Abstract

The Talbot effect is a well studied phenomenon by which grating pseudoimages appear at certain periodic distances when monochromatic light is used. Recently, numerical simulations have shown a new phenomenon; when a polychromatic light beam is used in a double grating system, the intensity of the pseudoimages presents a transverse-profile that remains unaffected over a wide range of propagation distances. This effect can be used to increase the tolerances of gratings based optical devices, such as displacement measurement systems, interferometers, and spectrometers. The pseudoimages formation with a polychromatic and finite extension light source is analytically and experimentally demonstrated. Relatively simple analytical expressions for the intensity and the contrast allow us to predict when pseudoimages present a constant contrast and when they disappear. Furthermore, we experimentally obtain the pseudoimages using the proposed configuration, corroborating the theoretical predictions.

© 2008 Optical Society of America

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  1. L. Liu, “Partially coherent diffraction effect between Lau and Talbot effects,” J. Opt. Soc. Am. A 5, 1709-1716 (1988).
    [CrossRef]
  2. K. V. Avudainayagam and S. Chitralekha, “Lau effect and beam collimation,” J. Mod. Opt. 44, 175-178 (1997).
    [CrossRef]
  3. M. Tebaldi, L. Angel Toro, and N. Bolognini, “Interferometry based on Lau effect with a grating registered in a photorefractive crystal,” Opt. Laser Technol. 31, 127-134 (1999).
    [CrossRef]
  4. D. Crespo, J. Alonso, T. Morlanes, and E. Bernabéu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817-824(2000).
    [CrossRef]
  5. J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263-267 (1979).
    [CrossRef]
  6. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, Vol. 27, E. Wolf, ed. (North Holland, 1989), pp. 3-108.
    [CrossRef]
  7. S. C. Som and A. Satpathi, “The generalized Lau effect,” J. Mod. Opt. 37, 1215-1226 (1990).
    [CrossRef]
  8. G. J. Swanson and E. N. Leith, “Analysis of the Lau effect and generalized grating imaging,” J. Opt. Soc. Am. A 2, 789-793(1985).
    [CrossRef]
  9. A. Olszak and L. Wronkowski, “Analysis of the Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. 36, 2149-2157 (1997).
    [CrossRef]
  10. J. Tu and L. Zhan, “Analysis of general double periodic structure diffraction phenomena based on the ambiguity function,” J. Opt. Soc. Am. A 9, 983-995 (1992).
    [CrossRef]
  11. D. Crespo, J. Alonso, and E. Bernabeu, “Experimental measurements of generalized grating images,” Appl. Opt. 41, 1223-1228 (2002).
    [CrossRef] [PubMed]
  12. S. Teng, L. Liu, J. Zu, Z. Luan, and D. Liu, “Uniform theory of the Talbot effect with partially coherent light illumination,” J. Opt. Soc. Am. A 20, 1747-1754 (2003).
    [CrossRef]
  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]
  14. L. Garcia-Rodriguez, J. Alonso, and E. Bernabeu, “Grating pseudo-imaging with polychromatic and finite extension sources” Opt. Express 12, 2529-2541 (2004).
    [CrossRef] [PubMed]
  15. 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]
  16. N. Guèrineau, E. di Mambro, J. Primot, and F. Alves, “Talbot experiment re-examined: study of the chromatic regime and application to spectrometry,” Opt. Express 11, 3310-3319(2003).
    [CrossRef] [PubMed]
  17. L. M. Sanchez-Brea, J. Alonso, and E. Bernabeu, “Continuous pseudoimages for sinusoidal grating imaging using an extended light source,” Opt. Commun. 236, 53-58 (2004).
    [CrossRef]
  18. L. M. Sanchez-Brea, J. Alonso, J. B. Saez-Landete, and E. Bernabeu, “Analytical model of a double grating system with partial temporal and spatial coherence,” in Nano- and Micro-Metrology, H. Ottevaere, P. De Wolf, and D. S. Wiersma, eds., Proc. SPIE 5858, 304-311 (2005).

2004 (2)

L. Garcia-Rodriguez, J. Alonso, and E. Bernabeu, “Grating pseudo-imaging with polychromatic and finite extension sources” Opt. Express 12, 2529-2541 (2004).
[CrossRef] [PubMed]

L. M. Sanchez-Brea, J. Alonso, and E. Bernabeu, “Continuous pseudoimages for sinusoidal grating imaging using an extended light source,” Opt. Commun. 236, 53-58 (2004).
[CrossRef]

2003 (2)

2002 (1)

2000 (3)

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]

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, T. Morlanes, and E. Bernabéu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817-824(2000).
[CrossRef]

1999 (1)

M. Tebaldi, L. Angel Toro, and N. Bolognini, “Interferometry based on Lau effect with a grating registered in a photorefractive crystal,” Opt. Laser Technol. 31, 127-134 (1999).
[CrossRef]

1997 (2)

K. V. Avudainayagam and S. Chitralekha, “Lau effect and beam collimation,” J. Mod. Opt. 44, 175-178 (1997).
[CrossRef]

A. Olszak and L. Wronkowski, “Analysis of the Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. 36, 2149-2157 (1997).
[CrossRef]

1992 (1)

1990 (1)

S. C. Som and A. Satpathi, “The generalized Lau effect,” J. Mod. Opt. 37, 1215-1226 (1990).
[CrossRef]

1988 (1)

1985 (1)

1979 (1)

J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263-267 (1979).
[CrossRef]

Alonso, J.

L. Garcia-Rodriguez, J. Alonso, and E. Bernabeu, “Grating pseudo-imaging with polychromatic and finite extension sources” Opt. Express 12, 2529-2541 (2004).
[CrossRef] [PubMed]

L. M. Sanchez-Brea, J. Alonso, and E. Bernabeu, “Continuous pseudoimages for sinusoidal grating imaging using an extended light source,” Opt. Commun. 236, 53-58 (2004).
[CrossRef]

D. Crespo, J. Alonso, and E. Bernabeu, “Experimental measurements of generalized grating images,” Appl. Opt. 41, 1223-1228 (2002).
[CrossRef] [PubMed]

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]

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabéu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817-824(2000).
[CrossRef]

L. M. Sanchez-Brea, J. Alonso, J. B. Saez-Landete, and E. Bernabeu, “Analytical model of a double grating system with partial temporal and spatial coherence,” in Nano- and Micro-Metrology, H. Ottevaere, P. De Wolf, and D. S. Wiersma, eds., Proc. SPIE 5858, 304-311 (2005).

Alves, F.

Angel Toro, L.

M. Tebaldi, L. Angel Toro, and N. Bolognini, “Interferometry based on Lau effect with a grating registered in a photorefractive crystal,” Opt. Laser Technol. 31, 127-134 (1999).
[CrossRef]

Avudainayagam, K. V.

K. V. Avudainayagam and S. Chitralekha, “Lau effect and beam collimation,” J. Mod. Opt. 44, 175-178 (1997).
[CrossRef]

Bernabeu, E.

L. Garcia-Rodriguez, J. Alonso, and E. Bernabeu, “Grating pseudo-imaging with polychromatic and finite extension sources” Opt. Express 12, 2529-2541 (2004).
[CrossRef] [PubMed]

L. M. Sanchez-Brea, J. Alonso, and E. Bernabeu, “Continuous pseudoimages for sinusoidal grating imaging using an extended light source,” Opt. Commun. 236, 53-58 (2004).
[CrossRef]

D. Crespo, J. Alonso, and E. Bernabeu, “Experimental measurements of generalized grating images,” Appl. Opt. 41, 1223-1228 (2002).
[CrossRef] [PubMed]

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]

L. M. Sanchez-Brea, J. Alonso, J. B. Saez-Landete, and E. Bernabeu, “Analytical model of a double grating system with partial temporal and spatial coherence,” in Nano- and Micro-Metrology, H. Ottevaere, P. De Wolf, and D. S. Wiersma, eds., Proc. SPIE 5858, 304-311 (2005).

Bernabéu, E.

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabéu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817-824(2000).
[CrossRef]

Bolognini, N.

M. Tebaldi, L. Angel Toro, and N. Bolognini, “Interferometry based on Lau effect with a grating registered in a photorefractive crystal,” Opt. Laser Technol. 31, 127-134 (1999).
[CrossRef]

Chitralekha, S.

K. V. Avudainayagam and S. Chitralekha, “Lau effect and beam collimation,” J. Mod. Opt. 44, 175-178 (1997).
[CrossRef]

Crespo, D.

di Mambro, E.

Garcia-Rodriguez, L.

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]

Guèrineau, N.

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.

J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263-267 (1979).
[CrossRef]

Leith, E. N.

Liu, D.

Liu, L.

Lohmann, A. W.

J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263-267 (1979).
[CrossRef]

Luan, Z.

Morlanes, T.

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabéu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817-824(2000).
[CrossRef]

Olszak, A.

A. Olszak and L. Wronkowski, “Analysis of the Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. 36, 2149-2157 (1997).
[CrossRef]

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, Vol. 27, E. Wolf, ed. (North Holland, 1989), pp. 3-108.
[CrossRef]

Primot, J.

N. Guèrineau, E. di Mambro, J. Primot, and F. Alves, “Talbot experiment re-examined: study of the chromatic regime and application to spectrometry,” Opt. Express 11, 3310-3319(2003).
[CrossRef] [PubMed]

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. B.

L. M. Sanchez-Brea, J. Alonso, J. B. Saez-Landete, and E. Bernabeu, “Analytical model of a double grating system with partial temporal and spatial coherence,” in Nano- and Micro-Metrology, H. Ottevaere, P. De Wolf, and D. S. Wiersma, eds., Proc. SPIE 5858, 304-311 (2005).

Sanchez-Brea, L. M.

L. M. Sanchez-Brea, J. Alonso, and E. Bernabeu, “Continuous pseudoimages for sinusoidal grating imaging using an extended light source,” Opt. Commun. 236, 53-58 (2004).
[CrossRef]

L. M. Sanchez-Brea, J. Alonso, J. B. Saez-Landete, and E. Bernabeu, “Analytical model of a double grating system with partial temporal and spatial coherence,” in Nano- and Micro-Metrology, H. Ottevaere, P. De Wolf, and D. S. Wiersma, eds., Proc. SPIE 5858, 304-311 (2005).

Satpathi, A.

S. C. Som and A. Satpathi, “The generalized Lau effect,” J. Mod. Opt. 37, 1215-1226 (1990).
[CrossRef]

Som, S. C.

S. C. Som and A. Satpathi, “The generalized Lau effect,” J. Mod. Opt. 37, 1215-1226 (1990).
[CrossRef]

Swanson, G. J.

Tebaldi, M.

M. Tebaldi, L. Angel Toro, and N. Bolognini, “Interferometry based on Lau effect with a grating registered in a photorefractive crystal,” Opt. Laser Technol. 31, 127-134 (1999).
[CrossRef]

Teng, S.

Tu, J.

Wronkowski, L.

A. Olszak and L. Wronkowski, “Analysis of the Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. 36, 2149-2157 (1997).
[CrossRef]

Zhan, L.

Zu, J.

Appl. Opt. (1)

J. Mod. Opt. (2)

K. V. Avudainayagam and S. Chitralekha, “Lau effect and beam collimation,” J. Mod. Opt. 44, 175-178 (1997).
[CrossRef]

S. C. Som and A. Satpathi, “The generalized Lau effect,” J. Mod. Opt. 37, 1215-1226 (1990).
[CrossRef]

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

Opt. Commun. (3)

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]

L. M. Sanchez-Brea, J. Alonso, and E. Bernabeu, “Continuous pseudoimages for sinusoidal grating imaging using an extended light source,” Opt. Commun. 236, 53-58 (2004).
[CrossRef]

J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263-267 (1979).
[CrossRef]

Opt. Eng. (2)

A. Olszak and L. Wronkowski, “Analysis of the Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. 36, 2149-2157 (1997).
[CrossRef]

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabéu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817-824(2000).
[CrossRef]

Opt. Express (2)

Opt. Laser Technol. (1)

M. Tebaldi, L. Angel Toro, and N. Bolognini, “Interferometry based on Lau effect with a grating registered in a photorefractive crystal,” Opt. Laser Technol. 31, 127-134 (1999).
[CrossRef]

Other (2)

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, Vol. 27, E. Wolf, ed. (North Holland, 1989), pp. 3-108.
[CrossRef]

L. M. Sanchez-Brea, J. Alonso, J. B. Saez-Landete, and E. Bernabeu, “Analytical model of a double grating system with partial temporal and spatial coherence,” in Nano- and Micro-Metrology, H. Ottevaere, P. De Wolf, and D. S. Wiersma, eds., Proc. SPIE 5858, 304-311 (2005).

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Figures (5)

Fig. 1
Fig. 1

Double grating configuration. The light source presents a finite size S, and it emits a polychromatic light distribution.

Fig. 2
Fig. 2

Fringes and contrast of pseudoimage ( 1 , 2 ) for monochromatic (a), (b) and polychromatic (c), (d) light obtained with Eq. (2). The mean wavelength is λ ¯ = 600 nm , the spectral width is Δ λ = 60 nm , the source size is S = 300 μm , and z 0 = 0 mm . Both gratings are amplitude Ronchi gratings, p 1 = p 2 = 20 μm . Fringes are obtained for z 1 = z 2 (exact location of the pseudoimage).

Fig. 3
Fig. 3

Contrast of pseudoimage ( 1 , 2 ) when the first grating G1 is an amplitude grating and the second grating G2 is a phase grating with periods p 1 = p 2 = 20 μm . The mean wavelength is λ ¯ = 600 nm , the spectral width is Δ λ = 50 nm , the source size is S = 300 μm , and z 0 = 0 mm . (a) Fill factor of the second grat ing is 0.5. (b) Fill factor of the second grating is 0.25. We see that the threshold distance z TH differs for these two cases as it is theoretically predicted. As shown in the text, the results obtained with Eq. (10) are very similar to those obtained with the simulations.

Fig. 4
Fig. 4

Experimental contrast of pseudoimage ( 1 , 2 ) for different LEDs and IREDs (a) HE8811, (b) 600-03V, and (c) HIR333 described in Table 1. We have used two amplitude Ronchi gratings (fill factor 50%) with periods p 1 = p 2 = 8 μm . The width of the pseudoimage depends mainly on the LED size. No other pseudoimages have been experimentally found since polychromatic light eliminates them.

Fig. 5
Fig. 5

Experimental contrast of the pseudoimage ( 1 , 2 ) for the LEDs described in Table 1: (1), LED 600-03V; (2), OPE5T85; (3), HE8811; (4), HIR333; (5), MARL. In all the cases, we have used two Ronchi gratings (fill factor 50%) with periods p 1 = p 2 = 8 μm . For clarity, the figures have been vertically shifted. Dashed–dotted lines represent the theoretical value (0.1513).

Tables (1)

Tables Icon

Table 1 LEDs and IREDs Used for Determining the Contrast of ( 1 , 2 ) Pseudoimages

Equations (16)

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I P ( x ) = I M ( x ) g ( λ ) d λ .
g ( λ ) = 1 2 π Δ λ exp [ 1 2 ( λ λ ¯ Δ λ ) 2 ] ,
I P ( x ) N M exp [ i ( N α 1 + M α 2 ) ] sinc ( N β 1 + M β 2 ) u v c 1 , u + N / 2 c 1 , u N / 2 * c 2 , v + M / 2 c 2 , v M / 2 * exp [ i π Γ N M u v ( λ ¯ ) ] exp [ 1 2 ( Δ λ λ ¯ ) 2 Γ N M u v 2 ( λ ¯ ) ] ,
I P ( x ) I 0 , 0 + k 0 exp [ i k ( N α 1 + M α 2 ) ] sinc [ k ( N β 1 + M β 2 ) ] u v c 1 , u + k N / 2 c 1 , u k N / 2 * c 2 , v + k M / 2 c 2 , v k M / 2 * exp [ i π Γ k N , k M , u , v ( λ ¯ ) ] exp [ 1 2 ( Δ λ λ ¯ ) 2 Γ k N , k M , u , v 2 ( λ ¯ ) ] ,
I P ( x ) I 0 , 0 + k 0 C 1 , k N / 2 e i k q 1 ( M R + N ) x v c 2 , v + k M / 2 c 2 , v k M / 2 * e 2 π i v k G λ e 2 ( k v G Δ λ ) 2 ,
where G = N z 1 p 1 p 2 N p 2 z 0 + M p 1 z 1 N p 2 z 0 + M p 1 ( z 0 + z 1 ) , C 1 , k N / 2 = u = c 1 , u + k N / 2 c 1 , u k N / 2 * .
I P ( x ) | M   even I 0 , 0 + k 0 C 1 , k N / 2 c 2 , k M / 2 c 2 , k M / 2 * exp [ i k q 1 ( M R + N ) x ] .
Contrast | M   even = 2 c 2 , M / 2 c 2 , M / 2 * u c 1 , u + N / 2 c 1 , u N / 2 * n | c 1 , n | 2 m | c 2 , m | 2 .
M p 2 z 1 2 + ( N p 1 z 0 M p 1 2 k N v Δ λ ) z 1 M p 1 + N p 2 2 k N v Δ λ z 0 0 ,
z 1 p 1 p 2 2 k N v Δ λ + z 0 .
z TH = p 1 p 2 2 N v n v Δ λ ,
I M ( x ) = N M exp [ i ( N α 1 + M α 2 ) ] sinc ( N β 1 + M β 2 ) × u v c 1 , u + N / 2 c 1 , u N / 2 * c 2 , v + M / 2 c 2 , v M / 2 * exp [ i π Γ ( λ ) N M u v ] ,
R = q 2 q 1 = p 1 p 2 , α 1 = q 1 z 0 z t x , α 2 = q 1 R z 01 z t x , β 1 = S 2 q 1 z 12 z t , β 2 = S 2 q 1 R z 2 z t , γ 11 = z 0 z 12 z λ z t , γ 22 = z 2 z 01 z λ z t R 2 , γ 12 = z 0 z 2 z λ z t R , z λ = p 1 2 λ , z t = z 0 + z 1 + z 2 , z 01 = z 0 + z 1 , z 12 = z 1 + z 2 ,
I P ( x ) I 0 , 0 + k 0 e i k ( N α 1 + M α 2 ) sinc [ k ( N β 1 + M β 2 ) ] u v c 1 , u + k N / 2 c 1 , u k N / 2 * c 2 , v + k M / 2 c 2 , v k M / 2 * e i π k Γ N M u v ( λ ) ,
z 2 = 1 R Q 1 z 1 ,
Δ z 2 2 p 1 S z 0 ( R M + N ) + z 1 R M ( R M + N ) 2 .

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