Abstract

In this paper we present a new kind of kinoform lenses in which the phase distribution is characterized by the “devil’s staircase” function. The focusing properties of these fractal DOEs coined devil’s lenses (DLs) are analytically studied and compared with conventional Fresnel kinoform lenses. It is shown that under monochromatic illumination a DL give rise a single fractal focus that axially replicates the self-similarity of the lens. Under broadband illumination the superposition of the different monochromatic foci produces an increase in the depth of focus and also a strong reduction in the chromaticity variation along the optical axis.

© 2007 Optical Society of America

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  1. J. Courtial and M. J. Padgett, "Monitor-outside-a-monitor effect and self-similar fractal structure in the eigenmodes of unstable optical resonators," Phys. Rev. Lett. 85, 5320-5323 (2000).
    [CrossRef]
  2. O. Trabocchi, S. Granieri, and W. D. Furlan, "Optical propagation of fractal fields. Experimental analysis in a single display," J. Mod. Opt. 48, 1247-1253 (2001).
  3. M. Lehman, "Fractal diffraction gratings built through rectangular domains," Opt. Commun. 195, 11-26 (2001).
    [CrossRef]
  4. J. G. Huang, J. M. Christian, and G. S. McDonald "Fresnel diffraction and fractal patterns from polygonal apertures," J. Opt. Soc. Am. A 23, 2768-2774 (2006).
    [CrossRef]
  5. G. Saavedra, W. D. Furlan, and J. A. Monsoriu, "Fractal zone plates," Opt. Lett. 28, 971-973 (2003).
    [CrossRef] [PubMed]
  6. J.A. Monsoriu, G. Saavedra, and W.D. Furlan, "Fractal zone plates with variable lacunarity," Opt. Express 12, 4227-4234 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-18-4227>
    [CrossRef] [PubMed]
  7. 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]
  8. H.-T. Dai, X. Wang, 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]
  9. 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]
  10. C. Martelli and J. Canning, "Fresnel Fibres with Omnidirectional Zone Cross-sections," Opt. Express 15, 4281-4286 (2007).
    [CrossRef] [PubMed]
  11. F. Giménez, J. A. Monsoriu, W. D. Furlan, and A. Pons, "Fractal Photon Sieves," Opt. Express 14, 11958-11963 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-11958>
    [CrossRef] [PubMed]
  12. J. A. Jordan, P. M. Hirsch, L. B. Lesem, and D. L. Van Rooy, "Kinoform lenses," Appl. Opt. 9, 1883-1887 (1970)
    [PubMed]
  13. D. R. Chalice, "A characterizationof the Cantor function," Amer. Math. Monthly 98, 255-258 (1991).
    [CrossRef]
  14. F. Doveil, A. Macor, and Y. Elskens, "Direct observation of a devil’s staircase in wave-particle interaction," Chaos 16, 033130 (2006).
    [CrossRef]
  15. M. Hupalo, J. Schamalian, and M. C. Tringides, "Devil’s staircase in Pb/Si(111) ordered phases," Phys. Rev. Lett. 90, 216106 (2003).
    [CrossRef] [PubMed]
  16. Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, "Devil’s staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces," Phys. Rev. Lett. 96, 213902 (2006).
    [CrossRef] [PubMed]
  17. Y. Han, L. N. Hazra, and C. A. Delisle, "Exact surface-relief profile of a kinoform lens from its phase function," J. Opt. Soc. Am. A 12, 524-529 (1995).
    [CrossRef]
  18. 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]

2007 (1)

2006 (5)

F. Giménez, J. A. Monsoriu, W. D. Furlan, and A. Pons, "Fractal Photon Sieves," Opt. Express 14, 11958-11963 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-11958>
[CrossRef] [PubMed]

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. Doveil, A. Macor, and Y. Elskens, "Direct observation of a devil’s staircase in wave-particle interaction," Chaos 16, 033130 (2006).
[CrossRef]

Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, "Devil’s staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces," Phys. Rev. Lett. 96, 213902 (2006).
[CrossRef] [PubMed]

J. G. Huang, J. M. Christian, and G. S. McDonald "Fresnel diffraction and fractal patterns from polygonal apertures," J. Opt. Soc. Am. A 23, 2768-2774 (2006).
[CrossRef]

2005 (1)

H.-T. Dai, X. Wang, 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 (2)

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

M. Hupalo, J. Schamalian, and M. C. Tringides, "Devil’s staircase in Pb/Si(111) ordered phases," Phys. Rev. Lett. 90, 216106 (2003).
[CrossRef] [PubMed]

2001 (2)

O. Trabocchi, S. Granieri, and W. D. Furlan, "Optical propagation of fractal fields. Experimental analysis in a single display," J. Mod. Opt. 48, 1247-1253 (2001).

M. Lehman, "Fractal diffraction gratings built through rectangular domains," Opt. Commun. 195, 11-26 (2001).
[CrossRef]

2000 (1)

J. Courtial and M. J. Padgett, "Monitor-outside-a-monitor effect and self-similar fractal structure in the eigenmodes of unstable optical resonators," Phys. Rev. Lett. 85, 5320-5323 (2000).
[CrossRef]

1995 (1)

1991 (1)

D. R. Chalice, "A characterizationof the Cantor function," Amer. Math. Monthly 98, 255-258 (1991).
[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]

1970 (1)

Alieva, T.

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]

Calvo, M. L.

Canning, J.

Chalice, D. R.

D. R. Chalice, "A characterizationof the Cantor function," Amer. Math. Monthly 98, 255-258 (1991).
[CrossRef]

Chen, Y. F.

Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, "Devil’s staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces," Phys. Rev. Lett. 96, 213902 (2006).
[CrossRef] [PubMed]

Christian, J. M.

Courtial, J.

J. Courtial and M. J. Padgett, "Monitor-outside-a-monitor effect and self-similar fractal structure in the eigenmodes of unstable optical resonators," Phys. Rev. Lett. 85, 5320-5323 (2000).
[CrossRef]

Dai, H.-T.

H.-T. Dai, X. Wang, 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.

Doveil, F.

F. Doveil, A. Macor, and Y. Elskens, "Direct observation of a devil’s staircase in wave-particle interaction," Chaos 16, 033130 (2006).
[CrossRef]

Elskens, Y.

F. Doveil, A. Macor, and Y. Elskens, "Direct observation of a devil’s staircase in wave-particle interaction," Chaos 16, 033130 (2006).
[CrossRef]

Furlan, W. D.

Furlan, W.D.

Giménez, F.

Granieri, S.

O. Trabocchi, S. Granieri, and W. D. Furlan, "Optical propagation of fractal fields. Experimental analysis in a single display," J. Mod. Opt. 48, 1247-1253 (2001).

Han, Y.

Hazra, L. N.

Hirsch, P. M.

Huang, J. G.

Huang, K. F.

Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, "Devil’s staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces," Phys. Rev. Lett. 96, 213902 (2006).
[CrossRef] [PubMed]

Hupalo, M.

M. Hupalo, J. Schamalian, and M. C. Tringides, "Devil’s staircase in Pb/Si(111) ordered phases," Phys. Rev. Lett. 90, 216106 (2003).
[CrossRef] [PubMed]

Jordan, J. A.

Lehman, M.

M. Lehman, "Fractal diffraction gratings built through rectangular domains," Opt. Commun. 195, 11-26 (2001).
[CrossRef]

Lesem, L. B.

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]

Lu, T. H.

Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, "Devil’s staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces," Phys. Rev. Lett. 96, 213902 (2006).
[CrossRef] [PubMed]

Macor, A.

F. Doveil, A. Macor, and Y. Elskens, "Direct observation of a devil’s staircase in wave-particle interaction," Chaos 16, 033130 (2006).
[CrossRef]

Martelli, C.

McDonald, G. S.

Monsoriu, J. A.

Monsoriu, J.A.

Padgett, M. J.

J. Courtial and M. J. Padgett, "Monitor-outside-a-monitor effect and self-similar fractal structure in the eigenmodes of unstable optical resonators," Phys. Rev. Lett. 85, 5320-5323 (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]

Schamalian, J.

M. Hupalo, J. Schamalian, and M. C. Tringides, "Devil’s staircase in Pb/Si(111) ordered phases," Phys. Rev. Lett. 90, 216106 (2003).
[CrossRef] [PubMed]

Su, K. W.

Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, "Devil’s staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces," Phys. Rev. Lett. 96, 213902 (2006).
[CrossRef] [PubMed]

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]

Trabocchi, O.

O. Trabocchi, S. Granieri, and W. D. Furlan, "Optical propagation of fractal fields. Experimental analysis in a single display," J. Mod. Opt. 48, 1247-1253 (2001).

Tringides, M. C.

M. Hupalo, J. Schamalian, and M. C. Tringides, "Devil’s staircase in Pb/Si(111) ordered phases," Phys. Rev. Lett. 90, 216106 (2003).
[CrossRef] [PubMed]

Van Rooy, D. L.

Wang, X.

H.-T. Dai, X. Wang, 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]

Xu, K.-S.

H.-T. Dai, X. Wang, 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]

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]

Amer. Math. Monthly (1)

D. R. Chalice, "A characterizationof the Cantor function," Amer. Math. Monthly 98, 255-258 (1991).
[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]

Chaos (1)

F. Doveil, A. Macor, and Y. Elskens, "Direct observation of a devil’s staircase in wave-particle interaction," Chaos 16, 033130 (2006).
[CrossRef]

Chin. Phys. Lett. (1)

H.-T. Dai, X. Wang, 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. Mod. Opt. (1)

O. Trabocchi, S. Granieri, and W. D. Furlan, "Optical propagation of fractal fields. Experimental analysis in a single display," J. Mod. Opt. 48, 1247-1253 (2001).

J. Opt. Soc. Am. A (2)

Opt. Acta (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]

Opt. Commun. (1)

M. Lehman, "Fractal diffraction gratings built through rectangular domains," Opt. Commun. 195, 11-26 (2001).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. Lett. (3)

J. Courtial and M. J. Padgett, "Monitor-outside-a-monitor effect and self-similar fractal structure in the eigenmodes of unstable optical resonators," Phys. Rev. Lett. 85, 5320-5323 (2000).
[CrossRef]

M. Hupalo, J. Schamalian, and M. C. Tringides, "Devil’s staircase in Pb/Si(111) ordered phases," Phys. Rev. Lett. 90, 216106 (2003).
[CrossRef] [PubMed]

Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, "Devil’s staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces," Phys. Rev. Lett. 96, 213902 (2006).
[CrossRef] [PubMed]

Supplementary Material (1)

» Media 1: MOV (1280 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

(a) Triadic Cantor set for S=1, S=2, and S=3. The structure for S=0 is the initiator and the one corresponding to S=1 is the generator. The Cantor function or Devil’s staircase, Fs (x), is shown under the corresponding Cantor set for S=3. (b) Convergent DL at stage of growth S=3 and the equivalent kinoform Fresnel lens.

Fig. 2.
Fig. 2.

Normalized irradiance vs. the axial coordinate u obtained for aDL at three stages of growth (upper part) and for its associated Fresnel kinoform lens (lower part).

Fig. 3.
Fig. 3.

(387 kB) Evolution of the transverse diffraction patterns generated by a S=2 DL, (leftt), and a Fresnel kinoform lens of the same focal lenght. [Media 1]

Fig. 4.
Fig. 4.

Transverse diffraction patterns around the principal focus of a DL with S=2. The normalized axial distance is given by z/f 2=32/u.

Fig. 5.
Fig. 5.

Normalized irradiance vs. the axial coordinate u obtained for a DL (left) and for its associated Fresnel kinoform lens for three wavelengths R=650 nm; G=550 nm; and B=480 nm. In all cases S=2.

Fig. 6.
Fig. 6.

Polychromatic axial illuminance computed (a) for a DL and (b) for its associated Fresnel kinoform lens. The chromaticity of both curves is shown in (c); the continuous line corresponds to the DL and the dashed line to the Fresnel kinoform.

Equations (11)

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F s ( x ) = { l 2 s if p s , l x q s , l 1 2 s x q s , l p s , l + 1 q s , l + 1 2 s if q s , l x p s , l + 1 ,
q ( ς ) = q DL ( ς , S ) = exp [ i 2 s + 1 π F s ( ς ) ] ,
ς = ( r a ) 2
h DL ( r ) = mod 2 π { 2 s + 1 π F s ( r 2 a 2 ) } λ 2 π ( n 1 ) ,
I ( z , r ) = ( 2 π λz ) 2 0 a p ( r o ) exp ( i π λz r o 2 ) J 0 ( 2 π r o r λz ) r o d r o 2 ;
I ( u , v ) = 4 π 2 u 2 0 1 q ( ς ) exp ( i 2 πuς ) J 0 ( 4 π ς uv ) d ς 2 ,
I ( u ) = 4 π 2 u 2 0 1 q ( ς ) exp ( i 2 πuς ) d ς 2 .
f s = a 2 2 λ 3 s .
X = λ 2 λ 1 I ( r = 0 , z ; λ ) S ( λ ) x ͂ ; Y = λ 2 λ 1 I ( r = 0 , z ; λ ) S ( λ ) y ͂ ;
Z = λ 2 λ 1 I ( r = 0 , z ; λ ) S ( λ ) z ͂ ;
x = X X + Y + Z ; y = Y X + Y + Z ;

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