Abstract

We study the mechanical properties of a broad class of multimode and polarization light patterns, resulting from the interference and superposition of waves in helical modes. General local and global properties of energy and angular momentum (AM) are identified, in order to define the conditions to optimize the AM with increasing beam complexity. We show the possibility to engineer independently the local densities of optical AM and energy, opening the possibility of an experimental demonstration of their respective effects in light-matter interaction. Multimode Laguerre-Gaussian beams also allows us to tailor the local spin AM through the Gouy phase.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
  3. D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15 - 28 (2005)
    [CrossRef]
  4. L. Allen, S. M. Barnett, M. J. Padgett, Optical angular momentum (Institute of Physics Publishing, Bristol, 2003)
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  5. F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini and C. Barbieri "Overcoming the Rayleigh Criterion Limit with Optical Vortices," Phys. Rev. Lett.  97, 163903 (2006); W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120 - 127 (2006).
    [CrossRef] [PubMed]
  6. C. Maurer, A Jesacher, S. Furhapter, S. Bernet and M. Ritsch-Marte, "Tailoring of arbitrary optical vector beams," New J. Phys. 9, 78 (2007) and references therein.
    [CrossRef]
  7. D. G. Grier, "A revolution in optical manipulation," Nature 424, 810 - 816 (2003);A. Jesacher, S. Frhapter, S. Bernet, and M. Ritsch-Marte, "Size selective trapping with optical cogwheel tweezers," Opt. Express 12, 4129 - 4135 (2004); S. H. Tao, X-C. Yuan, J. Lin, X. Peng, H. B. Niu, "Fractional optical vortex beam induced rotation of particles," Opt. Express 13, 7726 - 7631 (2005)
    [CrossRef] [PubMed]
  8. S. H. Tao, X. C. Yuan, J. Lin, and R. E. Burge, "Residue orbital angular momentum in interferenced double vortex beams with unequal topological charges," Opt. Express 14, 535 - 541 (2006)
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  11. R. Di Leonardo, J. Leach, H. Mushfique, J. M. Cooper, G. Ruocco and M. J. Padgett, "Multiport holographic velocimetry in microfluidic systems," Phys. Rev. Lett. 96, 134502 (2006)
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  23. R. Zambrini and S. M. Barnett, " Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum," Phys. Rev. Lett. 96, 113901 (2006)
    [CrossRef] [PubMed]
  24. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, "Topological charge and angular momentum of light beams carrying optical vortices," Phys. Rev. A 56, 4064 - 4074 (1997)
    [CrossRef]
  25. A. T. O’Neil, I. MacVicar, L. Allen, M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
    [CrossRef] [PubMed]
  26. R. Zambrini and S. M. Barnett, "Local transfer of angular momentum to matter," J. Mod. Opt. 52, 1045 - 1052 (2005).
    [CrossRef]
  27. R. Zambrini, L. C. Thomson, S. M. Barnett, M. Padgett, "Angular momentum paradox in a vortex core," J. Mod. Opt.,  52, 1135 - 1144 (2005).
    [CrossRef]
  28. S. M. Barnett, "Optical angular-momentum flux", J. Opt. B: Quantum Semiclass. Opt. 4, S7 - S16 (2002)
    [CrossRef]
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    [CrossRef]
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  33. The same periodicity was found for the energy and orbital AM of an interference field, because the imaginary part of the phase sensitive term UV appears in the spin AM of a superposition field while the real part appears in the orbital AM of the interference one.
  34. J. F. Nye, Natural focusing and the fine structure of light (Institute of Physics Publishing, Bristol, 1999)

2007 (3)

2006 (9)

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, S. Noda, "Lasers producing tailored beams," Nature 441, 946 (2006)
[CrossRef] [PubMed]

S. H. Tao, X. C. Yuan, J. Lin, and R. E. Burge, "Residue orbital angular momentum in interferenced double vortex beams with unequal topological charges," Opt. Express 14, 535 - 541 (2006)
[CrossRef] [PubMed]

J. Courtial, R. Zambrini, M. R. Dennis and M. Vasnetsov, "Angular momentum of optical vortex arrays," Opt. Express 14, 938 (2006)
[CrossRef] [PubMed]

K. O’Holleran, M. J. Padgett, and M. R. Dennis, "Topology of optical vortex lines formed by the interference of three, four, and five plane waves," Opt. Express 14, 3039-3044 (2006)
[CrossRef] [PubMed]

C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, "Tuning the orbital angular momentum in optical vortex beams," Opt. Express 14, 6604 - 6612 (2006)
[CrossRef] [PubMed]

R. Di Leonardo, J. Leach, H. Mushfique, J. M. Cooper, G. Ruocco and M. J. Padgett, "Multiport holographic velocimetry in microfluidic systems," Phys. Rev. Lett. 96, 134502 (2006)
[CrossRef] [PubMed]

W. Nasalski, "Polarization versus spatial characteristics of optical beams at a planar isotropic interface," Phys. Rev. E 74, 056613 (2006)
[CrossRef]

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini and C. Barbieri "Overcoming the Rayleigh Criterion Limit with Optical Vortices," Phys. Rev. Lett.  97, 163903 (2006); W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120 - 127 (2006).
[CrossRef] [PubMed]

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini and C. Barbieri "Overcoming the Rayleigh Criterion Limit with Optical Vortices," Phys. Rev. Lett.  97, 163903 (2006); W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120 - 127 (2006).
[CrossRef] [PubMed]

R. Zambrini and S. M. Barnett, " Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum," Phys. Rev. Lett. 96, 113901 (2006)
[CrossRef] [PubMed]

2005 (4)

D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15 - 28 (2005)
[CrossRef]

A. Ferrando, M. Zacares, and M.-A. Garcia-March, "Vorticity cutoff in Nonlinear Photonic Crystals," Phys. Rev. Lett. 95, 043901 (2005).
[CrossRef] [PubMed]

R. Zambrini and S. M. Barnett, "Local transfer of angular momentum to matter," J. Mod. Opt. 52, 1045 - 1052 (2005).
[CrossRef]

R. Zambrini, L. C. Thomson, S. M. Barnett, M. Padgett, "Angular momentum paradox in a vortex core," J. Mod. Opt.,  52, 1135 - 1144 (2005).
[CrossRef]

2004 (1)

2003 (1)

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810 - 816 (2003);A. Jesacher, S. Frhapter, S. Bernet, and M. Ritsch-Marte, "Size selective trapping with optical cogwheel tweezers," Opt. Express 12, 4129 - 4135 (2004); S. H. Tao, X-C. Yuan, J. Lin, X. Peng, H. B. Niu, "Fractional optical vortex beam induced rotation of particles," Opt. Express 13, 7726 - 7631 (2005)
[CrossRef] [PubMed]

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810 - 816 (2003);A. Jesacher, S. Frhapter, S. Bernet, and M. Ritsch-Marte, "Size selective trapping with optical cogwheel tweezers," Opt. Express 12, 4129 - 4135 (2004); S. H. Tao, X-C. Yuan, J. Lin, X. Peng, H. B. Niu, "Fractional optical vortex beam induced rotation of particles," Opt. Express 13, 7726 - 7631 (2005)
[CrossRef] [PubMed]

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810 - 816 (2003);A. Jesacher, S. Frhapter, S. Bernet, and M. Ritsch-Marte, "Size selective trapping with optical cogwheel tweezers," Opt. Express 12, 4129 - 4135 (2004); S. H. Tao, X-C. Yuan, J. Lin, X. Peng, H. B. Niu, "Fractional optical vortex beam induced rotation of particles," Opt. Express 13, 7726 - 7631 (2005)
[CrossRef] [PubMed]

2002 (2)

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

A. T. O’Neil, I. MacVicar, L. Allen, M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

2000 (1)

L. Allen and M. J. Padgett, "The Poynting vector in Laguerre-Gaussian beams and the interpretation of angular momentum density," Opt. Commun. 184, 67 - 71 (2000).
[CrossRef]

1998 (1)

M. Hoyuelos, P. Colet, M. San Miguel, and D. Walgraef, "Polarization patterns in Kerr media," Phys. Rev. E 58, 2992 - 3007 (1998);G.-L. Oppo, A. J. Scroggie, and W. J. Firth,"Characterization, dynamics and stabilization of diffractive domain walls and dark ring cavity solitons in parametric oscillators," Phys. Rev. E 63, 066209 (2001)
[CrossRef]

M. Hoyuelos, P. Colet, M. San Miguel, and D. Walgraef, "Polarization patterns in Kerr media," Phys. Rev. E 58, 2992 - 3007 (1998);G.-L. Oppo, A. J. Scroggie, and W. J. Firth,"Characterization, dynamics and stabilization of diffractive domain walls and dark ring cavity solitons in parametric oscillators," Phys. Rev. E 63, 066209 (2001)
[CrossRef]

1997 (1)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, "Topological charge and angular momentum of light beams carrying optical vortices," Phys. Rev. A 56, 4064 - 4074 (1997)
[CrossRef]

1996 (1)

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]

1936 (1)

R. A. Beth, "Mechanical Detection and Measurement of the Angular Momentum of Light," Phys. Rev. 50, 115 - 125 (1936)
[CrossRef]

Allen, L.

A. T. O’Neil, I. MacVicar, L. Allen, M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

L. Allen and M. J. Padgett, "The Poynting vector in Laguerre-Gaussian beams and the interpretation of angular momentum density," Opt. Commun. 184, 67 - 71 (2000).
[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]

Anzolin, G.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini and C. Barbieri "Overcoming the Rayleigh Criterion Limit with Optical Vortices," Phys. Rev. Lett.  97, 163903 (2006); W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120 - 127 (2006).
[CrossRef] [PubMed]

Barbieri, C.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini and C. Barbieri "Overcoming the Rayleigh Criterion Limit with Optical Vortices," Phys. Rev. Lett.  97, 163903 (2006); W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120 - 127 (2006).
[CrossRef] [PubMed]

Barnett, S. M.

R. Zambrini and S. M. Barnett, " Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum," Phys. Rev. Lett. 96, 113901 (2006)
[CrossRef] [PubMed]

R. Zambrini, L. C. Thomson, S. M. Barnett, M. Padgett, "Angular momentum paradox in a vortex core," J. Mod. Opt.,  52, 1135 - 1144 (2005).
[CrossRef]

R. Zambrini and S. M. Barnett, "Local transfer of angular momentum to matter," J. Mod. Opt. 52, 1045 - 1052 (2005).
[CrossRef]

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

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]

Beth, R.A.

R. A. Beth, "Mechanical Detection and Measurement of the Angular Momentum of Light," Phys. Rev. 50, 115 - 125 (1936)
[CrossRef]

Bianchini, A.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini and C. Barbieri "Overcoming the Rayleigh Criterion Limit with Optical Vortices," Phys. Rev. Lett.  97, 163903 (2006); W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120 - 127 (2006).
[CrossRef] [PubMed]

Boiko, D.

Burge, R. E.

Colet, P.

M. Hoyuelos, P. Colet, M. San Miguel, and D. Walgraef, "Polarization patterns in Kerr media," Phys. Rev. E 58, 2992 - 3007 (1998);G.-L. Oppo, A. J. Scroggie, and W. J. Firth,"Characterization, dynamics and stabilization of diffractive domain walls and dark ring cavity solitons in parametric oscillators," Phys. Rev. E 63, 066209 (2001)
[CrossRef]

Cooper, J. M.

R. Di Leonardo, J. Leach, H. Mushfique, J. M. Cooper, G. Ruocco and M. J. Padgett, "Multiport holographic velocimetry in microfluidic systems," Phys. Rev. Lett. 96, 134502 (2006)
[CrossRef] [PubMed]

Courtial, J.

Curtis, J. E.

Dennis, M. R.

Dholakia, K.

D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15 - 28 (2005)
[CrossRef]

Di Leonardo, R.

R. Di Leonardo, J. Leach, H. Mushfique, J. M. Cooper, G. Ruocco and M. J. Padgett, "Multiport holographic velocimetry in microfluidic systems," Phys. Rev. Lett. 96, 134502 (2006)
[CrossRef] [PubMed]

Ellinas, D.

Ferrando, A.

A. Ferrando, M. Zacares, and M.-A. Garcia-March, "Vorticity cutoff in Nonlinear Photonic Crystals," Phys. Rev. Lett. 95, 043901 (2005).
[CrossRef] [PubMed]

Firth, W. J.

M. Hoyuelos, P. Colet, M. San Miguel, and D. Walgraef, "Polarization patterns in Kerr media," Phys. Rev. E 58, 2992 - 3007 (1998);G.-L. Oppo, A. J. Scroggie, and W. J. Firth,"Characterization, dynamics and stabilization of diffractive domain walls and dark ring cavity solitons in parametric oscillators," Phys. Rev. E 63, 066209 (2001)
[CrossRef]

Franke-Arnold, S.

Gahagan, K. T.

Garcia-March, M.-A.

A. Ferrando, M. Zacares, and M.-A. Garcia-March, "Vorticity cutoff in Nonlinear Photonic Crystals," Phys. Rev. Lett. 95, 043901 (2005).
[CrossRef] [PubMed]

Girkin, J. M.

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, "Topological charge and angular momentum of light beams carrying optical vortices," Phys. Rev. A 56, 4064 - 4074 (1997)
[CrossRef]

Grier, D. G.

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810 - 816 (2003);A. Jesacher, S. Frhapter, S. Bernet, and M. Ritsch-Marte, "Size selective trapping with optical cogwheel tweezers," Opt. Express 12, 4129 - 4135 (2004); S. H. Tao, X-C. Yuan, J. Lin, X. Peng, H. B. Niu, "Fractional optical vortex beam induced rotation of particles," Opt. Express 13, 7726 - 7631 (2005)
[CrossRef] [PubMed]

Guerrero, G.

Hanson, S. G.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini and C. Barbieri "Overcoming the Rayleigh Criterion Limit with Optical Vortices," Phys. Rev. Lett.  97, 163903 (2006); W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120 - 127 (2006).
[CrossRef] [PubMed]

Heckenberg, N. R.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, "Topological charge and angular momentum of light beams carrying optical vortices," Phys. Rev. A 56, 4064 - 4074 (1997)
[CrossRef]

Hoyuelos, M.

M. Hoyuelos, P. Colet, M. San Miguel, and D. Walgraef, "Polarization patterns in Kerr media," Phys. Rev. E 58, 2992 - 3007 (1998);G.-L. Oppo, A. J. Scroggie, and W. J. Firth,"Characterization, dynamics and stabilization of diffractive domain walls and dark ring cavity solitons in parametric oscillators," Phys. Rev. E 63, 066209 (2001)
[CrossRef]

Ishijima, R.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini and C. Barbieri "Overcoming the Rayleigh Criterion Limit with Optical Vortices," Phys. Rev. Lett.  97, 163903 (2006); W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120 - 127 (2006).
[CrossRef] [PubMed]

Jesacher, A

C. Maurer, A Jesacher, S. Furhapter, S. Bernet and M. Ritsch-Marte, "Tailoring of arbitrary optical vector beams," New J. Phys. 9, 78 (2007) and references therein.
[CrossRef]

Jesacher, A.

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810 - 816 (2003);A. Jesacher, S. Frhapter, S. Bernet, and M. Ritsch-Marte, "Size selective trapping with optical cogwheel tweezers," Opt. Express 12, 4129 - 4135 (2004); S. H. Tao, X-C. Yuan, J. Lin, X. Peng, H. B. Niu, "Fractional optical vortex beam induced rotation of particles," Opt. Express 13, 7726 - 7631 (2005)
[CrossRef] [PubMed]

Kapon, E.

Kunishi, W.

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, S. Noda, "Lasers producing tailored beams," Nature 441, 946 (2006)
[CrossRef] [PubMed]

Leach, J.

S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg and A. S. Arnold, "Optical ferris wheel for ultracold atoms," Opt. Express 15, 8619 - 8625 (2007)
[CrossRef] [PubMed]

R. Di Leonardo, J. Leach, H. Mushfique, J. M. Cooper, G. Ruocco and M. J. Padgett, "Multiport holographic velocimetry in microfluidic systems," Phys. Rev. Lett. 96, 134502 (2006)
[CrossRef] [PubMed]

Lembessis, V. E.

Leniec, M.

Lin, J.

S. H. Tao, X. C. Yuan, J. Lin, and R. E. Burge, "Residue orbital angular momentum in interferenced double vortex beams with unequal topological charges," Opt. Express 14, 535 - 541 (2006)
[CrossRef] [PubMed]

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810 - 816 (2003);A. Jesacher, S. Frhapter, S. Bernet, and M. Ritsch-Marte, "Size selective trapping with optical cogwheel tweezers," Opt. Express 12, 4129 - 4135 (2004); S. H. Tao, X-C. Yuan, J. Lin, X. Peng, H. B. Niu, "Fractional optical vortex beam induced rotation of particles," Opt. Express 13, 7726 - 7631 (2005)
[CrossRef] [PubMed]

MacVicar, I.

A. T. O’Neil, I. MacVicar, L. Allen, M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

Malos, J. T.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, "Topological charge and angular momentum of light beams carrying optical vortices," Phys. Rev. A 56, 4064 - 4074 (1997)
[CrossRef]

Masajada, J.

Maurer, C.

C. Maurer, A Jesacher, S. Furhapter, S. Bernet and M. Ritsch-Marte, "Tailoring of arbitrary optical vector beams," New J. Phys. 9, 78 (2007) and references therein.
[CrossRef]

McGloin, D.

D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15 - 28 (2005)
[CrossRef]

Miyai, E.

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, S. Noda, "Lasers producing tailored beams," Nature 441, 946 (2006)
[CrossRef] [PubMed]

Miyamoto, Y.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini and C. Barbieri "Overcoming the Rayleigh Criterion Limit with Optical Vortices," Phys. Rev. Lett.  97, 163903 (2006); W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120 - 127 (2006).
[CrossRef] [PubMed]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres and L. Torner, "Twisted photons," Nature Phys. 3, 305 - 310 (2007) and refernces therein.
[CrossRef]

Mushfique, H.

R. Di Leonardo, J. Leach, H. Mushfique, J. M. Cooper, G. Ruocco and M. J. Padgett, "Multiport holographic velocimetry in microfluidic systems," Phys. Rev. Lett. 96, 134502 (2006)
[CrossRef] [PubMed]

Nasalski, W.

W. Nasalski, "Polarization versus spatial characteristics of optical beams at a planar isotropic interface," Phys. Rev. E 74, 056613 (2006)
[CrossRef]

Niu, H. B.

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810 - 816 (2003);A. Jesacher, S. Frhapter, S. Bernet, and M. Ritsch-Marte, "Size selective trapping with optical cogwheel tweezers," Opt. Express 12, 4129 - 4135 (2004); S. H. Tao, X-C. Yuan, J. Lin, X. Peng, H. B. Niu, "Fractional optical vortex beam induced rotation of particles," Opt. Express 13, 7726 - 7631 (2005)
[CrossRef] [PubMed]

Noda, S.

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, S. Noda, "Lasers producing tailored beams," Nature 441, 946 (2006)
[CrossRef] [PubMed]

O’Holleran, K.

O’Neil, A.T.

A. T. O’Neil, I. MacVicar, L. Allen, M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

Ohnishi, D.

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, S. Noda, "Lasers producing tailored beams," Nature 441, 946 (2006)
[CrossRef] [PubMed]

Okano, T.

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, S. Noda, "Lasers producing tailored beams," Nature 441, 946 (2006)
[CrossRef] [PubMed]

Oppo, G.-L.

M. Hoyuelos, P. Colet, M. San Miguel, and D. Walgraef, "Polarization patterns in Kerr media," Phys. Rev. E 58, 2992 - 3007 (1998);G.-L. Oppo, A. J. Scroggie, and W. J. Firth,"Characterization, dynamics and stabilization of diffractive domain walls and dark ring cavity solitons in parametric oscillators," Phys. Rev. E 63, 066209 (2001)
[CrossRef]

Padgett, M.

R. Zambrini, L. C. Thomson, S. M. Barnett, M. Padgett, "Angular momentum paradox in a vortex core," J. Mod. Opt.,  52, 1135 - 1144 (2005).
[CrossRef]

Padgett, M. J.

S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg and A. S. Arnold, "Optical ferris wheel for ultracold atoms," Opt. Express 15, 8619 - 8625 (2007)
[CrossRef] [PubMed]

K. O’Holleran, M. J. Padgett, and M. R. Dennis, "Topology of optical vortex lines formed by the interference of three, four, and five plane waves," Opt. Express 14, 3039-3044 (2006)
[CrossRef] [PubMed]

R. Di Leonardo, J. Leach, H. Mushfique, J. M. Cooper, G. Ruocco and M. J. Padgett, "Multiport holographic velocimetry in microfluidic systems," Phys. Rev. Lett. 96, 134502 (2006)
[CrossRef] [PubMed]

A. T. O’Neil, I. MacVicar, L. Allen, M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

L. Allen and M. J. Padgett, "The Poynting vector in Laguerre-Gaussian beams and the interpretation of angular momentum density," Opt. Commun. 184, 67 - 71 (2000).
[CrossRef]

Peng, X.

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810 - 816 (2003);A. Jesacher, S. Frhapter, S. Bernet, and M. Ritsch-Marte, "Size selective trapping with optical cogwheel tweezers," Opt. Express 12, 4129 - 4135 (2004); S. H. Tao, X-C. Yuan, J. Lin, X. Peng, H. B. Niu, "Fractional optical vortex beam induced rotation of particles," Opt. Express 13, 7726 - 7631 (2005)
[CrossRef] [PubMed]

Popiolek-Masajada, A.

Ruocco, G.

R. Di Leonardo, J. Leach, H. Mushfique, J. M. Cooper, G. Ruocco and M. J. Padgett, "Multiport holographic velocimetry in microfluidic systems," Phys. Rev. Lett. 96, 134502 (2006)
[CrossRef] [PubMed]

Sakai, K.

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, S. Noda, "Lasers producing tailored beams," Nature 441, 946 (2006)
[CrossRef] [PubMed]

San Miguel, M.

M. Hoyuelos, P. Colet, M. San Miguel, and D. Walgraef, "Polarization patterns in Kerr media," Phys. Rev. E 58, 2992 - 3007 (1998);G.-L. Oppo, A. J. Scroggie, and W. J. Firth,"Characterization, dynamics and stabilization of diffractive domain walls and dark ring cavity solitons in parametric oscillators," Phys. Rev. E 63, 066209 (2001)
[CrossRef]

Schmitz, C. H. J.

Scroggie, A. J.

M. Hoyuelos, P. Colet, M. San Miguel, and D. Walgraef, "Polarization patterns in Kerr media," Phys. Rev. E 58, 2992 - 3007 (1998);G.-L. Oppo, A. J. Scroggie, and W. J. Firth,"Characterization, dynamics and stabilization of diffractive domain walls and dark ring cavity solitons in parametric oscillators," Phys. Rev. E 63, 066209 (2001)
[CrossRef]

Senthilkumaran, P.

S. Vyas and P. Senthilkumaran, "Interferometric optical vortex array generator," App. Opt. 46, 2893 - 2898 (2007)
[CrossRef] [PubMed]

Soskin, M. S.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, "Topological charge and angular momentum of light beams carrying optical vortices," Phys. Rev. A 56, 4064 - 4074 (1997)
[CrossRef]

Spatz, J. P.

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]

Swartzlander, G. A.

Takeda, M.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini and C. Barbieri "Overcoming the Rayleigh Criterion Limit with Optical Vortices," Phys. Rev. Lett.  97, 163903 (2006); W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120 - 127 (2006).
[CrossRef] [PubMed]

Tamburini, F.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini and C. Barbieri "Overcoming the Rayleigh Criterion Limit with Optical Vortices," Phys. Rev. Lett.  97, 163903 (2006); W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120 - 127 (2006).
[CrossRef] [PubMed]

Tao, S. H.

S. H. Tao, X. C. Yuan, J. Lin, and R. E. Burge, "Residue orbital angular momentum in interferenced double vortex beams with unequal topological charges," Opt. Express 14, 535 - 541 (2006)
[CrossRef] [PubMed]

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810 - 816 (2003);A. Jesacher, S. Frhapter, S. Bernet, and M. Ritsch-Marte, "Size selective trapping with optical cogwheel tweezers," Opt. Express 12, 4129 - 4135 (2004); S. H. Tao, X-C. Yuan, J. Lin, X. Peng, H. B. Niu, "Fractional optical vortex beam induced rotation of particles," Opt. Express 13, 7726 - 7631 (2005)
[CrossRef] [PubMed]

Thomson, L. C.

R. Zambrini, L. C. Thomson, S. M. Barnett, M. Padgett, "Angular momentum paradox in a vortex core," J. Mod. Opt.,  52, 1135 - 1144 (2005).
[CrossRef]

Torner, L.

G. Molina-Terriza, J. P. Torres and L. Torner, "Twisted photons," Nature Phys. 3, 305 - 310 (2007) and refernces therein.
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. P. Torres and L. Torner, "Twisted photons," Nature Phys. 3, 305 - 310 (2007) and refernces therein.
[CrossRef]

Uhrig, K.

Umbriaco, G.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini and C. Barbieri "Overcoming the Rayleigh Criterion Limit with Optical Vortices," Phys. Rev. Lett.  97, 163903 (2006); W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120 - 127 (2006).
[CrossRef] [PubMed]

Vasnetsov, M.

Vasnetsov, M. V.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, "Topological charge and angular momentum of light beams carrying optical vortices," Phys. Rev. A 56, 4064 - 4074 (1997)
[CrossRef]

Vyas, S.

S. Vyas and P. Senthilkumaran, "Interferometric optical vortex array generator," App. Opt. 46, 2893 - 2898 (2007)
[CrossRef] [PubMed]

Wada, A.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini and C. Barbieri "Overcoming the Rayleigh Criterion Limit with Optical Vortices," Phys. Rev. Lett.  97, 163903 (2006); W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120 - 127 (2006).
[CrossRef] [PubMed]

Walgraef, D.

M. Hoyuelos, P. Colet, M. San Miguel, and D. Walgraef, "Polarization patterns in Kerr media," Phys. Rev. E 58, 2992 - 3007 (1998);G.-L. Oppo, A. J. Scroggie, and W. J. Firth,"Characterization, dynamics and stabilization of diffractive domain walls and dark ring cavity solitons in parametric oscillators," Phys. Rev. E 63, 066209 (2001)
[CrossRef]

Wang, W.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini and C. Barbieri "Overcoming the Rayleigh Criterion Limit with Optical Vortices," Phys. Rev. Lett.  97, 163903 (2006); W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120 - 127 (2006).
[CrossRef] [PubMed]

Woerdman, J. P.

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]

Wright, A. J.

Yokozeki, T.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini and C. Barbieri "Overcoming the Rayleigh Criterion Limit with Optical Vortices," Phys. Rev. Lett.  97, 163903 (2006); W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120 - 127 (2006).
[CrossRef] [PubMed]

Yuan, X. C.

Yuan, X-C.

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810 - 816 (2003);A. Jesacher, S. Frhapter, S. Bernet, and M. Ritsch-Marte, "Size selective trapping with optical cogwheel tweezers," Opt. Express 12, 4129 - 4135 (2004); S. H. Tao, X-C. Yuan, J. Lin, X. Peng, H. B. Niu, "Fractional optical vortex beam induced rotation of particles," Opt. Express 13, 7726 - 7631 (2005)
[CrossRef] [PubMed]

Zacares, M.

A. Ferrando, M. Zacares, and M.-A. Garcia-March, "Vorticity cutoff in Nonlinear Photonic Crystals," Phys. Rev. Lett. 95, 043901 (2005).
[CrossRef] [PubMed]

Zambrini, R.

R. Zambrini and S. M. Barnett, " Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum," Phys. Rev. Lett. 96, 113901 (2006)
[CrossRef] [PubMed]

J. Courtial, R. Zambrini, M. R. Dennis and M. Vasnetsov, "Angular momentum of optical vortex arrays," Opt. Express 14, 938 (2006)
[CrossRef] [PubMed]

R. Zambrini, L. C. Thomson, S. M. Barnett, M. Padgett, "Angular momentum paradox in a vortex core," J. Mod. Opt.,  52, 1135 - 1144 (2005).
[CrossRef]

R. Zambrini and S. M. Barnett, "Local transfer of angular momentum to matter," J. Mod. Opt. 52, 1045 - 1052 (2005).
[CrossRef]

Applied Optics (1)

S. Vyas and P. Senthilkumaran, "Interferometric optical vortex array generator," App. Opt. 46, 2893 - 2898 (2007)
[CrossRef] [PubMed]

Contemp. Phys. (1)

D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15 - 28 (2005)
[CrossRef]

J. Mod. Opt. (2)

R. Zambrini and S. M. Barnett, "Local transfer of angular momentum to matter," J. Mod. Opt. 52, 1045 - 1052 (2005).
[CrossRef]

R. Zambrini, L. C. Thomson, S. M. Barnett, M. Padgett, "Angular momentum paradox in a vortex core," J. Mod. Opt.,  52, 1135 - 1144 (2005).
[CrossRef]

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]

Nature (2)

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, S. Noda, "Lasers producing tailored beams," Nature 441, 946 (2006)
[CrossRef] [PubMed]

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810 - 816 (2003);A. Jesacher, S. Frhapter, S. Bernet, and M. Ritsch-Marte, "Size selective trapping with optical cogwheel tweezers," Opt. Express 12, 4129 - 4135 (2004); S. H. Tao, X-C. Yuan, J. Lin, X. Peng, H. B. Niu, "Fractional optical vortex beam induced rotation of particles," Opt. Express 13, 7726 - 7631 (2005)
[CrossRef] [PubMed]

Opt. Commun. (1)

L. Allen and M. J. Padgett, "The Poynting vector in Laguerre-Gaussian beams and the interpretation of angular momentum density," Opt. Commun. 184, 67 - 71 (2000).
[CrossRef]

Opt. Express (7)

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 (2)

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]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, "Topological charge and angular momentum of light beams carrying optical vortices," Phys. Rev. A 56, 4064 - 4074 (1997)
[CrossRef]

Phys. Rev. E (2)

M. Hoyuelos, P. Colet, M. San Miguel, and D. Walgraef, "Polarization patterns in Kerr media," Phys. Rev. E 58, 2992 - 3007 (1998);G.-L. Oppo, A. J. Scroggie, and W. J. Firth,"Characterization, dynamics and stabilization of diffractive domain walls and dark ring cavity solitons in parametric oscillators," Phys. Rev. E 63, 066209 (2001)
[CrossRef]

W. Nasalski, "Polarization versus spatial characteristics of optical beams at a planar isotropic interface," Phys. Rev. E 74, 056613 (2006)
[CrossRef]

Phys. Rev. Lett. (5)

A. Ferrando, M. Zacares, and M.-A. Garcia-March, "Vorticity cutoff in Nonlinear Photonic Crystals," Phys. Rev. Lett. 95, 043901 (2005).
[CrossRef] [PubMed]

R. Zambrini and S. M. Barnett, " Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum," Phys. Rev. Lett. 96, 113901 (2006)
[CrossRef] [PubMed]

R. Di Leonardo, J. Leach, H. Mushfique, J. M. Cooper, G. Ruocco and M. J. Padgett, "Multiport holographic velocimetry in microfluidic systems," Phys. Rev. Lett. 96, 134502 (2006)
[CrossRef] [PubMed]

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini and C. Barbieri "Overcoming the Rayleigh Criterion Limit with Optical Vortices," Phys. Rev. Lett.  97, 163903 (2006); W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120 - 127 (2006).
[CrossRef] [PubMed]

A. T. O’Neil, I. MacVicar, L. Allen, M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

Other (8)

The same periodicity was found for the energy and orbital AM of an interference field, because the imaginary part of the phase sensitive term UV appears in the spin AM of a superposition field while the real part appears in the orbital AM of the interference one.

J. F. Nye, Natural focusing and the fine structure of light (Institute of Physics Publishing, Bristol, 1999)

C. Maurer, A Jesacher, S. Furhapter, S. Bernet and M. Ritsch-Marte, "Tailoring of arbitrary optical vector beams," New J. Phys. 9, 78 (2007) and references therein.
[CrossRef]

G. Molina-Terriza, J. P. Torres and L. Torner, "Twisted photons," Nature Phys. 3, 305 - 310 (2007) and refernces therein.
[CrossRef]

L. Allen, S. M. Barnett, M. J. Padgett, Optical angular momentum (Institute of Physics Publishing, Bristol, 2003)
[CrossRef]

S. M. Barnett and R. Zambrini, "Orbital angular momentum of light" in Quantum Imaging, Mikhail I. Kolobov Ed., (Springer-Verlag New York, 2006) and references in Section 12.6

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

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

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Figures (6)

Fig. 1.
Fig. 1.

(a) Intensity distribution of the u 20(x,y,0) Laguerre-Gaussian mode. The orbital AM density (b) has the same profile of the intensity, the only difference being a scaling factor given by the helicity ℓ. If two orthogonally polarized beams in the same mode but with π/2 dephasing are superposed a circularly polarized wave is obtained. The spin of this wave for u 20 is represented in (c) and differs by the intensity distribution only by a scaling given by the degree of polarization. In this case σ = 1. In order to compare with the following pictures regions of the beam in which the intensity is above 80% of its maximum value are highlighted with dashed lines in (b-c).

Fig. 2.
Fig. 2.

AM densities obtained by interfering (a,e) constructively and by superposing (b-d,f-h) two p = 0 Laguerre-Gaussian modes with ℓ1 = 3,ℓ2 = 2 (a-d) and ℓ1 = 3,ℓ2 = 0 (f-h). Orbital AM0 (a,b,e,f), spin (c,g) and total(d,h) AM are represented and compared with the intensity highlighted regions (see Fig. 1).

Fig. 3.
Fig. 3.

AM densities as in Fig. 2 in the case of ℓ1 = 7,ℓ2 = -1, with p 1 = 0 = p 2.

Fig. 4.
Fig. 4.

a) Radial energy density (continuous line) and orbital AM density (dashed line) obtained superposing two beams with ℓ1 = 7,ℓ2 = -1 (see Fig. 3(b)). b) Average energy (continuous line) and orbital AM (dashed line) obtained by integral on a circular area of radius R in units of ε0 C 2 and ε0 C 2/ω, respectively.

Fig. 5.
Fig. 5.

Spin of the superposition of two orthogonal linearly polarized waves with same helicity ℓ = 2 and radial indexes p 1 = 0, p 2 = 1 at different planes along one Rayleigh range (zR ), for z = -0.5zR ,0,+0.5zR ,+zR . In the first panel the radial profile is represented to show the local sign of the spin density. The Gouy phase introduces local polarization gradients out of the focal plane.

Fig. 6.
Fig. 6.

Regions (shaded) in which there is more total AM (left) and more total AMPE (right) in superposition than in interference, as function of the angle θ and ℓ1, for a fixed ℓ2 = 2.

Equations (31)

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

e ( x , t ) = 1 2 [ E ( x ) e ikz iωt + c . c ]
p z ( x ) = ε 0 2 c ( E x ( x ) 2 + E y ( x ) 2 ) ,
W = ε 0 2 c d x d y ( E x 2 + E y 2 ) .
p x ( x ) = 0 4 ω [ ( E j x E j * ) y ( E x * E y ) c . c . ]
p y ( x ) = 0 4 ω [ ( E j y E j * ) + x ( E x * E y ) c . c . ] ,
P = d x d y ( ε 0 2 ω ) Im ( E j E j * ) ,
ε 0 2 ω ( y σ z ( x ) , x σ z ( x ) ) ,
j z ( x ) = 0 4 ω [ E j ϕ E j * + 2 E y * E x + x ( x E x * E y ) + y ( yE x * E y ) c . c . ] .
z = ε 0 2 ω Im ( E x * ϕ E x + E y * ϕ E y ) ,
s z = ε 0 ω Im ( E x * E y ) .
E s ( x ) = ( U ( x ) , V ( x ) )
E i ( x ) = ( U ( x ) + V ( x ) , 0 ) ,
J z s = 0 4 ω d x d y [ U ϕ U * + V ϕ V * + 2 V * U c . c . ] ,
J z i = 0 4 ω d x d y [ ( U + V ) ϕ ( U + V ) * c . c . ] .
L z i L z s = 0 4 ω d x d y [ U ϕ V * + V ϕ U * c . c . ] .
U ( x ) = C p , a p u p ( x ) , V ( x ) = C p , b p u p ( x ) ,
L z i W i = p , a p + b p 2 ω p , a p + b p 2 , L z s W s = p , ( a p 2 + b p 2 ) ω p , a p 2 + b p 2 .
p , Re ( a p b p * ) < 1 2 p , Re ( a p b p * ) p , ( a p 2 + b p 2 ) .
S z s W s = 1 ω p , Im ( a p b p * ) .
U ( x ) = Cu p 1 ( x ) , V ( x ) = Cu p 2 ( x )
W i = W s = ε 0 C 2 , L z i = L z s = ε 0 C 2 2 ω ( 1 + 2 ) .
z s ( x ) = ε 0 2 ω [ 1 U ( x ) 2 + 2 V ( x ) 2 ] ,
z i ( x ) = z s ( x ) ( 1 + 2 ) ε 0 2 ω Re ( U ( x ) V * ( x ) ) ,
U ( x ) = Cu p 1 ( x ) , V ( x ) = Cu p 2 ( x ) .
U ( x ) = Ce u 0 1 ( x ) , V ( x ) = C ( u 0 1 ( x ) + u 0 2 ( x ) ) 2 .
L z i = C 2 ε 0 2 ω [ ( 3 2 + 2 cos θ ) 1 + 1 2 2 ]
L z s = C 2 ε 0 2 ω [ 3 2 1 + 1 2 2 ] ,
L z i W i < L z s W s for 1 < 2 .
S z s = ε 0 C 2 2 ω sin θ .
J z s = C 2 ε 0 2 ω [ 3 2 1 + 1 2 2 2 sin θ ]
sin θ > 1 cos θ

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