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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  3. R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
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  4. C. G. Darwin, “Notes on the theory of radiation,” Proc. R. Soc. London A 136, 36–52 (1932).
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  5. 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]
  6. S. M. Barnett, L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994).
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  7. S. J. van Enk, G. Nienhuis, “Spin and orbital angular momentum of photons,” Europhys. Lett. 25, 497–501 (1994).
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  8. S. J. van Enk, G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41, 963–977 (1994).
    [CrossRef]
  9. S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclass. Opt. 4, S7–S16 (2002).
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  10. M. V. Berry, “Optical currents,” J. Opt. A Pure Appl. Opt. 11, 094001 (2009).
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  11. 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).
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  13. S. M. Barnett, “Rotation of electromagnetic fields and the nature of optical angular momentum,” J. Mod. Opt. 57, 1339–1343 (2010).
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  14. I. Bialynicki-Birula, Z. Bialynicki-Birula, “Canonical separation of angular momentum of light into its orbital and spin parts,” J. Opt. 13, 064014 (2011).
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  18. K. Y. Bliokh, 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).
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  19. 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).
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    [CrossRef]
  25. C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Photons and Atoms (Wiley-VCH, 2004), Chap. I.
  26. J. B. Götte, S. M. Barnett, “Light beams carrying orbital angular momentum,” in The Angular Momentum of Light, D. L. Andrews, M. Babiker, eds. (Cambridge University, 2012), 1–30.
  27. J. H. Crichton, 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, 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]
  30. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
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    [CrossRef]
  34. J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, 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, 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, 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, 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, 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, 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

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

2011

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

2010

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, 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, C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. 104, 253601 (2010).
[CrossRef] [PubMed]

P. Banzer, U. Peschel, S. Quabis, 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

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, 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

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

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

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

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

2007

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, 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

K. Y. Bliokh, 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

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, 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]

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

2003

2002

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

2000

J. H. Crichton, 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

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

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

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

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

1993

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

1992

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]

1975

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

1943

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

1936

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

1932

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

1909

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, 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, G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103, 100401 (2009).
[CrossRef] [PubMed]

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

Allen, L.

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

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

Alonso, M. A.

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, 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, 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, L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994).
[CrossRef]

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

Beijersbergen, M. W.

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]

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, 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, 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, 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, C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. 104, 253601 (2010).
[CrossRef] [PubMed]

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

Bliokh, Y. P.

K. Y. Bliokh, 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, 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, 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, G. Grynberg, Photons and Atoms (Wiley-VCH, 2004), Chap. I.

Courtial, J.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, 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, 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, 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, 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, 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, G. Grynberg, Photons and Atoms (Wiley-VCH, 2004), Chap. I.

Erikson, W. L.

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

Franke-Arnold, S.

S. Franke-Arnold, L. Allen, 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, 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, S. M. Barnett, “Light beams carrying orbital angular momentum,” in The Angular Momentum of Light, D. L. Andrews, M. Babiker, eds. (Cambridge University, 2012), 1–30.

Grier, D. G.

D. B. Ruffner, 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, G. Grynberg, Photons and Atoms (Wiley-VCH, 2004), Chap. I.

Haus, H. A.

H. A. Haus, 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, 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, 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, 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, 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, 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, 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, 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, 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.

P. Banzer, U. Peschel, S. Quabis, 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]

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

Loudon, R.

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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)

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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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