Abstract

We show that the orbital angular momentum (OAM) of evanescent light is drastically different from that of traveling light. Specifically, the paraxial contribution (typically the most significant part in a traveling wave) to the OAM vanishes in an evanescent Bessel wave when averaged over the azimuthal angle. Moreover, the OAM per unit energy for the evanescent Bessel field is reduced by a factor of (1+κ2k2) from the standard result for the corresponding traveling field, where k and κ are the wave number and the evanescent decay rate, respectively.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
  3. 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).
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  7. A. Lehmuskero, Y. Li, P. Johansson, and M. Kall, “Plasmonic particles set into fast orbital motion by an optical vortex beam,” Opt. Express 22, 4349–4356 (2014).
    [Crossref] [PubMed]
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  9. M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commmun. 5, 3115 (2014).
  10. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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2014 (3)

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commmun. 5, 3115 (2014).

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 3300 (2014).
[Crossref] [PubMed]

A. Lehmuskero, Y. Li, P. Johansson, and M. Kall, “Plasmonic particles set into fast orbital motion by an optical vortex beam,” Opt. Express 22, 4349–4356 (2014).
[Crossref] [PubMed]

2013 (2)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

M. Agnew, J. Z. Salvail, J. Leach, and R. W. Boyd, “Generation of orbital angular momentum Bell states and their verification via accessible nonlinear witnesses,” Phys. Rev. Lett. 111, 030402 (2013).
[Crossref] [PubMed]

2011 (2)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3, 161–204 (2011).
[Crossref]

V. E. Lembessis, S. Al-Awfi, M. Babiker, and D. L. Andrews, “Surface plasmon optical vortices and their influence on atoms,” J. Opt. 13, 064002 (2011).
[Crossref]

2010 (2)

S. N. Kurilkina, V. N. Belyi, and N. S. Kazak, “Features of evanescent Bessel light beams formed in structures containing a dielectric layer,” Opt. Commun. 238, 3860–3868 (2010).
[Crossref]

D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4, 211–217 (2010).
[Crossref]

2009 (2)

V. E. Lembessis, M. Babiker, and D. L. Andrews, “Surface optical vortices,” Phys. Rev. A 79, 011806 (2009).
[Crossref]

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[Crossref]

2008 (1)

2006 (1)

2005 (1)

V. Garces-Chavez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticles on a surface,” Appl. Phys. Lett. 86, 031106 (2005).
[Crossref]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

1999 (1)

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[Crossref]

1998 (1)

1997 (1)

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]

1979 (1)

L. W. Davis, “Theory of electromagnetics beams,” Phys. Rev. A 19, 1177–1179 (1979).
[Crossref]

1936 (1)

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

1909 (1)

J. 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]

Agnew, M.

M. Agnew, J. Z. Salvail, J. Leach, and R. W. Boyd, “Generation of orbital angular momentum Bell states and their verification via accessible nonlinear witnesses,” Phys. Rev. Lett. 111, 030402 (2013).
[Crossref] [PubMed]

Al-Awfi, S.

V. E. Lembessis, S. Al-Awfi, M. Babiker, and D. L. Andrews, “Surface plasmon optical vortices and their influence on atoms,” J. Opt. 13, 064002 (2011).
[Crossref]

Allen, L.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[Crossref]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
[Crossref] [PubMed]

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.

V. E. Lembessis, S. Al-Awfi, M. Babiker, and D. L. Andrews, “Surface plasmon optical vortices and their influence on atoms,” J. Opt. 13, 064002 (2011).
[Crossref]

V. E. Lembessis, M. Babiker, and D. L. Andrews, “Surface optical vortices,” Phys. Rev. A 79, 011806 (2009).
[Crossref]

Anetsberger, G.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[Crossref]

Arcizet, O.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[Crossref]

Babiker, M.

V. E. Lembessis, S. Al-Awfi, M. Babiker, and D. L. Andrews, “Surface plasmon optical vortices and their influence on atoms,” J. Opt. 13, 064002 (2011).
[Crossref]

V. E. Lembessis, M. Babiker, and D. L. Andrews, “Surface optical vortices,” Phys. Rev. A 79, 011806 (2009).
[Crossref]

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[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. Y.

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 3300 (2014).
[Crossref] [PubMed]

Belyi, V. N.

S. N. Kurilkina, V. N. Belyi, and N. S. Kazak, “Features of evanescent Bessel light beams formed in structures containing a dielectric layer,” Opt. Commun. 238, 3860–3868 (2010).
[Crossref]

Beth, R.

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

Bliokh, K. Y.

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 3300 (2014).
[Crossref] [PubMed]

Boyd, R. W.

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commmun. 5, 3115 (2014).

M. Agnew, J. Z. Salvail, J. Leach, and R. W. Boyd, “Generation of orbital angular momentum Bell states and their verification via accessible nonlinear witnesses,” Phys. Rev. Lett. 111, 030402 (2013).
[Crossref] [PubMed]

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Courtial, J.

Davis, L. W.

L. W. Davis, “Theory of electromagnetics beams,” Phys. Rev. A 19, 1177–1179 (1979).
[Crossref]

Dholakia, K.

V. Garces-Chavez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticles on a surface,” Appl. Phys. Lett. 86, 031106 (2005).
[Crossref]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
[Crossref] [PubMed]

Garces-Chavez, V.

V. Garces-Chavez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticles on a surface,” Appl. Phys. Lett. 86, 031106 (2005).
[Crossref]

Huang, H.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Johansson, P.

Kall, M.

Kazak, N. S.

S. N. Kurilkina, V. N. Belyi, and N. S. Kazak, “Features of evanescent Bessel light beams formed in structures containing a dielectric layer,” Opt. Commun. 238, 3860–3868 (2010).
[Crossref]

Kippenberg, T. J.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[Crossref]

Kotthaus, J. P.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[Crossref]

Kristensen, P.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Kurilkina, S. N.

S. N. Kurilkina, V. N. Belyi, and N. S. Kazak, “Features of evanescent Bessel light beams formed in structures containing a dielectric layer,” Opt. Commun. 238, 3860–3868 (2010).
[Crossref]

Lavery, M. P.

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commmun. 5, 3115 (2014).

Leach, J.

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commmun. 5, 3115 (2014).

M. Agnew, J. Z. Salvail, J. Leach, and R. W. Boyd, “Generation of orbital angular momentum Bell states and their verification via accessible nonlinear witnesses,” Phys. Rev. Lett. 111, 030402 (2013).
[Crossref] [PubMed]

Lehmuskero, A.

Leizer, A.

Lembessis, V. E.

V. E. Lembessis, S. Al-Awfi, M. Babiker, and D. L. Andrews, “Surface plasmon optical vortices and their influence on atoms,” J. Opt. 13, 064002 (2011).
[Crossref]

V. E. Lembessis, M. Babiker, and D. L. Andrews, “Surface optical vortices,” Phys. Rev. A 79, 011806 (2009).
[Crossref]

Li, Y.

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Malik, M.

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commmun. 5, 3115 (2014).

Mazilu, M.

Mirhosseini, M.

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commmun. 5, 3115 (2014).

Nori, F.

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 3300 (2014).
[Crossref] [PubMed]

Padgett, M. J.

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commmun. 5, 3115 (2014).

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3, 161–204 (2011).
[Crossref]

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[Crossref]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
[Crossref] [PubMed]

Poynting, J.

J. 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]

Ramachandran, S.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Ren, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Riviere, R.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[Crossref]

Roels, J.

D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4, 211–217 (2010).
[Crossref]

Ruschin, S.

Salvail, J. Z.

M. Agnew, J. Z. Salvail, J. Leach, and R. W. Boyd, “Generation of orbital angular momentum Bell states and their verification via accessible nonlinear witnesses,” Phys. Rev. Lett. 111, 030402 (2013).
[Crossref] [PubMed]

Schliesser, A.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[Crossref]

Simpson, N. B.

Spalding, G. C.

V. Garces-Chavez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticles on a surface,” Appl. Phys. Lett. 86, 031106 (2005).
[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]

Thomson, L. C.

Tur, M.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Unterreithmeier, Q. P.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[Crossref]

Van Thourhout, D.

D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4, 211–217 (2010).
[Crossref]

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Weig, E. M.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[Crossref]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Whyte, G.

Willner, A. E.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[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]

Yao, A. M.

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3, 161–204 (2011).
[Crossref]

Yue, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Zhan, Q.

Adv. Opt. Photonics (1)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3, 161–204 (2011).
[Crossref]

Appl. Phys. Lett. (1)

V. Garces-Chavez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticles on a surface,” Appl. Phys. Lett. 86, 031106 (2005).
[Crossref]

J. Opt. (1)

V. E. Lembessis, S. Al-Awfi, M. Babiker, and D. L. Andrews, “Surface plasmon optical vortices and their influence on atoms,” J. Opt. 13, 064002 (2011).
[Crossref]

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

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

Nat. Commmun. (1)

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commmun. 5, 3115 (2014).

Nat. Commun. (1)

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[Crossref]

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Figures (2)

Fig. 1
Fig. 1

m 2 ρ a 2 (ρ) [red dotted], μ2ρa2 (ρ) [black solid], ρa2 (ρ) [green dashed] as functions of ρ. Here a(ρ) = Jm (μρ), with μ = 2 k and m = 3.

Fig. 2
Fig. 2

0 ρ u p d ρ m 2 ρ a 2 ( ρ ) [red dotted], 0 ρ u p d ρ μ 2 ρ a 2 ( ρ ) [black solid], 0 ρ u p d ρ ρ a 2 ( ρ ) [green dashed] versus ρup. Here a(ρ) = Jm (μρ), with ρ 2 k and m = 3.

Equations (57)

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A ( r , t ) = A ( r ) e i ω t = A ( r ) ( α e x + β e y ) e i ω t , | α | 2 + | β | 2 = 1 ,
2 A ( r ) + k 2 A ( r ) = 0 , k = ω c ,
φ ( r ) = c 2 i ω A ( r ) .
2 A ( r ) + k 2 A ( r ) = 0 , 2 ϕ ( r ) + k 2 ϕ ( r ) = 0.
A ( r ) = u ( x , y ) e κ z ,
B ( r ) = × A ( r ) , E ( r ) = i c 2 ω × B ( r ) .
B = e κ z ( β κ u , α κ u , β u u α u y )
E = i ω k 2 e k z ( β 2 u x y α 2 u y 2 α κ 2 u , α 2 u x y β 2 u x 2 β κ 2 u , α κ u x β κ u y ) .
p = ε 0 E e i ω t + E * e i ω t 2 × B e i ω t + B * e i ω t 2 = ε 0 4 [ E × B * + c . c . ] ,
p = i ε 0 ω e 2 κ z 4 k 2 { κ 2 ( | α | 2 | β | 2 ) ( u ( ) u * c . c ) + 2 κ 2 Re ( α β * ) ( u ( ) u * c . c ) + | α | 2 ( u y u * y c . c ) + | β | 2 ( u x u * x c . c ) + [ ( α β * u * x u y + α * β u * y u x ) c . c ] } + i ε 0 ω e 2 κ z 4 k 2 e z [ | α | 2 κ y ( u * u y c . c ) + | β | 2 κ x ( u * u x c . c ) + Re ( α β * ) κ [ x ( u u * y c . c ) + y ( u u * x c . c ) ] ] ,
= e x x + e y y , ( ) = e x x e y y , ( ) = e x y + e y x .
p t r a = i ε 0 ω 4 k 2 { k z 2 ( u u * c . c ) + ( α β * α * β ) k z 2 | u | 2 × e z + | α | 2 ( u y u * y c . c ) + | β | 2 ( u x u * x c . c ) + [ ( α * β u * y u x + α β * u * x u y ) c . c ] } + ε 0 ω 4 k 2 e z [ 2 k z 3 | u | 2 + 2 | α | 2 k z | u y | 2 + 2 | β | 2 k z | u x | 2 + 2 Re ( α β * ) k z ( u * 2 u x y + c . c . ) | α | 2 k z y ( u * u y + c . c . ) | β | 2 k z x ( u * u x + c . c . ) ] ,
u ( ρ , ϕ ) = J m ( μ ρ ) e i m ϕ , μ = k 2 + κ 2 , m = 0 , ± 1 ,
p ϕ = m ε 0 ω e 2 κ z 2 k 2 ρ [ κ 2 a 2 ( ρ ) cos 2 ( ϕ ϕ 0 ) 1 ρ a ( ρ ) a ( ρ ) + a 2 ( ρ ) sin 2 ( ϕ ϕ 0 ) + m 2 ρ 2 a 2 ( ρ ) cos 2 ( ϕ ϕ 0 ) ] ,
cos ϕ 0 = α , sin ϕ 0 = β .
j z = m ε 0 ω e 2 κ z 2 k 2 [ κ 2 a 2 ( ρ ) cos 2 ( ϕ ϕ 0 ) 1 ρ a ( ρ ) a ( ρ ) + a 2 ( ρ ) sin 2 ( ϕ ϕ 0 ) + m 2 ρ 2 a 2 ( ρ ) cos 2 ( ϕ ϕ 0 ) ] .
j z ( t r a ) = m ε 0 ω 2 k 2 [ k z 2 a 2 ( ρ ) 1 ρ a ( ρ ) a ( ρ ) + a 2 ( ρ ) sin 2 ( ϕ ϕ 0 ) + m 2 ρ 2 a 2 ( ρ ) cos 2 ( ϕ ϕ 0 ) ] .
J ϕ ( t r a ) ( t r a ) = m ω ,
J z ( t r a ) = d x d y j z ( t r a ) ( x , y , z ) = 0 2 π d ϕ 0 d ρ ρ j z ( t r a ) ( ρ , ϕ , z ) ,
t r a = d x d y w t r a ( x , y , z ) = 0 2 π d ϕ 0 d ρ ρ w t r a ( ρ , ϕ , z ) ,
w t r a = ε 0 2 | E t r a ( r ) e i ω t + E t r a * ( r ) e i ω t 2 | 2 + 1 2 μ 0 | B t r a ( r ) e i ω t + B t r a * ( r ) e i ω t 2 | 2 = ε 0 4 | E t r a | 2 + 1 4 μ 0 | B t r a | 2 .
J z = ( 1 + κ 2 k 2 ) 1 m ω ,
J z = π m ε 0 ω e 2 κ z k 2 0 d ρ 1 2 [ ρ a 2 ( ρ ) + m 2 ρ a 2 ( ρ ) ] ,
0 d ρ a ( ρ ) a ( ρ ) = 1 2 0 d ρ d d ρ a 2 ( ρ ) = 0 ,
= π m ε 0 ω e 2 κ z k 2 0 d ρ { ( 1 + κ 2 k 2 ) κ 2 ρ a 2 ( ρ ) + 1 2 ( 1 κ 2 k 2 ) [ ρ a 2 ( ρ ) + m 2 ρ a 2 ( ρ ) ] } .
J m ( μ ρ ) 2 π μ ρ cos ( π ρ π 4 m π 2 ) ,
J m ( μ ρ ) ρ 2 π ρ sin ( μ ρ π 4 m π 2 ) .
ρ a 2 ( ρ ) 2 π μ cos 2 ( μ ρ π 4 m π 2 ) , ρ a 2 ( ρ ) 2 μ π sin 2 ( μ ρ π 4 m π 2 ) , m 2 ρ a 2 ( ρ ) 2 m 2 π μ ρ 2 cos 2 ( μ ρ π 4 m π 2 ) 2 m 2 π μ ρ 2 .
J z π m ε 0 ω e 2 κ z 2 k 2 d ρ ρ a 2 ( ρ ) π μ 2 m ε 0 ω 2 k 2 e 2 κ z d ρ ρ a 2 ( ρ )
π ε 0 ω 2 e 2 κ z k 2 d ρ { ( 1 + κ 2 k 2 ) κ 2 ρ a 2 ( ρ ) + μ 2 2 ( 1 κ 2 k 2 ) ρ a 2 ( ρ ) } ( 1 + κ 2 k 2 ) π μ 2 ε 0 ω 2 e 2 κ z 2 k 2 d ρ ρ a 2 ( ρ ) ,
J z = i ( α β * α * β ) π ε 0 ω e 2 κ z k 2 d ρ 1 2 ρ a ( ρ ) ,
= π ε 0 ω 2 e 2 κ z k 2 d ρ [ ( 1 + κ 2 k 2 ) κ 2 ρ a 2 ( ρ ) + 1 2 ( 1 κ 2 k 2 ) ρ a 2 ( ρ ) ] .
J z = ( 1 + κ 2 k 2 ) 1 σ ω ,
J z ( t r a ) ( t r a ) = σ ω .
( ) = e ρ ( cos 2 ϕ ρ 1 ρ sin 2 ϕ ϕ ) e ϕ ( sin 2 ϕ ρ + 1 ρ cos 2 ϕ ϕ ) , ( ) = e ρ ( sin 2 ϕ ρ + 1 ρ cos 2 ϕ ϕ ) + e ϕ ( cos 2 ϕ ρ 1 ρ sin 2 ϕ ϕ ) , = e ρ ρ + e ϕ 1 ρ ϕ , ( y x ) = ( cos ϕ 1 ρ sin ϕ sin ϕ 1 ρ cos ϕ ) ( ϕ ρ ) ,
p = i ε 0 ω e 2 κ z 4 k 2 { κ 2 ( | α | 2 | β | 2 ) ( u ( ) u * c . c ) + 2 κ 2 Re ( α β * ) ( u ( ) u * c . c ) + | α | 2 ( u y u * y c . c ) + | β | 2 ( u x u * x c . c ) + [ ( α β * u * x u y + α * β u * y u x ) c . c ] } .
( | α | 2 | β | 2 ) κ 2 ( u ( ) u * c . c . ) ϕ + 2 κ 2 Re ( α β * ) ( u ( ) u * c . c ) ϕ = 2 i ( α 2 β 2 ) κ 2 m ρ a 2 ( ρ ) cos 2 ϕ + 4 i α β κ 2 m ρ a 2 ( ρ ) sin 2 ϕ = 2 i κ 2 m ρ a 2 ( ρ ) [ ( α 2 β 2 ) cos 2 ϕ + 2 α β sin 2 ϕ ] .
cos ϕ 0 = α , sin ϕ 0 = β ,
α 2 β 2 = cos 2 ϕ 0 sin 2 ϕ 0 = cos 2 ϕ 0 , 2 α β = 2 sin ϕ 0 cos ϕ 0 = sin 2 ϕ 0 .
( | α | 2 | β | 2 ) κ 2 ( u ( ) u * u * ( ) u ) ϕ + 2 κ 2 Re ( α β * ) ( u ( ) u * c . c ) ϕ = 2 i κ 2 m ρ a 2 ( ρ ) [ cos 2 ϕ 0 cos 2 ϕ + sin 2 ϕ 0 sin 2 ϕ ] = 2 i κ 2 m ρ a 2 ( ρ ) cos 2 ( ϕ ϕ 0 ) .
| α | 2 ( u y u * y c . c . ) ϕ + | β | 2 ( u x u * x c . c . ) ϕ = 2 i m ρ 2 a ( ρ ) a ( ρ ) 2 i m ρ a 2 ( ρ ) [ α 2 sin 2 ϕ + β 2 cos 2 ϕ ] 2 i m 3 ρ 3 a 2 ( ρ ) [ α 2 cos 2 ϕ + β 2 sin 2 ϕ ]
[ ( α β * u * x u y + α * β u * y u x c . c . ) ] ϕ = 2 i m ρ a 2 ( ρ ) [ 2 α β sin ϕ cos ϕ ] 2 i m 3 ρ 3 a 2 ( ρ ) [ 2 α β cos ϕ sin ϕ ] .
| α | 2 ( u y u * y c . c . ) ϕ + | β | 2 ( u x u * x c . c . ) ϕ + [ ( α β * u * x u y + α * β u * y u x ) c . c . ] ϕ = 2 i [ m ρ a 2 ( ρ ) [ α sin ϕ β cos ϕ ] 2 + m 3 ρ 3 a 2 ( ρ ) [ α cos ϕ + β sin ϕ ] 2 m ρ 2 a ( ρ ) a ( ρ ) ] = 2 i [ m ρ a 2 ( ρ ) sin 2 ( ϕ ϕ 0 ) + m 3 ρ 3 a 2 ( ρ ) cos 2 ( ϕ ϕ 0 ) m ρ 2 a ( ρ ) a ( ρ ) ] .
p ϕ = m ε 0 ω e 2 κ z 2 k 2 ρ [ κ 2 a 2 ( ρ ) cos 2 ( ϕ ϕ 0 ) 1 ρ a ( ρ ) a ( ρ ) + a 2 ( ρ ) sin 2 ( ϕ ϕ 0 ) + m 2 ρ 2 a 2 ( ρ ) cos 2 ( ϕ ϕ 0 ) ] .
w = ε 0 4 | E ( r ) | 2 + 1 4 μ 0 | B ( r ) | 2 .
| B x | 2 + | B y | 2 = e 2 κ z κ 2 | u | 2 = e 2 κ z κ 2 a 2 ( ρ ) ,
| B z | 2 = e 2 κ z | β u x α u y | 2 = e 2 κ z { a 2 ( ρ ) [ β cos ϕ α sin ϕ ] 2 + m 2 ρ 2 a 2 ( ρ ) [ β sin ϕ + α cos ϕ ] 2 } = e 2 κ z { a 2 ( ρ ) sin 2 ( ϕ ϕ 0 ) + m 2 ρ 2 a 2 ( ρ ) cos 2 ( ϕ ϕ 0 ) } ,
d x d y 1 4 μ 0 | B | 2 1 4 μ 0 0 2 π d ϕ 0 d ρ ρ ( | B x | 2 + | B y | 2 + | B z | 2 ) = π ε 0 ω 2 e 2 κ z 2 k 2 0 d ρ ρ [ κ 2 a 2 ( ρ ) + 1 2 a 2 ( ρ ) + 1 2 m 2 ρ 2 a 2 ( ρ ) ] .
| E | 2 = c 4 ω 2 | × B | 2 = ω 2 k 4 ( × B ) . ( × B * ) .
b ( × a ) = ( a × b ) + a ( × b ) ,
( × B ) ( × B * ) = [ B * × ( × B ) ] + B * ( × [ × B ] ) = [ B * × ( × B ) ] + B * [ ( B ) 2 B ] = [ B * × ( × B ) ] + k 2 | B | 2 ,
( × B ) ( × B * ) = 1 2 { [ B * × ( × B ) ] + c . c . } + k 2 | B | 2 .
[ B * × ( × B ) ] = [ B * × ( × B ) ] + z [ B * × ( × B ) ] z ,
z [ B * × ( × B ) ] z + c . c . = z [ ( B x * B x z B x * B z x ) ( B y * B z y B y * B y z ) ] + c . c . = 2 ( | B x | 2 + | B y | 2 | B z | 2 ) z 2 z [ ( B x * B z ) x + ( B y * B z ) y + ( B x B z * ) x + ( B y B z * ) x ] ,
( × B ) ( × B * ) = k 2 | B | 2 + 1 2 2 z 2 ( | B x | 2 + | B y | 2 | B z | 2 ) 1 2 z [ ( B x * B z ) x + ( B y * B z ) y + ( B x B z * ) x + ( B y B z * ) y ] + 1 2 { [ B * × ( × B ) ] + c . c } .
d x d y ε 0 4 | E | 2 = d x d y ε 0 ω 2 4 k 4 ( × B ) ( × B * ) = d x d y [ ε 0 ω 2 4 k 2 | B | 2 + 1 2 ε 0 ω 2 4 k 4 2 z 2 ( | B x | 2 + | B y | 2 | B z | 2 ) ] = π ε 0 ω 2 e 2 κ z 2 k 2 0 d ρ ρ [ κ 2 a 2 ( ρ ) + 1 2 a 2 ( ρ ) + 1 2 m 2 ρ 2 a 2 ( ρ ) ] + π ε 0 ω 2 e 2 κ z k 2 κ 2 k 2 0 d ρ ρ [ κ 2 a 2 ( ρ ) 1 2 a 2 ( ρ ) 1 2 m 2 ρ 2 a 2 ( ρ ) ] .
= π ε 0 ω 2 e 2 κ z k 2 0 d ρ { ( 1 + κ 2 k 2 ) κ 2 ρ a 2 ( ρ ) + 1 2 ( 1 κ 2 k 2 ) [ ρ a 2 ( ρ ) + m 2 ρ a 2 ( ρ ) ] } ,

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