Abstract

In this paper we discuss the conservation of angular momentum of light in single scattering of circularly polarized light from a spherical, non-absorbing particle. We show that the angular momentum carried by the incident wave is distributed in the scattered waves between terms related to polarization or spin and to orbital angular momentum, respectively. We also show that, in all scattering directions, a constant ratio exists between the flux density of the total angular momentum and the intensity.

© 2006 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Dover Publications, New York, 1981).
  2. R. G. Newton, Scattering Theory of Waves and Particles, second edition (Dover Publications, New York, 2002).
  3. S. M Barnett, "Optical angular-momentum flux," J. Opt. B: Quantum Semiclassal Opt. 4S7-S16 (2002).
    [CrossRef]
  4. P. L. Marston and J. H. Crichton, "Radiation torque on a sphere caused by a circularly polarized electromagnetic wave," Phys. Rev. A 302508-2516 (1984).
    [CrossRef]
  5. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
    [CrossRef] [PubMed]
  6. J. D. Jackson, Classical Electrodynamics, (Wiely, New York, 1975).
  7. L. Allen, S. M. Barnett and M. J. Padgett, Optical Angular Momentum (Institute of Physics Pub., Bristol, 2003).
    [CrossRef]
  8. 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]
  9. G. Moe and W. Happer, "Conservation of angular-momentum for light propagating in a transparent anisotropic medium," J. Phys. B. 10, 1191-1208 (1977).
    [CrossRef]
  10. F. C. MacKintosh and S. John, "Diffusing-wave spectroscopy and multiple scattering of light in correlated random media," Phys. Rev. B 40,2383-2406 (1989).
    [CrossRef]
  11. W. Gouch, "The angualr momentum of radiation," Eur. J. Phys. 7, 81-87 (1986).

2002

S. M Barnett, "Optical angular-momentum flux," J. Opt. B: Quantum Semiclassal Opt. 4S7-S16 (2002).
[CrossRef]

1995

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

1992

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]

1989

F. C. MacKintosh and S. John, "Diffusing-wave spectroscopy and multiple scattering of light in correlated random media," Phys. Rev. B 40,2383-2406 (1989).
[CrossRef]

1986

W. Gouch, "The angualr momentum of radiation," Eur. J. Phys. 7, 81-87 (1986).

1984

P. L. Marston and J. H. Crichton, "Radiation torque on a sphere caused by a circularly polarized electromagnetic wave," Phys. Rev. A 302508-2516 (1984).
[CrossRef]

Barnett, S. M

S. M Barnett, "Optical angular-momentum flux," J. Opt. B: Quantum Semiclassal Opt. 4S7-S16 (2002).
[CrossRef]

Crichton, J. H.

P. L. Marston and J. H. Crichton, "Radiation torque on a sphere caused by a circularly polarized electromagnetic wave," Phys. Rev. A 302508-2516 (1984).
[CrossRef]

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Gouch, W.

W. Gouch, "The angualr momentum of radiation," Eur. J. Phys. 7, 81-87 (1986).

Happer, W.

G. Moe and W. Happer, "Conservation of angular-momentum for light propagating in a transparent anisotropic medium," J. Phys. B. 10, 1191-1208 (1977).
[CrossRef]

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

John, S.

F. C. MacKintosh and S. John, "Diffusing-wave spectroscopy and multiple scattering of light in correlated random media," Phys. Rev. B 40,2383-2406 (1989).
[CrossRef]

MacKintosh, F. C.

F. C. MacKintosh and S. John, "Diffusing-wave spectroscopy and multiple scattering of light in correlated random media," Phys. Rev. B 40,2383-2406 (1989).
[CrossRef]

Marston, P. L.

P. L. Marston and J. H. Crichton, "Radiation torque on a sphere caused by a circularly polarized electromagnetic wave," Phys. Rev. A 302508-2516 (1984).
[CrossRef]

Moe, G.

G. Moe and W. Happer, "Conservation of angular-momentum for light propagating in a transparent anisotropic medium," J. Phys. B. 10, 1191-1208 (1977).
[CrossRef]

Nienhuis, G.

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]

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

van Enk, S. J.

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]

Eur. J. Phys.

W. Gouch, "The angualr momentum of radiation," Eur. J. Phys. 7, 81-87 (1986).

J. Opt. B: Quantum Semiclassal Opt.

S. M Barnett, "Optical angular-momentum flux," J. Opt. B: Quantum Semiclassal Opt. 4S7-S16 (2002).
[CrossRef]

Opt. Commun.

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

P. L. Marston and J. H. Crichton, "Radiation torque on a sphere caused by a circularly polarized electromagnetic wave," Phys. Rev. A 302508-2516 (1984).
[CrossRef]

Phys. Rev. B

F. C. MacKintosh and S. John, "Diffusing-wave spectroscopy and multiple scattering of light in correlated random media," Phys. Rev. B 40,2383-2406 (1989).
[CrossRef]

Phys. Rev. Lett.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Other

J. D. Jackson, Classical Electrodynamics, (Wiely, New York, 1975).

L. Allen, S. M. Barnett and M. J. Padgett, Optical Angular Momentum (Institute of Physics Pub., Bristol, 2003).
[CrossRef]

G. Moe and W. Happer, "Conservation of angular-momentum for light propagating in a transparent anisotropic medium," J. Phys. B. 10, 1191-1208 (1977).
[CrossRef]

H. C. van de Hulst, Light Scattering by Small Particles (Dover Publications, New York, 1981).

R. G. Newton, Scattering Theory of Waves and Particles, second edition (Dover Publications, New York, 2002).

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Figures (5)

Fig. 1.
Fig. 1.

Illustration of the notation used, including the definitions of the angles in the spherical coordinates and the set of orthonormal unit vectors associated with the spherical coordinates.

Fig. 2.
Fig. 2.

The ratio of the spin flux to the total angular momentum flux as a function of the anisotropy parameter for several values of the relative index of refraction as indicated.

Fig. 3.
Fig. 3.

Mean helicity of the scattered field for the cases presented in Fig. 2. Below g=0.7 there is a good linear fit with a slope of about 1.3.

Fig. 4.
Fig. 4.

Three-dimensional angular distributions of the normalized spin term (angular momentum content of the spin term divided by the intensity) for several size parameters. The relative index of refraction was 1.09. Left circularly polarized light is incident along the z axis (the direction is indicated by the red arrow). The color bars represents the spin and a complementary illustration can be obtained for the OAM

Fig. 5.
Fig. 5.

Qualitative illustration of the Poynting vector direction for radiation scattered by a Rayleigh scatterer illuminated by a circularly polarized light which is perpendicular to the plane of the figure as indicated. The spiraling of the Poynting vector results in an apparent angular shift of the light at the far field.

Equations (23)

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E x y z = exp ( ikz ) F x y z
j z x y z = c ε 0 i 2 ω E * ( r ̂ × ) E z + c ε 0 i 2 ω E * × E z
= c ε 0 2 i ω k = x , y F k * ( x y y x ) F k + c ε 0 2 i ω ( F x * F y F y * F x ) .
F r ϕ = u ( r ) exp ( imϕ ) F ̂
j z r ϕ = c ε 0 2 ω ( m + σ ) u ( r ) 2 = ( m + σ ) I ω .
E ( θ ) = ( E L ( θ ) E R ( θ ) ) = exp ( i k r ) r ( S LL ( θ ) S RL ( θ ) S LR ( θ ) S RR ( θ ) ) ( E 0 0 ) ,
( S LL ( θ ) S RL ( θ ) S LR ( θ ) S RR ( θ ) ) = 1 2 ( S 2 ( θ ) + S 1 ( θ ) S 2 ( θ ) S 1 ( θ ) S 2 ( θ ) S 1 ( θ ) S 2 ( θ ) + S 1 ( θ ) ) .
E r ϕ = exp ( i k r ) r E 0 exp ( ) { S LL ( θ ) L ̂ + S LR ( θ ) R ̂ }
L ̂ = 1 2 ( θ ̂ + i ϕ ̂ )
R ̂ = 1 2 ( θ ̂ + i ϕ ̂ )
E θ ϕ = exp ( i k r ) r E 0 exp ( ) 1 2 { [ S LL ( θ ) + S LR ( θ ) ] θ ̂ + i [ S LL ( θ ) S LR ( θ ) ] ϕ ̂ }
= exp ( i k r ) r E 0 exp ( ) [ S θ ( θ ) θ ̂ + S ϕ ( θ ) ϕ ̂ ] ,
s ( θ ) = ε 0 c i 2 ω E s * × E s = ε 0 c i 2 ω E 0 2 r 2 [ S θ * ( θ ) S ϕ ( θ ) S ϕ * ( θ ) S θ ( θ ) ] r ̂
= ε 0 c 2 ω E 0 2 r 2 [ S LL ( θ ) 2 S LR ( θ ) 2 ] r ̂ .
s z ( θ ) = ε 0 c 2 ω E 0 2 r 2 [ S LL ( θ ) 2 S LR ( θ ) 2 ] cos ( θ ) .
s z ( θ ) = 3 ε 0 c 16 π ω E 0 2 r 2 σ sc cos 2 ( θ ) .
s z ¯ = 1 4 ε 0 c ω E 0 2 σ sc = 1 2 σ sc I 0 ω .
s z ( θ ) = 1 ω [ I L ( θ ) I R ( θ ) ] cos ( θ ) = 1 ω V ( θ ) cos ( θ ) ,
ω s z ¯ σ sc I 0 = 2 πω sin ( θ ) F 44 ( θ ) cos ( θ ) σ sc I 0 .
ω s z ¯ σ sc I 0 1 + g 2 2 ,
l ( θ ) = ε 0 c i 2 ω E * ( r ̂ × ) E .
l z ( θ ) = ε 0 c 2 ω E 0 2 r 2 { S LL ( θ ) 2 + S LR ( θ ) 2 [ S LL ( θ ) 2 S LR ( θ ) 2 ] cos ( θ ) } .
j z ¯ = s z ¯ + l z ¯ = σ sc I 0 ω z ̂ .

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