Abstract

We introduce the generalized devil’s lenses (GDLs) as a new family of diffractive kinoform lenses whose structure is based on the generalized Cantor set. The focusing properties of different members of this family are analyzed. It is shown that under plane wave illumination the GDLs give a single main focus surrounded by many subsidiary foci. It is shown that the total number of subsidiary foci is higher than the number of foci corresponding to conventional devil’s lenses; however, the self-similar behavior of the axial irradiance is preserved to some extent.

© 2011 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 plate,” Appl. Phys. Lett. 89, 031105 (2006).
[CrossRef]

J. A. Monsoriu, C. J. Zapata-Rodriguez, and W. D. Furlan, “Fractal axicon,” Opt. Commun. 263 (1), 1–5 (2006).
[CrossRef]

2005 (2)

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 (1)

2003 (2)

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

Y. Wang, W. Yun, and C. Jacobsen, “Achromatic Fresnel optics for wideband extreme-ultraviolet and x-ray imaging,” Nature (London) 424, 50–53 (2003).
[CrossRef]

2002 (1)

S. Wang and X. Zhang, “Terahertz tomographic imaging with a Fresnel lens,” Opt. Photonics News 13, 59–59(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 (London) 414, 184–188(2001).
[CrossRef]

1995 (1)

1991 (1)

D. R. Chalice, “A characterization of the Cantor function,” Am. Math. Mon. 98, 255–258 (1991).
[CrossRef]

1986 (1)

C. Allain and M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B 33, 3566–3569 (1986).
[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]

Alieva, T.

Allain, C.

C. Allain and M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B 33, 3566–3569 (1986).
[CrossRef]

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

Bolognini, N.

Burge, R.

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

Calvo, M. L.

Chalice, D. R.

D. R. Chalice, “A characterization of the Cantor function,” Am. Math. Mon. 98, 255–258 (1991).
[CrossRef]

Chen, Q.-D.

Climent, V.

Cloitre, M.

C. Allain and M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B 33, 3566–3569 (1986).
[CrossRef]

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.

Delisle, C. A.

Fernández-Alonso, M.

Furlan, W. D.

Giménez, F.

Han, Y.

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]

Hazra, L. N.

Jacobsen, C.

Y. Wang, W. Yun, and C. Jacobsen, “Achromatic Fresnel optics for wideband extreme-ultraviolet and x-ray imaging,” Nature (London) 424, 50–53 (2003).
[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]

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]

Lancis, J.

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 plate,” Appl. Phys. Lett. 89, 031105 (2006).
[CrossRef]

Mendoza-Yero, O.

Mínguez-Vega, G.

Monsoriu, J. A.

Niu, L.-G.

Ramirez, L.

Rodrigo Martín-Romo, J. A.

Saavedra, G.

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]

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]

Sun, H.-B.

Tajahuerce, E.

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 plate,” Appl. Phys. Lett. 89, 031105 (2006).
[CrossRef]

Tebaldi, M.

Torroba, R.

Wan, R.

Wang, S.

S. Wang and X. Zhang, “Terahertz tomographic imaging with a Fresnel lens,” Opt. Photonics News 13, 59–59(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 (London) 424, 50–53 (2003).
[CrossRef]

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 plate,” 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 (London) 424, 50–53 (2003).
[CrossRef]

Zapata-Rodriguez, C. J.

J. A. Monsoriu, C. J. Zapata-Rodriguez, and W. D. Furlan, “Fractal axicon,” Opt. Commun. 263 (1), 1–5 (2006).
[CrossRef]

Zhang, X.

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

Am. Math. Mon. (1)

D. R. Chalice, “A characterization of the Cantor function,” Am. Math. Mon. 98, 255–258 (1991).
[CrossRef]

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 plate,” 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 (2)

Nature (London) (2)

Y. Wang, W. Yun, and C. Jacobsen, “Achromatic Fresnel optics for wideband extreme-ultraviolet and x-ray imaging,” Nature (London) 424, 50–53 (2003).
[CrossRef]

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. (1)

J. A. Monsoriu, C. J. Zapata-Rodriguez, and W. D. Furlan, “Fractal axicon,” Opt. Commun. 263 (1), 1–5 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (7)

Opt. Photonics News (1)

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

Phys. Rev. B (1)

C. Allain and M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B 33, 3566–3569 (1986).
[CrossRef]

Other (1)

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

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Figures (5)

Fig. 1
Fig. 1

Generalized Cantor set for τ = 2 , 3, and 4 and growth levels S = 0 , 1, 2, and 3.

Fig. 2
Fig. 2

Generalized devil’s staircase function for τ = 3 and growth levels S = 0 , 1, 2, and 3. The gray bars represent the corresponding generalized Cantor set.

Fig. 3
Fig. 3

(a) Phase transmittance of a GDL for τ = 3 and S = 2 versus the quadratic radial variable. (b) Equivalent phase transmittance after taking modulus 2 π . (c) Phase transmittance versus the radial variable. (d) Phase transmittance of a kinoform lens of the same focal length.

Fig. 4
Fig. 4

Axial irradiances provided by GDL of orders S = 1 and S = 2 and generalization parameters τ = 2 , 3, and 4.

Fig. 5
Fig. 5

Self-similarity for the axial irradiance shown in Fig. 4 for S = 2 .

Equations (7)

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F τ , S ( x ) = { l 2 S if    p τ , S , l x q τ , S , l 1 2 S x q τ , S , l p τ , S , l + 1 q τ , S , l + l 2 S if     q τ , S , l x p τ , S , l + 1 ,
q ( ς ) = q GDL ( ς , τ , S ) = exp [ i ϕ τ , S ( ς ) ] , with     ϕ τ , S ( ς ) = 2 s + 1 π F τ , S ( ς ) ,
h GDL ( r ) = mod 2 π [ ϕ τ , S ( r 2 a 2 ) ] λ 2 π ( n 1 ) ,
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 ,
f τ , S = a 2 2 λ ( τ + 1 ) S .
C ( γ ) = 0 I ( u ) · I ( u 0 + u u 0 γ ) d u 0 I 2 ( u ) d u 0 I 2 ( u 0 + u u 0 γ ) d u .

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