Abstract

The reflectance of corrugated surfaces with a fractal distribution of grooves is investigated. Triadic and polyadic Cantor fractal distributions are considered, and the reflected intensity is compared with that of the corresponding periodic structure. The self-similarity property of the response is analyzed when varying the depth of the grooves and the lacunarity parameter. The results confirm that the response is self-similar for the whole range of depths considered, and this property is also maintained for all values of the lacunarity parameter.

© 2007 Optical Society of America

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References

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  1. R. Petit, “Diffraction gratings,” C. r. hebd. Seanc. Acad. Sci., Paris, 260, 4454 (1965).
  2. R. C. Hollins and D. L. Jordan, “Measurments of 10.6µm radiation scattered by a pseudo-random surface of rectangular grooves,” Optica Acta,  30, 1725–1734 (1983).
    [Crossref]
  3. J. R. Andrewartha, J. R. Fox, and I. J. Wilson, “Resonance anomalies in the lamellar grating,” Optica Acta,  26, 69–89 (1977).
    [Crossref]
  4. A. Wirgin and A. A. Maradudin, “Resonant enhancement of the electric field in the grooves of bare metallic gratings exposed to S-polarized light,” Phys. Rev. B,  31, 5573–5576 (1985).
    [Crossref]
  5. E. G. Loewen, M. Nevière, and D. Maystre, “Efficiency optimization of rectangular groove gratings for use in the visible and IR regions,” Appl. Opt. 18, 2262–2266 (1979).
    [Crossref] [PubMed]
  6. L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
    [Crossref]
  7. T. J. Park, H. J. Eom, and K. Yoshitomi, “Analysis of TM scattering from finite rectangular grooves in a conducting plane,” J. Opt. Soc. Am. A 10, 905–911 (1993).
    [Crossref]
  8. D. Maystre, “Rigorous theory of light scattering from rough surfaces,” J. Opt. 5, 43–51 (1984).
    [Crossref]
  9. Y. L. Kok, “A boundary value solution to electromagnetic scattering by a rectangular groove in a ground plane,” J. Opt. Soc. Am. A 9, 302–311 (1992).
    [Crossref]
  10. T.-M. Wang and H. Ling, “A connection algorithm on the problem of EM scattering from arbitrary cavities,” J. EM Waves and Applics. 5, 301–314 (1991).
  11. R. A. Depine and D. C. Skigin, “Scattering from metallic surfaces having a finite number of rectangular grooves,” J. Opt. Soc. Am. A 11, 2844–2850 (1994).
    [Crossref]
  12. D. C. Skigin, V. V. Veremey, and R. Mittra, “Superdirective radiation from finite gratings of rectangular grooves”, IEEE Trans. Antennas Propag. 47, 376–383 (1999).
    [Crossref]
  13. A. N. Fantino, S. I. Grosz, and D. C. Skigin, “Resonant effect in periodic gratings comprising a finite number of grooves in each period,” Phys. Rev. E 64, 016605 (2001).
    [Crossref]
  14. S. I. Grosz, D. C. Skigin, and A. N. Fantino, “Resonant effects in compound diffraction gratings: influence of the geometrical parameters of the surface,” Phys. Rev. E 65, 056619 (2002).
    [Crossref]
  15. D. C. Skigin and R. A. Depine, “Diffraction by dual-period gratings,” Appl. Opt. 46, 1385–1391 (2007).
    [Crossref] [PubMed]
  16. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1982.
  17. Y. Sakurada, J. Uozumi, and T Asakura, “Fresnel diffraction by 1-D regular fractals,” Pure Appl. Opt. 1, 29–40 (1992).
    [Crossref]
  18. 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).
  19. A. Lakhtakia, N. S. Holter, V. K. Varadan, and V. V. Varadan, “Self-similarity in diffraction by a self-similar fractal screen,” IEEE Transactions on Antennas and Propagation 35, 236–239 (1987).
    [Crossref]
  20. F. Giménez, J.A. Monsoriu, W.D. Furlan, and A. Pons, “Fractal photon sieves,” Opt. Express14, 11958–11963 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-11958
    [Crossref] [PubMed]
  21. S. A. Ledesma, C. C. Iemmi, and V. L. Brudny, “Scaling properties of the scattered field produced by fractal gratings,” Opt. Commun. 144, 292–298 (1997).
    [Crossref]
  22. A. D. Jaggard and D. L. Jaggard, “Scattering from fractal superlattices with variable lacunarity,” J. Opt. Soc. Am. A 15, 1626–1635 (1998).
    [Crossref]
  23. R. E. Plotnick, R. H. Gardner, W. W. Hargrove, K. Prestegaard, and M. Perlmutter, “Lacunarity analysis: A general technique for the analysis of spatial patterns,” Phys. Rev. E 53, 5461–5468 (1996).
    [Crossref]
  24. J. A. Monsoriu, G. Saavedra, and W. D. Furlan, “Fractal zone plates with variable lacunarity,” Opt. Express12, 4227–4234 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-18-4227
    [Crossref] [PubMed]

2007 (1)

2002 (1)

S. I. Grosz, D. C. Skigin, and A. N. Fantino, “Resonant effects in compound diffraction gratings: influence of the geometrical parameters of the surface,” Phys. Rev. E 65, 056619 (2002).
[Crossref]

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).

A. N. Fantino, S. I. Grosz, and D. C. Skigin, “Resonant effect in periodic gratings comprising a finite number of grooves in each period,” Phys. Rev. E 64, 016605 (2001).
[Crossref]

1999 (1)

D. C. Skigin, V. V. Veremey, and R. Mittra, “Superdirective radiation from finite gratings of rectangular grooves”, IEEE Trans. Antennas Propag. 47, 376–383 (1999).
[Crossref]

1998 (1)

1997 (1)

S. A. Ledesma, C. C. Iemmi, and V. L. Brudny, “Scaling properties of the scattered field produced by fractal gratings,” Opt. Commun. 144, 292–298 (1997).
[Crossref]

1996 (1)

R. E. Plotnick, R. H. Gardner, W. W. Hargrove, K. Prestegaard, and M. Perlmutter, “Lacunarity analysis: A general technique for the analysis of spatial patterns,” Phys. Rev. E 53, 5461–5468 (1996).
[Crossref]

1994 (1)

1993 (2)

1992 (2)

Y. L. Kok, “A boundary value solution to electromagnetic scattering by a rectangular groove in a ground plane,” J. Opt. Soc. Am. A 9, 302–311 (1992).
[Crossref]

Y. Sakurada, J. Uozumi, and T Asakura, “Fresnel diffraction by 1-D regular fractals,” Pure Appl. Opt. 1, 29–40 (1992).
[Crossref]

1991 (1)

T.-M. Wang and H. Ling, “A connection algorithm on the problem of EM scattering from arbitrary cavities,” J. EM Waves and Applics. 5, 301–314 (1991).

1987 (1)

A. Lakhtakia, N. S. Holter, V. K. Varadan, and V. V. Varadan, “Self-similarity in diffraction by a self-similar fractal screen,” IEEE Transactions on Antennas and Propagation 35, 236–239 (1987).
[Crossref]

1985 (1)

A. Wirgin and A. A. Maradudin, “Resonant enhancement of the electric field in the grooves of bare metallic gratings exposed to S-polarized light,” Phys. Rev. B,  31, 5573–5576 (1985).
[Crossref]

1984 (1)

D. Maystre, “Rigorous theory of light scattering from rough surfaces,” J. Opt. 5, 43–51 (1984).
[Crossref]

1983 (1)

R. C. Hollins and D. L. Jordan, “Measurments of 10.6µm radiation scattered by a pseudo-random surface of rectangular grooves,” Optica Acta,  30, 1725–1734 (1983).
[Crossref]

1979 (1)

1977 (1)

J. R. Andrewartha, J. R. Fox, and I. J. Wilson, “Resonance anomalies in the lamellar grating,” Optica Acta,  26, 69–89 (1977).
[Crossref]

1965 (1)

R. Petit, “Diffraction gratings,” C. r. hebd. Seanc. Acad. Sci., Paris, 260, 4454 (1965).

Andrewartha, J. R.

J. R. Andrewartha, J. R. Fox, and I. J. Wilson, “Resonance anomalies in the lamellar grating,” Optica Acta,  26, 69–89 (1977).
[Crossref]

Asakura, T

Y. Sakurada, J. Uozumi, and T Asakura, “Fresnel diffraction by 1-D regular fractals,” Pure Appl. Opt. 1, 29–40 (1992).
[Crossref]

Brudny, V. L.

S. A. Ledesma, C. C. Iemmi, and V. L. Brudny, “Scaling properties of the scattered field produced by fractal gratings,” Opt. Commun. 144, 292–298 (1997).
[Crossref]

Depine, R. A.

Eom, H. J.

Fantino, A. N.

S. I. Grosz, D. C. Skigin, and A. N. Fantino, “Resonant effects in compound diffraction gratings: influence of the geometrical parameters of the surface,” Phys. Rev. E 65, 056619 (2002).
[Crossref]

A. N. Fantino, S. I. Grosz, and D. C. Skigin, “Resonant effect in periodic gratings comprising a finite number of grooves in each period,” Phys. Rev. E 64, 016605 (2001).
[Crossref]

Fox, J. R.

J. R. Andrewartha, J. R. Fox, and I. J. Wilson, “Resonance anomalies in the lamellar grating,” Optica Acta,  26, 69–89 (1977).
[Crossref]

Furlan, W. D.

J. A. Monsoriu, G. Saavedra, and W. D. Furlan, “Fractal zone plates with variable lacunarity,” Opt. Express12, 4227–4234 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-18-4227
[Crossref] [PubMed]

Furlan, W.D.

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).

F. Giménez, J.A. Monsoriu, W.D. Furlan, and A. Pons, “Fractal photon sieves,” Opt. Express14, 11958–11963 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-11958
[Crossref] [PubMed]

Gardner, R. H.

R. E. Plotnick, R. H. Gardner, W. W. Hargrove, K. Prestegaard, and M. Perlmutter, “Lacunarity analysis: A general technique for the analysis of spatial patterns,” Phys. Rev. E 53, 5461–5468 (1996).
[Crossref]

Giménez, F.

F. Giménez, J.A. Monsoriu, W.D. Furlan, and A. Pons, “Fractal photon sieves,” Opt. Express14, 11958–11963 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-11958
[Crossref] [PubMed]

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).

Grosz, S. I.

S. I. Grosz, D. C. Skigin, and A. N. Fantino, “Resonant effects in compound diffraction gratings: influence of the geometrical parameters of the surface,” Phys. Rev. E 65, 056619 (2002).
[Crossref]

A. N. Fantino, S. I. Grosz, and D. C. Skigin, “Resonant effect in periodic gratings comprising a finite number of grooves in each period,” Phys. Rev. E 64, 016605 (2001).
[Crossref]

Hargrove, W. W.

R. E. Plotnick, R. H. Gardner, W. W. Hargrove, K. Prestegaard, and M. Perlmutter, “Lacunarity analysis: A general technique for the analysis of spatial patterns,” Phys. Rev. E 53, 5461–5468 (1996).
[Crossref]

Hollins, R. C.

R. C. Hollins and D. L. Jordan, “Measurments of 10.6µm radiation scattered by a pseudo-random surface of rectangular grooves,” Optica Acta,  30, 1725–1734 (1983).
[Crossref]

Holter, N. S.

A. Lakhtakia, N. S. Holter, V. K. Varadan, and V. V. Varadan, “Self-similarity in diffraction by a self-similar fractal screen,” IEEE Transactions on Antennas and Propagation 35, 236–239 (1987).
[Crossref]

Iemmi, C. C.

S. A. Ledesma, C. C. Iemmi, and V. L. Brudny, “Scaling properties of the scattered field produced by fractal gratings,” Opt. Commun. 144, 292–298 (1997).
[Crossref]

Jaggard, A. D.

Jaggard, D. L.

Jordan, D. L.

R. C. Hollins and D. L. Jordan, “Measurments of 10.6µm radiation scattered by a pseudo-random surface of rectangular grooves,” Optica Acta,  30, 1725–1734 (1983).
[Crossref]

Kok, Y. L.

Lakhtakia, A.

A. Lakhtakia, N. S. Holter, V. K. Varadan, and V. V. Varadan, “Self-similarity in diffraction by a self-similar fractal screen,” IEEE Transactions on Antennas and Propagation 35, 236–239 (1987).
[Crossref]

Ledesma, S. A.

S. A. Ledesma, C. C. Iemmi, and V. L. Brudny, “Scaling properties of the scattered field produced by fractal gratings,” Opt. Commun. 144, 292–298 (1997).
[Crossref]

Li, L.

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[Crossref]

Ling, H.

T.-M. Wang and H. Ling, “A connection algorithm on the problem of EM scattering from arbitrary cavities,” J. EM Waves and Applics. 5, 301–314 (1991).

Loewen, E. G.

Mandelbrot, B.

B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1982.

Maradudin, A. A.

A. Wirgin and A. A. Maradudin, “Resonant enhancement of the electric field in the grooves of bare metallic gratings exposed to S-polarized light,” Phys. Rev. B,  31, 5573–5576 (1985).
[Crossref]

Maystre, D.

Mittra, R.

D. C. Skigin, V. V. Veremey, and R. Mittra, “Superdirective radiation from finite gratings of rectangular grooves”, IEEE Trans. Antennas Propag. 47, 376–383 (1999).
[Crossref]

Monsoriu, J. A.

J. A. Monsoriu, G. Saavedra, and W. D. Furlan, “Fractal zone plates with variable lacunarity,” Opt. Express12, 4227–4234 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-18-4227
[Crossref] [PubMed]

Monsoriu, J.A.

F. Giménez, J.A. Monsoriu, W.D. Furlan, and A. Pons, “Fractal photon sieves,” Opt. Express14, 11958–11963 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-11958
[Crossref] [PubMed]

Nevière, M.

Park, T. J.

Perlmutter, M.

R. E. Plotnick, R. H. Gardner, W. W. Hargrove, K. Prestegaard, and M. Perlmutter, “Lacunarity analysis: A general technique for the analysis of spatial patterns,” Phys. Rev. E 53, 5461–5468 (1996).
[Crossref]

Petit, R.

R. Petit, “Diffraction gratings,” C. r. hebd. Seanc. Acad. Sci., Paris, 260, 4454 (1965).

Plotnick, R. E.

R. E. Plotnick, R. H. Gardner, W. W. Hargrove, K. Prestegaard, and M. Perlmutter, “Lacunarity analysis: A general technique for the analysis of spatial patterns,” Phys. Rev. E 53, 5461–5468 (1996).
[Crossref]

Pons, A.

F. Giménez, J.A. Monsoriu, W.D. Furlan, and A. Pons, “Fractal photon sieves,” Opt. Express14, 11958–11963 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-11958
[Crossref] [PubMed]

Prestegaard, K.

R. E. Plotnick, R. H. Gardner, W. W. Hargrove, K. Prestegaard, and M. Perlmutter, “Lacunarity analysis: A general technique for the analysis of spatial patterns,” Phys. Rev. E 53, 5461–5468 (1996).
[Crossref]

Saavedra, G.

J. A. Monsoriu, G. Saavedra, and W. D. Furlan, “Fractal zone plates with variable lacunarity,” Opt. Express12, 4227–4234 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-18-4227
[Crossref] [PubMed]

Sakurada, Y.

Y. Sakurada, J. Uozumi, and T Asakura, “Fresnel diffraction by 1-D regular fractals,” Pure Appl. Opt. 1, 29–40 (1992).
[Crossref]

Skigin, D. C.

D. C. Skigin and R. A. Depine, “Diffraction by dual-period gratings,” Appl. Opt. 46, 1385–1391 (2007).
[Crossref] [PubMed]

S. I. Grosz, D. C. Skigin, and A. N. Fantino, “Resonant effects in compound diffraction gratings: influence of the geometrical parameters of the surface,” Phys. Rev. E 65, 056619 (2002).
[Crossref]

A. N. Fantino, S. I. Grosz, and D. C. Skigin, “Resonant effect in periodic gratings comprising a finite number of grooves in each period,” Phys. Rev. E 64, 016605 (2001).
[Crossref]

D. C. Skigin, V. V. Veremey, and R. Mittra, “Superdirective radiation from finite gratings of rectangular grooves”, IEEE Trans. Antennas Propag. 47, 376–383 (1999).
[Crossref]

R. A. Depine and D. C. Skigin, “Scattering from metallic surfaces having a finite number of rectangular grooves,” J. Opt. Soc. Am. A 11, 2844–2850 (1994).
[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).

Uozumi, J.

Y. Sakurada, J. Uozumi, and T Asakura, “Fresnel diffraction by 1-D regular fractals,” Pure Appl. Opt. 1, 29–40 (1992).
[Crossref]

Varadan, V. K.

A. Lakhtakia, N. S. Holter, V. K. Varadan, and V. V. Varadan, “Self-similarity in diffraction by a self-similar fractal screen,” IEEE Transactions on Antennas and Propagation 35, 236–239 (1987).
[Crossref]

Varadan, V. V.

A. Lakhtakia, N. S. Holter, V. K. Varadan, and V. V. Varadan, “Self-similarity in diffraction by a self-similar fractal screen,” IEEE Transactions on Antennas and Propagation 35, 236–239 (1987).
[Crossref]

Veremey, V. V.

D. C. Skigin, V. V. Veremey, and R. Mittra, “Superdirective radiation from finite gratings of rectangular grooves”, IEEE Trans. Antennas Propag. 47, 376–383 (1999).
[Crossref]

Wang, T.-M.

T.-M. Wang and H. Ling, “A connection algorithm on the problem of EM scattering from arbitrary cavities,” J. EM Waves and Applics. 5, 301–314 (1991).

Wilson, I. J.

J. R. Andrewartha, J. R. Fox, and I. J. Wilson, “Resonance anomalies in the lamellar grating,” Optica Acta,  26, 69–89 (1977).
[Crossref]

Wirgin, A.

A. Wirgin and A. A. Maradudin, “Resonant enhancement of the electric field in the grooves of bare metallic gratings exposed to S-polarized light,” Phys. Rev. B,  31, 5573–5576 (1985).
[Crossref]

Yoshitomi, K.

Appl. Opt. (2)

C. r. hebd. Seanc. Acad. Sci., Paris (1)

R. Petit, “Diffraction gratings,” C. r. hebd. Seanc. Acad. Sci., Paris, 260, 4454 (1965).

IEEE Trans. Antennas Propag. (1)

D. C. Skigin, V. V. Veremey, and R. Mittra, “Superdirective radiation from finite gratings of rectangular grooves”, IEEE Trans. Antennas Propag. 47, 376–383 (1999).
[Crossref]

IEEE Transactions on Antennas and Propagation (1)

A. Lakhtakia, N. S. Holter, V. K. Varadan, and V. V. Varadan, “Self-similarity in diffraction by a self-similar fractal screen,” IEEE Transactions on Antennas and Propagation 35, 236–239 (1987).
[Crossref]

J. EM Waves and Applics. (1)

T.-M. Wang and H. Ling, “A connection algorithm on the problem of EM scattering from arbitrary cavities,” J. EM Waves and Applics. 5, 301–314 (1991).

J. Mod. Opt. (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).

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[Crossref]

J. Opt. (1)

D. Maystre, “Rigorous theory of light scattering from rough surfaces,” J. Opt. 5, 43–51 (1984).
[Crossref]

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

S. A. Ledesma, C. C. Iemmi, and V. L. Brudny, “Scaling properties of the scattered field produced by fractal gratings,” Opt. Commun. 144, 292–298 (1997).
[Crossref]

Optica Acta (2)

R. C. Hollins and D. L. Jordan, “Measurments of 10.6µm radiation scattered by a pseudo-random surface of rectangular grooves,” Optica Acta,  30, 1725–1734 (1983).
[Crossref]

J. R. Andrewartha, J. R. Fox, and I. J. Wilson, “Resonance anomalies in the lamellar grating,” Optica Acta,  26, 69–89 (1977).
[Crossref]

Phys. Rev. B (1)

A. Wirgin and A. A. Maradudin, “Resonant enhancement of the electric field in the grooves of bare metallic gratings exposed to S-polarized light,” Phys. Rev. B,  31, 5573–5576 (1985).
[Crossref]

Phys. Rev. E (3)

A. N. Fantino, S. I. Grosz, and D. C. Skigin, “Resonant effect in periodic gratings comprising a finite number of grooves in each period,” Phys. Rev. E 64, 016605 (2001).
[Crossref]

S. I. Grosz, D. C. Skigin, and A. N. Fantino, “Resonant effects in compound diffraction gratings: influence of the geometrical parameters of the surface,” Phys. Rev. E 65, 056619 (2002).
[Crossref]

R. E. Plotnick, R. H. Gardner, W. W. Hargrove, K. Prestegaard, and M. Perlmutter, “Lacunarity analysis: A general technique for the analysis of spatial patterns,” Phys. Rev. E 53, 5461–5468 (1996).
[Crossref]

Pure Appl. Opt. (1)

Y. Sakurada, J. Uozumi, and T Asakura, “Fresnel diffraction by 1-D regular fractals,” Pure Appl. Opt. 1, 29–40 (1992).
[Crossref]

Other (3)

B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1982.

J. A. Monsoriu, G. Saavedra, and W. D. Furlan, “Fractal zone plates with variable lacunarity,” Opt. Express12, 4227–4234 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-18-4227
[Crossref] [PubMed]

F. Giménez, J.A. Monsoriu, W.D. Furlan, and A. Pons, “Fractal photon sieves,” Opt. Express14, 11958–11963 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-11958
[Crossref] [PubMed]

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Figures (7)

Fig. 1.
Fig. 1.

Triadic Cantor set for the first levels of growth, S. The structure for S=0 is the initiator and the one corresponding to S=1 is the generator. Black regions correspond to the grooves etched in the fractal metallic grating (see Fig. 2)

Fig. 2.
Fig. 2.

(a) Fractal (S=3) and (b) periodic metallic gratings.

Fig. 3.
Fig. 3.

Angular reflected response of a metallic plane with grooves with fractal distribution for different steps S (top plots), and with the corresponding periodic distribution as shown in Fig. 1 (bottom plots).

Fig. 4.
Fig. 4.

Correlation coefficient as a function of log3(γ), for the same three fractal structures considered in Fig. 3.

Fig. 5.
Fig. 5.

Grey-scale maps of the system angular reflected intensity (in dBs) as a function of α and of the depth of the grooves h/a. (a) S=2; (b) S=3.

Fig. 6.
Fig. 6.

First steps of the development of polyadic, N=4, symmetrical generalized Cantor sets. The definitions of the scale factor γ and of the lacunarity parameter ε characterizing polyadic Cantor sets are also shown.

Fig. 7.
Fig. 7.

Twist plots, that is, gray-scale representations of the reflected intensity (in dB) as a function of α/k 0 and of the lacunarity parameter ε for the polyadic Cantor prefractal distributions for (a) S=1 and (b) S=2, for N=4 and γ=0.1.

Equations (11)

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

f inc ( x , y ) = e i ( α 0 x β 0 y ) ,
f spec μ ( x , y ) = ( 1 ) j e i ( α 0 x β 0 y ) ; j = { 1 for s polarization 0 for p polarization ,
f scatt μ ( x , y ) = R μ ( α ) e i ( α x + β y ) d α .
f s ( x , y ) = m = 1 a m , l sin [ μ m , l ( y + h ) ] sin [ m π c l ( x x l ) ] ,
f p ( x , y ) = m = 0 b m , l cos [ μ m , l ( y + h ) ] cos [ m π c l ( x x l ) ] .
μ m , l = { k 0 2 ( m π c l ) 2 if k 0 2 > ( m π c l ) 2 i ( m π c l ) 2 k 0 2 if k 0 2 < ( m π c l ) 2 ,
α k 0 = α 0 k 0 + n λ a 3 s 2 ,
C ( γ ) = ( I ( α ) I ̅ ) ( I ( α γ ) I ̅ γ ) d α [ ( I ( α ) I ̅ ) 2 d α ( I ( α γ ) I ̅ γ ) 2 d α ] 1 2 ,
ε max = { 1 N γ N 2 even N 1 N γ N 3 odd N ,
ε reg = 1 N γ N 1 .
α k 0 = m λ c , for m integer .

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