Abstract

Fractal zone plates (FZPs), i.e., zone plates with fractal structure, have been recently introduced in optics. These zone plates are distinguished by the fractal focusing structure they provide along the optical axis. In this paper we study the effects on this axial response of an important descriptor of fractals: the lacunarity. It is shown that this parameter drastically affects the profile of the irradiance response along the optical axis. In spite of this fact, the axial behavior always has the self-similarity characteristics of the FZP itself.

© 2004 Optical Society of America

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References

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  • |

  1. J. Ojeda-Castañeda and C. Gómez-Reino, Eds., Selected papers on zone plates (SPIE Optical Engineering Press, Washington, 1996).
  2. S. Wang, X. Zhang, �??Terahertz tomographic imaging with a Fresnel lens,�?? Opt. Photon. News 13, 59 (2002).
    [CrossRef]
  3. Y Wang, W. Yun, and C. Jacobsen, �??Achromatic Fresnel optics for wideband extreme-ultraviolet and X-ray imaging,�?? Nature 424, 50-53 (2003).
    [CrossRef] [PubMed]
  4. L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, �??Sharper images by focusing soft x-rays with photon sieves,�?? Nature 414, 184-188 (2001).
    [CrossRef] [PubMed]
  5. Q. Cao and J. Jahns, �??Modified Fresnel zone plates that produce sharp Gaussian focal spots,�?? J. Opt. Soc. Am. A 20, 1576-1581 (2003).
    [CrossRef]
  6. Q. Cao and J. Jahns, �??Comprehensive focusing analysis of various Fresnel zone plates,�?? J. Opt. Soc. Am. A 21, 561-571 (2004).
    [CrossRef]
  7. G. Saavedra, W.D. Furlan, and J.A. Monsoriu, �??Fractal zone plates,�?? Opt. Lett. 28, 971-973 (2003).
    [CrossRef] [PubMed]
  8. W.D. Furlan, G. Saavedra, and J.A. Monsoriu, �??Fractal zone plates produce axial irradiance with fractal profile,�?? Opt.& Photon. News 28, 971-973 (2003).
  9. J.A. Davis, L. Ramirez, J.A. Rodrigo Martín-Romo, T. Alieva, and M.L. Calvo, �??Focusing properties of fractal zone plates: experimental implementation with a liquid-crystal display,�?? Opt. Lett. 29, 1321-1323 (2004).
    [CrossRef] [PubMed]
  10. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1982).
  11. A.D. Jaggard and D.L. Jaggard, �??Cantor ring diffractals,�?? Opt. Commun. 158, 141�??148 (1998).
    [CrossRef]
  12. L. Zunino and M. Garavaglia, �??Fraunhofer diffraction by Cantor fractals with variable lacunarity,�?? J. Mod. Opt. 50, 717-727 (2003).
    [CrossRef]
  13. Y. Sakurada, J. Uozumi, and T Asakura, �??Fresnel diffraction by 1-D regular fractals,�?? Pure Appl. Opt. 1, 29�??40 (1992).
    [CrossRef]
  14. H. Melville and G. F. Milne, �??Optical trapping of three-dimensional structures using dynamic holograms,�?? Opt. Express 11, 3562-3567 (2003) <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3562">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3562</a>.
    [CrossRef] [PubMed]

J. Mod. Opt.

L. Zunino and M. Garavaglia, �??Fraunhofer diffraction by Cantor fractals with variable lacunarity,�?? J. Mod. Opt. 50, 717-727 (2003).
[CrossRef]

J. Opt. Soc. Am. A

Nature

Y Wang, W. Yun, and C. Jacobsen, �??Achromatic Fresnel optics for wideband extreme-ultraviolet and X-ray imaging,�?? Nature 424, 50-53 (2003).
[CrossRef] [PubMed]

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, �??Sharper images by focusing soft x-rays with photon sieves,�?? Nature 414, 184-188 (2001).
[CrossRef] [PubMed]

Opt. Commun.

A.D. Jaggard and D.L. Jaggard, �??Cantor ring diffractals,�?? Opt. Commun. 158, 141�??148 (1998).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Photon. News

S. Wang, X. Zhang, �??Terahertz tomographic imaging with a Fresnel lens,�?? Opt. Photon. News 13, 59 (2002).
[CrossRef]

Opt.& Photon. News

W.D. Furlan, G. Saavedra, and J.A. Monsoriu, �??Fractal zone plates produce axial irradiance with fractal profile,�?? Opt.& Photon. News 28, 971-973 (2003).

Pure Appl. Opt.

Y. Sakurada, J. Uozumi, and T Asakura, �??Fresnel diffraction by 1-D regular fractals,�?? Pure Appl. Opt. 1, 29�??40 (1992).
[CrossRef]

Other

J. Ojeda-Castañeda and C. Gómez-Reino, Eds., Selected papers on zone plates (SPIE Optical Engineering Press, Washington, 1996).

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

Supplementary Material (7)

» Media 1: GIF (332 KB)     
» Media 2: GIF (342 KB)     
» Media 3: GIF (324 KB)     
» Media 4: GIF (905 KB)     
» Media 5: GIF (310 KB)     
» Media 6: GIF (498 KB)     
» Media 7: GIF (519 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

Schemes for the generation of the FZP binary function q(ζ) for N=4 up to S=2.γ is the scale factor and ε is the parameter that characterizes the lacunarity.

Fig. 2.
Fig. 2.

FZPs generated with the following parameters: (a) γ=4/19, ε=1/19; (b) γ=1/7, ε=1/7; and (c) γ=1/16, ε=1/4. In all cases: N=4, S=1 and ε = εR . The animations fig2a.gif (332kB), Fig. 2(b).gif (341kB) and Fig. 2(c).gif (323kB) show the evolution of the resulting FZPs for a variable lacunarity, ε varying from zero to εmax . Note that εmax is different in each case (see Eq. (5)), being: a) 3/38; b) 3/14; and c) 3/8.

Fig. 3.
Fig. 3.

(a) FZP generated with the following parameters: γ=1/7, ε=εR =1/7, N=4, and S=2 (compare it with Fig. 2b). (b) Normalized axial irradiances obtained with the FZP in a) and with the FZP in Fig. 2b). The animation Fig. 3.gif (905kB) shows the evolution of the FZP for a variable lacunarity, ε varying from zero to ε3 and the corresponding axial irradiances for the above mentioned FZPs.

Fig. 4.
Fig. 4.

Gray–scale representation of the axial irradiance (in dB) plotted as a function of the normalized axial coordinate and the lacunarity (twist plots). Left and right correspond to the pupils shown in Fig. 2 and the corresponding ones with S=2, respectively.

Fig. 5.
Fig. 5.

Left: Autocorrelation function qε (ζ) ⊗ qε (ζ) for the FZPs shown in Fig 2, for ε=εR (black) and for ε=εmax (red). Right: C(ε) for the same FZPs. The animations Fig. 5(a).gif (311kB), Fig. 5(b). gif (498kB) and Fig. 5(c). gif (519kB), show the evolution of these functions for a variable ε.

Fig. 6.
Fig. 6.

C(ε) for the pupils shown in Fig. 2 (red) and the corresponding ones with S=2 (blue).

Equations (8)

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I ( R ) = ( 2 π λR ) 2 0 a p ( r o ) exp ( i π λR r o 2 ) r o d r o 2 .
ς = ( r o a ) 2 0.5 ,
I o ( u ) = 4 π 2 u 2 0.5 + 0.5 q ( ς ) exp ( i 2 πuς ) d ς 2 .
D = ln ( N ) / ln ( γ ) .
ε max = 1 N 2 .
ε R = 1 N 1 .
C ( ε ) = 0 I ε R ( u ) · I ε ( u ) du 0 I ε R 2 ( u ) du 0 I ε 2 ( u ) du .
C ( ε ) = 1 1 [ q ε R ( ς ) q ε R ( ς ) ] . [ q ε ( ς ) q ε ( ς ) ] 1 1 [ q ε R ( ς ) q ε R ( ς ) ] 2 1 1 [ q ε ( ς ) q ε ( ς ) ] 2 .

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