Abstract

A description is given of axicons obtained as a tandem of one diverging lens that has third-order spherical aberration and one perfect converging lens. The asymptotic representation obtained with the help of the nonuniform stationary phase method allows the determination of such principal features of the focal segment as the intensity distribution and the width of the central core along the optical axis. The obtained results are compared with the numerical evaluation of the corresponding diffraction integral. The analyzed system contains four special cases: the doublet of both lenses, the setup of the perfect converging lens placed in the front of the aberrated diverging lens, the setup of the aberrated diverging lens placed in the primary focal plane of the perfect converging lens, and the defocused Galilean telescope. A short discussion of the aberrated-lens bending factor as well as of choice of the most convenient system geometry is also included.

© 1998 Optical Society of America

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References

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    [CrossRef]
  2. L. M. Soroko, “Axicons and meso-optical imaging devices,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1989), Vol. 27, pp. 109–160.
  3. Z. Jaroszewicz, “Axicons, design and propagation properties,” Vol. 5 of Research and Development Treatises, M. Pluta, ed. (Polish Chapter of the International Society for Optical Engineering, Warsaw, 1997).
  4. J. Fujiwara, “Optical properties of conic surfaces. I. Reflecting cone,” J. Opt. Soc. Am. 52, 287–292 (1962).
    [CrossRef]
  5. M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
    [CrossRef]
  6. A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
    [CrossRef] [PubMed]
  7. J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, S. Bará, “Nonparaxial designing of generalized axicons,” Appl. Opt. 31, 5326–5330 (1992).
    [CrossRef] [PubMed]
  8. A. T. Friberg, “Stationary-phase analysis of generalized axicons,” J. Opt. Soc. Am. A 13, 743–750 (1996).
    [CrossRef]
  9. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  10. L. R. Staronski, J. Sochacki, Z. Jaroszewicz, A. Kolodziejczyk, “Lateral distribution and flow of energy in uniform-intensity axicons,” J. Opt. Soc. Am. A 9, 2091–2094 (1992).
    [CrossRef]
  11. A. C. S. van Heel, “Modern alignment devices,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1960), Vol. 1, pp. 288–329.
  12. R. B. Gwynn, D. A. Christensen, “Method for accurate optical alignment using diffraction rings from lenses with spherical aberration,” Appl. Opt. 32, 1210–1215 (1993).
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    [CrossRef]
  16. I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980), Chap. 6.
  17. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, 1986), pp. 91–135.
  18. 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).
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  19. M. Abramowitz, I. A. Stegun, eds. Handbook of Mathematical Functions (Dover, New York, 1972), pp. 355–389.
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    [CrossRef] [PubMed]

1997 (1)

1996 (2)

1995 (1)

1994 (1)

1993 (1)

1992 (2)

1991 (1)

1989 (1)

1987 (1)

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

1986 (1)

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

1962 (1)

1954 (1)

Aruga, T.

Bará, S.

Christensen, D. A.

Cuadrado, J. M.

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

Durnin, J.

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]

Friberg, A. T.

Fujiwara, J.

Gomez-Reino, C.

Gómez-Reino, C.

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

Gradshtein, I. S.

I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980), Chap. 6.

Gwynn, R. B.

Herman, R. M.

Jaroszewicz, Z.

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, London, 1984), pp. 152–162.

Kolodziejczyk, A.

McLeod, J. H.

Miceli, J. J.

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

Pérez, M. V.

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

Roman Dopazo, J. F.

Rosen, J.

Ryzhik, I. M.

I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980), Chap. 6.

Salik, B.

Sochacki, J.

Soroko, L. M.

L. M. Soroko, “Axicons and meso-optical imaging devices,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1989), Vol. 27, pp. 109–160.

Stamnes, J.

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

Staronski, L. R.

Turunen, J.

van Heel, A. C. S.

A. C. S. van Heel, “Modern alignment devices,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1960), Vol. 1, pp. 288–329.

Vasara, A.

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, London, 1984), pp. 152–162.

Wiggins, T. A.

Yariv, A.

Appl. Opt. (5)

J. Opt. Soc. Am. (2)

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

Opt. Acta (1)

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
[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 (7)

A. C. S. van Heel, “Modern alignment devices,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1960), Vol. 1, pp. 288–329.

L. M. Soroko, “Axicons and meso-optical imaging devices,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1989), Vol. 27, pp. 109–160.

Z. Jaroszewicz, “Axicons, design and propagation properties,” Vol. 5 of Research and Development Treatises, M. Pluta, ed. (Polish Chapter of the International Society for Optical Engineering, Warsaw, 1997).

I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980), Chap. 6.

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

M. Abramowitz, I. A. Stegun, eds. Handbook of Mathematical Functions (Dover, New York, 1972), pp. 355–389.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, London, 1984), pp. 152–162.

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Figures (4)

Fig. 1
Fig. 1

Illustration of the geometry of a system composed of an aberrated diverging lens placed in front of a perfect converging lens.

Fig. 2
Fig. 2

Illustration of the geometry of a system composed of a perfect converging lens placed in front of an aberrated diverging lens. The lower half refers to the lack of aberration and shows mutual position of both lenses, whereas the upper half illustrates the formation of the focal segment in the aberration’s presence.

Fig. 3
Fig. 3

Axial intensity of the doublet consisting of an aberrated diverging lens and a perfect converging lens (R1=2.5 mm, R2 =5 mm, β=6.667×10-5 mm-3, s=85.71 mm, d1 =100 mm, d2=200 mm, and λ0=632.8 nm): (a) numerical evaluation of the diffraction integral (the curve corresponds to the intensity of the stationary wave), (b) distribution resulting from the stationary phase method according to Eqs. (21)–(24), (c) difference between the two distributions.

Fig. 4
Fig. 4

Radius of the central spot along the focal segment of the doublet consisting of an aberrated diverging lens and a perfect converging lens (R1=2.5 mm, R2=5 mm, β=6.667 ×10-5 mm-3, s=85.71 mm, d1=100 mm, d2=200 mm, and λ0=632.8 nm): (a) for the stationary wave only, (b) numerical evaluation of the diffraction integral.

Equations (51)

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t(r)=exp[ik(r2/2 f1+βr4)]r(R1, R2)0otherwise.
U(r, z)=CU0kil0t(r)J0(krr/z)exp(ikr2/2l)rdr,
0J0(bx)J0(cx)exp(iax2)xdx=i2aJ0(bc/2a)exp[-i(b2+c2)/4a]
U(ρ, z)=U0kfi(lf+zf-zl)×expikl+z+ρ2(f-l)2(lf+zf-zl)×0t(r)J0krρflf+zf-zl×exp-ikr2(z-f)2(lf+zf-zl)rdr,
I(k)=R1R2g(r)exp[ikf(r)]dr,
g(r)=J0krρflf+zf-zlr,
f(r)=r221f1-(z-f)(lf+zf-zl)+βr4.
I(k)=2πk|f(rs)|1/2g(rs)exp(i{kf(rs)+sgn [f(rs)]π/4})+g(r)ikf(r)exp[ikf(r)]R1R2forrs(R1, R2),
I(k)=π2k|f(rs)|1/2g(rs)exp[i sgn f(rs)π/4]×exp[ikf(rs)]forrs=R1R2,
I(k)=g(r)ikf(r)exp[ikf(r)]R1R2forrsR1, R2,
f(rs)=0.
rs={[(z-f)/(lf+zf-zl)-1/f1]/4β}1/2;
ISt(ρ, z)=I0πkf24β(lf+zf-zl)2J02{kρf[(f1z-f1f-lf-zf+zl)/4βf1(lf+zf-zl)3]1/2}.
ISt(0, z)=I0πkf2/4β(z-t)2(f-l)2,
ρ0=cλ0[4βf1(f-l)3/f2(f1+l-f)]1/2×[(z-t)3/(z-s)]1/2,
ρ0,MIN=cλ0[27βf1(f-l)3(s-t)2/f2(f1+l-f)]1/2,
zMIN=3s/2-t/2.
dφdr=-sin θ-rz(r),
1/f1(r)=1/f1+4βr2,
1/z=1/f-1/[l+f1(r)].
1d1,2=f1+l-f+(l-f)4βf1R1,22f(f1+l)+lf4βf1R1,22.
R1=R2d1/d2
φ(r)=-r2/2s+βr4
1/s-1/d1,2=4βR1,22.
β=(d2-d1)/4d1d2(R22-R12),
s=d1d2(R22-R12)/(d2R22-d1R12).
rs=[(z-s)/4βzs]1/2.
I(ρ, z)IEl+IEu+ISt+2(IElIEu)1/2 cos ΦElEu+2(IStIEl)1/2 cos ΦStEl+2(IStIEu)1/2 cos ΦStEu,
IEu=I0d22(d2-z)2J02kρR2z,
IEl=I0d12(d1-z)2J02kρR1z,
ISt=I0πkJ02{kρ[(z-s)/4βsz3]1/2}/4βz2
ΦEuEl=k[(R22-R12)(1/2z-1/2s)+β(R24-R14)],
ΦStEl=k[R22(1/2z-1/2s)+βR24+(s-z)2/16βz2s2]-3π/4,
ΦStEu=k[R12(1/2z-1/2s)+βR14+(s-z)2/16βz2s2]-3π/4.
I(ρ, z)IEl+IEu+2(IElIEu)1/2 cos(ΦElEu-π).
I(ρ, z)=I0(k/z)2R1R2 exp[ikf(r)]J0(krρ/z)rdr2,
ρ0=cλ0[4βsz3/(z-s)]1/2.
ρ0,MIN=cλ0(27βs3)1/2,
zMIN=3s/2.
Inorm=1+A1/2/Δ2-(2A1,2)1/2/Δ,
Δ1,2MIN=(2A1,2)1/2=d1,228βπk1/2=K1,2d1,2d1,22πD1,2R1/2,
β=132 f13n(n-1)n+2n-1q2+4(n+1)pq+(3n+2)(n-1)p2+n3n-1,
q1,2=n-1n+2{2(n+1)±[32βf13n(n+2)-n2(4n-1)/(n-1)2]1/2}.
d1,2l=-d1,22(f-l)21+NA1,22n+1n(n-1)q+3n+22np.
β=(d2-d1)/4f2(R22-R12),
s=d1,2-aR1,22.
ρ0=cλ0f2[4β/(z-s)]1/2.
R1,2=R1,2d1,2/f.
Δf1,f,t,ls,z,d1,2.
ISt(0, z)I0πk4βz2ff12,
ρ0cλ0f1f2[4βsz3/(z-s)]1/2.

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