Abstract

Theoretic and experimental studies of the Talbot effect of quasi-periodic gratings are performed in this paper. The diffractions of periodic and quasi-periodic square aperture arrays in Fresnel fields are analyzed according to the scalar diffraction theory. The expressions of the diffraction intensities of two types of quasi-periodic gratings are deduced. Talbot images of the quasi-periodic gratings are predicted to appear at multiple certain distances. The quasi-periodic square aperture arrays are produced with the aid of a liquid crystal light modulator, and the self-images of the quasi-periodic gratings are measured successfully in the experiment. This study indicates that even a structure in short-range disorder may take on the self-imaging effect in a Fresnel field.

© 2013 Optical Society of America

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2012 (1)

S. Papanikolaou, D. M. Dimiduk, W. Choi, J. P. Sethna, M. D. Uchic, C. F. Woodward, and S. Zapperi, “Quasi-periodic events in crystal plasticity and the self-organized avalanche oscillator,” Nature 490, 517–521 (2012).
[CrossRef]

2011 (3)

2008 (3)

2007 (3)

S. Y. Teng, X. Y. Chen, T. J. Zhou, and C. F. Cheng, “Quasi-Talbot effect of a grating in the deep Fresnel diffraction region,” J. Opt. Soc. Am. A 24, 1656–1665 (2007).
[CrossRef]

Y. Bao, B. Zhang, Z. Wu, J. Si, M. Wang, R. Peng, X. Lu, J. Shao, Z. Li, X. Hao, and N. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90, 251914 (2007).
[CrossRef]

M. R. Dennis, N. I. Zheludev, and F. J. G. de Abajo, “The plasmon Talbot effect,” Opt. Express 15, 9692–9700 (2007).
[CrossRef]

2006 (1)

2004 (2)

2003 (1)

2002 (1)

Y. Yayon, M. Rappaport, V. Umansky, and I. Bar-Joseph, “Anisotropy and periodicity in the density distribution of electrons in a quantum well,” Phys. Rev. B 66, 033310 (2002).
[CrossRef]

1999 (1)

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

1992 (1)

1988 (1)

1984 (1)

D. S. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “A metallic phase with long-ranged orientational order and no translational symmetry,” Phys. Rev. Lett. 53, 1951–1953 (1984).
[CrossRef]

1836 (1)

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

Andrea, F.

Andrés, P.

Araiza-Esquivel, M. A.

Arie, A.

Atwater, H. A.

Bao, Y.

Y. Bao, B. Zhang, Z. Wu, J. Si, M. Wang, R. Peng, X. Lu, J. Shao, Z. Li, X. Hao, and N. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90, 251914 (2007).
[CrossRef]

Bar-Joseph, I.

Y. Yayon, M. Rappaport, V. Umansky, and I. Bar-Joseph, “Anisotropy and periodicity in the density distribution of electrons in a quantum well,” Phys. Rev. B 66, 033310 (2002).
[CrossRef]

Biener, G.

Blech, I.

D. S. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “A metallic phase with long-ranged orientational order and no translational symmetry,” Phys. Rev. Lett. 53, 1951–1953 (1984).
[CrossRef]

Bonod, N.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 2001).

Cahn, J. W.

D. S. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “A metallic phase with long-ranged orientational order and no translational symmetry,” Phys. Rev. Lett. 53, 1951–1953 (1984).
[CrossRef]

Chen, X. Y.

Cheng, C. F.

S. Y. Teng, T. J. Zhou, and C. F. Cheng, “Influence of the size of the grating on Talbot effect,” Optik 119, 695–699 (2008).
[CrossRef]

S. Y. Teng, X. Y. Chen, T. J. Zhou, and C. F. Cheng, “Quasi-Talbot effect of a grating in the deep Fresnel diffraction region,” J. Opt. Soc. Am. A 24, 1656–1665 (2007).
[CrossRef]

Choi, W.

S. Papanikolaou, D. M. Dimiduk, W. Choi, J. P. Sethna, M. D. Uchic, C. F. Woodward, and S. Zapperi, “Quasi-periodic events in crystal plasticity and the self-organized avalanche oscillator,” Nature 490, 517–521 (2012).
[CrossRef]

Crouse, R. F.

de Abajo, F. J. G.

Dennis, M. R.

Denz, C.

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

Dimiduk, D. M.

S. Papanikolaou, D. M. Dimiduk, W. Choi, J. P. Sethna, M. D. Uchic, C. F. Woodward, and S. Zapperi, “Quasi-periodic events in crystal plasticity and the self-organized avalanche oscillator,” Nature 490, 517–521 (2012).
[CrossRef]

Dolev, I.

Georges, P.

Giovanni, P.

Giuseppe, C.

Gorodetski, Y.

Gratias, D.

D. S. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “A metallic phase with long-ranged orientational order and no translational symmetry,” Phys. Rev. Lett. 53, 1951–1953 (1984).
[CrossRef]

Hao, X.

Y. Bao, B. Zhang, Z. Wu, J. Si, M. Wang, R. Peng, X. Lu, J. Shao, Z. Li, X. Hao, and N. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90, 251914 (2007).
[CrossRef]

Hasman, E.

Hou, B.

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]

Javidi, B.

Jensen, O. B.

Kleiner, V.

Lancis, J.

Larkins, E.

Latimer, P.

Lezec, H. J.

Li, Z.

Y. Bao, B. Zhang, Z. Wu, J. Si, M. Wang, R. Peng, X. Lu, J. Shao, Z. Li, X. Hao, and N. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90, 251914 (2007).
[CrossRef]

Lim, J.

Liu, D. A.

Liu, L. R.

Lu, X.

Y. Bao, B. Zhang, Z. Wu, J. Si, M. Wang, R. Peng, X. Lu, J. Shao, Z. Li, X. Hao, and N. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90, 251914 (2007).
[CrossRef]

Luan, Z.

Lucas-Leclin, G.

Martínez-León, L.

Ming, N.

Y. Bao, B. Zhang, Z. Wu, J. Si, M. Wang, R. Peng, X. Lu, J. Shao, Z. Li, X. Hao, and N. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90, 251914 (2007).
[CrossRef]

Neauport, J.

Niv, A.

Pabœuf, D.

Pacifici, D.

Papanikolaou, S.

S. Papanikolaou, D. M. Dimiduk, W. Choi, J. P. Sethna, M. D. Uchic, C. F. Woodward, and S. Zapperi, “Quasi-periodic events in crystal plasticity and the self-organized avalanche oscillator,” Nature 490, 517–521 (2012).
[CrossRef]

Peng, R.

Y. Bao, B. Zhang, Z. Wu, J. Si, M. Wang, R. Peng, X. Lu, J. Shao, Z. Li, X. Hao, and N. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90, 251914 (2007).
[CrossRef]

Pietro, F.

Porat, G.

Rappaport, M.

Y. Yayon, M. Rappaport, V. Umansky, and I. Bar-Joseph, “Anisotropy and periodicity in the density distribution of electrons in a quantum well,” Phys. Rev. B 66, 033310 (2002).
[CrossRef]

Sergio, D. N.

Sethna, J. P.

S. Papanikolaou, D. M. Dimiduk, W. Choi, J. P. Sethna, M. D. Uchic, C. F. Woodward, and S. Zapperi, “Quasi-periodic events in crystal plasticity and the self-organized avalanche oscillator,” Nature 490, 517–521 (2012).
[CrossRef]

Shao, J.

Y. Bao, B. Zhang, Z. Wu, J. Si, M. Wang, R. Peng, X. Lu, J. Shao, Z. Li, X. Hao, and N. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90, 251914 (2007).
[CrossRef]

Shechtman, D. S.

D. S. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “A metallic phase with long-ranged orientational order and no translational symmetry,” Phys. Rev. Lett. 53, 1951–1953 (1984).
[CrossRef]

Si, J.

Y. Bao, B. Zhang, Z. Wu, J. Si, M. Wang, R. Peng, X. Lu, J. Shao, Z. Li, X. Hao, and N. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90, 251914 (2007).
[CrossRef]

Simonetta, G.

Stankovic, S.

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

Sujecki, S.

Sweatlock, L. A.

Tajahuerce, E.

Talbot, W. H. F.

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

Teng, S. Y.

Thestrup, B.

Tschuli, T.

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

Uchic, M. D.

S. Papanikolaou, D. M. Dimiduk, W. Choi, J. P. Sethna, M. D. Uchic, C. F. Woodward, and S. Zapperi, “Quasi-periodic events in crystal plasticity and the self-organized avalanche oscillator,” Nature 490, 517–521 (2012).
[CrossRef]

Umansky, V.

Y. Yayon, M. Rappaport, V. Umansky, and I. Bar-Joseph, “Anisotropy and periodicity in the density distribution of electrons in a quantum well,” Phys. Rev. B 66, 033310 (2002).
[CrossRef]

Vijayakumar, D.

Volodarsky, M.

Walters, R. J.

Wang, M.

Y. Bao, B. Zhang, Z. Wu, J. Si, M. Wang, R. Peng, X. Lu, J. Shao, Z. Li, X. Hao, and N. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90, 251914 (2007).
[CrossRef]

Wen, W.

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]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 2001).

Wong, G. K. L.

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]

Woodward, C. F.

S. Papanikolaou, D. M. Dimiduk, W. Choi, J. P. Sethna, M. D. Uchic, C. F. Woodward, and S. Zapperi, “Quasi-periodic events in crystal plasticity and the self-organized avalanche oscillator,” Nature 490, 517–521 (2012).
[CrossRef]

Wu, Z.

Y. Bao, B. Zhang, Z. Wu, J. Si, M. Wang, R. Peng, X. Lu, J. Shao, Z. Li, X. Hao, and N. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90, 251914 (2007).
[CrossRef]

Xu, G.

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]

Yayon, Y.

Y. Yayon, M. Rappaport, V. Umansky, and I. Bar-Joseph, “Anisotropy and periodicity in the density distribution of electrons in a quantum well,” Phys. Rev. B 66, 033310 (2002).
[CrossRef]

Zapperi, S.

S. Papanikolaou, D. M. Dimiduk, W. Choi, J. P. Sethna, M. D. Uchic, C. F. Woodward, and S. Zapperi, “Quasi-periodic events in crystal plasticity and the self-organized avalanche oscillator,” Nature 490, 517–521 (2012).
[CrossRef]

Zhang, B.

Y. Bao, B. Zhang, Z. Wu, J. Si, M. Wang, R. Peng, X. Lu, J. Shao, Z. Li, X. Hao, and N. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90, 251914 (2007).
[CrossRef]

Zheludev, N. I.

Zhou, C.

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

Zhou, T. J.

S. Y. Teng, T. J. Zhou, and C. F. Cheng, “Influence of the size of the grating on Talbot effect,” Optik 119, 695–699 (2008).
[CrossRef]

S. Y. Teng, X. Y. Chen, T. J. Zhou, and C. F. Cheng, “Quasi-Talbot effect of a grating in the deep Fresnel diffraction region,” J. Opt. Soc. Am. A 24, 1656–1665 (2007).
[CrossRef]

Zu, J. F.

Appl. Opt. (2)

Appl. Phys. Lett. (2)

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]

Y. Bao, B. Zhang, Z. Wu, J. Si, M. Wang, R. Peng, X. Lu, J. Shao, Z. Li, X. Hao, and N. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90, 251914 (2007).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

Nature (1)

S. Papanikolaou, D. M. Dimiduk, W. Choi, J. P. Sethna, M. D. Uchic, C. F. Woodward, and S. Zapperi, “Quasi-periodic events in crystal plasticity and the self-organized avalanche oscillator,” Nature 490, 517–521 (2012).
[CrossRef]

Opt. Commun. (1)

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

Opt. Express (2)

Opt. Lett. (4)

Optik (1)

S. Y. Teng, T. J. Zhou, and C. F. Cheng, “Influence of the size of the grating on Talbot effect,” Optik 119, 695–699 (2008).
[CrossRef]

Philos. Mag. (1)

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

Phys. Rev. B (1)

Y. Yayon, M. Rappaport, V. Umansky, and I. Bar-Joseph, “Anisotropy and periodicity in the density distribution of electrons in a quantum well,” Phys. Rev. B 66, 033310 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

D. S. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “A metallic phase with long-ranged orientational order and no translational symmetry,” Phys. Rev. Lett. 53, 1951–1953 (1984).
[CrossRef]

Other (1)

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 2001).

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Figures (3)

Fig. 1.
Fig. 1.

Experimental configuration in the Fresnel diffraction of a quasi-periodic grating.

Fig. 2.
Fig. 2.

Experimental samples: (a) the grating with period d=0.54mm, (b) the grating with period d=0.44mm, (c) the superimposed quasi-period grating, and (d) the combination quasi-periodic grating.

Fig. 3.
Fig. 3.

Measured diffraction patterns for different grating structures at different distances. (a) The Talbot image of the grating with period d=0.54mm, (b) the Talbot image of the grating with period d=0.44mm, (c)–(e) the intensity distributions of the superimposed quasi-period grating at z=0.92, 0.61, and 1.83 m, and (f)–(h) the intensity distributions of the combination quasi-periodic grating at the same three distances.

Equations (4)

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

I(x,y,z)=A|mCmexp(i2πmx/d)exp(iπλm2z/d2)×nCnexp(i2πny/d)exp(iπλn2z/d2)|2.
t(x0,y0)=mnCmCnexp(i2πmx0/d1)exp(i2πny0/d1)+mnCmCnexp(i2πmx0/d2)exp(i2πny0/d2),
t(x0,y0)={mnCmCnexp(i2πmx0/d1)exp(i2πny0/d1)x0<0mnCmCnexp(i2πmx0/d2)exp(i2πny0/d2)x0>0.
I(x,y,z)=|x0<0mnCmCnexp(i2πmx0/d1)exp(i2πny0/d1)×exp{iπ[(xx0)2+(yy0)2]/λz}dx0dy0|2+|x0>0mnCmCnexp(i2πmx0/d2)exp(i2πny0/d2)×exp{iπ[(xx0)2+(yy0)2]/λz}dx0dy0|2.

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