Abstract

A particular class of nondiffracting beams is discussed called the nondiffracting arrays (NDA’s). Such beams are useful for applications that require the projection of arrays of achromatic narrow light spots with a long depth of field. Regarding their general expression, NDA’s can be seen as three-dimensional interference fields whose narrow fringes are parallel to the propagation axis. We propose to use a compact and achromatic N-wave interferometer to generate them. This original solution, based on the use of a spider’s-web-shaped diffractive pupil, is studied experimentally.

© 1999 Optical Society of America

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References

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  1. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
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    [CrossRef] [PubMed]
  3. J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954).
    [CrossRef]
  4. S. Fujiwara, “Fresnel conic mirror,” J. Opt. Soc. Am. 51, 1305 (1961).
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  5. G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6, 150–152 (1989).
    [CrossRef]
  6. J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
    [CrossRef] [PubMed]
  7. L. Niggl, T. Lanzl, M. Maier, “Properties of Bessel beams generated by periodic gratings of circular symmetry,” J. Opt. Soc. Am. A 14, 27–33 (1997).
    [CrossRef]
  8. 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]
  9. 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]
  10. J. Rosen, “Synthesis of nondiffracting beams in free space,” Opt. Lett. 19, 369–371 (1994).
    [PubMed]
  11. R. Piestun, J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. 19, 771–773 (1994).
    [CrossRef] [PubMed]
  12. J. Primot, M. Girard, M. Chambon, “Modulation transfer function for sampled imaging systems: a generalization of the line spread function,” J. Mod. Opt. 41, 1301–1306 (1994).
    [CrossRef]
  13. N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
    [CrossRef]
  14. J. Primot, L. Sogno, “Achromatic three-wave (or more) lateral shearing interferometer,” J. Opt. Soc. Am. A 12, 2679–2685 (1995).
    [CrossRef]

1997 (2)

1995 (1)

1994 (3)

J. Primot, M. Girard, M. Chambon, “Modulation transfer function for sampled imaging systems: a generalization of the line spread function,” J. Mod. Opt. 41, 1301–1306 (1994).
[CrossRef]

J. Rosen, “Synthesis of nondiffracting beams in free space,” Opt. Lett. 19, 369–371 (1994).
[PubMed]

R. Piestun, J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. 19, 771–773 (1994).
[CrossRef] [PubMed]

1989 (3)

1988 (1)

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]

1961 (1)

1954 (1)

Chambon, M.

J. Primot, M. Girard, M. Chambon, “Modulation transfer function for sampled imaging systems: a generalization of the line spread function,” J. Mod. Opt. 41, 1301–1306 (1994).
[CrossRef]

Durnin, J.

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]

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.

Fujiwara, S.

Girard, M.

J. Primot, M. Girard, M. Chambon, “Modulation transfer function for sampled imaging systems: a generalization of the line spread function,” J. Mod. Opt. 41, 1301–1306 (1994).
[CrossRef]

Indebetouw, G.

Kettunen, V.

Kuittinen, M.

Lanzl, T.

Maier, M.

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]

Niggl, L.

Piestun, R.

Primot, J.

J. Primot, L. Sogno, “Achromatic three-wave (or more) lateral shearing interferometer,” J. Opt. Soc. Am. A 12, 2679–2685 (1995).
[CrossRef]

J. Primot, M. Girard, M. Chambon, “Modulation transfer function for sampled imaging systems: a generalization of the line spread function,” J. Mod. Opt. 41, 1301–1306 (1994).
[CrossRef]

Rosen, J.

Shamir, J.

Sogno, L.

Streibl, N.

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

Turunen, J.

Vahimaa, P.

Vasara, A.

Appl. Opt. (1)

J. Mod. Opt. (2)

J. Primot, M. Girard, M. Chambon, “Modulation transfer function for sampled imaging systems: a generalization of the line spread function,” J. Mod. Opt. 41, 1301–1306 (1994).
[CrossRef]

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

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]

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Figures (9)

Fig. 1
Fig. 1

Construction in the Fourier plane of Cartesian NDA’s by selecting orders from a Cartesian grid with rings of different radii (three rings are represented that select 8, 12, and 24 orders of the grid).

Fig. 2
Fig. 2

Intensity pattern of (a) 8-, (b) 12-, and (c) 24-order Cartesian NDA’s (simulation).

Fig. 3
Fig. 3

Spider’s-web-shaped diffractive pupil of eight sectors.

Fig. 4
Fig. 4

Generation of interference fields diffracted by a pupil made of plane gratings.

Fig. 5
Fig. 5

Scheme of a spider’s web of 20 gratings (SW20G).

Fig. 6
Fig. 6

Predicted interferogram generated by a SW20G.

Fig. 7
Fig. 7

Recorded interferogram produced by the SW20G (under laser illumination).

Fig. 8
Fig. 8

Evolution of a light spot of the recorded interferogram along the propagation axis z.

Fig. 9
Fig. 9

Image of the interferogram delivered directly by the sensor (under white-light illumination).

Equations (16)

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E(x, y, z, t)=exp[i(βz-ωt)]×02πA(Φ)exp[iα(x cos Φ+y sin Φ)]dΦ,
E(r, z, t)=(1/α)exp[i(βz-ωt)]×02π0F(k)exp(ik·r)ρdρdΦ,
F(k)=A(Φ)ring(ρ-α),
A(Φ)=n=1pn exp(inΦ).
F(k)=n=1NA(Φn)δ(ρ-α, Φ-Φn),
E(r, z, t)=exp[i(βz-ωt)]×n=1NA(Φn)exp[iα(x cos Φn+y sin Φn)].
η=a0α/2π.
F(k)=grid1/a0(μ, ν)×δ(ρ-α),
E(r)=grida0(x, y)*J0(αr).
r02.405/α.
R1.1η2.
Epn(r, z)=Tp exp2iπpd(x cos Φn+y sin Φn)+2iπλz[1-(λp/d)2]1/2,
Ep(r, z)=Tp exp2iπλz[1-(λp/d)2]1/2×n=1N exp2iπpd(x cos Φn+y sin Φn).
α=2πp/d.
r00.38d/p.
a0=ηd/p=18d/p.

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