Abstract

We analyze the self-imaging process produced by a transmission grating whose strips present two different roughness levels. This kind of grating periodically modulates the transmitted light owing only to the different microtopographic properties of the strips. In spite of the fact that the grating is not purely periodic, it produces a kind of self-image at Talbot distances. These self-images gradually appear as light propagates, but they are not present just after the grating, as occurs in amplitude or phase gratings. There exists a distance from the grating, which depends on the stochastic properties of roughness, from which the contrast of the self-images becomes stable. Important cases are analyzed in detail, such as low- and high-roughness limits. We assume for the calculations that the grating can be used in a mobile system. Simulations using the Rayleigh–Sommerfeld regime have been performed, which confirm the validity of the theoretical approach proposed in this work

© 2008 Optical Society of America

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References

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  1. W. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401-407 (1836).
    [CrossRef]
  2. K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1-108 (1989).
    [CrossRef]
  3. E. Keren and O. Kafri, “Diffraction effects in moiré deflectometry,” J. Opt. Soc. Am. A 2, 111-120 (1985).
    [CrossRef]
  4. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).
  5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  6. E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).
  7. C. Palmer, Diffraction Grating Handbook (Richardson Grating Laboratory, New York, 2000).
  8. F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584-586 (1999).
    [CrossRef]
  9. C. G. Someda, “Far field of polarization gratings,” Opt. Lett. 24, 1657-1659 (1999).
    [CrossRef]
  10. 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]
  11. 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]
  12. 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]
  13. L. M. Sanchez-Brea, F. J. Torcal-Milla, and E. Bernabeu, “Far field of gratings with rough strips,” J. Opt. Soc. Am. A 25, 828-833 (2008).
    [CrossRef]
  14. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45, 1102-1110 (2006).
    [CrossRef] [PubMed]
  15. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
    [CrossRef]
  16. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987).
  17. J. W. Goodman, Statistical Optics (Wiley, 1985).

2008 (1)

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 (1)

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]

1999 (2)

1989 (1)

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

1985 (1)

1836 (1)

W. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401-407 (1836).
[CrossRef]

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]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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

Gori, F.

Kafri, O.

Keren, E.

Loewen, E. G.

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

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]

Palmer, C.

C. Palmer, Diffraction Grating Handbook (Richardson Grating Laboratory, New York, 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]

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.

Someda, C. G.

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).
[CrossRef]

Teich, M. C.

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

Torcal-Milla, F. J.

Wang, A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Appl. Opt. (2)

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

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]

Opt. Lett. (2)

Philos. Mag. (1)

W. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401-407 (1836).
[CrossRef]

Prog. Opt. (1)

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

Other (7)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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

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

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

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

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

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Figures (5)

Fig. 1
Fig. 1

Diffraction grating proposed in this paper. The strips present different microtopographic structures.

Fig. 2
Fig. 2

(a) Near-field intensity pattern using Eq. (9) for a grating with period p = 20 μ m when σ = 0.25 μ m , T 0 = 100 μ m , and n = 1.5 . The wavelength is λ = 0.68 μ m . (b) Amplitude of the self-images (solid curve) and terms of Eq. (9): first term (dashed-dot curve), second term (dashed curve).

Fig. 3
Fig. 3

Contrast of the self-images for a grating with parameters p = 20 μ m , n = 1.5 , l = l = 3 , T 0 = 100 μ m , and wavelength λ = 0.68 μ m for different values of σ. (a) σ = 0.05 μ m , (b) σ = 0.25 μ m , (c) σ = 1 μ m .

Fig. 4
Fig. 4

(a) Example of a grating with period p = 40 μ m , refractive index n = 1.5 , and roughness parameters σ = 0.25 μ m , T 0 = 1 μ m . (b) Near-field intensity pattern produced by this grating.

Fig. 5
Fig. 5

(a) Intensity pattern obtained using the Rayleigh–Sommerfeld formalism taking the average of 100 simulations with σ = 0.25 μ m , T 0 = 100 μ m , p = 20 μ m . (b) Contrast comparison between numerical simulation and theoretical results.

Equations (12)

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G 1 ( x ) = l a l exp ( i q l x ) ,
T ( x ) = 1 G 1 ( x ) [ 1 t ( x ) ] ,
t ( x ) = w ( z ) exp [ i k ( n 1 ) z ] d z = exp ( g 2 ) ,
w ( ζ ( x ) , ζ ( x ) ) = 1 2 π σ 1 C ( τ ) 2 exp [ ζ 2 ( x ) 2 C ( τ ) ζ ( x ) ζ ( x ) + ζ 2 ( x ) 2 σ 2 ( 1 C ( τ ) 2 ) ] ,
t ( x ) t * ( x ) = exp { g [ 1 C ( τ ) ] } = exp ( g ) m = 0 g m m ! exp [ m τ 2 T 0 2 ] .
J ( 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 ) ] ,
J ( x , x ) ¯ = 1 ( 1 e g 2 ) [ G 1 ( x ) + G 1 * ( x ) ] + G 1 ( x ) G 1 * ( x ) [ ( 1 2 e g 2 ) + e g m = 0 g m m ! exp ( m τ 2 T 0 2 ) ] .
J ( x 2 , x 2 , z ) ¯ = 1 λ z J ( x 1 , x 1 ) ¯ exp [ i k 2 z ( x 2 x 1 ) 2 ] exp [ i k 2 z ( x 2 x 1 ) 2 ] d x 1 d x 1 = [ 1 ( 1 e g 2 ) H ( x 2 , z ) ] [ 1 ( 1 e g 2 ) H * ( x 2 , z ) ] + e g m = 1 g m m ! l l a l a l * exp [ i q ( l x 2 l x 2 ) ] exp [ i ( l 2 l 2 ) z z T ] exp [ m [ ( l l ) q z k ( x 2 x 2 ) ] 2 ( k T 0 ) 2 ] ,
I ( x 2 , z ) ¯ = 1 ( 1 e g 2 ) H ( x 2 , z ) 2 + e g m = 1 g m m ! l , l a l a l * exp [ i ( l l ) q x 2 ] exp [ i π ( l 2 l 2 ) z z T ] exp [ m ( l l ) 2 ( z z C ) 2 ] ,
I ( x 2 , z ) ¯ = 1 ( 1 e g 2 ) H ( x 2 , z ) 2 + ( 1 e g ) κ .
I ( x 2 , z ) ¯ = 1 g l a l cos ( l q x 2 + l 2 z z T ) + g l , l a l a l * exp [ i ( l l ) q x 2 ] exp [ ( l l ) 2 ( z z C ) 2 ] exp [ i ( l 2 l 2 ) z z T ] .
I ( x 2 , z ) ¯ = 1 H ( x 2 , z ) 2 + κ .

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