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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 31, Iss. 7 — Jul. 1, 2014
  • pp: 1473–1480

Analysis of Fibonacci gratings and their diffraction patterns

Rupesh Verma, Manoj Kumar Sharma, Paramasivam Senthilkumaran, and Varsha Banerjee  »View Author Affiliations

JOSA A, Vol. 31, Issue 7, pp. 1473-1480 (2014)

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Aperiodic and fractal optical elements are proving to be promising candidates in image-forming devices. In this paper, we analyze the diffraction patterns of Fibonacci gratings (FbGs), which are prototypical examples of aperiodicity. They exhibit novel characteristics such as redundancy and robustness that keep their imaging characteristics intact even when there is significant loss of information. FbGs also contain fractal signatures and are characterized by a fractal dimension. Our study suggests that aperiodic gratings may be better than their fractal counterparts in technologies based on such architectures. We also identify the demarcating features of aperiodic and fractal diffraction, which have been rather fuzzy in the literature so far.

© 2014 Optical Society of America

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(050.1220) Diffraction and gratings : Apertures
(290.0290) Scattering : Scattering

ToC Category:
Diffraction and Gratings

Original Manuscript: March 31, 2014
Manuscript Accepted: May 8, 2014
Published: June 13, 2014

Rupesh Verma, Manoj Kumar Sharma, Paramasivam Senthilkumaran, and Varsha Banerjee, "Analysis of Fibonacci gratings and their diffraction patterns," J. Opt. Soc. Am. A 31, 1473-1480 (2014)

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  1. C. Palmer and E. Loewen, Diffraction Grating Handbook (Newport Corporation, 2005).
  2. M. Born, Principles of Optics (Pergamon, 1980).
  3. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  4. J. Zhang, Y. Cao, and J. Zheng, “Fibonacci quasi-periodic superstructure fiber Bragg gratings,” Optik 121, 417–421 (2010). [CrossRef]
  5. K. Wu and G. P. Wang, “One-dimensional Fibonacci grating for far field super-resolution imaging,” Opt. Lett. 38, 2032–2034 (2013). [CrossRef]
  6. A. Calatayud, V. Ferrando, L. Remon, W. D. Furlan, and J. A. Monsoriu, “Twin axial vortices generated by Fibonacci lenses,” Opt. Express 21, 10234–10239 (2013). [CrossRef]
  7. G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. 28, 971–973 (2003). [CrossRef]
  8. J. A. Monsoriu, G. Saavedra, and W. D. Furlan, “Fractal zone plates with variable lacunarity,” Opt. Express 12, 4227–4234 (2004). [CrossRef]
  9. W. D. Furlan, G. Saavedra, and J. A. Monsoriu, “White-light imaging with fractal zone plates,” Opt. Lett. 32, 2109–2111 (2007). [CrossRef]
  10. F. Gimenez, J. A. Monsoriu, W. D. Furlan, and A. Pons, “Fractal photon sieve,” Opt. Express 14, 11958–11963 (2006). [CrossRef]
  11. J. A. Monsoriu, C. J. Zapata-Rodriguez, and W. D. Furlan, “Fractal axicons,” Opt. Commun. 263, 1–5 (2006). [CrossRef]
  12. R. Verma, V. Banerjee, and P. Senthilkumaran, “Redundancy in Cantor diffractals,” Opt. Express 20, 8250–8255 (2012). [CrossRef]
  13. R. Verma, M. K. Sharma, V. Banerjee, and P. Senthilkumaran, “Robustness of Cantor diffractals,” Opt. Express 21, 7951–7956 (2013). [CrossRef]
  14. R. Verma, V. Banerjee, and P. Senthilkumaran, “Fractal signatures in the aperiodic Fibonacci grating,” Opt. Lett. 39, 2557–2560 (2014). [CrossRef]
  15. Z.-X. Wen and Z.-Y. Wen, “Some properties of the singular words of the Fibonacci word,” Eur. J. Combin. 15, 587–598 (1994). [CrossRef]
  16. A. Monnerot-Dumaine, “The Fibonacci word fractal,” 2009, http://hal.archives-ouvertes.fr/hal-00367972/fr .
  17. W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72, 633–636 (1994). [CrossRef]
  18. X. Yang, Y. Liu, and X. Fu, “Transmission properties of light through the Fibonacci-class multilayers,” Phys. Rev. B 59, 4545–4548 (1999). [CrossRef]
  19. N. V. Grushina, P. V. Korolenko, and S. N. Markova, “Special features of the diffraction of light on optical Fibonacci gratings,” Moscow Univ. Phys. Bull. 63, 123–126 (2008). [CrossRef]
  20. N. Gao, Y. Zhang, and C. Xie, “Circular Fibonacci gratings,” Appl. Opt. 50, G142–G148 (2011). [CrossRef]
  21. D. Levine and P. J. Steinhardt, “Quasicrystals. I. Definition and structure,” Phys. Rev. B 34, 596–616 (1986). [CrossRef]
  22. D. Levine and P. J. Steinhardt, “Quasicrystals: a new class of ordered structures,” Phys. Rev. Lett. 53, 2477–2480 (1984). [CrossRef]
  23. J. P. Lu, T. Odagaki, and J. L. Birman, “Properties of one-dimensional quasilattices,” Phys. Rev. B 33, 4809–4817 (1986). [CrossRef]
  24. J. M. Dubois, “The applied physics of quasicrystals,” Phys. Scr. T49, 17–23 (1993). [CrossRef]
  25. M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization in optics: quasiperiodic media,” Phys. Rev. Lett. 58, 2436–2438 (1987). [CrossRef]
  26. Y. Liu and R. Riklund, “Electronic properties of perfect and nonperfect one-dimensional quasicrystals,” Phys. Rev. B 35, 6034–6042 (1987). [CrossRef]
  27. N. Ferralis, A. W. Szmodis, and R. D. Diehl, “Diffraction from one- and two-dimensional quasicrystalline gratings,” Am. J. Phys. 72, 1241–1246 (2004). [CrossRef]
  28. R. A. Dunlap, The Golden Ratio and Fibonacci Numbers (World Scientific, 1997).
  29. 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]
  30. C. Allain and M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B 33, 3566–3569 (1986). [CrossRef]
  31. B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, and S. W. Zucker, “Evaluating the fractal dimension of profiles,” Phys. Rev. A 39, 1500–1512 (1989). [CrossRef]
  32. C. A. Guerin and M. Holschneider, “Scattering on fractal measures,” J. Phys. A 29, 7651–7667 (1996). [CrossRef]
  33. D. Damanik, M. Embree, A. Gorodetski, and S. Tcheremchantsev, “The fractal dimension of the spectrum of the Fibonacci Hamiltonian,” Commun. Math. Phys. 280, 499–516 (2008). [CrossRef]
  34. E. N. Leith and J. Upatnieks, “Wavefront reconstruction with diffused illumination and three dimensional objects,” J. Opt. Soc. Am. 54, 1295–1301 (1964). [CrossRef]
  35. H. J. Gerritsen, W. J. Hannan, and E. G. Ramberg, “Elimination of speckle noise in holograms with redundancy,” Appl. Opt. 7, 2301–2311 (1968). [CrossRef]

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