Abstract

Due to their immunity to diffraction, Bessel light modes potentially offer advantages in various applications. However, they do exhibit significant intensity variations along their axial propagation length which hampers their applicability. In this paper we present a technique to generate Bessel beams with a tunable axial intensity within the accessible range of spatial frequencies. The beam may be engineered to have a constant intensity along its propagation length. Finally, we demonstrate how one can form a Bessel beam with a varying propagation constant along its axial extent which results in a tunable scaling of its lateral cross-section.

© 2009 Optical Society of America

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  1. J. Durnin, J. J. Miceli, and J. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef] [PubMed]
  2. J. Durnin, "Exact solutions for nondifracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-641 (1987).
    [CrossRef]
  3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of Accelerating Airy Beams," Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
  4. J. Baumgartl, M. Mazilu, and K. Dholakia, "Optically mediated particle clearing using Airy wavepackets," Nature Photonics 2, 675-678 (2008).
    [CrossRef]
  5. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004).
    [CrossRef] [PubMed]
  6. T. Čižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, "Optical conveyor belt for delivery of submicron objects," Appl. Phys. Lett. 86, 174,101:1-3 (2005).
  7. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self- reconstructing light beam," Nature 419, 145-147 (2002).
    [CrossRef] [PubMed]
  8. V. Karásek T. Čižmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K . Dholakia, "Long-range onedimensional longitudinal optical binding," Phys. Rev. Lett. 101, 143601 (2008).
    [CrossRef] [PubMed]
  9. X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, "Femtosecond cellular transfection using a nondiffracting light beam," Appl. Phys. Lett. 91, 053902 (2007).
    [CrossRef]
  10. K.-S. Lee and J. P. Rolland, "Bessel beam spectral-domain high-resolution optical coherence tomography with micro-optic axicon providing extended focusing range," Opt. Lett. 33, 1696-1698 (2008).
    [CrossRef] [PubMed]
  11. P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, "Nonlinear Bessel beams," Opt. Commun. 222, 107-115 (2003).
    [CrossRef]
  12. P. Dufour, M. Piché, Y. D. Koninck, and N. McCarthy, "Two-photon excitation fluorescence microscopy with a high depth of field using an axicon," Appl. Opt. 45, 9246-9252 (2006).
    [CrossRef] [PubMed]
  13. Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
    [CrossRef]
  14. O. Brzobohatý, T. Čižmár, and P. Zemánek, "High quality quasi-Bessel beam generated by round-tip axicon," Opt. Express 16, 12688-12700 (2008).
  15. T. Čižmár, V. Kollárová, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, "Generation of multiple Bessel beams for a biophotonics workstation," Opt. Express 16, 14024-14035 (2008).
    [CrossRef]
  16. V. Jarutis, R. Paškauskas, and A. Stabinis, "Focusing of Laguerre-Gaussian beams by axicon," Opt. Commun. 184, 105-112 (2000).
    [CrossRef]
  17. M. Honkanen and J. Turunen, "Tandem systems for efficient generation of uniform-axial-intensity Bessel fields," Opt. Commun. 154, 368 - 375 (1998).
    [CrossRef]
  18. M. Pasienski and B. DeMarco, "A high-accuracy algorithm for designing arbitrary holographic atom traps," Opt. Express 16, 2176-2190 (2008).
    [CrossRef] [PubMed]
  19. R. Gerchberg and W. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).
  20. D. Palima, C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, "Generalized phase contrast matched to Gaussian illumination," Opt. Express 15, 11971-11977 (2007).
    [CrossRef] [PubMed]

2008 (6)

2007 (3)

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, "Femtosecond cellular transfection using a nondiffracting light beam," Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of Accelerating Airy Beams," Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

D. Palima, C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, "Generalized phase contrast matched to Gaussian illumination," Opt. Express 15, 11971-11977 (2007).
[CrossRef] [PubMed]

2006 (1)

2004 (1)

2003 (1)

P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, "Nonlinear Bessel beams," Opt. Commun. 222, 107-115 (2003).
[CrossRef]

2002 (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self- reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

2000 (1)

V. Jarutis, R. Paškauskas, and A. Stabinis, "Focusing of Laguerre-Gaussian beams by axicon," Opt. Commun. 184, 105-112 (2000).
[CrossRef]

1998 (2)

M. Honkanen and J. Turunen, "Tandem systems for efficient generation of uniform-axial-intensity Bessel fields," Opt. Commun. 154, 368 - 375 (1998).
[CrossRef]

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
[CrossRef]

1987 (2)

J. Durnin, J. J. Miceli, and J. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

J. Durnin, "Exact solutions for nondifracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-641 (1987).
[CrossRef]

1972 (1)

R. Gerchberg and W. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Agate, B.

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, "Femtosecond cellular transfection using a nondiffracting light beam," Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

Alonzo, C. A.

Anderson, D.

P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, "Nonlinear Bessel beams," Opt. Commun. 222, 107-115 (2003).
[CrossRef]

Bandres, M. A.

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, "Optically mediated particle clearing using Airy wavepackets," Nature Photonics 2, 675-678 (2008).
[CrossRef]

Bouchal, Z.

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of Accelerating Airy Beams," Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Brown, C. T. A.

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, "Femtosecond cellular transfection using a nondiffracting light beam," Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

Brzobohatý, O.

O. Brzobohatý, T. Čižmár, and P. Zemánek, "High quality quasi-Bessel beam generated by round-tip axicon," Opt. Express 16, 12688-12700 (2008).

V. Karásek T. Čižmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K . Dholakia, "Long-range onedimensional longitudinal optical binding," Phys. Rev. Lett. 101, 143601 (2008).
[CrossRef] [PubMed]

Chávez-Cerda, S.

Chlup, M.

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
[CrossRef]

Christodoulides, D. N.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of Accelerating Airy Beams," Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Cižmár, T.

Comrie, M.

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, "Femtosecond cellular transfection using a nondiffracting light beam," Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

DeMarco, B.

Dholakia, K

V. Karásek T. Čižmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K . Dholakia, "Long-range onedimensional longitudinal optical binding," Phys. Rev. Lett. 101, 143601 (2008).
[CrossRef] [PubMed]

Dholakia, K.

T. Čižmár, V. Kollárová, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, "Generation of multiple Bessel beams for a biophotonics workstation," Opt. Express 16, 14024-14035 (2008).
[CrossRef]

J. Baumgartl, M. Mazilu, and K. Dholakia, "Optically mediated particle clearing using Airy wavepackets," Nature Photonics 2, 675-678 (2008).
[CrossRef]

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, "Femtosecond cellular transfection using a nondiffracting light beam," Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self- reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of Accelerating Airy Beams," Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Dufour, P.

Durnin, J.

J. Durnin, J. J. Miceli, and J. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

J. Durnin, "Exact solutions for nondifracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-641 (1987).
[CrossRef]

Eberly, J.

J. Durnin, J. J. Miceli, and J. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Garcés-Chávez, V.

V. Karásek T. Čižmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K . Dholakia, "Long-range onedimensional longitudinal optical binding," Phys. Rev. Lett. 101, 143601 (2008).
[CrossRef] [PubMed]

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, "Femtosecond cellular transfection using a nondiffracting light beam," Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self- reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

Gerchberg, R.

R. Gerchberg and W. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Glückstad, J.

Gunn-Moore, F.

T. Čižmár, V. Kollárová, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, "Generation of multiple Bessel beams for a biophotonics workstation," Opt. Express 16, 14024-14035 (2008).
[CrossRef]

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, "Femtosecond cellular transfection using a nondiffracting light beam," Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

Gutiérrez-Vega, J. C.

Honkanen, M.

M. Honkanen and J. Turunen, "Tandem systems for efficient generation of uniform-axial-intensity Bessel fields," Opt. Commun. 154, 368 - 375 (1998).
[CrossRef]

Jarutis, V.

V. Jarutis, R. Paškauskas, and A. Stabinis, "Focusing of Laguerre-Gaussian beams by axicon," Opt. Commun. 184, 105-112 (2000).
[CrossRef]

Johannisson, P.

P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, "Nonlinear Bessel beams," Opt. Commun. 222, 107-115 (2003).
[CrossRef]

Karásek, V.

V. Karásek T. Čižmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K . Dholakia, "Long-range onedimensional longitudinal optical binding," Phys. Rev. Lett. 101, 143601 (2008).
[CrossRef] [PubMed]

Kollárová, V.

Koninck, Y. D.

Lee, K.-S.

Lisak, M.

P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, "Nonlinear Bessel beams," Opt. Commun. 222, 107-115 (2003).
[CrossRef]

Marklund, M.

P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, "Nonlinear Bessel beams," Opt. Commun. 222, 107-115 (2003).
[CrossRef]

Mazilu, M.

J. Baumgartl, M. Mazilu, and K. Dholakia, "Optically mediated particle clearing using Airy wavepackets," Nature Photonics 2, 675-678 (2008).
[CrossRef]

McCarthy, N.

McGloin, D.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self- reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

Melville, H.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self- reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Palima, D.

Pasienski, M.

Paškauskas, R.

V. Jarutis, R. Paškauskas, and A. Stabinis, "Focusing of Laguerre-Gaussian beams by axicon," Opt. Commun. 184, 105-112 (2000).
[CrossRef]

Piché, M.

Rodrigo, P. J.

Rolland, J. P.

Saxton, W.

R. Gerchberg and W. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Sibbett, W.

T. Čižmár, V. Kollárová, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, "Generation of multiple Bessel beams for a biophotonics workstation," Opt. Express 16, 14024-14035 (2008).
[CrossRef]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self- reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

Siviloglou, G. A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of Accelerating Airy Beams," Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Stabinis, A.

V. Jarutis, R. Paškauskas, and A. Stabinis, "Focusing of Laguerre-Gaussian beams by axicon," Opt. Commun. 184, 105-112 (2000).
[CrossRef]

Stevenson, D. J.

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, "Femtosecond cellular transfection using a nondiffracting light beam," Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

Tsampoula, X.

T. Čižmár, V. Kollárová, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, "Generation of multiple Bessel beams for a biophotonics workstation," Opt. Express 16, 14024-14035 (2008).
[CrossRef]

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, "Femtosecond cellular transfection using a nondiffracting light beam," Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

Turunen, J.

M. Honkanen and J. Turunen, "Tandem systems for efficient generation of uniform-axial-intensity Bessel fields," Opt. Commun. 154, 368 - 375 (1998).
[CrossRef]

Wagner, J.

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
[CrossRef]

Zemánek, P.

O. Brzobohatý, T. Čižmár, and P. Zemánek, "High quality quasi-Bessel beam generated by round-tip axicon," Opt. Express 16, 12688-12700 (2008).

V. Karásek T. Čižmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K . Dholakia, "Long-range onedimensional longitudinal optical binding," Phys. Rev. Lett. 101, 143601 (2008).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, "Femtosecond cellular transfection using a nondiffracting light beam," Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

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

Nature (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self- reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

Nature Photonics (1)

J. Baumgartl, M. Mazilu, and K. Dholakia, "Optically mediated particle clearing using Airy wavepackets," Nature Photonics 2, 675-678 (2008).
[CrossRef]

Opt. Commun. (4)

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
[CrossRef]

V. Jarutis, R. Paškauskas, and A. Stabinis, "Focusing of Laguerre-Gaussian beams by axicon," Opt. Commun. 184, 105-112 (2000).
[CrossRef]

M. Honkanen and J. Turunen, "Tandem systems for efficient generation of uniform-axial-intensity Bessel fields," Opt. Commun. 154, 368 - 375 (1998).
[CrossRef]

P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, "Nonlinear Bessel beams," Opt. Commun. 222, 107-115 (2003).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Optik (1)

R. Gerchberg and W. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Phys. Rev. Lett. (3)

V. Karásek T. Čižmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K . Dholakia, "Long-range onedimensional longitudinal optical binding," Phys. Rev. Lett. 101, 143601 (2008).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of Accelerating Airy Beams," Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

J. Durnin, J. J. Miceli, and J. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Other (1)

T. Čižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, "Optical conveyor belt for delivery of submicron objects," Appl. Phys. Lett. 86, 174,101:1-3 (2005).

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Figures (12)

Fig. 1.
Fig. 1.

A Bessel beam generated by an annular aperture in a back focal plane of a lens. R is the radius of the annular aperture, d is the annulus thickness and f is the focal length of the lens.

Fig. 2.
Fig. 2.

Axicon-generated Bessel beam. w is the half-width of the Gaussian beam and α is the angle under which the light is refracted by the axicon.

Fig. 3.
Fig. 3.

Bessel beam generated by an oblate-tipped axicon.

Fig. 4.
Fig. 4.

An example of the spatial spectrum obtained by equation (10) for the case of k r0/k=0.01 and k·zmax =1.5·106.

Fig. 5.
Fig. 5.

The retrieved on-axis intensity for a spatial spectrum truncated to the range of kr /k∊〈0,0.02〉.

Fig. 6.
Fig. 6.

The retrieved on-axis intensity for a spatial spectrum from the Fig. 4 enveloped by a Gaussian function centered at k r0 with a half-width of k r0/4.

Fig. 7.
Fig. 7.

Spatial spectrum of Bessel beam with the uniformly growing intensity over the interval z∊〈-zmax ,zmax 〉 for the same parameters as in the previous case. In this case the spectrum is complex and we show its absolute value.

Fig. 8.
Fig. 8.

The retrieved on-axis intensity for a spatial spectrum from the Fig. 7 enveloped by a Gaussian function for the same parameters as in the previous case.

Fig. 9.
Fig. 9.

Left: Hologram generated by the above described procedure described in 5.1. Right: resulting field in the image plane.

Fig. 10.
Fig. 10.

The comparison between the simulation and the experimentally obtained data for the case of the quasi-Bessel beams with the uniform intensity, uniformly increasing and uniformly decreasing intensity.

Fig. 11.
Fig. 11.

The comparison between the simulation and the experimentally obtained data for the case of the quasi-Bessel beams with a growing and a decreasing central core diameter and the beam with the central core reducing in diameter towards both axial directions from the center. The plotted curves showing the central core diameter were obtained by fitting the Bessel beam profile to the data.

Fig. 12.
Fig. 12.

The intensity of the same signal recorded at different places of the CCD chip.

Equations (11)

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

S(kx,ky,z=0)=U(x,y,z=0)ei(kxx+kyy)dxdy.
S(kr,z=0)=0U(r,z=0)J0(krr)rdr ,
U(r,z=0)=0S(kr,z=0)J0(krr)krdkr .
U (r=0,z)=0kkzS(k2kz2,z=0)eikzzdkz.
U (r=0,z)sin(bdz)z eiaz ,
U (r=0,z)ze(ztanαw)2+ikzcosα,
U (r=0,z)=A(z)eikz0z,
S(k2kz2,z=0)=12πkzA(z)eikz0zeikzzdz.
A (z)={1,ifzzmax0,ifz>zmax
S(kr,z=0)=sin[zmax(kzkz0)]πkz(kzkz0)
U(r=0,z)=A (z)eikz0(z)z

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