## Electromagnetic scattering by a fixed finite object embedded in an absorbing medium

Optics Express, Vol. 15, Issue 20, pp. 13188-13202 (2007)

http://dx.doi.org/10.1364/OE.15.013188

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### Abstract

This paper presents a general and systematic analysis of the problem of electromagnetic scattering by an arbitrary finite fixed object embedded in an absorbing, homogeneous, isotropic, and unbounded medium. The volume integral equation is used to derive generalized formulas of the far-field approximation. The latter serve to introduce direct optical observables such as the phase and extinction matrices. The differences between the generalized equations and their counterparts describing electromagnetic scattering by an object embedded in a non-absorbing medium are discussed.

© 2007 Optical Society of America

## 1. Introduction

15. M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium,” Opt. Express (in preparation). [PubMed]

15. M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium,” Opt. Express (in preparation). [PubMed]

15. M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium,” Opt. Express (in preparation). [PubMed]

13. 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.

13. 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.

## 2. Volume integral equation

*V*

_{INT}and is surrounded by the infinite exterior region

*V*

_{EXT}such that

*V*

_{INT}∪

*V*

_{EXT}= ℜ

^{3}, where ℜ

^{3}denotes the entire three-dimensional space. The interior region is filled with an isotropic, linear, and possibly inhomogeneous material. The scatterer can be either a single body or a cluster with touching and/or separated components. Point

*O*serves as the common origin of all position vectors

**r**and as the origin of the laboratory coordinate system (Fig. 1).

^{1/2},

**E**is the electric field,

**H**is the magnetic field,

*ω*is the angular frequency, μ

_{1}and ε

_{1}are the permeability and complex permittivity of the host medium, and μ

_{2}and ε

_{2}(

**r**) are the permeability and complex permittivity of the scattering object. Note that ε

_{2}is allowed to vary throughout the scattering object, whereas ε

_{1}, μ

_{1}, and μ

_{2}are assumed to be arbitrary but constant. Since the first relations in Eqs. (1) and (2) yield the magnetic field provided that the electric field is known everywhere, we will look for the solution of Eqs. (1) and (2) in terms of only the electric field. The latter satisfies the following vector wave equations:

*k*

_{1}=

*ω*[ε

_{1}μ

_{1}]

^{1/2}and

*k*

_{2}(

**r**) =

*ω*[ε

_{2}(

**r**)μ

_{2}]

^{1/2}, are, in general, complex-valued. It is convenient for our purposes to represent the wave number of the host medium in terms of its real and imaginary parts:

*k*

_{1}=

*k*

_{1}́ +

*ik*

_{1}″, where

*k*

_{1}´>0 and

*k*″

_{1}≥0. Equations (3) and (4) can be rewritten as a single inhomogeneous differential equation

*m*(

**r**) is the refractive index of the interior relative to that of the exterior. The complex wave numbers and the relative refractive index are, in general, frequency-dependent. It follows from Eq. (6) that the forcing function

**j**(

**r**) vanishes everywhere outside the interior region.

13. 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.

**E**

^{inc}(

**r**) is the incident field,

^{2}+

*k*

_{1}

^{2})

*g*(

**r**,

**r**´) = - δ(

**r**-

**r**´) with a complex

*k*

_{1}is

*g*(

**r**,

**r**´) = exp(

*ik*

_{1}∣

**r**-

**r**´´)/4

*π*∣

**r**-

**r**´∣ [18

18. E. J. Rothwell and M. J. Cloud, *Electromagnetics* (CRC Press, Boca Raton, Florida, 2001). [CrossRef]

**r**-

*r*´) is the three-dimensional delta function. The second term on the right-hand side of Eq. (8) satisfies the requisite Sommerfeld radiation condition at infinity [18

18. E. J. Rothwell and M. J. Cloud, *Electromagnetics* (CRC Press, Boca Raton, Florida, 2001). [CrossRef]

**E**

^{sca}(

**r**). The latter can be expressed in terms of the incident field as follows:

*Scattering, Absorption, and Emission of Light by Small Particles* (Cambridge U. Press, Cambridge, UK, 2002). http://www.giss.nasa.gov/~crmim/books.html.

## 3. Scattering in the far-field zone

*O*close to the geometrical center of the scattering object and assume that the distance

*r*from

*O*to the observation point

**r**(Fig. 1) is much greater than any linear dimension of the scattering object:

*r*= |

**r**| and

*r*´ = |

**r**´|. We also assume that

*r*but also by the exponential absorption factor exp(-

*k*

_{1}″

*r*).

*homogeneous*plane electromagnetic wave given by

**E**

^{inc}

_{0}is the electric field at the origin of the laboratory coordinate system. We then have

**n̂**

^{sca}=

**r̂**is the scattering direction (see Fig. 1),

*A↔*is the scattering dyadic such that

**0**is a zero vector. The expression for the scattering dyadic in terms of the dyadic transition operator follows from Eqs. (9) and (20):

*r*to the observation point. However,

*O*with respect to the scattering object. The dependence of the scattering dyadic on the absorption properties of the host medium enters through a non-zero value of

*k*

_{1}″ in Eqs. (21) and (10) and through the relative refractive index in Eq. (10).

**S**describing the transformation of the

*θ*- and

*φ*-components of the electric field vector of the incident plane wave into those of the scattered spherical wave:

**E**denotes a two-element column formed by the

*θ*- and

*φ*-components of the electric field vector:

*Scattering, Absorption, and Emission of Light by Small Particles* (Cambridge U. Press, Cambridge, UK, 2002). http://www.giss.nasa.gov/~crmim/books.html.

3. 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]

23. C. F. Bohren and D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. **72**, 215–221 (1979). [CrossRef]

## 4. Electromagnetic power

**n̂**

^{inc}. Polarization-sensitive detectors are located in the far-field zone of the object. The sensitive area of each detector is modeled as a plane circular surface element

*S*normal to and centered at a position vector

**r**. Each detector is assumed to be well collimated, which means that electromagnetic energy incident on any point of the respective sensitive area

*S*is detected only if the corresponding propagation direction falls within a narrow acceptance solid angle

*Ω*centered around

**r̂**. I also assume that the diameter of the sensitive area

*S*is significantly greater than any linear dimension

*a*of the scattering object:

*D*≪

*a*. This will ensure that the right-hand side of Eq. (37) below is positive, thereby making possible a meaningful measurement of extinction (see also [25

25. M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Extinction and the electromagnetic optical theorem,” 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. 9–12.

*S*propagates in approximately the same direction. This is equivalent to requiring that

*r*≫

*D*/2. Furthermore, it is assumed that the solid angle subtended by the sensitive area as viewed from the scattering object is smaller than the detector angular aperture:

*S*/

*r*

^{2}<

*Ω*. This ensures that all radiation scattered by the object in radial directions and impinging on

*S*is detected. Detector 1 in Fig. 2 is centered at the incidence direction, whereas detector 2 is oriented such that the incidence direction does not fall within its acceptance solid angle:

**n̂**

^{inc}∉

*Ω*

_{2}.

*(*

**S****r**´,

*t*)〉

_{t}at any point of the sensitive surface of a detector is the sum of three terms:

**r**´ =

**r**´

**r**´ is the corresponding position vector,

Ω ˜

_{l}is the solid angle centered around the direction

**n̂**

^{inc}and subtended by the detector 1 surface at the distance

*r*from the particle.

*S*and describes attenuation caused by interposing the object between the light source and the detector. Thus, the detector centered at the exact forward-scattering direction measures the power of the incident light attenuated by the interference of the incident and scattered fields as well as a relatively small contribution from the scattered light. The detector centered at any other direction registers only the scattered light.

## 5. Backscattering interference

**n̂**

^{inc}+

**r̂**´) in Eqs. (32) and (33) makes it instructive to also consider the flow of electromagnetic energy through a surface element S3 normal to the backscattering direction and centered at the position vector -

*r*

**n̂**

^{inc}. The orientation of the surface element is given by the unit vector -

**n̂**

^{inc}. It is straightforward to derive that the corresponding total electromagnetic power is given by

*S*

_{3}and is negative since the direction of the energy flow is opposite to the orientation of the surface element. The absolute value of this term increases exponentially with distance from the scattering object. The second term is the power backscattered by the object. The third term has no monotonous independence on

*r*and describes the interference of the incident and backscattered waves [26]; it vanishes if the host medium is non-absorbing [27

27. M. I. Mishchenko, “The electromagnetic optical theorem revisited,” J. Quant. Spectrosc. Radiat. Transfer **101**, 404–410 (2006). [CrossRef]

*k*

_{1}´(

*r*), does not cause a long-range transport of electromagnetic energy. However, there is a stationary local transport of energy within wavelength-long elementary volume elements centered at the straight line extending from the origin of the coordinate system in the direction -

**n̂**

^{inc}. Since the solution of the frequency-domain scattering problem is time-independent, there may not be infinite accumulation of energy at any point in the medium. Therefore, energy transported from one point of a wavelength-long volume element to another must be absorbed by the host medium. This means that the backscattering interference causes additional absorption of electromagnetic energy at points along the straight line extending in the exact backscattering direction.

*k*

_{1}″

*r*). In a weakly absorbing medium, the third term can usually be neglected because Im (

*k*

_{1}/μ

_{1}) is much smaller than Re (

*k*

_{1}/μ

_{1}).

## 6. Phase matrix

**J**, and Stokes column vectors,

**I**, as follows:

*I*, gives the total intensity of a wave. It is straightforward to show that

*S*(

_{ij}**n̂**

^{sca},

**n̂**

^{inc}) as follows:

**I**

^{inc}is the Stokes column vector of the incident wave at the origin of the laboratory coordinate system.

## 7. Extinction matrix

**r̂**=

**n̂**

^{inc}). Because now both the incident homogeneous plane wave and the scattered outgoing spherical wave propagate in the same direction and are transverse, their superposition is also a transverse wave propagating in the forward direction. Therefore, we can define the coherency column vector of the total field for propagation directions

**r̂**very close to

**n̂**

^{inc}as follows:

**J**(

*r*

**r̂**) over the surface of the collimated detector aligned normal to

**n̂**

^{inc}and using Eqs. (32) and (34), we derive for the coherency-vector representation of the polarized signal recorded by detector 1 in Fig. 2:

**Z**

^{J}(

**n̂**

^{inc},

**n̂**

^{inc}) is the forward-scattering coherency phase matrix, and the elements of the 4×4 coherency extinction matrix

**K**

^{J}(

**n̂**

^{inc}) are expressed in terms of the elements of the forward-scattering amplitude matrix

**S**(

**n̂**

^{inc},

**n̂**

^{inc}) as follows:

**K**(

**n̂**

^{inc}) is given by

**S**(

**n̂**

^{inc},

**n̂**

^{inc}) are as follows:

*r*to the observation point.

## 8. Reciprocity and backscattering symmetry

*Scattering, Absorption, and Emission of Light by Small Particles* (Cambridge U. Press, Cambridge, UK, 2002). http://www.giss.nasa.gov/~crmim/books.html.

*Scattering, Absorption, and Emission of Light by Small Particles* (Cambridge U. Press, Cambridge, UK, 2002). http://www.giss.nasa.gov/~crmim/books.html.

*k*

_{1}with

*k*

_{1}´. Therefore, the resulting reciprocity relations for the scattering dyadic [29

29. D. S. Saxon, “Tensor scattering matrix for the electromagnetic field,” Phys. Rev. **100**, 1771–1775 (1955). [CrossRef]

## 9. Discussion

*Scattering, Absorption, and Emission of Light by Small Particles* (Cambridge U. Press, Cambridge, UK, 2002). http://www.giss.nasa.gov/~crmim/books.html.

_{1}= Imμ

_{1}=

*k*″

_{1}= 0 and

*k*´

_{1}=

*k*

_{1}.

*T↔*is independent of the specific location and orientation of the scattering object with respect to the incident field. It depends, however, on the position and orientation of the object with respect to the laboratory coordinate system.

*k*″

_{1}remains sufficiently small.

*S*

_{11}(

**n̂**

^{inc},

**n̂**

^{inc}) =

*S*

_{22}(

**n̂**

^{inc},

**n̂**

^{inc}) ≡

*S*

_{11}(0°) and

*S*

_{12}(

**n̂**

^{inc},

**n̂**

^{inc}) =

*S*

_{21}(

**n̂**

^{inc},

**n̂**

^{inc}) ≡ 0. Therefore, the third term on the right-hand side of Eq. (69) reduces to –exp(-2

*k*″

_{1}

*r*)

*C*

_{ext}

**I**

^{inc}, where the spherical-particle extinction cross section is given by

23. C. F. Bohren and D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. **72**, 215–221 (1979). [CrossRef]

4. G. Videen and W. Sun, “Yet another look at light scattering from particles in absorbing media,” Appl. Opt. **42**, 6724–6727 (2003). [CrossRef] [PubMed]

23. C. F. Bohren and D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. **72**, 215–221 (1979). [CrossRef]

*k*

_{1}, Eq. (11) of [23

**72**, 215–221 (1979). [CrossRef]

*C*

_{ext}= 4

*π*Im[

*S*

_{11}(0°)/

*k*

_{1}], which is different from Eq. (87) above. The origin of the difference can be traced back to Eqs. (6) and (7) of [23

**72**, 215–221 (1979). [CrossRef]

**72**, 215–221 (1979). [CrossRef]

**n̂**

^{inc}appears to be unlimited. Giving the medium a boundary located far from the scattering object and assuming that the source of the incident wave is located outside of this boundary offers a simple practical way out of this seemingly unphysical situation: the energy flow would increase exponentially with distance from the object until the medium’s boundary is reached where it would then become constant.

## Acknowledgments

## References and links

1. | A. N. Lebedev, M. Gratz, U. Kreibig, and O. Stenzel, “Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,” Eur. Phys. J. D |

2. | I. W. Sudiarta and P. Chylek, “Mie scattering efficiency of a large spherical particle embedded in an absorbing medium,” J. Quant. Spectrosc. Radiat. Transfer |

3. | 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. |

4. | G. Videen and W. Sun, “Yet another look at light scattering from particles in absorbing media,” Appl. Opt. |

5. | Q. Fu and W. Sun, “Apparent optical properties of spherical particles in absorbing medium,” J. Quant. Spectrosc. Radiat. Transfer |

6. | J. Yin and L. Pilon, “Efficiency factors and radiation characteristics of spherical scatterers in an absorbing medium,” J. Opt. Soc. Am. A |

7. | 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 |

8. | H. C. van de Hulst, |

9. | H. C. van de Hulst, |

10. | C. F. Bohren and D. R. Huffman, |

11. | K. N. Liou, |

12. | J. W. Hovenier, C. van der Mee, and H. Domke, |

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

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

15. | M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium,” Opt. Express (in preparation). [PubMed] |

16. | J. D. Jackson, |

17. | D. S. Saxon, Lectures on the scattering of light (Scientific Report No. 9, Department of Meteorology, University of California at Los Angeles, 1955). |

18. | E. J. Rothwell and M. J. Cloud, |

19. | M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds., |

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

21. | A. Doicu, T. Wriedt, and Y. A. Eremin, |

22. | F. Borghese, P. Denti, and R. Saija, |

23. | C. F. Bohren and D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. |

24. | G. S. Sammelmann, “Electromagnetic scattering from large aspect ratio lossy dielectric solids in a conducting medium,” in |

25. | M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Extinction and the electromagnetic optical theorem,” in |

26. | P. Chýlek, “Light scattering by small particles in an absorbing medium,” J. Opt. Soc. Am. |

27. | M. I. Mishchenko, “The electromagnetic optical theorem revisited,” J. Quant. Spectrosc. Radiat. Transfer |

28. | M. Born and E. Wolf, |

29. | D. S. Saxon, “Tensor scattering matrix for the electromagnetic field,” Phys. Rev. |

30. | M. I. Mishchenko, “Far-field approximation in electromagnetic scattering,” J. Quant. Spectrosc. Radiat. Transfer |

**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: September 10, 2007

Revised Manuscript: September 24, 2007

Manuscript Accepted: September 24, 2007

Published: September 26, 2007

**Citation**

Michael I. Mishchenko, "Electromagnetic scattering by a fixed finite object embedded in an absorbing medium," Opt. Express **15**, 13188-13202 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-20-13188

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### References

- A. N. Lebedev, M. Gratz, U. Kreibig, and O. Stenzel, "Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films," Eur. Phys. J. D 6, 365-373 (1999).
- I. W. Sudiarta and P. Chylek, "Mie scattering efficiency of a large spherical particle embedded in an absorbing medium," J. Quant. Spectrosc. Radiat. Transfer 70, 709-714 (2001). [CrossRef]
- 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]
- G. Videen and W. Sun, "Yet another look at light scattering from particles in absorbing media," Appl. Opt. 42, 6724-6727 (2003). [CrossRef] [PubMed]
- Q. Fu and W. Sun, "Apparent optical properties of spherical particles in absorbing medium," J. Quant. Spectrosc. Radiat. Transfer 100, 137-142 (2006). [CrossRef]
- J. Yin and L. Pilon, "Efficiency factors and radiation characteristics of spherical scatterers in an absorbing medium," J. Opt. Soc. Am. A 23, 2784-2796 (2006). [CrossRef]
- 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]
- H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
- H. C. van de Hulst, Multiple Light Scattering (Academic Press, San Diego, 1980).
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
- K. N. Liou, An Introduction to Atmospheric Radiation (Academic Press, San Diego, 2002).
- 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, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002). http://www.giss.nasa.gov/~crmim/books.html.
- 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).
- M. I. Mishchenko, "Multiple scattering by particles embedded in an absorbing medium," Opt. Express (in preparation). [PubMed]
- J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).
- D. S. Saxon, Lectures on the scattering of light (Scientific Report No. 9, Department of Meteorology, University of California at Los Angeles, 1955).
- E. J. Rothwell and M. J. Cloud, Electromagnetics (CRC Press, Boca Raton, Florida, 2001). [CrossRef]
- M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds., Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic Press, San Diego, 2000).
- F. M. Kahnert, "Numerical methods in electromagnetic scattering theory," J. Quant. Spectrosc. Radiat. Transfer 79-80, 775-824 (2003). [CrossRef]
- A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, Berlin, 2006). [CrossRef]
- F. Borghese, P. Denti, and R. Saija, Scattering from Model Nonspherical Particles. Theory and Applications to Environmental Physics (Springer, Berlin, 2007).
- C. F. Bohren and D. P. Gilra, "Extinction by a spherical particle in an absorbing medium," J. Colloid Interface Sci. 72, 215-221 (1979). [CrossRef]
- G. S. Sammelmann, "Electromagnetic scattering from large aspect ratio lossy dielectric solids in a conducting medium," in OCEANS 2003 MTS/IEEE Proceedings (IEEE Service Center, Piscataway, New Jersey, 2003), pp. 2011-2016.
- M. J. Berg, C. M. Sorensen, and A. Chakrabarti, "Extinction and the electromagnetic optical theorem," 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. 9-12.
- P. Chýlek, "Light scattering by small particles in an absorbing medium," J. Opt. Soc. Am. 67, 561-563.
- M. I. Mishchenko, "The electromagnetic optical theorem revisited," J. Quant. Spectrosc. Radiat. Transfer 101, 404-410 (2006). [CrossRef]
- M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).
- D. S. Saxon, "Tensor scattering matrix for the electromagnetic field," Phys. Rev. 100, 1771-1775 (1955). [CrossRef]
- M. I. Mishchenko, "Far-field approximation in electromagnetic scattering," J. Quant. Spectrosc. Radiat. Transfer 100, 268-276 (2006). [CrossRef]

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