Abstract

The Talbot effect of a grating with different kinds of flaws is analyzed with the finite-difference time-domain (FDTD) method. The FDTD method can show the exact near-field distribution of different flaws in a high-density grating, which is impossible to obtain with the conventional Fourier transform method. The numerical results indicate that if a grating is perfect, its Talbot imaging should also be perfect; if the grating is distorted, its Talbot imaging will also be distorted. Furthermore, we evaluate high-density gratings by detecting the near-field distribution with the scanning near-field optical microscopy technique. Experimental results are also given.

© 2005 Optical Society of America

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  2. C. R. Fernández-Pousa, F. Mateos, L. Chantada, M. T. Flores-Arias, C. Bao, M. V. Pérez, C. Gómez-Reino, “Timing jitter smoothing by Talbot effect. I. Variance,” J. Opt. Soc. Am. A 21, 1170–1177 (2004).
    [CrossRef]
  3. A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).
  4. A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
    [CrossRef] [PubMed]
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    [CrossRef]
  6. C. Zhou, S. Stankovic, C. Denz, T. Tschudi, “Phase codes of Talbot array illumination for encoding holographic multiplexing storage,” Opt. Commun. 161, 209–211 (1999).
    [CrossRef]
  7. M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
    [CrossRef]
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    [CrossRef] [PubMed]
  9. C. Zhou, L. Wang, T. Tschudi, “Solutions and analyses of fractional-Talbot array illuminations,” Opt. Commun. 147, 224–228 (1998).
    [CrossRef]
  10. C. Zhou, S. Stankovic, T. Tschudi, “Analytic phase-factor equations for Talbot array illuminations,” Appl. Opt. 38, 284–290 (1999).
    [CrossRef]
  11. C. Zhou, H. Wang, S. Zhao, P. Xi, L. Liu, “Number of phase levels of a Talbot array illuminator,” Appl. Opt. 40, 607–613 (2001).
    [CrossRef]
  12. C. Zhou, W. Wang, E. Dai, L. Liu, “Simple principles of the Talbot effect,” Opt. Photonics News, Dec. 2004, pp. 46–50.
  13. 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).
  14. A. Taflove, S. Hagness, Computational Electromagnetics: The Finite-Difference Time Domain Method, 2nd ed. (Artech House, 2000).
  15. S. Sun, C. T.M. Choi, “A new subgridding scheme for two-dimensional FDTD and FDTD (2,4) methods,” IEEE Trans. Magn. 40, 1041–1044 (2004).
    [CrossRef]
  16. J. B. Cole, “High-accuracy FDTD solution of the absorbing wave equation, and conducting Maxwell’s equations based on a nonstandard finite-difference model,” IEEE Trans. Antennas Propag. 52, 725–729 (2004).
    [CrossRef]
  17. J. W. Wallance, M. A. Jensen, “Analysis of optical waveguide structures by use of a combined finite-difference/finite-difference time-domain method,” J. Opt. Soc. Am. A 19, 610–619 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  27. S. Wang, C. Zhou, H. Ru, Y. Zhang, “Optimized condition for etching fused-silica phase gratings with inductively coupled plasma technology,” Appl. Opt. 44, 4429–4434 (2005).
    [CrossRef] [PubMed]

2005 (2)

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

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

2004 (5)

C. R. Fernández-Pousa, F. Mateos, L. Chantada, M. T. Flores-Arias, C. Bao, M. V. Pérez, C. Gómez-Reino, “Timing jitter smoothing by Talbot effect. I. Variance,” J. Opt. Soc. Am. A 21, 1170–1177 (2004).
[CrossRef]

S. Sun, C. T.M. Choi, “A new subgridding scheme for two-dimensional FDTD and FDTD (2,4) methods,” IEEE Trans. Magn. 40, 1041–1044 (2004).
[CrossRef]

J. B. Cole, “High-accuracy FDTD solution of the absorbing wave equation, and conducting Maxwell’s equations based on a nonstandard finite-difference model,” IEEE Trans. Antennas Propag. 52, 725–729 (2004).
[CrossRef]

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, “A three-dimensional optical photonic crystal with designed point defects,” Nature 429, 538–542 (2004).
[CrossRef] [PubMed]

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

2002 (2)

J. W. Wallance, M. A. Jensen, “Analysis of optical waveguide structures by use of a combined finite-difference/finite-difference time-domain method,” J. Opt. Soc. Am. A 19, 610–619 (2002).
[CrossRef]

E. Miyai, M. Okano, M. Mochizuki, S. Noda, “Analysis of coupling between two-dimensional photonic crystal waveguide and external waveguide,” Appl. Phys. Lett. 81, 3729–3731 (2002).
[CrossRef]

2001 (1)

1999 (3)

1998 (3)

1997 (1)

1996 (1)

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

1995 (1)

1990 (2)

1988 (1)

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).

1971 (1)

1966 (1)

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).

1836 (1)

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

Bao, C.

C. R. Fernández-Pousa, F. Mateos, L. Chantada, M. T. Flores-Arias, C. Bao, M. V. Pérez, C. Gómez-Reino, “Timing jitter smoothing by Talbot effect. I. Variance,” J. Opt. Soc. Am. A 21, 1170–1177 (2004).
[CrossRef]

Berry, M. V.

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

Chantada, L.

C. R. Fernández-Pousa, F. Mateos, L. Chantada, M. T. Flores-Arias, C. Bao, M. V. Pérez, C. Gómez-Reino, “Timing jitter smoothing by Talbot effect. I. Variance,” J. Opt. Soc. Am. A 21, 1170–1177 (2004).
[CrossRef]

Chen, Y.

Choi, C. T.M.

S. Sun, C. T.M. Choi, “A new subgridding scheme for two-dimensional FDTD and FDTD (2,4) methods,” IEEE Trans. Magn. 40, 1041–1044 (2004).
[CrossRef]

Chou, H.

Cole, J. B.

J. B. Cole, “High-accuracy FDTD solution of the absorbing wave equation, and conducting Maxwell’s equations based on a nonstandard finite-difference model,” IEEE Trans. Antennas Propag. 52, 725–729 (2004).
[CrossRef]

Dai, E.

C. Zhou, W. Wang, E. Dai, L. Liu, “Simple principles of the Talbot effect,” Opt. Photonics News, Dec. 2004, pp. 46–50.

Dammann, H.

David, C.

Davis, C. C.

Denz, C.

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

Fernández-Pousa, C. R.

C. R. Fernández-Pousa, F. Mateos, L. Chantada, M. T. Flores-Arias, C. Bao, M. V. Pérez, C. Gómez-Reino, “Timing jitter smoothing by Talbot effect. I. Variance,” J. Opt. Soc. Am. A 21, 1170–1177 (2004).
[CrossRef]

Flores-Arias, M. T.

C. R. Fernández-Pousa, F. Mateos, L. Chantada, M. T. Flores-Arias, C. Bao, M. V. Pérez, C. Gómez-Reino, “Timing jitter smoothing by Talbot effect. I. Variance,” J. Opt. Soc. Am. A 21, 1170–1177 (2004).
[CrossRef]

Gómez-Reino, C.

C. R. Fernández-Pousa, F. Mateos, L. Chantada, M. T. Flores-Arias, C. Bao, M. V. Pérez, C. Gómez-Reino, “Timing jitter smoothing by Talbot effect. I. Variance,” J. Opt. Soc. Am. A 21, 1170–1177 (2004).
[CrossRef]

Groh, G.

Hagness, S.

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

Ichikawa, H.

Jensen, M. A.

Johnson, S. G.

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, “A three-dimensional optical photonic crystal with designed point defects,” Nature 429, 538–542 (2004).
[CrossRef] [PubMed]

Judkins, J. B.

Klein, S.

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

Kock, M.

Kurtsiefer, C.

Leger, J. R.

Lidorikis, E.

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, “A three-dimensional optical photonic crystal with designed point defects,” Nature 429, 538–542 (2004).
[CrossRef] [PubMed]

Liu, L.

C. Zhou, H. Wang, S. Zhao, P. Xi, L. Liu, “Number of phase levels of a Talbot array illuminator,” Appl. Opt. 40, 607–613 (2001).
[CrossRef]

C. Zhou, W. Wang, E. Dai, L. Liu, “Simple principles of the Talbot effect,” Opt. Photonics News, Dec. 2004, pp. 46–50.

Lohmann, A. W.

A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
[CrossRef] [PubMed]

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).

Lu, Y.

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

Luo, H.

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

Mateos, F.

C. R. Fernández-Pousa, F. Mateos, L. Chantada, M. T. Flores-Arias, C. Bao, M. V. Pérez, C. Gómez-Reino, “Timing jitter smoothing by Talbot effect. I. Variance,” J. Opt. Soc. Am. A 21, 1170–1177 (2004).
[CrossRef]

Miyai, E.

E. Miyai, M. Okano, M. Mochizuki, S. Noda, “Analysis of coupling between two-dimensional photonic crystal waveguide and external waveguide,” Appl. Phys. Lett. 81, 3729–3731 (2002).
[CrossRef]

Mochizuki, M.

E. Miyai, M. Okano, M. Mochizuki, S. Noda, “Analysis of coupling between two-dimensional photonic crystal waveguide and external waveguide,” Appl. Phys. Lett. 81, 3729–3731 (2002).
[CrossRef]

Noda, S.

E. Miyai, M. Okano, M. Mochizuki, S. Noda, “Analysis of coupling between two-dimensional photonic crystal waveguide and external waveguide,” Appl. Phys. Lett. 81, 3729–3731 (2002).
[CrossRef]

Nowak, S.

Okano, M.

E. Miyai, M. Okano, M. Mochizuki, S. Noda, “Analysis of coupling between two-dimensional photonic crystal waveguide and external waveguide,” Appl. Phys. Lett. 81, 3729–3731 (2002).
[CrossRef]

Pérez, M. V.

C. R. Fernández-Pousa, F. Mateos, L. Chantada, M. T. Flores-Arias, C. Bao, M. V. Pérez, C. Gómez-Reino, “Timing jitter smoothing by Talbot effect. I. Variance,” J. Opt. Soc. Am. A 21, 1170–1177 (2004).
[CrossRef]

Pfau, T.

Qi, M.

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, “A three-dimensional optical photonic crystal with designed point defects,” Nature 429, 538–542 (2004).
[CrossRef] [PubMed]

Rakich, P. T.

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, “A three-dimensional optical photonic crystal with designed point defects,” Nature 429, 538–542 (2004).
[CrossRef] [PubMed]

Ru, H.

Smolyaninov, I. I.

Stankovic, S.

C. Zhou, S. Stankovic, T. Tschudi, “Analytic phase-factor equations for Talbot array illuminations,” Appl. Opt. 38, 284–290 (1999).
[CrossRef]

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

Sun, S.

S. Sun, C. T.M. Choi, “A new subgridding scheme for two-dimensional FDTD and FDTD (2,4) methods,” IEEE Trans. Magn. 40, 1041–1044 (2004).
[CrossRef]

Swanson, G. J.

Taflove, A.

A. Taflove, S. Hagness, Computational Electromagnetics: The Finite-Difference Time Domain Method, 2nd ed. (Artech House, 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, T. Tschudi, “Phase codes of Talbot array illumination for encoding holographic multiplexing storage,” Opt. Commun. 161, 209–211 (1999).
[CrossRef]

C. Zhou, S. Stankovic, T. Tschudi, “Analytic phase-factor equations for Talbot array illuminations,” Appl. Opt. 38, 284–290 (1999).
[CrossRef]

C. Zhou, L. Wang, T. Tschudi, “Solutions and analyses of fractional-Talbot array illuminations,” Opt. Commun. 147, 224–228 (1998).
[CrossRef]

Wallance, J. W.

Wang, H.

Wang, L.

C. Zhou, L. Wang, T. Tschudi, “Solutions and analyses of fractional-Talbot array illuminations,” Opt. Commun. 147, 224–228 (1998).
[CrossRef]

Wang, S.

Wang, W.

C. Zhou, W. Wang, E. Dai, L. Liu, “Simple principles of the Talbot effect,” Opt. Photonics News, Dec. 2004, pp. 46–50.

Wei, P.

Xi, 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, S.

Zhou, C.

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

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

C. Zhou, H. Wang, S. Zhao, P. Xi, L. Liu, “Number of phase levels of a Talbot array illuminator,” Appl. Opt. 40, 607–613 (2001).
[CrossRef]

C. Zhou, S. Stankovic, T. Tschudi, “Analytic phase-factor equations for Talbot array illuminations,” Appl. Opt. 38, 284–290 (1999).
[CrossRef]

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

C. Zhou, L. Wang, T. Tschudi, “Solutions and analyses of fractional-Talbot array illuminations,” Opt. Commun. 147, 224–228 (1998).
[CrossRef]

C. Zhou, W. Wang, E. Dai, L. Liu, “Simple principles of the Talbot effect,” Opt. Photonics News, Dec. 2004, pp. 46–50.

Ziolkowski, R. W.

Zou, H.

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

Appl. Opt. (5)

Appl. Phys. Lett. (1)

E. Miyai, M. Okano, M. Mochizuki, S. Noda, “Analysis of coupling between two-dimensional photonic crystal waveguide and external waveguide,” Appl. Phys. Lett. 81, 3729–3731 (2002).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

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).

J. B. Cole, “High-accuracy FDTD solution of the absorbing wave equation, and conducting Maxwell’s equations based on a nonstandard finite-difference model,” IEEE Trans. Antennas Propag. 52, 725–729 (2004).
[CrossRef]

IEEE Trans. Magn. (1)

S. Sun, C. T.M. Choi, “A new subgridding scheme for two-dimensional FDTD and FDTD (2,4) methods,” IEEE Trans. Magn. 40, 1041–1044 (2004).
[CrossRef]

J. Mod. Opt. (1)

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

J. Opt. Soc. Am. A (5)

Nature (1)

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, “A three-dimensional optical photonic crystal with designed point defects,” Nature 429, 538–542 (2004).
[CrossRef] [PubMed]

Opt. Commun. (3)

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

C. Zhou, L. Wang, T. Tschudi, “Solutions and analyses of fractional-Talbot array illuminations,” Opt. Commun. 147, 224–228 (1998).
[CrossRef]

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

Opt. Lett. (4)

Optik (Stuttgart) (1)

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).

Philos. Mag. (1)

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

Other (2)

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

C. Zhou, W. Wang, E. Dai, L. Liu, “Simple principles of the Talbot effect,” Opt. Photonics News, Dec. 2004, pp. 46–50.

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Figures (8)

Fig. 1
Fig. 1

Grating structure employed here.

Fig. 2
Fig. 2

First-order diffraction efficiency η 1 as a function of the number of the grating period N. The horizontal line is the exact result obtained with the Fourier expansion method.

Fig. 3
Fig. 3

Numerical simulation of a perfect binary grating with a period of 3 λ and depth of 0.5 λ . (a) The near-field distribution along with the increased distance. (b) The distribution of the electric field intensity at three fourths of the Talbot distance.

Fig. 4
Fig. 4

Distorted grating with a slight random magnitude fluctuation of λ 10 . (a) Its near-field distribution along with the increased distance. (b) The distribution of the electric field intensity at three fourths of the Talbot distance, which is slightly different from the perfect Talbot imaging in Fig. 3b

Fig. 5
Fig. 5

Distorted grating with a larger random magnitude variation of λ 4 . (a) The near-field distribution along with the increased distance. (b) The distribution of the electric field intensity at three-fourths of the Talbot distance, which is obviously different from the perfect imaging in Fig. 3b.

Fig. 6
Fig. 6

Grating with half of the central period destroyed. (a) The near-field distribution along with the increased distance. (b) The distribution of the electric field intensity at three fourths of the Talbot distance, which is different from Fig. 3b. But it is hard to tell the difference between Figs. 5b, 6b.

Fig. 7
Fig. 7

Experimental setup of the Talbot–SNOM apparatus for scanning the Talbot image of a grating under test. PZT, piezoelectronic tube.

Fig. 8
Fig. 8

Talbot images of two kinds of grating obtained by the Talbot–SNOM apparatus. (a) and (b) A gelatin grating and a binary-phase fused-silica grating, respectively, both with the density of 600 l mm .

Equations (6)

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E z t = 1 ϵ ( H y x H x y σ E z ) ,
H x t = 1 μ ( E z y + σ * H x ) ,
H y t = 1 μ ( E z x σ * H y ) ,
E z n + 1 ( i , k ) = 2 ϵ ( i , k ) σ ( i , k ) Δ t 2 ϵ ( i , k ) + σ ( i , k ) Δ t E z n ( i , k ) + 2 Δ t 2 ϵ ( i , k ) + σ ( i , k ) Δ t × [ H y n + 1 ( i , k ) H y n + 1 ( i , k 1 ) Δ x H x n + 1 ( i , k ) H x n + 1 ( i 1 , k ) Δ y ] ,
H y n + 1 ( i , k ) = H y n ( i , k ) + Δ t μ ( i , k ) E z n ( i + 1 , k ) E z n ( i , k ) Δ x ,
H x n + 1 ( i , k ) = H x n ( i , k ) + Δ t μ ( i , k ) E z n ( i , k + 1 ) E z n ( i , k ) Δ y .

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