Abstract

We study the angular momentum (AM) of the arbitrary superposition of counterpropagating paraxial beams that have the same magnitude of the wavenumber. We derive compact analytical expressions for the total AM in a transverse cross section (linear AM density) and the total AM flux through the cross section. We demonstrate that whereas for the time-averaged linear AM density its separation into the spin and orbital parts is not, generally, observed, the total time-averaged AM flux is separated into well-identifiable spin and orbital constituents. Moreover, we show that such a flux is also naturally separated into the fluxes of forward- and backward-propagating beams.

© 2008 Optical Society of America

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References

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  1. L. Allen, M. J. Padgett, and M. Babiker, "The optical angular momentum of light," Prog. Opt. 39, 291-372 (1999).
    [CrossRef]
  2. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).
    [CrossRef]
  3. A. Bekshaev, M. Soskin, and M. Vasnetsov, Paraxial Light Beams with Angular Momentum (Nova, to be published); http://arXiv.org/abs/0801.2309.
  4. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
    [CrossRef] [PubMed]
  5. S. J. van Enk and G. Nienhuis, "Eigenfunction description of laser beams and orbital angular momentum of light," Opt. Commun. 94, 147-158 (1992).
    [CrossRef]
  6. M. V. Berry, "Paraxial beams of spinning light," Proc. SPIE 3487, 6-11 (1998).
    [CrossRef]
  7. S. M. Barnett and L. Allen, "Orbital angular momentum and nonparaxial light beams," Opt. Commun. 110, 670-678 (1994).
    [CrossRef]
  8. S. M. Barnett, "Optical angular-momentum flux," J. Opt. B: Quantum Semiclassical Opt. 4, S7-S17 (2002).
    [CrossRef]
  9. J. D. Jackson, Classical Electrodynamics (Wiley, 1975).
  10. L. Allen, V. E. Lembessis, and M. Babiker, "Spin-orbit coupling in free space Laguerre-Gaussian light beams," Phys. Rev. A 53, 2937-2945 (1995).
    [CrossRef]
  11. A. S. Davydov, Quantum Mechanics (Pergamon, 1976).

2002 (1)

S. M. Barnett, "Optical angular-momentum flux," J. Opt. B: Quantum Semiclassical Opt. 4, S7-S17 (2002).
[CrossRef]

1999 (1)

L. Allen, M. J. Padgett, and M. Babiker, "The optical angular momentum of light," Prog. Opt. 39, 291-372 (1999).
[CrossRef]

1998 (1)

M. V. Berry, "Paraxial beams of spinning light," Proc. SPIE 3487, 6-11 (1998).
[CrossRef]

1995 (1)

L. Allen, V. E. Lembessis, and M. Babiker, "Spin-orbit coupling in free space Laguerre-Gaussian light beams," Phys. Rev. A 53, 2937-2945 (1995).
[CrossRef]

1994 (1)

S. M. Barnett and L. Allen, "Orbital angular momentum and nonparaxial light beams," Opt. Commun. 110, 670-678 (1994).
[CrossRef]

1992 (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

S. J. van Enk and G. Nienhuis, "Eigenfunction description of laser beams and orbital angular momentum of light," Opt. Commun. 94, 147-158 (1992).
[CrossRef]

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

S. M. Barnett, "Optical angular-momentum flux," J. Opt. B: Quantum Semiclassical Opt. 4, S7-S17 (2002).
[CrossRef]

Opt. Commun. (2)

S. M. Barnett and L. Allen, "Orbital angular momentum and nonparaxial light beams," Opt. Commun. 110, 670-678 (1994).
[CrossRef]

S. J. van Enk and G. Nienhuis, "Eigenfunction description of laser beams and orbital angular momentum of light," Opt. Commun. 94, 147-158 (1992).
[CrossRef]

Phys. Rev. A (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

L. Allen, V. E. Lembessis, and M. Babiker, "Spin-orbit coupling in free space Laguerre-Gaussian light beams," Phys. Rev. A 53, 2937-2945 (1995).
[CrossRef]

Proc. SPIE (1)

M. V. Berry, "Paraxial beams of spinning light," Proc. SPIE 3487, 6-11 (1998).
[CrossRef]

Prog. Opt. (1)

L. Allen, M. J. Padgett, and M. Babiker, "The optical angular momentum of light," Prog. Opt. 39, 291-372 (1999).
[CrossRef]

Other (4)

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

A. Bekshaev, M. Soskin, and M. Vasnetsov, Paraxial Light Beams with Angular Momentum (Nova, to be published); http://arXiv.org/abs/0801.2309.

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

A. S. Davydov, Quantum Mechanics (Pergamon, 1976).

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Equations (27)

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M z lin W lin = 1 ω ψ ( l ̂ z + σ ̂ z ) ψ ψ ψ ,
E t = ( α β ) 0 k d t E ( t ) exp ( i l φ ) exp ( i z k 2 t 2 ) J l ( t r ) ,
M i k = ε i m n x m [ 1 2 δ n k ( ε 0 E 2 + μ 0 H 2 ) ε 0 E n E k μ 0 H n H k ] ,
E ( x , y , z ) = A ( x , y , z ) e i k z + B ( x , y , z ) e i k z ,
rot E = i k H ,
div E = 0 .
E z i k t A t e i k z i k t B t e i k z ,
H t k ω μ 0 ( n z × A t e i k z n z × B t e i k z ) ,
H z i ω μ 0 t × E t .
M = 1 c 2 r × [ E × H ] ,
M = 1 2 c 2 Re r × [ E × H * ] .
M 2 lin = S M z d x d y .
M z lin = 1 2 c 2 Re S r t × [ E t × H z * + E z × H t * ] d x d y .
M z lin = i 4 c 2 ω μ 0 S r t × { E t + × [ t × E t + * ] ( t E t ) E t + * + c.c. } d x d y ,
M z lin = 1 4 c 2 ω μ 0 [ 1 2 A J ̂ z A + 1 2 B J ̂ z B + 2 e 2 i k z B M ̂ A + c.c. ] ,
M ̂ = [ 0 e 2 i φ ( l ̂ z + r r ) e 2 i φ ( l ̂ z r r ) 0 ] .
W lin = 1 2 S ( ε 0 E 2 + μ 0 H 2 ) d x d y .
W lin = ε 0 2 ( A A + B B ) .
M z lin W lin = 1 ω A J ̂ z A + B J ̂ z B + ( e 2 i k z B M ̂ A + c.c. ) A A + B B .
M z z tot = 1 2 Re S [ y ( ε 0 E x E z * + μ 0 H x H z * ) x ( ε 0 E y E z * + μ 0 H y H z * ) ] d x d y .
M z z tot = ε 0 2 k ( A J ̂ z A B J ̂ z B ) .
N = 1 2 Re S E × H * d x d y ,
N = 1 2 ε 0 c ( A A B B ) .
M z z tot N = 1 ω A J ̂ z A B J ̂ z B A A B B .
M i t + M i k x k = 0 .
E x c μ 0 H y , E y c μ 0 H x .
P z k 2 ω μ 0 ( A t 2 B t 2 ) .

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