Abstract

Expressions describing the vortex beams that are generated by the process of Fresnel diffraction of a Gaussian beam incident out of waist on fork-shaped gratings of arbitrary integer charge p, and vortex spots in the case of Fraunhofer diffraction by these gratings, are deduced. The common general transmission function of the gratings is defined and specialized for the cases of amplitude holograms, binary amplitude gratings, and their phase versions. Optical vortex beams, or carriers of phase singularity with charges mp and mp, are the higher negative and positive diffraction-order beams. The radial part of their wave amplitudes is described by the product of the mpth-order Gauss-doughnut function and a Kummer function, or by the first-order Gauss-doughnut function and the difference of two modified Bessel functions whose orders do not match the singularity charge value. The wave amplitude and the intensity distributions are discussed for the near and far fields in the focal plane of a convergent lens, as well as the specialization of the results when the grating charge p=0; i.e., the grating turns from forked into rectilinear. The analytical expressions for the vortex radii are also discussed.

© 2008 Optical Society of America

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  1. S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147-1154 (1992).
    [CrossRef]
  2. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321-327 (1994).
    [CrossRef]
  3. V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Sofier, H. Elfstrom, and J. Turunen, “Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate,” J. Opt. Soc. Am. A 22, 849-861 (2005).
    [CrossRef]
  4. A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748-1754 (1989).
    [CrossRef] [PubMed]
  5. J. A. Davis, E. Carcole, and D. M. Cottrell, “Intensity and phase measurements of nondiffracting beams generated with a magneto-optic spatial light modulator,” Appl. Opt. 35, 593-598 (1996).
    [CrossRef] [PubMed]
  6. V. V. Kotlyar, A. A. Kovalev, R. V. Skidanov, O. Yu. Moiseev, and V. A. Soifer, “Diffraction of a finite-radius plane wave and a Gaussian beam by a helical axicon and a spiral phase plate,” J. Opt. Soc. Am. A 24, 1955-1964 (2007).
    [CrossRef]
  7. V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037-1039 (1990).
  8. N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951-S962 (1992).
    [CrossRef]
  9. G. F. Brand, “Phase singularities in beams,” Am. J. Phys. 67, 55-60 (1999).
    [CrossRef]
  10. H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217-223 (1995).
    [CrossRef]
  11. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221-223 (1992).
    [CrossRef] [PubMed]
  12. 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]
  13. V. V. Kotlyar and A. A. Kovalev, “Family of hypergeometric laser beams,” J. Opt. Soc. Am. A 25, 262-270 (2008).
    [CrossRef]
  14. N. Friedman, A. Kaplan, and N. Davidson, “Dark optical traps for cold atoms,” in Advances in Atomic, Molecular and Optical Physics (Elsevier Science, 2002), pp. 101-106.
  15. J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B: Lasers Opt. 71, 549-554 (2000).
    [CrossRef]
  16. J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297-301 (2000).
    [CrossRef]
  17. Lj. Janicijevic and S. Topuzoski, “Diffraction of nondiverging Bessel beams by fork-shaped and rectilinear grating,” in Proceedings of the Sixth International Conference of the Balkan Physical Union (American Institute of Physics, 2007), Vol. 899, pp. 333-334.
  18. Lj. Janicijevic, J. Mozer, and M. Jonoska, “Diffraction properties of circular and linear zone plates with trapezoid profile of the phase layer,” Bulletin des Sociétés des Physiciens de la Rep. Soc. de Macedoine 28, 23-29 (1978) (in Macedonian).
  19. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
  20. H. Bateman and A. Erdelyi, Higher Transcendental Functions II (Nauka, 1974).
  21. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964).
  22. D. Rozas, C. T. Law, and G. A. Swartzlander, Jr., “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054-3065 (1997).
    [CrossRef]
  23. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73-87 (1993).
    [CrossRef]

2008 (1)

2007 (1)

2005 (1)

2000 (2)

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B: Lasers Opt. 71, 549-554 (2000).
[CrossRef]

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297-301 (2000).
[CrossRef]

1999 (1)

G. F. Brand, “Phase singularities in beams,” Am. J. Phys. 67, 55-60 (1999).
[CrossRef]

1997 (1)

1996 (1)

1995 (1)

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217-223 (1995).
[CrossRef]

1994 (1)

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

1993 (1)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73-87 (1993).
[CrossRef]

1992 (4)

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

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

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221-223 (1992).
[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]

1990 (1)

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037-1039 (1990).

1989 (1)

1978 (1)

Lj. Janicijevic, J. Mozer, and M. Jonoska, “Diffraction properties of circular and linear zone plates with trapezoid profile of the phase layer,” Bulletin des Sociétés des Physiciens de la Rep. Soc. de Macedoine 28, 23-29 (1978) (in Macedonian).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964).

Allen, L.

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]

Almazov, A. A.

Arlt, J.

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B: Lasers Opt. 71, 549-554 (2000).
[CrossRef]

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297-301 (2000).
[CrossRef]

Bateman, H.

H. Bateman and A. Erdelyi, Higher Transcendental Functions II (Nauka, 1974).

Beijersbergen, M. W.

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

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Brand, G. F.

G. F. Brand, “Phase singularities in beams,” Am. J. Phys. 67, 55-60 (1999).
[CrossRef]

Carcole, E.

Coerwinkel, R. P. C.

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

Cottrell, D. M.

Davidson, N.

N. Friedman, A. Kaplan, and N. Davidson, “Dark optical traps for cold atoms,” in Advances in Atomic, Molecular and Optical Physics (Elsevier Science, 2002), pp. 101-106.

Davis, J. A.

Dholakia, K.

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B: Lasers Opt. 71, 549-554 (2000).
[CrossRef]

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297-301 (2000).
[CrossRef]

Elfstrom, H.

Erdelyi, A.

H. Bateman and A. Erdelyi, Higher Transcendental Functions II (Nauka, 1974).

Friberg, A. T.

Friedman, N.

N. Friedman, A. Kaplan, and N. Davidson, “Dark optical traps for cold atoms,” in Advances in Atomic, Molecular and Optical Physics (Elsevier Science, 2002), pp. 101-106.

He, H.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217-223 (1995).
[CrossRef]

Heckenberg, N. R.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217-223 (1995).
[CrossRef]

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

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221-223 (1992).
[CrossRef] [PubMed]

Hitomi, T.

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B: Lasers Opt. 71, 549-554 (2000).
[CrossRef]

Indebetouw, G.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73-87 (1993).
[CrossRef]

Janicijevic, Lj.

Lj. Janicijevic, J. Mozer, and M. Jonoska, “Diffraction properties of circular and linear zone plates with trapezoid profile of the phase layer,” Bulletin des Sociétés des Physiciens de la Rep. Soc. de Macedoine 28, 23-29 (1978) (in Macedonian).

Lj. Janicijevic and S. Topuzoski, “Diffraction of nondiverging Bessel beams by fork-shaped and rectilinear grating,” in Proceedings of the Sixth International Conference of the Balkan Physical Union (American Institute of Physics, 2007), Vol. 899, pp. 333-334.

Jonoska, M.

Lj. Janicijevic, J. Mozer, and M. Jonoska, “Diffraction properties of circular and linear zone plates with trapezoid profile of the phase layer,” Bulletin des Sociétés des Physiciens de la Rep. Soc. de Macedoine 28, 23-29 (1978) (in Macedonian).

Kaplan, A.

N. Friedman, A. Kaplan, and N. Davidson, “Dark optical traps for cold atoms,” in Advances in Atomic, Molecular and Optical Physics (Elsevier Science, 2002), pp. 101-106.

Khonina, S. N.

Kotlyar, V. V.

Kovalev, A. A.

Kristensen, M.

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

Law, C. T.

McDuff, R.

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221-223 (1992).
[CrossRef] [PubMed]

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

Moiseev, O. Yu.

Mozer, J.

Lj. Janicijevic, J. Mozer, and M. Jonoska, “Diffraction properties of circular and linear zone plates with trapezoid profile of the phase layer,” Bulletin des Sociétés des Physiciens de la Rep. Soc. de Macedoine 28, 23-29 (1978) (in Macedonian).

Rozas, D.

Rubinsztein-Dunlop, H.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217-223 (1995).
[CrossRef]

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

Shinkaryev, M. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

Skidanov, R. V.

Smith, C. P.

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

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221-223 (1992).
[CrossRef] [PubMed]

Sofier, V. A.

Soifer, V. A.

Soskin, M. S.

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037-1039 (1990).

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]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964).

Swartzlander, G. A.

Topuzoski, S.

Lj. Janicijevic and S. Topuzoski, “Diffraction of nondiverging Bessel beams by fork-shaped and rectilinear grating,” in Proceedings of the Sixth International Conference of the Balkan Physical Union (American Institute of Physics, 2007), Vol. 899, pp. 333-334.

Turunen, J.

Uspleniev, G. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

Vasara, A.

Vasnetsov, M. V.

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037-1039 (1990).

Wegener, M. J.

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

White, A. G.

Woerdman, J. P.

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

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Yu. Bazhenov, V.

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037-1039 (1990).

Am. J. Phys. (1)

G. F. Brand, “Phase singularities in beams,” Am. J. Phys. 67, 55-60 (1999).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B: Lasers Opt. (1)

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B: Lasers Opt. 71, 549-554 (2000).
[CrossRef]

Bulletin des Sociétés des Physiciens de la Rep. Soc. de Macedoine (1)

Lj. Janicijevic, J. Mozer, and M. Jonoska, “Diffraction properties of circular and linear zone plates with trapezoid profile of the phase layer,” Bulletin des Sociétés des Physiciens de la Rep. Soc. de Macedoine 28, 23-29 (1978) (in Macedonian).

J. Mod. Opt. (3)

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217-223 (1995).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73-87 (1993).
[CrossRef]

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

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

Opt. Commun. (2)

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297-301 (2000).
[CrossRef]

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

Opt. Lett. (1)

Opt. Quantum Electron. (1)

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

Phys. Rev. A (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]

Pis'ma Zh. Eksp. Teor. Fiz. (1)

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037-1039 (1990).

Other (5)

N. Friedman, A. Kaplan, and N. Davidson, “Dark optical traps for cold atoms,” in Advances in Atomic, Molecular and Optical Physics (Elsevier Science, 2002), pp. 101-106.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

H. Bateman and A. Erdelyi, Higher Transcendental Functions II (Nauka, 1974).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964).

Lj. Janicijevic and S. Topuzoski, “Diffraction of nondiverging Bessel beams by fork-shaped and rectilinear grating,” in Proceedings of the Sixth International Conference of the Balkan Physical Union (American Institute of Physics, 2007), Vol. 899, pp. 333-334.

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Figures (7)

Fig. 1
Fig. 1

(a) Forked grating with p = 2 . (b) Rectilinear grating when p = 0 .

Fig. 2
Fig. 2

Geometry of the problem.

Fig. 3
Fig. 3

Geometrical illustration of transformations (6).

Fig. 4
Fig. 4

Intensity distribution in the first diffraction order for diffraction of a Gaussian beam by a forked grating with singularity p = 1 (dotted–dashed curve), p = 2 (solid curve), p = 3 (dashed curve) at distance z ζ = 10 mm .

Fig. 5
Fig. 5

Intensity distribution in the first diffraction order for diffraction of a Gaussian beam by a forked grating with singularity p = 1 at distances: z ζ = 10 mm (dotted–dashed curve), z ζ = 50 mm (solid curve), z ζ = 150 mm (dashed curve).

Fig. 6
Fig. 6

Intensity distribution in the first diffraction order for diffraction of a Gaussian beam by a forked grating with singularity p = 1 (dotted–dashed curve), p = 2 (solid curve), p = 3 (dashed curve) at distance z = 1 m .

Fig. 7
Fig. 7

Intensity distribution in the first diffraction order for diffraction of a Gaussian beam by a forked grating with singularity p = 1 at distances z = 1 m (dotted–dashed curve), z = 3 m (solid curve), z = 5 m (dashed curve).

Equations (86)

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

U ( r ) ln [ 1 + I ( r ) F ] ,
T ( r , φ ) = m = t m exp [ i m ( 2 π D r cos φ p φ ) ] = t 0 + m = 1 t m exp [ i m ( 2 π D r cos φ p φ ) ] + m = 1 t m exp [ i m ( 2 π D r cos φ p φ ) ] ,
t 0 = χ exp ( i k α ) cos ( k β ) ; t ± m = ± ( 2 m 1 ) = ± χ exp ( i k α ) ( 2 π ) [ 1 ( 2 m 1 ) ] sin ( k β ) ; t ± 2 m = 0 ; ( m = 1 , 2 , 3 , ) ,
U ( i ) ( r , φ , ζ ) = q ( 0 ) q ( ζ ) exp [ i k ( ζ + r 2 2 q ( ζ ) ) ] ,
1 q ( ζ ) = 1 R ( ζ ) 2 i k w 2 ( ζ ) ,
U ( ρ , θ , z ) = i k 2 π ( z ζ ) exp { i k [ ( z ζ ) + ρ 2 2 ( z ζ ) ] } Δ T ( r , φ ) U ( i ) ( r , φ , ζ ) exp [ i k 2 ( r 2 z ζ 2 r ρ cos ( φ θ ) z ζ ) ] r d r d φ ,
U ( ρ , θ , z ) = i k 2 π ( z ζ ) q ( 0 ) q ( ζ ) exp [ i k ( z + ρ 2 2 ( z ζ ) ) ] { t 0 0 0 2 π exp [ i k 2 q ( z ) ( z ζ ) q ( ζ ) r 2 ] exp [ i k r ρ ( z ζ ) cos ( φ θ ) ] r d r d φ + m = 1 t m 0 0 2 π exp [ i k 2 q ( z ) ( z ζ ) q ( ζ ) r 2 ] exp [ i k r ρ ( z ζ ) sin θ sin φ ] exp [ i k r cos φ z ζ ( ρ cos θ m λ ( z ζ ) D ) ] exp ( i m p φ ) r d r d φ + m = 1 t m 0 0 2 π exp [ i k 2 q ( z ) ( z ζ ) q ( ζ ) r 2 ] exp [ i k r ρ ( z ζ ) sin θ sin φ ] exp [ i k r cos φ z ζ ( ρ cos θ + m λ ( z ζ ) D ) ] exp ( i m p φ ) r d r d φ } .
ρ cos θ m λ ( z ζ ) D = ρ ± m cos θ ± m ,
ρ sin θ = ρ ± m sin θ ± m ,
ρ ± m = ρ 2 + [ m λ ( z ζ ) D ] 2 2 m λ ρ ( z ζ ) D cos θ ;
tan θ ± m = ρ sin θ ρ cos θ m λ ( z ζ ) D .
Φ 0 = 0 2 π exp { i k ρ ( z ζ ) r cos ( φ θ ) } d φ ;
Φ ± m p = 0 2 π exp { i [ k ρ ± m ( z ζ ) r cos ( φ θ ± m ) ± m p φ ] } d φ ,
Φ 0 = 2 π J 0 ( k ρ z ζ r ) ;
Φ ± m p = 2 π J m p ( k ρ ± m z ζ r ) exp [ i m p ( π 2 ± θ ± m ) ] .
U ( ρ , θ , z ) = i k ( z ζ ) q ( 0 ) q ( ζ ) exp [ i k ( z + ρ 2 2 ( z ζ ) ) ] { t 0 0 J 0 ( k ρ r z ζ ) exp [ i k 2 q ( z ) ( z ζ ) q ( ζ ) r 2 ] r d r + m = 0 t m exp [ i m p ( π 2 + θ m ) ] 0 exp [ i k 2 q ( z ) ( z ζ ) q ( ζ ) r 2 ] J m p ( k ρ m r z ζ ) r d r + m = 0 t m exp [ i m p ( π 2 θ m ) ] 0 exp [ i k 2 q ( z ) ( z ζ ) q ( ζ ) r 2 ] J m p ( k ρ m r z ζ ) r d r } .
U ( ρ , θ , z ) = U 0 ( ρ , θ , z ) + m = 1 U m ( ρ m , θ m , z ) + m = 1 U m ( ρ m , θ m , z ) ,
U 0 ( ρ , θ , z ) = i k ( z ζ ) q ( 0 ) q ( ζ ) t 0 exp [ i k ( z + ρ 2 2 ( z ζ ) ) ] Y 0 ( ρ ) ,
U ± m ( ρ ± m , θ ± m , z ) = i k ( z ζ ) q ( 0 ) q ( ζ ) t ± m exp [ i k ( z + ρ 2 2 ( z ζ ) ) ] exp [ i m p ( π 2 ± θ ± m ) ] Y ± m ( ρ ± m ) .
Y 0 ( ρ ) = 0 J 0 ( k ρ r z ζ ) exp [ i k 2 q ( z ) ( z ζ ) q ( ζ ) r 2 ] r d r ,
Y ± m ( ρ ± m ) = 0 J m p ( k ρ ± m r z ζ ) exp [ i k 2 q ( z ) ( z ζ ) q ( ζ ) r 2 ] r d r ,
0 J l ( b 0 r ) exp ( a 2 r 2 ) r l + 1 d r = b 0 l ( 2 a 2 ) l + 1 exp ( b 0 2 4 a 2 ) ( Re ν > 1 , Re a 2 > 0 ) ,
0 J ν ( b ± m r ) exp ( a 2 r 2 ) r μ 1 d r = ( b ± m 2 4 a 2 ) ν 2 Γ ( ( μ + ν ) 2 ) 2 a μ Γ ( ν + 1 ) M ( μ + ν 2 , ν + 1 ; b ± m 2 4 a 2 ) ( Re ( μ + ν ) > 0 , Re a 2 > 0 ) .
U 0 ( ρ , θ , z ) = t 0 q ( 0 ) q ( z ) exp { i k [ z + ρ 2 2 q ( z ) ] } ,
U ± m ( ρ ± m , θ ± m , z ) = t ± m q ( 0 ) q ( z ) Γ ( m p 2 + 1 ) Γ ( m p + 1 ) [ i k q ( ζ ) 2 ( z ζ ) q ( z ) ] m p 2 exp { i k [ z + ρ 2 2 ( z ζ ) + ( 1 R ( z ) 1 z ζ ) ρ ± m 2 2 ] } ( ± 1 ) m p exp [ i m p ( π 2 ± θ ± m ) ] ρ ± m m p exp ( ρ ± m 2 w 2 ( z ) ) M ( m p 2 , m p + 1 ; i k q ( ζ ) 2 ( z ζ ) q ( z ) ρ ± m 2 ) ,
M ( m p 2 + 1 , m p + 1 ; x ) = exp ( x ) M ( m p 2 , m p + 1 ; x )
U ± m ( ρ ± m , θ ± m , z ) = t ± m q ( 0 ) q ( z ) [ i k π 4 Q ( z ) ] 1 2 ( ± 1 ) m p exp { i k [ z + ρ 2 2 ( z ζ ) + ρ ± m 2 2 Q ( z ) ] } exp [ i m p ( π 2 ± θ ± m ) ] ρ ± m [ I ( m p 1 ) 2 ( i k 2 Q ( z ) ρ ± m 2 ) I ( m p + 1 ) 2 ( i k 2 Q ( z ) ρ ± m 2 ) ] .
y ± m = i k 4 q ( ζ ) ( z ζ ) q ( z ) ρ ± m 2 = i k 2 { 1 2 [ 1 R ( z ) 1 ( z ζ ) ] i k w 2 ( z ) } ρ ± m 2 = i k 2 Q ( z ) ρ ± m 2 ,
1 Q ( z ) = 1 R ( z ) 2 i k w 2 ( z ) ,
1 R ( z ) = 1 2 [ 1 R ( z ) 1 z ζ ] ; w ( z ) = w ( z ) 2 .
U ± m ( ρ ± m , θ ± m , z ) = t ± m q ( 0 ) q ( z ) [ i k π 4 Q ( z ) ] 1 2 ( ± 1 ) m p i ( m p 1 ) 2 exp { i k [ z + ρ 2 2 ( z ζ ) + ρ ± m 2 2 Q ( z ) ] } exp [ i m p ( π 2 ± θ ± m ) ] ρ ± m [ J ( m p 1 ) 2 ( k 2 Q ( z ) ρ ± m 2 ) i J ( m p + 1 ) 2 ( k 2 Q ( z ) ρ ± m 2 ) ] .
ρ ± m exp [ i k 2 Q ( z ) ρ ± m 2 ] = ρ ± m exp [ ρ ± m 2 w 2 ( z ) ] exp [ i k 2 R ( z ) ρ ± m 2 ]
( ρ = m λ ( z ζ ) D , θ = 0 ) and ( ρ = m λ ( z ζ ) D , θ = π ) ,
k ρ ± m 2 2 Q ( z ) k ρ ± m 2 4 ( z ζ ) .
U ± m ( ρ ± m , θ ± m , z ) = t ± m 2 q ( 0 ) q ( ζ ) [ i k π 2 ( z ζ ) ] 1 2 ( ± 1 ) m p ( i ) ( m p 1 ) 2 exp { i k [ z + ρ 2 2 ( z ζ ) ρ ± m 2 4 ( z ζ ) ] } exp [ i m p ( π 2 ± θ ± m ) ] ρ ± m [ J ( m p 1 ) 2 ( k ρ ± m 2 4 ( z ζ ) ) + i J ( m p + 1 ) 2 ( k ρ ± m 2 4 ( z ζ ) ) ] ,
I ̂ ± m ( ρ ± m , θ ± m , z ) = t ± m 2 w 0 2 w 2 ( ζ ) k π 8 ( z ζ ) ρ ± m 2 [ J ( m p 1 ) 2 2 ( k ρ ± m 2 4 ( z ζ ) ) + J ( m p + 1 ) 2 2 ( k ρ ± m 2 4 ( z ζ ) ) ] .
J ( m p 1 ) 2 ( k ρ ± m 2 4 ( z ζ ) ) = J ( m p + 1 ) 2 ( k ρ ± m 2 4 ( z ζ ) ) .
( ρ ± m ) p [ ( m p 2 + 1 ) 2 λ ( z ζ ) π ] 1 2 .
J ν ( x ) = ( x 2 ) ν 1 Γ ( ν + 1 ) when x 0 ,
I ̂ ± m ( ρ ± m , θ ± m , z ) Γ 2 ( m p + 1 2 ) [ k ρ ± m 2 8 ( z ζ ) ] ( m p 1 ) [ 1 + k 2 ρ ± m 4 16 ( m p + 1 ) 2 ( z ζ ) 2 ] .
1 Q ( z ) = 1 2 { [ 1 R ( z ) 1 z ζ ] 2 i k w 2 ( z ) } i k w 2 ( z ) .
U ± m ( ρ ± m , θ ± m , z ) = t ± m 2 q ( 0 ) q ( z ) π w ( z ) ( ± 1 ) m p exp ( i k z ) exp [ ρ ± m 2 2 w 2 ( z ) ] exp [ i m p ( π 2 ± θ ± m ) ] × ρ ± m [ I ( m p 1 ) 2 ( ρ ± m 2 2 w 2 ( z ) ) I ( m p + 1 ) 2 ( ρ ± m 2 2 w 2 ( z ) ) ] ,
I ̂ ± m ( ρ ± m , θ ± m , z ) = w 0 2 w 2 ( z ) t ± m 2 π 4 ρ ± m 2 w 2 ( z ) exp [ ρ ± m 2 w 2 ( z ) ] [ I ( m p 1 ) 2 ( ρ ± m 2 2 w 2 ( z ) ) I ( m p + 1 ) 2 ( ρ ± m 2 2 w 2 ( z ) ) ] 2 .
I ̂ ± m ( ρ ± m , θ ± m , z ) Γ 2 ( m p + 1 2 ) [ 1 4 w 2 ( z ) ] m p ρ ± m 2 m p exp [ ρ ± m 2 w 2 ( z ) ] [ 1 ρ ± m 2 2 ( m p + 1 ) w 2 ( z ) ] 2 .
( ρ ± m ) p = w ( z ) m p ( m p + 1 ) m p + 2 .
I ν ( x ) 1 2 π x exp ( x ) ( 1 4 ν 2 1 8 x ) ,
I ̂ ± m ( ρ ± m , z ) t ± m 2 w 0 2 w 2 ( z ) 8 ( m p ) 2 ρ ± m 4 .
I ± 1 2 ( x ) = 2 π x { shx chx } = 2 π x { [ exp ( x ) exp ( x ) ] 2 [ exp ( x ) + exp ( x ) ] 2 } ,
U ± m ( ρ ± m , θ ± m , z ) = t ± m q ( 0 ) q ( z ) exp { i k [ z + ρ 2 ρ ± m 2 2 ( z ζ ) ] } exp [ i k 2 q ( z ) ρ ± m 2 ] ,
I ̂ ± m ( ρ ± m , θ ± m , z ) = t ± m 2 w 0 2 w 2 ( z ) exp [ 2 ρ ± m 2 w 2 ( z ) ] .
U 0 ( ρ , z ) = t 0 q ( 0 ) q ( z ) exp { i k [ z + ρ 2 2 q ( z ) ] } ,
I ̂ 0 ( ρ , z ) = t 0 2 w 0 2 w 2 ( z ) exp ( 2 ρ 2 w 2 ( z ) ) .
U ( i ) ( r , φ ) = A exp ( r 2 w o 2 ) ,
U ( ρ , θ , f ) = C σ T ( r , φ ) U ( i ) ( r , φ ) exp ( i k ρ f r cos ( φ θ ) ) r d r d φ ,
ρ cos θ m λ f D = ρ ± m cos θ ± m ; ρ sin θ = ρ ± m sin θ ± m ;
ρ ± m = ρ 2 + [ m λ f D ] 2 2 m λ ρ f D cos θ ;
tan θ ± m = ρ sin θ ρ cos θ m λ f D .
U 0 ( ρ , f ) = A i w 0 w f t 0 exp ( ρ 2 w f 2 )
U ± m ( ρ ± m , θ ± m , f ) = A ( ± 1 ) m p t ± m π 2 w 0 w f exp [ i m p ( π 2 ± θ ± m ) ] ρ ± m 2 w f exp [ ρ ± m 2 2 w f 2 ] [ I ( m p 1 ) 2 ( ρ ± m 2 2 w f 2 ) I ( m p + 1 ) 2 ( ρ ± m 2 2 w f 2 ) ]
w f = λ f w 0 π .
I ̂ 0 ( ρ , f ) = A 2 t 0 2 w 0 2 w f 2 exp ( 2 ρ 2 w f 2 ) .
I ̂ ± m ( ρ ± m , θ ± m , f ) = A 2 π 2 t ± m 2 ( w 0 w f ) 2 ( ρ ± m 2 w f ) 2 exp [ ρ ± m 2 w f 2 ] [ I ( m p 1 ) 2 ( ρ ± m 2 2 w f 2 ) I ( m p + 1 ) 2 ( ρ ± m 2 2 w f 2 ) ] 2 ,
ρ ± m = w f 2 m p ( m p + 1 ) m p + 2 = λ f w 0 π 2 m p ( m p + 1 ) m p + 2 .
2 π D ρ ± m cos θ ± m m p θ ± m = n π ; n = 0 , 1 , 2 , .
2 y ± m = i k 2 q ( ζ ) ( z ζ ) q ( z ) ρ ± m 2 .
( ν 1 ) M ( μ , ν 1 ; 2 y ± m ) = ( ν 1 ) M ( μ , ν ; 2 y ± m ) + 2 y ± m μ ν M ( μ + 1 , ν + 1 ; 2 y ± m ) ,
M ( m p 2 , m p + 1 ; 2 y ± m ) = M ( m p 2 , m p ; 2 y ± m ) y ± m m p + 1 M ( m p 2 + 1 , m p + 2 ; 2 y ± m ) .
I ( m p 1 ) 2 ( y ± m ) = Γ 1 ( m p + 1 2 ) ( y ± m 2 ) ( m p 1 ) 2 exp ( y ± m ) M ( m p 2 , m p ; 2 y ± m )
I ( m p + 1 ) 2 ( y ± m ) = Γ 1 ( m p + 1 2 ) ( y ± m 2 ) ( m p 1 ) 2 exp ( y ± m ) y ± m ( m p + 1 ) M ( m p 2 + 1 , m p + 2 ; 2 y ± m ) ,
M ( m p 2 , m p + 1 ; 2 y ± m ) = Γ ( m p + 1 2 ) ( y ± m 2 ) ( 1 m p ) 2 exp ( y ± m ) [ I ( m p 1 ) 2 ( y ± m ) I ( m p + 1 ) 2 ( y ± m ) ] .
I ̂ 0 ( ρ , z ) = U 0 ( ρ , z ) 2 = t 0 2 w 0 2 w 2 ( z ) exp [ 2 ρ 2 w 2 ( z ) ] ,
I ̂ ± m ( ρ ± m , θ ± m , z ) = U ± m ( ρ ± m , θ ± m , z ) 2 = t ± m 2 w 0 2 w 2 ( z ) K ( z ) ρ ± m 2 exp [ ρ ± m 2 w 2 ( z ) ] [ A m p 2 ( ρ ± m , z ) + B m p 2 ( ρ ± m , z ) ] .
K ( z ) = π 2 [ ( k 2 R ( z ) ) 2 + ( 1 w 2 ( z ) ) 2 ] ,
J ν ( u g ) = s = J ν + s ( u ) J s ( g ) when g u < 1 ,
J ν ( u g ) = ( 1 ) ν J ν ( g u ) = ( 1 ) ν s = J ν + s ( g ) J s ( u ) ,
when u g < 1 ,
k 2 R ( z ) < 1 w 2 ( z )
z ζ < k w 2 ( z ) k w 2 ( z ) R ( z ) 2 ,
A m p ( ρ ± m , z ) = s = ( 1 ) s J 2 s ( k ρ ± m 2 2 R ( z ) ) [ I ( m p 1 ) 2 + 2 s ( ρ ± m 2 w 2 ( z ) ) I ( m p + 1 ) 2 + 2 s ( ρ ± m 2 w 2 ( z ) ) ] ,
B m p ( ρ ± m , z ) = s = ( 1 ) s J 2 s + 1 ( k ρ ± m 2 2 R ( z ) ) [ I ( m p 1 ) 2 + 2 s + 1 ( ρ ± m 2 w 2 ( z ) ) I ( m p + 1 ) 2 + 2 s + 1 ( ρ ± m 2 w 2 ( z ) ) ] ,
k 2 R ( z ) > 1 w 2 ( z ) ,
z ζ > k w 2 ( z ) k w 2 ( z ) R ( z ) 2 ,
A m p ( ρ ± m , z ) = s = ( 1 ) s [ I 2 s ( ρ ± m 2 w 2 ( z ) ) J ( m p 1 ) 2 + 2 s ( k ρ ± m 2 2 R ( z ) ) + I 2 s + 1 ( ρ ± m 2 w 2 ( z ) ) J ( m p + 1 ) 2 + ( 2 s + 1 ) ( k ρ ± m 2 2 R ( z ) ) ] ,
B m p ( ρ ± m , z ) = s = ( 1 ) s [ I 2 s + 1 ( ρ ± m 2 w 2 ( z ) ) J ( m p 1 ) 2 + ( 2 s + 1 ) ( k ρ ± m 2 2 R ( z ) ) I 2 s ( ρ ± m 2 w 2 ( z ) ) J ( m p + 1 ) 2 + 2 s ( k ρ ± m 2 2 R ( z ) ) ] .
d I ̂ ± m ( ρ ± m , θ ± m , z ) d ρ ± m = 2 K ( z ) ρ ± m exp [ ρ ± m 2 w 2 ( z ) ] { ( 1 ρ ± m 2 w 2 ( z ) ) [ A m p 2 ( ρ ± m , z ) + B m p 2 ( ρ ± m , z ) ] + ρ ± m [ A m p ( ρ ± m , z ) d A m p ( ρ ± m , z ) d ρ ± m + B m p ( ρ ± m , z ) d B m p ( ρ ± m , z ) d ρ ± m ] } = 0 .
( 1 ρ ± m 2 w 2 ) [ A m p 2 ( ρ ± m , z ) + B m p 2 ( ρ ± m , z ) ] + ρ ± m d d ρ ± m [ A m p 2 ( ρ ± m , z ) + B m p 2 ( ρ ± m , z ) ] = 0 .

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