## Diffraction of gratings with rough edges

Optics Express, Vol. 16, Issue 24, pp. 19757-19769 (2008)

http://dx.doi.org/10.1364/OE.16.019757

Acrobat PDF (2099 KB)

### Abstract

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. Introduction

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

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]

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

*p*is the period of the grating. When the period of the grating is much larger than the wavelength then the Talbot distance simplifies to

*z*=2

_{T}*p*

^{2}/λ[16

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]

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]

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]

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]

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]

## 2. Far field approach

*k=2π/λ*,

*x*and

*y*the transversal coordinates at the observation plane,

*z*the propagation direction, and

*U*the incident field that, for simplicity, we will consider that it is a monochromatic plane wave in normal incidence,

_{i}(ξ,η)*U*. The transmission function only affects to the integration limits into eq. (2). Then, the integral is converted into a summatory of finite integrals. Every one of these integrals corresponds to the contribution to the intensity of every strip of the grating, which has rough edges. Besides, we have truncated the summatory which appears in

_{i}(ξ,η)=U_{0}*t(ξ,η)*to be able to give a clearer equation for the intensity pattern. Thus, the intensity results in

*N*is the number of strips and

*L*is their length. Performing the integrals in

*η*and

*η*

^{′}the intensity results

*θ*and

_{x}=x/z*θ*=

_{y}*y/z*. Assuming the stochastical description of the functions

*fn(ξ)*and

*g*given before, the characteristic functions for this distribution result in [23]

_{n}(ξ)*α=kθ*. Performing an averaging process in (4), using the relationships given in (5), and reorganizing the terms, then the average intensity is

_{y}*L*is large, although not infinite. Then, the integrals have a simple analytical solution and the average intensity results

*σ*=0,

*I*=[|

_{0}*U*

_{0}|

*L*(

_{p}*N*+1)/(2

*λz*)]

^{2}. Also, since is normally very high, the sinc functions are very narrow and have significant values only when their argument,

*θ*, is nearly zero. Then the mean intensity results

_{y}-jλ/p*a*=sinc(jπ/2). The first term of (9) corresponds to the diffraction pattern of an amplitude grating but multiplied by a factor which diminishes the intensity of the diffraction orders according to

_{j}*σ/p*.

*x*and

*y*. Roughness produces a Gaussian halo centred in the zero-th order. The width of the halo depends on

*T*along the

*x*direction and

*σ*along the

*y*direction. In Fig. 2, the far field diffraction pattern along the y-axis for different values of

*T*and

*σ*is shown. As it can be seen, when roughness increases, high diffraction orders disappear and the halo grows around the zero-th order. The width of the halo in the y-axis depends on

*σ*. For higher values of

*σ*, the width of the halo diminishes.

*σ*≫

*λ*. The characteristic functions in (5) are still valid, except the autocorrelation function which is now [23]

*T*=

_{F}*kT/σ*. With this substitution and performing the integrals, average intensity results

*σ*≪

*λ*. In this case, Eq. (5) is valid, except the autocorrelation function

## 3. Self-imaging process in the near field

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]

*σ/p*. Although the edges of the grating present a high roughness, the edges of the self-images are quite smooth. The reason is that Talbot effect is a cooperative effect since the intensity at a given point (

*x,y*) of the image is obtained as an integration of the amplitude at the diffraction grating. It performs an averaging in the intensity distribution. In addition, an interferential process happens and produces a kind of speckle in the fringes. For the simulation we have considered a grating with size 150

*µm*×300

*µm*. Since the algorithm does not consider that the grating is periodic, an edge effect is produced. We show the central region of the intensity pattern to avoid this edge effect.

*z*=0 and the first self-image is shown. The intensity distribution at fractional Talbot planes is also shown in Fig. 7 (Media 1) for distances

*z=z*/4,

_{T}*z*/3,

_{T}*z*/2, and also the average profile for these particular cases.

_{T}### 3.1. Average intensity distribution

*z=np*/λ, with

^{2}*n*=1,2,…. The average intensity of these self-images is shown in Fig. 8 for an ensemble of 100 images. For this case, the self-images are very smooth.

## 4. Experimental approach

*p*=100

*µm*. The grating is illuminated with a collimated laser diode whose wavelength is λ=0.65

*µm*. In the near field approximation, some self-images have been acquired. For this, we have used a CMOS camera (ueye, pixel size: 6×6 microns) and a microscope objective in order to get a better resolution. In Fig. 9 (Media 2) we can observe the image of the grating using an optical microscope and the first three self-images taken with the CMOS camera. These images correspond to just one realization. As it can be seen, experimental results are in total accordance with the numerical results. The shape of the self-images is quite smooth compared to the shape of the strips edges. In the self-images we can also see a defect in one of the strips (rectangle) which gradually disappears as the order of the self-image increases. In Fig. 9 (Media 2) a video with the experimental images is also shown.

*x*=200

*µm*distribution is very smooth except for a dust particle in the optics that we could not eliminate. Comparing this result with that shown in Fig. 6, we can validate the results given by the numerical analysis.

## 5. Conclusions

## Acknowledgments

## References and links

1. | M. Born and E. Wolf, |

2. | J. W. Goodman, |

3. | E. G. Loewen and E. Popov, |

4. | M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, “One-way diffraction grating,” Phys. Rev. E |

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. |

6. | C. Palmer, |

7. | R. Petit, |

8. | F. Gori, “Measuring Stokes parameters by means of a polarization grating,” |

9. | C. G. Someda, “ |

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. |

11. | F. J. Torcal-Milla, L. M. Sanchez-Brea, and E. Bernabeu, “Talbot effect with rough reflection gratings,” Appl. Opt. |

12. | L.M. Sanchez-Brea, F. J. Torcal-Milla, and E. Bernabeu, “Talbot effect in metallic gratings under Gaussian illumination,” Opt. Commun. |

13. | L. M. Sanchez-Brea, F. J. Torcal-Milla, and E. Bernabeu, “Far field of gratings with rough strips,” J. Opt. Soc. Am. A |

14. | W. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. |

15. | K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. |

16. | N. Guérineau, B. Harchaoui, and J. Primot, “Talbot effect re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. |

17. | Y. Lu, C. Zhou, and H. Luo, “Talbot effect of a grating with different kind of flaws,” J. Opt. Soc. Am. A |

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. |

19. | T. R. Michel, “Resonant light scattering from weakly rough random surfaces and imperfect gratings,” J. Opt. Soc. Am. A |

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 |

21. | M. V. Glazov. and S. N. Rashkeev, “Light scattering from rough surfaces with superimposed periodic structures,” Appl. Phys. B |

22. | F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. |

23. | P. Beckmann and A. Spizzichino, |

**OCIS Codes**

(030.0030) Coherence and statistical optics : Coherence and statistical optics

(030.5770) Coherence and statistical optics : Roughness

(050.0050) Diffraction and gratings : Diffraction and gratings

(050.2770) Diffraction and gratings : Gratings

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: July 11, 2008

Revised Manuscript: September 9, 2008

Manuscript Accepted: September 11, 2008

Published: November 14, 2008

**Citation**

Francisco Jose Torcal-Milla, Luis Miguel Sanchez-Brea, and Eusebio Bernabeu, "Diffraction of gratings with rough edges," Opt. Express **16**, 19757-19769 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-24-19757

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### References

- 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).
- M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, "One-way diffraction grating," Phys. Rev. E 74, 056611 (2006). [CrossRef]
- 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]
- C. Palmer, Diffraction Grating Handbook (Richardson Grating Laboratory, New York, 2000).
- R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- W. H. F. Talbot, "Facts relating to optical science," Philos. Mag. 9, 401-407 (1836).
- K. Patorski, "The self-imaging phenomenon and its applications," Prog. Opt. 27, 1-108 (1989). [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]
- 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]
- 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]
- T. R. Michel, "Resonant light scattering from weakly rough random surfaces and imperfect gratings," J. Opt. Soc. Am. A 11, 1874-1885 (1994). [CrossRef]
- 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)
- M. V. Glazov., and S. N. Rashkeev, "Light scattering from rough surfaces with superimposed periodic structures," Appl. Phys. B 66, 217-223 (1998) [CrossRef]
- 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]
- P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House Norwood, 1987).

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