## Fractal zone plates with variable lacunarity

Optics Express, Vol. 12, Issue 18, pp. 4227-4234 (2004)

http://dx.doi.org/10.1364/OPEX.12.004227

Acrobat PDF (1341 KB)

### 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

## 1. Introduction

2. S. Wang and X. Zhang, “Terahertz tomographic imaging with a Fresnel lens,” Opt. Photon. News **13**, 59 (2002). [CrossRef]

7. G. Saavedra, W.D. Furlan, and J.A. Monsoriu, “Fractal zone plates,” Opt. Lett. **28**, 971–973 (2003). [CrossRef] [PubMed]

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]

7. G. Saavedra, W.D. Furlan, and J.A. Monsoriu, “Fractal zone plates,” Opt. Lett. **28**, 971–973 (2003). [CrossRef] [PubMed]

## 2. Theory

*p(r)*, illuminated by a monochromatic plane wave. Within the Fresnel approximation, this magnitude is given as a function of the axial distance from the pupil plane

*R*, as

*a*is the maximum extent of the pupil function

*p(r)*and

*λ*is the wavelength of the light. For our purposes it is convenient to express the pupil transmittance as function of a new variable defined as

*q*(ζ)=

*p*(

*r*). By using the dimensionless reduced axial coordinate

_{o}*u*=

*a*

^{2}/2

*λR*, the irradiance along the optical axis can be expressed simply in terms of the Fourier transform of

*q*(

*ζ*) as

*F*as

*u*=

*F*/2.

*q*(

*ζ*) holds a fractal structure, it is well known that the Fourier transform preserves fractal properties [10], and then, it is direct to conclude that such pupil will provide irradiance along the optical axis also with a fractal profile.

*q*(

*ζ*) is a periodic function. In a similar way, a FZP results if

*q*(

*ζ*) represents any self-similar (fractal) 1-D function. In particular we will focus our attention in functions

*q*(

*ζ*) constructed from different levels of a polyadic Cantor set.

## 3. Cantor-like FZP design

*initiator*(stage

*S*=0). Next, at stage

*S*=1, the

*generator*of the set is constructed by

*N*(

*N*=4 in the figure) non-overlapping copies of the initiator, each one with a scale

*γ*<1, distributed in a particular way into the unit length segment. At the following stages of the construction of the set (

*S*=2,3,…), the generation process is repeated over and over again for each segment in the previous stage. To characterize the resulting Cantor set, as well as many other fractal structures, one of the most frequently used descriptors is the

*fractal dimension*, defined as

*N*copies into the unit length segment at

*S*=1. In other words, a parameter to specify the lacunarity (or “gapinness”) of the resulting structure is needed. In this work, as in previous papers dealing with Cantor fractals in optics [11

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]

*ε*(see Fig. 1) for this purpose.

*γ*, depends on the value of

_{max}*N*, i.e., 0 ≤

*γ*≤

_{max}*N*

^{-1}. On the other hand, for each value of

*N*and

*y*, there are two extreme values for

*ε*. For the first,

*ε*=0 , the result is the highest lacunar fractal, having the central gap very large while the outer ones become null. The other extreme value of

*ε*is

*ε*that gives the lowest lacunar (or

*regular*) fractal exists between

*zero*and

*. This value of*

*ε*_{max}*ε*, which is obtained by imposing bars and gaps to have the same size at the initiatior stage (as done in Ref. [7

7. G. Saavedra, W.D. Furlan, and J.A. Monsoriu, “Fractal zone plates,” Opt. Lett. **28**, 971–973 (2003). [CrossRef] [PubMed]

*N*=4 and

*S*=1, first by use of Eq. (2), and then by rotating the re-scaled bars around one of the extremes. FZPs for three different values of

*γ*are obtained. The animations in this figure show the evolution of the FZP for values of

*ε*ranging from zero to

*. According to our previous discussion, it can be seen that by changing*

*ε*_{max}*ε*, different structures with the same fractal dimension can be obtained.

## 4. Fractal behavior of the axial irradiance

**28**, 971–973 (2003). [CrossRef] [PubMed]

*axial scale property*to the fact (theoretically supported by Eq. (3) that the axial irradance reproduces the self-similarity of the FZP. In this section we will analyze the influence of the lacunarity on this property.

*u*with period

*u*=1/

_{p}*γ*[7

^{S}**28**, 971–973 (2003). [CrossRef] [PubMed]

*u*/

*u*=

_{p}*γ*. Figure 3(a) shows the FZP constructed with the same parameters as in Fig. 2(b), but for

^{S}u*S*=2. The normalized axial irradiances given by these two pupils (

*S*=1 and

*S*=2) for

*ε*=

*ε*are represented with different colors in Fig. 3(b). The self-similarity between these patterns can be clearly seen: the blue pattern is a magnified version of the red one, and the later is an envelope of the former. The animation in this figure shows the change the FZPs experiences for different values of the lacunarity (

_{r}*ε*ranging from

*ε*=0 to

*ε*), and at the same time, the evolution of the axial irradiance provided by this pupils. It can be seen that the optical irradiance produced by the FZPs is highly influenced by the lacunarity.

_{max}*S*=2 is still modulated by the irradiance for

*S*=1. Although it seems that the self-similarity observed for the regular FZP (

*ε*=

*ε*) is not supported by other values of

_{R}*ε*, this effect is an artifact due to the axial scale used in this figure. Another interesting result which is masked in Fig. 3 by the use of the normalized axial coordinate is the fact that there exists certain axial positions with zero axial irradiance for all values of

*ε*. These effects can be better seen in Fig.4.

*C*(

*ε*) is expected to be a continuous function, having an absolute maximum value at

*ε*=

*ε*. Since the infinite limits in the integrations in Eq. (7) would pose difficulties to the accurate numerical evaluation of

_{R}*C*(

*ε*), a more suitable expression can be obtained by using Eq. (3) and the Rayleigh’s theorem. In this way,

*C*(

*ε*) can be written as

## 5 Conclusions

14. H. Melville and G. F. Milne, “Optical trapping of three-dimensional structures using dynamic holograms,” Opt. Express **11**, 3562–3567 (2003). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | J. Ojeda-Castañeda and C. Gómez-Reino, Eds., |

2. | S. Wang and X. Zhang, “Terahertz tomographic imaging with a Fresnel lens,” Opt. Photon. News |

3. | Y Wang, W. Yun, and C. Jacobsen, “Achromatic Fresnel optics for wideband extreme-ultraviolet and X-ray imaging,” Nature |

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 |

5. | Q. Cao and J. Jahns, “Modified Fresnel zone plates that produce sharp Gaussian focal spots,” J. Opt. Soc. Am. A |

6. | Q. Cao and J. Jahns, “Comprehensive focusing analysis of various Fresnel zone plates,” J. Opt. Soc. Am. A |

7. | G. Saavedra, W.D. Furlan, and J.A. Monsoriu, “Fractal zone plates,” Opt. Lett. |

8. | W.D. Furlan, G. Saavedra, and J.A. Monsoriu, “Fractal zone plates produce axial irradiance with fractal profile,” Opt.& Photon. News |

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. |

10. | B. Mandelbrot, |

11. | A.D. Jaggard and D.L. Jaggard, “Cantor ring diffractals,” Opt. Commun. |

12. | L. Zunino and M. Garavaglia, “Fraunhofer diffraction by Cantor fractals with variable lacunarity,” J. Mod. Opt. |

13. | Y. Sakurada, J. Uozumi, and T Asakura, “Fresnel diffraction by 1-D regular fractals,” Pure Appl. Opt. |

14. | H. Melville and G. F. Milne, “Optical trapping of three-dimensional structures using dynamic holograms,” Opt. Express |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(050.1970) Diffraction and gratings : Diffractive optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 4, 2004

Revised Manuscript: August 24, 2004

Published: September 6, 2004

**Citation**

Juan Monsoriu, Genaro Saavedra, and Walter Furlan, "Fractal zone plates with variable lacunarity," Opt. Express **12**, 4227-4234 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-18-4227

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### References

- J. Ojeda-Castañeda and C. Gómez-Reino, Eds., Selected papers on zone plates (SPIE Optical Engineering Press, Washington, 1996).
- S. Wang, X. Zhang, �??Terahertz tomographic imaging with a Fresnel lens,�?? Opt. Photon. News 13, 59 (2002). [CrossRef]
- 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]
- 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]
- Q. Cao and J. Jahns, �??Comprehensive focusing analysis of various Fresnel zone plates,�?? J. Opt. Soc. Am. A 21, 561-571 (2004). [CrossRef]
- G. Saavedra, W.D. Furlan, and J.A. Monsoriu, �??Fractal zone plates,�?? Opt. Lett. 28, 971-973 (2003). [CrossRef] [PubMed]
- 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).
- 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]
- B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1982).
- A.D. Jaggard and D.L. Jaggard, �??Cantor ring diffractals,�?? Opt. Commun. 158, 141�??148 (1998). [CrossRef]
- L. Zunino and M. Garavaglia, �??Fraunhofer diffraction by Cantor fractals with variable lacunarity,�?? J. Mod. Opt. 50, 717-727 (2003). [CrossRef]
- Y. Sakurada, J. Uozumi, and T Asakura, �??Fresnel diffraction by 1-D regular fractals,�?? Pure Appl. Opt. 1, 29�??40 (1992). [CrossRef]
- 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]

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