We analyze the far field and near field diffraction pattern produced by an amplitude grating whose strips present rough edges. Due to the stochastic nature of the edges a statistical approach is performed. The grating with rough edges is not purely periodic, although it still divides the incident beam in diffracted orders. The intensity of each diffraction order is modified by the statistical properties of the irregular edges and it strongly decreases when roughness increases except for the zero-th diffraction order. This decreasing firstly affects to the higher orders. Then, it is possible to obtain an amplitude binary grating with only diffraction orders -1, 0 and +1. On the other hand, numerical simulations based on Rayleigh-Sommerfeld approach have been used for the case of near field. They show that the edges of the self-images are smoother than the edges of the grating. Finally, we fabricate gratings with rough edges and an experimental verification of the results is performed.

© 2008 Optical Society of America

1. M. Born, and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1980).
2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
3. E. G. Loewen and E. Popov, Diffraction gratings and applications (Marcel Dekker, New York, 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, Berlin, 1980). [CrossRef]
8. F. Gori, "Measuring Stokes parameters by means of a polarization grating," Opt. Lett. 24, 584-586 (1999).
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. W. H. F. Talbot, "Facts relating to optical science," Philos. Mag. 9, 401-407 (1836).
15. K. Patorski, "The self-imaging phenomenon and its applications," Prog. Opt. 27, 1-108 (1989). [CrossRef]
16. N. Guérineau, B. Harchaoui, and J. Primot, "Talbot effect re-examined: demonstration of an achromatic and continuous self-imaging regime," Opt. Commun. 180, 199-203 (2000). [CrossRef]
17. Y. Lu, C. Zhou, and H. Luo, "Talbot effect of a grating with different kind of flaws," J. Opt. Soc. Am. A 22, 2662-2667 (2005) [CrossRef]
18. P. P. Naulleau and G. M. Gallatin, "Line-edge roughness transfer function and its application to determining mask effects in EUV resist characterization," Appl. Opt. 42, 3390-3397 (2003). [CrossRef] [PubMed]
19. T. R. Michel, "Resonant light scattering from weakly rough random surfaces and imperfect gratings," J. Opt. Soc. Am. A 11, 1874-1885 (1994). [CrossRef]
20. V. A. Doroshenko, "Singular integral equations in the problem of wave diffraction by a grating of imperfect flat irregular strips," Telecommunications and Radio Engineering 57, 65-72 (2002)
21. M. V. Glazov., and S. N. Rashkeev, "Light scattering from rough surfaces with superimposed periodic structures," Appl. Phys. B 66, 217-223 (1998) [CrossRef]
22. 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]
23. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House Norwood, 1987).

Bernabeu, E.
Borghi, R.
Deshpande, A. J.
Doroshenko, V. A.
Gallatin, G. M.
Glazov, M. V.
Gori, F.
Guérineau, N.
Harchaoui, B.
Hibbins, A. P.
Lockyear, M. J.
Lu, Y.
Luo, H.
Michel, T. R.
Mondello, A.
Mueller, G.
Naulleau, P. P.
Patorski, K.
Piquero, G.
Primot, J.
Quetschke, V.
Rashkeev, S. N.
Reitze, D. H.
Sambles, J. R.
Sanchez-Brea, L. M.
Sanchez-Brea, L.M.
Santarsiero, M.
Shen, F.
Someda, C. G.
Talbot, W. H. F.
Tanner, D. B.
Torcal-Milla, F. J.
Wang, A.
White, K. R.
Whiting, B. F.
Wise, S.
Zhou, C.

Appl. Opt.

• 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]
• P. P. Naulleau and G. M. Gallatin, "Line-edge roughness transfer function and its application to determining mask effects in EUV resist characterization," Appl. Opt. 42, 3390-3397 (2003). [CrossRef] [PubMed]
• 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]

Appl. Phys. B

• M. V. Glazov., and S. N. Rashkeev, "Light scattering from rough surfaces with superimposed periodic structures," Appl. Phys. B 66, 217-223 (1998) [CrossRef]

J. Opt. Soc. Am. A

• Y. Lu, C. Zhou, and H. Luo, "Talbot effect of a grating with different kind of flaws," J. Opt. Soc. Am. A 22, 2662-2667 (2005) [CrossRef]
• T. R. Michel, "Resonant light scattering from weakly rough random surfaces and imperfect gratings," J. Opt. Soc. Am. A 11, 1874-1885 (1994). [CrossRef]
• 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]

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]
• N. Guérineau, B. Harchaoui, and J. Primot, "Talbot effect re-examined: demonstration of an achromatic and continuous self-imaging regime," Opt. Commun. 180, 199-203 (2000). [CrossRef]

Opt. Lett.

• F. Gori, "Measuring Stokes parameters by means of a polarization grating," Opt. Lett. 24, 584-586 (1999).
• C. G. Someda, "Far field of polarization gratings," Opt. Lett. 24, 1657-1659 (1999). [CrossRef]

Philos. Mag.

• W. H. F. Talbot, "Facts relating to optical science," Philos. Mag. 9, 401-407 (1836).

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]

Prog. Opt.

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

Telecommunications and Radio Engineering

• V. A. Doroshenko, "Singular integral equations in the problem of wave diffraction by a grating of imperfect flat irregular strips," Telecommunications and Radio Engineering 57, 65-72 (2002)

Other

• C. Palmer, Diffraction Grating Handbook (Richardson Grating Laboratory, New York, 2000).
• R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980). [CrossRef]
• M. Born, and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1980).
• J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
• E. G. Loewen and E. Popov, Diffraction gratings and applications (Marcel Dekker, New York, 1997).
• P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House Norwood, 1987).

2008, Sanchez-Brea, J. Opt. Soc. Am. A
2007, Torcal-Milla, Appl. Opt.
2007, Sanchez-Brea, Opt. Commun.
2006, Lockyear, Phys. Rev. E
2006, Shen, Appl. Opt.
2005, Lu, J. Opt. Soc. Am. A
2005, Wise, Phys. Rev. Lett.
2003, Naulleau, Appl. Opt.
2002, Doroshenko, Telecommunications and Radio Engineering
2001, Piquero, Opt. Commun.
2000, Guérineau, Opt. Commun.
1999, Gori, Opt. Lett.
1999, Someda, Opt. Lett.
1998, Glazov, Appl. Phys. B
1994, Michel, J. Opt. Soc. Am. A
1989, Patorski, Prog. Opt.
1836, Talbot, Philos. Mag.

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