## Measurement of surface parameters from autocorrelation function of speckles in deep Fresnel region with microscopic imaging system |

Optics Express, Vol. 22, Issue 2, pp. 1302-1312 (2014)

http://dx.doi.org/10.1364/OE.22.001302

Acrobat PDF (1301 KB)

### Abstract

The derived two-dimensional autocorrelation function of speckles in the deep Fresnel region shows that it is related to the scattering of rough surface with the scattered intensity profile acting as the aperture function. We propose the method that is convenient for measuring surface parameters from the normalized autocorrelation function of speckles acquired with a microscopic imaging system. In experiment, a multi-scale behavior of the speckles has been identified, which is compatible with fractal character. With the speckle intensity data, we calculate the normalized autocorrelation function of the speckles and extract the roughness, the lateral correlation length and the roughness exponent of the random surface samples by fitting the expression to the autocorrelation function data. Comparison of the results with an atomic force microscopic measurements shows that our method has a satisfying accuracy.

© 2014 Optical Society of America

## 1. Introduction

3. S. Zhang, L. Wu, R. Y. Yue, Z. K. Yan, H. R. Zhan, and Y. Xiang, “Effects of Sb-doping on the grain growth of Cu(In, Ga)Se2 thin films fabricated by means of single-target sputtering,” Thin Solid Films **527**, 137–140 (2013). [CrossRef]

4. J. G. Goodberlet and H. Kavak, “Patterning sub-50 nm features with near-field embedded-amplitude masks,” Appl. Phys. Lett. **81**(7), 1315–1317 (2002). [CrossRef]

5. K. Jang, Y. Ishibashi, D. Iwata, H. Suganuma, T. Yamada, and Y. Takemura, “Fabrication of ferromagnetic nanoconstriction using atomic force microscopy nanoscratching,” J. Nanosci. Nanotechnol. **11**(12), 10945–10948 (2011). [CrossRef] [PubMed]

6. Ö. F. Farsakoğlu, D. M. Zengin, and H. Kocabaş, “Grinding process for beveling and lapping operations in lens manufacturing,” Appl. Opt. **39**(10), 1541–1548 (2000). [CrossRef] [PubMed]

8. M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: a direct measurement of the density correlation function g(r),” Phys. Rev. Lett. **85**(7), 1416–1419 (2000). [CrossRef] [PubMed]

8. M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: a direct measurement of the density correlation function g(r),” Phys. Rev. Lett. **85**(7), 1416–1419 (2000). [CrossRef] [PubMed]

12. C. F. Cheng, M. Liu, N. Y. Zhang, S. Y. Teng, H. S. Song, and Z. Z. Xu, “Speckle intensity correlation in the diffraction region near rough surfaces and simulational experiments for extraction of surface parameters,” Europhys. Lett. **65**(6), 779–784 (2004). [CrossRef]

## 2. Theory

*D*of the illuminating beam is large, so that the scattered waves will spread and then enough scattering elements on the screen may contribute to the light field at an observation point. Figure 1(b) gives the schematic diagram of the superposition of the light waves scattered from an equivalent scattering aperture contributing to the intensity of a point on the observation plane, which represents the one-dimensional case shown in Fig. 1(a) with the red dashed line. From the scattering theory, we know that the light intensity scattered onto the observation plane from a small adjacent region of a point

*n*is the refractive index of the screen,

19. Y. P. Zhao, I. Wu, C. F. Cheng, U. Block, G. C. Wang, and T. M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. **84**(5), 2571–2582 (1998). [CrossRef]

20. C. F. Cheng, C. X. Liu, S. Y. Teng, N. Y. Zhang, and M. Liu, “Half-width of intensity profiles of light scattered from self-affine fractal random surfaces and simulational verifications,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **65**(6), 061104 (2002). [CrossRef] [PubMed]

20. C. F. Cheng, C. X. Liu, S. Y. Teng, N. Y. Zhang, and M. Liu, “Half-width of intensity profiles of light scattered from self-affine fractal random surfaces and simulational verifications,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **65**(6), 061104 (2002). [CrossRef] [PubMed]

## 3. Experimental study and discussion

### 3.1 Setup of the optical system and measurement of random surface samples with AFM

*D*of the incident beam is 6cm and the distance

^{2}. We need not a beam stop in the setup to dispose the non-diffused beam as in Ref [8

8. M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: a direct measurement of the density correlation function g(r),” Phys. Rev. Lett. **85**(7), 1416–1419 (2000). [CrossRef] [PubMed]

### 3.2 Measurement of surface parameters with the speckle method

*vs.*ρ with Eqs. (11) and (12). Figures 6(a)-6(c) show the data of normalized autocorrelation functions calculated from the speckle intensities and the fit curves in solid lines for samples No.1, No.2 and No.3, respectively. The extracted results of samples are

*vs.*ρ in Figs. 6(a)-6(c) with Eq. (15). The corresponding results of three samples are respectively

21. M. N. Zhang, Z. H. Li, X. Y. Chen, G. T. Liang, S. Y. Wang, S. Y. Teng, and C. F. Cheng, “Evolutions of speckles on rough glass/silver surfaces with film thickness,” Opt. Express **21**(7), 8831–8843 (2013). [CrossRef] [PubMed]

## 4. Discussion and conclusion

## Acknowledgments

## References and links

1. | P. Meakin, |

2. | P. Córdoba-Torres, T. J. Mesquita, I. N. Bastos, and R. P. Nogueira, “Complex dynamics during metal dissolution: from intrinsic to faceted anomalous scaling,” Phys. Rev. Lett. |

3. | S. Zhang, L. Wu, R. Y. Yue, Z. K. Yan, H. R. Zhan, and Y. Xiang, “Effects of Sb-doping on the grain growth of Cu(In, Ga)Se2 thin films fabricated by means of single-target sputtering,” Thin Solid Films |

4. | J. G. Goodberlet and H. Kavak, “Patterning sub-50 nm features with near-field embedded-amplitude masks,” Appl. Phys. Lett. |

5. | K. Jang, Y. Ishibashi, D. Iwata, H. Suganuma, T. Yamada, and Y. Takemura, “Fabrication of ferromagnetic nanoconstriction using atomic force microscopy nanoscratching,” J. Nanosci. Nanotechnol. |

6. | Ö. F. Farsakoğlu, D. M. Zengin, and H. Kocabaş, “Grinding process for beveling and lapping operations in lens manufacturing,” Appl. Opt. |

7. | J. C. Dainty, |

8. | M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: a direct measurement of the density correlation function g(r),” Phys. Rev. Lett. |

9. | M. Giglio, M. Carpineti, A. Vailati, and D. Brogioli, “Near-field intensity correlations of scattered light,” Appl. Opt. |

10. | R. Cerbino, “Correlations of light in the deep Fresnel region: An extended Van Cittert and Zernike theorem,” Phys. Rev. A |

11. | D. Brogioli, A. Vailati, and M. Giglio, “Heterodyne near-field scattering,” Appl. Phys. Lett. |

12. | C. F. Cheng, M. Liu, N. Y. Zhang, S. Y. Teng, H. S. Song, and Z. Z. Xu, “Speckle intensity correlation in the diffraction region near rough surfaces and simulational experiments for extraction of surface parameters,” Europhys. Lett. |

13. | G. T. Liang, X. Li, M. N. Zhang, Z. H. Li, C. X. Liu, and C. F. Cheng, “Experimental extraction of rough surface parameters from speckles in the deep Fresnel region with a scanning fibre-optic probe,” Eur. Phys. J. D |

14. | J. A. Ogilvy, |

15. | J. C. Dainty, |

16. | J. W. Goodman, |

17. | Y. P. Zhao, G. C. Wang, and T. H. Lu, |

18. | D. P. Qi, D. L. Liu, S. Y. Teng, N. Y. Zhang, and C. F. Cheng, “Morphology analysis atomic force microscope and light scattering study for random scattering screens,” Acta Phys. Sin. |

19. | Y. P. Zhao, I. Wu, C. F. Cheng, U. Block, G. C. Wang, and T. M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. |

20. | C. F. Cheng, C. X. Liu, S. Y. Teng, N. Y. Zhang, and M. Liu, “Half-width of intensity profiles of light scattered from self-affine fractal random surfaces and simulational verifications,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

21. | M. N. Zhang, Z. H. Li, X. Y. Chen, G. T. Liang, S. Y. Wang, S. Y. Teng, and C. F. Cheng, “Evolutions of speckles on rough glass/silver surfaces with film thickness,” Opt. Express |

**OCIS Codes**

(030.6140) Coherence and statistical optics : Speckle

(120.5820) Instrumentation, measurement, and metrology : Scattering measurements

(290.0290) Scattering : Scattering

**ToC Category:**

Scattering

**History**

Original Manuscript: December 6, 2013

Revised Manuscript: January 4, 2014

Manuscript Accepted: January 5, 2014

Published: January 13, 2014

**Virtual Issues**

Vol. 9, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Chunxiang Liu, Qingrui Dong, Haixia Li, Zhenhua Li, Xing Li, and Chuanfu Cheng, "Measurement of surface parameters from autocorrelation function of speckles in deep Fresnel region with microscopic imaging system," Opt. Express **22**, 1302-1312 (2014)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-22-2-1302

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### References

- P. Meakin, Fractal, Scaling and Growth Far From Equilibrium (Cambridge University, 1998).
- P. Córdoba-Torres, T. J. Mesquita, I. N. Bastos, R. P. Nogueira, “Complex dynamics during metal dissolution: from intrinsic to faceted anomalous scaling,” Phys. Rev. Lett. 102(5), 055504 (2009). [CrossRef] [PubMed]
- S. Zhang, L. Wu, R. Y. Yue, Z. K. Yan, H. R. Zhan, Y. Xiang, “Effects of Sb-doping on the grain growth of Cu(In, Ga)Se2 thin films fabricated by means of single-target sputtering,” Thin Solid Films 527, 137–140 (2013). [CrossRef]
- J. G. Goodberlet, H. Kavak, “Patterning sub-50 nm features with near-field embedded-amplitude masks,” Appl. Phys. Lett. 81(7), 1315–1317 (2002). [CrossRef]
- K. Jang, Y. Ishibashi, D. Iwata, H. Suganuma, T. Yamada, Y. Takemura, “Fabrication of ferromagnetic nanoconstriction using atomic force microscopy nanoscratching,” J. Nanosci. Nanotechnol. 11(12), 10945–10948 (2011). [CrossRef] [PubMed]
- Ö. F. Farsakoğlu, D. M. Zengin, H. Kocabaş, “Grinding process for beveling and lapping operations in lens manufacturing,” Appl. Opt. 39(10), 1541–1548 (2000). [CrossRef] [PubMed]
- J. C. Dainty, Laser Speckle and Related Phenomena 2nd edition (Springer Verlag, 1984).
- M. Giglio, M. Carpineti, A. Vailati, “Space intensity correlations in the near field of the scattered light: a direct measurement of the density correlation function g(r),” Phys. Rev. Lett. 85(7), 1416–1419 (2000). [CrossRef] [PubMed]
- M. Giglio, M. Carpineti, A. Vailati, D. Brogioli, “Near-field intensity correlations of scattered light,” Appl. Opt. 40(24), 4036–4040 (2001). [CrossRef] [PubMed]
- R. Cerbino, “Correlations of light in the deep Fresnel region: An extended Van Cittert and Zernike theorem,” Phys. Rev. A 75(5), 053815 (2007). [CrossRef]
- D. Brogioli, A. Vailati, M. Giglio, “Heterodyne near-field scattering,” Appl. Phys. Lett. 81(22), 4109–4111 (2002). [CrossRef]
- C. F. Cheng, M. Liu, N. Y. Zhang, S. Y. Teng, H. S. Song, Z. Z. Xu, “Speckle intensity correlation in the diffraction region near rough surfaces and simulational experiments for extraction of surface parameters,” Europhys. Lett. 65(6), 779–784 (2004). [CrossRef]
- G. T. Liang, X. Li, M. N. Zhang, Z. H. Li, C. X. Liu, C. F. Cheng, “Experimental extraction of rough surface parameters from speckles in the deep Fresnel region with a scanning fibre-optic probe,” Eur. Phys. J. D 67, 030498 (2013).
- J. A. Ogilvy, Theory of Wave Scattering from Rough Surfaces (Adam Hilger, 1991).
- J. C. Dainty, The Statistics of Speckle Patterns, in Progress in Optics Vol. XIV, (North-Holland, 1976).
- J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Ben Roberts & Company, 2007).
- Y. P. Zhao, G. C. Wang, and T. H. Lu, Characterization of Amorphous and Crystalline Rough Surfaces: Principles and Applications (Academic, 2001).
- D. P. Qi, D. L. Liu, S. Y. Teng, N. Y. Zhang, C. F. Cheng, “Morphology analysis atomic force microscope and light scattering study for random scattering screens,” Acta Phys. Sin. 49(7), 1260–1266 (2000).
- Y. P. Zhao, I. Wu, C. F. Cheng, U. Block, G. C. Wang, T. M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84(5), 2571–2582 (1998). [CrossRef]
- C. F. Cheng, C. X. Liu, S. Y. Teng, N. Y. Zhang, M. Liu, “Half-width of intensity profiles of light scattered from self-affine fractal random surfaces and simulational verifications,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(6), 061104 (2002). [CrossRef] [PubMed]
- M. N. Zhang, Z. H. Li, X. Y. Chen, G. T. Liang, S. Y. Wang, S. Y. Teng, C. F. Cheng, “Evolutions of speckles on rough glass/silver surfaces with film thickness,” Opt. Express 21(7), 8831–8843 (2013). [CrossRef] [PubMed]

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