Abstract

Analytical expressions are derived for a new set of optical beams, in which the radial dependence is described by a sum of Bessel distributions of different orders, modified by a flat-topped Gaussian function expressed in the form 1-[1-exp(-ξ2)]M, where ξ is a dimensionless parameter and M(1) is a scalar quantity. The flat-topped Gaussian function can be readily expanded into a series of the lowest-order Gaussian modes with different parameters; this situation makes it possible to express the optical beam as a series of conventional Bessel–Gaussian beams of different orders. The propagation features of this new set of optical beams are investigated to reveal how a windowed Bessel beam passes progressively from a smooth Gaussian window toward the hard-edge limit.

© 2004 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. R. M. Herman, T. A. Wiggins, “Apodization of diffractionless beams,” Appl. Opt. 31, 5913–5915 (1992).
    [CrossRef] [PubMed]
  4. F. Gori, G. Guattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  5. Z. Jiang, Q. Lu, Z. Liu, “Propagation of apertured Bessel beams,” Appl. Opt. 34, 7183–7185 (1995).
    [CrossRef] [PubMed]
  6. A. J. Cox, J. D’Anna, “Constant-axial-intensity nondiffracting beams,” Opt. Lett. 17, 232–234 (1992).
    [CrossRef] [PubMed]
  7. C. Palma, V. Bagini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (1994).
    [CrossRef]
  8. S. Rushin, “Modified Bessel nondiffracting beams,” J. Opt. Soc. Am. A 11, 3224–3228 (1994).
    [CrossRef]
  9. G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6, 150–152 (1989).
    [CrossRef]
  10. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Spagolos, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
  11. C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
    [CrossRef]
  12. R. Piestun, J. Shamir, “Generalized propagation-invariant fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998).
    [CrossRef]
  13. Y. Li, V. Gurevich, M. Krichever, J. Katz, E. Marom, “Propagation of anisotropic Bessel-Gaussian beams: sidelobe control, mode selection and field depth,” Appl. Opt. 40, 2709–2721 (2001).
    [CrossRef]
  14. Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206, 225–234 (2002).
    [CrossRef]
  15. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  16. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Sec. 8.2.2.

2002 (1)

Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206, 225–234 (2002).
[CrossRef]

2001 (1)

1998 (1)

1996 (2)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Spagolos, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

1995 (1)

1994 (3)

C. Palma, V. Bagini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (1994).
[CrossRef]

S. Rushin, “Modified Bessel nondiffracting beams,” J. Opt. Soc. Am. A 11, 3224–3228 (1994).
[CrossRef]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

1992 (2)

1989 (1)

1987 (3)

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

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]

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Spagolos, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

C. Palma, V. Bagini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (1994).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 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 (1996).
[CrossRef]

Cox, A. J.

D’Anna, J.

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]

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Spagolos, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Gori, F.

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

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

Guattari, G.

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

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

Gurevich, V.

Herman, R. M.

Indebetouw, G.

Jiang, Z.

Katz, J.

Krichever, M.

Li, Y.

Liu, Z.

Lu, Q.

Marom, E.

Miceli, J. J.

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

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 (1996).
[CrossRef]

C. Palma, V. Bagini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (1994).
[CrossRef]

Piestun, R.

Rushin, S.

Santarsiero, M.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Spagolos, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Spagolos, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Shamir, J.

Spagolos, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Spagolos, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Wiggins, T. A.

Wolf, E.

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

Appl. Opt. (3)

J. Mod. Opt. (2)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Spagolos, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Commun. (4)

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

C. Palma, V. Bagini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (1994).
[CrossRef]

Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206, 225–234 (2002).
[CrossRef]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[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 (1)

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

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Figures (6)

Fig. 1
Fig. 1

(a) Plot of flat-topped profiles FM(ξ), in Eq. (7), versus the normalized radial distance ξ from the axis of symmetry for the values of M given on each curve. (b) Rescaled and flat-topped profiles GM(ρ, 0), in Eq. (9).

Fig. 2
Fig. 2

Schematic diagram of the system configuration and illustration of notation.

Fig. 3
Fig. 3

Plot of normalized axial irradiance patterns IM(z), in Eq. (28), versus the normalized distance z/Zmax from the center of the source plane for the values of M given on each curve, and different values of κ. (a) κ=2, (b) κ=5, and (c) κ=20.

Fig. 4
Fig. 4

Normalized irradiance distributions along the radius of a J0 Bessel beam modified by a flat-topped Gaussian profile of M=5 in various planes perpendicular to the axis of the beam.

Fig. 5
Fig. 5

Irradiance distributions along the x axis (θ=0) and the y axis (θ=90°) of the elliptical Bessel–Gaussian beam with parameter given by Eq. (34) under the conditions λ=635 (nm) and Θ=0.5 (mrad).

Fig. 6
Fig. 6

Normalized irradiance distributions along the x and y axis of an elliptical Bessel beam modified by a flat-topped Gaussian profile of M=5 in various planes perpendicular to the axis of the beam.

Tables (1)

Tables Icon

Table 1 Scaling Factor γ for the Width of GM, Given by Eq. (9)

Equations (38)

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Vn,m(ρ, φ, 0)=AJn(αρ)exp(-ρ2/wm2)×exp(inφ) (n=0, 1, 2,),
Θ=arcsin(α/k)α/k,
VN,M(ρ, φ, 0)=BN(ρ, φ, 0)GM(ρ, 0),
BN(ρ, φ, 0)=nNBn(ρ, φ, 0),
Bn(ρ, φ, 0)=bnJn(αρ)exp(inφ),
B0(ρ, 0, 0)=b0J0(αρ)=J0(αρ),
FM(ξ)=1-[1-exp(-ξ2)]M,
GM(ρ, 0)FM(ρ/wS)=1-[1-exp(-ρ2/wS2)]M.
GM(ρ, 0)=1-1-exp-γρ2w12M.
γ=(w1/wS)2,
γ=m=1M1m.
GM(ρ, 0)=m=1MGm(ρ, 0),
Gm(ρ, 0)=gm exp-ρ2wm2=gm exp-mγρ2w12,
gm=(-1)m+1M(M-1)(M-m+1)m!.
VN,M(ρ, φ, 0)=nNm=1MVn,m(ρ, φ, 0).
Vn,m(ρ, φ, 0)=Bn(ρ, φ, 0)Gm(ρ, 0)=bngmJn(αρ)exp-mγρ2w12exp(inφ).
Vn,m(r, θ, z)=-iλzexpikz+r22zρ=0Gm(ρ, 0)×expikρ22zρdρφ=02πBn(ρ, φ, 0)×exp-ikρrzcos(θ-φ)dφ.
0exp(-τx2)Jn(x)Jn(δx)xdx=(-i)n2τexp-2+δ24τJniδ2τ.
Vn,m(r, θ, z)=gm1+imγ(z/zR)×expikz-r22z-iz2kα2+(kr/z)21+imγ(z/zR)×bnJnαr1+imγ(z/zR)exp(inθ),
Vn,m(r, θ, z)=exp(iβz)Bn(r, θ, z)Gm(r, z).
Gm(r, z)=gmw1wm(z)exp-iΦm(z)+[r2+(z sin Θ)2]×-1wm2(z)+ik2Rm(z),
wm(z)=w11+mγzzR21/2,
Φm(z)=arctanmγzzR,
Rm(z)=z+1(mγ)2zR2z.
Bn(r, θ, z)=bnJnαr1+imγ(z/zR)exp(inθ).
VN,M(r, θ, z)=nNm=1MVn,m(r, θ, z)=exp(iβz)nNm=1MGm(r, z)Bn(r, θ, z).
IN,M(r, θ, z)=nNm=1MVn,m(r, θ, z)2=nNm=1MGm(r, z)Bn(r, θ, z)2.
Bn(r, θ, z)bnJn(αr)exp(inθ).
Vn,M(r, θ, z)=exp(iβz)bnm=1MGm(r, z)×Jn(αr)exp(inθ),
IM(z)=m=1MGm(0, z)2=m=1Mgmmγ1+(mγz/zR)2exp-i arctanmγzzR-κ2mγ(z/zR)2[1-i(mγz/zR)]1+(mγz/zR)22,
κ=Θ/ΘG(αw1)/2,
ΘG=λ/(πw1).
Zmax=w1/ΘzR/κ
I1(z)=11+(z/zR)2.
I0,M(r, θ, z)=m=1MJ0(αrz)Gm(r, z)2.
rz=r1+imγ(z/zR).
z0.5Zmax.
I4,M(r, θ ,z)=m=1M[J0(αrz)-1.4J2(αrz)cos(2θ)+0.5J4(αrz)cos(4θ)]Gm(r, z)2.

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