Abstract

It is proposed to obtain intense optical fields, whose form shows little change in size over long paths, through the use of either conical lenses or spherical lenses showing spherical aberration together with a single projecting lens. The conical lens is shown to produce fields whose transverse structure is given by a zero-order Bessel function J0, while the spherical aberrating lens produces (real or virtual) J0-like transverse structures, provided that the central portion of the aberrating lens is occluded. In all cases projection gives a J0 real-image optical structure. Intensity, size of the transverse structure, and range considerations are developed, and some aspects of optimization are discussed. A negative aberrating lens gives a long range of nearly constant size in the image field, and a universal expression is presented to describe the image size as a function of image distance for this case. Projection with an aberrating projection lens is shown to improve the constancy of the final J0 pattern size dramatically. Typical photographic results are included for beams generated by using a low-power He–Ne laser. Brief considerations of practical uses of diffractionless beams are presented.

© 1991 Optical Society of America

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References

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  1. J. Durnin, “Continuously self-imaging fields of infinite aperture,” J. Opt. Soc. Am. A 2(13), P110 (1985); “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
  2. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction free beams,” J. Opt. Soc. Am. 3(13), P128 (1986).
  3. J. Durnin, J. J. Miceli, J. H. Eberly, “Experiments with nondiffracting needle beams,” J. Opt. Soc. Am. B 4(13), P230 (1987).
  4. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  5. J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
    [CrossRef] [PubMed]
  6. See M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).
  7. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  8. D. DeBeer, S. R. Hartmann, R. Friedberg, “Comment on ‘Diffraction-free beams,’” Phys. Rev. Lett. 59, 2611 (1987).
    [CrossRef] [PubMed]
  9. J. Durnin, J. J. Miceli, J. H. Eberly, “Reply to D. DeBeer, S. R. Hartmann and R. Friedberg,” Phys. Rev. Lett. 59, 2612 (1987).
    [CrossRef] [PubMed]
  10. J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954).
    [CrossRef]
  11. S. Fujiwara, “Optical properties of conic surfaces: I. Reflecting cone,” J. Opt. Soc. Am. 52, 287–292 (1962).
    [CrossRef]
  12. M. V. Perez, C. Gomez–Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1177 (1986).
    [CrossRef]
  13. J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
    [CrossRef] [PubMed]
  14. 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]
  15. G. Indebetouw, “Nondiffracting optical field: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6, 150–152 (1989).
    [CrossRef]
  16. See, for example, L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981).
  17. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  18. See, for example, Ref. 6, Sec. IX, and references therein.
  19. R. Cassidy, ed., “Method hits bulls-eye, eliminates diffraction,” Res. Dev.31(10), 166 (1989).

1989 (2)

1988 (2)

1987 (5)

J. Durnin, J. J. Miceli, J. H. Eberly, “Experiments with nondiffracting needle beams,” J. Opt. Soc. Am. B 4(13), P230 (1987).

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

D. DeBeer, S. R. Hartmann, R. Friedberg, “Comment on ‘Diffraction-free beams,’” Phys. Rev. Lett. 59, 2611 (1987).
[CrossRef] [PubMed]

J. Durnin, J. J. Miceli, J. H. Eberly, “Reply to D. DeBeer, S. R. Hartmann and R. Friedberg,” Phys. Rev. Lett. 59, 2612 (1987).
[CrossRef] [PubMed]

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

1986 (2)

M. V. Perez, C. Gomez–Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1177 (1986).
[CrossRef]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction free beams,” J. Opt. Soc. Am. 3(13), P128 (1986).

1985 (1)

J. Durnin, “Continuously self-imaging fields of infinite aperture,” J. Opt. Soc. Am. A 2(13), P110 (1985); “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).

1962 (1)

1954 (1)

Born, M.

See M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Cuadrado, J. M.

M. V. Perez, C. Gomez–Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1177 (1986).
[CrossRef]

DeBeer, D.

D. DeBeer, S. R. Hartmann, R. Friedberg, “Comment on ‘Diffraction-free beams,’” Phys. Rev. Lett. 59, 2611 (1987).
[CrossRef] [PubMed]

Durnin, 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, “Experiments with nondiffracting needle beams,” J. Opt. Soc. Am. B 4(13), P230 (1987).

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

J. Durnin, J. J. Miceli, J. H. Eberly, “Reply to D. DeBeer, S. R. Hartmann and R. Friedberg,” Phys. Rev. Lett. 59, 2612 (1987).
[CrossRef] [PubMed]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction free beams,” J. Opt. Soc. Am. 3(13), P128 (1986).

J. Durnin, “Continuously self-imaging fields of infinite aperture,” J. Opt. Soc. Am. A 2(13), P110 (1985); “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).

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, “Experiments with nondiffracting needle beams,” J. Opt. Soc. Am. B 4(13), P230 (1987).

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

J. Durnin, J. J. Miceli, J. H. Eberly, “Reply to D. DeBeer, S. R. Hartmann and R. Friedberg,” Phys. Rev. Lett. 59, 2612 (1987).
[CrossRef] [PubMed]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction free beams,” J. Opt. Soc. Am. 3(13), P128 (1986).

Friberg, A. T.

Friedberg, R.

D. DeBeer, S. R. Hartmann, R. Friedberg, “Comment on ‘Diffraction-free beams,’” Phys. Rev. Lett. 59, 2611 (1987).
[CrossRef] [PubMed]

Fujiwara, S.

Gomez–Reino, C.

M. V. Perez, C. Gomez–Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1177 (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]

Hartmann, S. R.

D. DeBeer, S. R. Hartmann, R. Friedberg, “Comment on ‘Diffraction-free beams,’” Phys. Rev. Lett. 59, 2611 (1987).
[CrossRef] [PubMed]

Indebetouw, G.

McLeod, J. H.

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]

J. Durnin, J. J. Miceli, J. H. Eberly, “Experiments with nondiffracting needle beams,” J. Opt. Soc. Am. B 4(13), P230 (1987).

J. Durnin, J. J. Miceli, J. H. Eberly, “Reply to D. DeBeer, S. R. Hartmann and R. Friedberg,” Phys. Rev. Lett. 59, 2612 (1987).
[CrossRef] [PubMed]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction free beams,” J. Opt. Soc. Am. 3(13), P128 (1986).

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. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1177 (1986).
[CrossRef]

Schulman, L. S.

See, for example, L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Turunen, J.

Vasara, A.

Wolf, E.

See M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

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]

G. Indebetouw, “Nondiffracting optical field: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6, 150–152 (1989).
[CrossRef]

J. Durnin, “Continuously self-imaging fields of infinite aperture,” J. Opt. Soc. Am. A 2(13), P110 (1985); “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).

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

J. Durnin, J. J. Miceli, J. H. Eberly, “Experiments with nondiffracting needle beams,” J. Opt. Soc. Am. B 4(13), P230 (1987).

Opt. Acta (1)

M. V. Perez, C. Gomez–Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1177 (1986).
[CrossRef]

Opt. Commun. (1)

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

Opt. Lett. (1)

Phys. Rev. Lett. (3)

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

D. DeBeer, S. R. Hartmann, R. Friedberg, “Comment on ‘Diffraction-free beams,’” Phys. Rev. Lett. 59, 2611 (1987).
[CrossRef] [PubMed]

J. Durnin, J. J. Miceli, J. H. Eberly, “Reply to D. DeBeer, S. R. Hartmann and R. Friedberg,” Phys. Rev. Lett. 59, 2612 (1987).
[CrossRef] [PubMed]

Other (5)

See M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

See, for example, Ref. 6, Sec. IX, and references therein.

R. Cassidy, ed., “Method hits bulls-eye, eliminates diffraction,” Res. Dev.31(10), 166 (1989).

See, for example, L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981).

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Figures (7)

Fig. 1
Fig. 1

Typical paths from the exit face of a conical lens of angle γ to appoint P. Z is the distance from the cone apex to the field plane, and y, in the ϕ = 0 plane, is the distance from the optical axis to P. The area element da is (ρdρdϕ)/cos γ.

Fig. 2
Fig. 2

Geometric illustration for determining the optical paths abP and abc′. The paths bc′ and bP are both at an angle β with respect to the optical axis and obey geometrical optics. A contributing (but not principal) path abP is included. The radii, ρβ of the path ab, ρ of the path ab′, and ρM of the marginal ray and z the distance from the cone apex to P are also shown. The plane AA is normal to the entrance rays, and the plane CC is normal to the rays bP and bc′.

Fig. 3
Fig. 3

Simple aberrating lens and associated geometrical quantities: R, the radius of the second lens surface; θ, the angle of incidence; fC, the focal point for central rays; fM, the focal point for marginal rays; z, the distance from the lens apex to the observing plane; β, the angle at which rays cross the axis at P as determined by geometrical optics; ρβ, the ring radius for illuminating the point P; ρM, the marginal radius; ρ, the ring radius for a contributing (but not principal) path, shown as a dashed curve; and a central stop C.

Fig. 4
Fig. 4

Curve A shows the radius of the central region of a J0-type beam formed by a negative spherically aberrating lens and a high-quality projector lens as a function of position from the high-quality lens. Curve B shows the radii as a function of position formed by spherically aberrating negative and positive lenses. The portion of this curve shown as a dashed curve can be removed by an aperture placed on either lens.

Fig. 5
Fig. 5

Photograph of the ring pattern at 5.8 m from a 20° conical lens projected by a 5.8-cm lens. The radius of the central spot on the original negative was 600 μm with a camera magnification of 5×.

Fig. 6
Fig. 6

Photograph of the ring pattern at 9.6 m from an aberrating lens (f = 9.0 cm, 3.6-cm aperture, 1.1-cm central diaphram) as projected by a 5.8-cm f/1.2 lens. The radius of the (overexposed) central spot was 200 μm.

Fig. 7
Fig. 7

A, The pattern produced at 24 m from a plano-concave lens (f = −6.0 cm, 2.8-cm aperture) and projected by a 13.5-cm f/3.5 lens. The radius of the central spot on the original negative is 600 μm. B, The pattern produced for the same conditions with a 1.4-cm central diaphragm on the negative lens.

Equations (40)

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E ( P ) = i k 2 π d aE ( d a ) exp { i [ Φ ( ρ ) + k l ( d a P ) ] } l ( d a P ) cos γ ,
sin ( γ + β ) = n sin γ .
E ( P ) = E ( ρ β ) i k 2 π l ( ρ β , z ) ρ d ρ d ϕ exp { i [ Φ ( ρ ) + k l ( d a P ) ] } .
Φ ( ρ ) = - n k ρ tan γ ,
l ( d a P ) = [ ( z + ρ tan γ ) 2 + ( ρ cos ϕ - y ) 2 + ( ρ sin ϕ ) 2 ] 1 / 2 .
E ( P ) = E ( ρ β ) i k ρ β l ( ρ β , z ) [ 0 ρ M d ρ exp ( i k { [ ( z + ρ tan γ ) 2 + ρ 2 ] 1 / 2 - n ρ tan γ } ) ] ( 1 2 π 0 2 π d ϕ exp { - i k ρ y cos ϕ [ ( z + ρ tan γ ) 2 + ρ 2 ] 1 / 2 } ) ,
{ ρ + tan γ ( z + ρ tan γ ) [ ( z + ρ tan γ ) 2 + ρ 2 ] 1 / 2 } ρ = ρ β = n tan γ .
sin β = { ρ [ ( z + ρ tan γ ) 2 + ρ 2 ] 1 / 2 } ρ = ρ β ,
sin β + cos β tan γ = n tan γ ,
{ [ ( z + ρ tan γ ) 2 + ρ 2 ] 1 / 2 - n ρ tan γ } ρ = ρ β = z cos β .
( b P ) - ( b c ) = { ( b c ) 2 + [ ( ρ - ρ β ) cos β ] 2 } 1 / 2 - ( b c ) .
[ ( z + ρ tan γ ) 2 + ρ 2 ] 1 / 2 - n ρ tan γ = z cos β + ( ρ - ρ β ) 2 cos 2 β 2 [ ( z + ρ β tan γ ) 2 + ρ β 2 ] 1 / 2
[ ( z + ρ tan γ ) 2 + ρ 2 ] 1 / 2 - n ρ tan γ = z cos β + ( ρ - ρ β ) 2 cos 2 β 2 l ( ρ β , z ) .
[ i k ρ β l ( ρ β , z ) ] exp ( i k cos β ) 0 ρ M d ρ exp [ i k ( ρ - ρ β ) 2 cos 2 β 2 l ( ρ β , z ) ] .
E ( P ) E ( ρ β ) ( 2 π k ρ β sin β cos 2 β ) 1 / 2 exp [ i ( k z cos β + π / 4 ) ] × [ 1 2 π d ϕ exp ( i k y sin β cos ϕ ) ] .
1 2 π d ϕ exp ( i k y sin β cos ϕ ) = J 0 ( k y sin β ) ,
E ( y , z ) = E ( ρ β ) ( 2 π k ρ β sin β cos 2 β ) 1 / 2 exp [ i ( k z cos β + π / 4 ) ] × J 0 ( k y sin β ) .
l opt = { [ z + R ( 1 - cos θ ) ] 2 + ( R sin θ ) 2 + y 2 - 2 R y sin θ cos ϕ } 1 / 2 - n R ( 1 - cos θ ) .
n sin θ β = sin ( θ β + β ) ,
exp ( i k l opt ) exp [ i k ( { [ z + R ( 1 + cos θ ) ] 2 + ( R sin θ ) 2 } 1 / 2 - n R ( 1 - cos θ ) ) ] exp ( - i k y sin β cos ϕ ) ,
E ( y , z ) ~ E 0 [ ρ β ( z ) ] exp [ i k 0 z cos β ( z ) d z + π / 4 ] × J 0 [ k y sin β ( z ) ] .
d β ( z ) k d z 1 ,
m = [ ( i / f ) - 1 ] ,
δ P = 2 π ρ δ ρ I 0 exp [ - 2 ( ρ / w ) 2 ] ,
P ring = 2 π ρ λ I 0 sin β exp [ - 2 ( ρ / w ) 2 ] .
P total = ( π / 2 ) w 2 I 0 .
con = P ring / P total = 4 ρ λ w 2 sin β exp [ - 2 ( ρ / w ) 2 ]
con ( opt ) = 2 e 1 / 2 λ w sin β ,
δ z 2 ( z / ρ β ) ( δ y ) .
( f C - z ) ( f C - f M ) = ( ρ β ρ M ) 2
δ z f C - f M = 2 ρ β δ ρ β ρ M 2 .
δ y = f C - z z δ ρ β .
f C - z z f C - f M 3 f C
sph = 6 ( f C f C - f M ) ρ β λ w 2 sin β exp [ - 2 ( ρ β / w ) 2 ] .
sph ( opt ) = 3 e 1 / 2 λ w sin β ( f C f C - f M ) .
N 4 27 ρ M sin β f C λ ( f C - f M ) ,
sph ( opt ) 0.2 N - 1 .
Λ = 2.408 λ 2 π ( f C ρ M ) ( f M - f C z f - f C ) 1 / 2 ( f z f - f C ) ( ξ 3 ξ - 1 ) 1 / 2 ,
ξ = ( i - f ) ( z f - f C ) f 2 ,
Λ = 2.408 λ 2 π ( f C ρ M ) ( f M - f C z f - f C ) 1 / 2 ( f z f - f C ) ( ξ 3 ξ - 1 ) 1 / 2 × [ 1 + ( z f - f C f C ) ( 1 - ξ - 1 ) ] - 1 .

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