Abstract

The propagation of Bessel–Gauss, generalized Bessel–Gauss, and modified Bessel–Gauss beams, for which the exact form of the optical fields is known, is analyzed according to the approximate theory developed previously by the authors [J. Opt. Soc. Am. 17, 1021 (2000)]. Approximations are developed for the fields themselves that are highly accurate and yet are simple in their form and physical description. A set of simple equations is developed, which directly give the parameters describing an image beam following passage through a perfect lens of focal length f, starting with any of the above-mentioned object beams. Ray propagation for these types of beams is described, and it is specifically noted that the intensity maxima do not follow straight paths, while the auxiliary F (ρ, z) function in fact does follow straight paths.

© 2001 Optical Society of America

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References

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  1. R. M. Herman, T. A. Wiggins, “Bessel-like beams modulated by arbitrary radial functions,” J. Opt. Soc. Am. A 17, 1021–1032 (2000).
    [CrossRef]
  2. V. Bagini, F. Frezza, M. Santarieso, G. Schettini, G. Spagnolos, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
  3. C. Palma, G. Cincotti, G. Guattari, M. Santorieso, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
    [CrossRef]
  4. S. A. Self, “Focusing of spherical Gaussian beams,” Appl. Opt. 22, 658–661 (1983).
    [CrossRef] [PubMed]
  5. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

2000 (1)

1996 (2)

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

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

1983 (1)

Bagini, V.

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

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

Cincotti, G.

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

Frezza, F.

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

Guattari, G.

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

Herman, R. M.

Palma, C.

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

Santarieso, M.

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

Santorieso, M.

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

Schettini, G.

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

Self, S. A.

Spagnolos, G.

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

Wiggins, T. A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

Appl. Opt. (1)

J. Mod. Opt. (2)

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

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

J. Opt. Soc. Am. A (1)

Other (1)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

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Figures (3)

Fig. 1
Fig. 1

Parameters describing the object and image fields for GBG beams passing through lens L of focal length f located at z=0.

Fig. 2
Fig. 2

Negative relative deviations (%) in the radial positions of maximum intensity compared with the corresponding straight rays followed by the maxima of |F(ρ, z)|2. The beam waist is set at z=0, with k=107 m-1 and w0=0.5 mm. (a) BG beam with kt=2×104 m-1, (b) GBG beam, with parameters as in (a), with t=-1.0 m (a=2 mm at z=0), (c) MBG beam, with parameters as in (b), except that kt=0 (a=a0=2 mm).

Fig. 3
Fig. 3

Negative relative deviations, as in Fig. 2, for (curve a) BG beam (N=2), with parameters as for Fig. 2(a), and (curve b) Bessel–super-Gauss beam, with N=5 and other parameters as in curve a.

Equations (36)

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E(ρ, z)12Fρ+ktkl z; z+F-ρ+ktkl z; z×J0(ktρ)-i Fρ+ktkl z; z-F-ρ+ktkl z; zN0(ktρ)×exp{i[kl(z-z0)-π/4]},
[(1/ktF)(F/ρ)]ρ=01.
|E(ρ, z)|2Fρ+ktkl z; zF*-ρ+ktkl z; z×J02(ktρ)+Fρ+ktkl z; z-F-ρ+ktkl z; z212πktρ.
F(ρ, z)=2πw021/41+i z-szR-1/2×exp-[ρ+(kt/k)(z-t)]2w02[1+i(z-s)/zR],
F(ρ, z)=2πw021/41+i z+szR-1/2×exp-[ρ+(kt/k)(z+t)]2w02[1+i(z+s)/zR].
F(ρ, z)=2πw021/41+i z-szR-1/2×exp-(ρ-a0)2w02[1+i(z-s)/zR],
zR=kw022,zR=kw022.
kt=ka0t-s,
kt=ka0s-t,
1/t+1/t=1/f.
kt/kt=-t/t,
s/f-1=s/f-1(s/f-1)2+(zR/f )2,
w0[1+(s/zR)2]1/2=w0[1+(s/zR)2]1/2.
J0(ktρ)2πktρ1/2cos(ktρ-π/4),
N0(ktρ)2πktρ1/2sin(ktρ-π/4).
E(ρ, z)=122πw021/4(1+iz/zR)-1/2π2 ktρ-1/2exp-ρ2+(kt/k)2(z-t)2w02(1+iz/zR)+i(klz-π/4)
×exp-2(kt/k)ρ(z-t)w02(1+iz/zR)[cos(ktρ-π/4)-i sin(ktρ-π/4)]+exp2(kt/k)ρ(z-t)w02(1+iz/zR)[cos(ktρ-π/4)+i sin(ktρ-π/4)].
exp-2(kt/k)ρ(z-t)w02(1+z2/zR2)exp-iktρ1+tz/zR21+z2/zR2+iπ/4+exp2(kt/k)ρ(z-t)w02(1+z2/zR2)expiktρ1+tz/zR21+z2/zR2-iπ/4.
2cosh2(kt/k)ρ(z-t)w02(1+z2/zR2)cosktρ1+zt/zR21+z2/zR2-π/4+i sinh2(kt/k)ρ(z-t)w02(1+z2/zR2)sinktρ1+zt/zR21+z2/zR2-π/4.
|E(ρ, z)|2=2πw021/2[1+(z/zR)2]-1/2π2 ktρ-1×exp-2[ρ2+(kt/k)2(z-t)2]w02[1+(z/zR)2]×cos2ktρ1+zt/zR21+z2/zR2-π/4+sinh22(kt/k)ρ(z-t)w02(1+z2/zR2).
|E(ρ, z)|22πw021/2[1+(z/zR)2]-1/2×exp-2[ρ2+(kt/k)2(z-t)2]w02[1+(z/zR)2]×1+zt/zR21+z2/zR2J02ktρ1+zt/zR21+z2/zR2+2πktρsinh22(kt/k)ρ(z-t)w02(1+z2/zR2).
ρmax=12 (kt/k)|z-t|1+1-w2(kt/k)2(z-t)21/2
ρmax=(a0/2){1+[1-(w/a0)2]1/2}
F(ρ, z=0)=const.×exp(-|ρ/w0|N).
2πktr1/2exp±iktr1+tz/zR21+z2/zR2-π/4
ktlocal=kt1+tz/zR21+z2/zR2kt.
kl=(k2-kt2)1/2,
-ik2πΔzr drdϕ2πktr1/2exp[±i(ktr-π/4)]×expikΔz+r2+ρ2-2rρ cos ϕ2Δz,
2π-ik2πΔz2πkt1/2i1/2expik-kt22kΔz
×expikρ22Δzr1/2dr expik2Δzr-ktk Δz2
×12π02πdϕ exp-ikrρ cos ϕΔz.
12πdϕ exp-ikrρ cos ϕΔz
12πdϕ exp(-iktρ cos ϕ)=J0(ktρ).
2(kt/kt)1/2J0(ktρ)exp(iklΔz)exp(ikρ2/2Δz)
×-ik2πΔz1/2dr expik2Δzr-ktk Δz2,
21+tz/zR21+z2/zR21/2J0kt1+tz/zR21+z2/zR2ρexp(iklΔz).

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