Abstract

Attention is called to a polarization basis formed by four Jones vectors that is reducible to two basic structures. In these states the field is linearly polarized at any point, but the polarization direction changes with the angular coordinate. Very simple equations hold for the field lines of these basis vectors. The adoption of this basis is of interest for beams possessing a single vortex of any order, say, m, as well as for beams with two vortices of opposite charges, m and -m. As an example, the application to vectorial Bessel–Gauss beams is briefly discussed.

© 2001 Optical Society of America

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  1. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Maxwell beams,” J. Opt. Soc. Am. A 3, 536–540 (1983).
    [CrossRef]
  2. R. H. Jordan, G. D. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel–Gauss beam solution,” Opt. Lett. 19, 427–429 (1994).
    [CrossRef] [PubMed]
  3. A. A. Tovar, G. H. Clark, “Concentric-circle-grating, surface-emitting laser beam propagation in complex optical systems,” J. Opt. Soc. Am. A 14, 3333–3340 (1997).
    [CrossRef]
  4. S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A 15, 2712–2719 (1998).
    [CrossRef]
  5. P. L. Greene, D. G. Hall, “Properties and diffraction of vector Bessel–Gauss beams,” J. Opt. Soc. Am. A 15, 3020–3027 (1998).
    [CrossRef]
  6. C. J. R. Sheppard, “Polarization of almost-plane waves,” J. Opt. Soc. Am. A 17, 335–341 (2000).
    [CrossRef]
  7. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
  8. D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11, 1641–1643 (1994).
    [CrossRef]
  9. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
    [CrossRef]
  10. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence–polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
    [CrossRef]
  11. G. Piquero, J. M. Movilla, P. M. Mejı́as, R. Martı́nez-Herrero, “Beam quality of partially polarized beams propagating through lens-like birefringent elements,” J. Opt. Soc. Am. A 16, 2666–2668 (1999).
    [CrossRef]
  12. S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (1999).
    [CrossRef]
  13. M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Balian, M. Kléman, J.-P. Poirer, eds. (North-Holland, Amsterdam, 1981), p. 454.
  14. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–80 (1993).
    [CrossRef]
  15. I. Freund, “Optical vortex trajectories,” Opt. Commun. 181, 19–33 (2000).
    [CrossRef]
  16. F. Vivanco, F. Melo, “Surface spiral waves in a filamentary vortex,” Phys. Rev. Lett. 85, 2116–2119 (2000).
    [CrossRef] [PubMed]
  17. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  18. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  19. E. Abramochkin, V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
    [CrossRef]
  20. 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]
  21. S. C. Tidwell, D. H. Ford, W. D. Kimura, “Generatingradially polarized beams interferometrically,” Appl. Opt. 29, 2234–2239 (1990).
    [CrossRef] [PubMed]
  22. S. C. Tidwell, G. H. Kim, W. D. Kimura, “Efficient radially polarized laser beam generation with a double interferometer,” Appl. Opt. 32, 5222–5229 (1993).
    [CrossRef] [PubMed]
  23. A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 15, 2705–2711 (1998).
    [CrossRef]
  24. R. Oron, N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
    [CrossRef]
  25. J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
    [CrossRef] [PubMed]
  26. R. M. Herman, T. A. Wiggins, “Bessel-like beams modulated by arbitrary radial functions,” J. Opt. Soc. Am. A 17, 1021–1032 (2000).
    [CrossRef]
  27. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980).
  28. T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
    [CrossRef]
  29. C. Brosseau, Fundamentals of Polarized Light (Wiley, New York, 1998).
  30. J. D. Lawrence, A Catalog of Special Plane Curves (Dover, New York, 1972).
  31. Jacques Bernoulli was so fond of this curve that, in accord with his desire, a spiral was engraved on his tombstone.
  32. C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” Microwaves Opt. Acoust. 2, 105–112 (1978).
    [CrossRef]
  33. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  34. R. H. Jordan, D. G. Hall, O. King, G. Wicks, S. Rishton, “Lasing behavior of circular grating surface-emitting semiconductor lasers,” J. Opt. Soc. Am. B 14, 449–453 (1997).
    [CrossRef]
  35. C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1268 (1998).
    [CrossRef]

2000 (5)

I. Freund, “Optical vortex trajectories,” 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]

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
[CrossRef]

C. J. R. Sheppard, “Polarization of almost-plane waves,” J. Opt. Soc. Am. A 17, 335–341 (2000).
[CrossRef]

R. M. Herman, T. A. Wiggins, “Bessel-like beams modulated by arbitrary radial functions,” J. Opt. Soc. Am. A 17, 1021–1032 (2000).
[CrossRef]

1999 (2)

1998 (6)

1997 (2)

1994 (2)

1993 (3)

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]

S. C. Tidwell, G. H. Kim, W. D. Kimura, “Efficient radially polarized laser beam generation with a double interferometer,” Appl. Opt. 32, 5222–5229 (1993).
[CrossRef] [PubMed]

1991 (1)

E. Abramochkin, V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

1990 (1)

1988 (1)

1987 (2)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

1985 (1)

1983 (1)

1978 (1)

C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” Microwaves Opt. Acoust. 2, 105–112 (1978).
[CrossRef]

Abramochkin, E.

E. Abramochkin, V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

Allen, L.

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]

Beijersbergen, M. W.

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]

Berry, M.

M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Balian, M. Kléman, J.-P. Poirer, eds. (North-Holland, Amsterdam, 1981), p. 454.

Borghi, R.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence–polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Brosseau, C.

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

Clark, G. H.

Davidson, N.

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
[CrossRef]

Durnin, J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Ford, D. H.

Freund, I.

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

Friberg, A. T.

Friesem, A. A.

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
[CrossRef]

Fukumitsu, O.

Gori, F.

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence–polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980).

Greene, P. L.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1268 (1998).
[CrossRef]

P. L. Greene, D. G. Hall, “Properties and diffraction of vector Bessel–Gauss beams,” J. Opt. Soc. Am. A 15, 3020–3027 (1998).
[CrossRef]

Guattari, G.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence–polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Hall, D. G.

Hall, G. D.

Hasman, E.

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
[CrossRef]

Herman, R. M.

Indebetouw, G.

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

James, D. F. V.

Jordan, R. H.

Kim, G. H.

Kimura, W. D.

King, O.

Lawrence, J. D.

J. D. Lawrence, A Catalog of Special Plane Curves (Dover, New York, 1972).

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).

Marti´nez-Herrero, R.

Meji´as, P. M.

Melo, F.

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

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Movilla, J. M.

Mukunda, N.

Olson, C.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1268 (1998).
[CrossRef]

Oron, R.

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Piquero, G.

Rishton, S.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980).

Santarsiero, M.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence–polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Seshadri, S. R.

Sheppard, C. J. R.

C. J. R. Sheppard, “Polarization of almost-plane waves,” J. Opt. Soc. Am. A 17, 335–341 (2000).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” Microwaves Opt. Acoust. 2, 105–112 (1978).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Simon, R.

Sudarshan, E. C. G.

Takenaka, T.

Tidwell, S. C.

Tovar, A. A.

Turunen, J.

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]

Vasara, A.

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence–polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Vivanco, F.

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

Volostnikov, V.

E. Abramochkin, V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

Wicks, G.

Wicks, G. W.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1268 (1998).
[CrossRef]

Wiggins, T. A.

Wilson, T.

C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” Microwaves Opt. Acoust. 2, 105–112 (1978).
[CrossRef]

Woerdman, J. P.

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]

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).

Yokota, M.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1268 (1998).
[CrossRef]

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. A (11)

C. J. R. Sheppard, “Polarization of almost-plane waves,” J. Opt. Soc. Am. A 17, 335–341 (2000).
[CrossRef]

R. M. Herman, T. A. Wiggins, “Bessel-like beams modulated by arbitrary radial functions,” J. Opt. Soc. Am. A 17, 1021–1032 (2000).
[CrossRef]

D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11, 1641–1643 (1994).
[CrossRef]

S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (1999).
[CrossRef]

G. Piquero, J. M. Movilla, P. M. Mejı́as, R. Martı́nez-Herrero, “Beam quality of partially polarized beams propagating through lens-like birefringent elements,” J. Opt. Soc. Am. A 16, 2666–2668 (1999).
[CrossRef]

A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 15, 2705–2711 (1998).
[CrossRef]

S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A 15, 2712–2719 (1998).
[CrossRef]

T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
[CrossRef]

A. A. Tovar, G. H. Clark, “Concentric-circle-grating, surface-emitting laser beam propagation in complex optical systems,” J. Opt. Soc. Am. A 14, 3333–3340 (1997).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Maxwell beams,” J. Opt. Soc. Am. A 3, 536–540 (1983).
[CrossRef]

P. L. Greene, D. G. Hall, “Properties and diffraction of vector Bessel–Gauss beams,” J. Opt. Soc. Am. A 15, 3020–3027 (1998).
[CrossRef]

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

Microwaves Opt. Acoust. (1)

C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” Microwaves Opt. Acoust. 2, 105–112 (1978).
[CrossRef]

Opt. Commun. (5)

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
[CrossRef]

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

E. Abramochkin, V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[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]

Opt. Lett. (2)

Phys. Rev. Lett. (2)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

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

Pure Appl. Opt. (1)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence–polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Other (7)

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).

M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Balian, M. Kléman, J.-P. Poirer, eds. (North-Holland, Amsterdam, 1981), p. 454.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980).

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

J. D. Lawrence, A Catalog of Special Plane Curves (Dover, New York, 1972).

Jacques Bernoulli was so fond of this curve that, in accord with his desire, a spiral was engraved on his tombstone.

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Figures (4)

Fig. 1
Fig. 1

Geometrical sketch of two mutually rotated frames.

Fig. 2
Fig. 2

Field lines for the polarization state described by Eq. (25) with m=1.

Fig. 3
Fig. 3

Field lines for the polarization state described by Eq. (26) with m=2.

Fig. 4
Fig. 4

Field lines for the polarization state described by Eq. (29) with α=π/2-0.1.

Equations (42)

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

V(ρ, φ, z)
=-i expikz+ρ22zλz002πV(ρ, φ, 0)×expik2z ρ2-ikz ρρcos(φ-φ)ρdρdφ,
V(p, φ, 0)=m=-v0m(ρ)exp(imφ),
v0m(ρ)=12π02πV(ρ, φ, 0)exp(-imφ)dφ.
V(ρ, φ, z)=-ik expikz+ρ22zz×m=-(-i)mexp(imφ)0v0m(ρ)×expi kρ22zJmkρρzρdρ,
Jm(t)=12π02πexp[i(mψ-t sin ψ)]dψ
V(ρ, φ, z)=m=-vzm(ρ)exp(imφ),
vzm(ρ)=(-i)m+1kzexp{ik[z+ρ2/(2z)]}×0v0m(ρ)exp[ikρ2/(2z)]Jmkρρzρdρ.
V(ρ, ϕ, 0)=v0m(ρ)[a+exp(imφ)+a-exp(-imφ)]=v0m(ρ)[(a++a-)Cm+i(a+-a-)Sm].
V(ρ, ϕ, z)=vzm(ρ)[(a++a-)Cm+i(a+-a-)Sm].
ExEy=vzm(ρ)c110exp(imφ)+c210exp(-imφ)+c301exp(imφ)+c401exp(-imφ),
v^1(m)=CmSm, v^2(m)=Cm-Sm,
v^3(m)=Sm-Cm,v^4(m)=SmCm.
u^1(m)(α)=cos(mφ+α)sin(mφ+α),
u^2(m)(α)=cos(mφ+α)-sin(mφ+α).
v^1(m)=u^1(m)(0),v^2(m)=u^2(m)(0),
v^3(m)=u^1(m)-π2,v^4(m)=u^2(m)-π2.
Ex=Excos γ+Eysin γ,
Ey=-Exsin γ+Eycos γ.
Ex=E cos(mφ+α),
Ey=E sin(mφ+α)
Ex=E cos[mφ+α+(m-1)γ],
Ey=E sin[mφ+α+(m-1)γ].
Ex=E cos[mφ+α+(m+1)γ],
Ey=E sin[mφ+α+(m+1)γ].
dydx=EyEx.
sin φdρ+ρ cos φdφcos φdρ-ρ sin φdφ=sin(mφ)cos(mφ).
dρρ=cos[(m-1)φ]sin[(m-1)φ]dφ.
ρm-1=Rm-1sin[(m-1)φ],
ρm+1=Rm+1cos[(m+1)φ]
sin φdρ+ρ cos φdφcos φdρ-ρ sin φdφ=sin φ cos α+cos φ sin αcos φ cos α-sin φ sin α,
dρρ=cos αsin αdφ.
ρ=R exp(φ cot α).
v0m(ρ)=AJm(βρ)exp-ρ2w02,
vzm(ρ)=A w0w(z)expik-β22kz-iΦ(z)×Jmβρ1+iz/L×exp-1w2(z)+ik2R(z)ρ2+β2z2k2,
w(z)=w01+zL21/2,
R(z)=z+L2z,
Φ(z)=tan-1zL.
fρ(ρ, φ, z)=amQ(z)[Jm-1(u)+Jm+1(u)]cos(mφ),
fφ(ρ, φ, z)=-amQ(z)[Jm-1(u)-Jm+1(u)]sin(mφ),
u=βρ1+iz/L,Q(z)=exp-iβ2z/(2k)1+iz/L.
ExEy=amQ(z)[Jm-1(u)v^2(m-1)+Jm+1(u)v^1(m+1)].

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