Abstract

In this paper we use the Cantor Dust to design zone plates based on a two-dimensional fractal for the first time. The pupil function that defines the coined Cantor Dust Zone Plates (CDZPs) can be written as a combination of rectangle functions. Thus CDZPs can be considered as photon sieves with rectangular holes. The axial irradiances produced by CDZPs of different fractal orders are obtained analitically and experimentally, analyzing the influence of the fractality. The transverse irradiance patterns generated by this kind of zone plates has been also investigated.

© 2013 OSA

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2013 (1)

A. Calatayud, V. Ferrando, F. Giménez, W. Furlan, G. Saavedra, and J. Monsoriu, “Fractal square zone plates,” Opt. Commun.286, 42–45 (2013).
[CrossRef]

2012 (3)

2010 (3)

2009 (1)

J. Alda, J. M. Rico-García, F. J. Salgado-Remacha, and L. M. Sanchez-Brea, “Diffractive performance of square Fresnel zone plates,” Opt. Commun.282, 3402–3407 (2009).
[CrossRef]

2007 (4)

2006 (1)

2005 (1)

D. Hai-Tao, W. Xin, and X. Ke-Shu, “Focusing properties of fractal zone plates with variable lacunarity: experimental studies based on liquid crystal on silicon,” Chinese Phys. Lett.22, 2851–2854 (2005).
[CrossRef]

2004 (3)

2003 (1)

2002 (1)

2001 (1)

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 (London)414, 184–188 (2001).
[CrossRef]

1982 (1)

L. J. Janicijevic, “Diffraction characteristics of square zone plates,” J. Opt.13, 199–206 (1982).
[CrossRef]

Adelung, R.

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 (London)414, 184–188 (2001).
[CrossRef]

Alda, J.

J. Alda, J. M. Rico-García, F. J. Salgado-Remacha, and L. M. Sanchez-Brea, “Diffractive performance of square Fresnel zone plates,” Opt. Commun.282, 3402–3407 (2009).
[CrossRef]

F. J. González, J. Alda, B. Ilic, and G. D. Boreman, “Infrared antennas coupled to lithographic Fresnel zone plate lenses,” Appl. Opt.43, 6067–6073 (2004).
[CrossRef] [PubMed]

Alieva, T.

Andersen, G.

G. Andersen, “Membrane photon sieve telescopes,” Appl. Phys.49, 63916394 (2010).

Berndt, R.

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 (London)414, 184–188 (2001).
[CrossRef]

Bomba, J.

Boreman, G. D.

Calatayud, A.

A. Calatayud, V. Ferrando, F. Giménez, W. Furlan, G. Saavedra, and J. Monsoriu, “Fractal square zone plates,” Opt. Commun.286, 42–45 (2013).
[CrossRef]

Calvo, M. L.

Cao, Q.

Chen, J.

X. Ge, Z. Wang, K. Gao, D. Wang, Z. Wu, J. Chen, Z. Pan, K. Zhang, Y. Hong, P. Zhu, and Z. Wu, “Use of fractal zone plates for transmission x-ray microscopy,” Anal. Bioanal. Chem.404, 1303–1309 (2012).
[CrossRef] [PubMed]

Coutaz, J.-L.

Czerwiski, A.

Davis, J. A.

Ferrando, V.

A. Calatayud, V. Ferrando, F. Giménez, W. Furlan, G. Saavedra, and J. Monsoriu, “Fractal square zone plates,” Opt. Commun.286, 42–45 (2013).
[CrossRef]

Furlan, W.

A. Calatayud, V. Ferrando, F. Giménez, W. Furlan, G. Saavedra, and J. Monsoriu, “Fractal square zone plates,” Opt. Commun.286, 42–45 (2013).
[CrossRef]

Furlan, W. D.

Gao, K.

X. Ge, Z. Wang, K. Gao, D. Wang, Z. Wu, J. Chen, Z. Pan, K. Zhang, Y. Hong, P. Zhu, and Z. Wu, “Use of fractal zone plates for transmission x-ray microscopy,” Anal. Bioanal. Chem.404, 1303–1309 (2012).
[CrossRef] [PubMed]

Garet, F.

Ge, X.

X. Ge, Z. Wang, K. Gao, D. Wang, Z. Wu, J. Chen, Z. Pan, K. Zhang, Y. Hong, P. Zhu, and Z. Wu, “Use of fractal zone plates for transmission x-ray microscopy,” Anal. Bioanal. Chem.404, 1303–1309 (2012).
[CrossRef] [PubMed]

Giménez, F.

A. Calatayud, V. Ferrando, F. Giménez, W. Furlan, G. Saavedra, and J. Monsoriu, “Fractal square zone plates,” Opt. Commun.286, 42–45 (2013).
[CrossRef]

F. Giménez, W. D. Furlan, and J. A. Monsoriu, “Lacunar fractal photon sieves,” Opt. Commun.277, 1–4 (2007).
[CrossRef]

F. Giménez, J. A. Monsoriu, W. D. Furlan, and A. Pons, “Fractal photon sieve,” Opt. Express14, 11958–11963 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-14-25-11958 .
[CrossRef] [PubMed]

González, F. J.

Hai-Tao, D.

D. Hai-Tao, W. Xin, and X. Ke-Shu, “Focusing properties of fractal zone plates with variable lacunarity: experimental studies based on liquid crystal on silicon,” Chinese Phys. Lett.22, 2851–2854 (2005).
[CrossRef]

Harm, S.

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 (London)414, 184–188 (2001).
[CrossRef]

Hong, Y.

X. Ge, Z. Wang, K. Gao, D. Wang, Z. Wu, J. Chen, Z. Pan, K. Zhang, Y. Hong, P. Zhu, and Z. Wu, “Use of fractal zone plates for transmission x-ray microscopy,” Anal. Bioanal. Chem.404, 1303–1309 (2012).
[CrossRef] [PubMed]

Hua, Y.

Ilic, B.

Jahns, J.

Janicijevic, L. J.

L. J. Janicijevic, “Diffraction characteristics of square zone plates,” J. Opt.13, 199–206 (1982).
[CrossRef]

Johnson, R. L.

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 (London)414, 184–188 (2001).
[CrossRef]

Kelemen, L.

Ke-Shu, X.

D. Hai-Tao, W. Xin, and X. Ke-Shu, “Focusing properties of fractal zone plates with variable lacunarity: experimental studies based on liquid crystal on silicon,” Chinese Phys. Lett.22, 2851–2854 (2005).
[CrossRef]

Kipp, L.

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 (London)414, 184–188 (2001).
[CrossRef]

Li, H.

Liu, M.

Liu, Y.

Makowski, M.

Monsoriu, J.

A. Calatayud, V. Ferrando, F. Giménez, W. Furlan, G. Saavedra, and J. Monsoriu, “Fractal square zone plates,” Opt. Commun.286, 42–45 (2013).
[CrossRef]

Monsoriu, J. A.

Ormos, P.

Pan, Z.

X. Ge, Z. Wang, K. Gao, D. Wang, Z. Wu, J. Chen, Z. Pan, K. Zhang, Y. Hong, P. Zhu, and Z. Wu, “Use of fractal zone plates for transmission x-ray microscopy,” Anal. Bioanal. Chem.404, 1303–1309 (2012).
[CrossRef] [PubMed]

Pons, A.

Ramirez, L.

Rico-García, J. M.

J. Alda, J. M. Rico-García, F. J. Salgado-Remacha, and L. M. Sanchez-Brea, “Diffractive performance of square Fresnel zone plates,” Opt. Commun.282, 3402–3407 (2009).
[CrossRef]

Rodrigo Martín-Romo, J. A.

Saavedra, G.

Sakdinawat, A.

Salgado-Remacha, F. J.

J. Alda, J. M. Rico-García, F. J. Salgado-Remacha, and L. M. Sanchez-Brea, “Diffractive performance of square Fresnel zone plates,” Opt. Commun.282, 3402–3407 (2009).
[CrossRef]

Sanchez-Brea, L. M.

J. Alda, J. M. Rico-García, F. J. Salgado-Remacha, and L. M. Sanchez-Brea, “Diffractive performance of square Fresnel zone plates,” Opt. Commun.282, 3402–3407 (2009).
[CrossRef]

Seemann, R.

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 (London)414, 184–188 (2001).
[CrossRef]

Shi, L.

Siemion, A.

Skibowski, M.

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 (London)414, 184–188 (2001).
[CrossRef]

Suszek, J.

Sypek, M.

Valkai, S.

Wang, D.

X. Ge, Z. Wang, K. Gao, D. Wang, Z. Wu, J. Chen, Z. Pan, K. Zhang, Y. Hong, P. Zhu, and Z. Wu, “Use of fractal zone plates for transmission x-ray microscopy,” Anal. Bioanal. Chem.404, 1303–1309 (2012).
[CrossRef] [PubMed]

Wang, Y.

Wang, Z.

X. Ge, Z. Wang, K. Gao, D. Wang, Z. Wu, J. Chen, Z. Pan, K. Zhang, Y. Hong, P. Zhu, and Z. Wu, “Use of fractal zone plates for transmission x-ray microscopy,” Anal. Bioanal. Chem.404, 1303–1309 (2012).
[CrossRef] [PubMed]

Wu, Z.

X. Ge, Z. Wang, K. Gao, D. Wang, Z. Wu, J. Chen, Z. Pan, K. Zhang, Y. Hong, P. Zhu, and Z. Wu, “Use of fractal zone plates for transmission x-ray microscopy,” Anal. Bioanal. Chem.404, 1303–1309 (2012).
[CrossRef] [PubMed]

X. Ge, Z. Wang, K. Gao, D. Wang, Z. Wu, J. Chen, Z. Pan, K. Zhang, Y. Hong, P. Zhu, and Z. Wu, “Use of fractal zone plates for transmission x-ray microscopy,” Anal. Bioanal. Chem.404, 1303–1309 (2012).
[CrossRef] [PubMed]

Xie, C.

Xin, W.

D. Hai-Tao, W. Xin, and X. Ke-Shu, “Focusing properties of fractal zone plates with variable lacunarity: experimental studies based on liquid crystal on silicon,” Chinese Phys. Lett.22, 2851–2854 (2005).
[CrossRef]

Zhang, B.

Zhang, K.

X. Ge, Z. Wang, K. Gao, D. Wang, Z. Wu, J. Chen, Z. Pan, K. Zhang, Y. Hong, P. Zhu, and Z. Wu, “Use of fractal zone plates for transmission x-ray microscopy,” Anal. Bioanal. Chem.404, 1303–1309 (2012).
[CrossRef] [PubMed]

Zhao, D.

Zhu, P.

X. Ge, Z. Wang, K. Gao, D. Wang, Z. Wu, J. Chen, Z. Pan, K. Zhang, Y. Hong, P. Zhu, and Z. Wu, “Use of fractal zone plates for transmission x-ray microscopy,” Anal. Bioanal. Chem.404, 1303–1309 (2012).
[CrossRef] [PubMed]

Zhu, X.

Anal. Bioanal. Chem. (1)

X. Ge, Z. Wang, K. Gao, D. Wang, Z. Wu, J. Chen, Z. Pan, K. Zhang, Y. Hong, P. Zhu, and Z. Wu, “Use of fractal zone plates for transmission x-ray microscopy,” Anal. Bioanal. Chem.404, 1303–1309 (2012).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. (1)

G. Andersen, “Membrane photon sieve telescopes,” Appl. Phys.49, 63916394 (2010).

Chinese Phys. Lett. (1)

D. Hai-Tao, W. Xin, and X. Ke-Shu, “Focusing properties of fractal zone plates with variable lacunarity: experimental studies based on liquid crystal on silicon,” Chinese Phys. Lett.22, 2851–2854 (2005).
[CrossRef]

J. Opt. (1)

L. J. Janicijevic, “Diffraction characteristics of square zone plates,” J. Opt.13, 199–206 (1982).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nature (London) (1)

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 (London)414, 184–188 (2001).
[CrossRef]

Opt. Commun. (3)

F. Giménez, W. D. Furlan, and J. A. Monsoriu, “Lacunar fractal photon sieves,” Opt. Commun.277, 1–4 (2007).
[CrossRef]

J. Alda, J. M. Rico-García, F. J. Salgado-Remacha, and L. M. Sanchez-Brea, “Diffractive performance of square Fresnel zone plates,” Opt. Commun.282, 3402–3407 (2009).
[CrossRef]

A. Calatayud, V. Ferrando, F. Giménez, W. Furlan, G. Saavedra, and J. Monsoriu, “Fractal square zone plates,” Opt. Commun.286, 42–45 (2013).
[CrossRef]

Opt. Express (3)

Opt. Lett. (8)

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Figures (5)

Fig. 1
Fig. 1

(a) Geometrical construction of the Cantor Dust distribution up to third order. (b) Cantor dust varying the distribution sides quadratically with respect to the transverse coordinates. (c) Pupil function q(0, ȳ0) —Eq. (2)— obtained from the order S = 3 Cantor Dust distribution.

Fig. 2
Fig. 2

Normalized axial irradiance for the CDZP and its associated FSZP.

Fig. 3
Fig. 3

Scheme of the experimental setup used for the simulation of the lenses.

Fig. 4
Fig. 4

a) Analytical (green) and experimental (red) irradiances provided by different fractal order CDZPs (first order focus) for S = 2. b) Same as in Fig. 4(b) for S = 3; F1, F2, F3 are three different maxima of the compound first order focus. (see Fig. 5)

Fig. 5
Fig. 5

Experimental transverse irradiance (top) compared with the analytically obtained irradiance (bottom) for a fractal order S = 3 CDZP. Transversal planes belong to the first order focus shown in Fig. 4(b) located at z = 10.4cm and two secondary foci at z = 11.3cm and z = 13.6 cm.

Equations (7)

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[ 1 ] [ 1 0 1 0 0 0 1 0 1 ] and [ 0 ] [ 0 0 0 0 0 0 0 0 0 ] .
q ( x ¯ 0 , y ¯ 0 ) = j = 1 N j = 1 N t Sj j RECT j ( x ¯ 0 ) RECT j ( y ¯ 0 ) ,
RECT j ( x ) = [ rect ( N j x ) rect ( N j 1 x ) ] .
I ( x ¯ , y ¯ , u ) = 4 u 2 | 1 1 1 1 q ( x ¯ 0 , y ¯ 0 ) exp { 2 π iu [ ( x 0 2 + y 0 2 ) ( 2 x ¯ x ¯ 0 + 2 y ¯ y ¯ 0 ) ] } d x ¯ 0 d y ¯ 0 | 2 ,
I ( x ¯ , y ¯ , u ) = 1 16 | j N { [ ERF j ( x ¯ , u ) ERF j 1 ( x ¯ , u ) ] × j N t S j j [ ERF j ( y ¯ , u ) ERF j 1 ( y ¯ , u ) ] } | 2 ,
ERF j ( x , u ) = Erf [ ( 1 + i ) π u ( j N + x ) ] + Erf [ ( 1 + i ) π u ( j N x ) ] ,
I ( 0 , 0 , u ) = | j N t S j j { Erf [ ( 1 + i ) π u j N ] Erf [ ( 1 + i ) π u j 1 N ] } | 4 .

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