Abstract

An exact Wigner representation of optical spatial modes carrying orbital angular momentum is found in closed form by exploiting the underlying SU(2) Lie-group algebra of their associated Poincaré sphere. Orthogonality relations and observables of these states are obtained within the phase space picture. Development of geometric phases on mode transformations is also elucidated.

© 2005 Optical Society of America

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References

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    [CrossRef]

2004 (1)

2003 (3)

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, J. Opt. Soc. Am. A 20, 1635 (2003).
[CrossRef]

M. J. Bastiaans and T. Alieva, Opt. Lett. 28, 2443 (2003).
[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]

2002 (1)

D. Dragoman, Prog. Opt. 43, 433 (2002).
[CrossRef]

2001 (2)

2000 (3)

1999 (2)

M. J. Padgett and J. Courtial, Opt. Lett. 24, 430 (1999).
[CrossRef]

G. S. Agarwal, J. Opt. Soc. Am. A 12, 2914 (1999).
[CrossRef]

1995 (1)

R. Gase, IEEE J. Quantum Electron. 31, 1811 (1995).
[CrossRef]

1993 (2)

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

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

1984 (1)

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

1979 (1)

Agarwal, G. S.

R. Simon and G. S. Agarwal, Opt. Lett. 25, 1313 (2000).
[CrossRef]

G. S. Agarwal, J. Opt. Soc. Am. A 12, 2914 (1999).
[CrossRef]

Alieva, T.

Allen, L.

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

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP Publishing, Bristol, UK 2003).
[CrossRef]

Barnett, S. M.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP Publishing, Bristol, UK 2003).
[CrossRef]

S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Clarendon, Oxford, UK 1997).

Bastiaans, M. J.

Bekshaev, A. Ya.

Berry, M. V.

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

Courtial, J.

A. T. O’Neil and J. Courtial, Opt. Commun. 181, 35 (2000).
[CrossRef]

M. J. Padgett and J. Courtial, Opt. Lett. 24, 430 (1999).
[CrossRef]

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]

Dragoman, D.

D. Dragoman, Prog. Opt. 43, 433 (2002).
[CrossRef]

Feng, S.

Galvez, E. 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]

Gase, R.

R. Gase, IEEE J. Quantum Electron. 31, 1811 (1995).
[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]

Movilla, J. M.

Mukunda, N.

Nienhuis, G.

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

O’Neil, A. T.

A. T. O’Neil and J. Courtial, Opt. Commun. 181, 35 (2000).
[CrossRef]

Padgett, M. J.

M. J. Padgett and J. Courtial, Opt. Lett. 24, 430 (1999).
[CrossRef]

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP Publishing, Bristol, UK 2003).
[CrossRef]

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]

Radmore, P. M.

S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Clarendon, Oxford, UK 1997).

Serna, J.

Simon, R.

Soskin, M. S.

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]

van Enk, S. J.

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

Vasnetsov, M. V.

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]

Winful, H. G.

IEEE J. Quantum Electron. (1)

R. Gase, IEEE J. Quantum Electron. 31, 1811 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (2)

A. T. O’Neil and J. Courtial, Opt. Commun. 181, 35 (2000).
[CrossRef]

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

Opt. Lett. (6)

Phys. Rev. A (1)

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

Phys. Rev. Lett. (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]

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

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

Prog. Opt. (1)

D. Dragoman, Prog. Opt. 43, 433 (2002).
[CrossRef]

Other (2)

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP Publishing, Bristol, UK 2003).
[CrossRef]

S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Clarendon, Oxford, UK 1997).

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Figures (1)

Fig. 1
Fig. 1

(a) Orbital Poincaré sphere of third-order modes. (b) Geometric phase development under mode transformations and its equivalent mode-converter setup.

Equations (13)

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ψ θ , φ = cos ( θ 2 ) LG 0 1 + exp ( i φ ) sin ( θ 2 ) LG 0 1 .
θ , φ l , p = exp ( i θ L ̂ u φ ) l , p U ̂ ( θ , φ ) l , p .
L ̂ x = x ̂ 2 y ̂ 2 2 w 2 + ( p ̂ x 2 p ̂ y 2 ) w 2 8 ƛ 2 ,
L ̂ y = x ̂ y ̂ w 2 + p ̂ x p ̂ y w 2 4 ƛ ,
L ̂ z = x ̂ p ̂ y y ̂ p ̂ x 2 ƛ .
S = [ c θ 0 z 0 s θ s φ z 0 s θ c φ 0 c θ z 0 s θ c φ z 0 s θ s φ s θ s φ z 0 s θ c φ z 0 c θ 0 s θ c φ z 0 s θ s φ z 0 0 c θ ] ,
W l , p ( ζ ; θ , φ ) = ( 1 ) N π 2 ƛ 2 exp ( Q 0 ) L N l 2 ( Q 0 4 Q u r ) L N + l 2 ( Q 0 + 4 Q u r ) ,
l , p θ , φ θ , φ l , p 2 = ( 2 π ƛ ) 2 d 4 ζ W l , p ( ζ ; θ , φ ) W l , p ( ζ ; θ , φ ) = [ k = 0 ( N l ) 2 ( 1 ) k ( N + l 2 k ) ( N l 2 k ) ( 1 ρ ρ ) k ] 2 × ρ N δ l , l δ p , p ,
u r s u v = z 0 u + v w r + s + u + v d 4 ζ x r y s p x u p y v W ( ζ ) ,
μ r s u v = 1 16 r + s u v r s + u v ( N + 1 ) + 1 4 ( r v s u ) l cos θ + 1 4 ( r s + u v ) l sin φ sin θ + 1 8 ( r u s v ) l cos φ sin θ .
L ̂ z = l cos θ ,
Φ B = i C θ , φ θ , φ θ , φ l , p l , p d l .
Φ B = ( l 2 ) Ω ,

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