Abstract

Theoretical topics encountered in engineering analysis of diffraction-free beams are addressed. These topics are the following: synthesizing a class of diffraction-free beams with noncircular transverse irradiance distributions, e.g., elliptical beams; the effect of Gaussian apodization on sidelobe control; and mode selection that allows one to synthesize a nearly diffraction-free beam with a certain depth of field. The focusing properties of Bessel–Gaussian beams will be investigated the future.

© 2001 Optical Society of America

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References

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  1. J. Durnin, “Exact solution 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. G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6, 150–152 (1989).
    [CrossRef]
  4. R. M. Herman, T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8, 932–942 (1991).
    [CrossRef]
  5. G. Scott, N. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 12, 2640–2646 (1992).
    [CrossRef]
  6. L. C. Laycock, S. C. Webster, “Bessel beams: their generation and application,” GEC J. Res. 10, 36–51 (1992).
  7. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  8. C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1992).
    [CrossRef]
  9. R. Piestun, J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998).
    [CrossRef]
  10. P. L. Overfelt, C. S. Kenney, “Comparison of propagation characteristics of Bessel, Bessel–Gaussian, and Gaussian beams diffracted by a circular aperture,” J. Opt. Soc. Am. A 8, 732–745 (1991).
    [CrossRef]
  11. E. Marom, N. Konforti, D. Mendlovic, “Beam with extended confinement for scanning purposes,” U.S. patent5,315,095 (24May1994).
  12. C. F. Du Toit, “The numerical computation of Bessel functions of the first and second kind for integer orders and complex arguments,” IEEE Trans. Antennas Propag. 38, 1341–1349 (1990).
    [CrossRef]
  13. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964).
  14. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999), Sec. 8.2.2.
  15. Y. Li, J. Katz, “Encircled energy of laser-diode beams,” Appl. Opt. 30, 4283–4284 (1991).
    [CrossRef] [PubMed]
  16. Y. Li, J. Katz, “Nonparaxial analysis of the far-field radiation patterns of double-heterostructure lasers,” Appl. Opt. 35, 1442–1451 (1996).
    [CrossRef] [PubMed]
  17. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Chap. 8.

1998 (1)

1996 (1)

1992 (3)

G. Scott, N. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 12, 2640–2646 (1992).
[CrossRef]

L. C. Laycock, S. C. Webster, “Bessel beams: their generation and application,” GEC J. Res. 10, 36–51 (1992).

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1992).
[CrossRef]

1991 (3)

1990 (1)

C. F. Du Toit, “The numerical computation of Bessel functions of the first and second kind for integer orders and complex arguments,” IEEE Trans. Antennas Propag. 38, 1341–1349 (1990).
[CrossRef]

1989 (1)

1987 (3)

J. Durnin, “Exact solution 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]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964).

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999), Sec. 8.2.2.

Cincotti, G.

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1992).
[CrossRef]

Du Toit, C. F.

C. F. Du Toit, “The numerical computation of Bessel functions of the first and second kind for integer orders and complex arguments,” IEEE Trans. Antennas Propag. 38, 1341–1349 (1990).
[CrossRef]

Durnin, J.

J. Durnin, “Exact solution 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]

Eberly, J. H.

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

Gori, F.

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

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Chap. 8.

Guattari, G.

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1992).
[CrossRef]

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

Herman, R. M.

Indebetouw, G.

Katz, J.

Kenney, C. S.

Konforti, N.

E. Marom, N. Konforti, D. Mendlovic, “Beam with extended confinement for scanning purposes,” U.S. patent5,315,095 (24May1994).

Laycock, L. C.

L. C. Laycock, S. C. Webster, “Bessel beams: their generation and application,” GEC J. Res. 10, 36–51 (1992).

Li, Y.

Marom, E.

E. Marom, N. Konforti, D. Mendlovic, “Beam with extended confinement for scanning purposes,” U.S. patent5,315,095 (24May1994).

McArdle, N.

G. Scott, N. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 12, 2640–2646 (1992).
[CrossRef]

Mendlovic, D.

E. Marom, N. Konforti, D. Mendlovic, “Beam with extended confinement for scanning purposes,” U.S. patent5,315,095 (24May1994).

Miceli, J. J.

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

Overfelt, P. L.

Padovani, C.

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

Palma, C.

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1992).
[CrossRef]

Piestun, R.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Chap. 8.

Santarsiero, M.

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1992).
[CrossRef]

Scott, G.

G. Scott, N. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 12, 2640–2646 (1992).
[CrossRef]

Shamir, J.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964).

Webster, S. C.

L. C. Laycock, S. C. Webster, “Bessel beams: their generation and application,” GEC J. Res. 10, 36–51 (1992).

Wiggins, T. A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999), Sec. 8.2.2.

Appl. Opt. (2)

GEC J. Res. (1)

L. C. Laycock, S. C. Webster, “Bessel beams: their generation and application,” GEC J. Res. 10, 36–51 (1992).

IEEE Trans. Antennas Propag. (1)

C. F. Du Toit, “The numerical computation of Bessel functions of the first and second kind for integer orders and complex arguments,” IEEE Trans. Antennas Propag. 38, 1341–1349 (1990).
[CrossRef]

J. Mod. Opt. (1)

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1992).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

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

Opt. Eng. (1)

G. Scott, N. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 12, 2640–2646 (1992).
[CrossRef]

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 (4)

E. Marom, N. Konforti, D. Mendlovic, “Beam with extended confinement for scanning purposes,” U.S. patent5,315,095 (24May1994).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999), Sec. 8.2.2.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Chap. 8.

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Figures (13)

Fig. 1
Fig. 1

(a) Synthesis of diffraction-free beams by superposition of plane waves whose wave vectors are distributed on a cone rotationally symmetric about the z axis and with an angular half-aperture Θ. (b) Decomposition of the wave vector into α and β components; α is the projection of the wave vector onto the plane vertical to the z axis.

Fig. 2
Fig. 2

Transverse irradiance patterns of the elliptical diffraction-free beam with parameters given by Eq. (2.9). (a) Distributions along the x axis (θ = 0) and the y axis (θ = 90°). (b) Longitudinal distribution showing the diffraction-free property in beam propagation.

Fig. 3
Fig. 3

Schematic diagram of the system configuration and illustration of notation used in Eqs. (3.1) and (3.2).

Fig. 4
Fig. 4

Sidelobe control by Gaussian apodization. Irradiance distributions of the elliptical diffraction-free beam with parameters given by Eq. (2.9) and apodized by a Gaussian profile of w 0 = 1.2 mm. (a) Transverse irradiance distributions along the x axis (θ = 0) and along the y axis (θ = 90°). (b) Longitudinal distribution showing the nearly diffraction-free property in beam propagation.

Fig. 5
Fig. 5

Axial irradiance distributions as a function of z/ z R and with κ as a parameter, where z R is the Rayleigh range and κ is the degree of Gaussian apodization.

Fig. 6
Fig. 6

Computer-simulated diffraction patterns between the Fresnel and the Fraunhofer diffraction regions for the Bessel–Gaussian beam with parameters given by Eq. (2.9). Solid curves, distribution along the x axis; dashed curves, distribution along the y axis.

Fig. 7
Fig. 7

Computer-simulated Fraunhofer diffraction patterns for the Bessel–Gaussian beam with parameters given by Eq. (2.9) with apodization by a Gaussian profile of w 0 = 1.2 (mm). Solid curves, distribution along the x axis; dashed curves, distribution along the y axis.

Fig. 8
Fig. 8

Schematic diagram to illustrate the decomposition of a Bessel–Gaussian beam into component Gaussian beams and the competition between two angles Θ and Θ G that dominates the changes of the beam profile in free-space propagation. Θ is the angular half-aperture of the cone on which the axes of the component Gaussian beams are uniformly distributed. Θ G is the divergence angle of a component Gaussian beam.

Fig. 9
Fig. 9

(a) Graphic solution of Eq. (5.1) for the depth of field (Δz) in systems with κ = 1 and κ = 2. (b) Depth of field normalized by (Δz)κ=1 and under the conditions that α = constant and λ = constant.

Fig. 10
Fig. 10

Dependence of the ellipticity of the main lobe on the ratio w y /w x of the elliptical Gaussian apodizer. Here w y and w x represent the lengths of the major and the minor axes, respectively.

Fig. 11
Fig. 11

Total energy carried by the Bessel–Gaussian beam as a function of κ, the degree of Gaussian apodization. The parameters of the beam are given by Eq. (2.9) with apodization by a Gaussian profile of w 0 = 1.2 mm.

Fig. 12
Fig. 12

Diagrams of isophotes of the x and y sections through the three-dimensional irradiance distribution in systems with three degrees of Gaussian apodization: (a) κ = 1, (b) κ = 2, and (c) κ = 4. The irradiance is normalized to unity at the center of the source plane. The beam parameters are given by Eq. (2.9) for apodization by a Gaussian profile of w 0 = 1.2 mm.

Fig. 13
Fig. 13

Geometrical shadow zone for an anisotropic Bessel–Gaussian beam.

Tables (1)

Tables Icon

Table 1 Fresnel Diffraction Patterns of Bessel-Guassian Beams with Parameters Given by Eq. (2.9) and w o = 1.2 mma

Equations (59)

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Ux, y, z=Aψexpiαx cos ψ+y sin ψ+iβz,
k2=α2+β2,
sin Θ=α/k.
Aψ=im2πexp-imψ m=0, 1, 2, .
μmr, θ, z=im2πexpiβz02πexp-imψ+iαx cos ψ+y sin ψdψ=expiβzJmαrexp-imθ,
r=x2+y2,  θ=arctany/x.
μ0r, z=expiβzJ0αr,
UMr, θ, z=mM cmμmr, θ, z=expiβzmM cmJmαrexp-imθ,
UMr, θ, z=expiβzc0J0αr+c2J2αrcos2θ+c4J4αrcos4θ+.
UMr, θ, z=expiβzc0J0αr-1.4J2αrcos2θ+0.5J4αrcos4θ
λ=635 nm,  Θ=0.5 mrad.
IM0, θ, z=|UM0, θ, z|2/c02=1
VMρ, ϕ, 0=UMρ, ϕ, 0exp-ρ/w02,
VMr, θ, z=-iλzexpikz+r22z×ρ=0ρ=expik ρ22zρdρ×θ=0θ=2π VMρ, ϕ, 0×exp-ik ρrzcosθ-ϕdϕ.
VMr, θ, z=expik-α2/2kz×Gr, z×UMrz, θ, 0.
expik-α2/2kzexpizk2-α2=expiβz,
Gr, z=w0wzexp-iΦz+r2+z sin Θ2×-1w2z+ik2Rz,
wz=w01+z/zR21/2,
Φz=arctanz/zR,
Rz=z+zR2/z,
zR=πw02/λ,
UMrz, θ, 0=m=0M cmJmαrzcosmθ,
rz=r1+iz/zR.
IMr, θ, z=|VMr, θ, z|2=|Gr, z|2|UMrz, θ, 0|2,
|Gr, z|2=11+z/zR2×exp-2 r2w2z-2κ2z/zR21+z/zR2,
|UMr, θ, z|2=m=0M cmJm2κ1+iz/zRrw0cosmθ2,
κ=αw0/2Θ/ΘG
ΘG=λ/πw0.
IM0, 0, z=11+z/zR2exp-2κ2z/zR21+z/zR2.
zzR.
rz=r1+iz/zRr.
IMrz, θ, zIMr, θ, 0=exp-2 r2w02×m=0M cmJm2κ rw0cosmθ2.
w0=1.2 mm,  zR=πw02/λ=7124 mm.
z0.20.3zR,
rzr1+i=-r×ii/2.
IMr, θ, z=exp-κ2exp-r2/w02×m=0M-1mcmberm2κ rw0×cosmθ2+m=0M-1mcmbeim2κ rw0×cosmθ2,
z  zR.
rz=r1+iz/zR-i rz/zR.
IMr, θ, z=1z/zR2exp-2 r/w02z/zR2-2κ2×m=0M-imcmIm2κz/zRrw0cosmθ2.
Imxex2πx
IMr, θ, z  exp-2 r2w02z/zR2exp2κz/zRrw0.
rp/w0=z/w0sin Θz/w0α/k=κz/zR.
αr1=2.405, 3.832, 5.136,.
κ=w0α/2<1.2.
w0r20=62.05/α, 63.61/α, 65.16/α,,
κ=w0α/235.
11+Δz/zR2exp-2κ2Δz/zR21+Δz/zR2=0.7.
exp-r2w02exp-r2wx2b0+b2 cos 2θ,
b0=1+1/ε2,  b2=1-1/ε2,  ε=wy/wx
Eκ=0 rdr 02π IMr, θ, 0dθ.
Eκ=κ22α2exp-κ2m=0M |cm|2Imκ2.
Zmaxw0/Θ=k×w0/α=zR/κ,
ΘS=zR/w0Θ=κ rad.
VMr, θ, z=-ikzexpikz+r22z×m=0M-imcmWmr, zcosmθ,
Wmr, z=0exp-1w02+i k2zρ2×JmαρJmkρr/zρdρ.
0exp-τx2JmγxJmδxxdx=12τexp-γ2+δ24τImγδ2τ
Imx=-imJmix.
Wmr, z=-im2w021+iz/zR×exp-w024α2+kr/z21+iz/zRJm-αr1+iz/zR,
VMr, θ, z=11+iz/zRexpikz+r22z-w024α2+kr/z21+iz/zR×m=0M cmJmαr1+iz/zRcosmθ.

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