Abstract

The pseudo-imaging process in a system consisting of two periodic gratings and illuminated by an incoherent polychromatic and finite extension source placed at a finite distance from the gratings is studied. An analytical expression of the irradiance distribution on a plane, also located at a finite distance from the gratings, has been obtained from previous results on monochromatic illumination. In the analysis presented, different imaging regimes are found and related to the parameters which characterize the double grating system. The pseudo-imaging phenomenon strongly depends on both the spatial and temporal coherence of the incident illuminating field. Certain pseudo-images are observed with polychromatic and incoherent incident light under some restrictions. On the other hand, pseudo-image process analogous to Talbot effect appears only by monochromatic and plane illuminating wavefront.

© 2004 Optical Society of America

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References

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Acta Opt. Sin.

L. Liu, �??Ambiguity function and general Talbot-Lau effects,�?? Acta Opt. Sin. 7, 501-510 (1987).

Ann. Phys.

E. Lau. �??Beugungserscheinungen an Doppelrastern,�?? Ann. Phys. (Leipzig) 6, 417-423 (1985).

Appl. Opt.

Eur. J. Phys.

J. Pomarico, R. Torroba, �??Colour image operations based on white light diffraction experiments (Lau effect),�?? Eur. J. Phys. 14, 114-120 (1993).
[CrossRef]

J. Mod. Opt.

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

L. Liu, �??Interferometry based on the partially coherent effect lying between the Talbot and Lau effects,�?? J. Mod. Opt. 35, 1605-1618 (1988).
[CrossRef]

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

K. Patorski, �??Incoherent superposition of multiple self-imaging Lau effect and moire fringes explanation,�?? Opt. Acta 30, 745-758 (1983).
[CrossRef]

Opt. Commun.

J. Tu and L. Zhan, �??Two-dimensional theory of the Lau-Talbot-Moiré effect under partially coherent illumination,�?? Opt. Commun. 82, 229-235 (1991).
[CrossRef]

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

F. Gori, �??Lau effect and coherence theory,�?? Opt. Commun. 31, 4-8 (1979).
[CrossRef]

R. Sudol and B.J. Thompson, �??An explanation of the Lau effect based on coherence theory,�?? Opt. Commun. 31, 105-110 (1979).
[CrossRef]

K.H. Brenner, A.W. Lohmann and Ojeda-Castaneda, �??Lau effect: OTF theory,�?? Opt. Commun. 46, 14-17 (1983).
[CrossRef]

Opt. Eng.

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

A. Olszak and L. Wronkowski, �??Analysis of 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]

Opt. Las. Technol.

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

Philos. Mag.

F. Talbot, �??Facts relating to optical science. No. IV,�?? Philos. Mag. 9, 401-407 (1836).

Proc. R. Soc. London Ser. B

J. M. Cowley and A. F. Moodie, J. M. Cowley and A. F. Moodie, Proc. R. Soc. London Ser. B 70, 486 (1957). 70, 486 (1957).
[CrossRef]

Other

K. Patorski, �??The self-imaging phenomenon and its applications,�?? in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam 1989) vol. 27, pp. 3-108.

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Figures (12)

Fig. 1.
Fig. 1.

Optical system studied.

Fig. 2.
Fig. 2.

Colormap used thorough this work for all the color plots of C(Z 1, Z 2).

Fig. 3.
Fig. 3.

Pseudo-image contrast for z 0=250 µm and S=0. (a) Δλ=0; (b) Δλ=400 nm.

Fig. 4.
Fig. 4.

Pseudo-image contrast for z 0→∞, S=0 and Δx=0. (a) Δλ=0; (b) Δλ=400 nm.

Fig. 5.
Fig. 5.

Pseudo-image contrast for z 0=250 µm and S→∞. (a) Δλ=0; (b) Δλ=400 nm.

Fig. 6.
Fig. 6.

Pseudo-image contrast for z 0→∞, S→∞ and S/z 0=0.1. (a) Δλ=0; (b) Δλ=400 nm.

Fig. 7.
Fig. 7.

Pseudo-image indexing. z 0=250 µm, S=300µm, Δλ=0 and λ=400 nm.

Fig. 8.
Fig. 8.

(a) Contrast of the pseudo-image (1, 2) with monochromatic (blue) and polychromatic (red) illumination. Parameters: p 1=p 2=10 µm, z 0=250 µm, S=300 µm, λ0=400 nm. The spectral width of the polychromatic source is 50 nm. (b) The same for the pseudo-image (1, 3).

Fig. 9.
Fig. 9.

Pseudo-images obtained with periods p 1=30 µm and p 2=10 µm. (a): Monochromatic light, parameters z 0=250 µm, S=300 µm, λ0=400 nm. (b): Same parameters than (a) except for Δλ=400 nm.

Fig. 10.
Fig. 10.

Effect of the source size. p 1=p 2=10 µm, z 0=250 µm, λ0=400 nm and Δλ=60 nm. (a): S=0; (b): S=10 µm; (c): S=18 µm; (d): S=116 µm.

Fig. 11.
Fig. 11.

Effect of the source location. p 1=p 2=10 µm, S=300 µm, λ0=400 nm and Δλ=60 nm. (a): z 0=1 mm; (b): z 0=10 mm; (c): z 0=20 mm; (d): z 0=104 mm.

Fig. 12.
Fig. 12.

Graphical summary of the distribution of pseudo-imaging into different regimes, and its relation with the coherence of the incident field.

Equations (17)

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t 1 ( x ) = n a n exp ( i 2 π nx p 1 ) and t 2 ( x ) = n b n exp ( i 2 π nx p 2 )
I λ ( x ) = nlmk A nlmk exp ( i B nlmk λ ) ,
A nlmk = a n a l * b m b k * exp { i 2 π x p 1 z t [ z 0 ( n l ) + Rz 01 ( m k ) ] }
× sinc { 2 π S p 1 z t [ z 12 ( n l ) + Rz 2 ( m k ) ] }
B nlmk = π p 1 2 z t [ z 0 z 12 ( n 2 l 2 ) + R 2 z 2 z 01 ( m 2 k 2 ) + 2 Rz 0 z 2 ( nm lk ) ] .
I ( x ) = + g ( λ ) I λ ( x ) d λ
= nlmk [ + g ( λ ) A nlmk ( λ ) exp ( iB nlmk λ ) d λ ] .
I ( x ) = nlmk [ A nlmk G ( B nlmk ) ] ,
g ( λ ) = 1 2 π ( Δ λ ) 2 exp [ ( λ λ 0 ) 2 2 ( Δ λ ) 2 ] .
I ( x ) = nlmk A nlmk exp [ B nlmk ( i λ 0 B nlmk ( Δ λ ) 2 2 ) ] .
C = max [ I ( x ) ] min [ I ( x ) ] max [ I ( x ) ] + min [ I ( x ) ] .
S z T > n p 2 2 z 2 and S z T > m p 1 2 ( z 1 + z 2 ) .
z 2 = ( m n p 1 p 2 1 ) 1 z 1 ,
( Z 1 ) nm min = 1 Rnm ( k + 1 ) ,
( Z 1 ) nm max = 1 2 Rnm ( 2 k + 1 ) , k = 0 , 1 , 2 ,
( Z 1 ) nm min = 1 2 Rnm ( 2 k + 1 ) ,
( Z 1 ) nm max = 1 Rnm k , k = 0 , 1 , 2 ,

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