Abstract

Coatings with graded transmission, placed near the rims of conical lenses and perhaps other axicons, provide apodization that removes the axial intensity variations in Bessel-type optical beams.

© 1992 Optical Society of America

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References

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  1. J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954).
    [Crossref]
  2. M. V. Perez, C. Gomez-Reino, J. M. Caudrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
    [Crossref]
  3. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [Crossref] [PubMed]
  4. A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer generated holograms,” J. Opt. Soc. Am. A6, 1748–1754 (1989).
    [Crossref]
  5. R. M. Herman, T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A8, 932–942 (1991).
    [Crossref]
  6. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [Crossref]
  7. P. L. Overfelt, C. S. Kenney, “Comparison of the propagation characteristics of Bessel, Bessel–Gauss, and Gaussian beams diffracted by a circular aperture,” J. Opt. Soc. Am. A 8, 732–745 (1991).
    [Crossref]
  8. N. Davidson, A. A. Friesem, E. Hasman, “Holographic axilens: high resolution and long focal depth,” Opt. Lett. 16, 523–525 (1991).
    [Crossref] [PubMed]
  9. V. V. Korobkin, L. Ya. Polonskii, V. P. Poponin, L. N. Pyatnitskii, “Focusing of Gaussian and super-Gaussian laser beams by axicons to obtain continuous laser sparks,” Sov. J. Quantum Electron. 16, 178–183 (1986).
    [Crossref]
  10. A. J. Cox, J. D’Anna, “Constant-axial-intensity beam,” Opt. Lett. 17, 232–234 (1992).
    [Crossref] [PubMed]

1992 (1)

1991 (3)

1989 (1)

A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer generated holograms,” J. Opt. Soc. Am. A6, 1748–1754 (1989).
[Crossref]

1987 (2)

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

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

1986 (2)

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

V. V. Korobkin, L. Ya. Polonskii, V. P. Poponin, L. N. Pyatnitskii, “Focusing of Gaussian and super-Gaussian laser beams by axicons to obtain continuous laser sparks,” Sov. J. Quantum Electron. 16, 178–183 (1986).
[Crossref]

1954 (1)

Caudrado, J. M.

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

Cox, A. J.

D’Anna, J.

Davidson, N.

Durnin, J.

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

Eberly, J. H.

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

Friberg, A. T.

A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer generated holograms,” J. Opt. Soc. Am. A6, 1748–1754 (1989).
[Crossref]

Friesem, A. A.

Gomez-Reino, C.

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

Gori, F.

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

Guattari, G.

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

Hasman, E.

Herman, R. M.

R. M. Herman, T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A8, 932–942 (1991).
[Crossref]

Kenney, C. S.

Korobkin, V. V.

V. V. Korobkin, L. Ya. Polonskii, V. P. Poponin, L. N. Pyatnitskii, “Focusing of Gaussian and super-Gaussian laser beams by axicons to obtain continuous laser sparks,” Sov. J. Quantum Electron. 16, 178–183 (1986).
[Crossref]

McLeod, J. H.

Miceli, J. J.

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

Overfelt, P. L.

Padovani, C.

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

Perez, M. V.

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

Polonskii, L. Ya.

V. V. Korobkin, L. Ya. Polonskii, V. P. Poponin, L. N. Pyatnitskii, “Focusing of Gaussian and super-Gaussian laser beams by axicons to obtain continuous laser sparks,” Sov. J. Quantum Electron. 16, 178–183 (1986).
[Crossref]

Poponin, V. P.

V. V. Korobkin, L. Ya. Polonskii, V. P. Poponin, L. N. Pyatnitskii, “Focusing of Gaussian and super-Gaussian laser beams by axicons to obtain continuous laser sparks,” Sov. J. Quantum Electron. 16, 178–183 (1986).
[Crossref]

Pyatnitskii, L. N.

V. V. Korobkin, L. Ya. Polonskii, V. P. Poponin, L. N. Pyatnitskii, “Focusing of Gaussian and super-Gaussian laser beams by axicons to obtain continuous laser sparks,” Sov. J. Quantum Electron. 16, 178–183 (1986).
[Crossref]

Turunen, J.

A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer generated holograms,” J. Opt. Soc. Am. A6, 1748–1754 (1989).
[Crossref]

Vasara, A.

A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer generated holograms,” J. Opt. Soc. Am. A6, 1748–1754 (1989).
[Crossref]

Wiggins, T. A.

R. M. Herman, T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A8, 932–942 (1991).
[Crossref]

J. Opt. Soc. Am. (3)

J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954).
[Crossref]

A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer generated holograms,” J. Opt. Soc. Am. A6, 1748–1754 (1989).
[Crossref]

R. M. Herman, T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A8, 932–942 (1991).
[Crossref]

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

Opt. Acta (1)

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

Opt. Commun. (1)

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

Opt. Lett. (2)

Phys. Rev. Lett. (1)

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

Sov. J. Quantum Electron. (1)

V. V. Korobkin, L. Ya. Polonskii, V. P. Poponin, L. N. Pyatnitskii, “Focusing of Gaussian and super-Gaussian laser beams by axicons to obtain continuous laser sparks,” Sov. J. Quantum Electron. 16, 178–183 (1986).
[Crossref]

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Figures (2)

Fig. 1
Fig. 1

Relative axial intensity as a function of distance, normalized to unity at the geometrical maximum for a cone with an angle of 6°, index = 1.51, and 3.2-cm aperture illuminated at 633 nm with a Gaussian beam with a characteristic radius of 1.6 cm. The dashed curves show the envelope of the oscillations whose details are included near the end of the range. The solid curve represents a similarly normalized intensity for the cone with a coating at the outer edge with a characteristic width of 0.05 cm.

Fig. 2
Fig. 2

Details of the relative axial intensity near the end of the range for the conditions and normalization of Fig. 1. Results are shown for several values of the characteristic width W of the apodization coating.

Equations (13)

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E ( P ) = 0 ρ m i k ρ E ( ρ ) l ( ρ , z ) exp [ i k μ ( p , z ) ] d ρ ,
μ ( ρ , z ) = [ ( z + ρ tan γ ) 2 + ρ 2 ] 1 / 2 - n ρ tan γ
μ ( ρ ) μ β + ( μ β / 2 ) ( ρ - ρ β ) 2 ,
[ d 2 μ ( ρ ) d ρ 2 ] ρ = ρ β ,
E ( P ) = [ 2 π k μ ( ρ β ) ] 1 / 2 ρ β E ( ρ β ) l ( ρ β , z ) exp [ i ( k μ β + π / 4 ) ] .
E ( P ) = E s ( P ) + E m ( P ) ,
E m ( P ) = - ρ m i k ρ E ( ρ ) l ( ρ , z ) exp [ i k μ ( ρ ) ] d ρ .
E m ( P ) - i k ρ m E ( ρ m ) l ( ρ m , z ) × ρ m exp [ i k μ m + μ m ( ρ - ρ m ) ] d ρ = ρ m E ( ρ m ) μ m l ( ρ m , z ) exp ( i k μ m ) ,
[ d μ ( ρ ) d ρ ] ρ = ρ m .
E m ( P ) - i k ρ m E ( ρ m ) l ( ρ m , z ) exp ( i k μ m ) × - g ( ρ , ρ m ) exp [ i k μ m ( ρ - ρ m ) ] d ρ .
E m ( P ) ρ m E ( ρ m ) μ m l ( ρ m , z ) exp ( i k μ m ) × - d g ( ρ , ρ m ) d ρ exp [ i k μ m ( ρ - ρ m ) ] d ρ .
d g ( ρ , ρ m ) d ρ = [ 1 / ( π W 2 ) ] 1 / 2 exp { - [ ( ρ - ρ m ) / W ] 2 } .
E m ( P ) ρ m E ( ρ m ) μ m l ( ρ m , z ) exp [ - ( k μ m W / 2 ) 2 + i k μ m ] .

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