Abstract

Based on the self-similarity property of fractal, two types of fractal gratings are produced according to the production and addition operations of multiple periodic gratings. Fresnel diffractions of fractal grating are analyzed theoretically, and the general mathematic expressions of the diffraction intensity distributions of fractal grating are deduced. The gray-scale patterns of the 2D diffraction distributions of fractal grating are provided through numerical calculations. The diffraction patterns take on the periodicity along the longitude and transverse directions. The 1D diffraction distribution at some certain distances shows the same structure as the fractal grating. This indicates that the self-image of fractal grating is really formed in the Fresnel diffraction region. The experimental measurement of the diffraction intensity distribution of fractal grating with different fractal dimensions and different fractal levels is performed, and the self-images of fractal grating are obtained successfully in experiments. The conclusions of this paper are helpful for the development of the application of fractal grating.

© 2014 Optical Society of America

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References

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2014

S. Y. Teng, J. H. Wang, F. R. Li, and W. Zhang, “Talbot image of two-dimensional fractal grating,” Opt. Commun. 315, 103–107 (2014).
[CrossRef]

2013

2011

2009

O. Mendoza-Yero, G. Mínguez-Vega, M. Fernández-Alonso, J. Lancis, E. Tajahuerce, V. Climent, and J. A. Monsoriu, “Optical filters with fractal transmission spectra based on diffractive optics,” Opt. Lett. 34, 560–562 (2009).
[CrossRef]

K. Singh, V. Grewal, and R. Saxena, “Fractal antennas: a novel miniaturization technique for wireless communications,” Internat. J. Recent Trends Engineer. 2, 172–176 (2009).

2008

2007

2005

C. F. Kao and M. H. Lu, “Optical encoder based on the fractional Talbot Effect,” Opt. Commun. 250, 16–23 (2005).
[CrossRef]

Y. L. Sheng and L. Sun, “Near-field diffraction of irregular phase gratings with multiple phase-shifts,” Opt. Express 13, 6111–6116 (2005).
[CrossRef]

2004

D. C. Méndez and M. Lehman, “Talbot effect with Cantor transmittances,” Optik 115, 439–442 (2004).
[CrossRef]

2002

J. P. Gianvittorio and Y. Rahmat-Samii, “Fractals antennas: a novel antenna miniaturization technique, and applications,” IEEE Antennas Propag. Mag. 44, 20–36 (2002).
[CrossRef]

P. Xi, C. H. Zhou, E. W. Dai, and L. R. Liu, “Generation of near-field hexagonal array illumination with a phase grating,” Opt. Lett. 27, 228–230 (2002).
[CrossRef]

2001

C. Aguirre Vélez, M. Lehman, and M. Garavaglia, “Two-dimensional fractal gratings with variable structure and their diffraction,” Optik 112, 209–217 (2001).
[CrossRef]

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

1999

C. Zhou, S. Stankovic, C. Denz, and T. Tschuli, “Phase codes of Talbot array illumination for encoding holographic multiplexing storage,” Opt. Commun. 161, 209–211 (1999).
[CrossRef]

1988

1836

W. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9(56), 401–407 (1836).
[CrossRef]

Aguirre Vélez, C.

C. Aguirre Vélez, M. Lehman, and M. Garavaglia, “Two-dimensional fractal gratings with variable structure and their diffraction,” Optik 112, 209–217 (2001).
[CrossRef]

Beermann, J.

Boltasseva, A.

Bonod, N.

Bozhevolnyi, S. I.

Brown, J. W.

J. W. Brown and R. V. Churchill, Fourier Series and Boundary Value Problems (McGraw-Hill, 1993).

Chen, X. Y.

Cheng, C. F.

Churchill, R. V.

J. W. Brown and R. V. Churchill, Fourier Series and Boundary Value Problems (McGraw-Hill, 1993).

Climent, V.

Dai, E. W.

Denz, C.

C. Zhou, S. Stankovic, C. Denz, and T. Tschuli, “Phase codes of Talbot array illumination for encoding holographic multiplexing storage,” Opt. Commun. 161, 209–211 (1999).
[CrossRef]

Dong, Q. R.

Fernández-Alonso, M.

Frankenhuysen, M. V.

M. L. Lapidus and M. V. Frankenhuysen, Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions (Birkhäuser, 2000).

Gao, N.

Garavaglia, M.

C. Aguirre Vélez, M. Lehman, and M. Garavaglia, “Two-dimensional fractal gratings with variable structure and their diffraction,” Optik 112, 209–217 (2001).
[CrossRef]

Gianvittorio, J. P.

J. P. Gianvittorio and Y. Rahmat-Samii, “Fractals antennas: a novel antenna miniaturization technique, and applications,” IEEE Antennas Propag. Mag. 44, 20–36 (2002).
[CrossRef]

Grewal, V.

K. Singh, V. Grewal, and R. Saxena, “Fractal antennas: a novel miniaturization technique for wireless communications,” Internat. J. Recent Trends Engineer. 2, 172–176 (2009).

Kao, C. F.

C. F. Kao and M. H. Lu, “Optical encoder based on the fractional Talbot Effect,” Opt. Commun. 250, 16–23 (2005).
[CrossRef]

Lancis, J.

Lapidus, M. L.

M. L. Lapidus and M. V. Frankenhuysen, Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions (Birkhäuser, 2000).

Lehman, M.

D. C. Méndez and M. Lehman, “Talbot effect with Cantor transmittances,” Optik 115, 439–442 (2004).
[CrossRef]

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

C. Aguirre Vélez, M. Lehman, and M. Garavaglia, “Two-dimensional fractal gratings with variable structure and their diffraction,” Optik 112, 209–217 (2001).
[CrossRef]

Li, F. R.

S. Y. Teng, J. H. Wang, F. R. Li, and W. Zhang, “Talbot image of two-dimensional fractal grating,” Opt. Commun. 315, 103–107 (2014).
[CrossRef]

C. Zhang, W. Zhang, F. R. Li, J. H. Wang, and S. Y. Teng, “Talbot effect of quasi-periodic grating,” Appl. Opt. 52, 5083–5087 (2013).
[CrossRef]

Liu, L. R.

Lu, M. H.

C. F. Kao and M. H. Lu, “Optical encoder based on the fractional Talbot Effect,” Opt. Commun. 250, 16–23 (2005).
[CrossRef]

Mandelbrot, B. B.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, 1982).

Méndez, D. C.

D. C. Méndez and M. Lehman, “Talbot effect with Cantor transmittances,” Optik 115, 439–442 (2004).
[CrossRef]

Mendoza-Yero, O.

Mínguez-Vega, G.

Monsoriu, J. A.

Neauport, J.

Radko, I. P.

Rahmat-Samii, Y.

J. P. Gianvittorio and Y. Rahmat-Samii, “Fractals antennas: a novel antenna miniaturization technique, and applications,” IEEE Antennas Propag. Mag. 44, 20–36 (2002).
[CrossRef]

Saxena, R.

K. Singh, V. Grewal, and R. Saxena, “Fractal antennas: a novel miniaturization technique for wireless communications,” Internat. J. Recent Trends Engineer. 2, 172–176 (2009).

Sheng, Y. L.

Singh, K.

K. Singh, V. Grewal, and R. Saxena, “Fractal antennas: a novel miniaturization technique for wireless communications,” Internat. J. Recent Trends Engineer. 2, 172–176 (2009).

Stankovic, S.

C. Zhou, S. Stankovic, C. Denz, and T. Tschuli, “Phase codes of Talbot array illumination for encoding holographic multiplexing storage,” Opt. Commun. 161, 209–211 (1999).
[CrossRef]

Sun, L.

Szwaykowski, P.

Tajahuerce, E.

Talbot, W. H. F.

W. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9(56), 401–407 (1836).
[CrossRef]

Teng, S. Y.

Tschuli, T.

C. Zhou, S. Stankovic, C. Denz, and T. Tschuli, “Phase codes of Talbot array illumination for encoding holographic multiplexing storage,” Opt. Commun. 161, 209–211 (1999).
[CrossRef]

Wang, J. H.

S. Y. Teng, J. H. Wang, F. R. Li, and W. Zhang, “Talbot image of two-dimensional fractal grating,” Opt. Commun. 315, 103–107 (2014).
[CrossRef]

C. Zhang, W. Zhang, F. R. Li, J. H. Wang, and S. Y. Teng, “Talbot effect of quasi-periodic grating,” Appl. Opt. 52, 5083–5087 (2013).
[CrossRef]

Xi, P.

Xie, C. Q.

Zhang, C.

Zhang, N. Y.

Zhang, W.

S. Y. Teng, J. H. Wang, F. R. Li, and W. Zhang, “Talbot image of two-dimensional fractal grating,” Opt. Commun. 315, 103–107 (2014).
[CrossRef]

C. Zhang, W. Zhang, F. R. Li, J. H. Wang, and S. Y. Teng, “Talbot effect of quasi-periodic grating,” Appl. Opt. 52, 5083–5087 (2013).
[CrossRef]

Zhang, Y. C.

Zhou, C.

C. Zhou, S. Stankovic, C. Denz, and T. Tschuli, “Phase codes of Talbot array illumination for encoding holographic multiplexing storage,” Opt. Commun. 161, 209–211 (1999).
[CrossRef]

Zhou, C. H.

Zhou, T. J.

Appl. Opt.

IEEE Antennas Propag. Mag.

J. P. Gianvittorio and Y. Rahmat-Samii, “Fractals antennas: a novel antenna miniaturization technique, and applications,” IEEE Antennas Propag. Mag. 44, 20–36 (2002).
[CrossRef]

Internat. J. Recent Trends Engineer.

K. Singh, V. Grewal, and R. Saxena, “Fractal antennas: a novel miniaturization technique for wireless communications,” Internat. J. Recent Trends Engineer. 2, 172–176 (2009).

J. Opt. Soc. Am. A

Opt. Commun.

C. Zhou, S. Stankovic, C. Denz, and T. Tschuli, “Phase codes of Talbot array illumination for encoding holographic multiplexing storage,” Opt. Commun. 161, 209–211 (1999).
[CrossRef]

C. F. Kao and M. H. Lu, “Optical encoder based on the fractional Talbot Effect,” Opt. Commun. 250, 16–23 (2005).
[CrossRef]

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

S. Y. Teng, J. H. Wang, F. R. Li, and W. Zhang, “Talbot image of two-dimensional fractal grating,” Opt. Commun. 315, 103–107 (2014).
[CrossRef]

Opt. Express

Opt. Lett.

Optik

D. C. Méndez and M. Lehman, “Talbot effect with Cantor transmittances,” Optik 115, 439–442 (2004).
[CrossRef]

C. Aguirre Vélez, M. Lehman, and M. Garavaglia, “Two-dimensional fractal gratings with variable structure and their diffraction,” Optik 112, 209–217 (2001).
[CrossRef]

Philos. Mag.

W. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9(56), 401–407 (1836).
[CrossRef]

Other

J. W. Brown and R. V. Churchill, Fourier Series and Boundary Value Problems (McGraw-Hill, 1993).

M. L. Lapidus and M. V. Frankenhuysen, Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions (Birkhäuser, 2000).

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, 1982).

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Figures (8)

Fig. 1.
Fig. 1.

Schematic diagrams of fractal grating. (a) The first type of fractal grating. (b) The second type of fractal grating.

Fig. 2.
Fig. 2.

Elementary grating G and fractal gratings F. Fa1, Fa2, Fb1, and Fb2 are the first types of fractal grating. The fractal dimension for the former two is D=0.6309, and that for the latter two is D=0.5000. Fc1, Fc2, Fd1, and Fd2 are the second types of fractal grating. The fractal dimension for the former two is D=1.0000 and that for the latter two is D=0.6309.

Fig. 3.
Fig. 3.

Diffraction patterns of fractal grating with 1-level fractal. (a)–(d) are for Fa1, Fb1, Fc1, and Fd1, respectively.

Fig. 4.
Fig. 4.

Diffraction patterns of fractal grating with 2-level fractal. (a)–(d) are for Fa2, Fb2, Fc2, and Fd2, respectively.

Fig. 5.
Fig. 5.

1D normalized diffraction distributions of fractal grating with 1-level fractal. (a) and (b) are for the first type of fractal grating with D=0.6309 and D=0.5000. (c) and (d) are for the second type of fractal grating with D=1.0000 and D=0.6309.

Fig. 6.
Fig. 6.

1D normalized diffraction distributions of fractal grating with 2-level fractal. (a) and (b) are for the first type of fractal grating with D=0.6309 and D=0.5000. (c) and (d) are for the second type of fractal grating with D=1.0000 and D=0.6309.

Fig. 7.
Fig. 7.

Experimental setup for the measurement of the diffraction intensity distribution of fractal grating.

Fig. 8.
Fig. 8.

Experimental results of the diffraction distributions of grating and fractal grating. Three patterns top to bottom in (a) are the diffractions of G1, F1, and F2. Three patterns top to bottom in (b) are the diffractions of G2, F3, and F4.

Equations (8)

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

t(x0,y0)=g(x0,y0)*nδ(x0nd,y0nd),
t(x0,y0)=mnAmAnexp(j2πmx0d)exp(j2πny0d),
t1(x0)=i=0qG1i(x0)=i=0q[mi=Amiexp(j2πmix0di)]
t2(x0)=i=0qG2i(x0)=i=0qmi=[Amiexpj2πmi(x0ai)d],
u(x,z)=Cjzλexp(jkz)t(x0)expjk(x0x)22zdx0,
u1(x,z)=C1jzλexp(jkz)n0={fn0n1=[fn1nk=fnk]},
fnk=Ankexp(j2πnkxdk)exp(jπλnk2zdk2)exp[j2πλz(l=0k1nldl)nkdk],
u2(x,z)=C1jzλexp(jkz)i=0qmi=Ami×expj2πmi(xai)dexpjπλmi2zd2.

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