## Multiple scattering by particles embedded in an absorbing medium. 1. Foldy—Lax equations, order-of-scattering expansion, and coherent field

Optics Express, Vol. 16, Issue 3, pp. 2288-2301 (2008)

http://dx.doi.org/10.1364/OE.16.002288

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### Abstract

This paper presents a systematic analysis of the problem of multiple scattering by a finite group of arbitrarily sized, shaped, and oriented particles embedded in an absorbing, homogeneous, isotropic, and unbounded medium. The volume integral equation is used to derive generalized Foldy—Lax equations and their order-of-scattering form. The far-field version of the Foldy—Lax equations is used to derive the transport equation for the so-called coherent field generated by a large group of sparsely, randomly, and uniformly distributed particles. The differences between the generalized equations and their counterparts describing multiple scattering by particles embedded in a non-absorbing medium are highlighted and discussed.

© 2008 Optical Society of America

## 1. Introduction

*et al.*[8

8. P. Yang, B.-C. Gao, W. J. Wiscombe, M. I. Mishchenko, S. E. Platnick, H.-L. Huang, B. A. Baum, Y. X. Hu, D. M. Winker, S.-C. Tsay, and S. K. Park, “Inherent and apparent scattering properties of coated or uncoated spheres embedded in an absorbing host medium,” Appl. Opt. **41**, 2740–2759 (2002). [CrossRef] [PubMed]

*et al.*[6

6. S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: diagrammatic expansion of the extinction coefficient,” J. Opt. Soc. Am. A **24**, 2943–2952 (2007). [CrossRef]

9. S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: full-wave Monte Carlo solutions of the extinction coefficient,” J. Opt. Soc. Am. A **24**, 2953–2962 (2007). [CrossRef]

10. M. I. Mishchenko, “Electromagnetic scattering by a fixed finite object embedded in an absorbing medium,” Opt. Express **15**, 13188–13202 (2007). [CrossRef] [PubMed]

10. M. I. Mishchenko, “Electromagnetic scattering by a fixed finite object embedded in an absorbing medium,” Opt. Express **15**, 13188–13202 (2007). [CrossRef] [PubMed]

10. M. I. Mishchenko, “Electromagnetic scattering by a fixed finite object embedded in an absorbing medium,” Opt. Express **15**, 13188–13202 (2007). [CrossRef] [PubMed]

## 2. Vector Foldy—Lax equations

*N*finite particles collectively occupying the interior region

*V*is the volume occupied by the ith particle (Fig. 1). The host medium can be absorbing, but otherwise it is assumed to be infinite, homogeneous, linear, and isotropic. The particles are assumed to have the same constant permeability, but are allowed to have different and spatially varying permittivities. The general volume integral equation describing the total electric field everywhere in space [10

_{i}**15**, 13188–13202 (2007). [CrossRef] [PubMed]

**E**

^{inc}(

**r**) is the incident field, the integration is performed over the entire space,

*k*

_{1}=

*k*′

_{1}+i

*k*″

_{1}is the (complex) wave number of the host medium,

*I⃡*is the identity dyadic,

*U*(

_{i}**r**) is the

*i*th-particle potential function. The latter is given by

*m*(

_{i}**r**)=

*k*

_{2i}(

**r**)/

*k*

_{1}is the refractive index of particle

*i*relative to that of the host medium. Importantly, all position vectors originate at the origin

*O*of the laboratory coordinate system, Fig. 1.

**E**

*(*

_{i}**r**) exciting particle

*i*is given by

**E**

^{exc}

*(*

_{ij}**r**) are partial exciting fields given by

*T⃡*is the solution of the integral equation

_{i}*i*th-particle dyadic transition operator with respect to the laboratory coordinate system.

*N*particles embedded in an absorbing medium. A fundamental property of the FLEs is that

*T⃡*is the dyadic transition operator of particle i in the absence of all the other particles (cf. Eq. (9) above and Eq. (10) in [10

_{i}**15**, 13188–13202 (2007). [CrossRef] [PubMed]

## 3. Multiple scattering

**15**, 13188–13202 (2007). [CrossRef] [PubMed]

*T⃡*for each

_{i}*i*is an individual property of the

*i*th particle computed as if this particle were alone allows one to introduce the mathematical concept of multiple scattering. Let us rewrite Eqs. (6)–(8) in a compact operator form:

*ĜT̂*dyadic according to Eq. (12), and the dashed curve indicates that both “scattering centers” are represented by the same particle.

## 4. Far-field Flody–Lax equations

- Each particle from the group is located in the far-field zones of all the other particles.
- The observation point is located in the far-field zone of any particle from the group.

**15**, 13188–13202 (2007). [CrossRef] [PubMed]

*j*in response to the incident field represented by

**E**

*(*

_{j}**r**). Since the resulting scattered field at any point is independent of the choice of coordinate system, it is convenient to evaluate the right-hand side of Eq. (8) in the far-field zone of particle

*j*using the local coordinate system centered at

*O*. This means that now the dyadic Green’s function, the dyadic transition operator, and the incident field

_{j}**E**

*are specified with respect to the*

_{j}*j*th local particle coordinate system. According to Section 3 of [10

**15**, 13188–13202 (2007). [CrossRef] [PubMed]

*O*:

_{j}**r**,

**r**

*,*

_{j}**R**

*,*

_{i}**R**

*and*

_{j}**R**

*are shown in Fig. 3(a). Note that we use a caret above a vector to denote a unit vector in the corresponding direction.*

_{ij}**E**

*in Eq. (16) is the value of the exciting field caused by particle*

_{ij}*j*at the origin of particle

*i*. Since the radius of curvature of the exciting wavelet generated by particle

*j*is much greater than the size of particle

*i*, Eqs. (7) and (16) show that each particle is excited by the external incident field and the superposition of

*locally*plane homogeneous waves with amplitudes exp(-i

*k*

_{1}

**RȂ**

*·*

_{ij}**R**

*)*

_{i}**E**

*and propagation directions*

_{ij}**R**̂

*:*

_{ij}**E**

^{inc}(

**r**) is a homogeneous plane electromagnetic wave propagating in the direction of the unit vector

**n̂**

^{inc}:

**15**, 13188–13202 (2007). [CrossRef] [PubMed]

*j*th particle in response to a plane-wave excitation of the form

**E**

^{inc}

_{0}exp(i

*k*

_{1}

**n̂**

^{inc}·

**r**

*) is given by*

_{j}*G*(

*r*)

_{j}*A⃡*

_{j}(

**r̂**

*,*

_{j}**n̂**

^{inc})·

**E**

^{inc}

_{0}, where

**r**

*originates at*

_{j}*O*,

_{j}**E**

^{inc}

_{0}is the incident field at

*O*, and

_{j}*A⃗*

*(*

_{j}**r̂**

*,*

_{j}**n̂**

^{inc}) is the

*j*th particle scattering dyadic centered at

*O*. To make use of this fact, we rewrite Eq. (21) for particle

_{j}*j*with respect to the

*j*th-particle coordinate system centered at

*O*, Fig. 3(a). Since

_{j}**r**=

**r**

*+*

_{j}**R**

*, we obtain*

_{j}*O*caused by particle

_{i}*j*in response to this excitation is given by

**r**=

**R**

*, we finally obtain a system of linear algebraic equations for the partial exciting fields*

_{i}**E**

*:*

_{ij}**r**″∊

*V*:

_{i}**r**, Fig. 3(b), is located in the far-field zone of any particle forming the group.

*B⃡*dyadic.

## 5. The Twersky approximation

*N*particles sparsely distributed throughout a finite macroscopic volume

*V*, Fig. 4. Assuming that

*N*is very large, we can keep in the far-field order-of-scattering expansion (28) only the terms corresponding to scattering paths going through a particle only once (so-called self-avoiding paths) [11]:

## 6. Coherent field

*N*particles are randomly moving and decompose the field

**E**(

**r**) at an internal point

**r**∊

*V*into the average (or coherent),

**E**

_{c}(

**r**), and fluctuating,

**E**

_{f}(

**r**), parts:

**R**) and states (subscript

*ξ*), we have

**0**is a zero vector. Furthermore, if all particles have the same statistical characteristics and the state and coordinates of each particle are independent of each other then we have from Eq. (34):

*p*

_{R}(

**R**) and

*p*(

_{ξ}*ξ*) are the corresponding probability density functions, and the spatial integrations are performed over the entire volume

*V*. Substituting Eqs. (30)–(33) yields

*A⃡*(

**m̂**,

**n̂**)〉

*ξ*is the average of the single-particle scattering dyadic over the particle states. Taking into account that

*p*

_{R}(

**R**)=

*n*

_{0}(

**R**)/

*N*, where

*n*

_{0}(

**R**) is the number of particles per unit volume, we finally derive in the limit

*N*→∞:

*V*is statistically uniform and introduce an s-axis parallel to the incidence direction and going through the observation point. This axis enters the volume

*V*at a point

*A*such that

*s*(

*A*)=0 and exits it at a point

*B*(Fig. 6). Let us evaluate the first integral on the right-hand side of Eq. (40):

*n*

_{0}=

*N/V*. The observation point is assumed to be in the far-field zone of any particle, which allows the use of the Saxon asymptotic expansion of a plane wave in spherical waves:

*z*-axis directed along the

*s*-axis. This gives

*V*are not perfectly fixed and can be expected to fluctuate during the time interval necessary to compute the coherent field according to Eq. (36). Averaging over these fluctuations does not affect the first term on the right-hand side of Eq. (43), but effectively extinguishes the second term proportional to the rapidly oscillating exponential exp{i2

*k*

_{1}[

*s*(

*B*)-

*s*(

**r**)]}. Taking the average of the coherent field over a small volume element centered at the observation point

**r**would have the same effect. Hence,

**R**

*=*

_{j}**r**+

**R**′

*+*

_{i}**R**

*, we have for the second integral on the right-hand side of Eq. (40):*

_{ji}*s*-axis contribute to

**I**

_{2}. Consequently,

**E**

_{c}(

*s*=0)=

**E**

^{inc}(

**r**

*) is the boundary value of the coherent field. Another form of Eq. (48) is*

_{A}**E**

_{c}(

**r**)·

**n̂**

^{inc}=0. Therefore, the electric vector of the coherent field can be written as the vector sum of the corresponding

*θ*- and

*φ*-components in the local coordinate system centered at the observation point:

**k**(

**n̂**

^{inc}) is the 2×2 matrix propagation constant with elements

**S**(

**n̂**

^{inc},

**n̂**

^{inc})〉

*is the forward-scattering amplitude matrix averaged over the particle states.*

_{ξ}**15**, 13188–13202 (2007). [CrossRef] [PubMed]

## 7. Transfer equation for the coherent field

*ω*is the angular frequency and µ

_{1}is the permeability of the host medium. As follows from Eqs. (56) and (61),

**J**

_{c}satisfies the transfer equation

**K**

*is the coherency extinction matrix given by Eq. (68) of [10*

^{J}**15**, 13188–13202 (2007). [CrossRef] [PubMed]

**K**is the Stokes extinction matrix given by Eqs. (71)–(78) of [10

**15**, 13188–13202 (2007). [CrossRef] [PubMed]

**15**, 13188–13202 (2007). [CrossRef] [PubMed]

**△**

_{3}=diag[1, 1, -1, 1].

## 8. Discussion

12. L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. **67**, 107–119 (1945). [CrossRef]

6. S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: diagrammatic expansion of the extinction coefficient,” J. Opt. Soc. Am. A **24**, 2943–2952 (2007). [CrossRef]

*k*

_{1}is replaced by

*k*′

_{1}. Furthermore, Eqs. (67), (69), and (71) contain additional, intuitively obvious terms which are proportional to

*k*″

_{1}and describe additional exponential attenuation due to true absorption of electromagnetic energy by the host medium. Importantly, these results have been derived directly from the Maxwell equations and involve no phenomenological assumptions or hypotheses.

*k*″

_{1}on the coherent propagation of an electromagnetic wave through a turbid medium is two-fold. First, it modifies the numerical values of the ensemble-averaged extinction matrix elements. Second, it causes an additional exponential-attenuation factor exp(-2

*k*″

_{1}

*s*). There is no doubt that the second manifestation of a non-vanishing absorptivity of the host medium is much more important than the first one since it affects directly the long-range transport of electromagnetic energy.

*k*

_{1}

*α*) (with a real-valued

*α*) as a product of a real-valued exponential exp(-

*k*″

_{1}

*α*) and a “purely complex” exponential exp(i

*k*′

_{1}

*α*) with a real-valued

*k*′

_{1}

*α*. It is important to remember that mathematical results such as the Jones lemma, the method of stationary phase, or the Saxon expansion of a plane wave in spherical waves [3, 13] are applicable only to situations involving purely complex exponentials of the type exp(i

*k*′

_{1}

*α*) with a real-valued

*k*′

_{1}

*α*. We note in this regard that Eq. (61), when applied to the case of spherical particles, appears to be inconsistent with Eq. (55) of [6

6. S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: diagrammatic expansion of the extinction coefficient,” J. Opt. Soc. Am. A **24**, 2943–2952 (2007). [CrossRef]

*k*′

_{1}rather than

*k*

_{1}. Although the origin of this discrepancy is not immediately obvious, it is likely to be the same as that of the discrepancy discussed in the penultimate paragraph of [10

**15**, 13188–13202 (2007). [CrossRef] [PubMed]

## Appendix

**15**, 13188–13202 (2007). [CrossRef] [PubMed]

## Acknowledgments

## References

1. | F. T. Ulaby and C. Elachi, eds., |

2. | L. Tsang and J. A. Kong, |

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

4. | P. A. Martin, |

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

6. | S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: diagrammatic expansion of the extinction coefficient,” J. Opt. Soc. Am. A |

7. | M. I. Mishchenko, “ |

8. | P. Yang, B.-C. Gao, W. J. Wiscombe, M. I. Mishchenko, S. E. Platnick, H.-L. Huang, B. A. Baum, Y. X. Hu, D. M. Winker, S.-C. Tsay, and S. K. Park, “Inherent and apparent scattering properties of coated or uncoated spheres embedded in an absorbing host medium,” Appl. Opt. |

9. | S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: full-wave Monte Carlo solutions of the extinction coefficient,” J. Opt. Soc. Am. A |

10. | M. I. Mishchenko, “Electromagnetic scattering by a fixed finite object embedded in an absorbing medium,” Opt. Express |

11. | V. Twersky, “On propagation in random media of discrete scatterers,” Proc. Symp. Appl. Math. |

12. | L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. |

13. | L. Mandel and E. Wolf, |

**OCIS Codes**

(030.5620) Coherence and statistical optics : Radiative transfer

(290.5850) Scattering : Scattering, particles

(290.5825) Scattering : Scattering theory

(290.5855) Scattering : Scattering, polarization

**ToC Category:**

Scattering

**History**

Original Manuscript: January 14, 2008

Revised Manuscript: January 24, 2008

Manuscript Accepted: January 29, 2008

Published: February 1, 2008

**Citation**

Michael I. Mishchenko, "Multiple scattering by particles embedded in an absorbing medium. 1. Foldy–Lax equations, order-of-scattering expansion, and coherent field," Opt. Express **16**, 2288-2301 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-3-2288

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### References

- F. T. Ulaby and C. Elachi, eds., Radar Polarimetry for Geoscience Applications (Artech House, Norwood, Mass., 1990).
- L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, New York, 2001).
- 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).
- P. A. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles (Cambridge U. Press, Cambridge, UK, 2006). [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]
- S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, "Light scattering by a random distribution of particles embedded in absorbing media: diagrammatic expansion of the extinction coefficient," J. Opt. Soc. Am. A 24, 2943-2952 (2007). [CrossRef]
- M. I. Mishchenko, "Multiple scattering, radiative transfer, and weak localization in discrete random media: the unified microphysical approach," Rev. Geophys. 46, doi:10.1029/2007RG000230.
- P. Yang, B.-C. Gao, W. J. Wiscombe, M. I. Mishchenko, S. E. Platnick, H.-L. Huang, B. A. Baum, Y. X. Hu, D. M. Winker, S.-C. Tsay, and S. K. Park, "Inherent and apparent scattering properties of coated or uncoated spheres embedded in an absorbing host medium," Appl. Opt. 41, 2740-2759 (2002). [CrossRef] [PubMed]
- S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, "Light scattering by a random distribution of particles embedded in absorbing media: full-wave Monte Carlo solutions of the extinction coefficient," J. Opt. Soc. Am. A 24, 2953-2962 (2007). [CrossRef]
- M. I. Mishchenko, "Electromagnetic scattering by a fixed finite object embedded in an absorbing medium," Opt. Express 15, 13188-13202 (2007). [CrossRef] [PubMed]
- V. Twersky, "On propagation in random media of discrete scatterers," Proc. Symp. Appl. Math. 16, 84-116 (1964).
- L. L. Foldy, "The multiple scattering of waves," Phys. Rev. 67, 107-119 (1945). [CrossRef]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

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