Abstract

Approximate analytic expressions are obtained for evaluating the axial intensity and the central-lobe diameter of J0 Bessel beams transmitted through a finite-aperture phase filter. A reasonable quality factor governing the axial-intensity behavior of a phase-undistorted truncated Bessel beam is found to be the inverse square root of the Fresnel number defined, for a given aperture, from the axial point of geometrical shadow. Additional drastic reduction of axial-intensity oscillations is accomplished by using marginal phase correction of the beam instead of the well-known amplitude apodization. A procedure for analytically calculating an optimal monotonic slowly varying correction phase function is described.

© 2000 Optical Society of America

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References

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  1. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  2. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  3. J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
    [CrossRef] [PubMed]
  4. Z. Jiang, Q. Lu, Z. Liu, “Propagation of apertured Bessel beams,” Appl. Opt. 34, 7183–7185 (1995).
    [CrossRef] [PubMed]
  5. R. Borghi, M. Santarsiero, F. Gori, “Axial intensity of apertured Bessel beams,” J. Opt. Soc. Am. A 14, 23–26 (1997).
    [CrossRef]
  6. J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
    [CrossRef] [PubMed]
  7. A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
    [CrossRef] [PubMed]
  8. A. J. Cox, D. C. Dibble, “Holographic reproduction of a diffraction-free beam,” Appl. Opt. 30, 1330–1332 (1991).
    [CrossRef] [PubMed]
  9. R. M. Herman, T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8, 932–942 (1991).
    [CrossRef]
  10. K. Thewes, M. A. Karim, A. A. S. Awwal, “Diffraction-free beam generation using a refracting system,” Opt. Laser Technol. 23, 105–108 (1991).
    [CrossRef]
  11. G. Scott, M. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31, 2640–2643 (1992).
    [CrossRef]
  12. N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
    [CrossRef]
  13. A. J. Cox, J. D’Anna, “Constant-axial-intensity nondiffracting beam,” Opt. Lett. 17, 232–234 (1992).
    [CrossRef] [PubMed]
  14. R. M. Herman, T. A. Wiggins, “Apodization of diffractionless beams,” Appl. Opt. 31, 5913–5915 (1992).
    [CrossRef] [PubMed]
  15. Z. Jaroszewicz, J. Sochacki, A. Kołodziejzcyk, L. R. Staroński, “Apodized annular-aperture logarithmic axicon: smoothness and uniformity of intensity distributions,” Opt. Lett. 18, 1893–1895 (1993).
    [CrossRef] [PubMed]
  16. J. Rosen, “Synthesis of nondiffracting beams in free space,” Opt. Lett. 19, 369–371 (1994).
    [PubMed]
  17. R. M. Herman, T. A. Wiggins, “High-efficiency diffractionless beams of constant size and intensity,” Appl. Opt. 33, 7297–7306 (1994).
    [CrossRef] [PubMed]
  18. A. T. Friberg, “Stationary-phase analysis of generalized axicons,” J. Opt. Soc. Am. A 13, 743–750 (1996).
    [CrossRef]
  19. K. Kono, M. Ire, T. Minemoto, “Generation of nearly diffraction-free beams using a new optical system,” Opt. Rev. 4, 423–428 (1997).
    [CrossRef]
  20. A. G. Sedukhin, “Two-component axicon focusing of light radiation,” Optoelectron. Instrum. Data Process. No. 5 (Sept.–Oct.), 28–40 (1997) (translation of Russian journal “Autometriya”).
  21. J. N. Provost, J. L. de Bougrenet de la Tocnaye, “Optimal trade-off pupil for synthesis of nondiffracting beams in free space,” J. Opt. Soc. Am. A 14, 2748–2757 (1997).
    [CrossRef]
  22. Z. L. Horvath, M. Erdelyi, G. Szabo, Zs. Bor, F. K. Tittel, J. R. Cavallaro, “Generation of nearly nondiffracting Bessel beams with a Fabry–Perot interferometer,” J. Opt. Soc. Am. A 14, 3009–3013 (1997).
    [CrossRef]
  23. S. Yu. Popov, A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
    [CrossRef]
  24. S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
    [CrossRef]
  25. M. Honkanen, J. Turunen, “Tandem systems for efficient generation of uniform-axial-intensity Bessel fields,” Opt. Commun. 154, 368–375 (1998).
    [CrossRef]
  26. W.-X. Cong, N.-X. Chen, B.-Yu. Gu, “Generation of nondiffracting beams by diffractive phase elements,” J. Opt. Soc. Am. A 15, 2362–2364 (1998).
    [CrossRef]
  27. Z. Jaroszewicz, J. Morales, “Lens axicons: systems composed of a diverging aberrated lens and a perfect converging lens,” J. Opt. Soc. Am. A 15, 2383–2390 (1998).
    [CrossRef]
  28. Z. Jaroszewicz, J. Morales, “Lens axicons: systems composed of a diverging aberrated lens and a converging aberrated lens,” J. Opt. Soc. Am. A 16, 191–197 (1999).
    [CrossRef]
  29. A. G. Sedukhin, “Beam-preshaping axicon focusing,” J. Opt. Soc. Am. A 15, 3057–3066 (1998).
    [CrossRef]
  30. A. Papoulis, Systems and Transforms with Applications in Optics, 1st ed. (McGraw-Hill, New York, 1968), Chap. 7.
  31. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Part III.
  32. M. Abramowitz, I. A. Stegun, eds. Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 9.
  33. M. V. Perez, C. Comes-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
    [CrossRef]

1999 (1)

1998 (6)

W.-X. Cong, N.-X. Chen, B.-Yu. Gu, “Generation of nondiffracting beams by diffractive phase elements,” J. Opt. Soc. Am. A 15, 2362–2364 (1998).
[CrossRef]

Z. Jaroszewicz, J. Morales, “Lens axicons: systems composed of a diverging aberrated lens and a perfect converging lens,” J. Opt. Soc. Am. A 15, 2383–2390 (1998).
[CrossRef]

S. Yu. Popov, A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[CrossRef]

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

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

A. G. Sedukhin, “Beam-preshaping axicon focusing,” J. Opt. Soc. Am. A 15, 3057–3066 (1998).
[CrossRef]

1997 (5)

1996 (1)

1995 (1)

1994 (2)

1993 (1)

1992 (4)

A. J. Cox, J. D’Anna, “Constant-axial-intensity nondiffracting beam,” Opt. Lett. 17, 232–234 (1992).
[CrossRef] [PubMed]

R. M. Herman, T. A. Wiggins, “Apodization of diffractionless beams,” Appl. Opt. 31, 5913–5915 (1992).
[CrossRef] [PubMed]

G. Scott, M. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31, 2640–2643 (1992).
[CrossRef]

N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[CrossRef]

1991 (3)

1989 (1)

1988 (2)

1987 (2)

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

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

1986 (1)

M. V. Perez, C. Comes-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

Awwal, A. A. S.

K. Thewes, M. A. Karim, A. A. S. Awwal, “Diffraction-free beam generation using a refracting system,” Opt. Laser Technol. 23, 105–108 (1991).
[CrossRef]

Bor, Zs.

Borghi, R.

Cavallaro, J. R.

Chen, N.-X.

Comes-Reino, C.

M. V. Perez, C. Comes-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

Cong, W.-X.

Cox, A. J.

Cuadrado, J. M.

M. V. Perez, C. Comes-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

D’Anna, J.

Davidson, N.

N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[CrossRef]

de Bougrenet de la Tocnaye, J. L.

Dibble, D. C.

Durnin, J.

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
[CrossRef] [PubMed]

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

Erdelyi, M.

Friberg, A. T.

S. Yu. Popov, A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[CrossRef]

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

A. T. Friberg, “Stationary-phase analysis of generalized axicons,” J. Opt. Soc. Am. A 13, 743–750 (1996).
[CrossRef]

A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
[CrossRef] [PubMed]

J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[CrossRef] [PubMed]

Friesem, A. A.

N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[CrossRef]

Gori, F.

Gu, B.-Yu.

Hasman, E.

N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[CrossRef]

Herman, R. M.

Honkanen, M.

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

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

Horvath, Z. L.

Ire, M.

K. Kono, M. Ire, T. Minemoto, “Generation of nearly diffraction-free beams using a new optical system,” Opt. Rev. 4, 423–428 (1997).
[CrossRef]

Jaroszewicz, Z.

Jiang, Z.

Karim, M. A.

K. Thewes, M. A. Karim, A. A. S. Awwal, “Diffraction-free beam generation using a refracting system,” Opt. Laser Technol. 23, 105–108 (1991).
[CrossRef]

Kolodziejzcyk, A.

Kono, K.

K. Kono, M. Ire, T. Minemoto, “Generation of nearly diffraction-free beams using a new optical system,” Opt. Rev. 4, 423–428 (1997).
[CrossRef]

Lautanen, J.

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

Liu, Z.

Lu, Q.

McArdle, M.

G. Scott, M. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31, 2640–2643 (1992).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
[CrossRef] [PubMed]

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

Minemoto, T.

K. Kono, M. Ire, T. Minemoto, “Generation of nearly diffraction-free beams using a new optical system,” Opt. Rev. 4, 423–428 (1997).
[CrossRef]

Morales, J.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics, 1st ed. (McGraw-Hill, New York, 1968), Chap. 7.

Perez, M. V.

M. V. Perez, C. Comes-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

Popov, S. Yu.

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

S. Yu. Popov, A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[CrossRef]

Provost, J. N.

Rosen, J.

Santarsiero, M.

Schnabel, B.

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

Scott, G.

G. Scott, M. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31, 2640–2643 (1992).
[CrossRef]

Sedukhin, A. G.

A. G. Sedukhin, “Beam-preshaping axicon focusing,” J. Opt. Soc. Am. A 15, 3057–3066 (1998).
[CrossRef]

A. G. Sedukhin, “Two-component axicon focusing of light radiation,” Optoelectron. Instrum. Data Process. No. 5 (Sept.–Oct.), 28–40 (1997) (translation of Russian journal “Autometriya”).

Sochacki, J.

Stamnes, J.

J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Part III.

Staronski, L. R.

Szabo, G.

Thewes, K.

K. Thewes, M. A. Karim, A. A. S. Awwal, “Diffraction-free beam generation using a refracting system,” Opt. Laser Technol. 23, 105–108 (1991).
[CrossRef]

Tittel, F. K.

Turunen, J.

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

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
[CrossRef] [PubMed]

J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[CrossRef] [PubMed]

Vasara, A.

Wiggins, T. A.

Appl. Opt. (5)

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

A. T. Friberg, “Stationary-phase analysis of generalized axicons,” J. Opt. Soc. Am. A 13, 743–750 (1996).
[CrossRef]

A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
[CrossRef] [PubMed]

A. G. Sedukhin, “Beam-preshaping axicon focusing,” J. Opt. Soc. Am. A 15, 3057–3066 (1998).
[CrossRef]

Z. Jaroszewicz, J. Morales, “Lens axicons: systems composed of a diverging aberrated lens and a converging aberrated lens,” J. Opt. Soc. Am. A 16, 191–197 (1999).
[CrossRef]

W.-X. Cong, N.-X. Chen, B.-Yu. Gu, “Generation of nondiffracting beams by diffractive phase elements,” J. Opt. Soc. Am. A 15, 2362–2364 (1998).
[CrossRef]

Z. Jaroszewicz, J. Morales, “Lens axicons: systems composed of a diverging aberrated lens and a perfect converging lens,” J. Opt. Soc. Am. A 15, 2383–2390 (1998).
[CrossRef]

R. Borghi, M. Santarsiero, F. Gori, “Axial intensity of apertured Bessel beams,” J. Opt. Soc. Am. A 14, 23–26 (1997).
[CrossRef]

J. N. Provost, J. L. de Bougrenet de la Tocnaye, “Optimal trade-off pupil for synthesis of nondiffracting beams in free space,” J. Opt. Soc. Am. A 14, 2748–2757 (1997).
[CrossRef]

Z. L. Horvath, M. Erdelyi, G. Szabo, Zs. Bor, F. K. Tittel, J. R. Cavallaro, “Generation of nearly nondiffracting Bessel beams with a Fabry–Perot interferometer,” J. Opt. Soc. Am. A 14, 3009–3013 (1997).
[CrossRef]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

R. M. Herman, T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8, 932–942 (1991).
[CrossRef]

Opt. Acta (1)

M. V. Perez, C. Comes-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

Opt. Commun. (3)

N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[CrossRef]

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

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

Opt. Eng. (1)

G. Scott, M. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31, 2640–2643 (1992).
[CrossRef]

Opt. Laser Technol. (1)

K. Thewes, M. A. Karim, A. A. S. Awwal, “Diffraction-free beam generation using a refracting system,” Opt. Laser Technol. 23, 105–108 (1991).
[CrossRef]

Opt. Lett. (4)

Opt. Rev. (1)

K. Kono, M. Ire, T. Minemoto, “Generation of nearly diffraction-free beams using a new optical system,” Opt. Rev. 4, 423–428 (1997).
[CrossRef]

Optoelectron. Instrum. Data Process. No. 5 (1)

A. G. Sedukhin, “Two-component axicon focusing of light radiation,” Optoelectron. Instrum. Data Process. No. 5 (Sept.–Oct.), 28–40 (1997) (translation of Russian journal “Autometriya”).

Phys. Rev. Lett. (1)

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

Pure Appl. Opt. (1)

S. Yu. Popov, A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[CrossRef]

Other (3)

A. Papoulis, Systems and Transforms with Applications in Optics, 1st ed. (McGraw-Hill, New York, 1968), Chap. 7.

J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Part III.

M. Abramowitz, I. A. Stegun, eds. Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 9.

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Figures (3)

Fig. 1
Fig. 1

(a) Pattern of the courses of converging rays for a phase-undistorted truncated J0 Bessel beam with k=9928.9 mm-1, α=40 mm-1, and rb=4 mm. For clarity, the scale along the r axis is enlarged with respect to the scale along the z axis by a factor of 50; the rays shown by heavy lines shade an approximate marginal region prefacing a region of geometrical shadow. (b) Relevant curves of the axial-intensity distribution: The dashed curve was obtained by numerically evaluating the Fresnel diffraction integral, and the solid curve was calculated following the stationary-phase approximation. (c) Variations of the axial-intensity components corresponding to the stationary-phase approximation.

Fig. 2
Fig. 2

(a) Pattern of the courses of converging rays for a truncated J0 Bessel beam with marginal phase correction introduced. The correction phase function has the form ϕ(r)=ϕb(r/rb)n with ϕb=(p+1)αrb/n, p=31 and n=27.80. Parameters k, α, and rb as well as comments relating to the scale of axes and to emphasizing the rays in (a) are the same as those in Fig. 1(a). (b) Relevant curves of the axial-intensity distribution and (c) variations of its components corresponding to the stationary-phase approximation. The dashed and solid curves in (b) are as in Fig. 1(b). Variation of the central-lobe diameter at half-maximum intensity level was obtained (d) by evaluating the diffraction integral and (e) by using the stationary-phase approximation.

Fig. 3
Fig. 3

(a) Evolution of a correction phase function of the kind ϕ(r)=ϕb(r/rb)n with ϕb=(p+1)αrb/n, p=31, and variable number n, which is indirectly defined through a radius ri and a phase level ΔΦi. Parameters k, α, and rb are the same as in Fig. 1(a). (b) Relevant evolution of the axial intensity for a phase-corrected truncated J0 Bessel beam. The dashed and solid curves are the same as in Fig. 1(b).

Equations (54)

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

U(ρ, z)=kiz expikz+ρ22z×0rbJ0(αr)expikr22z+ϕ(r)×J0krρzrdr.
U(ρ, z)  ϕ(r)=0rb =J0(αρ)expizk-α22k,
U(0, z)=kiz 0r0J0(αr)expikr22z+ϕ(r)rdr,
abf(t)exp[iKμ(t)]dt
f(ts)2πKμ(ts)1/2 expiKμ(ts)+π4+f(t)iKμ(t) exp[iKμ(t)]a b,ts(a, b),
μ(ts)=0.
J0(αr)=M0(αr){exp[-iΘ0(αr)]+exp[iΘ0(αr)]}/2.
U(0, z)=kiz 0rb M0(αr)2×expikr22z+ϕ (r)-Θ0(αr)rdr.
M0(αr)/2(2παr)-1/2,Θ0(αr)αr-π/4.
z(rs)=krs/α1-ϕ(rs)/α,α-1rs<rb.
U(0,rs)=Ug(rs)exp[iΦg(rs)]+Ub(rs)exp[iΦb(rs)].
Ug(rs)
=1-ϕ(rs)/α[1-ϕ(rs)/α+rsϕ(rs)/α]1/2,
Ub(rs)
=-1-ϕ(rs)/α(2παrb)1/2{1-ϕ(rs)/α-[1-ϕ(rb)/α]rs/rb},
Φg(rs)
=-[α+ϕ(rs)]rs/2+ϕ(rs),
Φb(rs)
=[α-ϕ(rs)]rb2/(2rs)-αrb+ϕ(rb)+π/4,
I(0, rs)=Ig(rs)+Ib(rs)+Ibg(rs).
Ig(rs)=Ug2(rs),
Ib(rs)=Ub2(rs),
Ibg(rs)=2Ug(rs)Ub(rs)cos[ΔΦ(rs)],
ΔΦ(rs)=Φb(rs)-Φg(rs)=α(rb-rs)2-ϕ(rs)(rb2-rs2)2rs+ϕ(rb)-ϕ(rs)+π/4,
I(z)=krs(z)α+12παrb(1-z/zb)2-[krs(z)/(2πrb)]1/2α(1-z/zb)×cos1-zzb k2rb2-α2zbz2kz+π4,
k/α2z<zb,zb>0,k/α2z<,zb<0,
I˜(z)=I˜g(z)+I˜b(z)+I˜bg(z),
I˜g(z)=1,
I˜b(z)=12παrb 1-αzkrb-2,
I˜bg(z)=-2[I˜g(z)I˜b(z)]1/2cos[ΔΦ˜(z)],
ΔΦ˜(z)=krb22z 1-αzkrb2+π4,
ΔΦ˜(z˜i)=ΔΦi,
δz=Δz/zmax=(2+2)1/2-,
=(ΔΦi-π/4)/(αrb)=(ΔΦi-π/4)/(2πN),
Nrb2/(λzmax)=α2zmax/(2πk)=αrb/(2π).
δz(2)1/2=[(ΔΦi-π/4)/(πN)]1/2.
|2I˜bg(z)/I˜g(z)|4/(2παrb)1/2=(2/π)N-1/2.
I˜(zp)/I˜g(zp)1+2/(31/2π),
δzp=(zmax-zp)/zmax[3π/(2αrb)]1/2=(31/2/2)N-1/2 .
|ϕ(rs)/α||rsϕ(rs)/α|1,
ϕ(rs)/α0,α-1rs<αzi/k,
Ig(z)1-zk ϕαzk,k/α2z<zi.
(zi/k)ϕ(αzi/k)δI.
Δλ=|ϕ(ri)/k|=λ/4.
ri=rbz˜i/zmax=rb(1-δz)=rb+(ΔΦi-π/4)/α-{[rb+(ΔΦi-π/4)/α]2-rb2}1/2.
Ib(z)12παrb{1+[ϕ(rb)/α-1]αz/(krb)}2,
k/α2rs<zi.
ϕ(rb)/α-1=p,p1,
γ=pθ,orzb=-zmax/p,p1,
p<πk/(10α).
ϕ(r)=ϕb(r/rb)n,
ϕ(rb)/α=ϕbn/(αrb)=p+1=32,
d(0.5)(rs)=2.253/[α-ϕ(rs)],
z(rs)=krs/[α-ϕ(rs)],

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