Abstract

An approximate method for determining the radial and axial intensity of a Bessel-like beam is presented for the general case in which a radial Bessel distribution of any order is modulated by an arbitrary function. For Bessel–Gauss, generalized Bessel–Gauss, and Bessel–super-Gauss beams, this simple approximation yields results that are very close to the exact values, while they are exact for Bessel beams. A practical beam that can be generated with a combination of simple lenses is also analyzed and illustrated.

© 2000 Optical Society of America

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References

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    [CrossRef]
  5. C. Palma, V. Bagini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (1994).
    [CrossRef]
  6. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Spagnolos, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  12. Bateman Manuscript Project, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2.
  13. C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
    [CrossRef]
  14. R. H. Jordan, D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel–Gauss beam solution,” Opt. Lett. 19, 427–429 (1994);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, 732–745 (1991).
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  15. Z.-P. Jiang, “Super Gaussian Bessel beam,” Opt. Commun. 125, 207–210 (1996).
    [CrossRef]
  16. J. Sochacki, Z. Jaroszewicz, L. R. Staroński, A. Kołodziejczyk, “Annular-aperture logorithmic axicon,” J. Opt. Soc. Am. A 10, 1765–1768 (1993).
    [CrossRef]
  17. H. B. Dwight, Tables of Integrals and Other Mathematical Data (Macmillan, New York, 1947), Eq. 808.

1996 (3)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Spagnolos, “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]

Z.-P. Jiang, “Super Gaussian Bessel beam,” Opt. Commun. 125, 207–210 (1996).
[CrossRef]

1994 (4)

1993 (1)

1992 (1)

1991 (1)

1988 (1)

1987 (2)

1986 (1)

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

1954 (1)

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Spagnolos, “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]

Cincotti, G.

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

Cuadrado, M. J.

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

Durnin, J.

Dwight, H. B.

H. B. Dwight, Tables of Integrals and Other Mathematical Data (Macmillan, New York, 1947), Eq. 808.

Frezza, F.

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

Friberg, A.

Gomez-Reino, C.

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

Gori, F.

F. Gori, G. Guattori, 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]

Guattori, G.

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

Hall, D. G.

Herman, R. M.

Jaroszewicz, Z.

Jiang, Z.-P.

Z.-P. Jiang, “Super Gaussian Bessel beam,” Opt. Commun. 125, 207–210 (1996).
[CrossRef]

Jordan, R. H.

Kolodziejczyk, A.

McLeod, J. H.

Padovani, C.

F. Gori, G. Guattori, 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]

Perez, M. V.

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

Ruschin, S.

Santarsiero, M.

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

Sochacki, J.

Spagnolos, G.

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

Staronski, L. R.

Turunen, J.

Vasara, A.

Wiggins, T. A.

Appl. Opt. (3)

J. Mod. Opt. (2)

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

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

J. Opt. Soc. Am. (1)

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

Opt. Acta (1)

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

Opt. Commun. (3)

F. Gori, G. Guattori, 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]

Z.-P. Jiang, “Super Gaussian Bessel beam,” Opt. Commun. 125, 207–210 (1996).
[CrossRef]

Opt. Lett. (1)

Other (2)

Bateman Manuscript Project, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2.

H. B. Dwight, Tables of Integrals and Other Mathematical Data (Macmillan, New York, 1947), Eq. 808.

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Figures (3)

Fig. 1
Fig. 1

Geometry of incoming optical fields for regions local to intersection point P, which then serves as the origin of coordinates (ρ, z) and (ρ, z). a, Output plane of axicon system (z=-z0); b, incoming wave front cone for intersection point P; c, local surface of constant phase; d, cone of incoming rays for P. The figure can be rotated about the optical (z) axis.

Fig. 2
Fig. 2

Radial intensity (intensity times radius) as a function of ρ for a J1 BG beam (solid curves) at distances (A) z=0, (B) z=25, (C) z=50, (D) z=100, (E) z=150 cm, as predicted by Eq. (18) for k=1.26×105 cm-1, kt=150 cm-1, w0=0.05 cm. All curves have the same normalization (equal to unity for the maximum radial intensity at z=0). Also shown (dotted curves) are 10× the differences between these results and the exact results from the theory of Bagini et al.6.

Fig. 3
Fig. 3

Envelope functions F(ρ, z) and F(-ρ, z) for λ=0.5 μm calculated by approximation (25), with the constants given, for z values (A) 0, (B) 16.5, (C) 20.25, (D) 24.1, (E) 27.75, (F) 31.5 m. Also shown are the resulting intensity distributions, with normalization such that in the geometric limit, F(ρ=0) at the midpoint of the pattern (near 24.1 m) would have the value unity. The integrated intensity remains constant throughout. Note approximate mirroring at 24.10 m. The oscillations dip to nearly zero because of the symmetry of F(ρ, z) about its midpoint.

Equations (78)

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ρ=ρ cos β+z sin β
z=z cos β-ρ sin β.
ρ=ρ cos β-z sin β,
z=z cos β+ρ sin β.
Ein(ρ, z)=E(ρ)z=-z0 exp[ik(z+z0)],
Ein(ρ, z)=E(ρ)z=-z0 exp{i[kl(z+z0)-ktρ]},
kl=k cos β
kt=k sin β,
Ein(ρ, z)=F(ρ, z)12πktρ1/2 exp{i[kl(z+z0)-ktρ]},
Ein(ρ, z)=12 F(ρ, z)H0(2)(ktρ)exp{i[kl(z+z0)-π/4]},
[H0(2)(ktρ)exp(iklz)]
i(^zkl-^ρkt)[H0(2)(ktρ)exp(iklz)]
E(ρ, z)12 Fρ+ktkl z; zH0(2)(ktρ)+F-ρ+ktkl z; zH0(1)(ktρ)×exp{i[kl(z+z0)-π/4]},
E(ρ, z)12 Fρ+ktkl z; z+F-ρ+ktkl z; zJ0(ktρ)-i Fρ+ktkl z; z-F-ρ+ktkl z; zN0(ktρ)×exp{i[kl(z+z0)-π/4]},
1ktF(ρ, z)max F(ρ, z)ρaxial1
ktwc1,
|E(ρ, z)|2 
Fρ+ktkl z; zF*-ρ+ktkl z; zJ02(ktρ)
+Fρ+ktkl z; z-F-ρ+ktkl z; z212πktρ.
E(ρ, z)=FJ0(ktρ)exp{i[kl(z+z0)-π/4]}
|E(ρ, z)|2
=(2πktρ)-1Fρ+ktkl z; z2,(raysconvergingtowardthe opticalaxis)(2πktρ)-1F-ρ+ktkl z; z2,(raysdivergingfromtheopticalaxis).
F(ρ)z=-z0=(1-iz0/zR)-1/2 exp-(ρ)2w02(1-iz0/zR),
Fρ+ktkl z; z
=(1+iz/zR)-1/2 exp-[ρ+(kt/kl)z]2w02(1+iz/zR),
F(ρ)z=0=exp[-(ρ/w0)2].
Fρ+ktkl z; z[1+(z/zR)2]-1/4×exp-[ρ+(kt/kl)z]2w02[1+(z/zR)2],
|E(ρ, z)|2 
[1+(z/zR)2]-1/2 exp-2[ρ2+(kt/kl)2z2]w02[1+(z/zR)2]×J02(ktρ)+2πktρsinh22ρ(kt/kl)zw02[1+(z/zR)2].
|E(ρ, z)|2=[1+(z/zR)2]-1/2 exp-2[ρ2+(kt/kl)2z2]w02[1+(z/zR)2]×[1+(z/zR)2]-1J02ktρ1+(z/zR)2+(2/πktρ)sinh22ρ(kt/kl)zw02[1+(z/zR)2]
F(ρ, z=0)=exp[-(ρ-ρ0)2/w02]
Fρ+ktkl z; z=(1+iz/zR)-1/2×exp-[ρ-ρ0+(kt/kl)z]2w02(1+iz/zR)
E(ρ)z0
 0ρ1.9 cmSoftboundaryat 1.9cmE0 exp{-[(ρ/w)2+iktρ]}1.9ρ3.4 cmwithw=5.3 cmandkt/k=1.1×10-3Softboundaryat3.4 cm0ρ3.4cm.,
F(ρ)z=0
E0(2πktρ)1/2exp [-(ρ/w)2],1.9ρ3.4 cm0otherwisewithsoftedges.
F(ρ)z=0=F0 exp-ρ-ρ0wN,
F(ρ, z)=-ik2πz1/2-F(r)z=0×expi k{[ρ+(ktz/k)2]-r}22zdr,
exp(-iktρ)(πktρ/2)1/2H0(2)(ktρ)exp(-iπ/4).
H0(1),(2)(ktρ)=J0(ktρ)±iN0(ktρ).
2z2+2ρ2+1ρ ρ+k2E(ρ, z)=0.
2z2+2ρ2+14ρ2+k2[ρE(ρ, z)]=0.
2πktρE(ρ, z)=F(ρ, z)(πktρ/2)1/2H0(2)(ktρ)exp(iklz),
2ρ2+14ρ2+kt2(πktρ/2)1/2H0(2)(ktρ)=0
2z2+kl2exp(iklz)=0
2z2+2ρ2+2ikl z+[(πktρ/2)1/2H0(2)(ktρ)]-1
× ddρ [(πktρ/2)1/2H0(2)(ktρ)]} ρF(ρ, z)=0.
[(πktρ/2)1/2H0(2)(ktρ)]-1 ddρ [(πktρ/2)1/2H0(2)(ktρ)]-ikt
2z2+2ρ2+2ikl z-kt ρF(ρ, z)0.
Eincoming(ρ, z)=F(ρ, z)H0(2)(ktρ)exp(iklz),
2z2+2ρ2+1ρ ρ+k2E(ρ, z)=0.
2z2+2ρ2+1ρ ρ+k2H0(2) exp(iklz)=0,
2z2+2ρ2+1ρ ρF(ρ, z)+2ikl F(ρ, z)zH0(2)(ktρ)
+2F(ρ, z)ρ H0(2)(ktρ)ρ=0.
H0(2)(x)dx=-H1(2)(x)
H1(2)(x)i+12xH0(2)(x).
-2ikt+1ρF(ρ, z)ρH0(2)(ktρ).
2z2+2ρ2+2ik zF(ρ, z)0
E(ρ, z)=(1/2)F(ρ; z)H0(2)(ktρ)×exp{i[kl(z+z0)-π/4]},
E(ρ, z)=12 Fρ+ktkl z; zH0(2)(ktρ)×exp{i[kl(z+z0)-π/4]}.
E(ρ, z)(1/2)G(ρ, z)H0(1)(ktρ)×exp{i[kl(z+z0)-π/4]},
ρ=ρ cos β-z sin β
z=z cos β+ρ sin β,
G(ρ, z)=F-ρ+ktkl z; z.
E(ρ, z)1/2Fρ+ktk z; zH0(2)(ktρ)+F-ρ+ktk z; zH0(1)(ktρ)×exp{i[kl(z+z0)-π/4]}
E(ρ, z)1/2Fρ+ktk z; z+F-ρ+ktk z; zJ0(ktρ)-iFρ+ktk z; z-F-ρ+ktk z; zN0(ktρ)×exp{i[kl(z+z0)-π/4]}.
E(ρ, z)(1/2)Fρ+ktk z; z+F-ρ+ktk z; zJ0(ktρ)-iFρ+ktk z; z-F-ρ+ktk z; z×(2/πktρ)1/2sin(ktρ-π/4)×exp{i[kl(z+z0)-π/4]}.
|E(ρ, z)|2Fρ+ktk z; z×F*-ρ+ktk z; zJ02(ktρ)+Fρ+ktk z; z-F-ρ+ktk z; z2(2πktρ)-1.
|E(ρ, z)|2(2πktρ)-1[1+(z/zR)2]-1/2×exp- 2[ρ-(kt/k)z]2w02[1+(z/zR)2],
|E(ρ, z)|exact2=[1+(z/zR)2]-1×exp- 2[ρ2+(ktz/k)2]w02[1+(z/zR)2]×Jmktρ1+i(z/zR)2.
Jmktρ1+i(z/zR)1+iz/zR2πktρ1/2 expiktρ1+i(z/zR)-(m+1/2)π/2[P(x)+iQ(x)]
P(x)=1-(4m2-1)(4m2-9)128x2+ ,
Q(x)=(4m2-1)8x+ ,
x=ktρ1+i(z/zR).
Jmktρ1+i(z/zR)2
[1+(z/zR)2]1/22πktρ×exp4ρ(kt/k)zw02[1+(z/zR)2]|P+iQ|2.
|E(ρ, z)|exact2|E(ρ, z)|2|P+iQ|2,
|P+iQ|2=1-4m2-14 zzRktρ+4m2-18×1ktρ-2+(4m2-1)(4m2-5)32×zzRktρ2+ .

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