Abstract

A new class of solutions to the scalar wave equation was introduced recently that represents transversely localized but totally nondiffracting fields. We show by the method of stationary phase that any of these wave fields can be realized approximately with a laser and a single computer-generated hologram. We briefly discuss various techniques for coding and fabrication of the required hologram and the associated diffraction efficiencies. Using both binary-amplitude and four-level phase holograms, we demonstrate experimentally the formation of arbitrary-order Bessel beams and rotationally nonsymmetric beams.

© 1989 Optical Society of America

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References

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  1. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  2. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  3. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  4. J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
    [CrossRef] [PubMed]
  5. J. H. McLeod, “The axicon: a new type of optical element,”J. Opt. Soc. Am. 44, 592–597 (1954).
    [CrossRef]
  6. M. V. Perez, C. Gomez-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
    [CrossRef]
  7. J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
    [CrossRef] [PubMed]
  8. Clearly, more-general (nonfactored) types of solutions to the scalar wave equation may exist that obey the nondiffractive intensity condition stated by Eq. (1). An obvious example would be E(r, t) = U(x, y)exp[iΘ(x, y, z)]exp(−iωt), where Θ(x, y, z) is real. We do not, however, consider these solutions here.
  9. P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), Sec. 14.14.
  10. E. T. Whittaker, “On the partial differential equations of mathematical physics,” Math. Ann. 57, 333–355 (1902).
    [CrossRef]
  11. T. H. Havelock, “Mathematical analysis of wave propagation in isotropic space of p dimensions,” Proc. London Math. Soc. 2, 122–137 (1904).
  12. G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970).
  13. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  14. W.-H. Lee, “Computer-generated holograms,” in Progress in Optics XVI, E. Wolf, ed. (North-Holland, Amsterdam, 1978).
    [CrossRef]
  15. J. R. Leger, M. L. Scott, P. Bundman, M. P. Griswold, “Astigmatic wavefront correction of a gain-guided laser diode array using anamorphic diffractive microlens,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum.884, 82–89 (1988).
    [CrossRef]
  16. M. Kajanto, E. Byckling, J. Fagerholm, J. Heikonen, J. Turunen, A. Vasara, A. Salin, “Photolithographic fabrication method of computer-generated holographic interferograms,” Appl. Opt. 28, 778–784 (1989).
    [CrossRef] [PubMed]
  17. A. Vasara, J. Turunen, A. T. Friberg, “General diffraction-free beams produced by computer-generated holograms,” in Holographic Systems, Components and Applications, J. C. Dainty, ed. (Institution of Electrical and Radio Engineers, London, to be published).

1989

1988

1987

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]

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

1986

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

1954

1904

T. H. Havelock, “Mathematical analysis of wave propagation in isotropic space of p dimensions,” Proc. London Math. Soc. 2, 122–137 (1904).

1902

E. T. Whittaker, “On the partial differential equations of mathematical physics,” Math. Ann. 57, 333–355 (1902).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970).

Bundman, P.

J. R. Leger, M. L. Scott, P. Bundman, M. P. Griswold, “Astigmatic wavefront correction of a gain-guided laser diode array using anamorphic diffractive microlens,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum.884, 82–89 (1988).
[CrossRef]

Byckling, E.

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, 1161–1176 (1986).
[CrossRef]

Durnin, J.

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

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), Sec. 14.14.

Fagerholm, J.

Friberg, A. T.

J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[CrossRef] [PubMed]

A. Vasara, J. Turunen, A. T. Friberg, “General diffraction-free beams produced by computer-generated holograms,” in Holographic Systems, Components and Applications, J. C. Dainty, ed. (Institution of Electrical and Radio Engineers, London, to be published).

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, 1161–1176 (1986).
[CrossRef]

Gori, F.

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

Griswold, M. P.

J. R. Leger, M. L. Scott, P. Bundman, M. P. Griswold, “Astigmatic wavefront correction of a gain-guided laser diode array using anamorphic diffractive microlens,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum.884, 82–89 (1988).
[CrossRef]

Guattari, G.

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

Havelock, T. H.

T. H. Havelock, “Mathematical analysis of wave propagation in isotropic space of p dimensions,” Proc. London Math. Soc. 2, 122–137 (1904).

Heikonen, J.

Kajanto, M.

Lee, W.-H.

W.-H. Lee, “Computer-generated holograms,” in Progress in Optics XVI, E. Wolf, ed. (North-Holland, Amsterdam, 1978).
[CrossRef]

Leger, J. R.

J. R. Leger, M. L. Scott, P. Bundman, M. P. Griswold, “Astigmatic wavefront correction of a gain-guided laser diode array using anamorphic diffractive microlens,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum.884, 82–89 (1988).
[CrossRef]

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]

Milonni, P. W.

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), Sec. 14.14.

Padovani, C.

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

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

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, 1161–1176 (1986).
[CrossRef]

Salin, A.

Scott, M. L.

J. R. Leger, M. L. Scott, P. Bundman, M. P. Griswold, “Astigmatic wavefront correction of a gain-guided laser diode array using anamorphic diffractive microlens,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum.884, 82–89 (1988).
[CrossRef]

Turunen, J.

M. Kajanto, E. Byckling, J. Fagerholm, J. Heikonen, J. Turunen, A. Vasara, A. Salin, “Photolithographic fabrication method of computer-generated holographic interferograms,” Appl. Opt. 28, 778–784 (1989).
[CrossRef] [PubMed]

J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[CrossRef] [PubMed]

A. Vasara, J. Turunen, A. T. Friberg, “General diffraction-free beams produced by computer-generated holograms,” in Holographic Systems, Components and Applications, J. C. Dainty, ed. (Institution of Electrical and Radio Engineers, London, to be published).

Vasara, A.

M. Kajanto, E. Byckling, J. Fagerholm, J. Heikonen, J. Turunen, A. Vasara, A. Salin, “Photolithographic fabrication method of computer-generated holographic interferograms,” Appl. Opt. 28, 778–784 (1989).
[CrossRef] [PubMed]

J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[CrossRef] [PubMed]

A. Vasara, J. Turunen, A. T. Friberg, “General diffraction-free beams produced by computer-generated holograms,” in Holographic Systems, Components and Applications, J. C. Dainty, ed. (Institution of Electrical and Radio Engineers, London, to be published).

Whittaker, E. T.

E. T. Whittaker, “On the partial differential equations of mathematical physics,” Math. Ann. 57, 333–355 (1902).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Math. Ann.

E. T. Whittaker, “On the partial differential equations of mathematical physics,” Math. Ann. 57, 333–355 (1902).
[CrossRef]

Opt. Acta

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

Opt. Commun.

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

Opt. Lett.

Phys. Rev. Lett.

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

Proc. London Math. Soc.

T. H. Havelock, “Mathematical analysis of wave propagation in isotropic space of p dimensions,” Proc. London Math. Soc. 2, 122–137 (1904).

Other

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

W.-H. Lee, “Computer-generated holograms,” in Progress in Optics XVI, E. Wolf, ed. (North-Holland, Amsterdam, 1978).
[CrossRef]

J. R. Leger, M. L. Scott, P. Bundman, M. P. Griswold, “Astigmatic wavefront correction of a gain-guided laser diode array using anamorphic diffractive microlens,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum.884, 82–89 (1988).
[CrossRef]

A. Vasara, J. Turunen, A. T. Friberg, “General diffraction-free beams produced by computer-generated holograms,” in Holographic Systems, Components and Applications, J. C. Dainty, ed. (Institution of Electrical and Radio Engineers, London, to be published).

Clearly, more-general (nonfactored) types of solutions to the scalar wave equation may exist that obey the nondiffractive intensity condition stated by Eq. (1). An obvious example would be E(r, t) = U(x, y)exp[iΘ(x, y, z)]exp(−iωt), where Θ(x, y, z) is real. We do not, however, consider these solutions here.

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), Sec. 14.14.

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Figures (7)

Fig. 1
Fig. 1

Normalized axial intensity of the holographically generated J0 beam as a function of the propagation distance z. The parameters are λ = 633 nm and (a) D = 10 mm, ρ0 = 1 mm, (b) D = 20 mm, ρ0 = 0.5 mm.

Fig. 2
Fig. 2

Radial profiles of the optical intensity distributions corresponding to the Jn(αρ) Bessel beams with (solid curve) n = 0, (dotted curve) n = 1, and (dashed curve) n = 6.

Fig. 3
Fig. 3

Binary-amplitude-coded transmission functions of holograms generating nondiffracting (higher-order) Bessel beams: (a) on-axis hologram for J0(αρ) beam, (b) off-axis version of the hologram in (a), (c) off-axis hologram for J1(αρ) beam, (d) off-axis hologram for J6(αρ) beam, (e) on-axis hologram generating a J1(αρ)cos ϕ beam, (f) off-axis version of the hologram in (e).

Fig. 4
Fig. 4

Fresnel diffraction patterns produced by J1 and J6 Bessel beam holograms: (a) J1 beam, z = 5 m; (b) J1 beam, z = 9 m; (c) J6 beam, z = 5 m; (d) J6 beam, z = 9 m.

Fig. 5
Fig. 5

Geometrical-optics predictions for the diffraction-free regions of beams produced by finite-aperture holograms: (a) off-axis hologram (only the first order is shown), (b) on-axis hologram producing the J0 beam (orders m = +1 and m = −1 are shown).

Fig. 6
Fig. 6

Fresnel diffraction pattern produced by the J1(αρ)cos ϕ beam hologram shown in Fig. 4(f) at a distance z = 5 m.

Fig. 7
Fig. 7

Fresnel diffraction pattern produced by a lithographically fabricated four-level phase-only hologram generating the fundamental Bessel beam.

Equations (25)

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I ( x , y , z 0 ) = I ( x , y , 0 ) ,
( ˜ 2 - 1 c 2 2 t 2 ) E ( r , t ) = 0
E ( r , t ) = U ( x , y ) Z ( z ) exp ( - i ω t )
d 2 Z d z 2 + β 2 Z = 0 ,
( 2 + α 2 ) U ( x , y ) = 0 ,
α 2 + β 2 = k 2 = ( ω / c ) 2 .
Z ( z ) = C exp ( i β z ) ,
U ( x , y ) = 0 2 π F ( u ) exp [ i α ( x cos u + y sin u ) ] d u ,
E ( r , t ) = exp [ i ( β z - ω t ) ] 0 2 π F ( u ) exp [ i α ( x cos u + y sin u ) ] d u ,
t ( ρ , ϕ ) = { A ( ϕ ) exp ( - i 2 π ρ / ρ 0 ) ρ D 0 ρ > D ,
U ( ρ , ϕ , z ) = 1 i λ z exp [ i k ( z + ρ 2 / 2 z ) ] × 0 D d ρ ρ exp [ i k ( ρ 2 / 2 z - 2 π ρ / γ ) ] × 0 2 π d ϕ A ( ϕ ) exp [ - i k ρ ρ cos ( ϕ - ϕ ) / z ] ,
A ( ϕ ) = n = - a n exp ( i n ϕ ) ,
U ( ρ , ϕ , z ) = 1 i λ z exp [ i k ( z + ρ 2 / 2 z ) ] n = - a n exp ( i n ϕ ) × 0 D f n ( ρ ) exp [ i k μ ( ρ ) ] d ρ ,
a n = 1 2 π 0 2 π A ( ϕ ) exp ( - i n ϕ ) d ϕ
f n ( ρ ) = 2 π ( - i ) n ρ J n ( k ρ ρ / z )
μ ( ρ ) = ρ 2 / 2 z - 2 π ρ / γ .
0 D f n ( ρ ) exp [ i k μ ( ρ ) ] d ρ f n ( ρ c ) exp [ i k μ ( ρ c ) ] [ k μ ( 2 ) ( ρ c ) ] 1 / 2 ,
I ( ρ , ϕ , z ) z | n = - 2 π ( - i ) n a n exp ( i n ϕ ) J n ( 2 π k ρ / γ ) | 2 .
I ( x , y , z ) z | 0 2 π A ( ϕ ) exp [ - i 2 π k ( x cos ϕ + y sin ϕ ) / γ ] d ϕ | 2 .
I ( x , y , z ) z U ( x , y ) 2 ,
T ( ρ , ϕ ) = ½ { 1 + a ( ϕ ) cos [ Ψ ( ρ , ϕ ) ] } ,
Ψ ( ρ , ϕ ) = 2 π ν ρ sin ϕ + ψ ( ϕ ) - 2 π ρ / ρ 0 .
T B ( ρ , ϕ ) = { 0 0 ( 1 / 2 ) { 1 + cos [ Ψ ( ρ , ϕ ) ] } b 1 b < ( 1 / 2 ) { 1 + cos [ Ψ ( ρ , ϕ ) ] } 1 ,
Ψ ( ρ , ϕ ) [ 2 π ( q - 1 ) / Q , 2 π q / Q ] Φ ( ρ , ϕ ) = 2 π q / Q ,             q = 1 , , Q .
η ( m ) = ( Q / π m ) 2 sin 2 ( π m / Q ) ,

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