## Robustness of Cantor Diffractals |

Optics Express, Vol. 21, Issue 7, pp. 7951-7956 (2013)

http://dx.doi.org/10.1364/OE.21.007951

Acrobat PDF (731 KB)

### 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

## 1. Introduction

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] .

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. Express **14**, 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. B **51**, 7621–7631 (1995) [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,” 1–7http://arxiv.org/abs/physics/9802007.

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

*d*≈ 0.63. Being the simplest to construct, it is not surprising that most manmade fractal creations are Cantor fractals [5

_{f}5. G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. **28**, 971–973 (2003) [CrossRef] [PubMed] .

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

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

*redundancy*[1

1. R. Verma, V. Banerjee, and P. Senthilkumaran, “Redundancy in cantor diffractals,” Opt. Express **20**, 8250–8255 (2012) [CrossRef] [PubMed] .

*arbitrary*band although it excluded the zero spatial frequency component and contained an insignificant energy content in it. Furthermore, the fractal generator could be obtained from yet smaller fragments of the arbitrary band.

*robustness*. Literally, robustness refers to a sturdiness in construction even under adverse conditions. The question we address is: How much malformation or disorder in the Cantor grating can be accommodated by the diffractals to continue to yield faithfully its fractal dimension and fractal generator? The answer is of consequence in a number of physical problems involving fractal organization. For example, it can provide a bound for the inherent manufacturing defects that can be tolerated in cantor irises, axicons and sieves, to keep their performance capabilities intact.

*transparencies*, or by opening a fraction of the randomly chosen

*opacities*, or by doing both simultaneously. The corresponding diffraction pattern also exhibits redundancy. We can therefore obtain the fractal dimension and the generator from an arbitrary band (primary or any of the secondary bands) and compare these with evaluations of the un-disordered Cantor aperture to quantify robustness to disorder. Our study indicates that unambiguous recovery is possible for as much as 50% disorder if the order of the grating is large.

*f*configuration” is also provided. Finally, we discuss the implications of the twin properties of redundancy and robustness and their applications in Section 4.

## 2. Mathematical formulation

*R*

_{0}(

*x*) = rect(

*ε*

_{0}= 2

*a*,

*x*= 0), where rect(

*w*,

*x*) is a rectangle function of width

*w*placed symmetrically about point

*x*. Defining

*R*(

_{n}*x*) = rect(

*ε*=

_{n}*ε*

_{n}_{−1}/3,

*x*= 0), the

*n*-th generation Cantor grating is given by [1

1. R. Verma, V. Banerjee, and P. Senthilkumaran, “Redundancy in cantor diffractals,” Opt. Express **20**, 8250–8255 (2012) [CrossRef] [PubMed] .

*x*= ± 2

_{i}*a*/3 ± 2

*a*/3

^{2}± ⋯ ± 2

*a*/3

*. The superscript 0 here refers to an un-disordered grating and*

^{n}*N*= 2

*.*

^{n}*t*of the

*N*transparencies in the

*n*-th generation Cantor grating are to be disordered or blocked. To do so, we generate random numbers {

*X*(

*i*);

*i*= 1, 2,··,

*N*} which take value 0 with probability

*t*and value 1 with probability 1 −

*t*by the following procedure:

**String**(

*N*,

*t*,

*X*(

*N*))

*i*= 1,

*N*

*r*< 1 using a standard random number generator

*r*<

*t*,

*X*(

*i*) = 0 else

*X*(

*i*) = 1

*t*, the aperture function in Eq. (1) therefore becomes

*p*of the

*M*= 2

^{n−1}opacities are replaced by a transparency. As in the preceding case,

**String**(

*M*,

*p*,

*Y*(

*N*)) can be used to generate {

*Y*(

*i*);

*i*= 1, 2,··,

*M*} which take values 0 and 1 with probabilities

*p*and 1 −

*p*respectively. The corresponding disordered aperture function is then given by

*q*of the total number

*L*=

*N*+

*M*is now disordered. Generating {

*Z*(

*i*);

*i*= 1, 2,··,

*L*} using

**String**(

*L*,

*q*,

*Z*(

*L*)), the aperture function can be written as where

*x*and

_{i}*y*are as defined in Eqs. (3) and (5) respectively.

_{i}*n*= 4 Cantor grating using Eq. (1). We have chosen 2

*a*= 1 here and in all our subsequent evaluations for convenience. Its disordered counterpart, obtained for

*q*= 1/4 in Eq. (6), is plotted in Fig. 1(b). To obtain reliable numerics, it is essential to average data over several realizations of disorder generated using

**String**.

1. R. Verma, V. Banerjee, and P. Senthilkumaran, “Redundancy in cantor diffractals,” Opt. Express **20**, 8250–8255 (2012) [CrossRef] [PubMed] .

*f*is the spatial frequency of the scattered wave having dimensions of inverse length and sinc(

*x*) = sin(

*x*)/

*x*. It is simple algebra to check that

**20**, 8250–8255 (2012) [CrossRef] [PubMed] .

^{2}in Eq. (8) is the structure factor

*S*(

*f*). The fractal dimension

*d*is customarily obtained from the integrated structure factor [20

_{f}20. C. Guerin and M. Holschneider, “Scattering on fractal measures,” J. Phys. A **29**, 7651–7667 (1996) [CrossRef] .

22. C. Allain and M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B **33**, 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. A **39**, 1500–1512 (1989) [CrossRef] [PubMed] .

*s*=

*t*,

*p*or

*q*as the case may be, can be obtained by substituting

*S*(

*f*), and consequently

*d*. We demonstrate now that the inherent self-similarity in the interference pattern compensates for this change in a major way.

_{f}## 3. Robustness of Diffractals

*d*to disorder in the Cantor aperture. We present results for the most general case when both transparencies and opacities are disordered. All the data that we present has been averaged over 30 disorder realizations generated using procedure

_{f}**String**. Table 1 presents the disorder averaged fractal dimension 〈

*d*〉 calculated from the first secondary band for different values of disorder

_{f}*q*and order

*n*. The angular brackets 〈⋯〉 indicate an averaging over disorder realizations. These data clearly reveal that the fractal dimension is correct to the first significant digit of the un-disordered value even if

*q*is as large as 50% for higher

*n*.

*n*length scales in the fractal grating. Further, the zeros of the

*m*-th cosine term are given by

*f*= ±(2

_{m}*m*+ 1)3

_{o}*/4;*

^{m}*m*= 0, 1, 2,... As

_{o}*f*= 3

_{m}*f*

_{m}_{−1}, the zeros of orders

*m*′ <

*m*are nested in those of order

*m*. The periods of all cosine terms are therefore commensurate. As a consequence, the energy content enclosed between zeros of cos(2

*f*/3

*), cos (2*

^{m}*f*/3

^{m}^{−1}),⋯,(cos 2

*f*/3)

^{2}is also sufficient to yield reconstructions of orders

*m*−1,

*m*−2,···,1 respectively [1

**20**, 8250–8255 (2012) [CrossRef] [PubMed] .

*n*= 1, can be reconstructed by Fourier transformation of a fragment of an arbitrary band.

*f*optical arrangement [1

**20**, 8250–8255 (2012) [CrossRef] [PubMed] .

*n*= 4 and

*q*= 0.25 is displayed on the Holoeye LCR-2500 reflective type LC Spatial Light Modulator (SLM), with pixel size 19

*μ*and resolution 1024×768, is illuminated with a coherent and collimated laser light of wavelength = 632.8 nm. The lens

*L*

_{1}(

*f*= 13.5 cm) Fourier transforms the displayed grating to yield the diffraction pattern on the Fourier plane

*F*

_{1}. An aperture A (of adjustable width) allows only a part of the diffraction pattern to be incident on the lens

*L*

_{2}(

*f*= 13.5 cm) which performs Fourier transformation once again to yield reconstruction of the grating. Figure 3(a) shows the disordered Cantor grating that was displayed on the SLM. The corresponding intensity profile observed on

*F*

_{1}is depicted in Fig. 3(b). Figures 3(c) – 3(f) are reconstructions via

*L*

_{2}from nested clips of first secondary band obtained using the description provided in the preceding paragraph. These reconstructions, in addition to yielding the generator (from Fig. 3(f)), also provide an experimental verification of redundancy in disordered diffractals. Higher generation gratings, which can accommodate larger disorder, could not be used due to issues of resolution. We have however been able to verify robustness theoretically for order 6 gratings with 50% disorder using Matlab simulations. But here too, resolution comes in the way when

*n*> 6.

## 4. Discussion

*parent*fractal! Said differently, Cantor diffractals can accommodate imperfections of the Cantor grating without significantly altering the diffraction field even if the disorder is as large as 50%.

5. G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. **28**, 971–973 (2003) [CrossRef] [PubMed] .

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

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] .

## Acknowledgments

## References and links

1. | R. Verma, V. Banerjee, and P. Senthilkumaran, “Redundancy in cantor diffractals,” Opt. Express |

2. | B. B. Mandelbrot, |

3. | A. L. Barabasi and H. E. Stanley, |

4. | Tamas Vicsek, |

5. | G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. |

6. | J. A. Monsoriu, G. Saavedra, and W. D. Furlan, “Fractal zone plates with variable lacunarity,” Opt. Express |

7. | W. D. Furlan, G. Saavedra, and J. A. Monsoriu, “White-light imaging with fractal zone plates,” Opt. Lett. |

8. | F. Gimenez, J. A. Monsoriu, W. D. Furlan, and Amparo Pons, “Fractal photon sieve,” Opt. Express |

9. | J. A. Monsoriu, C. J. Z. Rodriguez, and W. D. Furlan, “Fractal axicons,” Opt. Commun. |

10. | K. Jarrendahl, M. Dulea, J. Birch, and J.-E. Sundgren, “X-ray diffraction from amorphous Ge/Si Cantor superlattices,” Phys. Rev. B |

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

12. | H. Aubert and D. L. Jaggard, “Wavelet analysis of transients in fractal superlattices,” IEEE Trans. Antennas Propag. |

13. | G. H. C. New, M. A. Yates, J. P. Woerdman, and G. S. McDonald, “Diffractive origin of fractal resonator modes,” Opt. Commun. |

14. | M. Berry, C. Storm, and W. van Saarloos, “Theory of unstable laser modes: edge waves and fractality,” Opt. Commun. |

15. | M. V. Berry, “Diffractals,” J. Phys. A: Math. Gen. |

16. | D. Bak, S. P. Kim, S. K. Kim, K. -S. Soh, and J. H. Yee, “Fractal diffraction grating,” 1–7http://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. |

18. | C. Allain and M. Cloitre, “Spatial spectrum of a general family of self-similar arrays,” Phys. Rev. A |

19. | D. A. Hamburger-Lidar, “Elastic scattering by deterministic and random fractals: Self-affinity of the diffraction spectrum,” Phys. Rev. E |

20. | C. Guerin and M. Holschneider, “Scattering on fractal measures,” J. Phys. A |

21. | M. Lehman, “Fractal diffraction grating built through rectangular domains,” Opt. Commun. |

22. | C. Allain and M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B |

23. | B. Dubuc, J. F. Quiniou, C. R. Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimensions of profiles,” Phys. Rev. A |

24. | J. W. Goodman, |

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(050.1220) Diffraction and gratings : Apertures

(290.5880) Scattering : Scattering, rough surfaces

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: January 31, 2013

Revised Manuscript: March 12, 2013

Manuscript Accepted: March 14, 2013

Published: March 26, 2013

**Citation**

Rupesh Verma, Manoj Kumar Sharma, Varsha Banerjee, and Paramasivam Senthilkumaran, "Robustness of Cantor Diffractals," Opt. Express **21**, 7951-7956 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-7951

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### References

- R. Verma, V. Banerjee, and P. Senthilkumaran, “Redundancy in cantor diffractals,” Opt. Express20, 8250–8255 (2012). [CrossRef] [PubMed]
- 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]
- G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett.28, 971–973 (2003). [CrossRef] [PubMed]
- J. A. Monsoriu, G. Saavedra, and W. D. Furlan, “Fractal zone plates with variable lacunarity,” Opt. Express12, 4227–4234 (2004). [CrossRef] [PubMed]
- W. D. Furlan, G. Saavedra, and J. A. Monsoriu, “White-light imaging with fractal zone plates,” Opt. Lett.32, 2109–2111 (2007). [CrossRef] [PubMed]
- 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]
- 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]
- A. D. Jaggard and D. L. Jaggard, “Scattering from fractal superlattices with variable lacunarity,” J. Opt. Soc. Am. A15, 1626–1635 (1998). [CrossRef]
- H. Aubert and D. L. Jaggard, “Wavelet analysis of transients in fractal superlattices,” IEEE Trans. Antennas Propag.50, 338–345 (2002). [CrossRef]
- 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. V. Berry, “Diffractals,” J. Phys. A: Math. Gen.12, 781–797 (1979). [CrossRef]
- 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. Hou, G. Xu, W. Wen, and G. K. L. Wong, “Diffraction by an optical fractal grating,” Appl. Phys. Lett.85, 6125–6127 (2004). [CrossRef]
- C. Allain and M. Cloitre, “Spatial spectrum of a general family of self-similar arrays,” Phys. Rev. A36, 5751–5757 (1987). [CrossRef] [PubMed]
- 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]
- M. Lehman, “Fractal diffraction grating built through rectangular domains,” Opt. Commun.195, 11–26 (2001). [CrossRef]
- C. Allain and M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B33, 3566–3569 (1986). [CrossRef]
- 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]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

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