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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 18 — Sep. 6, 2004
  • pp: 4227–4234
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Fractal zone plates with variable lacunarity

Juan A. Monsoriu, Genaro Saavedra, and Walter D. Furlan  »View Author Affiliations


Optics Express, Vol. 12, Issue 18, pp. 4227-4234 (2004)
http://dx.doi.org/10.1364/OPEX.12.004227


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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

A renewed interest in zone plates [1

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

] has been experienced during the last years because they are becoming key elements used to obtain images in several scientific and technological areas such as, THz tomography and soft X-ray microscopy [2–6

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

]. With this motivation, we have recently proposed the fractal zone plates (FZPs) as new promising 2D photonic structures [7

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

,8

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).

]. Lately, FZPs were implemented experimentally with a liquid crystal display by Davis et al. [9

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]

]. A FZP can be thought as a conventional zone plate with certain missing zones. The resulting structure is characterized by its fractal profile along the square of the radial coordinate. The axial irradiance provided by a FZP when illuminated with a parallel wavefront presents multiple foci, the main lobe of which coincide with those of the associated conventional zone plate. However, the internal structure of each focus exhibits a characteristic fractal structure, reproducing the self-similarity of the originating FZP. In this way, synthesis of axial irradiances with fractal profile can be achieved easily given the simple theoretical relation between the transmittance of the FZP and their axial response [7

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

].

In this paper, we analyze the axial response of FZPs to a specific design parameter, frequently used as a measure of the “texture” of fractal structures: the lacunarity. In particular, we focus our attention on binary amplitude Cantor-like FZPs. First, some practical considerations about the design of this type of FZPs are investigated, taking into account the physical limits imposed by the different construction parameters. Finally, the axial irradiance provided by FZPs with variable lacunarity is numerically evaluated, and compared with the response of regular FZPs.

2. Theory

Let us consider the irradiance at a given point on the optical axis, provided by a rotationally invariant pupil with an amplitude transmittance 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

I(R)=(2πλR)20ap(ro)exp(iπλRro2)rodro2.
(1)

In Eq. (1), 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

ς=(roa)20.5,
(2)

in such a way that q(ζ)=p(ro). By using the dimensionless reduced axial coordinate u = a 2/2λR, the irradiance along the optical axis can be expressed simply in terms of the Fourier transform of q(ζ) as

Io(u)=4π2u20.5+0.5q(ς)exp(i2πuς)dς2.
(3)

Note that the reduced axial coordinate can also be expressed in terms of the Fresnel number F as u = F/2.

Now, if the pupil function q(ζ) holds a fractal structure, it is well known that the Fourier transform preserves fractal properties [10

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

], and then, it is direct to conclude that such pupil will provide irradiance along the optical axis also with a fractal profile.

The comparison between a conventional zone plate and a FZP can be done taking into account the change of variables in Eq. (2). For a conventional zone plate the function 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

D=ln(N)/ln(γ).
(4)

The construction parameters of a FZP are linked to each other, and also they must satisfy certain constraints. On the one hand, the maximum value of the scale, γmax, depends on the value of N, i.e., 0 ≤ γmaxN -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

εmax=1N2.
(5)

εR=1N1.
(6)

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. 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.

4. Fractal behavior of the axial irradiance

Interesting features about the axial irradiance provided by regular FZPs were previously reported in Ref. [7

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

]. In particular, we called 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.

Since for the regular case the axial irradiance is a periodic function of the coordinate u with period up=1/γS [7

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

], one way to observe the axial fractal behavior of the irradiance is by representing it as a function of the reduced axial coordinate as u/up = γSu. Figure 3(a) shows the FZP constructed with the same parameters as in Fig. 2(b), but for S=2. The normalized axial irradiances given by these two pupils (S=1 and S=2) for ε=εr 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 (ε ranging from ε=0 to εmax), 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.

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.

Nevertheless, it is clear that in all cases the irradiance for S=2 is still modulated by the irradiance for S=1. Although it seems that the self-similarity observed for the regular FZP (ε=εR) is not supported by other values of 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.

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.

To analyze how the axial irradiance changes with the lacunarity we used a generalization of the correlation coefficient defined by Sakurada et al. [13

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

] for measuring the self-similarity. In our case the axial irradiances for a variable lacunarity were correlated with the same function computed for ε=εR, since this particular value of ε gives the lowest lacunar FZP. Thus, the correlation coefficient we used is given by

C(ε)=0IεR(u)·Iε(u)du0IεR2(u)du0Iε2(u)du.
(7)

From its definition, the function C(ε) is expected to be a continuous function, having an absolute maximum value at ε = εR. Since the infinite limits in the integrations in Eq. (7) would pose difficulties to the accurate numerical evaluation of 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

C(ε)=11[qεR(ς)qεR(ς)].[qε(ς)qε(ς)]11[qεR(ς)qεR(ς)]211[qε(ς)qε(ς)]2.
(8)

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

5 Conclusions

FZPs with variable lacunarity has been extensively analyzed. The construction restrictions and the interrelations between the different parameters have been investigated. As a result, it was shown that the lacunarity has a dramatic effect on the axial irradiance provided by different FZPs with the same fractal dimension, but the essential aspects of the self-similarity are preserved. A new parameter that correlates the axial irradiances given by FZPs with different lacunarity was proposed and its fractal behavior was reported.

The present study brings new lights on the powerful potential applications of FZPs, especially in scientific and technological areas where conventional zone plates have been successfully applied. Particularly, recent proposals of optical tweezers use phase filters to facilitate the trapping of particles in three-dimensional structures [14

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

], spatial light modulators can be employed to display tunable FZPs producing focal spots that could be useful for this purposes. On the other hand, the non-uniform distribution of FZPs focal points along the optical axis could be exploited in the design of multifocal contact lenses for the correction of presbyopia. In this case a mechanism to control the diffraction efficiency of the FZP should be first developed. Currently we are investigating several properties of non-binary FZPs like the influence of optical aberrations, and polychromatic illumination. In particular, the attributes of rotationally non-symmetric FZPs, as elliptical or vortex FZPs, will be published in a forthcoming paper.

Acknowledgments

This research has been supported by the following grants: DPI 2003-04698 (Plan Nacional I + D + I Ministerio de Ciencia y Tecnología, Spain) and GV04B-186 (Generalitat Valenciana, Spain).

References and links

1.

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

2.

S. Wang and 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). [CrossRef] [PubMed]

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

  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]

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