Abstract

Traditionally, the angular momentum of light is calculated for “bullet-like” electromagnetic wave packets, although in actual optical experiments “pencil-like” beams of light are more commonly used. The fact that a wave packet is bounded transversely and longitudinally while a beam has, in principle, an infinite extent along the direction of propagation, renders incomplete the textbook calculation of the spin/orbital separation of the angular momentum of a light beam. In this work we demonstrate that a novel, extra surface part must be added in order to preserve the gauge invariance of the optical angular momentum per unit length. The impact of this extra term is quantified by means of two examples: a Laguerre-Gaussian and a Bessel beam, both circularly polarized.

© 2014 Optical Society of America

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    [Crossref] [PubMed]
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  9. S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclass. Opt. 4, S7–S16 (2002).
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    [Crossref]
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  27. J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electron. J. Diff. Eq. Conf. 04, 37–50 (2000).
  28. T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and A. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A Pure Appl. Opt. 10, 115005 (2008).
    [Crossref]
  29. J. Humblet, “Sur le moment d’impulsion d’une onde électromagnètique,” Physica 10, 585–603 (1943).
    [Crossref]
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    [Crossref]
  34. J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Energy and momentum flux in a high-numerical-aperture beam using the extended Nijboer-Zernike diffraction formalism,” J. Eur. Opt. Soc. Rapid 2, 07032 (2007).
    [Crossref]
  35. D. B. Ruffner and D. G. Grier, “Optical forces and torques in nonuniform beams of light,” Phys. Rev. Lett. 108, 173602 (2012).
    [Crossref] [PubMed]
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  37. The same “correction” factor β was found in [6]. However, the expression (4.4) in [6], when evaluated for l= 0 = p in the paraxial limit 1/(2kzR)=θ02/4≪1 gives 𝒥z/ℰ≃(σz/ω)(1+θ02/4) instead of 𝒥z/ℰ≃(σz/ω)(1−θ02/4), as given by Eq. (8) in [28]. The difference between the two results resides in the fact that different types of beams and geometries (planar versus spherical) are considered.
  38. W. L. Erikson and S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. A 49, 5778–5786 (1994).
  39. J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
    [Crossref] [PubMed]
  40. O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. 104, 253601 (2010).
    [Crossref] [PubMed]
  41. P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express 18, 10905–10923 (2010).
    [Crossref] [PubMed]

2012 (1)

D. B. Ruffner and D. G. Grier, “Optical forces and torques in nonuniform beams of light,” Phys. Rev. Lett. 108, 173602 (2012).
[Crossref] [PubMed]

2011 (1)

I. Bialynicki-Birula and Z. Bialynicki-Birula, “Canonical separation of angular momentum of light into its orbital and spin parts,” J. Opt. 13, 064014 (2011).
[Crossref]

2010 (4)

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[Crossref]

S. M. Barnett, “Rotation of electromagnetic fields and the nature of optical angular momentum,” J. Mod. Opt. 57, 1339–1343 (2010).
[Crossref]

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. 104, 253601 (2010).
[Crossref] [PubMed]

P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express 18, 10905–10923 (2010).
[Crossref] [PubMed]

2009 (4)

A. Y. Bekshaev, “Oblique section of a paraxial light beam: criteria for azimuthal energy flow and orbital angular momentum,” J. Opt. A Pure Appl. Opt. 11, 094003 (2009).
[Crossref]

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103, 100401 (2009).
[Crossref] [PubMed]

M. V. Berry, “Optical currents,” J. Opt. A Pure Appl. Opt. 11, 094001 (2009).
[Crossref]

C.-F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A 80, 063814 (2009).
[Crossref]

2008 (4)

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[Crossref] [PubMed]

A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. 33, 1437–1439 (2008).
[Crossref] [PubMed]

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. 2, 299–313 (2008).
[Crossref]

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and A. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A Pure Appl. Opt. 10, 115005 (2008).
[Crossref]

2007 (1)

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Energy and momentum flux in a high-numerical-aperture beam using the extended Nijboer-Zernike diffraction formalism,” J. Eur. Opt. Soc. Rapid 2, 07032 (2007).
[Crossref]

2006 (1)

K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
[Crossref] [PubMed]

2004 (2)

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
[Crossref] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref] [PubMed]

2003 (1)

2002 (1)

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclass. Opt. 4, S7–S16 (2002).
[Crossref]

2000 (1)

J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electron. J. Diff. Eq. Conf. 04, 37–50 (2000).

1994 (4)

W. L. Erikson and S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. A 49, 5778–5786 (1994).

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, “Spin and orbital angular momentum of photons,” Europhys. Lett. 25, 497–501 (1994).
[Crossref]

S. J. van Enk and G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41, 963–977 (1994).
[Crossref]

1993 (1)

H. A. Haus and J. L. Pan, “Photon spin and the paraxial wave equation,” Am. J. Phys. 61, 818–821 (1993).
[Crossref]

1992 (1)

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]

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

1943 (1)

J. Humblet, “Sur le moment d’impulsion d’une onde électromagnètique,” Physica 10, 585–603 (1943).
[Crossref]

1936 (1)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
[Crossref]

1932 (1)

C. G. Darwin, “Notes on the theory of radiation,” Proc. R. Soc. London A 136, 36–52 (1932).
[Crossref]

1909 (1)

J. H. Poynting, “The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light,” Proc. R. Soc. London Ser. A 82, 560–567 (1909).
[Crossref]

Aiello, A.

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[Crossref]

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103, 100401 (2009).
[Crossref] [PubMed]

A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. 33, 1437–1439 (2008).
[Crossref] [PubMed]

K. Y. Bliokh, A. Aiello, and M. Alonso, “Spin-orbit interactions of light in isotropic media,” in The Angular Momentum of Light, D. L. Andrews and M. Babiker, eds. (Cambridge University, 2012), 174–245.

Allen, L.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. 2, 299–313 (2008).
[Crossref]

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994).
[Crossref]

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]

Alonso, M.

K. Y. Bliokh, A. Aiello, and M. Alonso, “Spin-orbit interactions of light in isotropic media,” in The Angular Momentum of Light, D. L. Andrews and M. Babiker, eds. (Cambridge University, 2012), 174–245.

Alonso, M. A.

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[Crossref]

Banzer, P.

Barnett, S. M.

S. M. Barnett, “Rotation of electromagnetic fields and the nature of optical angular momentum,” J. Mod. Opt. 57, 1339–1343 (2010).
[Crossref]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref] [PubMed]

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclass. Opt. 4, S7–S16 (2002).
[Crossref]

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994).
[Crossref]

J. B. Götte and S. M. Barnett, “Light beams carrying orbital angular momentum,” in The Angular Momentum of Light, D. L. Andrews and M. Babiker, eds. (Cambridge University, 2012), 1–30.

Beijersbergen, M. W.

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]

Bekshaev, A. Y.

A. Y. Bekshaev, “Oblique section of a paraxial light beam: criteria for azimuthal energy flow and orbital angular momentum,” J. Opt. A Pure Appl. Opt. 11, 094003 (2009).
[Crossref]

Berry, M. V.

M. V. Berry, “Optical currents,” J. Opt. A Pure Appl. Opt. 11, 094001 (2009).
[Crossref]

Beth, R. A.

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
[Crossref]

Bialynicki-Birula, I.

I. Bialynicki-Birula and Z. Bialynicki-Birula, “Canonical separation of angular momentum of light into its orbital and spin parts,” J. Opt. 13, 064014 (2011).
[Crossref]

Bialynicki-Birula, Z.

I. Bialynicki-Birula and Z. Bialynicki-Birula, “Canonical separation of angular momentum of light into its orbital and spin parts,” J. Opt. 13, 064014 (2011).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[Crossref]

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. 104, 253601 (2010).
[Crossref] [PubMed]

K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
[Crossref] [PubMed]

K. Y. Bliokh, A. Aiello, and M. Alonso, “Spin-orbit interactions of light in isotropic media,” in The Angular Momentum of Light, D. L. Andrews and M. Babiker, eds. (Cambridge University, 2012), 174–245.

Bliokh, Y. P.

K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
[Crossref] [PubMed]

Braat, J. J. M.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Energy and momentum flux in a high-numerical-aperture beam using the extended Nijboer-Zernike diffraction formalism,” J. Eur. Opt. Soc. Rapid 2, 07032 (2007).
[Crossref]

J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, and A. S. van de Nes, “Extended Nijboer-Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A 20, 2281–2292 (2003).
[Crossref]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms (Wiley-VCH, 2004), Chap. I.

Courtial, J.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref] [PubMed]

Crichton, J. H.

J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electron. J. Diff. Eq. Conf. 04, 37–50 (2000).

Dainty, C.

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. 104, 253601 (2010).
[Crossref] [PubMed]

Darwin, C. G.

C. G. Darwin, “Notes on the theory of radiation,” Proc. R. Soc. London A 136, 36–52 (1932).
[Crossref]

Dirksen, P.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Energy and momentum flux in a high-numerical-aperture beam using the extended Nijboer-Zernike diffraction formalism,” J. Eur. Opt. Soc. Rapid 2, 07032 (2007).
[Crossref]

J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, and A. S. van de Nes, “Extended Nijboer-Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A 20, 2281–2292 (2003).
[Crossref]

Dupont-Roc, J.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms (Wiley-VCH, 2004), Chap. I.

Erikson, W. L.

W. L. Erikson and S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. A 49, 5778–5786 (1994).

Franke-Arnold, S.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. 2, 299–313 (2008).
[Crossref]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref] [PubMed]

Götte, J. B.

J. B. Götte and S. M. Barnett, “Light beams carrying orbital angular momentum,” in The Angular Momentum of Light, D. L. Andrews and M. Babiker, eds. (Cambridge University, 2012), 1–30.

Grier, D. G.

D. B. Ruffner and D. G. Grier, “Optical forces and torques in nonuniform beams of light,” Phys. Rev. Lett. 108, 173602 (2012).
[Crossref] [PubMed]

Grynberg, G.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms (Wiley-VCH, 2004), Chap. I.

Haus, H. A.

H. A. Haus and J. L. Pan, “Photon spin and the paraxial wave equation,” Am. J. Phys. 61, 818–821 (1993).
[Crossref]

Heckenberg, N. R.

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and A. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A Pure Appl. Opt. 10, 115005 (2008).
[Crossref]

Hosten, O.

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[Crossref] [PubMed]

Humblet, J.

J. Humblet, “Sur le moment d’impulsion d’une onde électromagnètique,” Physica 10, 585–603 (1943).
[Crossref]

Janssen, A. J. E. M.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Energy and momentum flux in a high-numerical-aperture beam using the extended Nijboer-Zernike diffraction formalism,” J. Eur. Opt. Soc. Rapid 2, 07032 (2007).
[Crossref]

J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, and A. S. van de Nes, “Extended Nijboer-Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A 20, 2281–2292 (2003).
[Crossref]

Kwiat, P.

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[Crossref] [PubMed]

Lara, D.

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. 104, 253601 (2010).
[Crossref] [PubMed]

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Leach, J.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref] [PubMed]

Leuchs, G.

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A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103, 100401 (2009).
[Crossref] [PubMed]

Li, C.-F.

C.-F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A 80, 063814 (2009).
[Crossref]

Lindlein, N.

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103, 100401 (2009).
[Crossref] [PubMed]

Loudon, R.

R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford University, 2000).

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Marquardt, C.

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103, 100401 (2009).
[Crossref] [PubMed]

Marston, P. L.

J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electron. J. Diff. Eq. Conf. 04, 37–50 (2000).

Maxwell, J. C.

J. C. Maxwell, A Treatise on Electricity and Magnetism(Dover, 1954).

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Murakami, S.

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
[Crossref] [PubMed]

Nagaosa, N.

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
[Crossref] [PubMed]

Nieminen, T. A.

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and A. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A Pure Appl. Opt. 10, 115005 (2008).
[Crossref]

Nienhuis, G.

S. J. van Enk and G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41, 963–977 (1994).
[Crossref]

S. J. van Enk and G. Nienhuis, “Spin and orbital angular momentum of photons,” Europhys. Lett. 25, 497–501 (1994).
[Crossref]

Onoda, M.

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
[Crossref] [PubMed]

Ostrovskaya, E. A.

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[Crossref]

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. 104, 253601 (2010).
[Crossref] [PubMed]

Padgett, M.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. 2, 299–313 (2008).
[Crossref]

Padgett, M. J.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref] [PubMed]

Pan, J. L.

H. A. Haus and J. L. Pan, “Photon spin and the paraxial wave equation,” Am. J. Phys. 61, 818–821 (1993).
[Crossref]

Peschel, U.

Poynting, J. H.

J. H. Poynting, “The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light,” Proc. R. Soc. London Ser. A 82, 560–567 (1909).
[Crossref]

Quabis, S.

Rodríguez-Herrera, O. G.

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. 104, 253601 (2010).
[Crossref] [PubMed]

Rubinsztein-Dunlop, A.

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and A. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A Pure Appl. Opt. 10, 115005 (2008).
[Crossref]

Ruffner, D. B.

D. B. Ruffner and D. G. Grier, “Optical forces and torques in nonuniform beams of light,” Phys. Rev. Lett. 108, 173602 (2012).
[Crossref] [PubMed]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Singh, S.

W. L. Erikson and S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. A 49, 5778–5786 (1994).

Skeldon, K.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref] [PubMed]

Spreeuw, R. J. C.

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]

Stilgoe, A. B.

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and A. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A Pure Appl. Opt. 10, 115005 (2008).
[Crossref]

van de Nes, A. S.

van Enk, S. J.

S. J. van Enk and G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41, 963–977 (1994).
[Crossref]

S. J. van Enk and G. Nienhuis, “Spin and orbital angular momentum of photons,” Europhys. Lett. 25, 497–501 (1994).
[Crossref]

van Haver, S.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Energy and momentum flux in a high-numerical-aperture beam using the extended Nijboer-Zernike diffraction formalism,” J. Eur. Opt. Soc. Rapid 2, 07032 (2007).
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A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. 33, 1437–1439 (2008).
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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]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Am. J. Phys. (1)

H. A. Haus and J. L. Pan, “Photon spin and the paraxial wave equation,” Am. J. Phys. 61, 818–821 (1993).
[Crossref]

Electron. J. Diff. Eq. (1)

J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electron. J. Diff. Eq. Conf. 04, 37–50 (2000).

Europhys. Lett. (1)

S. J. van Enk and G. Nienhuis, “Spin and orbital angular momentum of photons,” Europhys. Lett. 25, 497–501 (1994).
[Crossref]

J. Eur. Opt. Soc. Rapid (1)

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Energy and momentum flux in a high-numerical-aperture beam using the extended Nijboer-Zernike diffraction formalism,” J. Eur. Opt. Soc. Rapid 2, 07032 (2007).
[Crossref]

J. Mod. Opt. (2)

S. J. van Enk and G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41, 963–977 (1994).
[Crossref]

S. M. Barnett, “Rotation of electromagnetic fields and the nature of optical angular momentum,” J. Mod. Opt. 57, 1339–1343 (2010).
[Crossref]

J. Opt. (1)

I. Bialynicki-Birula and Z. Bialynicki-Birula, “Canonical separation of angular momentum of light into its orbital and spin parts,” J. Opt. 13, 064014 (2011).
[Crossref]

J. Opt. A Pure Appl. Opt. (3)

M. V. Berry, “Optical currents,” J. Opt. A Pure Appl. Opt. 11, 094001 (2009).
[Crossref]

A. Y. Bekshaev, “Oblique section of a paraxial light beam: criteria for azimuthal energy flow and orbital angular momentum,” J. Opt. A Pure Appl. Opt. 11, 094003 (2009).
[Crossref]

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and A. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A Pure Appl. Opt. 10, 115005 (2008).
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J. Opt. B Quantum Semiclass. Opt. (1)

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclass. Opt. 4, S7–S16 (2002).
[Crossref]

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

Laser Photonics Rev. (1)

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. 2, 299–313 (2008).
[Crossref]

Opt. Commun. (1)

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. (1)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
[Crossref]

Phys. Rev. A (5)

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]

C.-F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A 80, 063814 (2009).
[Crossref]

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[Crossref]

W. L. Erikson and S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. A 49, 5778–5786 (1994).

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Phys. Rev. Lett. (6)

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref] [PubMed]

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. 104, 253601 (2010).
[Crossref] [PubMed]

D. B. Ruffner and D. G. Grier, “Optical forces and torques in nonuniform beams of light,” Phys. Rev. Lett. 108, 173602 (2012).
[Crossref] [PubMed]

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
[Crossref] [PubMed]

K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
[Crossref] [PubMed]

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103, 100401 (2009).
[Crossref] [PubMed]

Physica (1)

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Proc. R. Soc. London A (1)

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

J. H. Poynting, “The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light,” Proc. R. Soc. London Ser. A 82, 560–567 (1909).
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Science (1)

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
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Other (9)

J. C. Maxwell, A Treatise on Electricity and Magnetism(Dover, 1954).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford University, 2000).

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

K. Y. Bliokh, A. Aiello, and M. Alonso, “Spin-orbit interactions of light in isotropic media,” in The Angular Momentum of Light, D. L. Andrews and M. Babiker, eds. (Cambridge University, 2012), 174–245.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms (Wiley-VCH, 2004), Chap. I.

J. B. Götte and S. M. Barnett, “Light beams carrying orbital angular momentum,” in The Angular Momentum of Light, D. L. Andrews and M. Babiker, eds. (Cambridge University, 2012), 1–30.

A. E. Siegman, Lasers (University Science Books, 1986).

The same “correction” factor β was found in [6]. However, the expression (4.4) in [6], when evaluated for l= 0 = p in the paraxial limit 1/(2kzR)=θ02/4≪1 gives 𝒥z/ℰ≃(σz/ω)(1+θ02/4) instead of 𝒥z/ℰ≃(σz/ω)(1−θ02/4), as given by Eq. (8) in [28]. The difference between the two results resides in the fact that different types of beams and geometries (planar versus spherical) are considered.

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Figures (1)

Fig. 1
Fig. 1

SuAM for a circularly polarized fundamental Gaussian beam (green line) and Bessel beam (blue line). At θ 0 = θ c 2 rad 81 ° the tails of the Gaussian angular spectrum distribution | A ˜ ( k ) 2 | exp [ 2 ( k x 2 + k y 2 ) / θ 0 2 ] becomes no longer negligible for k x 2 + k y 2 k 2 where evanescent waves occur [31]. The angular spectrum representation Eq. (9) with k z = + ( k 2 k x 2 k y 2 ) 1 / 2 does not account for evanescent waves, therefore it breaks down for θ0 > θc. This critical value is marked by a dashed vertical line. Such a problem does not occur for a Bessel beam because in this case the angular spectrum does not possess tails but is sharply peaked about ϑ0.

Equations (47)

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

𝒥 = ε 0 r × ( E × B ) d 3 r ,
𝒥 = ε 0 E × A d 3 r + ε 0 ξ E ξ ( r × ) A ξ d 3 r ε 0 ξ ξ ( E ξ r × A ) d 3 r ,
p ( r ) = ε 0 2 Re ( E * × B ) ,
j ( r ) = ε 0 2 r × Re ( E * × B ) ,
u ( r ) = ε 0 4 ( E E * + c 2 B B * ) ,
E = i ω A and  B = × A with  A = 0.
j ( r ) = ω ε 0 2 Re { A * ( i r × ) A i A * × A + i ξ { x , y , z } ξ [ A ξ * ( r × A ) ] } .
J = j ( r ) d 2 r , U = u ( r ) d 2 r ,
J ¯ = J U / ω .
J surf = z Re { i A z * ( r × A ) } d 2 r ,
A ( r ) = 1 2 π A ˜ ( k ) exp ( i r k + i z k z ) d 2 k ,
A ˜ ( k ) = A ˜ ( k ) ε ( σ , k ) , = | A ˜ ( k ) | exp [ i α ˜ ( k ) ] ε ( σ , k ) ,
ε ( σ , k ) = n σ k ( k n σ ) / k 2 = k × ( k × n σ ) / k 2 .
ε * ( σ , k ) ε ( σ , k ) = 1 | k n σ | 2 / k 2 = 1 k 2 2 k 2 1 ϑ 2 ,
ϑ 2 = k x 2 + k y 2 2 k 2 ,
J x orb = ε 0 ω 2 | A ˜ | 2 [ σ k x k z k 2 + k z 2 + k z α ˜ k y ] d 2 k ,
J y orb = ε 0 ω 2 | A ˜ | 2 [ σ k y k z k 2 + k z 2 k z α ˜ k y ] d 2 k ,
J z orb = ε 0 ω 2 | A ˜ | 2 [ σ ϑ 2 1 ϑ 2 + ( k x α ˜ k y k y α ˜ k y ) ] d 2 k ,
J x spin = ε 0 ω 2 2 σ | A ˜ | 2 k x k z k 2 + k z 2 d 2 k ,
J y spin = ε 0 ω 2 2 σ | A ˜ | 2 k y k z k 2 + k z 2 d 2 k ,
J z spin = ε 0 ω 2 σ | A ˜ | 2 1 2 ϑ 2 1 ϑ 2 d 2 k ,
J x surf = ε 0 ω 2 σ | A ˜ | 2 k x k z k 2 + k z 2 d 2 k ,
J x surf = ε 0 ω 2 σ | A ˜ | 2 k y k z k 2 + k z 2 d 2 k ,
J z surf = ε 0 ω 2 σ | A ˜ | 2 ϑ 2 1 ϑ 2 d 2 k .
U ω = ε 0 ω 2 | A | 2 d 2 k .
J z orb + J z spin + J z surf = ε 0 ω 2 | A ˜ | 2 [ σ 1 ϑ 2 + ( k x α ˜ k y k y α ˜ k x ) ] d 2 k .
zz = ε 0 c 2 2 | A ˜ | 2 k z [ σ + ( k x α ˜ k y k y α ˜ k x ) ] d 2 k ,
= c 2 ε 0 ω 2 | A ˜ | 2 k z d 2 k .
J z spin + J z surf = ε 0 ω 2 σ | A ˜ | 2 d 2 k ,
J z spin ¯ + J z surf ¯ = σ .
J surf ¯ = σ β 1 β e ^ z , J spin ¯ = σ 1 2 β 1 β e ^ z ,
E ( r ) = 1 2 π E ˜ ( k ) exp ( i r k + i z k z ) d 2 k ,
B ( r ) = 1 2 π B ˜ ( k ) exp ( i r k + i z k z ) d 2 k ,
k E ˜ = 0 ,
k B ˜ = 0 ,
c B ˜ = k k × E ˜ .
E ˜ ( k ) = E ˜ ( k ) ε ( σ , k ) = | E ˜ ( k ) | exp [ i θ ˜ ( k ) ] ε ( σ , k )
β ( σ , k ) = k × ε ( σ , k ) / k = ( k × n σ ) / k .
U ω = ε 0 2 ω | E ˜ | 2 d 2 k ,
S = c 2 ε 0 2 ω | E | 2 k d 2 k ,
J x = ε 0 2 ω | E ˜ | 2 k z θ ˜ k y d 2 k ,
J y = ε 0 2 ω | E ˜ | 2 k z θ ˜ k x d 2 k ,
J z = ε 0 2 ω | E ˜ | 2 [ σ 1 ϑ 2 + ( k x θ ˜ k y k y θ ˜ k x ) ] d 2 k .
zz = c 2 ε 0 2 ω 2 | E ˜ | 2 k z [ σ + ( k x θ ˜ k y k y θ ˜ k x ) ] d 2 k ,
J z orb + J z spin = ε 0 ω 2 | A ˜ | 2 [ σ + ( k x α ˜ k y k y α ˜ k x ) ] d 2 k ,
J z orb + J z spin + J z surf = ε 0 ω 2 | A ˜ | 2 [ σ 1 ϑ 2 + ( k x α ˜ k y k y α ˜ k x ) ] d 2 k .
J = J orb + J spin + J surf .

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