Abstract

We study geometric phases that arise from (cyclic) transformations of the transverse spatial structure of paraxial optical modes. Our approach involves bosonic ladder operators that, in the spirit of the quantum-mechanical harmonic oscillator, generate sets of transverse optical modes. It applies to modes of all orders in a very natural way and provides a universal geometric interpretation of the phase shifts acquired by nonastigmatic modes under typical experimental conditions.

© 2010 Optical Society of America

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  1. M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984).
    [CrossRef]
  2. S. Pancharatnam, Proc. Ind. Acad. Sci. A 44, 247 (1956).
  3. A. K. Jha, M. Malik, and R. W. Boyd, Phys. Rev. Lett. 101, 180405 (2008).
    [CrossRef] [PubMed]
  4. A. Tomita and R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
    [CrossRef] [PubMed]
  5. R. Y. Chiao and T. F. Jordan, Phys. Lett. A 132, 77 (1988).
    [CrossRef]
  6. S. J. van Enk, Opt. Commun. 102, 59 (1993).
    [CrossRef]
  7. M. Padgett and J. Courtial, Opt. Lett. 24, 430 (1999).
    [CrossRef]
  8. E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett. 90, 203901(2003).
    [CrossRef] [PubMed]
  9. L. Allen and M. J. Padgett, J. Mod. Opt. 54, 487 (2007).
    [CrossRef]
  10. E. J. Galvez and M. A. O’Connell, Proc. SPIE 5736, 166(2005).
    [CrossRef]
  11. M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
    [CrossRef]
  12. G. Nienhuis and L. Allen, Phys. Rev. A 48, 656 (1993).
    [CrossRef] [PubMed]
  13. J. Visser and G. Nienhuis, J. Opt. A 6, S248 (2004).
    [CrossRef]
  14. J. Visser and G. Nienhuis, Phys. Rev. A 70, 013809 (2004).
    [CrossRef]
  15. E. G. Abramochkin and V. G. Volostnikov, J. Opt. A 6, S157(2004).
    [CrossRef]
  16. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
    [CrossRef]
  17. S. Tiwari, J. Mod. Opt. 39, 1097 (1992).
    [CrossRef]
  18. S. J. M. Habraken and G. Nienhuis, J. Math. Phys. 51, 082702 (2010).
    [CrossRef]

2010 (1)

S. J. M. Habraken and G. Nienhuis, J. Math. Phys. 51, 082702 (2010).
[CrossRef]

2008 (1)

A. K. Jha, M. Malik, and R. W. Boyd, Phys. Rev. Lett. 101, 180405 (2008).
[CrossRef] [PubMed]

2007 (1)

L. Allen and M. J. Padgett, J. Mod. Opt. 54, 487 (2007).
[CrossRef]

2005 (1)

E. J. Galvez and M. A. O’Connell, Proc. SPIE 5736, 166(2005).
[CrossRef]

2004 (3)

J. Visser and G. Nienhuis, J. Opt. A 6, S248 (2004).
[CrossRef]

J. Visser and G. Nienhuis, Phys. Rev. A 70, 013809 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A 6, S157(2004).
[CrossRef]

2003 (1)

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett. 90, 203901(2003).
[CrossRef] [PubMed]

1999 (1)

1993 (3)

S. J. van Enk, Opt. Commun. 102, 59 (1993).
[CrossRef]

G. Nienhuis and L. Allen, Phys. Rev. A 48, 656 (1993).
[CrossRef] [PubMed]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

1992 (1)

S. Tiwari, J. Mod. Opt. 39, 1097 (1992).
[CrossRef]

1988 (1)

R. Y. Chiao and T. F. Jordan, Phys. Lett. A 132, 77 (1988).
[CrossRef]

1986 (1)

A. Tomita and R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
[CrossRef] [PubMed]

1984 (1)

M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984).
[CrossRef]

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

1956 (1)

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

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A 6, S157(2004).
[CrossRef]

Allen, L.

L. Allen and M. J. Padgett, J. Mod. Opt. 54, 487 (2007).
[CrossRef]

G. Nienhuis and L. Allen, Phys. Rev. A 48, 656 (1993).
[CrossRef] [PubMed]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

Berry, M. V.

M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984).
[CrossRef]

Boyd, R. W.

A. K. Jha, M. Malik, and R. W. Boyd, Phys. Rev. Lett. 101, 180405 (2008).
[CrossRef] [PubMed]

Chiao, R. Y.

R. Y. Chiao and T. F. Jordan, Phys. Lett. A 132, 77 (1988).
[CrossRef]

A. Tomita and R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
[CrossRef] [PubMed]

Courtial, J.

Crawford, P. R.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett. 90, 203901(2003).
[CrossRef] [PubMed]

Galvez, E. J.

E. J. Galvez and M. A. O’Connell, Proc. SPIE 5736, 166(2005).
[CrossRef]

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett. 90, 203901(2003).
[CrossRef] [PubMed]

Habraken, S. J. M.

S. J. M. Habraken and G. Nienhuis, J. Math. Phys. 51, 082702 (2010).
[CrossRef]

Haglin, P. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett. 90, 203901(2003).
[CrossRef] [PubMed]

Jha, A. K.

A. K. Jha, M. Malik, and R. W. Boyd, Phys. Rev. Lett. 101, 180405 (2008).
[CrossRef] [PubMed]

Jordan, T. F.

R. Y. Chiao and T. F. Jordan, Phys. Lett. A 132, 77 (1988).
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Malik, M.

A. K. Jha, M. Malik, and R. W. Boyd, Phys. Rev. Lett. 101, 180405 (2008).
[CrossRef] [PubMed]

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Nienhuis, G.

S. J. M. Habraken and G. Nienhuis, J. Math. Phys. 51, 082702 (2010).
[CrossRef]

J. Visser and G. Nienhuis, Phys. Rev. A 70, 013809 (2004).
[CrossRef]

J. Visser and G. Nienhuis, J. Opt. A 6, S248 (2004).
[CrossRef]

G. Nienhuis and L. Allen, Phys. Rev. A 48, 656 (1993).
[CrossRef] [PubMed]

O’Connell, M. A.

E. J. Galvez and M. A. O’Connell, Proc. SPIE 5736, 166(2005).
[CrossRef]

Padgett, M.

Padgett, M. J.

L. Allen and M. J. Padgett, J. Mod. Opt. 54, 487 (2007).
[CrossRef]

Pancharatnam, S.

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

Pysher, M. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett. 90, 203901(2003).
[CrossRef] [PubMed]

Sztul, H. I.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett. 90, 203901(2003).
[CrossRef] [PubMed]

Tiwari, S.

S. Tiwari, J. Mod. Opt. 39, 1097 (1992).
[CrossRef]

Tomita, A.

A. Tomita and R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
[CrossRef] [PubMed]

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

van Enk, S. J.

S. J. van Enk, Opt. Commun. 102, 59 (1993).
[CrossRef]

Visser, J.

J. Visser and G. Nienhuis, J. Opt. A 6, S248 (2004).
[CrossRef]

J. Visser and G. Nienhuis, Phys. Rev. A 70, 013809 (2004).
[CrossRef]

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A 6, S157(2004).
[CrossRef]

Williams, R. E.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett. 90, 203901(2003).
[CrossRef] [PubMed]

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

J. Math. Phys. (1)

S. J. M. Habraken and G. Nienhuis, J. Math. Phys. 51, 082702 (2010).
[CrossRef]

J. Mod. Opt. (2)

S. Tiwari, J. Mod. Opt. 39, 1097 (1992).
[CrossRef]

L. Allen and M. J. Padgett, J. Mod. Opt. 54, 487 (2007).
[CrossRef]

J. Opt. A (2)

J. Visser and G. Nienhuis, J. Opt. A 6, S248 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A 6, S157(2004).
[CrossRef]

Opt. Commun. (2)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

S. J. van Enk, Opt. Commun. 102, 59 (1993).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (1)

R. Y. Chiao and T. F. Jordan, Phys. Lett. A 132, 77 (1988).
[CrossRef]

Phys. Rev. A (3)

J. Visser and G. Nienhuis, Phys. Rev. A 70, 013809 (2004).
[CrossRef]

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

G. Nienhuis and L. Allen, Phys. Rev. A 48, 656 (1993).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

A. K. Jha, M. Malik, and R. W. Boyd, Phys. Rev. Lett. 101, 180405 (2008).
[CrossRef] [PubMed]

A. Tomita and R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
[CrossRef] [PubMed]

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett. 90, 203901(2003).
[CrossRef] [PubMed]

Proc. Ind. Acad. Sci. A (1)

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

Proc. R. Soc. Lond. A (1)

M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984).
[CrossRef]

Proc. SPIE (1)

E. J. Galvez and M. A. O’Connell, Proc. SPIE 5736, 166(2005).
[CrossRef]

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Equations (16)

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

( ρ 2 + 2 i k z ) u ( ρ , z ) = 0 ,
u n m ( ρ , z ) = 1 γ ψ n m ( x γ , y γ ) e i k ρ 2 2 R i ζ ( n + m + 1 ) ,
u n m ( ρ , z ) = 1 n ! m ! ( a ^ 1 ( z ) ) n ( a ^ 2 ( z ) ) m u 00 ( ρ , z ) ,
a ^ 1 = σ 1 a ^ x + σ 2 a ^ y and a ^ 2 = σ 2 * a ^ x σ 1 * a ^ y ,
( | σ 1 | 2 | σ 2 | 2 σ 1 σ 2 * + σ 1 * σ 2 i σ 1 σ 2 * + i σ 1 * σ 2 ) = ( cos φ sin ϑ sin φ sin ϑ cos ϑ ) ,
σ ( cos ( φ / 2 ) sin ( φ / 2 ) sin ( φ / 2 ) cos ( φ / 2 ) ) σ = e i φ τ 3 σ ,
τ 3 = 1 2 ( 0 i i 0 )
σ ( e i ϑ / 2 0 0 e i ϑ / 2 ) σ = e i ϑ τ 1 σ ,
τ 1 = 1 2 ( 1 0 0 1 )
σ e i ( π / 2 ) τ 3 e i ϑ τ 1 e i ( π / 2 ) τ 3 = ( cos ( ϑ / 2 ) i sin ( ϑ / 2 ) i sin ( ϑ / 2 ) cos ( ϑ / 2 ) ) σ = e i ϑ τ 2 σ ,
τ 2 = 1 2 ( 0 1 1 0 )
σ = e i χ υ ( ϑ , φ ) ,
σ t = i τ σ ,
χ ( t ) = ( υ τ υ ) t i 0 t d t υ υ t .
χ ( t ) = i C d t υ υ t = 1 2 S d Ω = 1 2 Ω ,
χ n m = ( n m ) χ .

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