Abstract

A simple and rigorous formalism is presented for describing the incoherent radiative properties of absorption, emission, and scattering of an opaque body in local thermodynamic equilibrium; polarization, inelastic scattering, and applied-magnetic-field effects are treated in full. The radiative behavior of such a body is shown to be completely characterized by the local bispectral bidirectional reflectivity matrix. Expressions for the emitted and reflected Stokes vectors of the source radiation are given in terms of this matrix. Use is made of the most general forms of the reciprocal relations and Kirchhoff’s law; derivations are also provided.

© 1991 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  2. This problem has received a great deal of partial treatment elsewhere. As good examples, Ref. 3 provides an extensive discussion of the radiance properties (without regard to polarization state) of elastically scattering, reciprocal media exposed to unpolarized radiation, and Ref. 4 provides an excellent discussion of the polarization properties of such media in the more general case of exposure to polarized radiation.
  3. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 2nd ed. (McGraw-Hill, New York, 1981).
  4. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vol. 1, Chap. 3, pp. 16–33.See also p. 173 of Ref. 1.
    [Crossref]
  5. W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).
  6. J. W. Hovenier, H. C. van de Hulst, C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).
  7. Systems whose microscopic evolution is independent of the sign of the applied magnetic field are termed reciprocal. This term is also used, however, to describe systems whose macroscopic properties of interest are independent of the sign of the applied magnetic field (even though aspects of the microscopic evolution may not be).
  8. A point perhaps worthy of some elaboration concerns the distinction between the applied magnetic field Bo and the magnetic field associated with the incident electromagnetic radiation. The distinction between these two fields is, in fact, solely a matter of separation of time scales: We are interested in describing the radiative behavior of material media on a time scale that is long compared with the wavelength period of the incident radiation but that is short compared with the characteristic time for variations in Bo. Indeed, the Stokes-vector formalism, as we have employed it in this paper, is inherently intended to describe the outcome of measurements that integrate over many oscillations of the incident electromagnetic field but that sample the applied field instantaneously.
  9. See Ref. 3, pp. 58 and 446–447.

1986 (1)

J. W. Hovenier, H. C. van de Hulst, C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

Chandrasekhar, S.

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

Hovenier, J. W.

J. W. Hovenier, H. C. van de Hulst, C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 2nd ed. (McGraw-Hill, New York, 1981).

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 2nd ed. (McGraw-Hill, New York, 1981).

van de Hulst, H. C.

J. W. Hovenier, H. C. van de Hulst, C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vol. 1, Chap. 3, pp. 16–33.See also p. 173 of Ref. 1.
[Crossref]

van der Mee, C. V. M.

J. W. Hovenier, H. C. van de Hulst, C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

Astron. Astrophys. (1)

J. W. Hovenier, H. C. van de Hulst, C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

Other (8)

Systems whose microscopic evolution is independent of the sign of the applied magnetic field are termed reciprocal. This term is also used, however, to describe systems whose macroscopic properties of interest are independent of the sign of the applied magnetic field (even though aspects of the microscopic evolution may not be).

A point perhaps worthy of some elaboration concerns the distinction between the applied magnetic field Bo and the magnetic field associated with the incident electromagnetic radiation. The distinction between these two fields is, in fact, solely a matter of separation of time scales: We are interested in describing the radiative behavior of material media on a time scale that is long compared with the wavelength period of the incident radiation but that is short compared with the characteristic time for variations in Bo. Indeed, the Stokes-vector formalism, as we have employed it in this paper, is inherently intended to describe the outcome of measurements that integrate over many oscillations of the incident electromagnetic field but that sample the applied field instantaneously.

See Ref. 3, pp. 58 and 446–447.

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

This problem has received a great deal of partial treatment elsewhere. As good examples, Ref. 3 provides an extensive discussion of the radiance properties (without regard to polarization state) of elastically scattering, reciprocal media exposed to unpolarized radiation, and Ref. 4 provides an excellent discussion of the polarization properties of such media in the more general case of exposure to polarized radiation.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 2nd ed. (McGraw-Hill, New York, 1981).

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vol. 1, Chap. 3, pp. 16–33.See also p. 173 of Ref. 1.
[Crossref]

W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).

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Figures (1)

Fig. 1
Fig. 1

Coordinate systems for the description of incoming (incident) and outgoing (emitted and reflected) radiation. The choice for the origin for the azimuthal coordinates ϕ and ϕr is arbitrary.

Equations (35)

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λ r L r ( R , Ω r , B o , λ r ) = ρ ( R , Ω r , Ω i , B o , λ r , λ i ) · λ i L i ( R , Ω i , λ i ) × cos β i d Ω i d λ i ,
L e ( R , Ω r , B o , λ r , T ) = ( R , Ω r , B o , λ r ) L b b ( λ r , T ) ,
L 0 a ( R , Ω r , B o , λ r ) = α ( R , Ω r , B o , λ r ) · L i ( R , Ω r , λ r ) .
d a = j = 0 3 α j ( Ω i , B o , λ i ) ( λ i h c ) L j i ( Ω i , λ i ) cos β i d A d Ω i d λ i .
d r = j = 0 3 ρ 0 j ( Ω r , Ω i , B o , λ r , λ i ) λ i L j i ( Ω i , λ i ) × cos β i d Ω i d λ i ( 1 / h c ) cos β r d A d Ω r d λ r .
L 0 i ( Ω i , λ i ) = j = 0 3 [ ρ 0 j ( Ω r , Ω i , B o , λ r , λ i ) L j i ( Ω i , λ i ) × cos β r d Ω r d λ r + α j ( Ω i , B o , λ i ) L j i ( Ω i , λ i ) ] .
1 α 0 ( Ω i , B o , λ i ) ρ 00 ( Ω r , Ω i , B o , λ r , λ i ) cos β r d Ω r d λ r = j = 1 3 L j i ( Ω i , λ i ) L 0 i ( Ω i , λ i ) [ α j ( Ω i , B o , λ i ) + ρ 0 j ( Ω r , Ω i , B o , λ r , λ i ) × cos β r d Ω r d λ r ] .
α 0 ( Ω i , B o , λ i ) = 1 ρ 00 ( Ω r , Ω i , B o , λ r , λ i ) cos β r d Ω r d λ r , α j ( Ω i , B o , λ i ) = ρ 0 j ( Ω r , Ω i , B o , λ r , λ i ) cos β r d Ω r d λ r , j = 1 , 2 , 3 .
λ i L b b ( λ i , T ) ρ j k ( Ω r , Ω i , B o , λ r , λ i ) = ( 1 ) δ 2 j + δ 2 k λ r L b b ( λ r , T ) ρ k j ( Ω i , Ω r , B o , λ i , λ r ) ,
α j ( Ω r , B o , λ r ) = ( 1 ) δ 2 j j ( Ω r , B o , λ r ) ,
0 ( Ω r , B o , λ r ) = 1 ρ 00 ( Ω r , Ω i , B o , λ r , λ i ) λ i L b b ( λ i , T ) λ i L b b ( λ r , T ) × cos β i d Ω i d λ i , j ( Ω r , B o , λ r ) = ρ j 0 ( Ω r , Ω i , B o , λ r , λ i ) λ i L b b ( λ i , T ) λ r L b b ( λ r , T ) × cos β i d Ω i d λ i , j = 1 , 2 , 3 .
L = L e + L r .
L = L L b b [ 1 0 0 0 ] , L i = L i L b b [ 1 0 0 0 ] ,
λ r L ( R , Ω r , B o , λ r , T ) = ρ ( R , Ω r , Ω i , B o , λ r , λ i ) · λ i L i ( R , Ω i , λ i , T ) cos β r d Ω i d λ i .
ρ ( R , Ω r , Ω i , B o , λ r , λ i ) = ρ ( R , Ω r , Ω i , B o , λ r ) δ ( λ r λ i ) ,
α 0 ( Ω r , λ ) = 1 ρ 00 ( Ω i , Ω r , λ ) cos β i d Ω i ,
ρ 00 ( Ω r , Ω i , λ ) = ρ 00 ( Ω i , Ω r , λ ) ,
α 0 ( Ω r , λ ) = 0 ( Ω r , λ ) ,
0 ( Ω r , λ ) = 1 ρ 00 ( Ω r , Ω i , λ ) cos β i d Ω i ,
L 0 ( Ω r , λ , T ) = L b b ( λ , T ) + ρ 00 ( Ω r , Ω i , λ ) × [ L 0 i ( Ω i , λ ) L b b ( λ , T ) ] cos β i d Ω i ,
d L 0 r = ( 1 / 4 ) ( O · ρ · I ) ( I · L ) ( λ i / λ r ) cos β i d Ω i d λ i .
0 ( Ω r , B o , λ r , T ) = 1 ρ 00 ( Ω r , Ω i , B o , λ r , λ i ) × λ i L b b ( λ i , T ) λ r L b b ( λ r , T ) cos β i d Ω i d λ i , α 0 ( Ω r , B o , λ r , T ) = 1 ρ 00 ( Ω i , Ω r , B o , λ i , λ r ) cos β i d Ω i d λ i ,
λ r L b b ( λ r , T ) α 0 ( Ω r , B o , λ r , T ) cos β r d Ω r d λ r = λ r L b b ( λ r , T ) 0 ( Ω r , B o , λ r , T ) cos β r d Ω r d λ r .
L 0 r ( Ω r , B o , λ ) = ρ 00 ( Ω r , Ω i , B o , λ ) L 0 i ( Ω i , λ ) cos β i d Ω i .
L 0 i , max ( λ ) ρ 00 ( Ω r , Ω i , B o , λ ) cos β i d Ω i L 0 r ( Ω r , B o , λ ) L 0 i , max ( λ ) ρ 00 ( Ω r , Ω i , B o , λ ) cos β i d Ω i ,
L 0 i , max ( λ ) [ 1 0 ( Ω r , B o , λ ) ] L 0 r ( Ω r , B o , λ ) L 0 i , min ( λ ) [ 1 0 ( Ω r , B o , λ ) ] .
L 0 i , max ( λ ) + [ L b b ( λ , T ) L 0 i , max ( λ ) ] 0 ( Ω r , B o , λ ) L 0 ( Ω r , B o , λ , T ) L 0 i , min ( λ ) + [ L b b ( λ , T ) L 0 i , min ( λ ) ] 0 ( Ω r , B o , λ ) .
α j ( Ω r , B o , λ r ) = ( 1 ) δ 2 j j ( Ω r , B o , λ r ) ,
d 2 = ( 1 / 4 ) [ O · ρ ( Ω r , Ω i , + B o , λ r , λ i ) · I ] ( λ i / h c ) L b b ( λ i , T ) × cos β i d Ω i d λ i cos β r d Ω r d λ r .
d 1 = ( 1 / 4 ) [ I * · ρ ( Ω i , Ω r B o , λ i , λ r ) · O * ] ( λ r / h c ) L b b ( λ r , T ) × cos β r d Ω r d λ r cos β i d Ω i d λ i .
λ i L b b ( λ i , T ) ρ j k ( Ω r , Ω i , B o , λ r , λ i ) = ( 1 ) δ 2 j + δ 2 k λ r L b b ( λ r , T ) ρ k j ( Ω i , Ω r , B o , λ i , λ r ) ,
α ( B o ) = [ 1 / 2 0 1 / 2 0 ] .
( B o ) = [ 1 0 0 0 ] .
( B o ) = [ 1 / 2 0 1 / 2 0 ] .
α j ( B o ) = ( 1 ) δ 2 j j ( B o ) j ( B o ) ,

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