Abstract

An SU(2) basis for the recently proposed analog of the Poincaré sphere for light beams with orbital angular momentum is presented. The work of Padgett and Courtial [Opt. Lett. 24, 430 (1999)] is generalized in several directions, including quantized as well as partially coherent beams with orbital angular momentum. The SU(N) version of the present theory can be used for higher-order Laguerre–Gauss beams.

© 1999 Optical Society of America

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References

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  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, 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]
  2. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [CrossRef]
  3. N. B. Simpson, L. Allen, M. J. Padgett, “Optical tweezers and optical spanners with Laguerre–Gaussian modes,” J. Mod. Opt. 43, 2485–2491 (1996).
    [CrossRef]
  4. J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144, 210–213 (1997).
    [CrossRef]
  5. H. He, M. E. J. Friese, N. R. Heckenberg, 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);M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Optical angular-momentum transfer to trapped observing particles,” Phys. Rev. A 54, 1593–1596 (1996).
    [CrossRef] [PubMed]
  6. S. J. van Enk, “Geometric phase, transformations of Gaussian light beams and angular momentum transfer,” Opt. Commun. 102, 59–64 (1993).
    [CrossRef]
  7. J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre–Gaussian modes,” Phys. Rev. A 56, 4193–4196 (1997).
    [CrossRef]
  8. J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Parametric down conversion for light beams possessing orbital angular momentum,” Phys. Rev. A 59, 3950–3952 (1999).
    [CrossRef]
  9. M. J. Padgett, J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24, 430–432 (1999).
    [CrossRef]
  10. J. Schwinger, in Quantum Theory of Angular Momentum, J. Schwinger, L. C. Biedenharn, H. van Dam, eds. (Academic, New York, 1965) p. 229.
  11. R. Simon, N. Mukunda, in “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993), have studied in detail the twisted Gaussian beams, which constitute a subclass of partially coherent beams with orbital angular momentum.
    [CrossRef]
  12. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959); see also the detailed discussion in Chap. 6 of L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
    [CrossRef]
  13. It can be borne in mind that the angular momentum aspect of SU(2) is relevant to the extent that an arbitrary state on the Poincaré sphere and its generalizations is obtained from another state by rotations.

1999 (2)

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Parametric down conversion for light beams possessing orbital angular momentum,” Phys. Rev. A 59, 3950–3952 (1999).
[CrossRef]

M. J. Padgett, J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24, 430–432 (1999).
[CrossRef]

1997 (2)

J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144, 210–213 (1997).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre–Gaussian modes,” Phys. Rev. A 56, 4193–4196 (1997).
[CrossRef]

1996 (1)

N. B. Simpson, L. Allen, M. J. Padgett, “Optical tweezers and optical spanners with Laguerre–Gaussian modes,” J. Mod. Opt. 43, 2485–2491 (1996).
[CrossRef]

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, 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);M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Optical angular-momentum transfer to trapped observing particles,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

1993 (3)

S. J. van Enk, “Geometric phase, transformations of Gaussian light beams and angular momentum transfer,” Opt. Commun. 102, 59–64 (1993).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

R. Simon, N. Mukunda, in “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993), have studied in detail the twisted Gaussian beams, which constitute a subclass of partially coherent beams with orbital angular momentum.
[CrossRef]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, 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]

1959 (1)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959); see also the detailed discussion in Chap. 6 of L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
[CrossRef]

Allen, L.

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Parametric down conversion for light beams possessing orbital angular momentum,” Phys. Rev. A 59, 3950–3952 (1999).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144, 210–213 (1997).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre–Gaussian modes,” Phys. Rev. A 56, 4193–4196 (1997).
[CrossRef]

N. B. Simpson, L. Allen, M. J. Padgett, “Optical tweezers and optical spanners with Laguerre–Gaussian modes,” J. Mod. Opt. 43, 2485–2491 (1996).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, 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]

Arlt, J.

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Parametric down conversion for light beams possessing orbital angular momentum,” Phys. Rev. A 59, 3950–3952 (1999).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, 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]

Courtial, J.

M. J. Padgett, J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24, 430–432 (1999).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144, 210–213 (1997).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre–Gaussian modes,” Phys. Rev. A 56, 4193–4196 (1997).
[CrossRef]

Dholakia, K.

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Parametric down conversion for light beams possessing orbital angular momentum,” Phys. Rev. A 59, 3950–3952 (1999).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre–Gaussian modes,” Phys. Rev. A 56, 4193–4196 (1997).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144, 210–213 (1997).
[CrossRef]

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, 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);M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Optical angular-momentum transfer to trapped observing particles,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, 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);M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Optical angular-momentum transfer to trapped observing particles,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, 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);M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Optical angular-momentum transfer to trapped observing particles,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

Mukunda, N.

Padgett, M. J.

M. J. Padgett, J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24, 430–432 (1999).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Parametric down conversion for light beams possessing orbital angular momentum,” Phys. Rev. A 59, 3950–3952 (1999).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144, 210–213 (1997).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre–Gaussian modes,” Phys. Rev. A 56, 4193–4196 (1997).
[CrossRef]

N. B. Simpson, L. Allen, M. J. Padgett, “Optical tweezers and optical spanners with Laguerre–Gaussian modes,” J. Mod. Opt. 43, 2485–2491 (1996).
[CrossRef]

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, 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);M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Optical angular-momentum transfer to trapped observing particles,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

Schwinger, J.

J. Schwinger, in Quantum Theory of Angular Momentum, J. Schwinger, L. C. Biedenharn, H. van Dam, eds. (Academic, New York, 1965) p. 229.

Simon, R.

Simpson, N. B.

N. B. Simpson, L. Allen, M. J. Padgett, “Optical tweezers and optical spanners with Laguerre–Gaussian modes,” J. Mod. Opt. 43, 2485–2491 (1996).
[CrossRef]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, 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]

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

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

van Enk, S. J.

S. J. van Enk, “Geometric phase, transformations of Gaussian light beams and angular momentum transfer,” Opt. Commun. 102, 59–64 (1993).
[CrossRef]

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, 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]

Wolf, E.

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959); see also the detailed discussion in Chap. 6 of L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
[CrossRef]

J. Mod. Opt. (1)

N. B. Simpson, L. Allen, M. J. Padgett, “Optical tweezers and optical spanners with Laguerre–Gaussian modes,” J. Mod. Opt. 43, 2485–2491 (1996).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nuovo Cimento (1)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959); see also the detailed discussion in Chap. 6 of L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
[CrossRef]

Opt. Commun. (3)

J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144, 210–213 (1997).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

S. J. van Enk, “Geometric phase, transformations of Gaussian light beams and angular momentum transfer,” Opt. Commun. 102, 59–64 (1993).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (3)

J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre–Gaussian modes,” Phys. Rev. A 56, 4193–4196 (1997).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Parametric down conversion for light beams possessing orbital angular momentum,” Phys. Rev. A 59, 3950–3952 (1999).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, 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]

Phys. Rev. Lett. (1)

H. He, M. E. J. Friese, N. R. Heckenberg, 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);M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Optical angular-momentum transfer to trapped observing particles,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

Other (2)

J. Schwinger, in Quantum Theory of Angular Momentum, J. Schwinger, L. C. Biedenharn, H. van Dam, eds. (Academic, New York, 1965) p. 229.

It can be borne in mind that the angular momentum aspect of SU(2) is relevant to the extent that an arbitrary state on the Poincaré sphere and its generalizations is obtained from another state by rotations.

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Equations (25)

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E±(x, y)=E0 exp(iθ)r exp(-r2/w2).
E=(E+b++E-b-)f exp(-iωt)+c.c.
[b+, b+]=1=[b-, b-];[b+, b-]=0etc.
E(E1b1+E2b2)f exp(-iωt)+c.c.
E±=(E1±iE2)2;b1=b++b-2,
b2=i(b+-b-)2;b±=b1ib22.
[bα, bβ]=δαβ.
S+=b+b-,S-=b-b+,
Sz=12 (b+b+-b-b-),
S2=N2/4+N/2;N(b+b++b-b-).
b±|β+, β-=β±|β+, β-,
S+=β+*β-,Sz=12(|β+|2-|β-|2),
|S|2=(|β+|2+|β-|2)24=const.
β1=|β+|expi(θ++θ-)22 cos(θ--θ+),
β±=|β±|exp(-iθ±),
β2=|β+|exp{i(θ++θ-)/2}2 sin(θ--θ+).
Sx=β+*β-+β-*β+=|β1|2-|β2|2,
Sy=β+*β--β-*β+i=β1*β2+β2*β1=β1+β222--β1+β222.
|ψβ+|1, 0+β-|0, 1,|β+|2+|β-|2=1.
S+=β+*β-, Sz=12(|β+|2-|β-|2).
L=b+b+b+b-b-b+b-b-N2+SzS+S-N2-Sz.
|b+|2|b-|2=|b+b-|2,
b+b-=0,b+b+=b-b-.
V=λ1-λ2λ1+λ2={(Tr L)2-4(det L)}1/2(Tr L),
(Tr L)2=4(det L).

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