## Experimental measurement of the statistics of the scattered intensity from particles on surfaces

Optics Express, Vol. 10, Issue 3, pp. 190-195 (2002)

http://dx.doi.org/10.1364/OE.10.000190

Acrobat PDF (388 KB)

### Abstract

We analyze the statistics of the co-polarized and cross-polarized scattered intensity from a flat substrate contaminated with spherical particles including multiple scattering between them. Both Gaussian and non-Gaussian regimes are considered. In particular, we focus on both the cross-polarized component and the probability of detecting zeros in the cross-polarized intensity, P(I_{cross}=0). As it is shown, the latter gives information about particle interaction and can be measured with higher accuracy than other statistical parameters. A theoretical model for P(I_{cross}=0) is presented for spherical Rayleigh scatterers. An scattering experiment was carried out to verify this model.

© 2002 Optical Society of America

## 1. Introduction

*I*> / <

^{m}*I*>

^{m}=

*m*!. When this happens, the spatial fluctuations of the scattered light are commonly referred to as “Gaussian speckle” [1]. However, when the “effective” number of scattering centers is very low, the statistics of the scattered radiation presents clear differences with respect to the Gaussian case and it is common to find in the literature the terms “non-Gaussian” speckle or “non-Gaussian” statistics [3

3. D.W. Schaefer and P.N. Pusey, “Statistics of Non-Gaussian scattered light,” Phys. Rev. Lett. **29**, 843 (1972) [CrossRef]

4. An excellent review can be found in E. Jakeman and R.J.A. Tough, “Non-Gaussian models for the statistics of scattered waves,” Advances in Physics **37**, 471–529 (1988) [CrossRef]

5. J. Ohtsubo, “Non-Gaussian speckle: a computer simulation,” Appl. Opt. **21**, 4167 (1982) [CrossRef] [PubMed]

10. M. Itoh and K. Takahashi, “Measurement of aerosol particles by dynamic light-scttering. I: Effects of Non-Gaussian concentration fluctuation in real time photon correlation spectroscopy,” J. Aerosol Sci. **22**, 815 (1991) [CrossRef]

11. N. García and A.Z. Genack, “Crossover to strong intensity correlations for microwave radiation in random media,” Phys. Rev. Lett. **63**, 1678 (1989) [CrossRef] [PubMed]

15. A. García-Martín, J.J. Sáenz, and M. Nieto-Vesperinas, “Spatial field distributions in the transition from ballistic to diffusive transport in randomly corrugated waveguides,” Phys. Rev. Lett. **84**, 3578 (2000) [CrossRef] [PubMed]

16. A. García-Martín, R. Gómez-Medina, J.J. Sáenz, and M. Nieto-Vesperinas, “Finite-size effects in the spatial distribution of the intensity reflected from disordered media,” Phys. Rev. B **62**, 9386 (2000) [CrossRef]

17. J.A. Sánchez-Gil, M. Nieto-Vesperinas, F. Moreno, and F. González, “Speckle statistics of electromagnetic waves scattered from perfectly conducting random rough surfaces,” J. Opt. Soc. Am A **10**, 2628 (1993) [CrossRef]

18. T.R. Watts, K.I. Hopcraft, and T.R. Faulkner, “Single measurement of probability density functions and their use in non-Gaussian light scattering,” J. Phys. A: Math. Gen. **29**, 7501 (1996) [CrossRef]

19. A.P. Bates, K.I. Hopcraft, and E. Jakeman, “Non-Gaussian fluctuations of Stokes parameters in scattering by small particles”, Waves in Random Media **8**, 1 (1998) [CrossRef]

## 2. Theoretical model

_{cross}=0) as a function of both particle surface density (which we consider constant) and the illuminated area. The geometry of the problem consists on a flat surface that separates two semi-infinite media of dielectric constants ε

_{1}and ε

_{2}. The substrate is seeded with particles of spherical symmetry. The particle distribution obeys a Poisson distribution where the particle density,

*ρ*, is constant over the substrate. The surface is illuminated by a top-hat profile beam (only for simplicity in the calculation process) which produces either a circular illumination area of radius w for normal incidence or an elliptical one of semiaxes a and b for other incidences. It is assumed that the detection system has a threshold intensity I

_{0}, which is the minimum scattered intensity it can detect (this is in fact the experimental case) and it can be considered zero for calculation purposes. Our theoretical model is based on the previous assumptions and the following hypothesis: If a particle scatters a crosspolarized intensity I

_{0}(or less) when its nearest neighbor is at a distance L, the probability of detecting zero for the scattered cross-polarized intensity is the probability of having the nearest neighbor at a distance l larger than L and is therefore given by

*ρ*

^{-1/2}. Therefore, the probability of detecting zero cross-polarized scattered intensity when the mean number of illuminated particles is

*N*̅ (= either πw

^{2}

*ρ*, or πab

*ρ*) is

_{cross}=0) the parameter

*γ*can be determined and it can be used to estimate the parameter d if L is known and vice-versa.

## 3. Experiment

21. M. Harris, G.N. Pearson, C.A. Hill, and J.M. Vaughan, “The fractal character of Gaussian-Lorentzian light,” Appl. Opt. **33**, 7226 (1994). [CrossRef] [PubMed]

22. F. Moreno, F. González, J.M. Saiz, P.J. Valle, and D.L. Jordan, “Experimental study of copolarized light scattering by spherical light scattering on conducting flat substrates,” J. Opt. Soc. Am. A **10**, 141 (1993) [CrossRef]

4. An excellent review can be found in E. Jakeman and R.J.A. Tough, “Non-Gaussian models for the statistics of scattered waves,” Advances in Physics **37**, 471–529 (1988) [CrossRef]

_{cross}=0). At that point the function can be determined with the smallest relative error. For spherical scatterers this parameter can be understood as a measurement of particle interaction (it decreases as interaction increases and vice-versa). The evolution of the measured values of P(I

_{cross}=0) with the radius of the illuminated area, w, is shown in Fig.3 for six different illuminating spot sizes (w≅ 7.4, 11, 15, 16.5, 17, 26.6 μm). The circles represent the experimental data, with their corresponding error bars. The fit to Eq.(2) is plotted as a continuous line. As can be seen, the agreement is quite good.

*γ*can be obtained (

*γ*= 8.6 10

^{-3}μm

^{-2}), so we can obtain d if L is known and vice-versa.. In our case, we can calculate the mean interparticle distance by two different ways: 1) From the photographs taken with the electronic microscope, d is 7.7 μm approximately. This corresponds to a value of L≅1.25 μm. 2) From the second normalized moment of the fluctuations of the copolarized scattered intensity [4

4. An excellent review can be found in E. Jakeman and R.J.A. Tough, “Non-Gaussian models for the statistics of scattered waves,” Advances in Physics **37**, 471–529 (1988) [CrossRef]

## 4. Conclusions

_{cross}=0), has been introduced. It has been shown that this function decreases exponentially, with a factor that depends on i) the area of the illuminating spot, ii) the particle surface density and iii) a parameter we have denoted by L. This can be physically interpreted as an effective minimum interparticle distance necessary to observe non-zero cross-polarized intensity, and therefore multiple scattering. An experiment based on a heterodyne detection system has been performed in order to confirm the proposed model. From the discussion of the experimental results we can conclude that the parameter L/d

^{3}can be assessed from the experimental measurement of P(I

_{cross}=0). This can be used either for determining particle surface density of samples, all with the same individual scatterer characteristics (same L) or for determining the interacting level of different samples with well known particle surface density.

## Acknowledgments

## References and links

1. | J.C. Dainty (Ed.), |

2. | J.W. Goodman, |

3. | D.W. Schaefer and P.N. Pusey, “Statistics of Non-Gaussian scattered light,” Phys. Rev. Lett. |

4. | An excellent review can be found in E. Jakeman and R.J.A. Tough, “Non-Gaussian models for the statistics of scattered waves,” Advances in Physics |

5. | J. Ohtsubo, “Non-Gaussian speckle: a computer simulation,” Appl. Opt. |

6. | E. Jakeman, R.C. Klewe, P.H. Richards, and J.G. Walker, “Application of Non-Gaussian scattering of laser light to measurements in a propane flame,” J. Phys. D: Appl. Phys. |

7. | E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng. |

8. | B.M. Levine, “Non-Gaussian speckle caused by thin phase screens of large root-mean-square phase variations and long single-scale autocorrelations,” J. Opt. Soc. Am. A |

9. | H.M. Escamilla and E.R. Méndez “Speckle statistics from gamma-distributed random-phase screens,” J. Opt. Soc. Am. A |

10. | M. Itoh and K. Takahashi, “Measurement of aerosol particles by dynamic light-scttering. I: Effects of Non-Gaussian concentration fluctuation in real time photon correlation spectroscopy,” J. Aerosol Sci. |

11. | N. García and A.Z. Genack, “Crossover to strong intensity correlations for microwave radiation in random media,” Phys. Rev. Lett. |

12. | J.F. de Boer, M.C.W. van Rossum, M.P. van Albada, T.M. Nieuwenhuizen, and A. Lagendijk, “Diffusion of waves in a layer with a rough interface,” Phys. Rev. Lett. |

13. | S. Feng in |

14. | P. Sebbah, R. Pnini, and A.Z. Genack, “Field and intensity correlations in random media,” Phys. Rev. E |

15. | A. García-Martín, J.J. Sáenz, and M. Nieto-Vesperinas, “Spatial field distributions in the transition from ballistic to diffusive transport in randomly corrugated waveguides,” Phys. Rev. Lett. |

16. | A. García-Martín, R. Gómez-Medina, J.J. Sáenz, and M. Nieto-Vesperinas, “Finite-size effects in the spatial distribution of the intensity reflected from disordered media,” Phys. Rev. B |

17. | J.A. Sánchez-Gil, M. Nieto-Vesperinas, F. Moreno, and F. González, “Speckle statistics of electromagnetic waves scattered from perfectly conducting random rough surfaces,” J. Opt. Soc. Am A |

18. | T.R. Watts, K.I. Hopcraft, and T.R. Faulkner, “Single measurement of probability density functions and their use in non-Gaussian light scattering,” J. Phys. A: Math. Gen. |

19. | A.P. Bates, K.I. Hopcraft, and E. Jakeman, “Non-Gaussian fluctuations of Stokes parameters in scattering by small particles”, Waves in Random Media |

20. | F. Moreno and F. González Eds., |

21. | M. Harris, G.N. Pearson, C.A. Hill, and J.M. Vaughan, “The fractal character of Gaussian-Lorentzian light,” Appl. Opt. |

22. | F. Moreno, F. González, J.M. Saiz, P.J. Valle, and D.L. Jordan, “Experimental study of copolarized light scattering by spherical light scattering on conducting flat substrates,” J. Opt. Soc. Am. A |

23. | F. González, J.M. Saiz, P.J. Valle, and F. Moreno, “Multiple scattering in particulate surfaces: Cross-polarization ratios and shadowing effects,” Opt. Comm. |

**OCIS Codes**

(030.6600) Coherence and statistical optics : Statistical optics

(290.4210) Scattering : Multiple scattering

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 22, 2002

Revised Manuscript: February 7, 2002

Published: February 11, 2002

**Citation**

E. Ortiz, F. Gonz�lez, J. Saiz, and F. Moreno, "Experimental measurement of the statistics of the scattered intensity from particles on surfaces," Opt. Express **10**, 190-195 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-3-190

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### References

- J. C. Dainty (Ed.), Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1984)
- J. W. Goodman, Statistical Optics (Wiley, New York, 1985)
- D. W. Schaefer and P. N. Pusey, ?Statistics of Non-Gaussian scattered light,? Phys. Rev. Lett. 29, 843 (1972) [CrossRef]
- An excellent review can be found in E. Jakeman and R.J.A. Tough, ?Non-Gaussian models for the statistics of scattered waves,? Advances in Physics 37, 471-529 (1988) [CrossRef]
- J. Ohtsubo, ?Non-Gaussian speckle: a computer simulation,? Appl. Opt. 21, 4167 (1982) [CrossRef] [PubMed]
- E. Jakeman, R. C. Klewe, P. H. Richards and J. G. Walker, ?Application of Non-Gaussian scattering of laser light to measurements in a propane flame,? J. Phys. D: Appl. Phys. 17, 1941 (1984) [CrossRef]
- E. Jakeman, ?Speckle statistics with a small number of scatterers,? Opt. Eng. 23, 453 (1984)
- B. M. Levine, ?Non-Gaussian speckle caused by thin phase screens of large root-mean-square phase variations and long single-scale autocorrelations,? J. Opt. Soc. Am. A 3, 1283 (1986) [CrossRef]
- H. M. Escamilla and E. R. M?ndez ?Speckle statistics from gamma-distributed random-phase screens,? J. Opt. Soc. Am. A 8, 1929 (1991) [CrossRef]
- M. Itoh and K. Takahashi, ?Measurement of aerosol particles by dynamic light-scttering. I: Effects of Non-Gaussian concentration fluctuation in real time photon correlation spectroscopy,? J. Aerosol Sci. 22, 815 (1991) [CrossRef]
- N. Garc?a and A. Z. Genack, ?Crossover to strong intensity correlations for microwave radiation in random media,? Phys. Rev. Lett. 63, 1678 (1989) [CrossRef] [PubMed]
- J. F. de Boer, M. C. W. van Rossum, M. P. van Albada, T. M. Nieuwenhuizen and A. Lagendijk, ?Diffusion of waves in a layer with a rough interface,? Phys. Rev. Lett. 73, 2567 (1994) [CrossRef] [PubMed]
- S. Feng in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1990), pp. 179-206 [CrossRef]
- P. Sebbah, R. Pnini and A. Z. Genack, ?Field and intensity correlations in random media,? Phys. Rev. E 62, 7348 (2000) [CrossRef]
- A. Garc?a-Mart?n, J. J. S?enz and M. Nieto-Vesperinas, ?Spatial field distributions in the transition from ballistic to diffusive transport in randomly corrugated waveguides,? Phys. Rev. Lett. 84, 3578 (2000) [CrossRef] [PubMed]
- A. Garc?a-Mart?n, R. G?mez-Medina, J. J. S?enz and M. Nieto-Vesperinas, ?Finite-size effects in the spatial distribution of the intensity reflected from disordered media,? Phys. Rev. B 62, 9386 (2000) [CrossRef]
- J. A. S?nchez-Gil, M. Nieto-Vesperinas, F. Moreno and F. Gonz?lez, ?Speckle statistics of electromagnetic waves scattered from perfectly conducting random rough surfaces,? J. Opt. Soc. Am A 10, 2628 (1993) [CrossRef]
- T. R. Watts, K. I. Hopcraft and T. R. Faulkner, ?Single measurement of probability density functions and their use in non-Gaussian light scattering,? J. Phys. A: Math. Gen. 29, 7501 (1996) [CrossRef]
- A. P. Bates, K. I. Hopcraft and E. Jakeman, ?Non-Gaussian fluctuations of Stokes parameters in scattering by small particles,? Waves in RandomMedia 8, 1 (1998) [CrossRef]
- F. Moreno and F. Gonz?lez Eds., Light Scattering from Microestructures (Springer-Verlag, Berlin, 2000) [CrossRef]
- M. Harris, G. N. Pearson, C. A. Hill and J. M. Vaughan, ?The fractal character of Gaussian-Lorentzian light,? Appl. Opt. 33, 7226 (1994). [CrossRef] [PubMed]
- F. Moreno, F. Gonz?lez, J. M. Saiz, P. J. Valle and D. L. Jordan, ?Experimental study of copolarized light scattering by spherical light scattering on conducting flat substrates,? J. Opt. Soc. Am. A 10, 141 (1993) [CrossRef]
- F. Gonz?lez, J. M. Saiz, P. J. Valle and F. Moreno, ?Multiple scattering in particulate surfaces: Cross-polarization ratios and shadowing effects,? Opt. Comm. 137, 359 (1997). [CrossRef]

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