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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 24 — Nov. 26, 2007
  • pp: 15628–15636
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Diffraction by fractal metallic supergratings

Diana C. Skigin, Ricardo A. Depine, Juan A. Monsoriu, and Walter D. Furlan  »View Author Affiliations


Optics Express, Vol. 15, Issue 24, pp. 15628-15636 (2007)
http://dx.doi.org/10.1364/OE.15.015628


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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

The electromagnetic diffraction from surfaces having rectangular profiles has been studied by many authors [1

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

]–[7

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]

]. This kind of surfaces is of interest because they can be manufactured quite easily and they permit an accurate control of their parameters, making it possible to compare theoretical with experimental data [2

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]

]. Most of the investigations have been devoted to ideal gratings, i.e., those with strictly periodic, unlimited boundaries separating two media. In particular, the case of perfectly conducting materials have been studied by Andrewartha et al. [3

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

] andWirgin and Maradudin [4

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]

].

An increasing interest in the usage of light diffracted as a means of measuring surface microgeometry lead many efforts to study the direct problem deeply. To compare experimental with theoretical results, Maystre [8

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

] developed a rigorous theory to study the reflected electromagnetic field from perfectly conducting finite gratings, that is, gratings having a finite number of grooves. Several methods for solving the scattering problem by grooves in a plane are mentioned in refs. [9

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]

] and [10

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).

].

In recent years the study of fractals has attracted the attention of researchers, encouraged by the fact that many physical phenomena, natural structures and statistical processes can be analyzed and described by using a fractal approach [16

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

]. From a mathematical point of view the concept of fractal is associated with a geometrical object which i) is self-similar (i.e., the object is exactly or approximately similar to a part of itself) and, ii) has a fractional (or noninteger) dimension. In optics, diffractive fractal structures, ranging from simple one-dimensional (1D) objects [17

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

, 18

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).

] to complex 2D systems [19

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

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]

] have been extensively analyzed. Ledesma 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]

] studied the diffraction properties of metallic fractal gratings with rectangular protuberances generated of the Cantor type, where the width and the depth of the corrugations is small compared with the incident wavelength. For this particular case, they showed that the spectral response of such structures exhibits self-similarity features.

2. Triadic fractal metallic gratings design

One of the classic and most well known fractals is the triadic Cantor set. As shown in Fig. 1, it can be constructed by an iterative process. The first step (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 2S segments of length a/3S with 2S-1 gaps in between. In Fig. 1, only the four first stages are shown for clarity.

Based on this scheme we propose a fractal metallic grating which, as it is shown in Fig. 2(a), is defined by the parameter S. At the S-th stage presents 2S 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 (3S+1)/2 segments at stage S each one of width a/3S, separated by gaps of the same length, so that the period of this finite structure is 2a/3S.

3. Modal theory

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

In the upper region, y≥0, we express the total field, fµ+(x,y), as the sum of the following terms: the incident field,

finc(x,y)=ei(α0xβ0y),
(1)

the speculary reflected field,

fspecμ(x,y)=(1)jei(α0xβ0y);j={1forspolarization0forppolarization,
(2)

and the scattered field,

fscattμ(x,y)=Rμ(α)ei(αx+βy)dα.
(3)

fs(x,y)=m=1am,lsin[μm,l(y+h)]sin[mπcl(xxl)],
(4)

and

fp(x,y)=m=0bm,lcos[μm,l(y+h)]cos[mπcl(xxl)].
(5)

where

μm,l={k02(mπcl)2ifk02>(mπcl)2i(mπcl)2k02ifk02<(mπcl)2,
(6)

Matching the fields at the plane 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 Rs(α) and Rp(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)=|Rµ(α)|2.

4. Self-similarity in fractal metallic gratings

αk0=α0k0+nλa3s2,
(7)

with 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.

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

An assesment of the self-similarity of the system response can be obtained by means of the correlation between the values of the scattered intensities produced by fractal gratings of different scales. This function is defined as [17]:

C(γ)=(I(α)I̅)(I(αγ)I̅γ)dα[(I(α)I̅)2dα(I(αγ)I̅γ)2dα]12,
(8)

where γ is a scale factor, and Ī and Īγ 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 3(γ), for the same three fractal structures considered in Fig. 3. This coefficient exhibits local maxima at 3m, being m a natural number, and the number of peaks increases with the stage S of the fractal.

In order to perform an exhaustive analysis, we have also computed the reflected intensities when the incident wave is 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/2a, 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).

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.

5. Lacunarity in fractal metallic supergratings

The construction parameter ε is arbitrary and it can be associated to the lacunarity [16

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

], 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]

]. Translational invariance is highly scale dependent because heterogeneous sets at small scales can be homogeneous on larger scales or vice versa. Actually, the lacunarity is a scale-dependent measure of the heterogeneity (or texture) of an object, whether or not it is fractal. For the purposes of this paper it is sufficient to consider 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

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]

].

Symmetrical polyadic Cantor fractals are characterized by the number of self-similar copies N, the scaling factor 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-Nγ, and for odd N, both large gaps surrounding the central well have a width (1-Nγ)/2. The other extreme value is

εmax={1NγN2evenN1NγN3oddN,
(9)

where for even (odd) N two (three) wells are joined together in the center and the central gap is missing. The width of the N-2 gaps in this case is equal to 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

εreg=1NγN1.
(10)

Note that 0<ε reg<ε max.

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. 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.

To show that the self-similar behavior of the reflected intensity for polyadic Cantor supergratings is retained even when the lacunarity parameter is varied, twist plots [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]

] may be used. These plots represent the intensity as a function of the normalized 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.

It can be observed that the re-scaled reflected intensity at step 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

αk0=mλc,forminteger.
(11)

For the value of 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

D. Skigin and R. Depine gratefully acknowledge support from Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Universidad de Buenos Aires (UBA, X150 and X286) and Agencia Nacional de Promoción Científica y Tecnológica (ANPCYT-BID 1728/OC-AR-PICT 14099). J.A. Monsoriu acknowledges the financial support from the Conselleria d’Empresa, Universitat i Ciència, Generalitat Valenciana (GV/2007/239), and “Programa de Incentivo a la Investigación UPV 2005”, Vicerrectorado de Innovación y Desarrollo, Universidad Politécnica de Valencia, Spain. W.D. Furlan also acknowledges the support from Plan Nacional I+D+I, Ministerio de Ciencia y Tecnología (DPI 2006-8309), Spain.

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 , 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]

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

  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 10.6 μm 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. Express 14,11958-11963 (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. Express 12,4227-4234 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-18-4227 [CrossRef] [PubMed]

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