Abstract

The term “polarization-dependent Talbot effect” means that the Talbot self-imaging intensity of a high-density grating is different for TE and TM polarization modes. Numerical simulations with the finite-difference time-domain method show that the polarization dependence of the Talbot images is obvious for gratings with period d between 2λ and 3λ. Such a polarization-dependent difference for TE and TM polarization of a high-density grating of 630linesmm (corresponding to dλ=2.5) is verified through experiments with the scanning near-field optical microscopy technique, in which a HeNe laser is used as its polarization is changed from the TE mode to the TM mode. The polarization-dependent Talbot effect should help us to understand more clearly the diffraction behavior of a high-density grating in nano-optics and contribute to wide application of the Talbot effect.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2006

2005

2004

B. Lehner and K. Hingerl, "The finite difference time domain method as a numerical tool for studying the polarization optical response of rough surface," Thin Solid Films 455-456, 462-467 (2004).
[CrossRef]

C. Zhou, W. Wang, E. Dai, and L. Liu, "Simple principles of the Talbot effect," Opt. Photon. News December 2004, pp. 46-50.

P. Wei, H. Chou, and Y. Chen, "Subwavelength focusing in the near field in mesoscale air-dielectric structures," Opt. Lett. 29, 433-435 (2004).
[CrossRef] [PubMed]

2002

2000

A. Taflove and S. Hagness, Computational Electromagnetics: The Finite-Difference Time Domain Method, 2nd ed. (Artech, 2000).

1999

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

1998

1997

1995

1993

E. Noponen and J. Turunen, "Electromagnetic theory of Talbot imaging," Opt. Commun. 98, 132-140 (1993).
[CrossRef]

1990

1981

G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

1966

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).

1881

Lord Rayleigh, "On copying diffraction-gratings, and on some phenomenon connected therewith," Philos. Mag. 11, 196-205 (1881).

1836

W. H. F. Talbot, "Facts relating to optical sciences. No. IV," Philos. Mag. 9, 401-407 (1836).

Biener, G.

Bomzon, Z.

Chen, Y.

Chou, H.

Dai, E.

C. Zhou, W. Wang, E. Dai, and L. Liu, "Simple principles of the Talbot effect," Opt. Photon. News December 2004, pp. 46-50.

David, C.

Davis, C. C.

Denz, C.

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

Fann, W.

Gaylord, T.

Grann, E.

Hagness, S.

A. Taflove and S. Hagness, Computational Electromagnetics: The Finite-Difference Time Domain Method, 2nd ed. (Artech, 2000).

Hasman, E.

Hingerl, K.

B. Lehner and K. Hingerl, "The finite difference time domain method as a numerical tool for studying the polarization optical response of rough surface," Thin Solid Films 455-456, 462-467 (2004).
[CrossRef]

Ichikawa, H.

Jeon, S.

Judkins, J. B.

Kleiner, V.

Kurtsiefer, C.

Lehner, B.

B. Lehner and K. Hingerl, "The finite difference time domain method as a numerical tool for studying the polarization optical response of rough surface," Thin Solid Films 455-456, 462-467 (2004).
[CrossRef]

Li, L.

Liu, L.

C. Zhou, W. Wang, E. Dai, and L. Liu, "Simple principles of the Talbot effect," Opt. Photon. News December 2004, pp. 46-50.

C. Zhou, X. Zhao, L. Liu, "Rediscovering waveguide beam splitter/combiner," in Proc. SPIE 4904, 500-505 (2002).
[CrossRef]

Lohmann, A. W.

Lu, Y.

Y. Lu, C. Zhou, and H. Luo, "Talbot effect of a grating with different kinds of flaws," J. Opt. Soc. Am. A 22, 2662-2667 (2005).
[CrossRef]

H. Luo, C. Zhou, H. Zou, and Y. Lu, "Talbot-SNOM method for non-contact evaluation of high-density gratings," Opt. Commun. 248, 97-103 (2005).
[CrossRef]

Luo, H.

Y. Lu, C. Zhou, and H. Luo, "Talbot effect of a grating with different kinds of flaws," J. Opt. Soc. Am. A 22, 2662-2667 (2005).
[CrossRef]

H. Luo, C. Zhou, H. Zou, and Y. Lu, "Talbot-SNOM method for non-contact evaluation of high-density gratings," Opt. Commun. 248, 97-103 (2005).
[CrossRef]

Malyarchuk, V.

Moharam, M.

Mur, G.

G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

Niv, A.

Noponen, E.

E. Noponen and J. Turunen, "Electromagnetic theory of Talbot imaging," Opt. Commun. 98, 132-140 (1993).
[CrossRef]

Nowak, S.

Pfau, T.

Pommet, D.

Rayleigh, Lord

Lord Rayleigh, "On copying diffraction-gratings, and on some phenomenon connected therewith," Philos. Mag. 11, 196-205 (1881).

Rogers, J.

Ru, H.

Smolyaninov, I. I.

Stankovic, S.

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

Taflove, A.

A. Taflove and S. Hagness, Computational Electromagnetics: The Finite-Difference Time Domain Method, 2nd ed. (Artech, 2000).

Talbot, W. H. F.

W. H. F. Talbot, "Facts relating to optical sciences. No. IV," Philos. Mag. 9, 401-407 (1836).

Thomas, J. A.

Tschudi, T.

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

Turunen, J.

E. Noponen and J. Turunen, "Electromagnetic theory of Talbot imaging," Opt. Commun. 98, 132-140 (1993).
[CrossRef]

Wang, S.

Wang, W.

C. Zhou, W. Wang, E. Dai, and L. Liu, "Simple principles of the Talbot effect," Opt. Photon. News December 2004, pp. 46-50.

Wei, P.

Wiederrecht, G. P.

Yee, K. S.

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).

Zhang, Y.

Zhao, X.

C. Zhou, X. Zhao, L. Liu, "Rediscovering waveguide beam splitter/combiner," in Proc. SPIE 4904, 500-505 (2002).
[CrossRef]

Zhou, C.

S. Wang, C. Zhou, Y. Zhang, and H. Ru, "Deep etched high-density fused silica transmission gratings with high efficiency at wavelength of 1550nm," Appl. Opt. 45, 2567-2571 (2006).
[CrossRef] [PubMed]

S. Wang, C. Zhou, H. Ru, and Y. Zhang, "Optimized condition for etching fused silica phase grating with inductively coupled plasma technology, " Appl. Opt. 44, 4429-4434 (2005).
[CrossRef] [PubMed]

Y. Lu, C. Zhou, and H. Luo, "Talbot effect of a grating with different kinds of flaws," J. Opt. Soc. Am. A 22, 2662-2667 (2005).
[CrossRef]

H. Luo, C. Zhou, H. Zou, and Y. Lu, "Talbot-SNOM method for non-contact evaluation of high-density gratings," Opt. Commun. 248, 97-103 (2005).
[CrossRef]

C. Zhou, W. Wang, E. Dai, and L. Liu, "Simple principles of the Talbot effect," Opt. Photon. News December 2004, pp. 46-50.

C. Zhou, X. Zhao, L. Liu, "Rediscovering waveguide beam splitter/combiner," in Proc. SPIE 4904, 500-505 (2002).
[CrossRef]

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

Ziolkowski, R. W.

Zou, H.

H. Luo, C. Zhou, H. Zou, and Y. Lu, "Talbot-SNOM method for non-contact evaluation of high-density gratings," Opt. Commun. 248, 97-103 (2005).
[CrossRef]

Appl. Opt.

IEEE Trans. Antennas Propag.

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).

IEEE Trans. Electromagn. Compat.

G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

H. Luo, C. Zhou, H. Zou, and Y. Lu, "Talbot-SNOM method for non-contact evaluation of high-density gratings," Opt. Commun. 248, 97-103 (2005).
[CrossRef]

E. Noponen and J. Turunen, "Electromagnetic theory of Talbot imaging," Opt. Commun. 98, 132-140 (1993).
[CrossRef]

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

Opt. Express

Opt. Lett.

Philos. Mag.

Lord Rayleigh, "On copying diffraction-gratings, and on some phenomenon connected therewith," Philos. Mag. 11, 196-205 (1881).

W. H. F. Talbot, "Facts relating to optical sciences. No. IV," Philos. Mag. 9, 401-407 (1836).

Proc. SPIE

C. Zhou, X. Zhao, L. Liu, "Rediscovering waveguide beam splitter/combiner," in Proc. SPIE 4904, 500-505 (2002).
[CrossRef]

Thin Solid Films

B. Lehner and K. Hingerl, "The finite difference time domain method as a numerical tool for studying the polarization optical response of rough surface," Thin Solid Films 455-456, 462-467 (2004).
[CrossRef]

Other

A. Taflove and S. Hagness, Computational Electromagnetics: The Finite-Difference Time Domain Method, 2nd ed. (Artech, 2000).

C. Zhou, W. Wang, E. Dai, and L. Liu, "Simple principles of the Talbot effect," Opt. Photon. News December 2004, pp. 46-50.

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Figures (8)

Fig. 1
Fig. 1

Grating structure employed here.

Fig. 2
Fig. 2

Theoretical ratio of the differing intensities of polarization modes (TE/TM) of the time-averaged Poynting vector at the middle of bright stripe of the Talbot images at 1/2 Talbot distance with various grating periods from λ to 4 λ . Note that at d λ = 2.5 , the difference in the polarization Talbot effect for TE and TM modes is significant. This is further illustrated in Fig. 3 and verified through experiments in Figs. 5, 7.

Fig. 3
Fig. 3

Numerical simulations of the near-field distribution of the time-averaged Poynting vector with increased distance is shown for (a) TE polarization and (b) TM polarization with d = 2.5 λ . The comparison of the intensity distribution of the time-averaged Poynting vector at 1/2 Talbot distance for TE and TM polarization is shown in (c).

Fig. 4
Fig. 4

Schematic illustration of experimental setup of the Talbot SNOM apparatus for scanning the Talbot images of a high-density grating for the different polarization modes. PZT, piezoelectric tube.

Fig. 5
Fig. 5

Talbot images of a grating of 630 lines mm ( 2.5 λ ) obtained by the Talbot SNOM apparatus are shown for (a) TE polarization and (b) TM polarization under illumination of the He Ne laser of λ = 0.6328 μ m . The gray value is averaged in the y direction to remove the stochastic error induced by the flaws in the grating. The comparison for both polarizations is shown in (c).

Fig. 6
Fig. 6

Numerical simulation of the intensity distribution of the time-averaged Poynting vector at the plane just 0.25 μ m in front of the 1 2 Talbot plane for both polarization modes. These results are in good agreement with those of Fig. 5c.

Fig. 7
Fig. 7

Experimental results of the Talbot images of a grating obtained by the Talbot SNOM are shown for (a) TE polarization and (b) TM polarization that were obtained with the grating on an electrically controlled stage. The gray value is averaged in the y direction to remove the stochastic error induced by the flaws in the grating. The comparison of the two polarizations is shown in (c). Note that the depressions in the middle of the stripes are commensurate with those in Fig. 3c, although there is still a small departure of the ratio of the magnitudes for different polarization modes between them.

Fig. 8
Fig. 8

Numerical simulations of the amplitudes of the propagation and evanescent orders { A ( m ) } generated by Fourier transformation of the grating of 630 lines mm show obvious differences for the two polarization states in the far field.

Equations (7)

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× E = μ H t σ * H ,
× H = ϵ E t + σ E ,
E y n + 1 ( i , k ) = 2 ϵ ( i , k ) σ ( i , k ) Δ t 2 ϵ ( i , k ) + σ ( i , k ) Δ t E y n ( i , k ) + 2 Δ t 2 ϵ ( i , k ) + σ ( i , k ) Δ t × [ H x n + 1 ( i , k ) H x n + 1 ( i , k 1 ) Δ z H z n + 1 ( i , k ) H z n + 1 ( i 1 , k ) Δ x ] ,
H x n + 1 ( i , k ) = H x n ( i , k ) + Δ t μ ( i , k ) E y n ( i + 1 , k ) E y n ( i , k ) Δ z ,
H z n + 1 ( i , k ) = H z n ( i , k ) + Δ t μ ( i , k ) E y n ( i , k + 1 ) E y n ( i , k ) Δ x .
TE TM = E ( x , z , t ) × H ( x , z , t ) t TE E ( x , z , t ) × H ( x , z , t ) t TM ( z = 0.5 Z t x = 0.75 d ) ,
I TE I TM I TE = 0.028 .

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