Abstract

We deduce the expressions for the two circularly polarized components of a paraxial beam propagating along the optical axis of a uniaxial crystal. We find that each of them is the sum of two contributions, the first being a free field and the second describing the interaction with the opposite component. Moreover, we expand both components as a superposition of vortices of any order, thus obtaining a complete physical picture of the interaction dynamics. Consequently, we argue that a left-hand circularly polarized incoming beam, endowed with a circular symmetric profile, gives rise, inside the crystal, to a right-hand circularly polarized vortex of order 2. The efficiency of this vortex generation is investigated by means of a power exchange analysis. The Gaussian case is fully discussed, showing the relevant features of the vortex generation.

© 2003 Optical Society of America

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References

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  1. C. Brosseau, Fundamentals of Polarized Light (Wiley, New York, 1998).
  2. H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, New York, 1983).
  3. R. M. Herrero, J. M. Movilla, P. M. Mejias, “Beam propagation through uniaxial anisotropic media:  global changes in the spatial profile,” J. Opt. Soc. Am. A 18, 2009–2014 (2001).
    [CrossRef]
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    [CrossRef]
  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. L. Allen, M. J. Padgett, M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–295 (1999).
    [CrossRef]
  7. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–80 (1993).
    [CrossRef]
  8. I. Freund, “Optical vortex trajectories,” J. Opt. Commun. 181, 19–33 (2000).
    [CrossRef]
  9. F. Vivanco, F. Melo, “Surface spiral waves in a filamentary vortex,” Phys. Rev. Lett. 85, 2116–2119 (2000).
    [CrossRef] [PubMed]
  10. J. M. Vaughan, D. V. Willetts, “Interference properties of a light beam having a helical wave surface,” Opt. Commun. 30, 263–267 (1979).
    [CrossRef]
  11. M. Harris, C. A. Hill, P. R. Tapster, J. M. Vaughan, “Lase modes with helical wave fronts,” Phys. Rev. A 49, 3119–3122 (1994).
    [CrossRef] [PubMed]
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    [CrossRef]
  13. C. Tamm, C. O. Weiss, “Bistability and optical switching of spatial patterns in a laser,” J. Opt. Soc. Am. B 7, 1034–1038 (1990).
    [CrossRef]
  14. G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
    [CrossRef]
  15. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristense, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
    [CrossRef]
  16. N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
    [CrossRef]
  17. J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
    [CrossRef]
  18. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. A. Ciattoni, G. Cincotti, C. Palma, H. Weber, “Energy exchange between the Cartesian components of a paraxial beam in a uniaxial crystal,” J. Opt. Soc. Am. A 19, 1894–1900 (2002).
    [CrossRef]
  22. G. Cincotti, A. Ciattoni, C. Palma, “Hermite–Gauss beams in uniaxially anisotropic crystal,” IEEE J. Quantum Electron. 12, 1517–1524 (2001).
    [CrossRef]

2002 (2)

2001 (3)

2000 (2)

I. Freund, “Optical vortex trajectories,” J. Opt. Commun. 181, 19–33 (2000).
[CrossRef]

F. Vivanco, F. Melo, “Surface spiral waves in a filamentary vortex,” Phys. Rev. Lett. 85, 2116–2119 (2000).
[CrossRef] [PubMed]

1999 (1)

L. Allen, M. J. Padgett, M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–295 (1999).
[CrossRef]

1998 (1)

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

1996 (1)

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

1994 (3)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristense, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

M. Harris, C. A. Hill, P. R. Tapster, J. M. Vaughan, “Lase modes with helical wave fronts,” Phys. Rev. A 49, 3119–3122 (1994).
[CrossRef] [PubMed]

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

1993 (2)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–80 (1993).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

1992 (2)

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[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]

1990 (1)

1983 (1)

1979 (1)

J. M. Vaughan, D. V. Willetts, “Interference properties of a light beam having a helical wave surface,” Opt. Commun. 30, 263–267 (1979).
[CrossRef]

Allen, L.

L. Allen, M. J. Padgett, M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–295 (1999).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[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]

Arlt, J.

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

Babiker, M.

L. Allen, M. J. Padgett, M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–295 (1999).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristense, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[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]

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light (Wiley, New York, 1998).

Chen, H. C.

H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, New York, 1983).

Ciattoni, A.

Cincotti, G.

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristense, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Crosignani, B.

Dholakia, K.

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

Di Porto, P.

Feit, M. D.

Fleck, J. A.

Freund, I.

I. Freund, “Optical vortex trajectories,” J. Opt. Commun. 181, 19–33 (2000).
[CrossRef]

Harris, M.

M. Harris, C. A. Hill, P. R. Tapster, J. M. Vaughan, “Lase modes with helical wave fronts,” Phys. Rev. A 49, 3119–3122 (1994).
[CrossRef] [PubMed]

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Heckenberg, N. R.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

Herrero, R. M.

Hill, C. A.

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

M. Harris, C. A. Hill, P. R. Tapster, J. M. Vaughan, “Lase modes with helical wave fronts,” Phys. Rev. A 49, 3119–3122 (1994).
[CrossRef] [PubMed]

Indebetouw, G.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–80 (1993).
[CrossRef]

Kristense, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristense, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

McDuff, R.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

Mejias, P. M.

Melo, F.

F. Vivanco, F. Melo, “Surface spiral waves in a filamentary vortex,” Phys. Rev. Lett. 85, 2116–2119 (2000).
[CrossRef] [PubMed]

Movilla, J. M.

Padgett, M. J.

L. Allen, M. J. Padgett, M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–295 (1999).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

Palma, C.

Robertson, D. A.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

Rubinsztein-Dunlop, H.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

Smith, C. P.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

Smith, G. M.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

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]

Tamm, C.

Tapster, P. R.

M. Harris, C. A. Hill, P. R. Tapster, J. M. Vaughan, “Lase modes with helical wave fronts,” Phys. Rev. A 49, 3119–3122 (1994).
[CrossRef] [PubMed]

Turnbull, G. A.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Vaughan, J. M.

M. Harris, C. A. Hill, P. R. Tapster, J. M. Vaughan, “Lase modes with helical wave fronts,” Phys. Rev. A 49, 3119–3122 (1994).
[CrossRef] [PubMed]

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

J. M. Vaughan, D. V. Willetts, “Interference properties of a light beam having a helical wave surface,” Opt. Commun. 30, 263–267 (1979).
[CrossRef]

Vivanco, F.

F. Vivanco, F. Melo, “Surface spiral waves in a filamentary vortex,” Phys. Rev. Lett. 85, 2116–2119 (2000).
[CrossRef] [PubMed]

Weber, H.

Wegener, M. J.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

Weiss, C. O.

Willetts, D. V.

J. M. Vaughan, D. V. Willetts, “Interference properties of a light beam having a helical wave surface,” Opt. Commun. 30, 263–267 (1979).
[CrossRef]

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristense, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[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]

IEEE J. Quantum Electron. (1)

G. Cincotti, A. Ciattoni, C. Palma, “Hermite–Gauss beams in uniaxially anisotropic crystal,” IEEE J. Quantum Electron. 12, 1517–1524 (2001).
[CrossRef]

J. Mod. Opt. (2)

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–80 (1993).
[CrossRef]

J. Opt. Commun. (1)

I. Freund, “Optical vortex trajectories,” J. Opt. Commun. 181, 19–33 (2000).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

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

Opt. Commun. (5)

J. M. Vaughan, D. V. Willetts, “Interference properties of a light beam having a helical wave surface,” Opt. Commun. 30, 263–267 (1979).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristense, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Opt. Quantum Electron. (1)

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[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. Harris, C. A. Hill, P. R. Tapster, J. M. Vaughan, “Lase modes with helical wave fronts,” Phys. Rev. A 49, 3119–3122 (1994).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

F. Vivanco, F. Melo, “Surface spiral waves in a filamentary vortex,” Phys. Rev. Lett. 85, 2116–2119 (2000).
[CrossRef] [PubMed]

Prog. Opt. (1)

L. Allen, M. J. Padgett, M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–295 (1999).
[CrossRef]

Other (2)

C. Brosseau, Fundamentals of Polarized Light (Wiley, New York, 1998).

H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, New York, 1983).

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Figures (3)

Fig. 1
Fig. 1

Normalized moduli of (a) A+ and (b) A- (plotted at z=0 µm, z=1500 µm, z=3000 µm, z=4500 µm, and z=6000 µm) of a Gaussian beam of waist s=10 µm at λ=0.5 µm, propagating in a calcite crystal (no=1.658 and ne=1.486).

Fig. 2
Fig. 2

Interference pattern |A-(r, ϕ, z)+E0 exp(ik0noz)|2/|E0|2 (for the same Gaussian beam as that of Fig. 1) at the planes (a) z=1500 µm, (b) z=3000 µm, (c) z=4500 µm, and (d) z=6000 µm.

Fig. 3
Fig. 3

Normalized powers W+(z)/W+(0) and W-(z)/W+(0) for a Gaussian beam. Both powers exhibit a Lorentzian saturation toward their common asymptotic value.

Equations (45)

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

Ao(r, z)= d2k expik·r-ik22k0no z×1k2 ky2-kxky-kxkykx2·E˜(k),Ae(r, z)== d2k expik·r-inok22k0ne2 z×1k2 kx2kxkykxkyky2·E˜(k),
E˜(k)=1(2π)2 = d2r exp(-ik·r)E(r, 0).
eˆ+=12 (eˆx+ieˆy),eˆ-=12 (eˆx-ieˆy),
V+V-=12 1-i1iVxVy,
VxVy=12 11i-iV+V-.
Ao+(r, z)Ao-(r, z)=12  d2k expik·r-ik22k0no z×1-(kx-iky)2k2-(kx+iky)2k21×E˜+(k)E˜-(k),
Ae+(r, z)Ae-(r, z)=12  d2k expik·r-inok22k0ne2 z×1(kx-iky)2k2(kx+iky)2k21×E˜+(k)E˜-(k),
E˜±(k)=1(2π)2  d2r exp(-ik·r)E±(r, 0)
A+(r, z)=F+(r, z)+G-(r, z),A-(r, z)=F-(r, z)+G+(r, z),
F±(r, z)=12  d2k exp(ik·r)×exp-ik22k0no z+exp-inok22k0ne2 zE˜±(k),G±(r, z)=12  d2k exp(ik·r) (kx±iky)2k2×-exp-ik22k0no z+exp-inok22k0ne2 zE˜±(k).
r=r(cos ϕ eˆx+sin ϕ eˆy),k=k(cos θ eˆx+sin θ eˆy).
F±(r, ϕ, z)=12 0dkkexp-ik22k0no z+exp-inok22k0ne2 z×02πdθ exp[ikr cos(θ-ϕ)]E˜±(k, θ),G±(r, ϕ, z)=12 0dkk-exp-ik22k0no z+exp-inok22k0ne2 z 02πdθ×exp[ikr cos(θ-ϕ)±i2θ]E˜±(k, θ),
E˜±(k, θ)=1(2π)2 0drr 02πdϕ×exp[-ikr cos(θ-ϕ)]E±(r, ϕ, 0).
F±(r, ϕ, z)=n=-+ exp(inϕ)F±(n)(r, z),
G±(r, ϕ, z)=n=-+ exp(inϕ)G±(n)(r, z),
F±(n)(r, z)=π 0dkkexp-ik22k0no z+exp-inok22k0ne2 zJn(kr)E˜±(n)(k),G±(n)=π 0dkkexp-ik22k0no z-exp-inok22k0ne2 zJn(kr)E˜±(n2)(k),
E˜±(n)(k)=1(2π)2 0drrJn(kr) 02πdϕ×exp(-inϕ)E±(r, ϕ, 0).
A+(r, ϕ, z)=n=-+ exp(inϕ)[F+(n)(r, z)+G-(n)(r, z)],A-(r, ϕ, z)=n=-+ exp(inϕ)[F-(n)(r, z)+G+(n)(r, z)],
E(r, ϕ, 0)=exp(imϕ)E+(r)eˆ+,
A+(r, ϕ, z)=exp(imϕ)F+(m)(r, z),A-(r, ϕ, z)=exp[i(m+2)ϕ]G+(m+2)(r, z).
A+(r, ϕ, z)=π 0dkkexp-ik22k0no z+exp-inok22k0ne2 zJ0(kr)E˜+(k),A-(r, ϕ, z)=exp(i2ϕ)π 0dkkexp-ik22k0no z-exp-inok22k0ne2 zJ2(kr)E˜+(k),
E˜+(k)=12π 0drrJ0(kr)E+(r, 0).
W±(z)=0drr 02πdϕ|A±(r, ϕ, z)|2.
W±(z)=12 W+(0)±4π30dkk coszΔ2k0no k2|E˜+(k)|2,
W+()=W-()=12 W+(0);
E(r, ϕ, 0)=E0 exp-r22s2eˆ+,
A+(r, ϕ, z)=12 E0s2exp-r22so2(z)so2(z)+exp-r22se2(z)se2(z),
A-(r, ϕ, z)=-12 E0s2 exp(i2ϕ)×exp-r22so2(z)so2(z)-exp-r22se2(z)se2(z)+2 exp-r22so2(z)-exp-r22se2(z)r2,
so2(z)=s2+izk0no,se2(z)=s2+inozk0ne2.
W±(z)=12 W+(0)1±11+(z/L)2,
F±(n)(r, z)=12π 02πdϕ exp(-inϕ)F±(r, ϕ, z),G±(n)(r, z)=12π 02πdϕ exp(-inϕ)G±(r, ϕ, z).
12π 02πdα exp[ikr cos(α-β)+iqα]=iq exp(iqβ)Jq(kr),
F±(n)(r, z)=π 0dkkexp-ik22k0no z+exp-inok22k0ne2 zJn(kr)inI±(n)(k),G±(n)(r, z)=π 0dkk-exp-ik22k0no z+exp-inok22k0ne2 zJn(kr)inI±(n2)(k),
I±(n)(k)=12π 02πdθ exp(-inθ)E˜±(k, θ).
I±(n)(k)=i-n 1(2π)2 0drrJn(kr)×02πdϕ exp(-inϕ)E±(r, ϕ, 0)i-nE˜±(n)(k),
W+(z)=2π3 0dkk 0dkkexp-ik22k0no z+exp-inok22k0ne2 z×expik22k0no z+expinok22k0ne2 z×0drrJ0(kr)J0(kr)E˜+(k)E˜+*(k),W-(z)=2π3 0dkk 0dkkexp-ik22k0no z-exp-inok22k0ne2 z×expik22k0no z-expinok22k0ne2 z×0drrJ2(kr)J2(kr)E˜+(k)E˜+*(k).
0drrJn(kr)Jn(kr)=1k δ(k-k),
W+(z)=4π3 0dkk1+coszΔ2k0no k2|E˜+(k)|2,W-(z)=4π3 0dkk1-coszΔ2k0no k2|E˜+(k)|2.
W+(0)=8π3 0dkk|E˜+(k)|2,
0dξξJ0(bξ)exp(-aξ2)=12a exp-b24a,
E˜+(k)=E0s22π exp-k2s22.
A+(r, ϕ, z)=E0s22 0dkkJ0(kr)×exp-s2+izk0no k22+exp-s2+inozk0ne2 k22,A-(r, ϕ, z)=exp(i2ϕ) E0s22 0dkkJ2(kr)×exp-s2+izk0no k22-exp-s2+inozk0ne2 k22.
0dξξJ2(bξ)exp(-aξ2)=-12a exp-b24a+2r2 1-exp-b24a,
W+(0)=πE02s2.
W±(z)=W+(0)s2 0dkk exp(-s2k2)±s2 Re 0dkk exp-s2+izΔ2k0nok2,

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