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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 25 — Dec. 11, 2006
  • pp: 11958–11963
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Fractal photon sieve

Fernando Giménez, Juan A. Monsoriu, Walter D. Furlan, and Amparo Pons  »View Author Affiliations


Optics Express, Vol. 14, Issue 25, pp. 11958-11963 (2006)
http://dx.doi.org/10.1364/OE.14.011958


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Abstract

A novel focusing structure with fractal properties is presented. It is a photon sieve in which the pinholes are appropriately distributed over the zones of a fractal zone plate. The focusing properties of the fractal photon sieve are analyzed. The good performance of our proposal is demonstrated experimentally with a series of images obtained under white light illumination. It is shown that compared with a conventional photon sieve, the fractal photon sieve exhibits an extended depth of field and a reduced chromatic aberration.

© 2006 Optical Society of America

1. Introduction

A renewed interest in diffractive focusing elements has been experienced by the scientific community in the last years because these elements are essential in image forming setups that are used in THz tomography [1

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

], soft X-ray microscopy [2

2. 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]

, 3

3. 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]

], astronomy [4

4. R. Hyde, “Eyeglass. 1. Very large aperture diffractive telescopes,” Appl. Opt. 38, 4198–4212 (1999). [CrossRef]

, 5

5. G. Andersen, “Large optical photon sieve,” Opt. Lett. 30, 2976–2978 (2005). [CrossRef] [PubMed]

], and lithography [6

6. R. Menon, D. Gil, G. Barbastathis, and H. Smith, “Photon sieve lithography,” J. Opt. Soc. Am. A 22, 342–345 (2005). [CrossRef]

]. Following this trend, our group has recently introduced a new type of 2D photonic-image-forming structures: the Fractal Zone Plates (FZPs) [7–9

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

]. When illuminated by a plane wavefront, a FZP produces multiple foci along the optical axis. The internal structure of each focus exhibits a characteristic fractal structure reproducing the self-similarity of the originating FZP. We have shown that the number of foci and their relative amplitude can be modified with the FZP design [9

9. J. A. Monsoriu, G. Saavedra, and W. D. Furlan, “Fractal zone plates with variable lacunarity,” Opt. Express 12, 4227–4234 (2004). [CrossRef] [PubMed]

]. It has been suggested, and proved very recently [10

10. W. D. Furlan, G. Saavedra, and J. A. Monsoriu are preparing a paper to be called “Imaging with fractal zone plates.”

], that this property can be profited in image forming systems to obtain an enhancement of the depth of field.

In this paper a new diffractive element: the fractal photon sieve (FPS) is presented. It consists of hundreds of small circular holes distributed over the zones of a FZP. We show that FPS potentially improves the performance of FZP in several aspects. Two features that greatly reduce the fabrication constraints and allow fractal focusing of electromagnetic waves in a wide range of the spectrum are the following: 1) The hole diameter can be bigger than the width of the underlying zone, allowing a better resolution as compared with a FZP made with the same technology. 2) A FPS has no connected regions and then, it can be fabricated in a single surface without any substrate. In addition, the number and the distribution of holes per zone can be modified to improve the suppression of secondary maxima and higher orders of diffraction.

2. Fractal photon sieve design

As was discussed in Ref. [7

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

], a FZP can be constructed following the same procedure performed to design conventional Fresnel zone plates. Let us review the concept: As it is well known, a Fresnel zone plate consists of alternately transparent and opaque zones whose radii are proportional to the square root of the natural numbers, thus it can be generated from a 1-D structure (see Fig. 1, upper part) defined by the periodic function q(ς), by performing a change of coordinates ς=(r 0/a)2 and by rotating the transformed 1-D function around one of its extremes. The result is a Fresnel zone plate having a radial coordinate r o and an outermost ring of radius a [see Fig. 2(a)]. In a similar way a FZP is constructed by replacing the periodic function used in the generation of a Fresnel zone plate, by a 1-D fractal structure, as for example the triadic Cantor set shown in Fig. 1 (lower part). The corresponding zone plate with fractal profile is represented in Fig. 2(b). It has been shown [7

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

, 10

10. W. D. Furlan, G. Saavedra, and J. A. Monsoriu are preparing a paper to be called “Imaging with fractal zone plates.”

] that the irradiance along the optical axis produced by a FZP presents multiple foci with a distinctive fractal structure. The position, size and depth of the foci depend on the fractal level and on the lacunarity of the encoded fractal structure. Based on a FZP, the FPS here proposed combines the features of FZP with the concept of photon sieve. A FPS has essentially the same structure of a FZP but instead of transparent rings the corresponding zones have been broken up into isolated circular holes. The result is shown in Fig. 2(c). In the construction procedure we adopted the results reported in Ref [3

3. 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]

] where it has been shown that for a photon sieve constructed with a Fresnel zone plate structure, the diameter d of the holes in each ring of width w of has an optimum value for the effective contribution to the focus. This value is given by d=1.53w.

Fig. 1. Periodic and fractal 1-D structures to be used to generate zone plates.
Fig. 2. Comparison between (a) Fresnel zone plate, (b) FZP, and (c) FPS.

3. Focusing and imaging properties of FPS

Let us consider the irradiance along the optical axis, z, given by an optical system having a 2-D pupil function p(ro, ϕ), expressed in canonical polar coordinates, when it is illuminated by a plane wave of wavelength λ:

I(z)=(2πλz)20a02πp(ro,ϕ)exp(iπλzro2)rodrodϕ2
(1)

The axial irradiance in Eq. (1) can be conveniently expressed in terms of a single radial integral, by performing first the azimuthal average of the pupil function p(ro, ϕ):

po(ro)=12π0ap(ro,ϕ)dϕ.
(2)

Then Eq. (1) can be rewritten as

I(z)=(2πλz)20apo(ro)exp(iπλzro2)rodro2.
(3)

It is important to note that the azimuthal average of the pupil is the final responsible of the behavior of the axial irradiance, instead of the pupil itself. Therefore, from this point of view a photon sieve can be designed to achieve any convenient apodization by a proper modulation of the density of pinholes in each zone in which the distribution of holes can be either regular or random.

To compare the performance of a FPS with the conventional FZP [7

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

] we used Eq. (2) and Eq. (3) to compute the axial irradiances provided by the corresponding pupil functions [see Figs. 2(b) and 2(c)]. The result is shown in Fig. 3. The parameters used to calculate these plots were a=2mm and λ=632.8nm. The number of holes in the FPS was 650 and the density of holes per zone (i.e. the ratio between the area covered by the holes and the total area of the zone) was aproximately 90%. The minimun diameter in the outermost ring was d=1.53w=0.0572mm. As shown in Ref. [7

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

] the focal structure of a FZP along the optical axis is characterized by the coincidence of the central peak with the one obtained for a conventional Fresnel zone plate, but the internal structure of the focus reproduce the self-similarity of the zone plate.

Fig. 3. Normalized irradiance distributions along the optical axis produced by the zone plates in Figs. 2(b) and 2(c), (a) FZP, (b) and FPS (both computed for S=3).

Fig. 4. Images obtained with FPS (left) and with Fresnel photon sieve (right) in a 4 f setup (λ=568nm). Images (a) and (b) were obtained at the green image plane. In (c) and (d) the object-image distance was kept constant and the defocus was obtained by moving the photon sieve 45 mm towards the CCD.
Fig. 5. Axial irradiances computed for (a) the Fresnel photon sieve and (b) the FPS used in Fig. 4 for λ=647nm (red line, R), 568nm (green line, G), and 488nm (blue line, B).

4. Conclusions

We have proposed a photon sieve with fractal focusing properties. The structure of the sieve is based on FZP, and therefore, their behavior for the first diffraction order is similar. The main difference is that the higher-orders obtained with the FZP are highly reduced with the sieve. The numerical and experimental results provided in this paper show the focusing and image-forming properties of FPS. With a FPS a substantial increase in the depth of field and a noticeable reduction in the chromatic aberration can be obtained with respect to a Fresnel photon sieve of the same focal distance. In our opinion the FPS offer a great versatility in design pupils for particular focusing properties because of there are many parameters, as the fractal dimension, the number of zones, the diameter and density of holes per zone, that can be varied conveniently to obtain a specific result. Another advantage of FPS over the FZP arises from the fabrication point of view: the FPS can be constructed in a single structure without any supporting substrate. Furthermore, there are potential improvements in the design of FPS such as increasing the efficiency by the use of composite zones [14

14. M. J. Simpson and A. G. Michette, “Imaging properties of modified Fresnel zone plates,” Opt. Acta 31, 403–413 (1984). [CrossRef]

]. Therefore applications in optical and non-optical wavelength range in which graded amplitude pupils are difficult or even impossible to construct will greatly will benefit from these new focusing structures.

Acknowledgments

This research has been supported by the following grants:

- DPI 2003-04698, Plan Nacional I+D+I, Ministerio de Ciencia y Tecnología. Spain.

- Programa de Incentivo a la Investigación de la UPV 2005, Vicerrectorado de Innovación y Desarrollo, Universidad Politécnica de Valencia, Spain.

- MTM 2004-06015-C02-01, DGI (Spain) and FEDER Project.

References and links

1.

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

2.

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]

3.

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]

4.

R. Hyde, “Eyeglass. 1. Very large aperture diffractive telescopes,” Appl. Opt. 38, 4198–4212 (1999). [CrossRef]

5.

G. Andersen, “Large optical photon sieve,” Opt. Lett. 30, 2976–2978 (2005). [CrossRef] [PubMed]

6.

R. Menon, D. Gil, G. Barbastathis, and H. Smith, “Photon sieve lithography,” J. Opt. Soc. Am. A 22, 342–345 (2005). [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. and Photon. News 14, 31 (2003). [CrossRef]

9.

J. A. Monsoriu, G. Saavedra, and W. D. Furlan, “Fractal zone plates with variable lacunarity,” Opt. Express 12, 4227–4234 (2004). [CrossRef] [PubMed]

10.

W. D. Furlan, G. Saavedra, and J. A. Monsoriu are preparing a paper to be called “Imaging with fractal zone plates.”

11.

Q. Cao and J. Jahns, “Focusing analysis of the pinhole photon sieve: individual far field model,” J. Opt. Soc. Am. A 19, 2387–2393 (2002). [CrossRef]

12.

Q. Cao and J. Jahns, “Non paraxial model for the focusing of high-numerical-aperture- photon sieves,” J. Opt. Soc. Am. A 20, 1005–1012 (2003). [CrossRef]

13.

The spectral sensitivity of the camera used in the experiment can be looked up at the site http://astrosurf.com/buil/350d/350d.htm.

14.

M. J. Simpson and A. G. Michette, “Imaging properties of modified Fresnel zone plates,” Opt. Acta 31, 403–413 (1984). [CrossRef]

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(050.1970) Diffraction and gratings : Diffractive optics
(220.1000) Optical design and fabrication : Aberration compensation

ToC Category:
Diffraction and Gratings

History
Original Manuscript: July 7, 2006
Revised Manuscript: October 16, 2006
Manuscript Accepted: November 5, 2006
Published: December 11, 2006

Citation
Fernando Giménez, Juan A. Monsoriu, Walter D. Furlan, and Amparo Pons, "Fractal photon sieve," Opt. Express 14, 11958-11963 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-11958


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References

  1. S. Wang and X. Zhang, "Terahertz tomographic imaging with a Fresnel lens," Opt. Photonics News 13, 59 (2002). [CrossRef]
  2. 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]
  3. 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]
  4. R. Hyde, "Eyeglass. 1. Very large aperture diffractive telescopes," Appl. Opt. 38, 4198-4212 (1999). [CrossRef]
  5. G. Andersen, "Large optical photon sieve," Opt. Lett. 30, 2976-2978 (2005). [CrossRef] [PubMed]
  6. R. Menon, D. Gil, G. Barbastathis, and H. Smith, "Photon sieve lithography," J. Opt. Soc. Am. A 22, 342-345 (2005). [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. and Photon. News 14, 31 (2003). [CrossRef]
  9. J. A. Monsoriu, G. Saavedra, and W. D. Furlan, "Fractal zone plates with variable lacunarity," Opt. Express 12, 4227-4234 (2004). [CrossRef] [PubMed]
  10. W. D. Furlan, G. Saavedra and J. A. Monsoriu are preparing a paper to be called "Imaging with fractal zone plates."
  11. Q. Cao and J. Jahns, "Focusing analysis of the pinhole photon sieve: individual far field model," J. Opt. Soc. Am. A 19, 2387-2393 (2002). [CrossRef]
  12. Q. Cao and J. Jahns, "Non paraxial model for the focusing of high-numerical-aperture- photon sieves," J. Opt. Soc. Am. A 20, 1005-1012 (2003). [CrossRef]
  13. The spectral sensitivity of the camera used in the experiment can be looked up at the site http://astrosurf.com/buil/350d/350d.htm.
  14. M. J. Simpson and A. G. Michette, "Imaging properties of modified Fresnel zone plates," Opt. Acta 31, 403-413 (1984). [CrossRef]

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