#### For citation:

Alonova M. V., Ushakova O. V., Zimnyakov D. A., Baiburin V. B. A Hybrid Approach in Modeling of Statistical Characteristics of Multiple Scattered Light. *Izvestiya of Sarat. Univ. Physics. *, 2018, vol. 18, iss. 4, pp. 242-252. DOI: 10.18500/1817-3020-2018-18-4-242-252

# A Hybrid Approach in Modeling of Statistical Characteristics of Multiple Scattered Light

Background and Objectives: A hybrid approach to modeling of the statistical characteristics of multiple scattered light in application to optical probes of random media is considered. The approach is based on recovery of the probability density of path lengths for partial components of a scattered light field in a probed medium using approximate analytical methods or the numerical simulation. Further, the statistical characteristics of scattered radiation (the temporal correlation function, the contrast, the average intensity, etc.) are calculated as the integral transforms of the recovered probability density function of the path lengths. Materials and Methods: The most appropriate approach for the recovery of the probability density function of the path length is the numerical solution of the radiative transfer equation using the statistical (Monte-Carlo) modeling of the temporal response of a probed medium by a short light pulse. The shape of the medium response can be easily transformed to the recovered probability density using the linear relationship between the path length and the propagation time for an arbitrarily chosen partial component of the multiply scattered light field in the medium. After the Monte-Carlo simulation of light pulse propagation in the medium, the frequency analysis of the accumulated path lengths of propagating photons is carried out. The number of bins used for estimates of the relative frequencies of detection of partial components with the path lengths in a given range must be chosen as a square root of the total number of accumulated components. Results: Application of this approach to optical diagnostics of relaxation processes in random media using the diffusing-wave spectroscopy is discussed. Viscoelastic relaxation of deformable model porous media saturated by near-critical carbon dioxide was examined using the technique of multi-speckle diffusing-wave spectroscopy. The relaxation is caused by a step-wise pressure drop in the system “porous layer – saturating agent” in the isothermal mode. 100 μm-thick layers of filter paper and Teflon with the fibrillar structure were used as the model samples in the experimental study. These layers were saturated by near-critical carbon dioxide and probed by He-Ne laser beam in the trans-illumination mode in the course of pressure drop. During the experiment, the examined layers were probed at various temperatures below and above the critical point of saturating carbon dioxide. The speckle dynamics in the small-angle forward scattered light was analyzed using principles of the multi-speckle diffusing-wave spectroscopy. The hybrid approach was applied for establishing the relationships between the correlation time of speckle intensity fluctuations and the average deformation rate of the relaxed probed samples. It has been found that the characteristic time of viscoelastic relaxation increases in the vicinity of the critical temperature of the saturating agent, whereas the correlation time falls down. These peculiarities can be explained in terms of remarkable increase of the compressibility of the saturating agent in the case of approaching to the critical point. Conclusion: The obtained results and discussed examples illustrate the efficiency of the hybrid approach to characterization of the structure and dynamics of complex media using the diffusing light technologies.

1. Pine D. J., Weitz D. A., Chaikin P. M., Herbolzheimer E. Diffusing wave spectroscopy. Phys. Rev. Lett., 1988, vol. 60, pp. 1134–1137.

2. Thompson C. A., Webb K. J., Weiner A. M. Imaging in scattering media by use of laser speckle. J. Opt. Soc. Am. A, 1997, vol. 14, pp. 2269–2277.

3. Zimnyakov D. A., Oh J.-T., Sinichkin Yu. P., Trifonov V. A., Gurianov E. V. Polarization-sensitive speckle spectroscopy of random media beyond the diffusion limit. J. Opt. Soc. Am. A, 2004, vol. 21, pp. 59–70.

4. Ishimaru A. Wave Propagation and Scattering in Random Media. New York, Wiley-IEEE, 1999. 600 p.

5. Zimnyakov D. A., Sina J. S., Yuvchenko S. A., Isaeva E. A., Chekmasov S. P., Ushakova O. V. Low-coherence interferometry as a method for assessing the transport parameters in randomly inhomogeneous media. Quantum Electronics, 2014, vol. 44, no. 1, pp. 59–64.

6. Twersky V. On propagation in random media of discrete scatterers. Proc. Symp. Appl. Math., 1964, vol. 16, pp. 84–116.

7. Foldy L. L. The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers. Physical Review, 1945, vol. 67, pp. 107–119.

8. Barabanenkov Yu. N. Multiple scattering of waves by ensembles of particles and the theory of radiation transport. Sov. Phys. Usp., 1975, vol.18, pp. 673–689.

9. Cummins H. Z., Pike E. R. Photon Correlation and Light Beating Spectroscopy. New York, Plenum Press, 1974. 504 p.

10. Zimnyakov D. A., Yuvchenko S. A., Pavlova M. V., Alonova M. V. Reference-free path length interferometry of random media with the intensity moments analysis. Optics Express, 2017, vol. 25, no. 13, pp. 13953–13972.

11. Novickij P. V., Zograf I. A. Ocenka pogreshnostej rezul’-tatov izmerenij [An estimate of measurement data errors]. Leningrad, Jenergoatomizdat, 1991. 304 p. (in Russian).

12. Zimnyakov D. A., Chekmasov S. P., Ushakova O. V., Isaeva E. A., Bagratashvili V. N., Yermolenko S. B. Laser speckle probes of relaxation dynamics in soft porous media saturated by near-critical fl uids. Applied Optics, 2014, vol. 53, no. 10, pp. B12–B21.

13. Viasnoff V., Lequeux F., Pine D. J. Multispeckle diffusingwave spectroscopy: A tool to study slow relaxation and time-dependent dynamics. Review of scientifi c instruments, 2002, vol. 73, no. 6, pp. 2336–2344.

14. Lemmon E. W., McLinden M. O., Friend D. G. Thermophysical Properties of Fluid Systems in NIST Chemistry WebBook, NIST Standard Reference Database Number 69. Eds. P. J. Linstrom, W. G. Mallard. National Institute of Standards and Technology. Gaithersburg MD, 20899, 2012. Available at: http://webbook.nist.gov/chemistry/fluid (accessed 11 June 2018).

15. Scherer G. W. Structure and properties of gels. Cement and Concrete Research, 1999, vol. 29, no. 8, pp. 1149–1157.

16. Scherer G. W. Dynamic pressurization method for measuring permeability and modulus: I. Theory. Materials and Structures, 2006, vol. 39, no. 10, pp. 1041–1057.

- 988 reads