Abstract

A novel phenomenon was reported recently that the “local optical spin density” based on the Poynting vector might be counter-intuitively opposite to the integrated spin orientation while the one related to the gauge-invariant canonical expression might not [Phys. Lett. B 779, 385 (2018)]. However, the “local optical spin density” of the gauge-invariant canonical expression can also be counter-intuitively opposite to the integrated spin orientation under the interference of plane waves, even if all of the plane waves possess the same polarization state. Moreover, the interference fields might acquire a transverse spin density (perpendicular to the propagation plane), which can have more well-controlled relations with the polarization. Additionally, the Poynting vector shows counter-intuitive back-flow and a circular motion (vortex) in the propagation plane locally, which implies a transverse local “orbital” angular momentum density related to the polarization.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  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]
  2. L. Allen, M. J. Padgett, and M. Babiker, “The Orbital Angular Momentum of Light,” Prog. Opt. 39, 291–372(1999).
    [Crossref]
  3. S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser & Photon. Rev. 2, 299–313 (2008).
    [Crossref]
  4. D. L. Andrews and M. Babiker, The Angular Momentum of Light(Cambridge University, 2012).
    [Crossref]
  5. J. D. Jackson, Classical Electrodynamics(John Wiley & Sons, 2012).
  6. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics(Wiley, New York, 1989).
  7. S. J. V. 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]
  8. X.-S. Chen, X.-F. Lü, W.-M. Sun, F. Wang, and T. Goldman, “Spin and Orbital Angular Momentum in Gauge Theories: Nucleon Spin Structure and Multipole Radiation Revisited,” Phys. Rev. Lett. 100, 232002 (2008).
    [Crossref] [PubMed]
  9. E. Leader, “A proposed measurement of optical orbital and spin angular momentum and its implications for photon angular momentum,” Phys. Lett. B 779, 385–387 (2018).
    [Crossref]
  10. R. P. Cameron and S. M. Barnett, “Electric-magnetic symmetry and Noether’s theorem,” New J. Phys. 14, 123019 (2012).
    [Crossref]
  11. K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum and angular momentum,” New J. Phys. 15, 033026 (2013).
    [Crossref]
  12. A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse Spin and Momentum in Two-Wave Interference,” Phys. Rev. X 5, 011039 (2015).
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  19. A. Canaguier-Durand, A. Cuche, C. Genet, and T. W. Ebbesen, “Force and torque on an electric dipole by spinning light fields,” Phys. Rev. A 88, 033831 (2013).
    [Crossref]
  20. A. Y. Bekshaev, “Subwavelength particles in an inhomogeneous light field: optical forces associated with the spin and orbital energy flows,” J. Opt. 15, 044004 (2013).
    [Crossref]
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    [Crossref]
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    [Crossref]
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2018 (1)

E. Leader, “A proposed measurement of optical orbital and spin angular momentum and its implications for photon angular momentum,” Phys. Lett. B 779, 385–387 (2018).
[Crossref]

2016 (1)

E. Leader, “The photon angular momentum controversy: Resolution of a conflict between laser optics and particle physics,” Phys. Lett. B 756, 303–308 (2016).
[Crossref]

2015 (2)

A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse Spin and Momentum in Two-Wave Interference,” Phys. Rev. X 5, 011039 (2015).

K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rept. 592, 1–38 (2015).
[Crossref]

2014 (1)

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 4300 (2014).
[Crossref]

2013 (3)

A. Canaguier-Durand, A. Cuche, C. Genet, and T. W. Ebbesen, “Force and torque on an electric dipole by spinning light fields,” Phys. Rev. A 88, 033831 (2013).
[Crossref]

A. Y. Bekshaev, “Subwavelength particles in an inhomogeneous light field: optical forces associated with the spin and orbital energy flows,” J. Opt. 15, 044004 (2013).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum and angular momentum,” New J. Phys. 15, 033026 (2013).
[Crossref]

2012 (1)

R. P. Cameron and S. M. Barnett, “Electric-magnetic symmetry and Noether’s theorem,” New J. Phys. 14, 123019 (2012).
[Crossref]

2011 (1)

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13, 053001 (2011).
[Crossref]

2009 (2)

S. Albaladejo, M. I. Marqés, M. Laroche, and J. J. Sáenz, “Scattering Forces from the Curl of the Spin Angular Momentum of a Light Field,” Phys. Rev. Lett. 102, 113602 (2009).
[Crossref] [PubMed]

A. V. Novitsky and L. M. Barkovsky, “Poynting singularities in optical dynamic systems,” Phys. Rev. A 79, 033821 (2009).
[Crossref]

2008 (2)

X.-S. Chen, X.-F. Lü, W.-M. Sun, F. Wang, and T. Goldman, “Spin and Orbital Angular Momentum in Gauge Theories: Nucleon Spin Structure and Multipole Radiation Revisited,” Phys. Rev. Lett. 100, 232002 (2008).
[Crossref] [PubMed]

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

2000 (1)

1999 (1)

L. Allen, M. J. Padgett, and M. Babiker, “The Orbital Angular Momentum of Light,” Prog. Opt. 39, 291–372(1999).
[Crossref]

1994 (1)

S. J. V. 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]

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]

1983 (1)

Albaladejo, S.

S. Albaladejo, M. I. Marqés, M. Laroche, and J. J. Sáenz, “Scattering Forces from the Curl of the Spin Angular Momentum of a Light Field,” Phys. Rev. Lett. 102, 113602 (2009).
[Crossref] [PubMed]

Allen, L.

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

L. Allen, M. J. Padgett, and M. Babiker, “The Orbital Angular Momentum of Light,” Prog. Opt. 39, 291–372(1999).
[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]

Andrews, D. L.

D. L. Andrews and M. Babiker, The Angular Momentum of Light(Cambridge University, 2012).
[Crossref]

Ashkin, A.

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, “The Orbital Angular Momentum of Light,” Prog. Opt. 39, 291–372(1999).
[Crossref]

D. L. Andrews and M. Babiker, The Angular Momentum of Light(Cambridge University, 2012).
[Crossref]

Barkovsky, L. M.

A. V. Novitsky and L. M. Barkovsky, “Poynting singularities in optical dynamic systems,” Phys. Rev. A 79, 033821 (2009).
[Crossref]

Barnett, S. M.

R. P. Cameron and S. M. Barnett, “Electric-magnetic symmetry and Noether’s theorem,” New J. Phys. 14, 123019 (2012).
[Crossref]

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.

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13, 053001 (2011).
[Crossref]

Bekshaev, A. Y.

A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse Spin and Momentum in Two-Wave Interference,” Phys. Rev. X 5, 011039 (2015).

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 4300 (2014).
[Crossref]

A. Y. Bekshaev, “Subwavelength particles in an inhomogeneous light field: optical forces associated with the spin and orbital energy flows,” J. Opt. 15, 044004 (2013).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum and angular momentum,” New J. Phys. 15, 033026 (2013).
[Crossref]

Bliokh, K. Y.

A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse Spin and Momentum in Two-Wave Interference,” Phys. Rev. X 5, 011039 (2015).

K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rept. 592, 1–38 (2015).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 4300 (2014).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum and angular momentum,” New J. Phys. 15, 033026 (2013).
[Crossref]

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13, 053001 (2011).
[Crossref]

Cameron, R. P.

R. P. Cameron and S. M. Barnett, “Electric-magnetic symmetry and Noether’s theorem,” New J. Phys. 14, 123019 (2012).
[Crossref]

Canaguier-Durand, A.

A. Canaguier-Durand, A. Cuche, C. Genet, and T. W. Ebbesen, “Force and torque on an electric dipole by spinning light fields,” Phys. Rev. A 88, 033831 (2013).
[Crossref]

Chaumet, P. C.

Chen, X.-S.

X.-S. Chen, X.-F. Lü, W.-M. Sun, F. Wang, and T. Goldman, “Spin and Orbital Angular Momentum in Gauge Theories: Nucleon Spin Structure and Multipole Radiation Revisited,” Phys. Rev. Lett. 100, 232002 (2008).
[Crossref] [PubMed]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics(Wiley, New York, 1989).

Cuche, A.

A. Canaguier-Durand, A. Cuche, C. Genet, and T. W. Ebbesen, “Force and torque on an electric dipole by spinning light fields,” Phys. Rev. A 88, 033831 (2013).
[Crossref]

Dupont-Roc, J.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics(Wiley, New York, 1989).

Ebbesen, T. W.

A. Canaguier-Durand, A. Cuche, C. Genet, and T. W. Ebbesen, “Force and torque on an electric dipole by spinning light fields,” Phys. Rev. A 88, 033831 (2013).
[Crossref]

Enk, S. J. V.

S. J. V. 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]

Franke-Arnold, S.

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

Genet, C.

A. Canaguier-Durand, A. Cuche, C. Genet, and T. W. Ebbesen, “Force and torque on an electric dipole by spinning light fields,” Phys. Rev. A 88, 033831 (2013).
[Crossref]

Goldman, T.

X.-S. Chen, X.-F. Lü, W.-M. Sun, F. Wang, and T. Goldman, “Spin and Orbital Angular Momentum in Gauge Theories: Nucleon Spin Structure and Multipole Radiation Revisited,” Phys. Rev. Lett. 100, 232002 (2008).
[Crossref] [PubMed]

Gordon, J. P.

Grynberg, G.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics(Wiley, New York, 1989).

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics(Cambridge University, 2013).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics(John Wiley & Sons, 2012).

Laroche, M.

S. Albaladejo, M. I. Marqés, M. Laroche, and J. J. Sáenz, “Scattering Forces from the Curl of the Spin Angular Momentum of a Light Field,” Phys. Rev. Lett. 102, 113602 (2009).
[Crossref] [PubMed]

Leader, E.

E. Leader, “A proposed measurement of optical orbital and spin angular momentum and its implications for photon angular momentum,” Phys. Lett. B 779, 385–387 (2018).
[Crossref]

E. Leader, “The photon angular momentum controversy: Resolution of a conflict between laser optics and particle physics,” Phys. Lett. B 756, 303–308 (2016).
[Crossref]

Lü, X.-F.

X.-S. Chen, X.-F. Lü, W.-M. Sun, F. Wang, and T. Goldman, “Spin and Orbital Angular Momentum in Gauge Theories: Nucleon Spin Structure and Multipole Radiation Revisited,” Phys. Rev. Lett. 100, 232002 (2008).
[Crossref] [PubMed]

Marqés, M. I.

S. Albaladejo, M. I. Marqés, M. Laroche, and J. J. Sáenz, “Scattering Forces from the Curl of the Spin Angular Momentum of a Light Field,” Phys. Rev. Lett. 102, 113602 (2009).
[Crossref] [PubMed]

Nienhuis, G.

S. J. V. 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]

Nieto-Vesperinas, M.

Nori, F.

K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rept. 592, 1–38 (2015).
[Crossref]

A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse Spin and Momentum in Two-Wave Interference,” Phys. Rev. X 5, 011039 (2015).

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 4300 (2014).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum and angular momentum,” New J. Phys. 15, 033026 (2013).
[Crossref]

Novitsky, A. V.

A. V. Novitsky and L. M. Barkovsky, “Poynting singularities in optical dynamic systems,” Phys. Rev. A 79, 033821 (2009).
[Crossref]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics(Cambridge University, 2013).

Padgett, M.

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

Padgett, M. J.

L. Allen, M. J. Padgett, and M. Babiker, “The Orbital Angular Momentum of Light,” Prog. Opt. 39, 291–372(1999).
[Crossref]

Sáenz, J. J.

S. Albaladejo, M. I. Marqés, M. Laroche, and J. J. Sáenz, “Scattering Forces from the Curl of the Spin Angular Momentum of a Light Field,” Phys. Rev. Lett. 102, 113602 (2009).
[Crossref] [PubMed]

Soskin, M.

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13, 053001 (2011).
[Crossref]

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]

Sun, W.-M.

X.-S. Chen, X.-F. Lü, W.-M. Sun, F. Wang, and T. Goldman, “Spin and Orbital Angular Momentum in Gauge Theories: Nucleon Spin Structure and Multipole Radiation Revisited,” Phys. Rev. Lett. 100, 232002 (2008).
[Crossref] [PubMed]

Wang, F.

X.-S. Chen, X.-F. Lü, W.-M. Sun, F. Wang, and T. Goldman, “Spin and Orbital Angular Momentum in Gauge Theories: Nucleon Spin Structure and Multipole Radiation Revisited,” Phys. Rev. Lett. 100, 232002 (2008).
[Crossref] [PubMed]

Woerdman, J. P.

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]

J. Mod. Opt. (1)

S. J. V. 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]

J. Opt. (2)

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13, 053001 (2011).
[Crossref]

A. Y. Bekshaev, “Subwavelength particles in an inhomogeneous light field: optical forces associated with the spin and orbital energy flows,” J. Opt. 15, 044004 (2013).
[Crossref]

Laser & Photon. Rev. (1)

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

Nat. Commun. (1)

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 4300 (2014).
[Crossref]

New J. Phys. (2)

R. P. Cameron and S. M. Barnett, “Electric-magnetic symmetry and Noether’s theorem,” New J. Phys. 14, 123019 (2012).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum and angular momentum,” New J. Phys. 15, 033026 (2013).
[Crossref]

Opt. Lett. (2)

Phys. Lett. B (2)

E. Leader, “The photon angular momentum controversy: Resolution of a conflict between laser optics and particle physics,” Phys. Lett. B 756, 303–308 (2016).
[Crossref]

E. Leader, “A proposed measurement of optical orbital and spin angular momentum and its implications for photon angular momentum,” Phys. Lett. B 779, 385–387 (2018).
[Crossref]

Phys. Rept. (1)

K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rept. 592, 1–38 (2015).
[Crossref]

Phys. Rev. A (3)

A. V. Novitsky and L. M. Barkovsky, “Poynting singularities in optical dynamic systems,” Phys. Rev. A 79, 033821 (2009).
[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]

A. Canaguier-Durand, A. Cuche, C. Genet, and T. W. Ebbesen, “Force and torque on an electric dipole by spinning light fields,” Phys. Rev. A 88, 033831 (2013).
[Crossref]

Phys. Rev. Lett. (2)

X.-S. Chen, X.-F. Lü, W.-M. Sun, F. Wang, and T. Goldman, “Spin and Orbital Angular Momentum in Gauge Theories: Nucleon Spin Structure and Multipole Radiation Revisited,” Phys. Rev. Lett. 100, 232002 (2008).
[Crossref] [PubMed]

S. Albaladejo, M. I. Marqés, M. Laroche, and J. J. Sáenz, “Scattering Forces from the Curl of the Spin Angular Momentum of a Light Field,” Phys. Rev. Lett. 102, 113602 (2009).
[Crossref] [PubMed]

Phys. Rev. X (1)

A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse Spin and Momentum in Two-Wave Interference,” Phys. Rev. X 5, 011039 (2015).

Prog. Opt. (1)

L. Allen, M. J. Padgett, and M. Babiker, “The Orbital Angular Momentum of Light,” Prog. Opt. 39, 291–372(1999).
[Crossref]

Other (4)

D. L. Andrews and M. Babiker, The Angular Momentum of Light(Cambridge University, 2012).
[Crossref]

J. D. Jackson, Classical Electrodynamics(John Wiley & Sons, 2012).

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics(Wiley, New York, 1989).

L. Novotny and B. Hecht, Principles of Nano-Optics(Cambridge University, 2013).

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Figures (2)

Fig. 1
Fig. 1 Interference of three plane waves all propagating in the x-z plane with equal amplitudes and different wave vectors k1, k2 and k3. The angle θ denotes the angle between k1,2 and k3.
Fig. 2
Fig. 2 p ¯, s ¯ (short for s ¯ gic, s ¯ e and s ¯ m) and p ¯ gic projected in the x-z plane when all σa = 1. (a) If all σa = 1, the Poynting vector and spin density have the same structure; omitting their constant factors and plotting the distribution of the 2-D vectors of ( p ¯ x , p ¯ z ) and ( s ¯ x , s ¯ z ) with θ = 1.4 and k = 2π. The red and green points are the singularities of ( p ¯ x , p ¯ z ) and ( s ¯ x , s ¯ z ). The circular motions take place around the red points. (b) The distribution of the 2-D vector of ( p ¯ gic , x , p ¯ gic , z ) with θ = 1.4 and k = 2π.

Equations (20)

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

J = d 3 x [ r × ( E × B ) ] ,
p = E × B .
J = d 3 x E × A + d 3 x E i ( r × ) A i ,
J = d 3 x E × A + d 3 x E i ( r × ) A i ,
s gic = E × A
l gic = E i ( r × ) A i = r × p gic ,
p gic = E i A i .
E a ( r , t ) = Re [ a ( r ) e i ω t ] , ( a = 1 , 2 , 3 )
a ( r ) = E 0 e i ϕ a ( r ) χ a , a ( r ) = k a ω × a ( r )
k ^ a e y = 0 , χ a = [ e y + i σ a ( k ^ a × e y ) ] / 1 + σ a 2
k 1 , 2 = k ( cos θ e z ± sin θ e x ) , k 3 = k e z ,
p ¯ = E × B ¯ = 1 2 Re [ * × ] ,
p ¯ gic = E i A i ¯ = 1 2 ω Im [ i * i ] ,
s ¯ gic = 2 s ¯ e = 1 2 ω Im [ * × ] .
s ¯ dua = s ¯ e + s ¯ m = 1 4 ω Im [ * × + * × ] .
p ¯ = a , b = 1 3 E 0 2 2 ( 1 + σ a 2 ) ( 1 + σ b 2 ) [ ( 1 + σ a σ b ) cos ( δ a b ) k ^ a σ a sin ( δ a b ) [ ( k ^ a × k ^ b ) e y ] e y ] ,
p ¯ gic = a , b = 1 3 E 0 2 2 ( 1 + σ a 2 ) ( 1 + σ b 2 ) ( 1 + σ a σ b k ^ a k ^ b ) cos ( δ a b ) k ^ a ,
s ¯ e = a , b = 1 3 E 0 2 4 ω ( 1 + σ a 2 ) ( 1 + σ b 2 ) [ 2 σ a cos ( δ a b ) k ^ a σ a σ b sin ( δ a b ) [ ( k ^ a × k ^ b ) e y ] e y ] ,
s ¯ m = a , b = 1 3 E 0 2 4 ω ( 1 + σ a 2 ) ( 1 + σ b 2 ) [ 2 σ a cos ( δ a b ) k ^ b sin ( δ a b ) [ ( k ^ a × k ^ b ) e y ] e y ] .
f = Re ( α ) 2 E 2 ¯ + ω Im ( α ) p ¯ gic , τ = ω Im ( α ) s ¯ gic .

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