Abstract

Solutions of the scalar-wave equation that are based on modified Bessel functions are introduced. These functions are radially nonoscillating, unbound, and nondiffractive. Their propagation constant is larger than that of vacuum, meaning that there is an inverse Guoy effect or a phase velocity smaller than c. The beams are physically realized by apodization of the modified Bessel profiles by means of suitable window functions. When Gaussian apodization functions are used, axial decay is slower than in the Gaussian case and the case of ordinary Bessel counterparts. Super-Gaussian and circular apodization are also examined. In the latter case a persistent narrow axial lobe that also has a nondecaying character is encountered.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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. A. E. Siegman, “Defining the radius of curvature of a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
    [CrossRef]
  4. S. Ruschin, “Phase velocity of light modification by means of axially displaced Gaussian modes,” Nonlinear Optics (to be published).
  5. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  6. P. L. Overfelt, C. S. Kenney, “Comparison of the propagation characteristics of Bessel, Bessel–Gauss, and Gaussian beams diffracted by a circular aperture,” J. Opt. Soc. Am. A 8, 732–744 (1991).
    [CrossRef]
  7. S. DeSilvestri, P. Laporta, V. Magni, O. Svelto, “Unstable resonators with super-Gaussian mirrors,” Opt. Lett. 13, 201–203 (1988).
    [CrossRef]
  8. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986) Chap. 18.
  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. K. Uehara, H. Kikuchi, “Generation of nearly diffraction-free laser beams,” Appl. Phys. B 48, 125–129 (1989).
    [CrossRef]
  11. S. Ruschin, “Laser beam shaping by nonconventional laser cavity configurations,” Bull. Am. Phys. Soc. 36, 1990 (1991).
  12. M. Morin, P. A. Belanger, “Diffractive analysis of annular resonators,” Appl. Opt. 31, 1942–1947 (1992).
    [CrossRef] [PubMed]

1992 (1)

1991 (3)

A. E. Siegman, “Defining the radius of curvature of a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

S. Ruschin, “Laser beam shaping by nonconventional laser cavity configurations,” Bull. Am. Phys. Soc. 36, 1990 (1991).

P. L. Overfelt, C. S. Kenney, “Comparison of the propagation characteristics of Bessel, Bessel–Gauss, and Gaussian beams diffracted by a circular aperture,” J. Opt. Soc. Am. A 8, 732–744 (1991).
[CrossRef]

1989 (2)

1988 (1)

1987 (3)

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]

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

Belanger, P. A.

DeSilvestri, S.

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]

Friberg, A. T.

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]

Kenney, C. S.

Kikuchi, H.

K. Uehara, H. Kikuchi, “Generation of nearly diffraction-free laser beams,” Appl. Phys. B 48, 125–129 (1989).
[CrossRef]

Laporta, P.

Magni, V.

Miceli, J. J.

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

Morin, M.

Overfelt, P. L.

Padovani, C.

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

Ruschin, S.

S. Ruschin, “Laser beam shaping by nonconventional laser cavity configurations,” Bull. Am. Phys. Soc. 36, 1990 (1991).

S. Ruschin, “Phase velocity of light modification by means of axially displaced Gaussian modes,” Nonlinear Optics (to be published).

Siegman, A. E.

A. E. Siegman, “Defining the radius of curvature of a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986) Chap. 18.

Svelto, O.

Turunen, J.

Uehara, K.

K. Uehara, H. Kikuchi, “Generation of nearly diffraction-free laser beams,” Appl. Phys. B 48, 125–129 (1989).
[CrossRef]

Vasara, A.

Appl. Opt. (1)

Appl. Phys. B (1)

K. Uehara, H. Kikuchi, “Generation of nearly diffraction-free laser beams,” Appl. Phys. B 48, 125–129 (1989).
[CrossRef]

Bull. Am. Phys. Soc. (1)

S. Ruschin, “Laser beam shaping by nonconventional laser cavity configurations,” Bull. Am. Phys. Soc. 36, 1990 (1991).

IEEE J. Quantum Electron. (1)

A. E. Siegman, “Defining the radius of curvature of a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

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

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. (1)

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

Other (2)

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986) Chap. 18.

S. Ruschin, “Phase velocity of light modification by means of axially displaced Gaussian modes,” Nonlinear Optics (to be published).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Absolute value of the amplitude for (a) an ordinary (J-type) Bessel beam and (b) a modified (I-type) Bessel beam. Lighter shadowing denotes higher amplitude values.

Fig. 2
Fig. 2

Axial intensity as a function of propagation distance for Bessel–Gauss beams. Curve labels denote the value of w0γ. Real values of this parameter correspond to modified Bessel profiles, imaginary values correspond to the ordinary Bessel case, and zero corresponds to a pure Gaussian.

Fig. 3
Fig. 3

(a) Phase lag and (b) phase-velocity deviation as a function of propagating distance for Bessel–Gauss beams for different values of w0γ. Phase velocities are in units of c/(kzR).

Fig. 4
Fig. 4

Axial intensity as a function of propagating distance for a modified Bessel beam apodizied by the super-Gaussian functions exp[−(r/w0)2n]. Curve labels denote the value of n. All beams are characterized by the parameter w 0 γ = 3.

Fig. 5
Fig. 5

Transverse profile (solid curve) after propagation of a modified Bessel beam apodized at the entrance plane by a circular aperture. Dashed curve, profile at the entrance plane. Calculation parameters: λ = 0.5 μm, γ = 5, window radius 2 mm, propagation distance 1 m.

Fig. 6
Fig. 6

(a) Comparison of the calculated transverse profile of modified (solid curve) and ordinary (dashed curve) Bessel beams apodized by a circular aperture after 1-m propagation. Both beams have a maximum amplitude of unity at the entrance plane, where a circular window of 4 mm diameter is located. λ = 0.5 μm, γ = 10 mm−1, α = 24 mm−1. (b) Axial intensity profiles for the same beams as in (a), showing that the modified case (solid curve), is free of oscillations and has a longer decay range.

Fig. 7
Fig. 7

Far-field patterns of the examples of Fig. 6, obtained by calculation of the field at the back focal plane of a lens of 1-m focal length, located at a distance of 1 m from the entrance window. Solid curve, I-type beam; dashed curve, J-type beam.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

E ( x , y , z , t ) = ( 1 / 2 π ) exp [ i ( ω t - β z ) ] × 0 A ( ϕ ) exp [ i α ( x cos ϕ + y sin ϕ ) ] d ϕ ,
E ( x , y , z , t ) = exp [ i ( ω t - β z ) ] I 0 ( γ r ) ,
β 2 = ( ω / c ) 2 + γ 2 .
U ( r , z ) = i k z 0 U ( r 0 , 0 ) exp [ - i k 2 z ( r 2 + r 0 2 ) ] × J 0 [ ( k / z ) r r 0 ] r 0 d r 0 ,
U ( r 0 , 0 ) = I 0 ( γ r 0 ) exp [ - ( r 0 / w 0 ) 2 ] .
U ( 0 , z ) = A g ( z ) exp [ ( γ z / k w 0 ) 2 A g ( z ) ] × exp [ - i ( γ 2 z / 2 k ) A g ( z ) + i ψ g ( z ) ] ,
A g ( z ) = [ 1 + ( z / z R ) 2 ] - 1 ,             ψ g ( z ) = arctan ( z / z R ) , z R = k w 0 2 / 2.
A = { max [ U ( 0 , z ) ] } 0 { max [ U ( r , 0 ) ] } 0 .
U ( 0 , z ) = exp [ - i k a 2 / ( 2 z ) ] n = 1 ( i k a γ z ) n I n ( γ a ) ,
U ( 0 , z ) = exp [ i γ 2 z / ( 2 k ) ] - exp [ - i k a 2 / ( 2 z ) ] × n = 0 ( - i γ z a k ) n I n ( γ a ) .
U ( 0 , z ) exp [ i γ 2 z / ( 2 k ) ] - exp [ - i k a 2 / ( 2 z ) ] I 0 ( γ a ) .
U ( r , z ) I 0 ( a γ ) exp [ - i k a 2 / ( 2 z ) ] × exp [ - i k r 2 / ( 2 z ) ] J 0 ( a k r / z ) .
G I ( ρ ) = a γ 2 + ( 2 π ρ ) 2 [ 2 π ρ I 0 ( γ a ) J 1 ( 2 π ρ a ) - γ I 1 ( γ a ) J 0 ( 2 π ρ a ) ] ,

Metrics