## Effects of absorption on multiple scattering by random particulate media: exact results

Optics Express, Vol. 15, Issue 20, pp. 13182-13187 (2007)

http://dx.doi.org/10.1364/OE.15.013182

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### Abstract

We employ the numerically exact superposition *T*-matrix method to perform extensive computations of electromagnetic scattering by a volume of discrete random medium densely filled with increasingly absorbing as well as non-absorbing particles. Our numerical data demonstrate that increasing absorption diminishes and nearly extinguishes certain optical effects such as depolarization and coherent backscattering and increases the angular width of coherent backscattering patterns. This result corroborates the multiple-scattering origin of such effects and further demonstrates the heuristic value of the concept of multiple scattering even in application to densely packed particulate media.

© 2007 Optical Society of America

## 1. Introduction

1. S. H. Tseng, A. Taflove, D. Maitland, and V. Backman, “Pseudospectral time domain simulations of multiple light scattering in three-dimensional macroscopic random media,” Radio Sci. **41**, RS4009 (2006). [CrossRef]

4. M. I. Mishchenko, L. Liu, D. W. Mackowski, B. Cairns, and G. Videen, “Multiple scattering by random particulate media: exact 3D results,” Opt. Express **15**, 2822–2836 (2007). [CrossRef] [PubMed]

*N*, in a statistically homogeneous volume of discrete random medium. We argued that if specific scattering features either intensify or weaken with

*N*in an expected way [6, 7] then these features can be attributed to the increasing effect of multiple scattering. The particles in [4

4. M. I. Mishchenko, L. Liu, D. W. Mackowski, B. Cairns, and G. Videen, “Multiple scattering by random particulate media: exact 3D results,” Opt. Express **15**, 2822–2836 (2007). [CrossRef] [PubMed]

9. S. Etemad, R. Thompson, M. J. Andrejco, S. John, and F. C. MacKintosh, “Weak localization of photons: termination of coherent random walks by absorption and confined geometry,” Phys. Rev. Lett. **59**, 1420–1423 (1987). [CrossRef] [PubMed]

4. M. I. Mishchenko, L. Liu, D. W. Mackowski, B. Cairns, and G. Videen, “Multiple scattering by random particulate media: exact 3D results,” Opt. Express **15**, 2822–2836 (2007). [CrossRef] [PubMed]

## 2. Concept of multiple scattering

10. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, *Scattering, Absorption, and Emission of Light by Small Particles* (Cambridge U. Press, Cambridge, UK, 2002). http://www.giss.nasa.gov/~crmim/books.html.

11. F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer **79–80**, 775–824 (2003). [CrossRef]

**r**as a superposition of partial fields scattered by the individual particles, i.e.,

*N*is the number of particles in the group,

**E**(

**r**) is the total field,

**E**

^{inc}(

**r**) is the incident field, and

**E**

_{sca}

_{i}(

**r**) is the

*i*th partial scattered field. The partial scattered fields can be found by solving the vector Foldy–Lax equations (FLEs) [7]. Specifically, the

*i*th partial scattered field is given by

*V*is the volume occupied by the

_{i}*i*th particle,

**r**,

**r′**) is the free-space dyadic Green’s function, and

**E**

_{i}(

**r**′′) is the electric field “exciting” particle

*i*. The

*N*dyadics

*i*separately:

*k*

_{1}is the wave number in the host medium,

*m*(

_{i}**r**) is the relative refractive index, and

*i*with respect to the fixed laboratory coordinate system computed in the absence of all the other particles. In other words, the

*N*dyadic transition operators are totally independent of each other. However, the exciting fields are interdependent and must be found by solving the following system of

*N*linear integral equations:

*Ĝ*

*T̂*

_{i}*E*

_{inc}can be interpreted as the partial scattered field at the observation point generated by particle

*i*in response to the excitation by the incident field only,

*Ĝ*

*T̂*

_{i}*Ĝ*

*T̂*

_{j}*E*

_{inc}is the partial field generated by the same particle in response to the excitation caused by particle

*j*in response to the excitation by the incident field, etc. This order-of-scattering interpretation of Eq. (9) becomes even more transparent when the particles are widely separated, and single-scattering dyadics replace the dyadic transition operators as complete electromagnetic descriptors of the individual particles [7].

**15**, 2822–2836 (2007). [CrossRef] [PubMed]

## 3. Numerical results

*n*times to the total specific intensity is proportional to the

*n*th power of the single-scattering albedo. The single-scattering albedo is equal to unity for Im(

*m*) = 0 but can decrease significantly as Im(

*m*) increases, thereby suppressing various manifestations of multiple scattering. The radiative transfer theory and the concept of the single-scattering albedo may not be applicable directly to densely packed particles. Still, the various effects identified in [4

**15**, 2822–2836 (2007). [CrossRef] [PubMed]

*n*-th order of scattering would suffer

*n*times in succession from such a diminished response, Eq. (9).

*k*

_{1}

*R*= 40 and

*k*

_{1}

*r*= 4, respectively, which means that the particle volume concentration is equal to 16%. The real part of the particle refractive index is fixed at 1.32, while the imaginary part is varied from 0 to 0.3. To perform averaging over particle positions, we use only one randomly configured 160-particle group and average over all possible orientations of this configuration with respect to the laboratory coordinate system [4

**15**, 2822–2836 (2007). [CrossRef] [PubMed]

*T*-matrix method [10

10. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, *Scattering, Absorption, and Emission of Light by Small Particles* (Cambridge U. Press, Cambridge, UK, 2002). http://www.giss.nasa.gov/~crmim/books.html.

12. D. W. Mackowski and M. I. Mishchenko, “Calculation of the *T* matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A **13**, 2266–2278 (1996). [CrossRef]

13. D. Mackowski, K. Fuller, and M. Mishchenko, “Codes for calculation of scattering by clusters of spheres.” ftp://ftp.eng.auburn.edu/pub/dmckwski/scatcodes/index.html.

**n**̂

^{inc}(Fig. 1). The observation direction is specified by the unit vector

**n**̂

^{sca}. Since all scattering properties of the volume are averaged over the uniform orientation distribution of the multi-particle group, we can simplify the discussion by using the scattering plane for defining the Stokes parameters of the incident and scattered light. The transformation of the Stokes parameters in the far-field zone of the volume is then written in terms of the normalized Stokes scattering matrix [4

**15**, 2822–2836 (2007). [CrossRef] [PubMed]

*k*

_{1}

*r*= 4 and

*m*= 1.32 + i0.3.

*m*) above a certain threshold can cause decreasing rather than increasing absorption since the scatterer eventually behaves as a metallic object. Therefore, we have verified specifically that the single-scattering albedo of the entire scattering volume decreases and the total absorption cross section increases as Im(

*m*) is increased from 0 to 0.3.

## 4. Discussion

*a*

_{2}(Θ)

*a*

_{1}(Θ) is identically equal to unity for scattering by a single sphere. Therefore, the large deviation of this ratio from 100% for a scattering volume comprising non-absorbing particles was interpreted in [4

**15**, 2822–2836 (2007). [CrossRef] [PubMed]

*a*

_{2}(Θ)/

*a*

_{1}(Θ) from 100% decreases with increasing Im(

*m*) quite significantly, even though it does not completely vanish even for Im(

*m*) = 0.3 thereby revealing a residual influence of multiple scattering. The deviation of the ratio

*a*

_{3}(180°)

*a*

_{1}(180°) from –100% behaves quite similarly. With increasing absorption, the ratios

*a*

_{2}(Θ)/

*a*

_{1}(Θ),

*a*

_{3}(Θ)/

*a*

_{1}(Θ), and

*b*

_{2}(Θ)/

*a*

_{1}(Θ) for the entire scattering volume should be expected to approach those for a single constituent sphere. The left-hand panels of Fig. 2 clearly exhibit this tendency, although the residual multiple-scattering effect diminishes noticeably the large amplitude of the interference features typical of monodisperse spheres. The increasing amplitude of oscillations of the ratio

*b*

_{2}(Θ)/

*a*

_{1}(Θ) with increasing Im(

*m*) and the similar behavior of the ratio –

*b*

_{1}(Θ)

*a*

_{1}(Θ) (not shown) obviously represent what is known in astrophysics of planetary surfaces as the Umov effect.

**15**, 2822–2836 (2007). [CrossRef] [PubMed]

14. Y. Kuga and A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A **1**, 831–835 (1984). [CrossRef]

9. S. Etemad, R. Thompson, M. J. Andrejco, S. John, and F. C. MacKintosh, “Weak localization of photons: termination of coherent random walks by absorption and confined geometry,” Phys. Rev. Lett. **59**, 1420–1423 (1987). [CrossRef] [PubMed]

*μ*

_{L}and

*μ*

_{C}.

## 5. Conclusion

3. M. I. Mishchenko and L. Liu, “Weak localization of electromagnetic waves by densely packed many-particle groups: exact 3D results,” J. Quant. Spectrosc. Radiat. Transfer **106**, 616–621 (2007). [CrossRef]

**15**, 2822–2836 (2007). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | S. H. Tseng, A. Taflove, D. Maitland, and V. Backman, “Pseudospectral time domain simulations of multiple light scattering in three-dimensional macroscopic random media,” Radio Sci. |

2. | D. W. Mackowski, “Direct simulation of scattering and absorption by particle deposits,” Proc. IMECE |

3. | M. I. Mishchenko and L. Liu, “Weak localization of electromagnetic waves by densely packed many-particle groups: exact 3D results,” J. Quant. Spectrosc. Radiat. Transfer |

4. | M. I. Mishchenko, L. Liu, D. W. Mackowski, B. Cairns, and G. Videen, “Multiple scattering by random particulate media: exact 3D results,” Opt. Express |

5. | A. Penttilä and K. Lumme, “Coherent backscattering effects with Discrete Dipole Approximation method,” in |

6. | J. W. Hovenier, C. van der Mee, and H. Domke, Transfer of Polarized Light in Planetary Atmospheres – Basic Concepts and Practical Methods (Springer, Berlin, 2004). |

7. | M. I. Mishchenko, L. D. Travis, and A. A. Lacis, |

8. | H. C. van de Hulst, Multiple Light Scattering. Tables, Formulas, and Applications (Academic Press, San Diego, 1980). |

9. | S. Etemad, R. Thompson, M. J. Andrejco, S. John, and F. C. MacKintosh, “Weak localization of photons: termination of coherent random walks by absorption and confined geometry,” Phys. Rev. Lett. |

10. | M. I. Mishchenko, L. D. Travis, and A. A. Lacis, |

11. | F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer |

12. | D. W. Mackowski and M. I. Mishchenko, “Calculation of the |

13. | D. Mackowski, K. Fuller, and M. Mishchenko, “Codes for calculation of scattering by clusters of spheres.” ftp://ftp.eng.auburn.edu/pub/dmckwski/scatcodes/index.html. |

14. | Y. Kuga and A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A |

15. | L. Tsang and A. Ishimaru, “Backscattering enhancement of random discrete scatterers,” J. Opt. Soc. Am. A |

16. | V. L. Kuz’min and V. P. Romanov, “Coherent phenomena in light scattering from disordered systems,” Phys.-Uspekhi |

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(030.5620) Coherence and statistical optics : Radiative transfer

(290.4210) Scattering : Multiple scattering

(290.5850) Scattering : Scattering, particles

**ToC Category:**

Scattering

**History**

Original Manuscript: August 20, 2007

Manuscript Accepted: September 18, 2007

Published: September 26, 2007

**Virtual Issues**

Vol. 2, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Michael I. Mishchenko, Li Liu, and Joop W. Hovenier, "Effects of absorption on multiple scattering by random particulate media: exact results," Opt. Express **15**, 13182-13187 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-20-13182

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### References

- S. H. Tseng, A. Taflove, D. Maitland, and V. Backman, "Pseudospectral time domain simulations of multiple light scattering in three-dimensional macroscopic random media," Radio Sci. 41, RS4009 (2006). [CrossRef]
- D. W. Mackowski, "Direct simulation of scattering and absorption by particle deposits," Proc. IMECE 2006, 14615 (2006).
- M. I. Mishchenko and L. Liu, "Weak localization of electromagnetic waves by densely packed many-particle groups: exact 3D results," J. Quant. Spectrosc. Radiat. Transfer 106, 616-621 (2007). [CrossRef]
- M. I. Mishchenko, L. Liu, D. W. Mackowski, B. Cairns, and G. Videen, "Multiple scattering by random particulate media: exact 3D results," Opt. Express 15, 2822-2836 (2007). [CrossRef] [PubMed]
- A. Penttilä and K. Lumme, "Coherent backscattering effects with Discrete Dipole Approximation method," in Peer-Reviewed Abstracts of the Tenth Conference on Electromagnetic & Light Scattering, G. Videen, M. Mishchenko, M. P. Mengüç, and N. Zakharova, eds. (http://www.giss.nasa.gov/~crmim/, 2007), pp. 157-160.
- J. W. Hovenier, C. van der Mee, and H. Domke, Transfer of Polarized Light in Planetary Atmospheres - Basic Concepts and Practical Methods (Springer, Berlin, 2004).
- M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, Cambridge, UK, 2006).
- H. C. van de Hulst, Multiple Light Scattering. Tables, Formulas, and Applications (Academic Press, San Diego, 1980).
- S. Etemad, R. Thompson, M. J. Andrejco, S. John, and F. C. MacKintosh, "Weak localization of photons: termination of coherent random walks by absorption and confined geometry," Phys. Rev. Lett. 59, 1420-1423 (1987). [CrossRef] [PubMed]
- M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002). http://www.giss.nasa.gov/~crmim/books.html.
- F. M. Kahnert, "Numerical methods in electromagnetic scattering theory," J. Quant. Spectrosc. Radiat. Transfer 79-80, 775-824 (2003). [CrossRef]
- D. W. Mackowski and M. I. Mishchenko, "Calculation of the T matrix and the scattering matrix for ensembles of spheres," J. Opt. Soc. Am. A 13, 2266-2278 (1996). [CrossRef]
- D. Mackowski, K. Fuller, and M. Mishchenko, "Codes for calculation of scattering by clusters of spheres." ftp://ftp.eng.auburn.edu/pub/dmckwski/scatcodes/index.html.
- Y. Kuga and A. Ishimaru, "Retroreflectance from a dense distribution of spherical particles," J. Opt. Soc. Am. A 1, 831-835 (1984). [CrossRef]
- L. Tsang and A. Ishimaru, "Backscattering enhancement of random discrete scatterers," J. Opt. Soc. Am. A 1, 836-839 (1984). [CrossRef]
- V. L. Kuz’min and V. P. Romanov, "Coherent phenomena in light scattering from disordered systems," Phys.-Uspekhi 39, 231-260 (1996). [CrossRef]

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