## Diffraction by fractal metallic supergratings

Optics Express, Vol. 15, Issue 24, pp. 15628-15636 (2007)

http://dx.doi.org/10.1364/OE.15.015628

Acrobat PDF (543 KB)

### 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

## 1. Introduction

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]

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]

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]

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]

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

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]

*et al*. [21

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]

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]

## 2. Triadic fractal metallic gratings design

*S*=0) is to take a segment of length

*a*. The second step (

*S*=1) is to divide the segment in three equal parts of length

*a*/3 and remove the central one. Then, on each of these segments this procedure is repeated, and so on. The Cantor-set is the set of segments remaining. In general, at stage

*S*, there are 2

*segments of length*

^{S}*a*/3

*with 2*

^{S}*-1 gaps in between. In Fig. 1, only the four first stages are shown for clarity.*

^{S}*S*. At the

*S*-th stage presents 2

*grooves corresponding to the black regions of the Cantor set. Note that this structure can be interpreted as a quasiperiodic metallic grating which can be obtained by “filling in” some grooves of a finite periodic grating as the one shown in Fig. 2(b). This distribution has (3*

^{S}*+1)/2 segments at stage*

^{S}*S*each one of width

*a*/3

*, separated by gaps of the same length, so that the period of this finite structure is 2*

^{S}*a*/3

*.*

^{S}## 3. Modal theory

*L*grooves of the same height

*h*and widths

*c*(

_{l}*l*=1, 2, …,

*L*) as shown in Fig. 2(a). A plane wave of wavelength

*λ*is incident upon the surface from the region

*y*>0 forming an angle θ

_{0}with the

*y*-axis, being the

*x*,

*y*-plane the plane of incidence. Assuming an harmonic time dependence in the form exp(-

*iωt*), where

*ω*is the frequency of the incident light, Maxwell’s equations are combined to get the Helmholtz equation that must satisfy the fields

*s*-polarization (electric field parallel to the grooves) and

*p*-polarization (magnetic field parallel to the grooves). We call

*f*to the

^{µ}*z*-component of the electric (magnetic) field in the case of

*s*- (

*p*-) polarization (

*µ*=

*s*,

*p*). We separate the space into two regions (see Fig. 2): the region

*y*≥0(+), in which the scattered fields are represented by a continuous superposition of plane waves (Rayleigh expansion), and the region inside the grooves, -

*h*≤

*y*≤0(-), where the fields are represented using modal expansions.

*y*≥0, we express the total field,

*f*+(

^{µ}*x*,

*y*), as the sum of the following terms: the incident field,

*α*=

*k*

_{0}sin

*θ*and

*x*- and

*y*-components of the wavevector (

*k*

_{0}=

*ω*/

*c*=2

*π*/

*λ*) in the

*θ*direction, respectively. To obtain the Rayleigh amplitudes,

*R*(

^{µ}*α*), corresponding to the scattered field, we must consider the fields inside the grooves which are expressed in terms of the corresponding modal functions for each polarization. For the

*l*-th groove we have:

*a*

_{m,l}and

*b*

_{m,l}are the modal amplitudes corresponding to

*s*- and

*p*- polarization, respectively.

*y*=0 we obtain a system of coupled equations for each polarization mode, which are projected in convenient bases to drop the

*x*dependence. The projected equations are then combined to yield an integral equation for the Rayleigh amplitudes

*R*(

^{s}*α*) and

*R*(

^{p}*α*), for

*s*and

*p*polarization, respectively. Once these amplitudes are found, the normalized intensity of the reflected field in the

*θ*direction is calculated by

*I*(

*α*)=|

*R*(

^{µ}*α*)|

^{2}.

## 4. Self-similarity in fractal metallic gratings

*S*(top plots), with those of the corresponding periodic grating (bottom plots). The width of the grooves in each case is

*c*/

*a*=1/3

*, the depth is*

^{S}*h*/

*a*=0.00077125. The incident wave is

*s*-polarized, with a wavelength

*λ*/

*a*=0.00617, and

*θ*

_{0}=0°. Figures 3(a), 3(b) and 3(c) correspond to different stages of the fractal, and then each one of the surfaces have 4 (

*S*=2), 8 (

*S*=3), and 16 (

*S*=4) grooves, respectively. The corresponding periodic surfaces have 5, 14, and 41 grooves, respectively. It can be observed that the position of the main maxima, which is given by the periodicity of the structure, is the same for the periodic as well as for the fractal structures. Taking into account that the period of the structure is 2

*a*/3

*, the directions*

^{S}*α*/

*k*

_{0}at which the reflectance is maximum are given by:

*n*integer. For the parameters of Fig. 3, the first maximum (

*n*=1) for

*S*=2 should be at

*α*/

*k*

_{0}=0.027765, for

*S*=3 at

*α*/

*k*

_{0}=0.083295, and for

*S*=4 at

*α*/

*k*

_{0}=0.249885. The positions of the maxima agree very well with these calculated values.

*γ*is a scale factor, and

*I*̄ and

*I*̄

*are the mean values of the function and of its scaled version, respectively. In Fig. 4 we plot the self-similarity coefficient as a function of log*

_{γ}_{3}(

*γ*), for the same three fractal structures considered in Fig. 3. This coefficient exhibits local maxima at 3

*, being*

^{m}*m*a natural number, and the number of peaks increases with the stage

*S*of the fractal.

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]

*α*and of the depth of the grooves, for

*S*=2 [Fig. 5(a)] and for

*S*=3 [Fig. 5(b)]. The parameters of the structures and of the incident wave are the same as those in Fig. 3. It can be observed that as the depth of the grooves is increased, the response of the fractal system keeps the same characteristics, with its maxima and minima located in the same positions as for the small depth case [Figs. 3(a) and 3(b)], what corresponds to peaks at the same scale factors

*γ*in the self-similarity coefficient, as those found for the previous case [Figs. 4(a) and 4(b)]. It can be seen that for both cases (

*S*=2 and

*S*=3) the reflected intensity is a periodic function of

*h*/

*a*as predicted by Eqs. (4) and (5). A rough estimate of this period can be done assuming that the fundamental mode in each groove is the only relevant mode, and then the period

*p*-polarized. We have found that the self similarity behavior is pre-served for this polarization, as well as the periodic behavior with the depth of the grooves. Taking into account the fundamental mode, the period in h/a for the

*p*-case can be estimated as

*λ*/2

*a*, which for the parameters studied in this paper gives roughly the same value as for the

*s*-case (these results have not been included for brevity).

## 5. Lacunarity in fractal metallic supergratings

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]

*S*=0) is again a straight line segment of unit length. At step

*S*=1 the initiator is replaced by N nonoverlapping copies of the initiator, each one scaled by a factor

*γ*<1. For even N, as shown in Fig. 6, one half of the copies are placed to the left of the interval and the other half to its right, each copy being separated by a fixed distance

*ε*. For odd N, not shown in Fig. 6, one copy lies centered in the interval and the rest are distributed as for even N, that is, [N/2] copies are placed to the left of the interval and the other [N/2] copies to its right, where [N/2] is the greatest integer less than or equal to N/2. At each step of the construction, the generation process is repeated over and over again for each segment in the previous step.

*ε*is arbitrary and it can be associated to the lacunarity [16], which may be defined as the deviation of a fractal from translational invariance [23

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]

*ε*as an indication of the lacunarity of the generalized Cantor set, as shown in Fig. 6 [22

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]

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]

*γ*, and the lacunarity parameter

*ε*. The last two parameters must satisfy certain constraints to avoid overlapping between the copies. The maximum value of the scaling factor depends on the value of N, such that 0<

*γ*<

*γ*

_{max}=1/N. For each value of N and

*γ*, there are two extreme values for

*ε*. One is

*ε*

_{min}=0, for which the highest lacunar fractal is obtained, that is, one with the largest possible gap. For even N, the central gap has a width of 1-N

*γ*, and for odd N, both large gaps surrounding the central well have a width (1-N

*γ*)/2. The other extreme value is

*ε*

_{max}. Thus the corresponding lacunarity is smaller than that for

*ε*=0, but not the smallest one, which is obtained for the most regular distribution, where the gaps and wells have the same width at the first step (

*S*=1) given by

*ε*

_{reg}<

*ε*

_{max}.

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]

*x*-component of the wavevector (

*α*/

*k*

_{0}) and the lacunarity parameter

*ε*. Figure 7 shows twist plots for tetradic (N=4) Cantor supergratings for

*S*=1 (a) and

*S*=2 (b). In these plots a logarithmic gray scale was used for the reflection coefficient, from black for zero values to white for the maximum value equal to unity.

*S*=1 forms an envelope for the (unscaled) reflected intensity at step

*S*=2, showing that the response is self-similar for any value of the lacunarity parameter

*ε*. Notice that in each plot there is a vertical dark line (no reflected intensity) that corresponds to

*α*/

*k*

_{0}=0.0617 and 0.617 for

*S*=1 and

*S*=2, respectively. It is remarkable that this null does not depend on the value of

*ε*. The origin of these bands can be understood as diffraction minima, produced by the diffraction by a single groove. The diffraction minima produced by a single groove are in the observation directions given by

*γ*considered in these figures,

*c*/

*a*=0.1 (

*S*=1) and

*c*/

*a*=0.01 (

*S*=2), in which case eq. (11) gives

*α*/

*k*

_{0}=0.0617 (

*S*=1) and 0.617 (

*S*=2), exactly the positions at which the vertical nulls are found. The other nulls in these plots can be understood as multiple cross-interferences between different grooves of the fractal reflecting structure.

## 6. Conclusion

## Acknowledgements

## References and links

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 , |

3. | J. R. Andrewartha, J. R. Fox, and I. J. Wilson, “Resonance anomalies in the lamellar grating,” Optica Acta , |

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 , |

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. |

6. | L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. |

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 |

8. | D. Maystre, “Rigorous theory of light scattering from rough surfaces,” J. Opt. |

9. | Y. L. Kok, “A boundary value solution to electromagnetic scattering by a rectangular groove in a ground plane,” J. Opt. Soc. Am. A |

10. | T.-M. Wang and H. Ling, “A connection algorithm on the problem of EM scattering from arbitrary cavities,” J. EM Waves and Applics. |

11. | R. A. Depine and D. C. Skigin, “Scattering from metallic surfaces having a finite number of rectangular grooves,” J. Opt. Soc. Am. A |

12. | D. C. Skigin, V. V. Veremey, and R. Mittra, “Superdirective radiation from finite gratings of rectangular grooves”, IEEE Trans. Antennas Propag. |

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 |

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 |

15. | D. C. Skigin and R. A. Depine, “Diffraction by dual-period gratings,” Appl. Opt. |

16. | B. Mandelbrot, |

17. | Y. Sakurada, J. Uozumi, and T Asakura, “Fresnel diffraction by 1-D regular fractals,” Pure Appl. Opt. |

18. | O. Trabocchi, S. Granieri, and W.D. Furlan, “Optical propagation of fractal fields. Experimental analysis in a single display,” J. Mod. Opt. |

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 |

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. |

22. | A. D. Jaggard and D. L. Jaggard, “Scattering from fractal superlattices with variable lacunarity,” J. Opt. Soc. Am. A |

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 |

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] |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(050.1970) Diffraction and gratings : Diffractive optics

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: June 18, 2007

Revised Manuscript: September 21, 2007

Manuscript Accepted: September 22, 2007

Published: November 12, 2007

**Citation**

Diana C. Skigin, Ricardo A. Depine, Juan A. Monsoriu, and Walter D. Furlan, "Diffraction by fractal metallic supergratings," Opt. Express **15**, 15628-15636 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-15628

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### References

- R. Petit, "Diffraction gratings," C. r. hebd. Seanc. Acad. Sci., Paris, 260, 4454 (1965).
- R. C. Hollins and D. L. Jordan, "Measurments of 10.6 μm 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]
- 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]
- 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]
- L. Li, "A modal analysis of lamellar diffraction gratings in conical mountings," J. Mod. Opt. 40,553-573 (1993). [CrossRef]
- 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]
- D. Maystre, "Rigorous theory of light scattering from rough surfaces," J. Opt. 5,43-51 (1984). [CrossRef]
- 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]
- 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).
- 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]
- 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]
- 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]
- D. C. Skigin and R. A. Depine, "Diffraction by dual-period gratings," Appl. Opt. 46,1385-1391 (2007). [CrossRef] [PubMed]
- B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1982.
- Y. Sakurada, J. Uozumi, and T Asakura, "Fresnel diffraction by 1-D regular fractals," Pure Appl. Opt. 1,29-40 (1992). [CrossRef]
- 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. 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]
- F. Giménez, J.A. Monsoriu, W.D. Furlan, and A. Pons, "Fractal photon sieves," Opt. Express 14,11958-11963 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-11958 [CrossRef] [PubMed]
- 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]
- A. D. Jaggard and D. L. Jaggard, "Scattering from fractal superlattices with variable lacunarity," J. Opt. Soc. Am. A 15,1626-1635 (1998). [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]
- J. A. Monsoriu, G. Saavedra, and W. D. Furlan, "Fractal zone plates with variable lacunarity," Opt. Express 12,4227-4234 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-18-4227 [CrossRef] [PubMed]

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