Abstract

This paper is a continuation of our previous publication on the stationary-phase-method analysis of lens axicons [J. Opt. Soc. Am. A 152383 (1998)]. Systems with spherical aberration up to the fifth order are studied. Such lens axicons in their simplest versions can be made either as a setup composed of two separated third-order spherical-aberration lenses of opposite powers or as a doublet consisting of one third-order diverging element and one fifth-order converging element. The axial intensity distribution and the central core width turn out to be improved and become almost constant. The results obtained are compared with the numerical evaluation of the corresponding diffraction integral.

© 1999 Optical Society of America

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References

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  1. R. M. Herman, T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8, 932–942 (1991).
    [CrossRef]
  2. R. M. Herman, T. A. Wiggins, “High-efficiency diffractionless beams of constant size and intensity,” Appl. Opt. 33, 7297–7306 (1994).
    [CrossRef] [PubMed]
  3. Z. Jaroszewicz, “Axicons. Design and propagation properties,” in Research and Development Treatises, M. Pluta, ed. (Polish Chapter of SPIE, Warsaw, 1997), Vol. 5.
  4. T. Aruga, “Generation of long-range nondiffracting narrow light beams,” Appl. Opt. 36, 3762–3768 (1997).
    [CrossRef] [PubMed]
  5. Z. Jaroszewicz, J. Morales, “Lens axicons: systems composed of a diverging aberrated lens and a converging perfect lens,” J. Opt. Soc. Am. A 15, 2383–2390 (1998).
    [CrossRef]
  6. A. T. Friberg, “Stationary-phase analysis of generalized axicons,” J. Opt. Soc. Am. A 13, 743–750 (1996).
    [CrossRef]
  7. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), pp. 91–135.
  8. M. Abramowitz, I. A. Stegun, eds. Handbook of Mathematical Functions (Dover, New York, 1972), pp. 355–389.
  9. Z. Jaroszewicz, J. F. Roman Dopazo, C. Gomez-Reino, “Uniformization of the axial intensity of diffraction axicons by polychromatic illumination,” Appl. Opt. 35, 1025–1031 (1996).
    [CrossRef] [PubMed]

1998

1997

1996

1994

1991

Aruga, T.

Friberg, A. T.

Gomez-Reino, C.

Herman, R. M.

Jaroszewicz, Z.

Morales, J.

Roman Dopazo, J. F.

Stamnes, J.

J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), pp. 91–135.

Wiggins, T. A.

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Figures (8)

Fig. 1
Fig. 1

Geometry of a doublet composed of an aberrated diverging lens and an aberrated converging lens.

Fig. 2
Fig. 2

Central-core-width distribution along the focal segment of a doublet consisting of an aberrated diverging lens and an aberrated converging lens according to Eq. (7) (R1=2.5 mm, R2=5 mm, d1=100 mm, d2=200 mm, dmax=214.286 mm, and λ=632.8 nm) for various values of the paraxial focus (s=50 mm, s=76.923 mm, s=95.24 mm, and s=120 mm). The upper part of the plot limited by lines L1 (s=0) and L2 (s=dmax) is the region of central-core-width distributions for r<rmax (forward case). The curves labeled A and B delimit the part of region within which the curves with the best characteristics can be found. The lower part of the plot limited by lines L1 (s=0) and L2 (s=dmax) is the region of central-core-width distributions for r>rmax (backward case). The central-core-width distribution for curve A is shown for illustration.

Fig. 3
Fig. 3

Axial intensity distribution of the lens axicon according to Eq. (8) (R1=2.5 mm, R2=5 mm, d1=100 mm, d2=200 mm, dmax=214.286 mm, and λ=632.8 nm). The smooth curve corresponds to the intensity of the stationary wave, the oscillating plot is the numerical evaluation of the diffraction integral made for axicon with paraxial focus s=76.923 mm (corresponding to the curve A in Fig. 2), and the top of the vertical line is the value of the axial intensity distribution in dmax equal to 3.006 determined according to the modified SPM version [Eq. (15)].

Fig. 4
Fig. 4

Axial intensity of a doublet consisting of an aberrated diverging lens and an aberrated converging lens corresponding to curve A in Fig. 2 (R1=2.5 mm, R2=5 mm, d1=100 mm, d2=200 mm, dmax=214.286 mm, s=76.923 mm, and λ=632.8 nm): (a) curve resulting from the stationary-phase method according to Eq. (8); for comparison, the curve for the analogous third-order lens axicon is shown (dashed curve); (b) numerical evaluation of the diffraction integral.

Fig. 5
Fig. 5

Central-core-width distribution along the focal segment of a doublet consisting of an aberrated diverging lens and an aberrated converging lens corresponding to curve A in Fig. 2 (R1=2.5 mm, R2=5 mm, d1=100 mm, d2=200 mm, dmax=214.286 mm, s=76.923 mm, and λ=632.8 nm): (a) for the stationary wave only, (b) numerical evaluation of the diffraction integral.

Fig. 6
Fig. 6

Axial intensity of a doublet consisting of an aberrated diverging lens and an aberrated converging lens corresponding to curve B of Fig. 2 (R1=2.5 mm, R2=5 mm, d1=100 mm, d2=200 mm, dmax=214.286 mm, s=95.24 mm, and λ=632.8 nm): (a) curve resulting from the stationary-phase method according to Eq. (8), (b) numerical evaluation of the diffraction integral.

Fig. 7
Fig. 7

Central-core-width distribution along the focal segment of a doublet consisting of an aberrated diverging lens and an aberrated converging lens corresponding to curve B of Fig. 2 (R1=2.5 mm, R2=5 mm, d1=100 mm, d2=200 mm, dmax=214.286 mm, s=95.24 mm, and λ=632.8 nm): (a) for the stationary wave only, (b) numerical evaluation of the diffraction integral.

Fig. 8
Fig. 8

Geometry of third-order-spherical-aberration wave-front propagation at a distance d.

Equations (26)

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ϕ(r)=-r2/2s+βr4-γr6,1/s=1/f1-1/f2,
I(ρ, z)=I0(k/z)2R1R2 exp[ikf(r)]J0(krρ/z)rdr2,
f(r)=ϕ(r)+r2/2z,
I(ρ, z)2πkrs2J02(krsρ/z)/z2|f(rs)|,
f(rs)=0.
rs1,2=rmax[1±1-L(z)/Lmax]1/2,
L(z)=1/s-1/z,Lmax=1/s-1/dmax=2β2/3γ,
I(ρ, z)πkrmax22z2{Lmax[Lmax-L(z)]}1/2×J02kρrmaxz{[Lmax-Lmax-L(z)]/Lmax}1/2.
ρ0(z)=cλz/rmax[1-s(dmax-z)/z(dmax-s)]1/2,
Inorm(0, z)=I(0, z)/I(0, d1)=(d1/z)3/2[(dmax-d1)/(dmax-z)]1/2,
dmax=(d24-d14)/(d23-d13).
Lmax=(d12+d22)/d1d2(d1+d2).
rmax=R1,2/{[1-1-L(d1,2)/Lmax]}1/2
β=Lmax/2rm2,γ=Lmax/6rm4.
z1,2=zc[1±(1-25sdmax/16zc2)1/2],
zc=(3s/4+dmax/2).
s=4dmax/9,zinf=5dmax/6.
I(0, dmax)/I(0, d2)=πC12k1/3211/3d22|f(rs)|-2/3×{Lmax[Lmax-L(d2)]}1/2/dmax2,
ϕ(r)=-r2/2f1-βr4.
dϕdrr-rd.
r=r(1-d/f1)-4βdr3.
r=r/(1-d/f1)+Ar3+Br5.
A=4βd/(1-d/f1)4,B=48β2d2/(1-d/f1)7.
dϕdrr-rd,
ϕ(r)=-r2/2(f1-d)-βr4-γr6,
β=β/(1-d/f1)4,γ=8β2d/(1-d/f1)7.

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