Abstract

Recently Durnin pointed out the existence of nondiffracting beams that propagate in free space with their energy confined around their axis but that experience no spread or divergence [ J. Opt. Soc. Am. A 4, 651 ( 1987)]. This Communication provides an alternative way of interpreting these results. The approach is based on the McCutchen theorem from which the necessary conditions for a nondiffracting field are derived. This description may be useful in suggesting several different means of synthesizing such fields and in providing a convenient way of estimating their practical limitations. Some examples, inspired by known techniques studied for depth of field enhancement, are briefly mentioned.

© 1989 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]
  2. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  3. C. W. McCutchen, “Generalized aperture and three-dimensional diffraction image,”J. Opt. Soc. Am. 54, 240–244 (1964).
    [CrossRef]
  4. W. D. Montgomery, “Self-imaging objects of infinite aperture,”J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]
  5. G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
    [CrossRef]
  6. W. T. Wilford, “Use of annular aperture to increase focal depth,”J. Opt. Soc. Am. 50, 749–753 (1960).
    [CrossRef]
  7. J. T. McCrickerd, “Coherent processing and depth of focus of annular aperture imaging,” Appl. Opt. 10, 2226–2230 (1971).
    [CrossRef] [PubMed]
  8. J. H. McLead, “Axicon: a new type of optical element,”J. Opt. Soc. Am. 44, 592–597 (1954).
    [CrossRef]
  9. G. Bickel, G. Häusler, M. Maul, “Triangulation with expanded range depth,” Opt. Eng. 24, 975–977 (1985).
    [CrossRef]

1988 (1)

G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
[CrossRef]

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]

1985 (1)

G. Bickel, G. Häusler, M. Maul, “Triangulation with expanded range depth,” Opt. Eng. 24, 975–977 (1985).
[CrossRef]

1971 (1)

1967 (1)

1964 (1)

1960 (1)

1954 (1)

Bickel, G.

G. Bickel, G. Häusler, M. Maul, “Triangulation with expanded range depth,” Opt. Eng. 24, 975–977 (1985).
[CrossRef]

Durnin, J.

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

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

Eberly, J. H.

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

Häusler, G.

G. Bickel, G. Häusler, M. Maul, “Triangulation with expanded range depth,” Opt. Eng. 24, 975–977 (1985).
[CrossRef]

Indebetouw, G.

G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
[CrossRef]

Maul, M.

G. Bickel, G. Häusler, M. Maul, “Triangulation with expanded range depth,” Opt. Eng. 24, 975–977 (1985).
[CrossRef]

McCrickerd, J. T.

McCutchen, C. W.

McLead, 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]

Montgomery, W. D.

Wilford, W. T.

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Figures (5)

Fig. 1
Fig. 1

Synthesis of a nondiffracting field in the far field of a thin annular aperture of radius R0 and width Δ. If the aperture has a uniform transmittance, the approximate zero-order nondiffracting field, in the back focal plane of a lens of focal length f, has a longitudinal extend δz ≈ λf2/R0Δ.

Fig. 2
Fig. 2

The axial projection of the generalized pupil stretched over the unit sphere is the Fourier transform of the axial field distribution. The size of this distribution is inversely proportional to the size δβ of the pupil projection. The case shown is that of a diffracting annular aperture of numerical aperture NA and angular width δθ.

Fig. 3
Fig. 3

Synthesis of a polychromatic nondiffracting field by using a Fabry–Perot étalon with a mirror spacing d and a finesse F. The longitudinal extend of the field is δz ≈ 2dF.

Fig. 4
Fig. 4

Axial projection of the generalized pupil for the case of an interferential pupil consisting of a Fabry–Perot étalon with mirror spacing d and finesse F. The selected spatial frequency is independent of the wavelength.

Fig. 5
Fig. 5

Synthesis of an approximately nondiffracting beam by using an axicon of radius Ra, index of refraction n, and cone angle α. For small angles, the longitudinal extent of the field is δzRa/(n − 1)α.

Equations (8)

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V ( r ) = P ( q ) exp ( i k q · r ) d 3 q ,
P ( q ) = A ( k q ) δ ( q 2 + q z 2 - 1 ) .
q z = β = constant .
q = ( 1 - β 2 ) 1 / 2 = ρ
A ( k q ) = P ( θ ) δ ( q - ρ ) .
P ( θ ) = n p n     exp ( i n θ ) .
V ( r ) = exp [ i k β z ] n p n exp ( - i n ϕ ) J n ( k ρ r ) ,
δ z δ β ~ λ .

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