Abstract

In this paper we present a new kind of vortex lenses in which the radial phase distribution is characterized by the “devil’s staircase” function. The focusing properties of these fractal DOEs coined Devil’s vortex-lenses are analytically studied and the influence of the topological charge is investigated. It is shown that under monochromatic illumination a vortex devil’s lens give rise a focal volume containing a delimited chain of vortices that are axially distributed according to the self-similarity of the lens.

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References

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2008 (1)

2007 (2)

2006 (5)

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

F. Doveil, A. Macor, and Y. Elskens, “Direct observation of a devil’s staircase in wave-particle interaction,” Chaos 16(3), 033103 (2006).
[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(21), 213902 (2006).
[CrossRef] [PubMed]

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259(2), 428–435 (2006).
[CrossRef]

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(3), 031105 (2006).
[CrossRef]

2005 (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(11), 2851–2854 (2005).
[CrossRef]

2004 (5)

2003 (2)

G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. 28(12), 971–973 (2003).
[CrossRef] [PubMed]

M. Hupalo, J. Schmalian, and M. C. Tringides, “Devil’s staircase” in Pb/Si(111) ordered phases,” Phys. Rev. Lett. 90(21), 216106 (2003).
[CrossRef] [PubMed]

2001 (1)

1992 (1)

1991 (1)

D. R. Chalice, “A characterization of the Cantor function,” Am. Math. Mon. 98(3), 255–258 (1991).
[CrossRef]

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(3), 031105 (2006).
[CrossRef]

Calvo, M. L.

Canning, J.

Chalice, D. R.

D. R. Chalice, “A characterization of the Cantor function,” Am. Math. Mon. 98(3), 255–258 (1991).
[CrossRef]

Chen, Q.-D.

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(21), 213902 (2006).
[CrossRef] [PubMed]

Cheong, W. C.

Crabtree, K.

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(11), 2851–2854 (2005).
[CrossRef]

Davis, J. A.

Doveil, F.

F. Doveil, A. Macor, and Y. Elskens, “Direct observation of a devil’s staircase in wave-particle interaction,” Chaos 16(3), 033103 (2006).
[CrossRef] [PubMed]

Elskens, Y.

F. Doveil, A. Macor, and Y. Elskens, “Direct observation of a devil’s staircase in wave-particle interaction,” Chaos 16(3), 033103 (2006).
[CrossRef] [PubMed]

Furlan, W. D.

Gbur, G.

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259(2), 428–435 (2006).
[CrossRef]

Giménez, F.

Heckenberg, N. R.

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(21), 213902 (2006).
[CrossRef] [PubMed]

Hupalo, M.

M. Hupalo, J. Schmalian, and M. C. Tringides, “Devil’s staircase” in Pb/Si(111) ordered phases,” Phys. Rev. Lett. 90(21), 216106 (2003).
[CrossRef] [PubMed]

Lee, W. M.

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(3), 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(21), 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(3), 033103 (2006).
[CrossRef] [PubMed]

Martelli, C.

Martín-Romo, J. A.

McDuff, R.

Monsoriu, J. A.

Moreno, I.

Niu, L.-G.

Pons, A.

Ramirez, L.

Roux, F. S.

F. S. Roux, “Distribution of angular momentum and vortex morphology in optical beams,” Opt. Commun. 242(1–3), 45–55 (2004).
[CrossRef]

Saavedra, G.

Schmalian, J.

M. Hupalo, J. Schmalian, and M. C. Tringides, “Devil’s staircase” in Pb/Si(111) ordered phases,” Phys. Rev. Lett. 90(21), 216106 (2003).
[CrossRef] [PubMed]

Smith, C. P.

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(21), 213902 (2006).
[CrossRef] [PubMed]

Sun, H.-B.

Swartzlander, G. A.

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(3), 031105 (2006).
[CrossRef]

Tringides, M. C.

M. Hupalo, J. Schmalian, and M. C. Tringides, “Devil’s staircase” in Pb/Si(111) ordered phases,” Phys. Rev. Lett. 90(21), 216106 (2003).
[CrossRef] [PubMed]

Visser, T. D.

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259(2), 428–435 (2006).
[CrossRef]

Wang, R.

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(11), 2851–2854 (2005).
[CrossRef]

White, A. G.

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(11), 2851–2854 (2005).
[CrossRef]

Yuan, X. C.

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(3), 031105 (2006).
[CrossRef]

Am. Math. Mon. (1)

D. R. Chalice, “A characterization of the Cantor function,” Am. Math. Mon. 98(3), 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(3), 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(3), 033103 (2006).
[CrossRef] [PubMed]

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(11), 2851–2854 (2005).
[CrossRef]

Opt. Commun. (2)

F. S. Roux, “Distribution of angular momentum and vortex morphology in optical beams,” Opt. Commun. 242(1–3), 45–55 (2004).
[CrossRef]

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259(2), 428–435 (2006).
[CrossRef]

Opt. Express (4)

Opt. Lett. (6)

Phys. Rev. Lett. (2)

M. Hupalo, J. Schmalian, and M. C. Tringides, “Devil’s staircase” in Pb/Si(111) ordered phases,” Phys. Rev. Lett. 90(21), 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(21), 213902 (2006).
[CrossRef] [PubMed]

Supplementary Material (2)

» Media 1: AVI (1661 KB)     
» Media 2: AVI (2233 KB)     

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Figures (4)

Fig. 1
Fig. 1

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.

Fig. 2
Fig. 2

(a) Phase variation as gray levels for a DL (S = 2), and for DVLs with topological charge (b) m = 1, and (c) m = 3.

Fig. 3
Fig. 3

Normalized irradiance contours computed for the lenses in Fig. 2. (a) m = 0, (b) m = 1, and (c) m = 3.

Fig. 4
Fig. 4

Transverse field maps (as the product of the irradiance times phase) at (a) u = 9 and (b) u = 9.8 computed for the lens in Fig. 1(b) with m = 1. The animations Fig. 4a.avi (Media 1, 1.62 MB) and Fig. 4b.avi (Media 2, 2.18 MB) show the evolution of the vortices as they propagate along the optical axis.

Equations (8)

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F s ( x ) = { 1 2 S                 if     p S , l x q S , l 1 2 S x q s , l p s , l + 1 q s , l + l 2 S         if     q S , l x p S , l + 1 ,
q ( ς ) = exp [ i Φ DL ] = exp [ 2 s + 1 π F S ( ς ) ] ,
ς = ( r / a ) 2
t ( r , θ ) = p ( r ) exp [ i m θ ] .
I ( z , r ) = ( 2 π λ z ) 2 | 0 a p ( r o ) exp ( i π λ z r o 2 ) J m ( 2 π r o r λ z ) r o d r o | 2 ;
Φ ( z , r , θ ) = m ( θ + π 2 ) 2 π λ z π r 2 λ z π 2 ;
I ( u , v ) = 4 π 2 u 2 | 0 1 q ( ς ) exp ( i 2 π u ς ) J m ( 4 π ς u v ) d ς | 2 ,
Φ ( u , v , θ ) = m ( θ + π 2 ) π a 2 λ 2 u 2 π u v 2 π 2 ;

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