Abstract

The Talbot effect is analyzed when steel tape gratings are used. These gratings are made on a steel substrate, and, because of the manufacture process, both levels of the grating are rough with different roughness parameters. A theoretical analysis based on Fresnel regime, which considers the statistical properties of roughness, is developed. Analytical formulas that show a decreasing exponential dependence on the intensity in terms of the distance between the grating and the observation plane are obtained, and an experimental verification is also performed.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  4. A. W. Lohmann and D. E. Silva, "An interferometer based on the Talbot effect," Opt. Commun. 2, 413-415 (1971).
    [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]
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    [CrossRef] [PubMed]
  7. S. Wei, S. Wu, I. Kao, and F. P. Chiang, "Measurement of wafer surface using shadow moiré technique with Talbot effect," Trans. ASME J. Electron. Packag. 120, 166-170 (1998).
    [CrossRef]
  8. M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, "Talbot effect for oblique angle of light propagation," Opt. Commun. 129, 167-172 (1996).
    [CrossRef]
  9. 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]
  10. S. Teng, L. Liu, J. Zu, Z. Luan, and De'an, "Uniform theory of the Talbot effect with partially coherent light illumination," J. Opt. Soc. Am. A 20, 1747-1754 (2003).
    [CrossRef]
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    [CrossRef]
  12. Y. Lu, C. Zhou, S. Wang, and B. Wang, "Polarization-dependent Talbot effect," J. Opt. Soc. Am. A 23, 2154-2160 (2006).
    [CrossRef]
  13. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987).
  14. J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, 1984).
  15. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Institute of Physics, 1991).
  16. F. Perez-Quintián, A. Lutenberg, and M. A. Rebollo, "Linear displacement measurement with a grating and speckle pattern illumination," Appl. Opt. 45, 4821-4825 (2006).
    [CrossRef] [PubMed]

2006 (2)

2005 (1)

2003 (1)

2002 (1)

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 (1)

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]

1998 (1)

S. Wei, S. Wu, I. Kao, and F. P. Chiang, "Measurement of wafer surface using shadow moiré technique with Talbot effect," Trans. ASME J. Electron. Packag. 120, 166-170 (1998).
[CrossRef]

1996 (1)

M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, "Talbot effect for oblique angle of light propagation," Opt. Commun. 129, 167-172 (1996).
[CrossRef]

1994 (1)

1989 (1)

K. Patorski, "The self-imaging phenomenon and its applications," Prog. Opt. 27, 1-108 (1989).
[CrossRef]

1985 (1)

1971 (1)

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

1836 (1)

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

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]

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987).

Chiang, F. P.

S. Wei, S. Wu, I. Kao, and F. P. Chiang, "Measurement of wafer surface using shadow moiré technique with Talbot effect," Trans. ASME J. Electron. Packag. 120, 166-170 (1998).
[CrossRef]

Dainty, J. C.

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, 1984).

De'an,

Dorsch, R. G.

Goncharenko, A. M.

M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, "Talbot effect for oblique angle of light propagation," Opt. Commun. 129, 167-172 (1996).
[CrossRef]

Guérineau, N.

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]

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]

Jahns, J.

M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, "Talbot effect for oblique angle of light propagation," Opt. Commun. 129, 167-172 (1996).
[CrossRef]

Kafri, O.

Kao, I.

S. Wei, S. Wu, I. Kao, and F. P. Chiang, "Measurement of wafer surface using shadow moiré technique with Talbot effect," Trans. ASME J. Electron. Packag. 120, 166-170 (1998).
[CrossRef]

Keren, E.

Khilo, N. A.

M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, "Talbot effect for oblique angle of light propagation," Opt. Commun. 129, 167-172 (1996).
[CrossRef]

Liu, L.

Lohmann, A. W.

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

Lu, Y.

Luan, Z.

Luo, H.

Lutenberg, A.

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Institute of Physics, 1991).

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," Prog. Opt. 27, 1-108 (1989).
[CrossRef]

Perez-Quintián, F.

Primot, J.

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]

Rebollo, M. A.

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]

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987).

Talbot, W. H. F.

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

Teng, S.

Testorf, M.

M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, "Talbot effect for oblique angle of light propagation," Opt. Commun. 129, 167-172 (1996).
[CrossRef]

Wang, B.

Wang, S.

Wei, S.

S. Wei, S. Wu, I. Kao, and F. P. Chiang, "Measurement of wafer surface using shadow moiré technique with Talbot effect," Trans. ASME J. Electron. Packag. 120, 166-170 (1998).
[CrossRef]

Wu, S.

S. Wei, S. Wu, I. Kao, and F. P. Chiang, "Measurement of wafer surface using shadow moiré technique with Talbot effect," Trans. ASME J. Electron. Packag. 120, 166-170 (1998).
[CrossRef]

Zhou, C.

Zu, J.

Appl. Opt. (2)

J. Opt. A (1)

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 (4)

Opt. Commun. (3)

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

M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, "Talbot effect for oblique angle of light propagation," Opt. Commun. 129, 167-172 (1996).
[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]

Philos. Mag. (1)

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

Prog. Opt. (1)

K. Patorski, "The self-imaging phenomenon and its applications," Prog. Opt. 27, 1-108 (1989).
[CrossRef]

Trans. ASME J. Electron. Packag. (1)

S. Wei, S. Wu, I. Kao, and F. P. Chiang, "Measurement of wafer surface using shadow moiré technique with Talbot effect," Trans. ASME J. Electron. Packag. 120, 166-170 (1998).
[CrossRef]

Other (3)

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987).

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, 1984).

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Institute of Physics, 1991).

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Figures (8)

Fig. 1
Fig. 1

(a) Experimental contrast of fringes for a 100 μm steel grating in terms of the distance between the grating and the observation plane. (b) Image of the grating using the experimental setup used in Section 3. The levels of the grating with a shorter correlation length present a low-intensity since light is scattered in all directions.

Fig. 2
Fig. 2

Scheme showing the notation used.

Fig. 3
Fig. 3

Experimental setup for reflection configuration.

Fig. 4
Fig. 4

Self-images obtained for steel (a), (b) and glass (c), (d) gratings. (a) and (c) are the first self-images of the gratings obtained with the microscope, and (b) and (d) are the third self-images.

Fig. 5
Fig. 5

Fringes in terms of the distance between the grating and the observation plane for (a) the steel rough grating and (b) the chrome on glass grating.

Fig. 6
Fig. 6

Experimental contrast in terms of the distance for the steel rough surface (solid) and theoretical contrast obtained with Eq. (6) (dashed). In the numerical approach, we have truncated the infinite series to the interval n = n = 5 ,  ,  5 .

Fig. 7
Fig. 7

(a) Image obtained with confocal microscopy of the steel rough grating. (b) Same steel grating showing that the statistical parameters of level B are the same as the zone of the steel tape where the grating is not engraved.

Fig. 8
Fig. 8

Experimental fringes obtained for the third self-image of the steel reflection grating and fit to a sinusoidal function.

Equations (7)

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

U ( x 1 ) = A 0 t ( x 1 ) n a n exp ( i q n x 1 ) .
U 2 ( x 2 ) = A 0 n a n t ( x 1 ) exp ( i q n x 1 ) × exp [ ( i k / 2 z 1 ) ( x 2 x 1 ) 2 ] d x 1 ,
I 2 ( z 1 ) = | A 0 | 2 n n a n a n * exp ( i q n x 1 ) exp ( i q n x 1 ) × t ( x 1 ) t * ( x 1 ) exp [ ( i k / 2 z 1 ) ( x 2 x 1 ) 2 ] × exp [ ( i k / 2 z 1 ) ( x 2 x 1 ) 2 ] d x 1 d x 1 ,
t ( x 1 ) t * ( x 1 ) = exp [ | x 1 x 1 | T 0 ] ,
T 0 = λ T 4 π σ .
I 2 ( z 1 ) = | A 0 | 2 n n a n a n * exp [ i q x 2 ( n n ) ] × exp [ i q 2 2 k ( n 2 n 2 ) z 1 ] exp [ λ z 1 p T 0 | n n | ] .
I 2 = | A 0 | 2 n n a n a n * exp [ i q x 2 ( n n ) ] × exp [ i q 2 2 k ( n 2 n 2 ) z 1 ] exp [ ( ( n n ) λ z 1 p T 0 ) 2 ] .

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