Abstract

We analyze the Talbot effect produced by a mask composed of two diffraction gratings. Combinations with phase and amplitude gratings have been studied in the near-field regime. For a two-phase-gratings configuration, the Talbot effect is canceled, even when using monochromatic light; that is, the intensity distribution is nearly independent of the distance from the mask to the observation plane. Therefore, the mechanical tolerances of devices that use the Talbot effect may be improved. In addition, the spatial frequency of the fringes is quadrupled, which improves the accuracy of devices that employ this mask. An experimental verification for the best case two phase gratings, has also been performed, validating the theoretical results.

© 2009 Optical Society of America

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References

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  1. E. Keren and O. Kafri, “Diffraction effects in moiré deflectometry,” J. Opt. Soc. Am. A 2 (2), 111-120 (1985).
    [CrossRef]
  2. A. W. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413-415 (1971).
    [CrossRef]
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  4. S. Wei, S. Wu, I. Kao, and F. P. Chiang “Measurement of wafer surface using shadow moire technique with Talbot effect,” J. Electron. Packag. 120166-170 (1998).
    [CrossRef]
  5. G. Schirripa Spagnolo, D. Ambrosini, and D. Paoletti, “Displacement measurement using the Talbot effect with a Ronchi grating,” J. Opt. A Pure Appl. Opt. 4, S376-S380 (2002).
    [CrossRef]
  6. W. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401-407 (1836).
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    [CrossRef]
  8. N. Guérineau, B. Harchaoui, and J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. 180, 199-203(2000).
    [CrossRef]
  9. L. M.Sanchez-Brea, J. Saez-Landete, J. Alonso, and E. Bernabeu “Invariant grating pseudo-imaging using polychromatic light and finite extension source,” Appl. Opt. 47, 1470-1477(2008).
    [CrossRef] [PubMed]
  10. L. M. Sanchez-Brea, J. Alonso, and E. Bernabeu “Quasicontinuous pseudoimages for sinusoidal grating imaging using an extended light source,” Opt. Commun. 23653-58 (2004).
    [CrossRef]
  11. G. Vincent, R. Haidar, S. Collin, N. Guérineau, J. Primot, E. Cambril, and J. L. Pelouard “Realization of sinusoidal transmittance with subwavelength metallic structures,” J. Opt. Soc. Am. B 25, 834-840 (2008).
    [CrossRef]
  12. K. Patorsky, Handbook of the Moiré Fringe Technique (Elsevier, 1993).
  13. D. Crespo, J. Alonso, and E. Bernabeu, “Generalized grating imaging using an extended monochromatic light source,” J. Opt. Soc. Am. A 17, 1231-1240 (2000).
    [CrossRef]

2008

2004

L. M. Sanchez-Brea, J. Alonso, and E. Bernabeu “Quasicontinuous pseudoimages for sinusoidal grating imaging using an extended light source,” Opt. Commun. 23653-58 (2004).
[CrossRef]

2002

G. Schirripa Spagnolo, D. Ambrosini, and D. Paoletti, “Displacement measurement using the Talbot effect with a Ronchi grating,” J. Opt. A Pure Appl. Opt. 4, S376-S380 (2002).
[CrossRef]

2000

N. Guérineau, B. Harchaoui, and J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. 180, 199-203(2000).
[CrossRef]

D. Crespo, J. Alonso, and E. Bernabeu, “Generalized grating imaging using an extended monochromatic light source,” J. Opt. Soc. Am. A 17, 1231-1240 (2000).
[CrossRef]

1998

S. Wei, S. Wu, I. Kao, and F. P. Chiang “Measurement of wafer surface using shadow moire technique with Talbot effect,” J. Electron. Packag. 120166-170 (1998).
[CrossRef]

1994

1985

1971

A. W. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413-415 (1971).
[CrossRef]

1836

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

Alonso, J.

Ambrosini, D.

G. Schirripa Spagnolo, D. Ambrosini, and D. Paoletti, “Displacement measurement using the Talbot effect with a Ronchi grating,” J. Opt. A Pure Appl. Opt. 4, S376-S380 (2002).
[CrossRef]

Bernabeu, E.

Cambril, E.

Chiang, F. P.

S. Wei, S. Wu, I. Kao, and F. P. Chiang “Measurement of wafer surface using shadow moire technique with Talbot effect,” J. Electron. Packag. 120166-170 (1998).
[CrossRef]

Collin, S.

Crespo, D.

Dorsch, R. G.

Guérineau, N.

G. Vincent, R. Haidar, S. Collin, N. Guérineau, J. Primot, E. Cambril, and J. L. Pelouard “Realization of sinusoidal transmittance with subwavelength metallic structures,” J. Opt. Soc. Am. B 25, 834-840 (2008).
[CrossRef]

N. Guérineau, B. Harchaoui, and J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. 180, 199-203(2000).
[CrossRef]

Haidar, R.

Harchaoui, B.

N. Guérineau, B. Harchaoui, and J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. 180, 199-203(2000).
[CrossRef]

Kafri, O.

Kao, I.

S. Wei, S. Wu, I. Kao, and F. P. Chiang “Measurement of wafer surface using shadow moire technique with Talbot effect,” J. Electron. Packag. 120166-170 (1998).
[CrossRef]

Keren, E.

Lohmann, A. W.

A. W. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413-415 (1971).
[CrossRef]

Oreb, B. F.

Paoletti, D.

G. Schirripa Spagnolo, D. Ambrosini, and D. Paoletti, “Displacement measurement using the Talbot effect with a Ronchi grating,” J. Opt. A Pure Appl. Opt. 4, S376-S380 (2002).
[CrossRef]

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1989), Vol. 27, pp. 1-108.
[CrossRef]

Patorsky, K.

K. Patorsky, Handbook of the Moiré Fringe Technique (Elsevier, 1993).

Pelouard, J. L.

Primot, J.

G. Vincent, R. Haidar, S. Collin, N. Guérineau, J. Primot, E. Cambril, and J. L. Pelouard “Realization of sinusoidal transmittance with subwavelength metallic structures,” J. Opt. Soc. Am. B 25, 834-840 (2008).
[CrossRef]

N. Guérineau, B. Harchaoui, and J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. 180, 199-203(2000).
[CrossRef]

Saez-Landete, J.

Sanchez-Brea, L. M.

L. M.Sanchez-Brea, J. Saez-Landete, J. Alonso, and E. Bernabeu “Invariant grating pseudo-imaging using polychromatic light and finite extension source,” Appl. Opt. 47, 1470-1477(2008).
[CrossRef] [PubMed]

L. M. Sanchez-Brea, J. Alonso, and E. Bernabeu “Quasicontinuous pseudoimages for sinusoidal grating imaging using an extended light source,” Opt. Commun. 23653-58 (2004).
[CrossRef]

Silva, D. E.

A. W. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413-415 (1971).
[CrossRef]

Spagnolo, G. Schirripa

G. Schirripa Spagnolo, D. Ambrosini, and D. Paoletti, “Displacement measurement using the Talbot effect with a Ronchi grating,” J. Opt. A Pure Appl. Opt. 4, S376-S380 (2002).
[CrossRef]

Talbot, W. H. F.

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

Vincent, G.

Wei, S.

S. Wei, S. Wu, I. Kao, and F. P. Chiang “Measurement of wafer surface using shadow moire technique with Talbot effect,” J. Electron. Packag. 120166-170 (1998).
[CrossRef]

Wu, S.

S. Wei, S. Wu, I. Kao, and F. P. Chiang “Measurement of wafer surface using shadow moire technique with Talbot effect,” J. Electron. Packag. 120166-170 (1998).
[CrossRef]

Appl. Opt.

J. Electron. Packag.

S. Wei, S. Wu, I. Kao, and F. P. Chiang “Measurement of wafer surface using shadow moire technique with Talbot effect,” J. Electron. Packag. 120166-170 (1998).
[CrossRef]

J. Opt. A Pure Appl. Opt.

G. Schirripa Spagnolo, D. Ambrosini, and D. Paoletti, “Displacement measurement using the Talbot effect with a Ronchi grating,” J. Opt. A Pure Appl. Opt. 4, S376-S380 (2002).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

N. Guérineau, B. Harchaoui, and J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. 180, 199-203(2000).
[CrossRef]

A. W. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413-415 (1971).
[CrossRef]

L. M. Sanchez-Brea, J. Alonso, and E. Bernabeu “Quasicontinuous pseudoimages for sinusoidal grating imaging using an extended light source,” Opt. Commun. 23653-58 (2004).
[CrossRef]

Philos. Mag.

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

Other

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1989), Vol. 27, pp. 1-108.
[CrossRef]

K. Patorsky, Handbook of the Moiré Fringe Technique (Elsevier, 1993).

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Figures (6)

Fig. 1
Fig. 1

General scheme of the mask.

Fig. 2
Fig. 2

Intensity distribution produced by the double-grating mask considering only 0 and ± 1 orders: (a) amplitude–phase mask, (b) phase–amplitude mask, (c) amplitude–amplitude mask, and (d) phase–phase mask.

Fig. 3
Fig. 3

Intensity distribution produced by the phase–phase mask when 11 , , 11 orders have been considered. The period of the gratings is p = 20 μm . A slight dependence with the distance between the mask and the observation plane is observed, but continuous self-imaging is still obtained.

Fig. 4
Fig. 4

Scheme of the experimental setup.

Fig. 5
Fig. 5

(a) Experimental intensity distribution of the fringes in terms of the distance z between the mask and the observation plane. (b) Experimental contrast along the z axis.

Fig. 6
Fig. 6

(a) Experimental contrast of the fringes for the double-grating mask with the phase–phase configuration in terms of the relative displacement Δ x between gratings for different values of the distance between the mask and the observation plane. (b) Maximum and minimum experimental contrast for the different relative displacement Δ x between gratings.

Equations (8)

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I ( x , z 1 ) = n , n , m , m a n a n * b m b m * exp { i q x [ ( n n ) + ( m m ) ] } exp [ i q 2 2 k ( n 2 n 2 ) z 1 ] ,
I ( x , z 1 , z 2 ) = n , n , m , m a n a n * b m b m * exp { i q x [ ( n n ) + ( m m ) ] } exp [ i q 2 2 k ( n 2 n 2 ) ( z 1 + z 2 ) ] × exp [ i q 2 2 k ( m 2 m 2 ) z 2 ] exp [ i q 2 k ( n m n m ) z 2 ] .
I ( x , z 1 , z 2 ) = N , M , u , v a u + N / 2 a u N / 2 * b v + M / 2 b v M / 2 * exp [ i q x ( N + M ) ] exp ( i M q Δ x ) × exp ( 2 π i u N z 1 z T ) exp [ i 2 π ( N + M ) ( u + v ) z 2 z T ] ,
I AP ( x , Z 2 ) = a 0 2 ( b 0 2 + 2 b 1 2 ) + 2 a 1 2 ( b 0 2 + b 1 2 ) 4 a 0 a 1 cos ( q x ) [ b 0 2 sin ( π Z 2 ) + b 1 2 sin ( 3 π Z 2 ) ] + 4 sin ( q x ) Re ( b 1 b 0 * ) [ a 0 2 cos ( π Z 2 ) + a 1 2 cos ( 3 π Z 2 ) ] + 2 ( a 1 2 b 0 2 a 0 2 b 1 2 ) cos ( 2 q x ) 4 I m ( b 1 b 0 * ) sin ( 2 q x ) a 0 a 1 [ 1 sin ( 4 π Z 2 ) ] + 4 a 0 a 1 b 1 2 cos ( 3 q x ) sin ( 3 π Z 2 ) + 4 a 1 2 Re ( b 1 b 0 * ) sin ( 3 q x ) cos ( 3 π Z 2 ) 2 a 1 2 b 1 2 cos ( 4 q x ) ,
I AP ( x , Z 2 ) = 8 π 4 [ ( 1 + π 2 ) π 2 cos ( 2 q x ) 2 π cos ( q x ) sin 2 ( q x ) sin ( 3 π Z 2 ) cos ( 4 q x ) ] .
I PA ( x , Z 2 ) = 8 π 4 [ ( 1 + π 2 ) + π 2 cos ( 2 q x ) + 4 π cos ( q x ) cos ( 2 q x ) cos ( 3 π Z 2 ) cos ( 4 q x ) ] .
I AA ( x , Z 2 ) = 1 π 4 { ( 2 + 4 π 2 + π 4 ) 2 cos ( 4 q x ) + 4 π sin ( 2 q x ) [ 2 cos ( q x + 3 π Z 2 ) π sin ( 4 π Z 2 ) ] + 4 π 3 sin ( q x π Z 2 ) } .
I PP ( x , Z 2 ) = 32 π 4 [ 1 cos ( 4 q x ) ] = 64 π 4 sin 2 ( 2 q x ) .

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