Abstract

We present simple closed-form expressions for evaluating the overall and the Pancharatnam–Berry phase introduced by an optical system with either orthogonal or nonorthogonal eigenpolarizations. The formulas provide a meaningful connection with the Pancharatnam–Berry phase associated with nonclosed paths on the Poincaré sphere.

© 2011 Optical Society of America

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References

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  1. S. Pancharatnam, Proc. Indian Acad. Sci. A 44, 247 (1956).
  2. M. V. Berry, J. Mod. Opt. 34, 1401 (1987).
    [CrossRef]
  3. T. F. Jordan, Phys. Rev. A 38, 1590 (1988).
    [CrossRef] [PubMed]
  4. T. van Dijk, H. F. Schouten, W. Ubachs, and T. D. Visser, Opt. Express 18, 10796 (2010).
    [CrossRef] [PubMed]
  5. T. van Dijk, H. F. Schouten, and T. D. Visser, J. Opt. Soc. Am. A 27, 1972 (2010).
    [CrossRef]
  6. M. A. Bandres and J. C. Gutiérrez-Vega, Opt. Lett. 30, 2155 (2005).
    [CrossRef] [PubMed]
  7. A. M. Beckley, T. G. Brown, and M. A. Alonso, Opt. Express 18, 10777 (2010).
    [CrossRef] [PubMed]
  8. Z. Bomzon, V. Kleiner, and E. Hasman, Opt. Lett. 26, 1424 (2001).
    [CrossRef]
  9. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, Opt. Lett. 27, 1141 (2002).
    [CrossRef]

2010

2005

2002

2001

1988

T. F. Jordan, Phys. Rev. A 38, 1590 (1988).
[CrossRef] [PubMed]

1987

M. V. Berry, J. Mod. Opt. 34, 1401 (1987).
[CrossRef]

1956

S. Pancharatnam, Proc. Indian Acad. Sci. A 44, 247 (1956).

Alonso, M. A.

Bandres, M. A.

Beckley, A. M.

Berry, M. V.

M. V. Berry, J. Mod. Opt. 34, 1401 (1987).
[CrossRef]

Biener, G.

Bomzon, Z.

Brown, T. G.

Gutiérrez-Vega, J. C.

Hasman, E.

Jordan, T. F.

T. F. Jordan, Phys. Rev. A 38, 1590 (1988).
[CrossRef] [PubMed]

Kleiner, V.

Pancharatnam, S.

S. Pancharatnam, Proc. Indian Acad. Sci. A 44, 247 (1956).

Schouten, H. F.

Ubachs, W.

van Dijk, T.

Visser, T. D.

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Figures (2)

Fig. 1
Fig. 1

(a) Spherical lune formed by two great circles connecting the orthogonal states Q and Q and the geodesic triangle A B Q on the Poincaré sphere. (b) Curves iso-PB-phase on the Poincaré sphere. (c) Plot of μ 1 + μ 2 + ( μ 1 μ 2 ) Q · A in the complex plane.

Fig. 2
Fig. 2

(a) Nonclosed path on the Poincaré sphere for a set of polarization devices. (b) Triangles defined by the initial and final states with the eigenpolarizations P and Q of the optical system.

Equations (18)

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| a = a x a y , a x , a y C , | a x | 2 + | a y | 2 = 1 ,
A = A 1 A 2 A 3 = a | σ 1 | a a | σ 2 | a a | σ 3 | a = | a x | 2 | a y | 2 2 Re ( a x * a y ) 2 Im ( a x * a y ) R 3 ,
σ 1 1 0 0 1 , σ 2 0 1 1 0 , σ 3 0 i i 0 .
| q 1 = q x q y , | q 2 = q y * q x * , | q x | 2 + | q y | 2 = 1 ,
J = μ 1 | q x | 2 + μ 2 | q y | 2 ( μ 1 μ 2 ) q x q y * ( μ 1 μ 2 ) q x * q y μ 2 | q x | 2 + μ 1 | q y | 2 .
ϕ = arg a | b = arg a | J | a .
ϕ = arg [ μ 1 + μ 2 + ( μ 1 μ 2 ) Q · A ] .
ϕ = arg ( det J ) / 2 + [ Ω A B Q Ω B A ( Q ) ] / 4 ) .
Ω A B Q = 2 arctan [ A · ( B × Q ) 1 + A · B + B · Q + Q · A ] .
γ = arg μ 2 arg μ 1 = arg ( μ 1 * μ 2 )
| a N = J N J 2 J 1 | a 0 = M | a 0 ,
| a N = exp ( i n = 1 N ϕ n 1 , n ) | a N .
Φ P B = n = 1 N ϕ n 1 , n + n = 2 N ϕ A n A n 1 A 0 .
M = 1 Δ μ q q x p y μ p p x q y ( μ p μ q ) q x p x ( μ q μ p ) q y p y μ p q x p y μ q p x q y ,
Φ = arg [ μ q + μ p + ( μ q μ p ) p * | σ 3 ( σ · A 0 | q p * | σ 3 | q ] ,
Φ = arg μ q + ϕ A 0 A N ( P ) = arg μ p + ϕ A 0 A N ( Q ) .
Φ = arg ( det M ) / 2 + [ Ω A 0 A N ( P ) Ω A N A 0 ( Q ) ] / 4 ,
Φ P B [ Ω A O A N ( P ) Ω A N A O ( Q ) ] / 4

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