Abstract

The Fibonacci grating (FbG) is an archetypal example of aperiodicity and self-similarity. While aperiodicity distinguishes it from a fractal, self-similarity identifies it with a fractal. Our paper investigates the outcome of these complementary features on the FbG diffraction profile (FbGDP). We find that the FbGDP has unique characteristics (e.g., no reduction in intensity with increasing generations), in addition to fractal signatures (e.g., a non-integer fractal dimension). These make the Fibonacci architecture potentially useful in image forming devices and other emerging technologies.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Z.-X. Wen and Z.-Y. Wen, Eur. J. Combin. 15, 587 (1994).
    [CrossRef]
  2. R. A. Dunlap, The Golden Ratio and Fibonacci Numbers (World Scientific, 1997).
  3. D. Levine and P. J. Steinhardt, Phys. Rev. Lett. 53, 2477 (1984).
    [CrossRef]
  4. D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984).
    [CrossRef]
  5. N. Gao, Y. Zhang, and C. Xie, Appl. Opt. 50, G142 (2011).
    [CrossRef]
  6. A. Calatayud, V. Ferrando, L. Remon, W. D. Furlan, and J. A. Monsoriu, Opt. Express 21, 10234 (2013).
    [CrossRef]
  7. J. Zhang, Y. Cao, and J. Zheng, Optik 121, 417 (2010).
    [CrossRef]
  8. K. Wu and G. P. Wang, Opt. Lett. 38, 2032 (2013).
    [CrossRef]
  9. Bo. Hou, Gu. Xu, W. Wen, and G. K. L. Wong, Appl. Phys. Lett. 85, 6125 (2004).
    [CrossRef]
  10. D. A. Hamburger-Lidar, Phys. Rev. E 54, 354 (1996).
    [CrossRef]
  11. R. Verma, V. Banerjee, and P. Senthilkumaran, Opt. Express 20, 8250 (2012).
    [CrossRef]
  12. R. Verma, M. K. Sharma, V. Banerjee, and P. Senthilkumaran, Opt. Express 21, 7951 (2013).
    [CrossRef]
  13. G. Saavedra, W. D. Furlan, and J. A. Monsoriu, Opt. Lett. 28, 971 (2003).
    [CrossRef]
  14. J. A. Monsoriu, G. Saavedra, and W. D. Furlan, Opt. Express 12, 4227 (2004).
    [CrossRef]
  15. W. D. Furlan, G. Saavedra, and J. A. Monsoriu, Opt. Lett. 32, 2109 (2007).
    [CrossRef]
  16. Q. Zhang, J. Wang, M. Wang, J. Bu, S. Zhu, B. Z. Gao, and X. Yuan, Opt. Laser Technol. 44, 2140 (2012).
    [CrossRef]
  17. N. Ferralis, A. W. Szmodis, and R. D. Diehl, Am. J. Phys. 72, 1241 (2004).
    [CrossRef]
  18. N. V. Grushina, P. V. Korolenko, and S. N. Markova, Moscow Univ. Phys. Bull. 63, 123 (2008).
    [CrossRef]
  19. B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, and S. W. Zucker, Phys. Rev. A 39, 1500 (1989).
    [CrossRef]
  20. C. Guerin and M. Holschneider, J. Phys. A 29, 7651 (1996).
    [CrossRef]
  21. C. Allain and M. Cloitre, Phys. Rev. B 33, 3566 (1986).
    [CrossRef]
  22. P. Cvitanovic, M. H. Jensen, L. P. Kadanoff, and I. Procaccia, Phys. Rev. Lett. 55, 343 (1985).
    [CrossRef]
  23. D. Damanik, M. Embree, A. Gorodetski, and S. Tcheremchantsev, Commun. Math. Phys. 280, 499 (2008).
    [CrossRef]
  24. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

2013 (3)

2012 (2)

R. Verma, V. Banerjee, and P. Senthilkumaran, Opt. Express 20, 8250 (2012).
[CrossRef]

Q. Zhang, J. Wang, M. Wang, J. Bu, S. Zhu, B. Z. Gao, and X. Yuan, Opt. Laser Technol. 44, 2140 (2012).
[CrossRef]

2011 (1)

2010 (1)

J. Zhang, Y. Cao, and J. Zheng, Optik 121, 417 (2010).
[CrossRef]

2008 (2)

D. Damanik, M. Embree, A. Gorodetski, and S. Tcheremchantsev, Commun. Math. Phys. 280, 499 (2008).
[CrossRef]

N. V. Grushina, P. V. Korolenko, and S. N. Markova, Moscow Univ. Phys. Bull. 63, 123 (2008).
[CrossRef]

2007 (1)

2004 (3)

J. A. Monsoriu, G. Saavedra, and W. D. Furlan, Opt. Express 12, 4227 (2004).
[CrossRef]

N. Ferralis, A. W. Szmodis, and R. D. Diehl, Am. J. Phys. 72, 1241 (2004).
[CrossRef]

Bo. Hou, Gu. Xu, W. Wen, and G. K. L. Wong, Appl. Phys. Lett. 85, 6125 (2004).
[CrossRef]

2003 (1)

1996 (2)

C. Guerin and M. Holschneider, J. Phys. A 29, 7651 (1996).
[CrossRef]

D. A. Hamburger-Lidar, Phys. Rev. E 54, 354 (1996).
[CrossRef]

1994 (1)

Z.-X. Wen and Z.-Y. Wen, Eur. J. Combin. 15, 587 (1994).
[CrossRef]

1989 (1)

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, and S. W. Zucker, Phys. Rev. A 39, 1500 (1989).
[CrossRef]

1986 (1)

C. Allain and M. Cloitre, Phys. Rev. B 33, 3566 (1986).
[CrossRef]

1985 (1)

P. Cvitanovic, M. H. Jensen, L. P. Kadanoff, and I. Procaccia, Phys. Rev. Lett. 55, 343 (1985).
[CrossRef]

1984 (2)

D. Levine and P. J. Steinhardt, Phys. Rev. Lett. 53, 2477 (1984).
[CrossRef]

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

Allain, C.

C. Allain and M. Cloitre, Phys. Rev. B 33, 3566 (1986).
[CrossRef]

Banerjee, V.

Blech, I.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

Bu, J.

Q. Zhang, J. Wang, M. Wang, J. Bu, S. Zhu, B. Z. Gao, and X. Yuan, Opt. Laser Technol. 44, 2140 (2012).
[CrossRef]

Cahn, J. W.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

Calatayud, A.

Cao, Y.

J. Zhang, Y. Cao, and J. Zheng, Optik 121, 417 (2010).
[CrossRef]

Cloitre, M.

C. Allain and M. Cloitre, Phys. Rev. B 33, 3566 (1986).
[CrossRef]

Cvitanovic, P.

P. Cvitanovic, M. H. Jensen, L. P. Kadanoff, and I. Procaccia, Phys. Rev. Lett. 55, 343 (1985).
[CrossRef]

Damanik, D.

D. Damanik, M. Embree, A. Gorodetski, and S. Tcheremchantsev, Commun. Math. Phys. 280, 499 (2008).
[CrossRef]

Diehl, R. D.

N. Ferralis, A. W. Szmodis, and R. D. Diehl, Am. J. Phys. 72, 1241 (2004).
[CrossRef]

Dubuc, B.

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, and S. W. Zucker, Phys. Rev. A 39, 1500 (1989).
[CrossRef]

Dunlap, R. A.

R. A. Dunlap, The Golden Ratio and Fibonacci Numbers (World Scientific, 1997).

Embree, M.

D. Damanik, M. Embree, A. Gorodetski, and S. Tcheremchantsev, Commun. Math. Phys. 280, 499 (2008).
[CrossRef]

Ferralis, N.

N. Ferralis, A. W. Szmodis, and R. D. Diehl, Am. J. Phys. 72, 1241 (2004).
[CrossRef]

Ferrando, V.

Furlan, W. D.

Gao, B. Z.

Q. Zhang, J. Wang, M. Wang, J. Bu, S. Zhu, B. Z. Gao, and X. Yuan, Opt. Laser Technol. 44, 2140 (2012).
[CrossRef]

Gao, N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Gorodetski, A.

D. Damanik, M. Embree, A. Gorodetski, and S. Tcheremchantsev, Commun. Math. Phys. 280, 499 (2008).
[CrossRef]

Gratias, D.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

Grushina, N. V.

N. V. Grushina, P. V. Korolenko, and S. N. Markova, Moscow Univ. Phys. Bull. 63, 123 (2008).
[CrossRef]

Guerin, C.

C. Guerin and M. Holschneider, J. Phys. A 29, 7651 (1996).
[CrossRef]

Hamburger-Lidar, D. A.

D. A. Hamburger-Lidar, Phys. Rev. E 54, 354 (1996).
[CrossRef]

Holschneider, M.

C. Guerin and M. Holschneider, J. Phys. A 29, 7651 (1996).
[CrossRef]

Hou, Bo.

Bo. Hou, Gu. Xu, W. Wen, and G. K. L. Wong, Appl. Phys. Lett. 85, 6125 (2004).
[CrossRef]

Jensen, M. H.

P. Cvitanovic, M. H. Jensen, L. P. Kadanoff, and I. Procaccia, Phys. Rev. Lett. 55, 343 (1985).
[CrossRef]

Kadanoff, L. P.

P. Cvitanovic, M. H. Jensen, L. P. Kadanoff, and I. Procaccia, Phys. Rev. Lett. 55, 343 (1985).
[CrossRef]

Korolenko, P. V.

N. V. Grushina, P. V. Korolenko, and S. N. Markova, Moscow Univ. Phys. Bull. 63, 123 (2008).
[CrossRef]

Levine, D.

D. Levine and P. J. Steinhardt, Phys. Rev. Lett. 53, 2477 (1984).
[CrossRef]

Markova, S. N.

N. V. Grushina, P. V. Korolenko, and S. N. Markova, Moscow Univ. Phys. Bull. 63, 123 (2008).
[CrossRef]

Monsoriu, J. A.

Procaccia, I.

P. Cvitanovic, M. H. Jensen, L. P. Kadanoff, and I. Procaccia, Phys. Rev. Lett. 55, 343 (1985).
[CrossRef]

Quiniou, J. F.

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, and S. W. Zucker, Phys. Rev. A 39, 1500 (1989).
[CrossRef]

Remon, L.

Roques-Carmes, C.

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, and S. W. Zucker, Phys. Rev. A 39, 1500 (1989).
[CrossRef]

Saavedra, G.

Senthilkumaran, P.

Sharma, M. K.

Shechtman, D.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

Steinhardt, P. J.

D. Levine and P. J. Steinhardt, Phys. Rev. Lett. 53, 2477 (1984).
[CrossRef]

Szmodis, A. W.

N. Ferralis, A. W. Szmodis, and R. D. Diehl, Am. J. Phys. 72, 1241 (2004).
[CrossRef]

Tcheremchantsev, S.

D. Damanik, M. Embree, A. Gorodetski, and S. Tcheremchantsev, Commun. Math. Phys. 280, 499 (2008).
[CrossRef]

Tricot, C.

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, and S. W. Zucker, Phys. Rev. A 39, 1500 (1989).
[CrossRef]

Verma, R.

Wang, G. P.

Wang, J.

Q. Zhang, J. Wang, M. Wang, J. Bu, S. Zhu, B. Z. Gao, and X. Yuan, Opt. Laser Technol. 44, 2140 (2012).
[CrossRef]

Wang, M.

Q. Zhang, J. Wang, M. Wang, J. Bu, S. Zhu, B. Z. Gao, and X. Yuan, Opt. Laser Technol. 44, 2140 (2012).
[CrossRef]

Wen, W.

Bo. Hou, Gu. Xu, W. Wen, and G. K. L. Wong, Appl. Phys. Lett. 85, 6125 (2004).
[CrossRef]

Wen, Z.-X.

Z.-X. Wen and Z.-Y. Wen, Eur. J. Combin. 15, 587 (1994).
[CrossRef]

Wen, Z.-Y.

Z.-X. Wen and Z.-Y. Wen, Eur. J. Combin. 15, 587 (1994).
[CrossRef]

Wong, G. K. L.

Bo. Hou, Gu. Xu, W. Wen, and G. K. L. Wong, Appl. Phys. Lett. 85, 6125 (2004).
[CrossRef]

Wu, K.

Xie, C.

Xu, Gu.

Bo. Hou, Gu. Xu, W. Wen, and G. K. L. Wong, Appl. Phys. Lett. 85, 6125 (2004).
[CrossRef]

Yuan, X.

Q. Zhang, J. Wang, M. Wang, J. Bu, S. Zhu, B. Z. Gao, and X. Yuan, Opt. Laser Technol. 44, 2140 (2012).
[CrossRef]

Zhang, J.

J. Zhang, Y. Cao, and J. Zheng, Optik 121, 417 (2010).
[CrossRef]

Zhang, Q.

Q. Zhang, J. Wang, M. Wang, J. Bu, S. Zhu, B. Z. Gao, and X. Yuan, Opt. Laser Technol. 44, 2140 (2012).
[CrossRef]

Zhang, Y.

Zheng, J.

J. Zhang, Y. Cao, and J. Zheng, Optik 121, 417 (2010).
[CrossRef]

Zhu, S.

Q. Zhang, J. Wang, M. Wang, J. Bu, S. Zhu, B. Z. Gao, and X. Yuan, Opt. Laser Technol. 44, 2140 (2012).
[CrossRef]

Zucker, S. W.

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, and S. W. Zucker, Phys. Rev. A 39, 1500 (1989).
[CrossRef]

Am. J. Phys. (1)

N. Ferralis, A. W. Szmodis, and R. D. Diehl, Am. J. Phys. 72, 1241 (2004).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

Bo. Hou, Gu. Xu, W. Wen, and G. K. L. Wong, Appl. Phys. Lett. 85, 6125 (2004).
[CrossRef]

Commun. Math. Phys. (1)

D. Damanik, M. Embree, A. Gorodetski, and S. Tcheremchantsev, Commun. Math. Phys. 280, 499 (2008).
[CrossRef]

Eur. J. Combin. (1)

Z.-X. Wen and Z.-Y. Wen, Eur. J. Combin. 15, 587 (1994).
[CrossRef]

J. Phys. A (1)

C. Guerin and M. Holschneider, J. Phys. A 29, 7651 (1996).
[CrossRef]

Moscow Univ. Phys. Bull. (1)

N. V. Grushina, P. V. Korolenko, and S. N. Markova, Moscow Univ. Phys. Bull. 63, 123 (2008).
[CrossRef]

Opt. Express (4)

Opt. Laser Technol. (1)

Q. Zhang, J. Wang, M. Wang, J. Bu, S. Zhu, B. Z. Gao, and X. Yuan, Opt. Laser Technol. 44, 2140 (2012).
[CrossRef]

Opt. Lett. (3)

Optik (1)

J. Zhang, Y. Cao, and J. Zheng, Optik 121, 417 (2010).
[CrossRef]

Phys. Rev. A (1)

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, and S. W. Zucker, Phys. Rev. A 39, 1500 (1989).
[CrossRef]

Phys. Rev. B (1)

C. Allain and M. Cloitre, Phys. Rev. B 33, 3566 (1986).
[CrossRef]

Phys. Rev. E (1)

D. A. Hamburger-Lidar, Phys. Rev. E 54, 354 (1996).
[CrossRef]

Phys. Rev. Lett. (3)

D. Levine and P. J. Steinhardt, Phys. Rev. Lett. 53, 2477 (1984).
[CrossRef]

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

P. Cvitanovic, M. H. Jensen, L. P. Kadanoff, and I. Procaccia, Phys. Rev. Lett. 55, 343 (1985).
[CrossRef]

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

R. A. Dunlap, The Golden Ratio and Fibonacci Numbers (World Scientific, 1997).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1.
Fig. 1.

FbG for generations n=6, 7, 8.

Fig. 2.
Fig. 2.

Amplitude of the diffraction pattern of the FbG for n=6, 7, and 8. We plotted the modulus of An(f) because of its complex nature.

Fig. 3.
Fig. 3.

Plot of scaled intensity profiles in the FbG. The zoomed portion illustrates self-similarity.

Fig. 4.
Fig. 4.

Plot of the integrated structure factor versus frequency. The slope of the best fit line yields df0.88.

Fig. 5.
Fig. 5.

Schematic representation of the 4F arrangement of the experimental setup.

Fig. 6.
Fig. 6.

Reconstructions using the 4F experimental setup. Top row, FbG for n=6 and 7; middle row, reconstruction from the complete profile; bottom row, reconstruction from a secondary band.

Fig. 7.
Fig. 7.

(Top) FbG and (middle and bottom) two distinct disorder realizations with D=50% (n=8).

Tables (1)

Tables Icon

Table 1. Fractal Dimension df as a Function of Disorder D

Equations (5)

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

G2(x)=R2(x)*δ(xa2),G3(x)=R3(x)*δ(x0),G4(x)=R4(x)*[δ(x+2a5)+δ(x4a5)],
Gn(x)=Rn(x)*Δn(x),n5,
Δ5(x)=k=1F3δ(xxk)=k=13δ(xxk)
An(f)=dxei2πfxGn(x),
=2aFnsinc(2πafFn)k=1Fn2ei2πfxk,

Metrics