Abstract

We present exact, nonsingular solutions of the scalar-wave equation for beams that are nondiffracting. This means that the intensity pattern in a transverse plane is unaltered by propagating in free space. These beams can have extremely narrow intensity profiles with effective widths as small as several wavelengths and yet possess an infinite depth of field. We further show (by using numerical simulations based on scalar diffraction theory) that physically realizable finite-aperture approximations to the exact solutions can also possess an extremely large depth of field.

© 1987 Optical Society of America

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References

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  1. See, for example, H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE, 54, 1312–1392 (1966).
    [CrossRef]
  2. See, for example, C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys., 17, 35–100 (1954).
    [CrossRef]
  3. A brief report on some of our initial experimental results was presented at the 1986 Annual Meeting of the Optical Society of America: J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” J. Opt. Soc. Am. A 3, P128 (1986).

1986 (1)

A brief report on some of our initial experimental results was presented at the 1986 Annual Meeting of the Optical Society of America: J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” J. Opt. Soc. Am. A 3, P128 (1986).

1966 (1)

See, for example, H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE, 54, 1312–1392 (1966).
[CrossRef]

1954 (1)

See, for example, C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys., 17, 35–100 (1954).
[CrossRef]

Bouwkamp, C. J.

See, for example, C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys., 17, 35–100 (1954).
[CrossRef]

Durnin, J.

A brief report on some of our initial experimental results was presented at the 1986 Annual Meeting of the Optical Society of America: J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” J. Opt. Soc. Am. A 3, P128 (1986).

Eberly, J. H.

A brief report on some of our initial experimental results was presented at the 1986 Annual Meeting of the Optical Society of America: J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” J. Opt. Soc. Am. A 3, P128 (1986).

Kogelnik, H.

See, for example, H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE, 54, 1312–1392 (1966).
[CrossRef]

Li, T.

See, for example, H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE, 54, 1312–1392 (1966).
[CrossRef]

Miceli, J. J.

A brief report on some of our initial experimental results was presented at the 1986 Annual Meeting of the Optical Society of America: J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” J. Opt. Soc. Am. A 3, P128 (1986).

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

A brief report on some of our initial experimental results was presented at the 1986 Annual Meeting of the Optical Society of America: J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” J. Opt. Soc. Am. A 3, P128 (1986).

Proc. IEEE (1)

See, for example, H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE, 54, 1312–1392 (1966).
[CrossRef]

Rep. Prog. Phys. (1)

See, for example, C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys., 17, 35–100 (1954).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Intensity distribution J02(αρ) (—) and its envelope function 2/παρ(- - - -).

Fig. 2
Fig. 2

Intensity distributions for a J0 beam (—) and a Gaussian beam (- - - -): (a) when z = 0 (i.e., in the initial plane where the beams are assumed to be formed), (b) when z = 25 cm, (c) when z = 75 cm, (d) when z = 100 cm, and (e) when z = 120 cm, assuming that λ = 0.5 μm. Note in 2(b)–2(e) the intensity of the Gaussian beam has been multiplied by 10.

Fig. 3
Fig. 3

Intensities I(ρ = 0, z) at beam center, as a function of distance, of the J0 (—) and Gaussian (- - - -) beams whose initial intensity distributions at z = 0 are shown in Fig. 2(a).

Fig. 4
Fig. 4

Intensity distributions for a J0 beam: (a) when z = 0, (b) when z = 2 m, (c) when z = 4 m, and (d) when z = 5.5 m. The aperture at z = 0 has a radius of 1 cm, but only the central 4 mm of the beam is plotted in (b)–(d) in order to show clearly that the central spot diameter has not changed.

Fig. 5
Fig. 5

Propagation of the central peak intensity I(ρ = 0, z) of the J0 beam shown in Fig. 4(a).

Fig. 6
Fig. 6

Geometrical shadow zone for J0 beams of finite aperture. A conical shadow zone begins at the distance z = r/tan θ, where r is the radius of the limiting aperture at z = 0, θ = sin−1(αλ/2π), and the diameter of the central maximum of the J0 beam is approximately 3π/2α.

Equations (5)

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( 2 - 1 c 2 2 t 2 ) E ( r , t ) = 0.
E ( x , y , z 0 , t ) = exp [ i ( β z - ω t ) ] 0 2 π A ( ϕ ) exp [ i α ( x cos ϕ + y sin ϕ ) ] d ϕ ,
I ( x , y , z 0 ) = ½ E ( r , t ) 2 = I ( x , y , z = 0 ) ,
E ( r , t ) = exp [ i ( β z - ω t ) ] 0 2 π exp [ i α ( x cos ϕ + y sin ϕ ) ] d ϕ 2 π = exp [ i ( β z - ω t ) ] J 0 ( α ρ ) .
z max = r / tan θ = r [ ( 2 π / α λ ) 2 - 1 ] 1 / 2 ,

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