Abstract

Diffraction gratings produce self-images in the near field. Defects on the surface of the grating may occur due to the manufacturing process. These devices are often placed in dirty industrial environments. Dust particles or drops of liquid can be deposited over their surface. In this work, we analyze the effect of surface defects placed over the grating on the self-imaging process. We analytically show how the self-images gradually recover as we separate from the grating when one defect is present. Also a random distribution of surface defects over the grating is analyzed. In particular, we focus on how the contrast of the self-images decreases in terms of the density of the defects. Analytical expressions for the near field are derived, considering a stochastic description of the spatial distribution of defects. In addition, numerical simulations based on the Rayleigh–Sommerfeld formulation are performed to validate the analytical results.

© 2010 Optical Society of America

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References

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  1. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E.Wolf, ed. (North-Holland, 1989), Vol. 27.
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2009 (1)

2008 (2)

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]

A. Sauceda-Carvajal, J. Ibarra-Galitzia, Gustavo Ramírez-Zavaleta, and E. Tepichín-Rodríguez, “Dynamic noise elimination on 2-D periodic structures using a liquid crystal display as an incoherent reconfigurable spatial source,” Opt. Eng. 46, 058001 (2007).
[CrossRef]

2006 (1)

2005 (2)

1999 (1)

1998 (1)

J. Ojeda-Castañeda and C. Frausto, “Multiplicative noise reduction using the Lau effect,” Opt. Commun. 154, 130-136(1998).
[CrossRef]

1990 (1)

1971 (1)

Bernabeu, E.

Dammann, H.

Frausto, C.

J. Ojeda-Castañeda and C. Frausto, “Multiplicative noise reduction using the Lau effect,” Opt. Commun. 154, 130-136(1998).
[CrossRef]

Garavaglia, M.

Groh, G.

He, S.

Ibarra-Galitzia, J.

A. Sauceda-Carvajal, J. Ibarra-Galitzia, Gustavo Ramírez-Zavaleta, and E. Tepichín-Rodríguez, “Dynamic noise elimination on 2-D periodic structures using a liquid crystal display as an incoherent reconfigurable spatial source,” Opt. Eng. 46, 058001 (2007).
[CrossRef]

Kock, M.

Lohmann, A. W.

Lu, Y.

Luo, H.

Ojeda-Castañeda, J.

J. Ojeda-Castañeda and C. Frausto, “Multiplicative noise reduction using the Lau effect,” Opt. Commun. 154, 130-136(1998).
[CrossRef]

Patorski, K.

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

Pérez, D. G.

Ramírez-Zavaleta, Gustavo

A. Sauceda-Carvajal, J. Ibarra-Galitzia, Gustavo Ramírez-Zavaleta, and E. Tepichín-Rodríguez, “Dynamic noise elimination on 2-D periodic structures using a liquid crystal display as an incoherent reconfigurable spatial source,” Opt. Eng. 46, 058001 (2007).
[CrossRef]

Sanchez-Brea, L. M.

Sauceda-Carvajal, A.

A. Sauceda-Carvajal, J. Ibarra-Galitzia, Gustavo Ramírez-Zavaleta, and E. Tepichín-Rodríguez, “Dynamic noise elimination on 2-D periodic structures using a liquid crystal display as an incoherent reconfigurable spatial source,” Opt. Eng. 46, 058001 (2007).
[CrossRef]

Shen, F.

Song, J.

Sun, J.

Tepichín-Rodríguez, E.

A. Sauceda-Carvajal, J. Ibarra-Galitzia, Gustavo Ramírez-Zavaleta, and E. Tepichín-Rodríguez, “Dynamic noise elimination on 2-D periodic structures using a liquid crystal display as an incoherent reconfigurable spatial source,” Opt. Eng. 46, 058001 (2007).
[CrossRef]

Thomas, J. A.

Torcal-Milla, F. J.

Wang, A.

Zheng, C.

Zhou, C.

Zhu, N.

Appl. Opt. (4)

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

Opt. Commun. (1)

J. Ojeda-Castañeda and C. Frausto, “Multiplicative noise reduction using the Lau effect,” Opt. Commun. 154, 130-136(1998).
[CrossRef]

Opt. Eng. (1)

A. Sauceda-Carvajal, J. Ibarra-Galitzia, Gustavo Ramírez-Zavaleta, and E. Tepichín-Rodríguez, “Dynamic noise elimination on 2-D periodic structures using a liquid crystal display as an incoherent reconfigurable spatial source,” Opt. Eng. 46, 058001 (2007).
[CrossRef]

Opt. Express (1)

Other (1)

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

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Figures (6)

Fig. 1
Fig. 1

Analytical self-images produced by an amplitude grating with p = 10 μm and a Gaussian defect placed at the origin whose width is ω = 15 μm . The wavelength is λ = 0.6328 μm . (a) Intensity distribution just after the grating, (b) second self-image, (c) fourth self-image, and (d) sixth self-image.

Fig. 2
Fig. 2

(a) Mean intensity I ( x , z ) and (b) contrast for a grating with a period p = 20 μm , density of defects ρ = 0.2 , and radius of the defects ω = 15 μm , illuminated with a plane wave λ = 0.6328 μm .

Fig. 3
Fig. 3

(a) Example of amplitude diffraction grating with amplitude surface defects (as dust particles) used for the simulations. The period of the grating is 40 μm , the wavelength is λ = 0.6328 μm , and the defects are simulated by circles with radius 10 μm . (b) First self-image, (c) second self-image, and (d) third self-image.

Fig. 4
Fig. 4

First self-image for (a) amplitude grating and amplitude defects, (b) amplitude grating and phase defects, (c) phase grating and amplitude defects, and (d) phase grating and phase defects ( p = 10 μm and λ = 0.6328 μm ). The radius of defects is ω = 5 μm . The density is ρ = 0.2 . The phase gratings present a phase shift of φ = π / 2 , and the phase defects present a random phase between 0 and 2 π . White color in the plots corresponds to null intensity, and black color corresponds to intensity equal to 2.

Fig. 5
Fig. 5

Mean intensity profiles corresponding to the zeroth, first, second, and third self-images (down to up) for the cases depicted in Fig. 4.

Fig. 6
Fig. 6

Theoretical (smooth curves) and numerical (rough curves) contrast for the first four self-images, first (solid), second (dashed), third (dashed-dotted), and fourth (dotted): (a) amplitude grating and amplitude defects, (b) amplitude grating and phase defects, (c) phase gratings and amplitude defects, and (d) phase grating and phase defects. The period of the grating is p = 10 μm , wavelength is λ = 0.6328 μm , and size of the defect is ω = 5 μm . The phase shift for phase gratings is φ = π / 2 and for phase defects is also δ = π / 2 .

Equations (26)

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t 1 ( ξ ) = n a n exp ( i n q ξ ) ,
U ( x , y , z ) = e i k z i λ z U 0 ( ξ , η ) t 1 ( ξ ) e i k 2 z [ ( x ξ ) 2 + ( y η ) 2 ] d ξ d η ,
U ( x , z ) = A 0 n a n exp ( i n q x ) exp ( 2 π in 2 z / z T ) ,
t ( ξ ) = t 1 ( ξ ) t 2 ( ξ , η ) ,
U 2 ( x , y , z ) = A 0 i λ z e i k z n a n { exp ( i q n x ) exp ( i q 2 2 k n 2 z ) 1 i z / k ω 2 + 1 exp [ k ( x 2 + y 2 ) 2 ( i z + k ω 2 ) ] exp [ ω 2 2 n q ( 2 k x + n q z ) ( z i k ω 2 ) ] } .
t 2 ( ξ , η ) = i = 1 N [ 1 f ( ξ ξ i , η η i ) ] ,
t 2 ( ξ , η ) = 1 i = 1 N f ( ξ ξ i , η η i ) .
I ( x , y , z ) = t 1 ( ξ ) t 1 * ( ξ ) t 2 ( ξ , η ) t 2 * ( ξ , η ) H ( ξ , η ) H * ( ξ , η ) d ξ d η d ξ d η ,
t 2 ( ξ , η ) t 2 * ( ξ , η ) = [ 1 i = 1 N f ( ξ ξ i , η η i ) ] [ 1 j = 1 N f * ( ξ ξ j , η η j ) ] = 1 i = 1 N f ( ξ ξ i , η η i ) j = 1 N f * ( ξ ξ j , η η j ) + i = 1 N j = 1 N f ( ξ ξ i , η η i ) f * ( ξ ξ j , η η j ) .
i = 1 N f ( ξ ξ i , η η i ) = ϒ A = { ρ / 2 Amplitude grating ρ Phase grating ,
i = 1 N j = 1 N f ( ξ ξ i , η η i ) f * ( ξ ξ j , η η j ) = i = 1 N f ( ξ ξ i , η η i ) f * ( ξ ξ i , η η i ) = N f ( ξ ξ 0 , η η 0 ) f * ( ξ ξ 0 , η η 0 ) .
t 2 ( ξ , η ) t 2 * ( ξ , η ) = 1 2 ϒ A + ρ R f f ( ξ ξ , η η ) .
R f f ( ξ , η ) = exp ( 1 2 ξ 2 + η 2 ω 2 ) .
I ( x , z ) = [ I 1 ( x , z ) + I 2 ( x , z ) ] / ( A 0 / λ z ) 2 ,
I 1 ( x , z ) = ( 1 2 ϒ A ) n , n a n a n * exp [ i q x ( n n ) ] exp [ i q 2 2 k ( n 2 n 2 ) z ] , I 2 ( x , z ) = ρ n , n a n a n * exp [ i q x ( n n ) ] exp [ i q 2 2 k ( n 2 n 2 ) z ] exp { [ ( n n ) z z ω ] 2 } ,
I 2 ( x , z ) = ρ n | a n | 2 = ρ I k ,
I ( x , z ) = ρ I k + ( 1 2 ϒ A ) n , n a n a n * exp [ i ( n n ) q x ] exp [ i q 2 2 k ( n 2 n 2 ) z ] .
C ρ ( z ) = I max ( z ) I min ( z ) I max ( z ) + I min ( z ) ,
I max = ( 1 2 ϒ A ) I max 0 + ρ I k I min = ( 1 2 ϒ A ) I min 0 + ρ I k ,
C A ρ = ( I max 0 I min 0 ) ( I max 0 + I min 0 ) + 2 ρ 1 2 ϒ A I k .
i = 1 N f ( ξ ξ i , η η i ) = ϒ P = { ρ ( 1 e i δ ) / 2 Amplitude grating ρ ( 1 e i δ ) Phase grating ,
t 2 ( ξ , η ) t 2 * ( ξ , η ) = 1 2 ϒ P ( 1 cos δ ) + 2 ρ ( 1 cos δ ) R f f ( ξ ξ , η η ) .
I 1 ( x , z ) = [ 1 2 ϒ P ( 1 cos δ ) ] n , n a n a n * exp [ i q x ( n n ) ] exp [ i q 2 2 k ( n 2 n 2 ) z ] , I 2 ( x , z ) = 2 ρ ( 1 cos δ ) n , n a n a n * exp [ i q x ( n n ) ] exp [ i q 2 2 k ( n 2 n 2 ) z ] exp { [ ( n n ) z z ω ] 2 } .
I 2 ( x , z ) = 2 ρ ( 1 + cos δ ) n | a n | 2 = 2 ρ ( 1 + cos δ ) I k ,
C P ρ = ( I max 0 I min 0 ) ( I max 0 + I min 0 ) + 2 ρ ( 1 cos δ ) 1 2 ϒ P ( 1 cos δ ) I k .
U ( x , y , z ) = U ( ξ , η , 0 ) exp ( i k r ) 2 π r z r ( 1 r i k ) d ξ d η ,

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