Abstract

In this work we analyze the near-field intensity distribution produced by a rough grating illuminated with a Gaussian–Schell-model beam. This kind of grating is formed by rough and smooth slits. Statistical techniques are used to describe the grating, and the Fresnel approach is used to perform the propagation of light. Two kinds of coherence affect the light propagation. One of them comes from the light beam, since it is not totally coherent. The other one comes from the rough topography of the grating surface. We have found that the Talbot effect is not present just after the grating, but it gradually increases. In addition, the contrast of the self-images decreases from a certain distance due to the coherence properties of the illumination beam. Then, the self-imaging process is only present between two specific distances from the grating. To corroborate the analytical results, we have performed numerical simulations for the mean intensity distribution based on the Sommerfeld–Rayleigh approach, showing their validity.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2009 (1)

2008 (3)

2007 (2)

F. J. Torcal-Milla, L. M. Sanchez-Brea, and E. Bernabeu, “Talbot effect with rough reflection gratings,” Appl. Opt. 46, 3668-3673(2007).
[CrossRef] [PubMed]

L. M. Sanchez-Brea, F. J. Torcal-Milla, and E. Bernabeu, “Talbot effect in metallic gratings under Gaussian illumination,” Opt. Commun. 278, 23-27 (2007).
[CrossRef]

2006 (2)

2004 (1)

U. Levy, E. Marom, and D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: simplified model and validity limits,” Opt. Commun. 229, 11-21 (2004).
[CrossRef]

2003 (1)

V. Ya. Mendeleev and S. N. Skovorod'ko, “Relation for estimating the reflectance of a very rough surface with an approximately one-dimensional distribution of roughness,” Opt. Spectrosc. 94, 437-443 (2003).
[CrossRef]

2001 (1)

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339-350 (2001).
[CrossRef]

2000 (1)

E. Wolf, “Coherence of two interfering beams modulated by a uniformly moving diffuser,” J. Mod. Opt. 47, 1569-1573(2000).

1999 (2)

1991 (1)

1989 (1)

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

1985 (1)

Beckmann, P.

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

Bernabeu, E.

Borghi, R.

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339-350 (2001).
[CrossRef]

Cairns, B.

Celli, V.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Gori, F.

Levy, U.

U. Levy, E. Marom, and D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: simplified model and validity limits,” Opt. Commun. 229, 11-21 (2004).
[CrossRef]

Loewen, E. G.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Maradudin, A. A.

Marom, E.

U. Levy, E. Marom, and D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: simplified model and validity limits,” Opt. Commun. 229, 11-21 (2004).
[CrossRef]

Marvin, A. M.

McGurn, A. R.

Mendeleev, V. Ya.

V. Ya. Mendeleev and S. N. Skovorod'ko, “Relation for estimating the reflectance of a very rough surface with an approximately one-dimensional distribution of roughness,” Opt. Spectrosc. 94, 437-443 (2003).
[CrossRef]

Mendlovic, D.

U. Levy, E. Marom, and D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: simplified model and validity limits,” Opt. Commun. 229, 11-21 (2004).
[CrossRef]

Mondello, A.

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339-350 (2001).
[CrossRef]

Ogilvy, J. A.

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

Palmer, C.

C. Palmer, Diffraction Grating Handbook (Richardson Grating Laboratory, 2000).

Patorski, K.

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

Piquero, G.

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339-350 (2001).
[CrossRef]

Popov, E.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991).
[CrossRef]

Salgado-Remacha, F. J.

Sanchez-Brea, L. M.

Santarsiero, M.

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339-350 (2001).
[CrossRef]

Shen, F.

Skovorod'ko, S. N.

V. Ya. Mendeleev and S. N. Skovorod'ko, “Relation for estimating the reflectance of a very rough surface with an approximately one-dimensional distribution of roughness,” Opt. Spectrosc. 94, 437-443 (2003).
[CrossRef]

Someda, C. G.

Spizzichino, A.

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

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991).
[CrossRef]

Torcal-Milla, F. J.

Voelz, D.

Wang, A.

Wolf, E.

E. Wolf, “Coherence of two interfering beams modulated by a uniformly moving diffuser,” J. Mod. Opt. 47, 1569-1573(2000).

B. Cairns and E. Wolf, “Changes in the spectrum of light scattered by a moving diffuser plate,” J. Opt. Soc. Am. A 8, 1922-1928 (1991).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Xiao, X.

Appl. Opt. (3)

J. Mod. Opt. (1)

E. Wolf, “Coherence of two interfering beams modulated by a uniformly moving diffuser,” J. Mod. Opt. 47, 1569-1573(2000).

J. Opt. Soc. Am. A (4)

Opt. Commun. (4)

U. Levy, E. Marom, and D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: simplified model and validity limits,” Opt. Commun. 229, 11-21 (2004).
[CrossRef]

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339-350 (2001).
[CrossRef]

L. M. Sanchez-Brea, F. J. Torcal-Milla, and E. Bernabeu, “Talbot effect in metallic gratings under Gaussian illumination,” Opt. Commun. 278, 23-27 (2007).
[CrossRef]

F. J. Torcal-Milla, L. M. Sanchez-Brea, and E. Bernabeu, “Double grating systems with one steel tape grating,” Opt. Commun. 281, 5647-5652 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Opt. Spectrosc. (1)

V. Ya. Mendeleev and S. N. Skovorod'ko, “Relation for estimating the reflectance of a very rough surface with an approximately one-dimensional distribution of roughness,” Opt. Spectrosc. 94, 437-443 (2003).
[CrossRef]

Prog. Opt. (1)

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

Other (7)

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).

C. Palmer, Diffraction Grating Handbook (Richardson Grating Laboratory, 2000).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

J. W. Goodman, Statistical Optics (Wiley, 1985).

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

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

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Figures (6)

Fig. 1
Fig. 1

Topography for a rough grating with period p = 20 μm , and roughness parameters σ = 0.25 μm , T 0 = 1 μm .

Fig. 2
Fig. 2

Average intensity calculated using Eq. (8) for different values of the coherence length: (a) σ μ = 1000 μm , (b) σ μ = 200 μm , (c) σ μ = 100 μm , and (d) σ μ = 50 μm . The beam width is σ i = 200 μm , the wavelength λ = 0.65 μm , the period of the grating p = 40 μm , the refraction index n = 1.5 , and the roughness parameters of the grating σ = 0.25 μm and T 0 = 100 μm . The sums runs are l, l = 7 , , 7 , and m = 1 , 2 , , 5 .

Fig. 3
Fig. 3

Contrast of the self-images for same conditions as in Fig. 2.

Fig. 4
Fig. 4

Numerical simulations: (a) same conditions as Fig. 2a; (b) the grating presents a period p = 20 μm , the refractive index n = 1.5 , and the roughness parameters of the grating σ = 0.5 μm and T 0 = 10 μm . The GSM beam presents a wavelength λ = 0.65 μm , a beam width σ i = 100 μm , and a coherence length σ μ = 100 μm . The number of samples used for the averaging is 200.

Fig. 5
Fig. 5

Comparison between theoretical, Eq. (8), solid curve, and numerical, dashed curve, results for the first two self-images of a grating with period p = 40 μm , refractive index n = 1.5 , and the roughness parameters of the grating σ = 0.25 μm and T 0 = 100 μm . The GSM beam presents a wavelength λ = 0.65 μm , a beam width σ i = 100 μm , and a coherence length σ μ = 1000 μm . The number of samples used for the averaging is 200. (a) First self-image, (b) second self-image.

Fig. 6
Fig. 6

(a) Numerical average intensity distribution I num ( x 1 , z ) , (b) theoretical intensity distribution obtained with Eq. (8), (c) difference between theoretical and numerical results, and (d) maximum (solid curve) and average (dashed curve) differences between the theoretical and numerical results for each location z. The conditions are the same as in Fig. 5.

Equations (10)

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T ( x ) = 1 G 1 ( x ) [ 1 t ( x ) ] ,
U 0 ( x ; τ ) U 0 * ( x ; τ ) = J GSM ( x , x ) = I 0 exp [ ( x 2 + x 2 4 σ i 2 ) ] exp [ ( x x ) 2 2 σ μ 2 ] ,
J G ( x , x ) = 1 + G 1 ( x ) [ t ( x ) 1 ] + G 1 * ( x ) [ t * ( x ) 1 ] + G 1 ( x ) G 1 * ( x ) [ 1 t ( x ) t * ( x ) + t ( x ) t * ( x ) ] .
t ( x ) = w ( z ) exp [ i k ( n 1 ) z ] d z = exp ( g / 2 ) ,
t ( x ) t * ( x ) = e g m = 0 g m m ! exp [ m ( x x ) 2 / T 0 2 ] ,
J ( x , x ) = J GSM ( x , x ) J G ( x , x ) ,
J 1 ( x 1 , x 1 , z ) = 1 λ z J ( x , x ) e i k 2 z ( x 1 x ) 2 e i k 2 z ( x 1 x ) 2 d x d x .
I ¯ 1 ( x 1 , z ) = 1 β 1 e 1 2 ( x 2 β 1 σ i ) 2 1 e g / 2 β 1 l a l ( e i l q x 2 β 1 2 e i π l 2 z / z T β 1 2 + e i l q x 2 β 1 2 e i π l 2 z / z T β 1 2 ) e ( l q z / k x 1 2 σ i β 1 ) 2 e 1 2 ( l q z k σ μ β 1 ) 2 e ( x 1 2 σ i β 1 ) 2 + ( 1 e g / 2 ) 2 β 1 l l a l a l e i ( l l ) q x 1 β 1 2 e i π ( l 2 l 2 ) z / z T β 1 2 e ( l q z / k x 1 2 σ i β 1 ) 2 e ( l q z / k x 1 2 σ i β 1 ) 2 e ( l l ) 2 2 ( q z k σ μ β 1 ) 2 + e g m = 1 g m m ! β 2 ( m ) l l a l a l e i ( l l ) q x 1 β 2 ( m ) 2 e i π ( l 2 l 2 ) z / z T β 2 ( m ) 2 e ( l q z / k x 1 2 σ i β 2 ( m ) ) 2 e ( l q z / k x 1 2 σ i β 2 ( m ) ) 2 e ( l l ) 2 2 ( q z k σ μ β 2 ( m ) ) 2 e m ( q z ( l l ) k T 0 β 2 ( m ) ) 2 ,
I ¯ 1 ( x 1 , z ) = 1 β i 1 e 1 2 ( x 1 β i 1 σ i ) 2 1 e g / 2 β i 1 l a l ( e i l q x 1 β i 1 2 e i π l 2 z / z T β i 1 2 + e i l q x 1 β i 1 2 e i π l 2 z / z T β i 1 2 ) e ( l q z / k x 1 2 σ i β i 1 2 ) 2 e ( x 1 2 σ i β i 1 2 ) 2 + ( 1 e g / 2 ) 2 β i 1 l l a l a l e i ( l l ) q x 1 β i 1 2 e i π ( l 2 l 2 ) z / z T β i 1 2 e ( l q z / k x 1 2 σ i β i 1 ) 2 e ( l q z / k x 1 2 σ i β i 1 ) 2 + e g m = 1 g m m ! β i 2 ( m ) l l a l a l e i ( l l ) q x 1 β i 2 ( m ) 2 e i π ( l 2 l 2 ) z / z T β i 2 ( m ) 2 e ( l q z / k x 1 2 σ i β i 2 ( m ) ) 2 e ( l q z / k x 1 2 σ i β i 2 ( m ) ) 2 e m ( q z ( l l ) k T 0 β i 2 ( m ) ) 2 ,
I ¯ 1 ( x 1 , z ) = 1 ( 1 e g / 2 ) l a l ( e i π l 2 z / z T e i l q x 1 + e i π l 2 z / z T e i l q x 1 ) + l , l a l a l * e i q ( l l ) x 1 e i π ( l 2 l 2 ) z / z T [ ( 1 e g / 2 ) 2 + e g m = 1 g m m ! e m [ q z ( l l ) k T 0 ] 2 ] ,

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