Abstract

Fractal zone plates (FZPs), i.e., zone plates with a fractal structure, are described. The focusing properties of this new type of zone plate are compared with those of conventional Fresnel zone plates. It is shown that the axial irradiance exhibited by the FZP has self-similarity properties that can be correlated to those of the diffracting aperture.

© 2003 Optical Society of America

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References

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  1. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1982).
  2. C. Allain and M. Coiltre, Phys. Rev. B 33, 3566 (1986).
    [CrossRef]
  3. J. Uozumi and T. Asakura, in Current Trends in Optics, J. C. Dainty, ed. (Academic, London, 1994), pp. 189–196.
  4. G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woederman, Nature 402, 138 (1999).
    [CrossRef]
  5. G. H. C. New, M. A. Yates, J. P. Woederman, and G. S. McDonald, Opt. Commun. 193, 261 (2001).
    [CrossRef]
  6. J. Courtial and M. J. Padgett, Phys. Rev. Lett. 85, 5320 (2000).
    [CrossRef]
  7. O. Trabocchi, S. Granieri, and W. D. Furlan, J. Mod. Opt. 48, 1247 (2001).
    [CrossRef]
  8. A. D. Jaggard and D. L. Jaggard, Opt. Commun. 158, 141 (1998).
    [CrossRef]
  9. M. Berry and S. Klein, J. Mod. Opt. 43, 2139 (1996).
    [CrossRef]
  10. A. Boidin, J. Opt. Soc. Am. 42, 60 (1952).

2001 (2)

G. H. C. New, M. A. Yates, J. P. Woederman, and G. S. McDonald, Opt. Commun. 193, 261 (2001).
[CrossRef]

O. Trabocchi, S. Granieri, and W. D. Furlan, J. Mod. Opt. 48, 1247 (2001).
[CrossRef]

2000 (1)

J. Courtial and M. J. Padgett, Phys. Rev. Lett. 85, 5320 (2000).
[CrossRef]

1999 (1)

G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woederman, Nature 402, 138 (1999).
[CrossRef]

1998 (1)

A. D. Jaggard and D. L. Jaggard, Opt. Commun. 158, 141 (1998).
[CrossRef]

1996 (1)

M. Berry and S. Klein, J. Mod. Opt. 43, 2139 (1996).
[CrossRef]

1986 (1)

C. Allain and M. Coiltre, Phys. Rev. B 33, 3566 (1986).
[CrossRef]

1952 (1)

Allain, C.

C. Allain and M. Coiltre, Phys. Rev. B 33, 3566 (1986).
[CrossRef]

Asakura, T.

J. Uozumi and T. Asakura, in Current Trends in Optics, J. C. Dainty, ed. (Academic, London, 1994), pp. 189–196.

Berry, M.

M. Berry and S. Klein, J. Mod. Opt. 43, 2139 (1996).
[CrossRef]

Boidin, A.

Coiltre, M.

C. Allain and M. Coiltre, Phys. Rev. B 33, 3566 (1986).
[CrossRef]

Courtial, J.

J. Courtial and M. J. Padgett, Phys. Rev. Lett. 85, 5320 (2000).
[CrossRef]

Furlan, W. D.

O. Trabocchi, S. Granieri, and W. D. Furlan, J. Mod. Opt. 48, 1247 (2001).
[CrossRef]

Granieri, S.

O. Trabocchi, S. Granieri, and W. D. Furlan, J. Mod. Opt. 48, 1247 (2001).
[CrossRef]

Jaggard, A. D.

A. D. Jaggard and D. L. Jaggard, Opt. Commun. 158, 141 (1998).
[CrossRef]

Jaggard, D. L.

A. D. Jaggard and D. L. Jaggard, Opt. Commun. 158, 141 (1998).
[CrossRef]

Karman, G. P.

G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woederman, Nature 402, 138 (1999).
[CrossRef]

Klein, S.

M. Berry and S. Klein, J. Mod. Opt. 43, 2139 (1996).
[CrossRef]

Mandelbrot, B. B.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1982).

McDonald, G. S.

G. H. C. New, M. A. Yates, J. P. Woederman, and G. S. McDonald, Opt. Commun. 193, 261 (2001).
[CrossRef]

G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woederman, Nature 402, 138 (1999).
[CrossRef]

New, G. H. C.

G. H. C. New, M. A. Yates, J. P. Woederman, and G. S. McDonald, Opt. Commun. 193, 261 (2001).
[CrossRef]

G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woederman, Nature 402, 138 (1999).
[CrossRef]

Padgett, M. J.

J. Courtial and M. J. Padgett, Phys. Rev. Lett. 85, 5320 (2000).
[CrossRef]

Trabocchi, O.

O. Trabocchi, S. Granieri, and W. D. Furlan, J. Mod. Opt. 48, 1247 (2001).
[CrossRef]

Uozumi, J.

J. Uozumi and T. Asakura, in Current Trends in Optics, J. C. Dainty, ed. (Academic, London, 1994), pp. 189–196.

Woederman, J. P.

G. H. C. New, M. A. Yates, J. P. Woederman, and G. S. McDonald, Opt. Commun. 193, 261 (2001).
[CrossRef]

G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woederman, Nature 402, 138 (1999).
[CrossRef]

Yates, M. A.

G. H. C. New, M. A. Yates, J. P. Woederman, and G. S. McDonald, Opt. Commun. 193, 261 (2001).
[CrossRef]

J. Mod. Opt. (2)

O. Trabocchi, S. Granieri, and W. D. Furlan, J. Mod. Opt. 48, 1247 (2001).
[CrossRef]

M. Berry and S. Klein, J. Mod. Opt. 43, 2139 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

Nature (1)

G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woederman, Nature 402, 138 (1999).
[CrossRef]

Opt. Commun. (2)

G. H. C. New, M. A. Yates, J. P. Woederman, and G. S. McDonald, Opt. Commun. 193, 261 (2001).
[CrossRef]

A. D. Jaggard and D. L. Jaggard, Opt. Commun. 158, 141 (1998).
[CrossRef]

Phys. Rev. B (1)

C. Allain and M. Coiltre, Phys. Rev. B 33, 3566 (1986).
[CrossRef]

Phys. Rev. Lett. (1)

J. Courtial and M. J. Padgett, Phys. Rev. Lett. 85, 5320 (2000).
[CrossRef]

Other (2)

J. Uozumi and T. Asakura, in Current Trends in Optics, J. C. Dainty, ed. (Academic, London, 1994), pp. 189–196.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1982).

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Figures (4)

Fig. 1
Fig. 1

Diagrams of the generation of binary function qζ for (a) a Fresnel zone plate with periods ps=pN,S for N=2 and several values of S and (b) its associated FZP. In this representation open and filled segments correspond to the values 1 and 0, respectively, of the generating binary function.

Fig. 2
Fig. 2

(a) Fresnel zone plate and (b) the associated FZP generated from the 1D functions in Fig. 1 for S=3. The generating process consists in rotating the whole structure around one extreme after the change of variables in Eq. (2).

Fig. 3
Fig. 3

Normalized irradiance versus axial coordinate u obtained top for a FZP at four stages of growth and bottom, for its associated Fresnel zone plate. In all cases N=3.

Fig. 4
Fig. 4

Log plot of axial irradiances versus reduced axial coordinate uN obtained from the upper part of Fig. 3.

Equations (9)

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IR=2πλR20apr0exp-iπλRr02r0dr02,
ς=r0/a2-0.5,
I0u=4π2u2-0.5+0.5qςexp-i2πuςdς2.
qς=qZPς,p=rectςrectmodς+p-1/2,p/p,
qς=qFZPς,N,S=i=0SqZPς,22N-1i.
pN,S=22N-1S
I0FZPu,N,S=4 sin2πu2N-1S×i=1Ssin22πNu/2N-1isin22πu/2N-1i.
I0ZPu,N,S=4 sin2πu2N-1S×sin2π2N-1S+1u/2N-1Susin22πu/2N-1S.
I0NFZPuN,N,S=4 sin2πuN×i=1Ssin22πN2N-1S-iuNsin22π2N-1S-iuN.

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