. 173-289.

Haglin, P. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 1-4 (2003).
[CrossRef]

Hamacher, K.

A. G. Wagh, V. C. Rakhecha, J. Summhammer, G. Badurek, H. Weinfurter, B. E. Allman, H. Kaiser, K. Hamacher, D. L. Jacobson, and S. A. Werner, “Experimental separation of geometric and dynamical phases using neutron interferometry,” Phys. Rev. Lett. 78, 755-759 (1997).
[CrossRef]

Hariharan, P.

P. Hariharan, S. Mujumdar, and H. Ramachandran,” A simple demonstration of Pancharatnam phase as a geometric phase,” J. Mod. Opt. 46, 1443-1446 (1999).
[CrossRef]

P. Hariharan, “The geometric phase,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2005), Vol. 48, pp. 149-201.
[CrossRef]

Hasman, E.

Ishigaki, T.

N. Murakami, Y. Kato, N. Baba, and T. Ishigaki, “Geometric phase modulation for the separate arms in nulling interferometer,” Opt. Commun. 237, 9-15 (2004).
[CrossRef]

Jacobson, D. L.

A. G. Wagh, V. C. Rakhecha, J. Summhammer, G. Badurek, H. Weinfurter, B. E. Allman, H. Kaiser, K. Hamacher, D. L. Jacobson, and S. A. Werner, “Experimental separation of geometric and dynamical phases using neutron interferometry,” Phys. Rev. Lett. 78, 755-759 (1997).
[CrossRef]

Kaiser, H.

A. G. Wagh, V. C. Rakhecha, J. Summhammer, G. Badurek, H. Weinfurter, B. E. Allman, H. Kaiser, K. Hamacher, D. L. Jacobson, and S. A. Werner, “Experimental separation of geometric and dynamical phases using neutron interferometry,” Phys. Rev. Lett. 78, 755-759 (1997).
[CrossRef]

Kato, Y.

N. Murakami, Y. Kato, N. Baba, and T. Ishigaki, “Geometric phase modulation for the separate arms in nulling interferometer,” Opt. Commun. 237, 9-15 (2004).
[CrossRef]

Klein, S.

H. Schmitzer, S. Klein, and W. Dultz, “An experimental test of the path dependency of Pancharatnam's geometric phase,” J. Mod. Opt. 45, 1039-1047 (1998).
[CrossRef]

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

Kleiner, V.

Kurzynowski, P.

Love, G. D.

G. D. Love, “The unbounded nature of geometrical and dynamical phases in polarization optics,” Opt. Commun. 131, 236-240(1996).
[CrossRef]

Mandel, L.

Martinelli, M.

M. Martinelli and P. Vavassori, “A geometric (Pancharatnam) phase approach to the polarization and phase control in the coherent optics circuits,” Opt. Commun. 80, 166-176(1990).
[CrossRef]

Mujumdar, S.

P. Hariharan, S. Mujumdar, and H. Ramachandran,” A simple demonstration of Pancharatnam phase as a geometric phase,” J. Mod. Opt. 46, 1443-1446 (1999).
[CrossRef]

Murakami, N.

N. Murakami, Y. Kato, N. Baba, and T. Ishigaki, “Geometric phase modulation for the separate arms in nulling interferometer,” Opt. Commun. 237, 9-15 (2004).
[CrossRef]

Niv, A.

Pancharatnam, S.

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

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

Pysher, M. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 1-4 (2003).
[CrossRef]

Rakhecha, V. C.

A. G. Wagh, V. C. Rakhecha, J. Summhammer, G. Badurek, H. Weinfurter, B. E. Allman, H. Kaiser, K. Hamacher, D. L. Jacobson, and S. A. Werner, “Experimental separation of geometric and dynamical phases using neutron interferometry,” Phys. Rev. Lett. 78, 755-759 (1997).
[CrossRef]

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

Ramachandran, H.

P. Hariharan, S. Mujumdar, and H. Ramachandran,” A simple demonstration of Pancharatnam phase as a geometric phase,” J. Mod. Opt. 46, 1443-1446 (1999).
[CrossRef]

Ramashesan, S.

S. Ramashesan, “The Poincaré sphere and the Pancharatnam phase--some historical remarks,” Curr. Sci. 59, 1154-1158 (1990).

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]

Schmitzer, H.

H. Schmitzer, S. Klein, and W. Dultz, “An experimental test of the path dependency of Pancharatnam's geometric phase,” J. Mod. Opt. 45, 1039-1047 (1998).
[CrossRef]

Schouten, H. F.

Simon, R.

Summhammer, J.

A. G. Wagh, V. C. Rakhecha, J. Summhammer, G. Badurek, H. Weinfurter, B. E. Allman, H. Kaiser, K. Hamacher, D. L. Jacobson, and S. A. Werner, “Experimental separation of geometric and dynamical phases using neutron interferometry,” Phys. Rev. Lett. 78, 755-759 (1997).
[CrossRef]

Sztul, H. I.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 1-4 (2003).
[CrossRef]

Ubachs, W.

van Dijk, T.

Vavassori, P.

M. Martinelli and P. Vavassori, “A geometric (Pancharatnam) phase approach to the polarization and phase control in the coherent optics circuits,” Opt. Commun. 80, 166-176(1990).
[CrossRef]

Visser, T. D.

Wagh, A. G.

A. G. Wagh, V. C. Rakhecha, J. Summhammer, G. Badurek, H. Weinfurter, B. E. Allman, H. Kaiser, K. Hamacher, D. L. Jacobson, and S. A. Werner, “Experimental separation of geometric and dynamical phases using neutron interferometry,” Phys. Rev. Lett. 78, 755-759 (1997).
[CrossRef]

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.

Weinfurter, H.

A. G. Wagh, V. C. Rakhecha, J. Summhammer, G. Badurek, H. Weinfurter, B. E. Allman, H. Kaiser, K. Hamacher, D. L. Jacobson, and S. A. Werner, “Experimental separation of geometric and dynamical phases using neutron interferometry,” Phys. Rev. Lett. 78, 755-759 (1997).
[CrossRef]

Werner, S. A.

A. G. Wagh, V. C. Rakhecha, J. Summhammer, G. Badurek, H. Weinfurter, B. E. Allman, H. Kaiser, K. Hamacher, D. L. Jacobson, and S. A. Werner, “Experimental separation of geometric and dynamical phases using neutron interferometry,” Phys. Rev. Lett. 78, 755-759 (1997).
[CrossRef]

Williams, R. E.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 1-4 (2003).
[CrossRef]

Wozniak, W. A.

Yeh, P.

P. Yeh and C. Gu, “Jones matrix formulation,” in Optics of Liquid Crystal Displays (Wiley, 2010), Chap. 4.1, pp. 173-289.

Appl. Opt. (1)

Curr. Sci. (1)

S. Ramashesan, “The Poincaré sphere and the Pancharatnam phase--some historical remarks,” Curr. Sci. 59, 1154-1158 (1990).

J. Mod. Opt. (4)

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

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

P. Hariharan, S. Mujumdar, and H. Ramachandran,” A simple demonstration of Pancharatnam phase as a geometric phase,” J. Mod. Opt. 46, 1443-1446 (1999).
[CrossRef]

H. Schmitzer, S. Klein, and W. Dultz, “An experimental test of the path dependency of Pancharatnam's geometric phase,” J. Mod. Opt. 45, 1039-1047 (1998).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

K. Y. Bliokh, “Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium,” J. Opt. A Pure Appl. Opt. 11, 094009 (2009).
[CrossRef]

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

Y. Ben-Aryeh, “Berry and Pancharatnam topological phases of atomic and optical systems,” J. Opt. B: Quantum Semiclass. Opt. 6, R1-R18 (2004).
[CrossRef]

Opt. Commun. (4)

N. Murakami, Y. Kato, N. Baba, and T. Ishigaki, “Geometric phase modulation for the separate arms in nulling interferometer,” Opt. Commun. 237, 9-15 (2004).
[CrossRef]

J. Courtial, “Wave plates and the Pancharatnam phase,” Opt. Commun. 171, 179-183 (1999).
[CrossRef]

G. D. Love, “The unbounded nature of geometrical and dynamical phases in polarization optics,” Opt. Commun. 131, 236-240(1996).
[CrossRef]

M. Martinelli and P. Vavassori, “A geometric (Pancharatnam) phase approach to the polarization and phase control in the coherent optics circuits,” Opt. Commun. 80, 166-176(1990).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Lett. A (2)

R. Bhandari, “SU(2) phase jump and geometrical phases,” Phys. Lett. A 157, 221-225 (1991).
[CrossRef]

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

Phys. Rep. (1)

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

Phys. Rev. Lett. (3)

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 1-4 (2003).
[CrossRef]

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

A. G. Wagh, V. C. Rakhecha, J. Summhammer, G. Badurek, H. Weinfurter, B. E. Allman, H. Kaiser, K. Hamacher, D. L. Jacobson, and S. A. Werner, “Experimental separation of geometric and dynamical phases using neutron interferometry,” Phys. Rev. Lett. 78, 755-759 (1997).
[CrossRef]

Phys. Today (1)

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

Proc. Indian Acad. Sci. A (1)

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

Other (3)

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

P. Hariharan, “The geometric phase,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2005), Vol. 48, pp. 149-201.
[CrossRef]

P. Yeh and C. Gu, “Jones matrix formulation,” in Optics of Liquid Crystal Displays (Wiley, 2010), Chap. 4.1, pp. 173-289.

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Figures (7)

Fig. 1
Fig. 1

(a) Set of birefringent media in a Mach–Zehnder setup and (b) proper construction of the Pancharatnam’s triangle to calculate the geometrical phase.

Fig. 2
Fig. 2

(a) Single birefringent medium in a Mach–Zehnder setup and (b) Courtial’s idea of calculating the Pancharatnam’s phase in this case.

Fig. 3
Fig. 3

(a) Idea of construction of two triangles on the Poincaré sphere using the points representing both eigenwaves of the birefringent medium; (b) difference of two triangles forming a quadrangle ABCD with the area proportional to the desired geometrical phase.

Fig. 4
Fig. 4

(a) Polariscope setup with an elliptically birefringent Wollaston compensator; (b) construction of the characteristic lune on the Poincaré sphere, which explains the geometric phase calculating method.

Fig. 5
Fig. 5

(a) Polariscope setup with an equivalent elliptically birefringent medium: linearly birefringent Wollaston compensator combined with linear quarter-wave plate; (b) construction of the characteristic lune on the Poincaré sphere in this case.

Fig. 6
Fig. 6

Comparison of the theoretical and experimental results obtained from polariscope experiments for the variable analyzer azimuth angle Δ α A and some chosen ellipticity angles ϑ F = α q of the medium: (a)  ϑ F = 45 ° , (b)  ϑ F = 25 ° , (c)  ϑ F = 5 ° , (d)  ϑ F = 1 ° . Solid curves, theoretical curves; dots, experimental results.

Fig. 7
Fig. 7

Comparison of the theoretical and experimental results obtained from polariscope experiments for the variable ellipticity angle ϑ F = α q of the medium and two chosen azimuth angles α A of the analyzer: (a)  α A = 0 ° , (b)  α A = 45 ° . Solid curves, theoretical curves; dots, experimental results.

Equations (17)

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φ arg ( E A · E B ) ,
E A · E B = E A · ( M · E A ) = exp ( i δ ¯ ) · ( cos γ 2 + i sin γ 2 cos 2 α ) ,
φ = δ ¯ + arctan ( tan γ 2 · cos 2 α ) .
φ = φ D + φ G ,
φ D = δ ¯ ,
φ G = arctan ( tan γ 2 · cos 2 α ) .
Ω F 2 = γ 2 arctan ( tan γ 2 · cos 2 α ) ,
Ω S 2 = γ 2 + arctan ( tan γ 2 · cos 2 α ) ,
φ = δ F Ω F 2 = δ S + Ω S 2 .
φ D = δ F and φ G = Ω F 2
φ D = δ S and φ G = + Ω S 2 ,
φ D = δ ¯ = δ F + δ S 2 ,
φ G = arctan ( tan γ 2 · cos 2 α ) = Ω S / 2 Ω F / 2 2 .
I ( x ) 1 + V · cos [ γ ( x ) + φ G ] ,
φ G = arctan ( tan 2 α A · sin 2 ϑ F ) .
Ω 2 = Ω F 2 + Ω S 2 = arctan ( tan 2 α A · sin 2 ϑ F ) ,
φ G = arctan ( tan 2 Δ α A · sin 2 α q ) ,

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