Abstract

We present multifractal zone plates (MFZPs) as what is to our knowledge a new family of diffractive lenses whose structure is based on the combination of fractal zone plates (FZPs) of different orders. The typical result is a composite of two FZPs with the central one having a first-order focal length f surrounded by outer zones with a third-order focal length f. The focusing properties of different members of this family are examined and compared with conventional composite Fresnel zone plates. It is shown that MFZPs improve the axial resolution and also give better performance under polychromatic illumination.

© 2010 Optical Society of America

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2009 (3)

2008 (1)

2007 (2)

2006 (2)

S. H. Tao, X.-C. Yuan, J. Lin, and R. Burge, “Sequence of focused optical vortices generated by a spiral fractal zone plates,” Appl. Phys. Lett. 89, 031105 (2006).
[Crossref]

F. Giménez, J. A. Monsoriu, W. D. Furlan, and A. Pons, “Fractal photon sieves,” Opt. Express 14, 11958–11963 (2006).
[Crossref] [PubMed]

2005 (3)

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

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

H.-T. Dai, X. Wang, and K.-S. Xu, “Focusing properties of fractal zone plates with variable lacunarity: experimental studies based on liquid crystal on silicon,” Chin. Phys. Lett. 22, 2851–2854 (2005).
[Crossref]

2004 (2)

2003 (3)

2002 (1)

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

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 414, 184–188 (2001).
[Crossref] [PubMed]

2000 (1)

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[Crossref]

1999 (1)

1995 (1)

E. H. Anderson, V. Boegli, and L. P. Muray, “Electron beam lithography digital pattern generator and electronics for generalized curvilinear structures,” J. Vac. Sci. Technol. B 13, 2529–2534 (1995).
[Crossref]

1984 (1)

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

1975 (1)

M. J. Yzuel and J. Santamaria, “Polychromatic optical image. Diffraction limited system and influence of the longitudinal chromatic aberration,” Opt. Acta 22, 673–690 (1975).
[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 414, 184–188 (2001).
[Crossref] [PubMed]

Alieva, T.

Andersen, G.

Anderson, E. H.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[Crossref]

E. H. Anderson, V. Boegli, and L. P. Muray, “Electron beam lithography digital pattern generator and electronics for generalized curvilinear structures,” J. Vac. Sci. Technol. B 13, 2529–2534 (1995).
[Crossref]

Attwood, D.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[Crossref]

Barbastathis, G.

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 414, 184–188 (2001).
[Crossref] [PubMed]

Boegli, V.

E. H. Anderson, V. Boegli, and L. P. Muray, “Electron beam lithography digital pattern generator and electronics for generalized curvilinear structures,” J. Vac. Sci. Technol. B 13, 2529–2534 (1995).
[Crossref]

Burge, R.

S. H. Tao, X.-C. Yuan, J. Lin, and R. Burge, “Sequence of focused optical vortices generated by a spiral fractal zone plates,” Appl. Phys. Lett. 89, 031105 (2006).
[Crossref]

Calatayud, A.

Calvo, M. L.

Cao, Q.

Chao, W.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[Crossref]

Chen, Q. -D.

Climent, V.

Dai, H. T.

Dai, H. -T.

H.-T. Dai, X. Wang, and K.-S. Xu, “Focusing properties of fractal zone plates with variable lacunarity: experimental studies based on liquid crystal on silicon,” Chin. Phys. Lett. 22, 2851–2854 (2005).
[Crossref]

Davis, J. A.

Denbeaux, G.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[Crossref]

Fernández-Alonso, M.

Furlan, W. D.

Gil, D.

Giménez, F.

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 414, 184–188 (2001).
[Crossref] [PubMed]

Harteneck, B.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[Crossref]

Huang, T. J.

Hyde, R.

Jacobsen, C.

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]

Jahns, J.

Johnson, L.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[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 414, 184–188 (2001).
[Crossref] [PubMed]

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 414, 184–188 (2001).
[Crossref] [PubMed]

Láncis, J.

Lin, J.

S. H. Tao, X.-C. Yuan, J. Lin, and R. Burge, “Sequence of focused optical vortices generated by a spiral fractal zone plates,” Appl. Phys. Lett. 89, 031105 (2006).
[Crossref]

Liu, Y. J.

Lucero, A.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[Crossref]

Mendoza-Yero, O.

Menon, R.

Michette, A. G.

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

Mínguez-Vega, G.

Monsoriu, J. A.

Muray, L. P.

E. H. Anderson, V. Boegli, and L. P. Muray, “Electron beam lithography digital pattern generator and electronics for generalized curvilinear structures,” J. Vac. Sci. Technol. B 13, 2529–2534 (1995).
[Crossref]

Niu, L. -G.

Olynick, D. L.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[Crossref]

Pons, A.

Ramirez, L.

Rodrigo Martín-Romo, J. A.

Saavedra, G.

Santamaria, J.

M. J. Yzuel and J. Santamaria, “Polychromatic optical image. Diffraction limited system and influence of the longitudinal chromatic aberration,” Opt. Acta 22, 673–690 (1975).
[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 414, 184–188 (2001).
[Crossref] [PubMed]

Simpson, M. J.

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

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 414, 184–188 (2001).
[Crossref] [PubMed]

Smith, H.

Sun, H. -B.

Sun, X. W.

Tao, S. H.

S. H. Tao, X.-C. Yuan, J. Lin, and R. Burge, “Sequence of focused optical vortices generated by a spiral fractal zone plates,” Appl. Phys. Lett. 89, 031105 (2006).
[Crossref]

Veklerov, E.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[Crossref]

Wan, R.

Wang, S.

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

Wang, X.

H.-T. Dai, X. Wang, and K.-S. Xu, “Focusing properties of fractal zone plates with variable lacunarity: experimental studies based on liquid crystal on silicon,” Chin. Phys. Lett. 22, 2851–2854 (2005).
[Crossref]

Wang, Y.

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]

Wu, D.

Xu, K. -S.

H.-T. Dai, X. Wang, and K.-S. Xu, “Focusing properties of fractal zone plates with variable lacunarity: experimental studies based on liquid crystal on silicon,” Chin. Phys. Lett. 22, 2851–2854 (2005).
[Crossref]

Yuan, X. -C.

S. H. Tao, X.-C. Yuan, J. Lin, and R. Burge, “Sequence of focused optical vortices generated by a spiral fractal zone plates,” Appl. Phys. Lett. 89, 031105 (2006).
[Crossref]

Yun, W.

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]

Yzuel, M. J.

M. J. Yzuel and J. Santamaria, “Polychromatic optical image. Diffraction limited system and influence of the longitudinal chromatic aberration,” Opt. Acta 22, 673–690 (1975).
[Crossref]

Zhang, X.

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

Appl. Opt. (1)

Appl. Phys. Lett. (1)

S. H. Tao, X.-C. Yuan, J. Lin, and R. Burge, “Sequence of focused optical vortices generated by a spiral fractal zone plates,” Appl. Phys. Lett. 89, 031105 (2006).
[Crossref]

Chin. Phys. Lett. (1)

H.-T. Dai, X. Wang, and K.-S. Xu, “Focusing properties of fractal zone plates with variable lacunarity: experimental studies based on liquid crystal on silicon,” Chin. Phys. Lett. 22, 2851–2854 (2005).
[Crossref]

J. Opt. Soc. Am. A (3)

J. Vac. Sci. Technol. B (2)

E. H. Anderson, V. Boegli, and L. P. Muray, “Electron beam lithography digital pattern generator and electronics for generalized curvilinear structures,” J. Vac. Sci. Technol. B 13, 2529–2534 (1995).
[Crossref]

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[Crossref]

Nature (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]

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. Acta (2)

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

M. J. Yzuel and J. Santamaria, “Polychromatic optical image. Diffraction limited system and influence of the longitudinal chromatic aberration,” Opt. Acta 22, 673–690 (1975).
[Crossref]

Opt. Express (5)

Opt. Lett. (5)

Opt. Photonics News (1)

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

Other (1)

J.Ojeda-Castañeda and C.Gómez-Reino, eds., Selected Papers on Zone Plates (SPIE Optical Engineering, 1996).

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Figures (5)

Fig. 1
Fig. 1

(a) Triadic fractal Cantor set for the levels S = 0 , 1, 2, 3, and 4. The structures for S = 0 and S = 1 are the initiator and generator, respectively. Note that the fractal set at stage S can be considered as a composite of two equivalent fractals at the previous stage S 1 . (b) Multifractals as a composite of two different fractals.

Fig. 2
Fig. 2

(a) MFZP generated at M = { 3 , 2 } and (b) the equivalent CZP.

Fig. 3
Fig. 3

Axial irradiances provided by different MFZPs shown in the inset. Note that in (c) {3,2} is the composite of {0,2} and {3,0}.

Fig. 4
Fig. 4

Self-similar behavior of the axial irradiances provided by two MFZPs around the corresponding main focus.

Fig. 5
Fig. 5

(a) Polychromatic axial illuminance computed for the zone plates in Fig. 2. The chromaticity of both curves is shown in (b); the continuous line corresponds to the MFZP and the dashed line corresponds to the CZP.

Equations (6)

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I ( z ) = ( 2 π λ z ) 2 | 0 a p ( r o ) exp ( i π λ z r o 2 ) r o d r o | 2 .
I ( u ) = 4 π 2 u 2 | 0 1 q ( ς ) exp ( i 2 π u ς ) d ς | 2 ,
X = λ 2 λ 1 I ( r = 0 , z ; λ ) S ( λ ) x ̃ d λ ,
Y = λ 2 λ 1 I ( r = 0 , z ; λ ) S ( λ ) y ̃ d λ ,
Z = λ 2 λ 1 I ( r = 0 , z ; λ ) S ( λ ) z ̃ d λ ,
x = X X + Y + Z ,     y = Y X + Y + Z .

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