Abstract

We report on a Fourier series approach that predicts the focal points and intensities produced by fractal zone plate lenses. This approach allows us to separate the effects of the fractal order from those of the lens aperture. We implement these fractal lenses onto a liquid-crystal display and show experimental verification of our theory.

© 2006 Optical Society of America

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References

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  1. G. Saavedra, W. D. Furlan, and J. A. Monsoriu, "Fractal zone plates," Opt. Lett. 28, 971-973 (2003).
    [CrossRef] [PubMed]
  2. J. A. Davis, L. Ramirez, J. A. R. Martin-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]
  3. L. Zunino and M. Garavaglia, "Frauhofer diffraction by Cantor fractals with variable lacunarity," J. Mod. Opt. 50, 717-728 (2003).
    [CrossRef]
  4. J. A. Monsoriu, G. Saavedra, and W. D. Furlan, "Fractal zone plates with variable lacunarity," Opt. Express 12, 4227-4234 (2004).
    [CrossRef] [PubMed]
  5. D. L. Jaggard and A. D. Jaggard, "Polyadic Cantor superlattices with variable lacunarity," Opt. Lett. 22, 145-147 (1997).
    [CrossRef] [PubMed]
  6. A. D. Jaggard and D. L. Jaggard, "Scattering from fractal superlattices with variable lacunarity," J. Opt. Soc. Am. A 15, 1626-1635 (1998).
    [CrossRef]
  7. J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, "Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects," Opt. Eng. 38, 1051-1057 (1999).
    [CrossRef]

2004 (2)

2003 (2)

L. Zunino and M. Garavaglia, "Frauhofer diffraction by Cantor fractals with variable lacunarity," J. Mod. Opt. 50, 717-728 (2003).
[CrossRef]

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

1999 (1)

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, "Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects," Opt. Eng. 38, 1051-1057 (1999).
[CrossRef]

1998 (1)

1997 (1)

Alieva, T.

Amako, J.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, "Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects," Opt. Eng. 38, 1051-1057 (1999).
[CrossRef]

Calvo, M. L.

Cottrell, D. M.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, "Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects," Opt. Eng. 38, 1051-1057 (1999).
[CrossRef]

Davis, J. A.

J. A. Davis, L. Ramirez, J. A. R. Martin-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]

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, "Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects," Opt. Eng. 38, 1051-1057 (1999).
[CrossRef]

Furlan, W. D.

Garavaglia, M.

L. Zunino and M. Garavaglia, "Frauhofer diffraction by Cantor fractals with variable lacunarity," J. Mod. Opt. 50, 717-728 (2003).
[CrossRef]

Jaggard, A. D.

Jaggard, D. L.

Martin-Romo, J. A. R.

Monsoriu, J. A.

Ramirez, L.

Saavedra, G.

Sonehara, T.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, "Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects," Opt. Eng. 38, 1051-1057 (1999).
[CrossRef]

Tsai, P.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, "Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects," Opt. Eng. 38, 1051-1057 (1999).
[CrossRef]

Zunino, L.

L. Zunino and M. Garavaglia, "Frauhofer diffraction by Cantor fractals with variable lacunarity," J. Mod. Opt. 50, 717-728 (2003).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Sequence showing (a) a one-dimensional linear phase function, (b) a one-dimensional lens phase pattern, (c) a binarized one-dimensional linear phase function, and (d) a binarized one-dimensional lens pattern.

Fig. 2
Fig. 2

Sequence showing generation of (a) Fresnel zone plates for N = 3 and several values of S and (b) fractal zone plates for N = 3 and several values of S. Transparent and opaque segments are represented by white and black, respectively.

Fig. 3
Fig. 3

Fractal lens patterns for S = 2 and N = (a) 3, (b) 5, (c) 7, (d) 9.

Fig. 4
Fig. 4

Measured and theoretical positions of primary focal points corresponding to f 1 , - 1 (squares), f 1 , 0 (circles), and f 1 , 1 (diamonds) for N = 2 9 and S = 2 .

Tables (1)

Tables Icon

Table 1 Experimental Data for Intensities and Focal Point Locations Corresponding to a Fractal Lens for N = 7 and S = 2

Equations (21)

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E ( x 2 = 0 , y 2 = 0 ) = 1 λ z 0 t ( r 1 ) exp ( i k r 1 2 2 z ) 2 π r 1 d r 1 .
g ( x ) = exp ( i 2 π x / d ) .
t ( f , r 1 ) = exp ( i k r 1 2 2 f ) .
E ( x 2 = 0 , y 2 = 0 ) = 1 λ z 0 a exp ( i k r 1 2 2 f ) exp ( i k r 1 2 2 z ) 2 π r 1 d r 1 .
E ( x 2 = 0 , y 2 = 0 ) = 2 π u 0.5 0.5 exp ( 2 π u 0 s ) exp ( i 2 π u s ) d s .
E ( u ) = 2 π u rect ( s ) exp ( i 2 π u 0 s ) exp ( i 2 π u s ) d s .
E ( u ) = 2 π u sinc ( u ) δ ( u - u 0 ) = 2 π u sinc [ ( u u 0 ) ] .
t ( x ) = l = c l exp ( i l 2 π x d ) .
c l = ( γ d ) sinc ( l πγ d ) .
t ( f , r 1 ) = l = c l exp ( i l k r 1 2 2 f ) .
E ( u ) l = c l sinc [ ( u l u 0 ) ] .
t ( f , N , S = 2 , r 1 ) = l = c l exp ( i l k r 1 2 2 f ) m = exp [ i m k r 1 2 2 ( 2 N 1 ) f ] .
c l c m exp ( i l k r 1 2 2 f ) exp [ i m k r 1 2 2 ( 2 N 1 ) f ] = c l c m exp ( i k r 1 2 2 f lm ) .    
1 f l m = l f + m ( 2 N 1 ) f .
E ( u ) l = m = c l c m sinc [ ( u u l m ) ] .
I l m c l 2 c m 2 .
t ( f , N , S = 3 , r 1 ) = l = c l exp ( i l k r 1 2 2 f ) m = c m exp [ i m k r 1 2 2 ( 2 N 1 ) f ] n = c n exp [ i n k r 1 2 2 ( 2 N - 1 ) 2 f ] .
c l c m c n exp ( i l k r 1 2 2 f ) exp [ i m k r 1 2 2 ( 2 N 1 ) f ] exp [ i n k r 1 2 2 ( 2 N 1 ) 2 f ] = c l c m c n exp ( i l k r 1 2 2 f lmn ) .
1 f lmn = l f + m ( 2 N 1 ) f + n ( 2 N 1 ) 2 f .
D = ln N ln ( 1 / γ ) .
I 1 , 0 I 1 , ± 1 = c 1 2 c 0 2 c 1 2 c 1 2 = π 2 4 2.5.

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