Abstract

In this work, we analyze the far-field pattern produced by a grating made of strips with two different random roughness levels. The efficiency and shape of the diffraction orders is obtained, which are shown to depend on the statistical properties of roughness. We assume for the calculations that the grating can be used in a mobile mechanical system. A preliminary experimental approach which partially corroborates the theoretical results is also performed.

© 2008 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  3. E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).
  4. M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, “One-way diffraction grating,” Phys. Rev. E 74, 056611 (2006).
    [CrossRef]
  5. S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, and B. F. Whiting, “Phase effect in the diffraction of light: beyond the grating equation,” Phys. Rev. Lett. 95, 013901 (2005).
    [CrossRef] [PubMed]
  6. C. Palmer, Diffraction Grating Handbook (Richardson Grating Laboratory, New York, 2000).
  7. R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
    [CrossRef]
  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. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
    [CrossRef]
  14. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House Norwood, 1987).
  15. J. W. Goodman, Statistical Optics (Wiley, 1985).

2007

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

M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, “One-way diffraction grating,” Phys. Rev. E 74, 056611 (2006).
[CrossRef]

2005

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, and B. F. Whiting, “Phase effect in the diffraction of light: beyond the grating equation,” Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef] [PubMed]

2001

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

Appl. Opt.

Opt. Commun.

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]

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]

Opt. Lett.

Phys. Rev. E

M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, “One-way diffraction grating,” Phys. Rev. E 74, 056611 (2006).
[CrossRef]

Phys. Rev. Lett.

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, and B. F. Whiting, “Phase effect in the diffraction of light: beyond the grating equation,” Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef] [PubMed]

Other

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

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
[CrossRef]

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).

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 Norwood, 1987).

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

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Figures (5)

Fig. 1
Fig. 1

Diffraction grating formed as a sum of two amplitude gratings. One of them presents a rough surface.

Fig. 2
Fig. 2

Mutual intensity J ( x , x , 0 , 0 ) just after the diffraction grating when p = 20 μ m , λ = 0.68 μ m , κ = 0.5 , R x = 0.1 mm , R y = 0.1 mm , and n = 1.5 for four cases: (a) σ = 0.1 μ m , T 0 = 10 μ m ; (b) σ = 0.1 μ m , T 0 = 50 μ m ; (c) σ = 0.5 μ m , T 0 = 10 μ m ; and (d) σ = 0.5 μ m , T 0 = 50 μ m .

Fig. 3
Fig. 3

Intensity at the far field for the situations depicted in Fig. 2.

Fig. 4
Fig. 4

Confocal microscopy image of the grating used in the experiment.

Fig. 5
Fig. 5

(a) Average of an ensemble of experimental diffraction patterns of the grating, showing the different diffraction orders and the halo produced by scattering. In order to see the halo, the integration time of the camera needs to be increased so the pixels at the location of the diffraction peaks are saturated. (b) Theoretical fit (dashed curve) to the experimental diffraction pattern (solid curve).

Equations (20)

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G 1 ( x ) = l a l exp ( i q l x ) ,
T ( x , y ) = 1 G 1 ( x ) [ 1 t ( x , y ) ] ,
t ( x , y ) = d z w ( z ) exp [ i k ( n 1 ) z ] = exp ( g 2 ) ,
J ( x , x , y , y ) = U 1 ( x , y ) U 1 * ( x , y ) = A 0 2 T ( x , y ) T * ( x , y ) ,
J ( x , x , y , y ) A 0 2 = 1 + [ G 1 ( x ) + G 1 * ( x ) ] ( t ( x , y ) 1 ) + G 1 ( x ) G 1 * ( x ) [ t ( x , y ) t * ( x , y ) 2 t ( x , y ) + 1 ] ,
w ( z 1 , z 2 ) = 1 2 π σ 1 C ( τ , η ) 2 exp [ z 1 2 2 C ( τ , η ) z 1 z 2 + z 2 2 2 σ 2 ( 1 C ( τ , η ) 2 ) ] ,
t ( x 1 , y 1 ) t * ( x 1 , y 1 ) = exp { g [ 1 C ( τ , η ) ] } = e g m = 0 g m m ! e m ( τ 2 + η 2 ) T 0 2 .
J ( x , x , y , y ) A 0 2 = 1 [ G 1 ( x ) + G 1 * ( x ) ] [ 1 exp ( g 2 ) ] + G 1 ( x ) G 1 * ( x ) { [ 1 exp ( g 2 ) ] 2 + e g m = 1 g m m ! e m ( τ 2 + η 2 ) T 0 2 } .
U 1 ( x , y ) = A 0 T ( x , y ) ( x R x ) ( y R y ) ,
( x R ) = { 1 x R 2 0 x > R 2 } .
U 2 ( x 2 , y 2 ) = e i k [ z + ( x 2 2 + y 2 2 ) 2 z ] i λ z U 1 ( x 1 , y 1 ) exp [ i k z ( x 1 x 2 + y 1 y 2 ) ] d x 1 d y 1 ,
J ( x 2 , x 2 , y 2 , y 2 ) = K R y 2 R y 2 R y 2 R y 2 R x 2 R x 2 R x 2 R x 2 J ( x 1 , x 1 , y 1 , y 1 ) e i k z ( x 2 x 1 + y 2 y 1 x 2 x 1 y 2 y 1 ) d x 1 d x 1 d y 1 d y 1 ,
J ( θ x , θ x , θ y , θ y ) K A 0 2 R x 2 R y 2 sinc ( k R y 2 θ y ) sinc ( k R y 2 θ y ) { sinc ( k R x 2 θ x ) sinc ( k R x 2 θ x ) + ( e g 2 1 ) sinc ( k R x 2 θ x ) l a l sinc [ R x 2 ( k θ x l q ) ] + ( e g 2 1 ) sinc ( k R x 2 θ x ) l a l * sinc [ R x 2 ( k θ x l q ) ] + ( 1 e g 2 ) 2 l , l a l a l * sinc [ R x 2 ( k θ x l q ) ] sinc [ R x 2 ( k θ x l q ) ] } + 4 T 0 2 R x R y e g sinc [ k R y 2 ( θ y θ y ) ] l , l a l a l * sinc { R x 2 [ k ( θ x θ x ) ( l l ) q ] } m = 1 m 2 g m m ! [ 1 m 2 + ( k T 0 θ y ) 2 + 1 m 2 + ( k T 0 θ y ) 2 ] [ 1 m 2 + T 0 2 ( k θ y l q ) 2 + 1 m 2 + T 0 2 ( k θ y l q ) 2 ] ,
I ( θ x , θ y ) ¯ = sinc 2 ( k R y 2 θ y ) × { sinc 2 ( k R x 2 θ x ) + 2 ( e g 2 1 ) sinc ( k R x 2 θ x ) l Re ( a l ) sinc [ R x 2 ( k θ x l q ) ] + ( 1 e g 2 ) 2 l , l a l a l * sinc [ R x 2 ( k θ x l q ) ] sinc [ R x 2 ( k θ x l q ) ] } + 8 T 0 2 R x R y e g l , l a l a l * sinc [ R x 2 ( l l ) q ] 1 m 2 + ( k T 0 θ y ) 2 m = 1 m 2 g m m ! [ 1 m 2 + T 0 2 ( k θ x l q ) 2 + 1 m 2 + T 0 2 ( k θ x l q ) 2 ] ,
I ( θ x , θ y ) ¯ = sinc 2 ( k R y 2 θ y ) { [ 1 + 2 Re ( a 0 ) ( e g 2 1 ) ] sinc 2 ( k R x 2 θ x ) + ( 1 e g 2 ) 2 l a l 2 sinc 2 [ R x 2 ( k θ x l q ) ] } + 16 T 0 2 R x R y e g m = 1 g m m ! m 2 m 2 + ( k T 0 θ y ) 2 l a l 2 m 2 + T 0 2 ( k θ x l q ) 2 .
I l ( θ x , θ y ) ¯ = sinc 2 ( k R y 2 θ y ) { h 0 + ( 1 e g 2 ) 2 a l 2 sinc 2 [ R x 2 ( k θ x l q ) ] } + 16 T 0 2 R x R y e g m = 1 g m m ! m 2 m 2 + ( k T 0 θ y ) 2 a l 2 m 2 + T 0 2 ( k θ x l q ) 2 ,
η l = I l ( θ x , θ y ) ¯ d θ x d θ y l I l ( θ x , θ y ) ¯ d θ x d θ y .
I ( θ x , θ y ) ¯ = sinc 2 ( 1 2 k R x θ x ) sinc 2 ( 1 2 k R y θ y ) [ 1 g Re ( a 0 ) ] + 16 g T 0 2 R x R y ( 1 + k 2 T 0 2 θ y 2 ) l a l 2 1 + T 0 2 ( k θ x l q ) 2 .
I ( θ x , θ y ) ¯ = A 0 2 R x 2 R y 2 ( λ z ) 2 sinc 2 ( 1 2 k R x θ x ) sinc 2 ( 1 2 k R y θ y ) ,
I ( θ x , θ y ) ¯ = sinc 2 ( 1 2 k R y θ y ) { [ 1 2 Re ( a 0 ) ] sinc 2 ( 1 2 k R x θ x ) + l a l 2 sinc 2 [ 1 2 R x ( l q k θ x ) ] } + 16 T 0 2 R x R y ( 1 + k 2 T 0 2 θ y 2 ) l a l 2 1 + T 0 2 ( l q k θ x ) 2 .

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