Abstract

This paper solves the long-standing problem of establishing the fundamental physical link between the radiative transfer theory and macroscopic electromagnetics in the case of elastic scattering by a sparse discrete random medium. The radiative transfer equation (RTE) is derived directly from the macroscopic Maxwell equations by computing theoretically the appropriately defined so-called Poynting–Stokes tensor carrying information on both the direction, magnitude, and polarization characteristics of local electromagnetic energy flow. Our derivation from first principles shows that to compute the local Poynting vector averaged over a sufficiently long period of time, one can solve the RTE for the direction-dependent specific intensity column vector and then integrate the direction-weighted specific intensity over all directions. Furthermore, we demonstrate that the specific intensity (or specific intensity column vector) can be measured with a well-collimated radiometer (photopolarimeter), which provides the ultimate physical justification for the use of such instruments in radiation-budget and particle-characterization applications. However, the specific intensity cannot be interpreted in phenomenological terms as signifying the amount of electromagnetic energy transported in a given direction per unit area normal to this direction per unit time per unit solid angle. Also, in the case of a densely packed scattering medium the relation of the measurement with a well-collimated radiometer to the time-averaged local Poynting vector remains uncertain, and the theoretical modeling of this measurement is likely to require a much more complicated approach than solving an RTE.

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References

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  1. 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(6), 2822–2836 (2007).
    [CrossRef] [PubMed]
  2. M. I. Mishchenko, J. M. Dlugach, and L. Liu, “Azimuthal asymmetry of the coherent backscattering cone: theoretical results,” Phys. Rev. A 80(5), 053824 (2009).
    [CrossRef]
  3. M. I. Mishchenko, J. M. Dlugach, L. Liu, V. K. Rosenbush, N. N. Kiselev, and Yu. G. Shkuratov, “Direct solutions of the Maxwell equations explain opposition phenomena observed for high-albedo solar system objects,” Astrophys. J. 705(2), L118–L122 (2009).
    [CrossRef]
  4. M. I. Mishchenko, V. K. Rosenbush, N. N. Kiselev, et al., Polarimetric Remote Sensing of Solar System Bodies (Akedemperoidyka, Kyiv, 2010). http://www.giss.nasa.gov/staff/mmishchenko/books.html
  5. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  6. J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16(4), 527–610 (1974).
    [CrossRef]
  7. V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, Oxford, 1975).
  8. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, New York, 1978).
  9. H. C. van de Hulst, Multiple Light Scattering (Academic Press, New York, 1980).
  10. J. Lenoble, ed., Radiative Transfer in Scattering and Absorbing Atmospheres (A. Deepak, Hampton, Va., 1985).
  11. R. M. Goody, and Y. L. Yung, Atmospheric Radiation: Theoretical Basis (Oxford U. Press, Oxford, 1989).
  12. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic Press, San Diego, Ca., 1994).
  13. G. L. Stephens, Remote Sensing of the Lower Atmosphere (Oxford U. Press, New York, 1994).
  14. E. G. Yanovitskij, Light Scattering in Inhomogeneous Atmospheres (Springer, Berlin, 1997).
  15. G. E. Thomas, and K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, New York, 1999).
  16. K. N. Liou, An Introduction to Atmospheric Radiation (Academic Press, San Diego, 2002).
  17. M. Modest, Radiative Heat Transfer (Academic Press, San Diego, Ca., 2003).
  18. 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).
  19. A. Marshak, and A. B. Davis, eds., 3D Radiative Transfer in Cloudy Atmospheres (Springer, Berlin, 2005).
  20. 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).
  21. G. W. Petty, A First Course in Atmospheric Radiation (Sundog Publishing, Madison, Wi., 2006).
  22. W. Zdunkowski, T. Trautmann, and A. Bott, Radiation in the Atmosphere (Cambridge U. Press, Cambridge, UK, 2007).
  23. A. B. Davis and A. Marshak, “Solar radiation transport in the cloudy atmosphere: a 3D perspective on observations and climate impacts,” Rep. Prog. Phys. 73(2), 026801 (2010).
    [CrossRef]
  24. Yu. N. Barabanenkov, “Multiple scattering of waves by ensembles of particles and the theory of radiation transport,” Sov. Phys. Usp. 18(9), 673–689 (1975).
    [CrossRef]
  25. L. A. Apresyan, and Yu. A. Kravtsov, Radiation Transfer (Gordon and Breach, Basel, 1996).
  26. M. I. Mishchenko, “Multiple scattering, radiative transfer, and weak localization in discrete random media: the unified microphysical approach,” Rev. Geophys. 46(2), RG2003 (2008).
    [CrossRef]
  27. M. I. Mishchenko, “Gustav Mie and the fundamental concept of electromagnetic scattering by particles: a perspective,” J. Quant. Spectrosc. Radiat. Transf. 110(14-16), 1210–1222 (2009).
    [CrossRef]
  28. M. I. Mishchenko, “Vector radiative transfer equation for arbitrarily shaped and arbitrarily oriented particles: a microphysical derivation from statistical electromagnetics,” Appl. Opt. 41(33), 7114–7134 (2002).
    [CrossRef] [PubMed]
  29. M. I. Mishchenko, “Microphysical approach to polarized radiative transfer: extension to the case of an external observation point,” Appl. Opt. 42(24), 4963–4967 (2003).
    [CrossRef] [PubMed]
  30. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  31. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).
  32. E. J. Rothwell, and M. J. Cloud, Electromagnetics (CRC Press, Boca-Raton, Fl., 2009).
  33. R. W. Preisendorfer, Radiative Transfer on Discrete Spaces (Pergamon Press, Oxford, 1965).
  34. E. A. Milne, “Thermodynamics of the stars,” Handbuch der Astrophysik 3, 65–255 (1930).
  35. 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/staff/mmishchenko/books.html
  36. M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium. 2. Radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 109(14), 2386–2390 (2008).
    [CrossRef]
  37. S. Silver, ed., Microwave Antenna Theory and Design (McGraw-Hill, New York, 1949).
  38. C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer, Berlin, 1969).
  39. G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399–416 (1852).
  40. A. G. Borovoy, ““Method of iterations in multiple scattering: the transfer equation,” Izv. Vuzov Fizika, 6, 50–54 (1966).
  41. A. G. Borovoi, “Multiple scattering of short waves by uncorrelated and correlated scatterers,” Light Scattering Rev. 1, 181–252 (2006).
    [CrossRef]
  42. V. Twersky, “On propagation in random media of discrete scatterers,” Proc. Symp. Appl. Math. 16, 84–116 (1964).
  43. C. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34(11), 5072–5088 (1993).
    [CrossRef]
  44. J. W. Hovenier and C. V. M. van der Mee, “Testing scattering matrices: a compendium of recipes,” J. 
Quantum. Spectrosc. Radiat. Transf. 55(5), 649–661 (1996).
    [CrossRef]

2010

A. B. Davis and A. Marshak, “Solar radiation transport in the cloudy atmosphere: a 3D perspective on observations and climate impacts,” Rep. Prog. Phys. 73(2), 026801 (2010).
[CrossRef]

2009

M. I. Mishchenko, J. M. Dlugach, and L. Liu, “Azimuthal asymmetry of the coherent backscattering cone: theoretical results,” Phys. Rev. A 80(5), 053824 (2009).
[CrossRef]

M. I. Mishchenko, J. M. Dlugach, L. Liu, V. K. Rosenbush, N. N. Kiselev, and Yu. G. Shkuratov, “Direct solutions of the Maxwell equations explain opposition phenomena observed for high-albedo solar system objects,” Astrophys. J. 705(2), L118–L122 (2009).
[CrossRef]

M. I. Mishchenko, “Gustav Mie and the fundamental concept of electromagnetic scattering by particles: a perspective,” J. Quant. Spectrosc. Radiat. Transf. 110(14-16), 1210–1222 (2009).
[CrossRef]

2008

M. I. Mishchenko, “Multiple scattering, radiative transfer, and weak localization in discrete random media: the unified microphysical approach,” Rev. Geophys. 46(2), RG2003 (2008).
[CrossRef]

M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium. 2. Radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 109(14), 2386–2390 (2008).
[CrossRef]

2007

2006

A. G. Borovoi, “Multiple scattering of short waves by uncorrelated and correlated scatterers,” Light Scattering Rev. 1, 181–252 (2006).
[CrossRef]

2003

2002

1996

J. W. Hovenier and C. V. M. van der Mee, “Testing scattering matrices: a compendium of recipes,” J. 
Quantum. Spectrosc. Radiat. Transf. 55(5), 649–661 (1996).
[CrossRef]

1993

C. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34(11), 5072–5088 (1993).
[CrossRef]

1975

Yu. N. Barabanenkov, “Multiple scattering of waves by ensembles of particles and the theory of radiation transport,” Sov. Phys. Usp. 18(9), 673–689 (1975).
[CrossRef]

1974

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16(4), 527–610 (1974).
[CrossRef]

1966

A. G. Borovoy, ““Method of iterations in multiple scattering: the transfer equation,” Izv. Vuzov Fizika, 6, 50–54 (1966).

1930

E. A. Milne, “Thermodynamics of the stars,” Handbuch der Astrophysik 3, 65–255 (1930).

1852

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399–416 (1852).

Barabanenkov, Yu. N.

Yu. N. Barabanenkov, “Multiple scattering of waves by ensembles of particles and the theory of radiation transport,” Sov. Phys. Usp. 18(9), 673–689 (1975).
[CrossRef]

Borovoi, A. G.

A. G. Borovoi, “Multiple scattering of short waves by uncorrelated and correlated scatterers,” Light Scattering Rev. 1, 181–252 (2006).
[CrossRef]

Borovoy, A. G.

A. G. Borovoy, ““Method of iterations in multiple scattering: the transfer equation,” Izv. Vuzov Fizika, 6, 50–54 (1966).

Cairns, B.

Davis, A. B.

A. B. Davis and A. Marshak, “Solar radiation transport in the cloudy atmosphere: a 3D perspective on observations and climate impacts,” Rep. Prog. Phys. 73(2), 026801 (2010).
[CrossRef]

Dlugach, J. M.

M. I. Mishchenko, J. M. Dlugach, L. Liu, V. K. Rosenbush, N. N. Kiselev, and Yu. G. Shkuratov, “Direct solutions of the Maxwell equations explain opposition phenomena observed for high-albedo solar system objects,” Astrophys. J. 705(2), L118–L122 (2009).
[CrossRef]

M. I. Mishchenko, J. M. Dlugach, and L. Liu, “Azimuthal asymmetry of the coherent backscattering cone: theoretical results,” Phys. Rev. A 80(5), 053824 (2009).
[CrossRef]

Hansen, J. E.

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16(4), 527–610 (1974).
[CrossRef]

Hovenier, J. W.

J. W. Hovenier and C. V. M. van der Mee, “Testing scattering matrices: a compendium of recipes,” J. 
Quantum. Spectrosc. Radiat. Transf. 55(5), 649–661 (1996).
[CrossRef]

Kiselev, N. N.

M. I. Mishchenko, J. M. Dlugach, L. Liu, V. K. Rosenbush, N. N. Kiselev, and Yu. G. Shkuratov, “Direct solutions of the Maxwell equations explain opposition phenomena observed for high-albedo solar system objects,” Astrophys. J. 705(2), L118–L122 (2009).
[CrossRef]

Liu, L.

M. I. Mishchenko, J. M. Dlugach, L. Liu, V. K. Rosenbush, N. N. Kiselev, and Yu. G. Shkuratov, “Direct solutions of the Maxwell equations explain opposition phenomena observed for high-albedo solar system objects,” Astrophys. J. 705(2), L118–L122 (2009).
[CrossRef]

M. I. Mishchenko, J. M. Dlugach, and L. Liu, “Azimuthal asymmetry of the coherent backscattering cone: theoretical results,” Phys. Rev. A 80(5), 053824 (2009).
[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(6), 2822–2836 (2007).
[CrossRef] [PubMed]

Mackowski, D. W.

Marshak, A.

A. B. Davis and A. Marshak, “Solar radiation transport in the cloudy atmosphere: a 3D perspective on observations and climate impacts,” Rep. Prog. Phys. 73(2), 026801 (2010).
[CrossRef]

Milne, E. A.

E. A. Milne, “Thermodynamics of the stars,” Handbuch der Astrophysik 3, 65–255 (1930).

Mishchenko, M. I.

M. I. Mishchenko, “Gustav Mie and the fundamental concept of electromagnetic scattering by particles: a perspective,” J. Quant. Spectrosc. Radiat. Transf. 110(14-16), 1210–1222 (2009).
[CrossRef]

M. I. Mishchenko, J. M. Dlugach, L. Liu, V. K. Rosenbush, N. N. Kiselev, and Yu. G. Shkuratov, “Direct solutions of the Maxwell equations explain opposition phenomena observed for high-albedo solar system objects,” Astrophys. J. 705(2), L118–L122 (2009).
[CrossRef]

M. I. Mishchenko, J. M. Dlugach, and L. Liu, “Azimuthal asymmetry of the coherent backscattering cone: theoretical results,” Phys. Rev. A 80(5), 053824 (2009).
[CrossRef]

M. I. Mishchenko, “Multiple scattering, radiative transfer, and weak localization in discrete random media: the unified microphysical approach,” Rev. Geophys. 46(2), RG2003 (2008).
[CrossRef]

M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium. 2. Radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 109(14), 2386–2390 (2008).
[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(6), 2822–2836 (2007).
[CrossRef] [PubMed]

M. I. Mishchenko, “Microphysical approach to polarized radiative transfer: extension to the case of an external observation point,” Appl. Opt. 42(24), 4963–4967 (2003).
[CrossRef] [PubMed]

M. I. Mishchenko, “Vector radiative transfer equation for arbitrarily shaped and arbitrarily oriented particles: a microphysical derivation from statistical electromagnetics,” Appl. Opt. 41(33), 7114–7134 (2002).
[CrossRef] [PubMed]

Rosenbush, V. K.

M. I. Mishchenko, J. M. Dlugach, L. Liu, V. K. Rosenbush, N. N. Kiselev, and Yu. G. Shkuratov, “Direct solutions of the Maxwell equations explain opposition phenomena observed for high-albedo solar system objects,” Astrophys. J. 705(2), L118–L122 (2009).
[CrossRef]

Shkuratov, Yu. G.

M. I. Mishchenko, J. M. Dlugach, L. Liu, V. K. Rosenbush, N. N. Kiselev, and Yu. G. Shkuratov, “Direct solutions of the Maxwell equations explain opposition phenomena observed for high-albedo solar system objects,” Astrophys. J. 705(2), L118–L122 (2009).
[CrossRef]

Stokes, G. G.

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399–416 (1852).

Travis, L. D.

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16(4), 527–610 (1974).
[CrossRef]

van der Mee, C. V. M.

J. W. Hovenier and C. V. M. van der Mee, “Testing scattering matrices: a compendium of recipes,” J. 
Quantum. Spectrosc. Radiat. Transf. 55(5), 649–661 (1996).
[CrossRef]

C. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34(11), 5072–5088 (1993).
[CrossRef]

Videen, G.

Appl. Opt.

Astrophys. J.

M. I. Mishchenko, J. M. Dlugach, L. Liu, V. K. Rosenbush, N. N. Kiselev, and Yu. G. Shkuratov, “Direct solutions of the Maxwell equations explain opposition phenomena observed for high-albedo solar system objects,” Astrophys. J. 705(2), L118–L122 (2009).
[CrossRef]

Handbuch der Astrophysik

E. A. Milne, “Thermodynamics of the stars,” Handbuch der Astrophysik 3, 65–255 (1930).

J. ?Quantum. Spectrosc. Radiat. Transf.

J. W. Hovenier and C. V. M. van der Mee, “Testing scattering matrices: a compendium of recipes,” J. 
Quantum. Spectrosc. Radiat. Transf. 55(5), 649–661 (1996).
[CrossRef]

J. Math. Phys.

C. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34(11), 5072–5088 (1993).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf.

M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium. 2. Radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 109(14), 2386–2390 (2008).
[CrossRef]

M. I. Mishchenko, “Gustav Mie and the fundamental concept of electromagnetic scattering by particles: a perspective,” J. Quant. Spectrosc. Radiat. Transf. 110(14-16), 1210–1222 (2009).
[CrossRef]

Light Scattering Rev.

A. G. Borovoi, “Multiple scattering of short waves by uncorrelated and correlated scatterers,” Light Scattering Rev. 1, 181–252 (2006).
[CrossRef]

Opt. Express

Phys. Rev. A

M. I. Mishchenko, J. M. Dlugach, and L. Liu, “Azimuthal asymmetry of the coherent backscattering cone: theoretical results,” Phys. Rev. A 80(5), 053824 (2009).
[CrossRef]

Rep. Prog. Phys.

A. B. Davis and A. Marshak, “Solar radiation transport in the cloudy atmosphere: a 3D perspective on observations and climate impacts,” Rep. Prog. Phys. 73(2), 026801 (2010).
[CrossRef]

Rev. Geophys.

M. I. Mishchenko, “Multiple scattering, radiative transfer, and weak localization in discrete random media: the unified microphysical approach,” Rev. Geophys. 46(2), RG2003 (2008).
[CrossRef]

Sov. Phys. Usp.

Yu. N. Barabanenkov, “Multiple scattering of waves by ensembles of particles and the theory of radiation transport,” Sov. Phys. Usp. 18(9), 673–689 (1975).
[CrossRef]

Space Sci. Rev.

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16(4), 527–610 (1974).
[CrossRef]

Trans. Cambridge Philos. Soc.

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399–416 (1852).

Other

A. G. Borovoy, ““Method of iterations in multiple scattering: the transfer equation,” Izv. Vuzov Fizika, 6, 50–54 (1966).

V. Twersky, “On propagation in random media of discrete scatterers,” Proc. Symp. Appl. Math. 16, 84–116 (1964).

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, Oxford, 1975).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, New York, 1978).

H. C. van de Hulst, Multiple Light Scattering (Academic Press, New York, 1980).

J. Lenoble, ed., Radiative Transfer in Scattering and Absorbing Atmospheres (A. Deepak, Hampton, Va., 1985).

R. M. Goody, and Y. L. Yung, Atmospheric Radiation: Theoretical Basis (Oxford U. Press, Oxford, 1989).

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic Press, San Diego, Ca., 1994).

G. L. Stephens, Remote Sensing of the Lower Atmosphere (Oxford U. Press, New York, 1994).

E. G. Yanovitskij, Light Scattering in Inhomogeneous Atmospheres (Springer, Berlin, 1997).

G. E. Thomas, and K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, New York, 1999).

K. N. Liou, An Introduction to Atmospheric Radiation (Academic Press, San Diego, 2002).

M. Modest, Radiative Heat Transfer (Academic Press, San Diego, Ca., 2003).

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).

A. Marshak, and A. B. Davis, eds., 3D Radiative Transfer in Cloudy Atmospheres (Springer, Berlin, 2005).

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).

G. W. Petty, A First Course in Atmospheric Radiation (Sundog Publishing, Madison, Wi., 2006).

W. Zdunkowski, T. Trautmann, and A. Bott, Radiation in the Atmosphere (Cambridge U. Press, Cambridge, UK, 2007).

L. A. Apresyan, and Yu. A. Kravtsov, Radiation Transfer (Gordon and Breach, Basel, 1996).

M. I. Mishchenko, V. K. Rosenbush, N. N. Kiselev, et al., Polarimetric Remote Sensing of Solar System Bodies (Akedemperoidyka, Kyiv, 2010). http://www.giss.nasa.gov/staff/mmishchenko/books.html

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

S. Silver, ed., Microwave Antenna Theory and Design (McGraw-Hill, New York, 1949).

C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer, Berlin, 1969).

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/staff/mmishchenko/books.html

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).

E. J. Rothwell, and M. J. Cloud, Electromagnetics (CRC Press, Boca-Raton, Fl., 2009).

R. W. Preisendorfer, Radiative Transfer on Discrete Spaces (Pergamon Press, Oxford, 1965).

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Figures (8)

Fig. 1
Fig. 1

A cloud consisting of N particles and illuminated by a plane electromagnetic wave propagating in the direction of the unit vector s ^ .

Fig. 2
Fig. 2

(a) Quasi-instantaneous radiation budget of a volume element Δ V bounded by a closed surface Δ S . The arrows represent the distribution of Re[S(r)] over the closed boundary Δ S corresponding to the specific multi-particle configuration. (b) Configuration-averaged radiation budget of the same volume element evaluated for a statistically uniform spatial distribution of particle positions inside V.

Fig. 3
Fig. 3

Local spherical coordinates. The local Cartesian frame { x , y , z } has the same spatial orientation as the laboratory Cartesian frame {x, y, z}.

Fig. 4
Fig. 4

Geometry showing the quantities appearing in Eq. (15).

Fig. 5
Fig. 5

(a) Optical scheme of a well-collimated detector of electromagnetic energy flux. (b) The well-collimated detector does not always respond to the Poynting vector directed along the optical axis of the instrument.

Fig. 6
Fig. 6

Response of a well-collimated radiometer to multidirectional illumination.

Fig. 7
Fig. 7

A well-collimated radiometer placed inside a random particulate medium. The sizes of the detector and the particles relative to that of the medium are exaggerated for demonstration purposes.

Fig. 8
Fig. 8

A hypothetical radiance flux meter accumulating local Poynting vectors with directions falling within the acceptance solid angle Δ Ω q ^ .

Equations (55)

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

    × E ( r )    =   i ω μ 0 H ( r ) × H ( r )    =    i ω ε 1 E ( r ) }    r         V EXT ,
× E ( r )    =   i ω μ 0 H ( r ) × H ( r )    =    i ω ε 2 ( r ,   ω ) E ( r ) }    r         V INT .
O   sca = T ^ O   inc .
O   sca ¯ = T ^ O   inc ¯ .
O   sca ¯ R , ξ = T ^ R , ξ O   inc ¯ ,
S ( r )    =    1 2 E ( r ) × [ H ( r ) ] ,
W Δ S     =     Re     Δ S d S S ( r ) n ^ ( r ) ,
P ( r )    =    1 2 H ( r ) [ E ( r ) ] .
S     =     ( P z y     P y z ) x ^     +     ( P x z     P z x ) y ^     +     ( P y x     P x y ) z ^ ,
S     =     ( P ϕ θ     P θ ϕ ) q ^ ,
J     =     1 2   ε   1 μ 0 [ E θ E θ E θ E ϕ E ϕ E θ E ϕ E ϕ ]     =     [ P ϕ θ P θ θ P ϕ ϕ P θ ϕ ] ,
I     =     1 2   ε   1 μ 0   [ E θ E θ    +    E ϕ E ϕ E θ E θ       E ϕ E ϕ E θ E ϕ       E ϕ E θ i ( E ϕ E θ       E θ E ϕ ) ]     =     [ P ϕ θ         P θ ϕ P ϕ θ     +     P θ ϕ P θ θ         P ϕ ϕ i   ( P θ θ     +     P ϕ ϕ ) ] ,
P ( r ) R , ξ    =    1 2 H ( r ) [ E ( r ) ] R , ξ
P ( r ) R , ξ    =    1 2   i ω μ 0 [ r × C ( r , r ) ] | r = r ,
C ( r , r ) = E ( r ) [ E ( r ) ] R , ξ
C ( r , r )     =     E c ( r ) [ E c ( r ) ]
+     n 0   d R 1   d ξ 1 η   ( r ^ 1 , r 1 ) r 1 A 1 ( r ^ 1 , s ^ )
C c ( R 1 ) [ A 1 ( r ^ 1 , s ^ ) ] T [ η   ( r ^ 1 , r 1 ) ] T r 1
+     n 0 2   d R 1   d ξ 1   d R 2   d ξ 2 η   ( r ^ 1 , r 1 ) r 1 A 1 ( r ^ 1 , R ^ 12 ) η   ( R ^ 12 , R 12 ) R 12 A 2 ( R ^ 12 , s ^ )
C c ( R 2 ) [ A 2 ( R ^ 12 , s ^ ) ] T [ η   ( R ^ 12 , R 12 ) ] T R 12 [ A 1 ( r ^ 1 , R ^ 12 ) ] T [ η   ( r ^ 1 , r 1 ) ] T r 1
+     n 0 3   d R 1   d ξ 1   d R 2   d ξ 2   d R 3   d ξ 3 η   ( r ^ 1 , r 1 ) r 1 A 1 ( r ^ 1 , R ^ 12 ) η   ( R ^ 12 , R 12 ) R 12
A 2 ( R ^ 12 , R ^ 23 ) η   ( R ^ 23 , R 23 ) R 23 A 3 ( R ^ 23 , s ^ ) C c ( R 3 ) [ A 3 ( R ^ 23 , s ^ ) ] T
[ η   ( R ^ 23 , R 23 ) ] T R 23 [ A 2 ( R ^ 12 , R ^ 23 ) ] T [ η   ( R ^ 12 , R 12 ) ] T R 12 [ A 1 ( r ^ 1 , R ^ 12 ) ] T   [ η   ( r ^ 1 , r 1 ) ] T r 1
+     ,
r 1 × η   ( r ^ 1 , r 1 ) r 1 A 1 ( r ^ 1 , q ^ )
i   k 1 r ^ 1 × η   ( r ^ 1 , r 1 ) r 1 A 1 ( r ^ 1 , q ^ ) ,
[ r 1 exp ( i k r ) ]     =
( i k r 1 ) r 1 exp ( i k r ) r ^ k r > > 1   i   k r 1 exp ( i k r ) r ^ .
P ( r ) R , ξ     =     1 2 ε   1 / μ 0    4 π d p ^ p ^ × Σ   L ( r , p ^ ) ,
Σ   L ( r , p ^ )     =     δ   ( p ^ + s ^ ) C c ( r )
+     n 0   d p   d ξ 1 η   ( p ^ , p ) A 1 ( p ^ , s ^ ) C c ( r + p ) [ A 1 ( p ^ , s ^ ) ] T [ η   ( p ^ , p ) ] T
+     n 0 2   d p   d ξ 1   d R 21   d R ^ 21   d ξ 2 η   ( p ^ , p ) A 1 ( p ^ , R ^ 21 ) η   ( R ^ 21 , R 21 )
A 2 ( R ^ 21 , s ^ ) C c ( r + p + R 21 ) [ A 2 ( R ^ 21 , s ^ ) ] T [ η   ( R ^ 21 , R 21 ) ] T
[ A 1 ( p ^ , R ^ 21 ) ] T [ η   ( p ^ , p ) ] T
+     n 0 3   d p   d ξ 1   d R 21 d R ^ 21   d ξ 2   d R 32   d R ^ 32   d ξ 3 η   ( p ^ , p ) A 1 ( p ^ , R ^ 21 )
η   ( R ^ 21 , R 21 ) A 2 ( R ^ 21 , R ^ 32 ) η   ( R ^ 32 , R 32 ) A 3 ( R ^ 32 , s ^ )
C c ( r + p + R 21 + R 32 ) [ A 3 ( R ^ 32 , s ^ ) ] T [ η   ( R ^ 32 , R 32 ) ] T
[ A 2 ( R ^ 21 , R ^ 32 ) ] T [ η   ( R ^ 21 , R 21 ) ] T [ A 1 ( p ^ , R ^ 21 ) ] T [ η   ( p ^ , p ) ] T
+       .
Σ L ( r , p ^ )     =     δ ( p ^ + s ^ ) C c ( r )     +     n 0   d p d p ^   d ξ η   ( p ^ , p ) A ( ξ ; p ^ , p ^ )
Σ   L ( r + p , p ^ ) [ A ( ξ ; p ^ , p ^ ) ] T [ η   ( p ^ , p ) ] T .
J ˜ ( r , q ^ )     =     1 2   ε   1 μ 0 [ J ˜ 1 ( r , q ^ ) J ˜ 2 ( r , q ^ ) J ˜ 3 ( r , q ^ ) J ˜ 4 ( r , q ^ ) ]     =     [ θ ^ ( q ^ ) Σ   L ( r , q ^ ) θ ^ ( q ^ ) θ ^ ( q ^ ) Σ   L ( r , q ^ ) φ ^ ( q ^ ) φ ^ ( q ^ ) Σ   L ( r , q ^ ) θ ^ ( q ^ ) φ ^ ( q ^ ) Σ   L ( r , q ^ ) φ ^ ( q ^ ) ] ,
I ˜ ( r , q ^ )     =     [ I ˜ ( r , q ^ ) Q ˜ ( r , q ^ ) U ˜ ( r , q ^ ) V ˜ ( r , q ^ ) ]     =     [ J ˜ 1 ( r , q ^ ) + J ˜ 4 ( r , q ^ ) J ˜ 1 ( r , q ^ ) J ˜ 4 ( r , q ^ ) J ˜ 2 ( r , q ^ ) J ˜ 3 ( r , q ^ ) i   [ J ˜ 3 ( r , q ^ ) J ˜ 2 ( r , q ^ ) ] ] ,
q ^ I ˜ ( r , q ^ )     =     n 0 K ( q ^ ) ξ I ˜ ( r , q ^ )     +     n 0    4 π   d q ^ Z ( q ^ , q ^ ) ξ I ˜ ( r , q ^ ) ,
P ( r ) R , ξ     =     1 2 ε   1 / μ 0    4 π d q ^ q ^ × Σ   L ( r , q ^ ) ,
S ( r ) R , ξ     =         4 π d q ^ q ^ I ˜ ( r , q ^ ) .
E ( r , t )     =     E inc ( r , t ) + i = 1 N E i ( r , t ) ,       H ( r , t )     =     H inc ( r , t ) + i = 1 N H i ( r , t ) ,
l m E l ( r , t ) × H m ( r , t ) ¯     =     1 2 Re l m E l ( r ) × [ H m ( r ) ] R , ξ ,
P ( r ;   Δ V q ^ )     =     1 2 l m H l ( r ) [ E m ( r ) ] R , ξ .
P ( r ;   Δ V q ^ )     =     Δ   Ω ε   1 / μ 0 q ^ × Σ   L ( r , q ^ ) ,
C ( r ) R , ξ     =        4 π d p ^ Σ   L ( r , p ^ ) ,
N ˜ ( r , q ^ )     =     1 S d lim Δ   Ω q ^ 0     1 Δ   Ω q ^     lim T     1 T     S d d S     t     t + T d t | S ( r , t ) | χ [   Δ   Ω q ^ , s ^ ( r , t ) ]
    =     lim Δ   Ω q ^ 0     1 Δ   Ω q ^     lim T     1 T     t     t + T d t | S ( r , t ) | χ [   Δ   Ω q ^ , s ^ ( r , t ) ] ,
χ [   Δ   Ω q ^ , s ^ ( r , t ) ]     =     { 1     if     s ^ ( r , t ) Δ   Ω q ^ , 0  otherwise .            
S ( r , t ) ¯     =         4 π       d q ^ q ^ N ˜ ( r , q ^ )     =     S ( r ) R , ξ     =         4 π       d q ^ q ^ I ˜ ( r , q ^ ) .

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