Abstract

Diffractals are electromagnetic waves diffracted by a fractal aperture. In an earlier paper, we reported an important property of Cantor diffractals, that of redundancy [R. Verma et. al., Opt. Express 20, 8250 (2012)]. In this paper, we report another important property, that of robustness. The question we address is: How much disorder in the Cantor grating can be accommodated by diffractals to continue to yield faithfully its fractal dimension and generator? This answer is of consequence in a number of physical problems involving fractal architecture.

© 2013 OSA

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References

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  1. R. Verma, V. Banerjee, and P. Senthilkumaran, “Redundancy in cantor diffractals,” Opt. Express20, 8250–8255 (2012).
    [CrossRef] [PubMed]
  2. B. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman, 1982).
  3. A. L. Barabasi and H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge University, 1995).
    [CrossRef]
  4. Tamas Vicsek, Fractal Growth Phenomena (World Scientific, 1992).
    [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. Express12, 4227–4234 (2004).
    [CrossRef] [PubMed]
  7. W. D. Furlan, G. Saavedra, and J. A. Monsoriu, “White-light imaging with fractal zone plates,” Opt. Lett.32, 2109–2111 (2007).
    [CrossRef] [PubMed]
  8. F. Gimenez, J. A. Monsoriu, W. D. Furlan, and Amparo Pons, “Fractal photon sieve,” Opt. Express14, 11958–11963 (2006).
    [CrossRef] [PubMed]
  9. J. A. Monsoriu, C. J. Z. Rodriguez, and W. D. Furlan, “Fractal axicons,” Opt. Commun.263, 1–5 (2006).
    [CrossRef]
  10. K. Jarrendahl, M. Dulea, J. Birch, and J.-E. Sundgren, “X-ray diffraction from amorphous Ge/Si Cantor superlattices,” Phys. Rev. B51, 7621–7631 (1995).
    [CrossRef]
  11. A. D. Jaggard and D. L. Jaggard, “Scattering from fractal superlattices with variable lacunarity,” J. Opt. Soc. Am. A15, 1626–1635 (1998).
    [CrossRef]
  12. H. Aubert and D. L. Jaggard, “Wavelet analysis of transients in fractal superlattices,” IEEE Trans. Antennas Propag.50, 338–345 (2002).
    [CrossRef]
  13. G. H. C. New, M. A. Yates, J. P. Woerdman, and G. S. McDonald, “Diffractive origin of fractal resonator modes,” Opt. Commun.193, 261–266 (2001).
    [CrossRef]
  14. M. Berry, C. Storm, and W. van Saarloos, “Theory of unstable laser modes: edge waves and fractality,” Opt. Commun.197, 393–402 (2001).
    [CrossRef]
  15. M. V. Berry, “Diffractals,” J. Phys. A: Math. Gen.12, 781–797 (1979).
    [CrossRef]
  16. D. Bak, S. P. Kim, S. K. Kim, K. -S. Soh, and J. H. Yee, “Fractal diffraction grating,” 1–7 http://arxiv.org/abs/physics/9802007 .
  17. B. Hou, G. Xu, W. Wen, and G. K. L. Wong, “Diffraction by an optical fractal grating,” Appl. Phys. Lett.85, 6125–6127 (2004).
    [CrossRef]
  18. C. Allain and M. Cloitre, “Spatial spectrum of a general family of self-similar arrays,” Phys. Rev. A36, 5751–5757 (1987).
    [CrossRef] [PubMed]
  19. D. A. Hamburger-Lidar, “Elastic scattering by deterministic and random fractals: Self-affinity of the diffraction spectrum,” Phys. Rev. E54, 354–370 (1996).
    [CrossRef]
  20. C. Guerin and M. Holschneider, “Scattering on fractal measures,” J. Phys. A29, 7651–7667 (1996).
    [CrossRef]
  21. M. Lehman, “Fractal diffraction grating built through rectangular domains,” Opt. Commun.195, 11–26 (2001).
    [CrossRef]
  22. C. Allain and M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B33, 3566–3569 (1986).
    [CrossRef]
  23. B. Dubuc, J. F. Quiniou, C. R. Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimensions of profiles,” Phys. Rev. A39, 1500–1512 (1989).
    [CrossRef] [PubMed]
  24. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

2012 (1)

2007 (1)

2006 (2)

F. Gimenez, J. A. Monsoriu, W. D. Furlan, and Amparo Pons, “Fractal photon sieve,” Opt. Express14, 11958–11963 (2006).
[CrossRef] [PubMed]

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

2004 (2)

J. A. Monsoriu, G. Saavedra, and W. D. Furlan, “Fractal zone plates with variable lacunarity,” Opt. Express12, 4227–4234 (2004).
[CrossRef] [PubMed]

B. Hou, G. Xu, W. Wen, and G. K. L. Wong, “Diffraction by an optical fractal grating,” Appl. Phys. Lett.85, 6125–6127 (2004).
[CrossRef]

2003 (1)

2002 (1)

H. Aubert and D. L. Jaggard, “Wavelet analysis of transients in fractal superlattices,” IEEE Trans. Antennas Propag.50, 338–345 (2002).
[CrossRef]

2001 (3)

G. H. C. New, M. A. Yates, J. P. Woerdman, and G. S. McDonald, “Diffractive origin of fractal resonator modes,” Opt. Commun.193, 261–266 (2001).
[CrossRef]

M. Berry, C. Storm, and W. van Saarloos, “Theory of unstable laser modes: edge waves and fractality,” Opt. Commun.197, 393–402 (2001).
[CrossRef]

M. Lehman, “Fractal diffraction grating built through rectangular domains,” Opt. Commun.195, 11–26 (2001).
[CrossRef]

1998 (1)

1996 (2)

D. A. Hamburger-Lidar, “Elastic scattering by deterministic and random fractals: Self-affinity of the diffraction spectrum,” Phys. Rev. E54, 354–370 (1996).
[CrossRef]

C. Guerin and M. Holschneider, “Scattering on fractal measures,” J. Phys. A29, 7651–7667 (1996).
[CrossRef]

1995 (1)

K. Jarrendahl, M. Dulea, J. Birch, and J.-E. Sundgren, “X-ray diffraction from amorphous Ge/Si Cantor superlattices,” Phys. Rev. B51, 7621–7631 (1995).
[CrossRef]

1989 (1)

B. Dubuc, J. F. Quiniou, C. R. Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimensions of profiles,” Phys. Rev. A39, 1500–1512 (1989).
[CrossRef] [PubMed]

1987 (1)

C. Allain and M. Cloitre, “Spatial spectrum of a general family of self-similar arrays,” Phys. Rev. A36, 5751–5757 (1987).
[CrossRef] [PubMed]

1986 (1)

C. Allain and M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B33, 3566–3569 (1986).
[CrossRef]

1979 (1)

M. V. Berry, “Diffractals,” J. Phys. A: Math. Gen.12, 781–797 (1979).
[CrossRef]

Allain, C.

C. Allain and M. Cloitre, “Spatial spectrum of a general family of self-similar arrays,” Phys. Rev. A36, 5751–5757 (1987).
[CrossRef] [PubMed]

C. Allain and M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B33, 3566–3569 (1986).
[CrossRef]

Aubert, H.

H. Aubert and D. L. Jaggard, “Wavelet analysis of transients in fractal superlattices,” IEEE Trans. Antennas Propag.50, 338–345 (2002).
[CrossRef]

Banerjee, V.

Barabasi, A. L.

A. L. Barabasi and H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge University, 1995).
[CrossRef]

Berry, M.

M. Berry, C. Storm, and W. van Saarloos, “Theory of unstable laser modes: edge waves and fractality,” Opt. Commun.197, 393–402 (2001).
[CrossRef]

Berry, M. V.

M. V. Berry, “Diffractals,” J. Phys. A: Math. Gen.12, 781–797 (1979).
[CrossRef]

Birch, J.

K. Jarrendahl, M. Dulea, J. Birch, and J.-E. Sundgren, “X-ray diffraction from amorphous Ge/Si Cantor superlattices,” Phys. Rev. B51, 7621–7631 (1995).
[CrossRef]

Carmes, C. R.

B. Dubuc, J. F. Quiniou, C. R. Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimensions of profiles,” Phys. Rev. A39, 1500–1512 (1989).
[CrossRef] [PubMed]

Cloitre, M.

C. Allain and M. Cloitre, “Spatial spectrum of a general family of self-similar arrays,” Phys. Rev. A36, 5751–5757 (1987).
[CrossRef] [PubMed]

C. Allain and M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B33, 3566–3569 (1986).
[CrossRef]

Dubuc, B.

B. Dubuc, J. F. Quiniou, C. R. Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimensions of profiles,” Phys. Rev. A39, 1500–1512 (1989).
[CrossRef] [PubMed]

Dulea, M.

K. Jarrendahl, M. Dulea, J. Birch, and J.-E. Sundgren, “X-ray diffraction from amorphous Ge/Si Cantor superlattices,” Phys. Rev. B51, 7621–7631 (1995).
[CrossRef]

Furlan, W. D.

Gimenez, F.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Guerin, C.

C. Guerin and M. Holschneider, “Scattering on fractal measures,” J. Phys. A29, 7651–7667 (1996).
[CrossRef]

Hamburger-Lidar, D. A.

D. A. Hamburger-Lidar, “Elastic scattering by deterministic and random fractals: Self-affinity of the diffraction spectrum,” Phys. Rev. E54, 354–370 (1996).
[CrossRef]

Holschneider, M.

C. Guerin and M. Holschneider, “Scattering on fractal measures,” J. Phys. A29, 7651–7667 (1996).
[CrossRef]

Hou, B.

B. Hou, G. Xu, W. Wen, and G. K. L. Wong, “Diffraction by an optical fractal grating,” Appl. Phys. Lett.85, 6125–6127 (2004).
[CrossRef]

Jaggard, A. D.

Jaggard, D. L.

H. Aubert and D. L. Jaggard, “Wavelet analysis of transients in fractal superlattices,” IEEE Trans. Antennas Propag.50, 338–345 (2002).
[CrossRef]

A. D. Jaggard and D. L. Jaggard, “Scattering from fractal superlattices with variable lacunarity,” J. Opt. Soc. Am. A15, 1626–1635 (1998).
[CrossRef]

Jarrendahl, K.

K. Jarrendahl, M. Dulea, J. Birch, and J.-E. Sundgren, “X-ray diffraction from amorphous Ge/Si Cantor superlattices,” Phys. Rev. B51, 7621–7631 (1995).
[CrossRef]

Lehman, M.

M. Lehman, “Fractal diffraction grating built through rectangular domains,” Opt. Commun.195, 11–26 (2001).
[CrossRef]

Mandelbrot, B. B.

B. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman, 1982).

McDonald, G. S.

G. H. C. New, M. A. Yates, J. P. Woerdman, and G. S. McDonald, “Diffractive origin of fractal resonator modes,” Opt. Commun.193, 261–266 (2001).
[CrossRef]

Monsoriu, J. A.

New, G. H. C.

G. H. C. New, M. A. Yates, J. P. Woerdman, and G. S. McDonald, “Diffractive origin of fractal resonator modes,” Opt. Commun.193, 261–266 (2001).
[CrossRef]

Pons, Amparo

Quiniou, J. F.

B. Dubuc, J. F. Quiniou, C. R. Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimensions of profiles,” Phys. Rev. A39, 1500–1512 (1989).
[CrossRef] [PubMed]

Rodriguez, C. J. Z.

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

Saavedra, G.

Senthilkumaran, P.

Stanley, H. E.

A. L. Barabasi and H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge University, 1995).
[CrossRef]

Storm, C.

M. Berry, C. Storm, and W. van Saarloos, “Theory of unstable laser modes: edge waves and fractality,” Opt. Commun.197, 393–402 (2001).
[CrossRef]

Sundgren, J.-E.

K. Jarrendahl, M. Dulea, J. Birch, and J.-E. Sundgren, “X-ray diffraction from amorphous Ge/Si Cantor superlattices,” Phys. Rev. B51, 7621–7631 (1995).
[CrossRef]

Tricot, C.

B. Dubuc, J. F. Quiniou, C. R. Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimensions of profiles,” Phys. Rev. A39, 1500–1512 (1989).
[CrossRef] [PubMed]

van Saarloos, W.

M. Berry, C. Storm, and W. van Saarloos, “Theory of unstable laser modes: edge waves and fractality,” Opt. Commun.197, 393–402 (2001).
[CrossRef]

Verma, R.

Vicsek, Tamas

Tamas Vicsek, Fractal Growth Phenomena (World Scientific, 1992).
[CrossRef]

Wen, W.

B. Hou, G. Xu, W. Wen, and G. K. L. Wong, “Diffraction by an optical fractal grating,” Appl. Phys. Lett.85, 6125–6127 (2004).
[CrossRef]

Woerdman, J. P.

G. H. C. New, M. A. Yates, J. P. Woerdman, and G. S. McDonald, “Diffractive origin of fractal resonator modes,” Opt. Commun.193, 261–266 (2001).
[CrossRef]

Wong, G. K. L.

B. Hou, G. Xu, W. Wen, and G. K. L. Wong, “Diffraction by an optical fractal grating,” Appl. Phys. Lett.85, 6125–6127 (2004).
[CrossRef]

Xu, G.

B. Hou, G. Xu, W. Wen, and G. K. L. Wong, “Diffraction by an optical fractal grating,” Appl. Phys. Lett.85, 6125–6127 (2004).
[CrossRef]

Yates, M. A.

G. H. C. New, M. A. Yates, J. P. Woerdman, and G. S. McDonald, “Diffractive origin of fractal resonator modes,” Opt. Commun.193, 261–266 (2001).
[CrossRef]

Zucker, S. W.

B. Dubuc, J. F. Quiniou, C. R. Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimensions of profiles,” Phys. Rev. A39, 1500–1512 (1989).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

B. Hou, G. Xu, W. Wen, and G. K. L. Wong, “Diffraction by an optical fractal grating,” Appl. Phys. Lett.85, 6125–6127 (2004).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

H. Aubert and D. L. Jaggard, “Wavelet analysis of transients in fractal superlattices,” IEEE Trans. Antennas Propag.50, 338–345 (2002).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. A (1)

C. Guerin and M. Holschneider, “Scattering on fractal measures,” J. Phys. A29, 7651–7667 (1996).
[CrossRef]

J. Phys. A: Math. Gen. (1)

M. V. Berry, “Diffractals,” J. Phys. A: Math. Gen.12, 781–797 (1979).
[CrossRef]

Opt. Commun. (4)

G. H. C. New, M. A. Yates, J. P. Woerdman, and G. S. McDonald, “Diffractive origin of fractal resonator modes,” Opt. Commun.193, 261–266 (2001).
[CrossRef]

M. Berry, C. Storm, and W. van Saarloos, “Theory of unstable laser modes: edge waves and fractality,” Opt. Commun.197, 393–402 (2001).
[CrossRef]

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

M. Lehman, “Fractal diffraction grating built through rectangular domains,” Opt. Commun.195, 11–26 (2001).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. A (2)

C. Allain and M. Cloitre, “Spatial spectrum of a general family of self-similar arrays,” Phys. Rev. A36, 5751–5757 (1987).
[CrossRef] [PubMed]

B. Dubuc, J. F. Quiniou, C. R. Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimensions of profiles,” Phys. Rev. A39, 1500–1512 (1989).
[CrossRef] [PubMed]

Phys. Rev. B (2)

C. Allain and M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B33, 3566–3569 (1986).
[CrossRef]

K. Jarrendahl, M. Dulea, J. Birch, and J.-E. Sundgren, “X-ray diffraction from amorphous Ge/Si Cantor superlattices,” Phys. Rev. B51, 7621–7631 (1995).
[CrossRef]

Phys. Rev. E (1)

D. A. Hamburger-Lidar, “Elastic scattering by deterministic and random fractals: Self-affinity of the diffraction spectrum,” Phys. Rev. E54, 354–370 (1996).
[CrossRef]

Other (5)

D. Bak, S. P. Kim, S. K. Kim, K. -S. Soh, and J. H. Yee, “Fractal diffraction grating,” 1–7 http://arxiv.org/abs/physics/9802007 .

B. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman, 1982).

A. L. Barabasi and H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge University, 1995).
[CrossRef]

Tamas Vicsek, Fractal Growth Phenomena (World Scientific, 1992).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

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Figures (3)

Fig. 1
Fig. 1

n = 4 Cantor grating (a) without disorder and (b) with q = 1/4 or 25 % disorder.

Fig. 2
Fig. 2

Schematic representation of the 4f arrangement.

Fig. 3
Fig. 3

(a) n = 4 Cantor grating with 25 % disorder; (b) Corresponding diffraction pattern; (c) – (f): Reconstructions from nested clips of first secondary band (refer text for details).

Tables (1)

Tables Icon

Table 1df〉 as a function of disorder q for Cantor gratings of order n.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

G n ( 0 ) ( x ) = R n ( x ) * Δ n ( 0 ) ( x ) , n 1 ,
G n t ( x ) = R n ( x ) * Δ n t ( x ) , n 1 ,
= R n ( x ) * i = 1 N X ( i ) δ ( x x i ) x i = ± 2 a / 3 ± 2 a / 3 2 + ± 2 a / 3 n .
G n p ( x ) = R n ( x ) * Δ n 1 p ( x ) , n 1 ,
= R n ( x ) * i = 1 M ( Y ( i ) 1 ) δ ( x y i ) , y i = ± 2 a / 3 ± 2 a / 3 2 ± ± 2 a / 3 n 1 .
G n q ( x ) = R n ( x ) * ( Δ n q ( x ) + Δ n 1 q ( x ) ) , n 1 ,
= R n ( x ) * ( i = 1 N Z ( i ) δ ( x x i ) + ( Z ( i ) 1 ) δ ( x y i ) ) ,
I n 0 ( f ) = | a a d x e i 2 π f x G n 0 ( x ) | 2 = [ ( 2 3 ) n 2 a sinc ( 2 π a f 3 n ) ] 2 { m = 1 n cos ( 4 π a f 3 m ) } 2 ,
S ( f ) ¯ = 1 Δ f f f + Δ f d q S ( q ) f d f .

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