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

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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 6, 365-373 (1999).
  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 70, 709-714 (2001).
    [CrossRef]
  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]
  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]
  5. Q. Fu and W. Sun, "Apparent optical properties of spherical particles in absorbing medium," J. Quant. Spectrosc. Radiat. Transfer 100, 137-142 (2006).
    [CrossRef]
  6. 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]
  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 24, 2943-2952 (2007).
    [CrossRef]
  8. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  9. H. C. van de Hulst, Multiple Light Scattering (Academic Press, San Diego, 1980).
  10. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  11. K. N. Liou, An Introduction to Atmospheric Radiation (Academic Press, San Diego, 2002).
  12. 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).
  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.
  14. 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).
  15. M. I. Mishchenko, "Multiple scattering by particles embedded in an absorbing medium," Opt. Express (in preparation).
    [PubMed]
  16. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).
  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, Electromagnetics (CRC Press, Boca Raton, Florida, 2001).
    [CrossRef]
  19. 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).
  20. F. M. Kahnert, "Numerical methods in electromagnetic scattering theory," J. Quant. Spectrosc. Radiat. Transfer 79-80, 775-824 (2003).
    [CrossRef]
  21. A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, Berlin, 2006).
    [CrossRef]
  22. F. Borghese, P. Denti, and R. Saija, Scattering from Model Nonspherical Particles. Theory and Applications to Environmental Physics (Springer, Berlin, 2007).
  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]
  24. 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.
  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.
  26. P. Chýlek, "Light scattering by small particles in an absorbing medium," J. Opt. Soc. Am. 67, 561-563.
  27. M. I. Mishchenko, "The electromagnetic optical theorem revisited," J. Quant. Spectrosc. Radiat. Transfer 101, 404-410 (2006).
    [CrossRef]
  28. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).
  29. D. S. Saxon, "Tensor scattering matrix for the electromagnetic field," Phys. Rev. 100, 1771-1775 (1955).
    [CrossRef]
  30. M. I. Mishchenko, "Far-field approximation in electromagnetic scattering," J. Quant. Spectrosc. Radiat. Transfer 100, 268-276 (2006).
    [CrossRef]

2007

2006

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]

M. I. Mishchenko, "The electromagnetic optical theorem revisited," J. Quant. Spectrosc. Radiat. Transfer 101, 404-410 (2006).
[CrossRef]

M. I. Mishchenko, "Far-field approximation in electromagnetic scattering," J. Quant. Spectrosc. Radiat. Transfer 100, 268-276 (2006).
[CrossRef]

Q. Fu and W. Sun, "Apparent optical properties of spherical particles in absorbing medium," J. Quant. Spectrosc. Radiat. Transfer 100, 137-142 (2006).
[CrossRef]

2003

F. M. Kahnert, "Numerical methods in electromagnetic scattering theory," J. Quant. Spectrosc. Radiat. Transfer 79-80, 775-824 (2003).
[CrossRef]

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

2002

2001

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]

1999

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).

1979

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]

1955

D. S. Saxon, "Tensor scattering matrix for the electromagnetic field," Phys. Rev. 100, 1771-1775 (1955).
[CrossRef]

Baum, B. A.

Bohren, C. F.

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]

Calvo-Perez, O.

Chylek, P.

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]

Chýlek, P.

Durant, S.

Fu, Q.

Q. Fu and W. Sun, "Apparent optical properties of spherical particles in absorbing medium," J. Quant. Spectrosc. Radiat. Transfer 100, 137-142 (2006).
[CrossRef]

Gao, B.-C.

Gilra, D. P.

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]

Gratz, M.

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).

Greffet, J.-J.

Hu, Y. X.

Huang, H.-L.

Kahnert, F. M.

F. M. Kahnert, "Numerical methods in electromagnetic scattering theory," J. Quant. Spectrosc. Radiat. Transfer 79-80, 775-824 (2003).
[CrossRef]

Kreibig, U.

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).

Lebedev, A. N.

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).

Mishchenko, M. I.

M. I. Mishchenko, "The electromagnetic optical theorem revisited," J. Quant. Spectrosc. Radiat. Transfer 101, 404-410 (2006).
[CrossRef]

M. I. Mishchenko, "Far-field approximation in electromagnetic scattering," J. Quant. Spectrosc. Radiat. Transfer 100, 268-276 (2006).
[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]

Park, S. K.

Pilon, L.

Platnick, S. E.

Saxon, D. S.

D. S. Saxon, "Tensor scattering matrix for the electromagnetic field," Phys. Rev. 100, 1771-1775 (1955).
[CrossRef]

Stenzel, O.

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).

Sudiarta, I. W.

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]

Sun, W.

Q. Fu and W. Sun, "Apparent optical properties of spherical particles in absorbing medium," J. Quant. Spectrosc. Radiat. Transfer 100, 137-142 (2006).
[CrossRef]

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

Tsay, S.-C.

Videen, G.

Vukadinovic, N.

Winker, D. M.

Wiscombe, W. J.

Yang, P.

Yin, J.

Appl. Opt.

Eur. Phys. J. D

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).

J. Colloid Interface Sci.

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]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

F. M. Kahnert, "Numerical methods in electromagnetic scattering theory," J. Quant. Spectrosc. Radiat. Transfer 79-80, 775-824 (2003).
[CrossRef]

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]

Q. Fu and W. Sun, "Apparent optical properties of spherical particles in absorbing medium," J. Quant. Spectrosc. Radiat. Transfer 100, 137-142 (2006).
[CrossRef]

M. I. Mishchenko, "The electromagnetic optical theorem revisited," J. Quant. Spectrosc. Radiat. Transfer 101, 404-410 (2006).
[CrossRef]

M. I. Mishchenko, "Far-field approximation in electromagnetic scattering," J. Quant. Spectrosc. Radiat. Transfer 100, 268-276 (2006).
[CrossRef]

Phys. Rev.

D. S. Saxon, "Tensor scattering matrix for the electromagnetic field," Phys. Rev. 100, 1771-1775 (1955).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).

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).

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.

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).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1.
Fig. 1.

Electromagnetic scattering by a fixed object.

Fig. 2.
Fig. 2.

Far-field optical observables.

Equations (98)

Equations on this page are rendered with MathJax. Learn more.

× E ( r ) = i ω μ 1 H ( r ) × H ( r ) = i ω ε 1 E ( r ) } r V EXT ,
× E ( r ) = i ω μ 2 H ( r ) × H ( r ) = i ω ε 2 ( r ) E ( r ) } r V INT ,
× × E ( r ) k 1 2 E ( r ) = 0 , r V EXT ,
× × E ( r ) k 2 2 ( r ) E ( r ) = 0 , r V INT ,
× × E ( r ) k 1 2 E ( r ) = j ( r ) , r 3 ,
j ( r ) = k 1 2 [ m ˜ 2 ( r ) 1 ] E ( r ) ,
m ˜ ( r ) = { 1 , r V INT m ( r ) = k 2 ( r ) k 1 , r V INT ,
E ( r ) = E inc ( r ) + k 1 2 ( I + 1 k 1 2 ) V INT d r ´ [ m 2 ( r ´ ) 1 ] E ( r ´ ) exp ( i k 1 r r ´ ) 4 π r r ´ , r 3 ,
E sca ( r ) = ( I + 1 k 1 2 ) V INT d r ´ exp ( i k 1 r r ´ ) 4 π r r ´ V INT d r T ( r ´ , r ) E inc ( r ) , r 3 ,
T ( r , r ´ ) = k 1 2 [ m 2 ( r ) 1 ] δ ( r r ´ ) I + k 1 2 [ m 2 ( r ) 1 ] ( I + 1 k 1 2 ) V INT d r exp ( i k 1 r r ) 4 π r r T ( r , r ´ ) , r , r ´ V INT .
r r ´ for any r ´ V INT ,
k 1 r ´ 2 2 r 1 for any r ´ V INT .
r r ´ = r 1 2 r ̂ r ´ r + r ´ 2 r 2 r r ̂ r ´ + r ´ 2 2 r ,
exp ( i k 1 r r ´ ) 4 π r r ´ exp ( i k 1 r i k 1 r ̂ r ´ ) 4 πr .
k 1 r 1 ,
E sca ( r ) exp ( i k 1 r ) r E 1 sca ( r ̂ )
= exp ( k 1 r ) exp ( i k 1 ´ r ) r E 1 sca ( r ̂ ) ,
E 1 sca ( r ̂ ) = k 1 2 4 π ( I r ̂ r ̂ ) V INT d r ´ [ m 2 ( r ´ ) 1 ] E ( r ´ ) exp ( i k 1 r ̂ r ´ ) , r ̂ E 1 sca ( r ̂ ) = 0 .
E inc ( r ) = exp ( i k 1 n ̂ inc r ) E 0 inc , E 0 inc . n ̂ inc = 0 .
E 1 sca ( n ̂ sca ) = A ( n ̂ sca , n ̂ inc ) E 0 inc ,
n ̂ sca A ( n ̂ sca , n ̂ inc ) = 0 and A ( n ̂ sca , n ̂ inc ) n ̂ inc = 0 ,
A ( n ̂ sca , n ̂ inc ) = 1 4 π ( I n ̂ sca n ̂ sca ) V INT d r ´ exp ( i k 1 n ̂ sca r ´ ) × V INT d r T ( r ´ , r ) exp ( i k 1 n ̂ inc r ) ( I n ̂ inc n ̂ inc ) .
E sca ( r n ̂ sca ) = exp ( i k 1 r ) r S ( n ̂ sca , n ̂ inc ) E 0 inc ,
E = [ E θ E φ ] .
S 11 = θ ̂ sca A θ ̂ inc ,
S 12 = θ ̂ sca A φ ̂ inc ,
S 21 = φ ̂ sca A θ ̂ inc ,
S 22 = φ ̂ sca A φ ̂ inc .
S ( r ´ , t ) t = 1 2 Re { E ( r ´ ) × [ H ( r ´ ) ] * }
= S inc ( r ´ , t ) t + S sca ( r ´ , t ) t + S ext ( r ´ , t ) t ,
S inc ( r ´ , t ) t = 1 2 Re { E inc ( r ´ ) × [ H inc ( r ´ ) ] * }
S sca ( r ´ , t ) t = 1 2 Re { E sca ( r ´ ) × [ H sca ( r ´ ) ] * }
S ext ( r ´ , t ) t = 1 2 Re { E inc ( r ´ ) × [ H sca ( r ´ ) ] * + E sca ( r ´ ) × [ H inc ( r ´ ) ] * }
E inc ( r ´ ) = exp ( i k 1 n ̂ inc r ´ ) E 0 inc
= k 1 ´ r ´ exp ( k 1 n ̂ inc r ´ ) i 2 π k 1 ´ [ δ ( n ̂ inc + r ̂ ´ ) exp ( i k 1 ´ r ´ ) r ´ δ ( n ̂ inc r ̂ ´ ) exp ( i k 1 ´ r ´ ) r ´ ] E 0 inc ,
H inc ( r ´ ) = k 1 ω μ 1 exp ( i k 1 n ̂ inc r ´ ) n ̂ inc × E 0 inc
= k 1 ´ r ´ exp ( k 1 n ̂ inc r ´ ) i 2 π k 1 ´ [ δ ( n ̂ inc + r ̂ ´ ) exp ( i k 1 ´ r ´ ) r ´ δ ( n ̂ inc r ̂ ´ ) exp ( i k 1 ´ r ´ ) r ´ ] × k 1 ω μ 1 n ̂ inc × E 0 inc ,
E sca ( r ´ ) = exp ( i k 1 r ´ ) r ´ E 1 sca ( r ̂ ´ ) ,
H sca ( r ´ ) = k 1 ω μ 1 exp ( i k 1 r ´ ) r ´ r ̂ ´ × E 1 sca ( r ̂ ´ ) .
W 2 ( r ̂ ) = S 2 d S r ̂ S ( r ´ , t ) t S Re ( k 1 2 ω μ 1 ) exp ( 2 k 1 r ) r 2 E 1 sca ( r ̂ ) 2 ,
W 1 ( n ̂ inc ) = S 1 d S n ̂ inc S ( r ´ , t ) t
= S Re ( k 1 2 ω μ 1 ) exp ( 2 k 1 r ) E 0 inc 2 + S 1 d S n ̂ inc [ S sca ( r ´ , t ) t + S ext ( r ´ , t ) t ]
S Re ( k 1 2 ω μ 1 ) exp ( 2 k 1 r ) E 0 inc 2 + S Re ( k 1 2 ω μ 1 ) exp ( 2 k 1 r ) r 2 E 1 sca ( n ̂ inc ) 2 + r 2 Ω ˜ 1 d r ̂ ´ n ̂ inc S ext ( r r ´ , t ) t
S Re ( k 1 2 ω μ 1 ) exp ( 2 k 1 r ) E 0 inc 2 + S Re ( k 1 2 ω μ 1 ) exp ( 2 k 1 r ) r 2 E 1 sca ( n ̂ inc ) 2 exp ( 2 k 1 r ) Re ( k 1 ω μ 1 ) 2 π k 1 ´ Im [ E 1 sca ( n ̂ inc ) ( E 0 inc ) * ] ,
W 3 ( n ̂ inc ) = S 3 d S n ̂ inc S ( r ´ , t ) t
S Re ( k 1 2 ω μ 1 ) exp ( 2 k 1 r ) E 0 inc 2 + S Re ( k 1 2 ω μ 1 ) exp ( 2 k 1 r ) r 2 E 1 sca ( n ̂ inc ) 2 + Im ( k 1 ω μ 1 ) 2 π k 1 ´ Re [ exp ( i 2 k 1 ´ r ) E 1 sca ( n ̂ inc ) ( E 0 inc ) * ] .
J inc = Re ( k 1 2 ω μ 1 ) [ E 0 θ inc ( E 0 θ inc ) * E 0 θ inc ( E 0 φ inc ) * E 0 φ inc ( E 0 θ inc ) * E 0 φ inc ( E 0 φ inc ) * ] ,
J sca ( r n ̂ sca ) = Re ( k 1 2 ω μ 1 ) [ E θ sca ( r n ̂ sca ) [ E θ sca ( r n ̂ sca ) ] * E θ sca ( r n ̂ sca ) [ E φ sca ( r n ̂ sca ) ] * E φ sca ( r n ̂ sca ) [ E θ sca ( r n ̂ sca ) ] * E φ sca ( r n ̂ sca ) [ E φ sca ( r n ̂ sca ) ] * ]
= exp ( 2 k 1 r ) r 2 Re ( k 1 2 ω μ 1 ) [ E 1 θ sca ( n ̂ sca ) [ E 1 θ sca ( n ̂ sca ) ] * E 1 θ sca ( n ̂ sca ) [ E 1 φ sca ( n ̂ sca ) ] * E 1 φ sca ( n ̂ sca ) [ E 1 θ sca ( n ̂ sca ) ] * E 1 φ sca ( n ̂ sca ) [ E 1 φ sca ( n ̂ sca ) ] * ] ,
I inc = DJ inc = Re ( k 1 2 ω μ 1 ) [ E 0 θ inc ( E 0 θ inc ) * + E 0 φ inc ( E 0 φ inc ) * E 0 θ inc ( E 0 φ inc ) * E 0 φ inc ( E 0 φ inc ) * E 0 θ inc ( E 0 φ inc ) * E 0 φ inc ( E 0 θ inc ) * i [ E 0 φ inc ( E 0 θ inc ) * E 0 θ inc ( E 0 φ inc ) * ] ] ,
I sca ( r n ̂ sca ) = DJ sca ( r n ̂ sca ) = exp ( 2 k 1 r ) r 2 Re ( k 1 2 ω μ 1 ) [ E 1 θ sca ( E 1 θ sca ) * + E 1 φ sca ( E 1 φ sca ) * E 1 θ sca ( E 1 φ sca ) * E 1 φ sca ( E 1 φ sca ) * E 1 θ sca ( E 1 φ sca ) * E 1 φ sca ( E 1 θ sca ) * i [ E 1 φ sca ( E 1 θ sca ) * E 1 θ sca ( E 1 φ sca ) * ] ] ,
D = [ 1 0 0 1 1 0 0 1 0 1 1 0 0 i i 0 ] .
J sca ( r n ̂ sca ) = exp ( 2 k 1 r ) r 2 Z J ( n ̂ sca , n ̂ inc ) J inc
I sca ( r n ̂ sca ) = exp ( 2 k 1 r ) r 2 Z ( n ̂ sca , n ̂ inc ) I inc ,
Z J = [ S 11 2 S 11 S 12 * S 12 S 11 * S 12 2 S 11 S 21 * S 11 S 22 * S 12 S 21 * S 12 S 22 * S 21 S 11 * S 21 S 12 * S 22 S 11 * S 22 S 12 * S 21 2 S 21 S 22 * S 22 S 21 * S 22 2 ] ,
Z ( n ̂ sca , n ̂ inc ) = DZ J ( n ̂ sca , n ̂ inc ) D 1
Z 11 = 1 2 ( S 11 2 + S 12 2 + S 21 2 + S 22 2 ) ,
Z 12 = 1 2 ( S 11 2 - S 12 2 + S 21 2 - S 22 2 ) ,
Z 13 = Re ( S 11 S 12 * + S 22 S 21 * ) ,
Z 14 = Im ( S 11 S 12 * S 22 S 21 * ) ,
Z 21 = 1 2 ( S 11 2 + S 12 2 S 21 2 - S 22 2 ) ,
Z 22 = 1 2 ( S 11 2 S 12 2 S 21 2 + S 22 2 ) ,
Z 23 = Re ( S 11 S 12 * S 22 S 21 * ) ,
Z 24 = Im ( S 11 S 12 * + S 22 S 21 * ) ,
Z 31 = Re ( S 11 S 21 * + S 22 S 12 * ) ,
Z 32 = Re ( S 11 S 21 * + S 22 S 12 * ) ,
Z 33 = Re ( S 11 S 22 * + S 12 S 21 * ) ,
Z 34 = Im ( S 11 S 22 * + S 21 S 12 * ) ,
Z 41 = Im ( S 21 S 11 * + S 22 S 12 * ) ,
Z 42 = Im ( S 21 S 11 * S 22 S 12 * ) ,
Z 43 = Im ( S 22 S 11 * S 12 S 21 * ) ,
Z 44 = Re ( S 22 S 11 * S 12 S 21 * ) .
Signal 2 Δ S I sca ( r n ̂ sca ) = exp ( 2 k 1 r ) r 2 Z ( n ̂ sca , n ̂ inc ) I inc .
J ( r r ̂ ) = Re ( k 1 2 ω μ 1 ) [ E θ ( r r ̂ ) [ E θ ( r r ̂ ) ] * E θ ( r r ̂ ) [ E φ ( r r ̂ ) ] * E φ ( r r ̂ ) [ E θ ( r r ̂ ) ] * E φ ( r r ̂ ) [ E φ ( r r ̂ ) ] * ] ,
E ( r r ̂ ) = E inc ( r r ̂ ) + E sca ( r r ̂ ) .
( Signal 1 ) J = S 1 d S J ( r r ̂ )
S exp ( 2 k 1 r ) J inc + S exp ( 2 k 1 r ) r 2 Z J ( n ̂ inc , n ̂ inc ) J inc exp ( 2 k 1 r ) K J ( n ̂ inc ) J inc ,
K J = i 2 π k 1 ´ [ S 11 * S 11 S 12 * S 12 0 S 21 * S 22 * S 11 0 S 12 S 12 0 S 11 * S 22 S 12 * 0 S 21 S 21 * S 22 * S 22 ]
( Signal 1 ) J = S 1 d S I ( r r ̂ )
S exp ( 2 k 1 r ) I inc + S exp ( 2 k 1 r ) r 2 Z ( n ̂ inc , n ̂ inc ) I inc exp ( 2 k 1 r ) K ( n ̂ inc ) I inc ,
I ( r n ̂ inc ) = DJ ( r n ̂ inc ) .
K ( n ̂ inc ) = DK J ( n ̂ inc ) D 1 .
K jj = 2 π k 1 ´ Im ( S 11 + S 22 ) , j = 1 , , 4 ,
K 12 = K 21 = 2 π k 1 ´ Im ( S 11 S 22 ) ,
K 13 = K 31 = 2 π k 1 ´ Im ( S 12 S 21 ) ,
K 14 = K 41 = 2 π k 1 ´ Re ( S 21 S 12 ) ,
K 23 = K 32 = 2 π k 1 ´ Im ( S 21 S 12 ) ,
K 24 = K 42 = 2 π k 1 ´ Re ( S 12 + S 21 ) ,
K 34 = K 43 = 2 π k 1 ´ Re ( S 22 S 11 ) .
A ( n ̂ inc , n ̂ sca ) = [ A ( n ̂ sca , n ̂ inc ) ] T ,
S ( n ̂ inc , n ̂ sca ) = [ S 11 ( n ̂ sca , n ̂ inc ) S 21 ( n ̂ sca , n ̂ inc ) S 12 ( n ̂ sca , n ̂ inc ) S 22 ( n ̂ sca , n ̂ inc ) ] ,
Z ( n ̂ inc , n ̂ sca ) = Δ 3 [ Z ( n ̂ sca , n ̂ inc ) ] T Δ 3 ,
K ( n ̂ inc ) = Δ 3 [ K ( n ̂ inc ) ] T Δ 3 ,
Δ 3 = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ]
S 21 ( n ̂ , n ̂ ) = S 12 ( n ̂ , n ̂ ) ,
Z 11 ( n ̂ , n ̂ ) Z 22 ( n ̂ , n ̂ ) + Z 33 ( n ̂ , n ̂ ) Z 44 ( n ̂ , n ̂ ) = 0 ,
K ( n ̂ inc ) = [ K 11 ( n ̂ inc ) K 12 ( n ̂ inc ) K 13 ( n ̂ inc ) K 14 ( n ̂ inc ) K 21 ( n ̂ inc ) K 22 ( n ̂ inc ) K 23 ( n ̂ inc ) K 24 ( n ̂ inc ) K 31 ( n ̂ inc ) K 32 ( n ̂ inc ) K 33 ( n ̂ inc ) K 34 ( n ̂ inc ) K 41 ( n ̂ inc ) K 42 ( n ̂ inc ) K 43 ( n ̂ inc ) K 44 ( n ̂ inc ) ]
C ext = 4 π k 1 ´ Im S 11 ( ) .

Metrics