Abstract

Cantor diffractals are waves that have encountered a Cantor grating. In this paper, we report an important property of Cantor diffractals, namely that of redundancy. We observe that the Fraunhofer diffraction pattern comprises of several bands, each containing complete information about the fractal aperture. This redundancy allows for a faithful reconstruction of the Cantor grating by an inverse Fourier transformation of an arbitrary band.

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  1. B. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman, New York, 1982).
  2. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications (Wiley, 2003).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  7. 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 .
  8. Hou Bo, Gu Xu, W. Wen, and G. K. L. Wong, “Diffraction by an optical fractal grating,” Appl. Phys. Lett. 85, 6125–6127 (2004).
    [CrossRef]
  9. G. Chabassier, B. Angeli, F. Heliodore, and A. Le Mehaute, “Optical wave diffraction on fractal objects,” Pure Appl. Opt. 1, 41–54 (1992).
    [CrossRef]
  10. M. Born, Principles of Optics (Pergamon Press, Oxford, 1980).
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  12. C. Allain and M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B 33, 3566–3569 (1986).
    [CrossRef]
  13. C. Allain and M. Cloitre, “Spatial spectrum of a general family of self-similar arrays,” Phys. Rev. A 36, 5751–5757 (1987).
    [CrossRef] [PubMed]
  14. D. A. Hamburger-Lidar, “Elastic scattering by deterministic and random fractals: Self-affinity of the diffraction spectrum,” Phys. Rev. E 54, 354–370 (1996).
    [CrossRef]
  15. M. Lehman, “Fractal diffraction grating built through rectangular domains,” Opt. Commun. 195, 11–26 (2001).
    [CrossRef]
  16. C. Guerin and M. Holschneider, “Scattering on fractal measures,” J. Phys. A 29, 7651–7667 (1996).
    [CrossRef]
  17. B. Dubuc, J. F. Quiniou, C. R. Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimensions of profiles,” Phys. Rev. A 39, 1500–1512 (1989).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  19. A. Ghatak, Optics (Tata McGraw-Hill, 2007).
  20. See “Application scenarios, tutorials, modules, and snippets” provided by LightTrans VirtualLab.
  21. We have observed redundancy in other deterministic fractals such as the Sierpinski carpet and the Gosper curve.
  22. J. A. Monsoriu, G. Saavedra, and W. D. Furlan, “Fractal zone plates with variable lacunarity,” Opt. Express 12, 4227–4234 (2004).
    [CrossRef] [PubMed]
  23. F. Gimenez, J. A. Monsoriu, W. D. Furlan, and Amparo Pons, “Fractal photon sieve,” Opt. Express 14, 11958–11963 (2006).
    [CrossRef] [PubMed]
  24. G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. 28, 971–973 (2003).
    [CrossRef] [PubMed]

2007 (1)

2006 (1)

2004 (2)

J. A. Monsoriu, G. Saavedra, and W. D. Furlan, “Fractal zone plates with variable lacunarity,” Opt. Express 12, 4227–4234 (2004).
[CrossRef] [PubMed]

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

2003 (1)

2001 (1)

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

1996 (2)

C. Guerin and M. Holschneider, “Scattering on fractal measures,” J. Phys. A 29, 7651–7667 (1996).
[CrossRef]

D. A. Hamburger-Lidar, “Elastic scattering by deterministic and random fractals: Self-affinity of the diffraction spectrum,” Phys. Rev. E 54, 354–370 (1996).
[CrossRef]

1992 (1)

G. Chabassier, B. Angeli, F. Heliodore, and A. Le Mehaute, “Optical wave diffraction on fractal objects,” Pure Appl. Opt. 1, 41–54 (1992).
[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. A 39, 1500–1512 (1989).
[CrossRef] [PubMed]

1987 (1)

C. Allain and M. Cloitre, “Spatial spectrum of a general family of self-similar arrays,” Phys. Rev. A 36, 5751–5757 (1987).
[CrossRef] [PubMed]

1986 (1)

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

1979 (1)

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

1968 (1)

1964 (1)

Allain, C.

C. Allain and M. Cloitre, “Spatial spectrum of a general family of self-similar arrays,” Phys. Rev. A 36, 5751–5757 (1987).
[CrossRef] [PubMed]

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

Angeli, B.

G. Chabassier, B. Angeli, F. Heliodore, and A. Le Mehaute, “Optical wave diffraction on fractal objects,” Pure Appl. Opt. 1, 41–54 (1992).
[CrossRef]

Berry, M. V.

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

Bo, Hou

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

Born, M.

M. Born, Principles of Optics (Pergamon Press, Oxford, 1980).

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. A 39, 1500–1512 (1989).
[CrossRef] [PubMed]

Chabassier, G.

G. Chabassier, B. Angeli, F. Heliodore, and A. Le Mehaute, “Optical wave diffraction on fractal objects,” Pure Appl. Opt. 1, 41–54 (1992).
[CrossRef]

Cloitre, M.

C. Allain and M. Cloitre, “Spatial spectrum of a general family of self-similar arrays,” Phys. Rev. A 36, 5751–5757 (1987).
[CrossRef] [PubMed]

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

Depine, R. A.

Dubuc, B.

B. Dubuc, J. F. Quiniou, C. R. Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimensions of profiles,” Phys. Rev. A 39, 1500–1512 (1989).
[CrossRef] [PubMed]

Falconer, K.

K. Falconer, Fractal Geometry: Mathematical Foundations and Applications (Wiley, 2003).

Furlan, W. D.

Gerritsen, H. J.

Ghatak, A.

A. Ghatak, Optics (Tata McGraw-Hill, 2007).

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. A 29, 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. E 54, 354–370 (1996).
[CrossRef]

Hannan, W. J.

Heliodore, F.

G. Chabassier, B. Angeli, F. Heliodore, and A. Le Mehaute, “Optical wave diffraction on fractal objects,” Pure Appl. Opt. 1, 41–54 (1992).
[CrossRef]

Holschneider, M.

C. Guerin and M. Holschneider, “Scattering on fractal measures,” J. Phys. A 29, 7651–7667 (1996).
[CrossRef]

Jurgens, H.

H. O. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fractals (New Frontiers of Science, Springer, 2004).

Le Mehaute, A.

G. Chabassier, B. Angeli, F. Heliodore, and A. Le Mehaute, “Optical wave diffraction on fractal objects,” Pure Appl. Opt. 1, 41–54 (1992).
[CrossRef]

Lehman, M.

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

Leith, E. N.

Mandelbrot, B. B.

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

Monsoriu, J. A.

Peitgen, H. O.

H. O. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fractals (New Frontiers of Science, Springer, 2004).

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. A 39, 1500–1512 (1989).
[CrossRef] [PubMed]

Ramberg, E. G.

Saavedra, G.

Saupe, D.

H. O. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fractals (New Frontiers of Science, Springer, 2004).

Skigin, D. C.

Tricot, C.

B. Dubuc, J. F. Quiniou, C. R. Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimensions of profiles,” Phys. Rev. A 39, 1500–1512 (1989).
[CrossRef] [PubMed]

Upatnieks, J.

Wen, W.

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

Wong, G. K. L.

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

Xu, Gu

Hou Bo, Gu Xu, W. Wen, and G. K. L. Wong, “Diffraction by an optical fractal grating,” Appl. Phys. Lett. 85, 6125–6127 (2004).
[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. A 39, 1500–1512 (1989).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

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

J. Opt. Soc. Am. (1)

J. Phys. A (1)

C. Guerin and M. Holschneider, “Scattering on fractal measures,” J. Phys. A 29, 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. (1)

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

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. A (2)

B. Dubuc, J. F. Quiniou, C. R. Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimensions of profiles,” Phys. Rev. A 39, 1500–1512 (1989).
[CrossRef] [PubMed]

C. Allain and M. Cloitre, “Spatial spectrum of a general family of self-similar arrays,” Phys. Rev. A 36, 5751–5757 (1987).
[CrossRef] [PubMed]

Phys. Rev. B (1)

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

Phys. Rev. E (1)

D. A. Hamburger-Lidar, “Elastic scattering by deterministic and random fractals: Self-affinity of the diffraction spectrum,” Phys. Rev. E 54, 354–370 (1996).
[CrossRef]

Pure Appl. Opt. (1)

G. Chabassier, B. Angeli, F. Heliodore, and A. Le Mehaute, “Optical wave diffraction on fractal objects,” Pure Appl. Opt. 1, 41–54 (1992).
[CrossRef]

Other (9)

M. Born, Principles of Optics (Pergamon Press, Oxford, 1980).

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

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, New York, 1982).

K. Falconer, Fractal Geometry: Mathematical Foundations and Applications (Wiley, 2003).

H. O. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fractals (New Frontiers of Science, Springer, 2004).

A. Ghatak, Optics (Tata McGraw-Hill, 2007).

See “Application scenarios, tutorials, modules, and snippets” provided by LightTrans VirtualLab.

We have observed redundancy in other deterministic fractals such as the Sierpinski carpet and the Gosper curve.

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