Abstract

We study the nonparaxial propagation of Bessel–Gauss beams of any order. Closed-form expressions of all corrections to be added to the solution that is pertinent to the corresponding paraxial problem are found. Such corrections are expressed in terms of two families of polynomials, defined through recurrence rules, that encompass the Laguerre–Gauss polynomials for the particular case of a fundamental Gaussian beam. Numerical examples are shown.

© 2001 Optical Society of America

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    [CrossRef]
  28. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
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    [CrossRef]
  30. R. Borghi, M. Santarsiero, “M2 factor of Bessel–Gauss beams,” Opt. Lett. 22, 262–264 (1997).
    [CrossRef] [PubMed]
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    [CrossRef]
  32. C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
    [CrossRef]
  33. R. Borghi, “Superposition scheme for J0-correlated partially coherent sources,” IEEE J. Quantum Electron. 35, 849–856 (1999).
    [CrossRef]
  34. S. R. Seshadri, “Average characteristics of a partially coherent Bessel–Gauss optical beam,” J. Opt. Soc. Am. A 16, 2917–2927 (1999).
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    [CrossRef] [PubMed]
  37. D. G. Hall, “Vector-beam solution of Maxwell’s wave equation,” Opt. Lett. 21, 9–12 (1996).
    [CrossRef] [PubMed]
  38. C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (1998).
    [CrossRef]
  39. P. L. Greene, D. G. Hall, “Properties and diffraction of vector Bessel–Gauss beams,” J. Opt. Soc. Am. A 15, 3020–3027 (1998).
    [CrossRef]
  40. P. Pääkkönen, J. Turunen, “Resonators with Bessel–Gauss modes,” Opt. Commun. 156, 359–366 (1998).
    [CrossRef]
  41. C. F. R. Caron, R. M. Potvliege, “Phase matching and harmonic generation in Bessel–Gauss beams,” J. Opt. Soc. Am. B 15, 1096–1106 (1998).
    [CrossRef]
  42. J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
    [CrossRef]
  43. C. Altucci, R. Bruzzese, D. D’Antuoni, C. de Lisio, S. Solimeno, “Harmonic generation in gases by use of Bessel–Gauss laser beams,” J. Opt. Soc. Am. B 17, 34–42 (2000).
    [CrossRef]

2000

1999

1998

Q. Cao, X. Deng, “Corrections to the paraxial approximation of an arbitrary free-propagation beam,” J. Opt. Soc. Am. A 15, 1144–1148 (1998).
[CrossRef]

H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147, 1–4 (1998).
[CrossRef]

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (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]

P. Pääkkönen, J. Turunen, “Resonators with Bessel–Gauss modes,” Opt. Commun. 156, 359–366 (1998).
[CrossRef]

C. F. R. Caron, R. M. Potvliege, “Phase matching and harmonic generation in Bessel–Gauss beams,” J. Opt. Soc. Am. B 15, 1096–1106 (1998).
[CrossRef]

1997

1996

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

M. Santarsiero, “Propagation of generalized Bessel–Gauss beams through ABCD optical systems,” Opt. Commun. 132, 1–7 (1996).
[CrossRef]

G. W. Forbes, “Validity of the Fresnel approximation in the diffraction of collimated beams,” J. Opt. Soc. Am. A 13, 1816–1826 (1996).
[CrossRef]

D. G. Hall, “Vector-beam solution of Maxwell’s wave equation,” Opt. Lett. 21, 9–12 (1996).
[CrossRef] [PubMed]

1995

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

S. Chi, Q. Guo, “Vector theory of self-focusing of an optical beam in Kerr media,” Opt. Lett. 20, 1598–1600 (1995).
[CrossRef] [PubMed]

1994

1992

A. T. Friberg, T. Jakkola, J. Tuovinen, “Electromagnetic Gaussian beam beyond the paraxial regime,” IEEE Trans. Antennas Propag. 40, 984–989 (1992).
[CrossRef]

A. Wünsche, “Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams,” J. Opt. Soc. Am. A 9, 765–774 (1992).
[CrossRef]

1990

1987

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

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

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

1985

1983

G. P. Agrawal, M. Lax, “Free-space propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

1981

M. Couture, P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

1979

1978

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

1975

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, M. Lax, “Free-space propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
[CrossRef]

Allen, L.

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
[CrossRef]

Alonso, M. A.

Altucci, C.

Arlt, J.

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
[CrossRef]

Asatryan, A. A.

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Belanger, P. A.

M. Couture, P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

Borghi, R.

R. Borghi, “Superposition scheme for J0-correlated partially coherent sources,” IEEE J. Quantum Electron. 35, 849–856 (1999).
[CrossRef]

R. Borghi, M. Santarsiero, “M2 factor of Bessel–Gauss beams,” Opt. Lett. 22, 262–264 (1997).
[CrossRef] [PubMed]

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Bouchal, Z.

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

Bruzzese, R.

Butler, D. J.

Cao, Q.

Caron, C. F. R.

Chi, S.

Ciattoni, A.

A. Ciattoni, B. Crosignani, P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
[CrossRef]

A. Ciattoni, P. Di Porto, B. Crosignani, A. Yariv, “Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation,” J. Opt. Soc. Am. B 17, 809–819 (2000).
[CrossRef]

Cincotti, G.

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Couture, M.

M. Couture, P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

Crosignani, B.

A. Ciattoni, P. Di Porto, B. Crosignani, A. Yariv, “Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation,” J. Opt. Soc. Am. B 17, 809–819 (2000).
[CrossRef]

A. Ciattoni, B. Crosignani, P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
[CrossRef]

D’Antuoni, D.

de Lisio, C.

Deng, X.

Dholakia, K.

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
[CrossRef]

Di Porto, P.

A. Ciattoni, B. Crosignani, P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
[CrossRef]

A. Ciattoni, P. Di Porto, B. Crosignani, A. Yariv, “Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation,” J. Opt. Soc. Am. B 17, 809–819 (2000).
[CrossRef]

Durnin, J.

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

Eberly, J. H.

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

Forbes, G. W.

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Friberg, A. T.

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

P. Vahimaa, V. Kettunen, M. Kuittinen, J. Turunen, A. T. Friberg, “Electromagnetic analysis of nonparaxial Bessel beams generated by diffractive axicons,” J. Opt. Soc. Am. A 14, 1817–1824 (1997).
[CrossRef]

A. T. Friberg, T. Jakkola, J. Tuovinen, “Electromagnetic Gaussian beam beyond the paraxial regime,” IEEE Trans. Antennas Propag. 40, 984–989 (1992).
[CrossRef]

Fukumitsu, O.

Glasner, M.

Gordon, R. L.

Gori, F.

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

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Greene, P. L.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (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, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

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

Guo, Q.

Hall, D. G.

Jakkola, T.

A. T. Friberg, T. Jakkola, J. Tuovinen, “Electromagnetic Gaussian beam beyond the paraxial regime,” IEEE Trans. Antennas Propag. 40, 984–989 (1992).
[CrossRef]

Jordan, R. H.

Kettunen, V.

Kuittinen, M.

Laabs, H.

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147, 1–4 (1998).
[CrossRef]

Lax, M.

G. P. Agrawal, M. Lax, “Free-space propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Miceli, J. J.

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

Nemoto, S.

Olivi´k, M.

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

Olson, C.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (1998).
[CrossRef]

Pääkkönen, P.

P. Pääkkönen, J. Turunen, “Resonators with Bessel–Gauss modes,” Opt. Commun. 156, 359–366 (1998).
[CrossRef]

Padgett, M. J.

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
[CrossRef]

Padovani, C.

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

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Palma, C.

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Pattanayak, D. N.

Potvliege, R. M.

Rishton, S.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (1998).
[CrossRef]

Saghafi, S.

Santarsiero, M.

R. Borghi, M. Santarsiero, “M2 factor of Bessel–Gauss beams,” Opt. Lett. 22, 262–264 (1997).
[CrossRef] [PubMed]

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

M. Santarsiero, “Propagation of generalized Bessel–Gauss beams through ABCD optical systems,” Opt. Commun. 132, 1–7 (1996).
[CrossRef]

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Schirripa Spagnolo, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Seshadri, S. R.

Shchegrov, A. V.

Sheppard, C. J. R.

Siegman, A. E.

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

Solimeno, S.

Takenaka, T.

Tuovinen, J.

A. T. Friberg, T. Jakkola, J. Tuovinen, “Electromagnetic Gaussian beam beyond the paraxial regime,” IEEE Trans. Antennas Propag. 40, 984–989 (1992).
[CrossRef]

Turunen, J.

Vahimaa, P.

Wicks, G. W.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (1998).
[CrossRef]

Wilson, T.

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

Wolf, E.

Wünsche, A.

Yariv, A.

Yerick, D.

Yokota, M.

Appl. Opt.

Appl. Phys. Lett.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (1998).
[CrossRef]

IEEE J. Quantum Electron.

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

R. Borghi, “Superposition scheme for J0-correlated partially coherent sources,” IEEE J. Quantum Electron. 35, 849–856 (1999).
[CrossRef]

IEEE Trans. Antennas Propag.

A. T. Friberg, T. Jakkola, J. Tuovinen, “Electromagnetic Gaussian beam beyond the paraxial regime,” IEEE Trans. Antennas Propag. 40, 984–989 (1992).
[CrossRef]

J. Mod. Opt.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

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

P. Vahimaa, V. Kettunen, M. Kuittinen, J. Turunen, A. T. Friberg, “Electromagnetic analysis of nonparaxial Bessel beams generated by diffractive axicons,” J. Opt. Soc. Am. A 14, 1817–1824 (1997).
[CrossRef]

A. Wünsche, “Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams,” J. Opt. Soc. Am. A 9, 765–774 (1992).
[CrossRef]

G. W. Forbes, “Validity of the Fresnel approximation in the diffraction of collimated beams,” J. Opt. Soc. Am. A 13, 1816–1826 (1996).
[CrossRef]

G. W. Forbes, D. J. Butler, R. L. Gordon, A. A. Asatryan, “Algebraic corrections for paraxial wave fields,” J. Opt. Soc. Am. A 14, 3300–3315 (1997).
[CrossRef]

M. A. Alonso, A. A. Asatryan, G. W. Forbes, “Beyond the Fresnel approximation for focused waves,” J. Opt. Soc. Am. A 16, 1958–1969 (1999).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Superposition scheme for BG beams.

Fig. 2
Fig. 2

On-axis field modulus plotted as a function of the normalized distance z/D for a BG beam (solid curve), its first-order (m=1) nonparaxial correction (dashed curve), and its second-order (m=2) nonparaxial correction (dotted curve) for (a) β=2×103 mm-1, (b) β=2.5×103 mm-1, (c) β=3×103 mm-1 with parameters of w0=20 μm and λ=2π×10-4 mm.

Fig. 3
Fig. 3

Three-dimensional plots of the transverse modulus distribution of the propagation of (a) a BG beam, (b) its first-order nonparaxial corrections, and (c) its second-order nonparaxial corrections as functions of r/w0 and z/D, with parameters of w0=20 μm, λ=2π×10-4 mm, and β=2.5×103 mm-1.

Fig. 4
Fig. 4

Behaviors of the transverse intensity profiles plotted versus the dimensionless variable βr of a Bessel beam (solid curve) with β=3×103 mm-1, a BG beam (dashed curve), and its second-order nonparaxial version (dotted curve) with the same value of β and with w0=70 μm at propagation distances of (a) z=D/4, (b) z=D/2, (c) z=D, (d) z=2D, where D=w0k/β and k=2π/λ, with λ=2π×10-4 mm.

Equations (58)

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ϕ0(r)=Jl(βr)exp(ilφ).
ϕ0(r)=Jl(βr)exp(-r2/w02)exp(ilφ),
(2+z2+k2)ψnp=0,
ψnp(r, 0)=ϕ0(r),
ψnp(r, z)=exp(ikz)m=0ψ(2m)(r, z).
2ikzψ(0)+2ψ(0)=0,
ψ(2m)(r, z)=i2kmr=1mcr(m)zrzm+rψ(0)(r, z),
cr(m)=(2m)!m(r-1)!(m-r)!(m+r)!,
m=1, 2,,r=1, 2,, m
ψ(2m)(r, 0)=0,m=1, 2, .
2ikzψ(2m)+2ψ(2m)=-z2ψ(2m-2),m1.
ψ(0)(r, z)=A exp(ilφ) expik2z (r2+δ2)z Jlkδz r,
ψ(0)(r, z)=exp(ilφ) exp(-β2L/2k)1+iz/L×exp-k2Lr2-β2L2/k21+iz/LJlβr1+iz/L,
A=-iL exp(-β2L/2k),δ=βL/ik.
ψ(0)(r, 0)=Jl(βr)exp(-r2/w02)exp(ilφ),
β=k sin ϑ.
α=kr22L-β2L2k,γ=βr,ξ=11+iz/L.
z=-iL ξ2ξ,
ψ(2m)=f2mexpilφ-β2ω024×r=1mcr(m)-izLrFm+r(α, γ; ξ),
Fn(α, γ; ξ)=(ξ2ξ)n[ξ exp(-αξ)Jl(γξ)].
Fn(α, γ; ξ)=ξn+1exp(-αξ)[Jl(γξ)Qn(l)(ξ)-γξJl+1(γξ)Rn(l)(ξ)],
Qn+1(l)(ξ)=(n+l+1-αξ)Qn(l)(ξ)+ξ dQn(l)(ξ)dξ-γ2ξ2Rn(l)(ξ),
Rn+1(l)(ξ)=(n-l+1-αξ)Rn(l)(ξ)+ξ dRn(l)(ξ)dξ+Qn(l)(ξ),
Q0(l)(ξ)=1,
R0(l)(ξ)=0.
ψ(2m)(r, z)
=ψ(0)(r, z)f 21+iz/Lmr=1mcr(m)-iz/L1+iz/Lr×Qm+r(l)11+iz/L-βr1+iz/L×Jl+1βr1+iz/LJlβr1+iz/L Rm+r(l)11+iz/L,
zψ(2m)=i2kmr=1mcr(m)z[zrzm+rψ(0)]=i2kmr=1mcr(m)[rzr-1zm+rψ(0)+zrzm+r+1ψ(0)],
2ψ(2m)=i2kmr=1mcr(m)zrrm+r2ψ(0)=i2km-1r=1mcr(m)zrzm+r+1ψ(0),
2ikzψ(2m)+2ψ(2m)=-i2km-1r=1mcr(m)rzr-1zm+rψ(0).
z2ψ(2m-2)=i2km-1r=2m-1cr(m-1)r(r-2)zr-2zm+r-1ψ(0)+r=1m-12cr(m-1)rzr-1zm+rψ(0)+r=1m-1cr(m-1)zrzm+r+1ψ(0).
z2ψ(2m-2)=i2km-1r=1m-2cr+1(m-1)(r+1)rzr-1zm+rψ(0)+r=1m-12cr(m-1)rzr-1zm+rψ(0)+r=2mcr+1(m-1)zr-1zm+rψ(0),
cr(m)=2[c1(m-1)+c2(m-1)]r=1cm-1(m-1)m,r=m.(r+1)cr+1(m-1)+2cr(m-1)+cr-1(m-1)r1<r<m
Fn+1(α, γ; ξ)=(ξ2ξ)n+1[ξ exp(-αξ)Jl(γξ)]=(ξ2ξ){ξn+1exp(-αξ)[JlQn(l)-γξJl+1Rn(l)]},
Jl,l+1=Jl,l+1(γξ),
Qn(l)=Qn(l)(ξ),
Rn(l)=Rn(l)(ξ).
Fn+1(α, γ; ξ)=ξn+2exp(-αξ)(n+1-α)[JlQn(l)-γξJl+1Rn(l)]+ξ dQn(l)(ξ)dξ Jl-γξRn(l)+ξ dRn(l)(ξ)dξJl+1+γξ dJl(t)dt Qn(l)-γ2ξ2dJl+1(t)dt Rn(l).
t dJl(t)dt=lJl(t)-tJl+1(t),
t dJl+1(t)dt=tJl(t)l-(l+1)Jl+1(t).
Fn+1(α, γ; ξ)=ξn+2exp(-αξ)(n+1+l-αξ)Qn(l)+ξ dQn(l)(ξ)dξ-γ2ξ2Rn(l)Jl- γξ(n+1+l-αξ)Rn(l)+ξ dRn(l)(ξ)dξ+Qn(l)Jl+1,
Qn+1(l)(ξ)=(n+l+1+αξ)Qn(l)(ξ)+ξ dQn(l)(ξ)dξ-γ2ξ2Rn(l)(ξ),
Rn+1(l)(ξ)=(n-l+1+αξ)Rn(l)(ξ)+ξ dRn(l)(ξ)dξ+Qn(l)(ξ),
ψ(2m)(r, z)=ψ(0)(r, z)f 21+iz/Lmr=1mcr(m)×-iz/L1+iz/LrQm+r(0)11+iz/L,
Qn+1(0)(ξ)=(n+1-αξ)Qn(0)(ξ)+ξ dQn(0)(ξ)dξ,
Q0(l)(ξ)=1,
Qn(0)(ξ)=n!Ln(αξ),
ψ(2m)(r, z)
=12kLm(2m)!m!11+iz/Lm×r=1m(m-1)!(r-1)!(m-r)!-iz/L1+iz/Lr×Lm+rr2w0211+iz/Lexp-r2w0211+iz/L.
ψnp(r, z)=Jl(βr)exp(ilφ)exp[ikz(1-β2/k2)1/2].
11+iz/L1,
Qn(l)11+iz/L(-α)n(β2L/2k)n.
ψ(0)(r,z)Jl(βr)exp(ilθ)exp-i β2z2k.
ψ(2m)(r, z)12kmr=1mcr(m)(-iz)rβ22km+rψ(0)(r, z),
ψ(0)(r, z)=β24k2mr=1mcr(m)-i β2z2krψ(0)(r, z).
ψnp(r, z)=Jl(βr)exp(ilθ)expikz1-β22k2×m=0β24k2mr=1mcr(m)-iβ2z2kr.
m=04mr=1mcr(m)-iq2r
=exp{iq[(1-)1/2+/2-1]}

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