Abstract

For the reduction of the equations of radiative transfer, the most convenient representation of the state of polarization is by a one-column intensity matrix of the four coherency-matrix elements or their linear combinations, derived from the electric-field components parallel and perpendicular to the plane through the local vertical and the direction of the light beam. A particular law of scattering is then expressed by a four-by-four (Mueller) scattering matrix which connects the intensity matrices of the incident and scattered light. The form of such scattering matrices for various representations of the state of polarization, as well as the corresponding reciprocity relationships, are derived with the use of the matrix formalism recently introduced by Marathay. The reciprocity relationships, i.e., the pertinent relationships when the incident and scattered beams are interchanged or when their directions of propagations are reversed, can be expressed in terms of a transposition of the original scattering matrix accompanied by pre- and post-multiplication by certain diagonal matrices that incorporate the changes in the sign of some elements of the scattering matrix after transposition. The derived reciprocity relationships are new except for those which govern the reversal of directions when the state of polarization is represented by the Stokes parameters and their modifications (used by Chandrasekhar) and which have been given earlier by Chandrasekhar and van de Hulst.

© 1966 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, New York, 1957), p. 44.
  2. S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, England, 1950), p. 39.
  3. Z. Sekera, “Scattering Matrix for Spherical Particles and Its Transformation,” , Dept. of Meteorology, University of California, Los Angeles, (1955), Appendix D.
  4. Chandrasekhar, Ref. 2, p. 41.
  5. I. Kuščer and M. Ribarić, Optica Acta 6, 42 (1959).
    [CrossRef]
  6. A. S. Marathay, J. Opt. Soc. Am. 55, 969 (1965); or E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Reading, Mass., 1963), p. 139.
  7. J. Headings, Matrix Theory for Physicists (Longmans Green and Co., London, 1958).
  8. G. G. Stokes, Trans. Camb. Philos. Soc. 9, 399 (1852).
  9. Chandrasekhar, Ref. 2, p. 35.
  10. Chandrasekhar, Ref. 2, p. 42.
  11. Cf. Eq. (3) of Kuščer and Ribarić, Ref. 5.
  12. H. C. van de Hulst, Bull. Astr. Inst. Netherlands 18, 1 (1965).
  13. I. M. Gel’fand and Z. Ya. Shapiro, Uspekhi Mat. Nauk 7, 3 (1952); Amer. Math. Soc. Translation,  2, 207 (1956). See also I. M. Gel’fand, R. A. Menlos, and Z. Ya. Shapiro, Representations of the Rotation and Lorentz Groups and their Applications, English translation by G. Cummings and T. Boddington (The Macmillan Company, New York, 1963), pp. 89–92.
  14. Cf. Eq. (20) of Kuščer and Ribarić, Ref. 5.
  15. Headings, Ref. 9, p. 3.
  16. Chandrasekhar, Ref. 2, p. 177.
  17. With the use of matrix algebra, Eq. (51) follows directly from Eq. (49), if it is realized that with respect to Eq. (40)trSS(Ω,Ω′)=trTS−1·trS(Ω,Ω′)trTS.(51a)Writing Eq. (49) in the formSS(Ω′,Ω)=TS·trTS·trTS−1S(Ω,Ω′)trTStrTS−1·TS−1=TS·trTStrSS(Ω,Ω′)·trTS−1·TS−1, we get, after multiplication,TS·trTS=2q4,trTS−1TS−1=12q4, and after the substitution above, Eq. (51) directly.
  18. Chandrasekhar, Ref. 2, p. 173.

1965 (2)

1959 (1)

I. Kuščer and M. Ribarić, Optica Acta 6, 42 (1959).
[CrossRef]

1952 (1)

I. M. Gel’fand and Z. Ya. Shapiro, Uspekhi Mat. Nauk 7, 3 (1952); Amer. Math. Soc. Translation,  2, 207 (1956). See also I. M. Gel’fand, R. A. Menlos, and Z. Ya. Shapiro, Representations of the Rotation and Lorentz Groups and their Applications, English translation by G. Cummings and T. Boddington (The Macmillan Company, New York, 1963), pp. 89–92.

1852 (1)

G. G. Stokes, Trans. Camb. Philos. Soc. 9, 399 (1852).

Chandrasekhar,

Chandrasekhar, Ref. 2, p. 41.

Chandrasekhar, Ref. 2, p. 35.

Chandrasekhar, Ref. 2, p. 42.

Chandrasekhar, Ref. 2, p. 177.

Chandrasekhar, Ref. 2, p. 173.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, England, 1950), p. 39.

Gel’fand, I. M.

I. M. Gel’fand and Z. Ya. Shapiro, Uspekhi Mat. Nauk 7, 3 (1952); Amer. Math. Soc. Translation,  2, 207 (1956). See also I. M. Gel’fand, R. A. Menlos, and Z. Ya. Shapiro, Representations of the Rotation and Lorentz Groups and their Applications, English translation by G. Cummings and T. Boddington (The Macmillan Company, New York, 1963), pp. 89–92.

Headings, J.

J. Headings, Matrix Theory for Physicists (Longmans Green and Co., London, 1958).

Kušcer, I.

I. Kuščer and M. Ribarić, Optica Acta 6, 42 (1959).
[CrossRef]

Marathay, A. S.

Ribaric, M.

I. Kuščer and M. Ribarić, Optica Acta 6, 42 (1959).
[CrossRef]

Sekera, Z.

Z. Sekera, “Scattering Matrix for Spherical Particles and Its Transformation,” , Dept. of Meteorology, University of California, Los Angeles, (1955), Appendix D.

Shapiro, Z. Ya.

I. M. Gel’fand and Z. Ya. Shapiro, Uspekhi Mat. Nauk 7, 3 (1952); Amer. Math. Soc. Translation,  2, 207 (1956). See also I. M. Gel’fand, R. A. Menlos, and Z. Ya. Shapiro, Representations of the Rotation and Lorentz Groups and their Applications, English translation by G. Cummings and T. Boddington (The Macmillan Company, New York, 1963), pp. 89–92.

Stokes, G. G.

G. G. Stokes, Trans. Camb. Philos. Soc. 9, 399 (1852).

van de Hulst, H. C.

H. C. van de Hulst, Bull. Astr. Inst. Netherlands 18, 1 (1965).

H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, New York, 1957), p. 44.

Bull. Astr. Inst. Netherlands (1)

H. C. van de Hulst, Bull. Astr. Inst. Netherlands 18, 1 (1965).

J. Opt. Soc. Am. (1)

Optica Acta (1)

I. Kuščer and M. Ribarić, Optica Acta 6, 42 (1959).
[CrossRef]

Trans. Camb. Philos. Soc. (1)

G. G. Stokes, Trans. Camb. Philos. Soc. 9, 399 (1852).

Uspekhi Mat. Nauk (1)

I. M. Gel’fand and Z. Ya. Shapiro, Uspekhi Mat. Nauk 7, 3 (1952); Amer. Math. Soc. Translation,  2, 207 (1956). See also I. M. Gel’fand, R. A. Menlos, and Z. Ya. Shapiro, Representations of the Rotation and Lorentz Groups and their Applications, English translation by G. Cummings and T. Boddington (The Macmillan Company, New York, 1963), pp. 89–92.

Other (13)

Cf. Eq. (20) of Kuščer and Ribarić, Ref. 5.

Headings, Ref. 9, p. 3.

Chandrasekhar, Ref. 2, p. 177.

With the use of matrix algebra, Eq. (51) follows directly from Eq. (49), if it is realized that with respect to Eq. (40)trSS(Ω,Ω′)=trTS−1·trS(Ω,Ω′)trTS.(51a)Writing Eq. (49) in the formSS(Ω′,Ω)=TS·trTS·trTS−1S(Ω,Ω′)trTStrTS−1·TS−1=TS·trTStrSS(Ω,Ω′)·trTS−1·TS−1, we get, after multiplication,TS·trTS=2q4,trTS−1TS−1=12q4, and after the substitution above, Eq. (51) directly.

Chandrasekhar, Ref. 2, p. 173.

J. Headings, Matrix Theory for Physicists (Longmans Green and Co., London, 1958).

Chandrasekhar, Ref. 2, p. 35.

Chandrasekhar, Ref. 2, p. 42.

Cf. Eq. (3) of Kuščer and Ribarić, Ref. 5.

H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, New York, 1957), p. 44.

S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, England, 1950), p. 39.

Z. Sekera, “Scattering Matrix for Spherical Particles and Its Transformation,” , Dept. of Meteorology, University of California, Los Angeles, (1955), Appendix D.

Chandrasekhar, Ref. 2, p. 41.

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Figures (2)

Fig. 1
Fig. 1

Orientation of the directions of the incident and scattered, beams and the angles between the meridional planes through these directions (planes containing k ˆ and n ˆ or n ˆ , respectively) and the scattering plane (through the points BC).

Fig. 2
Fig. 2

Orientation of the directions of the incident and scattered beams and the angles between the meridional planes through these directions and the scattering plane, before and after reversal of these directions.

Equations (86)

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E scat ( n ˆ ) = A ( n ˆ , n ˆ ) E inc ( n ˆ ) ,
E inc ( n ˆ ) = A r ( n ˆ , n ˆ ) · E scat ( n ˆ )
A r ( n ˆ , n ˆ ) = [ A l l A r l A l r A r r ] .
R ( α ) [ cos α , sin α sin α , cos α ]
E scat ( Ω ) = A ( Ω , Ω ) · E inc ( Ω ) ,
A ( Ω , Ω ) = R ( π + γ ) · A ( cos θ ) · R ( + β )
A 11 = ( s , s ) A r r ( c , c ) A l l , A 12 = ( c , s ) A r r ( s , c ) A l l , A 21 = ( s , c ) A r r + ( c , s ) A l l , A 22 = ( c , c ) A r r + ( s , s ) A l l ,
( s , s ) = sin β sin γ , ( s , c ) = sin β cos γ , ( c , s ) = cos β sin γ , ( c , c ) = cos β cos γ .
A ( Ω , Ω ) = A r r ( x ) a 1 ( β , γ ) + A l l ( x ) a d j a 1 ( β , γ ) ,
a 1 ( β , γ ) [ ( s , s ) ( c , s ) ( s , c ) ( c , c ) ] , x = cos θ .
A l r a 2 ( β , γ ) A r l a d j a 2 ( β , γ ) ,
a 2 ( β , γ ) [ ( s , c ) ( c , c ) ( s , s ) ( c , s ) ] .
( l , l ) = [ ( 1 μ 2 ) ( 1 μ 2 ) ] 1 2 + μ μ cos ( φ φ ) , ( l , r ) = μ sin ( φ φ ) , ( r , l ) = μ sin ( φ φ ) , ( r , r ) = c o s ( φ φ ) ,
( c , c ) = [ ( r , r ) + x ( l , l ) ] / ( 1 x 2 ) , ( c , s ) = [ ( r , l ) + x ( l , r ) ] / ( 1 x 2 ) , ( s , c ) = [ ( l , r ) x ( r , l ) ] / ( 1 x 2 ) , ( s , s ) = [ ( l , l ) x ( r , r ) ] / ( 1 x 2 ) ,
A 11 = ( l , l ) T 1 + ( r , r ) T 2 , A 12 = ( r , l ) T 1 + ( l , r ) T 2 , A 21 = ( l , r ) T 1 + ( r , l ) T 2 , A 22 = ( r , r ) T 1 + ( l , l ) T 2 ,
T 1 ( cos θ ) = [ A r r ( x ) x A l l ( x ) ] / ( 1 x 2 ) ,
T 2 ( cos θ ) = [ A l l ( x ) x A r r ( x ) ] / ( 1 x 2 ) .
E CP [ E + E ] = ( 2 ) 1 2 [ 1 i 1 + i ] [ E l E r ] = T CP · E LP .
R CP ( α ) = T CP · R LP ( α ) · T CP 1 = [ e i α 0 0 e i α ] .
T CP A LP = [ e i γ 0 0 e i γ ] T CP · A · T CP 1 · [ e i β 0 0 e i β ] T CP .
1 , 2 = 1 2 [ A l l + A r r ± i ( A l r A r l ) ] ,
Δ 1 , 2 = 1 2 [ A l l A r r ± i ( A l r + A r l ) ] ,
T CP · A · T CP 1 = [ 1 Δ 2 Δ 1 2 ] ,
A CP ( Ω , Ω ) = [ 1 e i ( β + γ ) Δ 2 e i ( β γ ) Δ 1 e i ( β γ ) 2 e i ( β + γ ) ] .
c E l E l * + E r E r *
J = c E × E = [ J l l J l r J r l J r r ] ,
J = c E × E * = ( J l l J l r J r l J r r ) ,
J i j = c E i E j * ( i , j = l , r ) ,
I = c E l E l * + E r E r * = J l l + J r r Q = c E l E l * E r E r * = J l l J r r U = 2 c Re E l E r * = J l r + J r l V = 2 c Im E l E r * = i J l r + i J r l ,
T S ( 1 0 0 1 1 0 0 1 0 1 1 0 0 i i 0 ) ,
T c ( 1 0 0 0 0 0 0 1 0 1 1 0 0 i i 0 ) .
T S = ( 2 ) 1 2 T S
T C = Q · T C ,
T C ( 1 0 0 0 0 0 0 1 0 1 / ( 2 ) 1 2 1 / ( 2 ) 1 2 0 0 i / ( 2 ) 1 2 i / ( 2 ) 1 2 0 )
J CP ( I 2 , I 0 , I 0 , I 2 ) = T 0 · c E CP × E CP * ,
T 0 ( 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 ) ,
J CP = T K R J ,
T K R = T 0 · T CP × T CP * = 1 2 ( 1 i i 1 1 i i 1 1 i i 1 1 i i 1 ) .
I 2 = 1 2 ( Q i U ) , I 0 = 1 2 ( I V ) , I 0 = 1 2 ( I + V ) , I 2 = 1 2 ( Q + i U ) .
J scat ( Ω ) = A ( Ω , Ω ) · J inc ( Ω ) · A ( Ω , Ω )
J scat ( Ω ) = S ( Ω , Ω ) · J inc ( Ω ) ,
S ( Ω , Ω ) = [ A ( Ω , Ω ) × A * ( Ω , Ω ) ] ,
S ( Ω , Ω ) ( A 11 A 11 * , A 11 A 12 * , A 12 A 11 * , A 12 A 12 * , A 11 A 21 * , A 11 A 22 * , A 12 A 21 * , A 12 A 22 * , A 21 A 11 * , A 21 A 12 * , A 22 A 11 * , A 22 A 12 * , A 21 A 21 * , A 21 A 22 * , A 22 A 21 * , A 22 A 22 * , ) .
S ( Ω , Ω ) = β scat P ( Ω , Ω ) ,
I = T · J .
I scat ( Ω ) = T · S ( Ω , Ω ) J ( Ω ) = T · S ( Ω , Ω ) · T 1 · T J ( Ω ) = S ( Ω , Ω ) I inc ( Ω ) ;
S ( Ω , Ω ) = T · S ( Ω , Ω ) · T 1 .
A 11 = A 2 , A 12 = A 3 , A 21 = A 4 , A 22 = A 1 ,
S CP ( Ω , Ω ) = T 0 ( A CP × A CP * ) T 0 .
S CP ( Ω , Ω ) = R ( γ ) [ 1 Δ 2 Δ 1 2 ] × [ 2 * Δ 1 * Δ 2 * 1 * ] R ( β ) ,
S CP ( Ω , Ω ) ( 1 2 * e i 2 ( β , γ ) 1 Δ 1 * e i 2 γ Δ 2 2 * e i 2 γ Δ 2 Δ 1 * e i 2 ( β , γ ) 1 Δ 2 * e i 2 β 1 1 * Δ 2 Δ 2 * Δ 2 1 * e i 2 β Δ 1 2 * e i 2 β Δ 1 Δ 1 * 2 2 * 2 Δ 1 * e i 2 β Δ 1 Δ 2 * e i 2 ( β γ ) Δ 1 1 * e i 2 γ 2 Δ 2 * e i 2 γ 2 1 * e i 2 ( β + γ ) ) .
S m n ( Ω , Ω ) = s m n ( cos θ ) e i m χ + i n χ ( m , n = 2 , 0 , 0 , 2 ) .
s m n ( cos θ ) = sup ( | m | · | n | ) p m n l P m n l ( cos θ ) ,
S m n ( Ω , Ω ) = sup ( | m | · s = l | n | ) + l ( ) s p m l P m s l ( μ ) P s n l ( μ ) e i s ( φ φ ) .
s 22 = s 00 = s 0 0 = s 2 2 = | | 2 s 20 = s 02 = s 2 0 = s 0 2 = Δ * s 2 0 = s 0 2 = s 02 = s 20 = Δ * s 2 2 = s 00 = s 0 0 = s 22 = | Δ | 2 .
( S CP ( Ω , Ω ) ) β = 0 γ = 0 = [ 1 Δ 2 Δ 1 2 ] × [ 2 * Δ 1 * Δ 2 * 1 * ]
S ( Ω , Ω ) = tr S ( Ω , Ω ) ;
S S ( Ω , Ω ) = T S tr S ( Ω , Ω ) · T S 1 .
S i k = c i c k S k i ,
c i = 1 for i = 1 , 2 , 3 c i = 1 i = 4 ;
S S ( Ω , Ω ) = q 4 · tr S ( Ω , Ω ) · q 4 .
S C = T C · S ( Ω , Ω ) · T C 1 ,
S C = Q · S ( n ) .
S C ( n ) = ( Q ) 1 T C · S ( Ω , Ω ) · ( T C ) 1 ( Q ) 1 .
S C ( n ) = ( M 2 M 3 S 23 D 23 M 4 M 1 S 41 D 41 S 24 S 31 1 2 ( S 21 + S 34 ) 1 2 ( D 21 D 34 ) D 24 D 31 1 2 ( D 21 + D 34 ) 1 2 ( S 21 S 34 ) ) = Q 1 · F .
S C ( n ) ( Ω , Ω ) = q 4 · tr S C ( n ) ( Ω , Ω ) · q 4 .
S m n ( Ω , Ω ) = s m n ( cos θ ) e i ( m β + n γ ) ( m , n = 2 , 0 0 , 2 ) .
S ( Ω ˆ , Ω ˆ ) = tr * S ( Ω , Ω ) ,
S i k = d i d k Ω ˆ k i ,
d i = 1 for i = 2 , 3 d i = 1 for i = 1 , 4.
tr * S q * · tr S · q * .
S ( Ω ˆ , Ω ˆ ) = tr * S ( Ω , Ω ) = q * · tr S ( Ω , Ω ) · q * .
S S ( Ω ˆ , Ω ˆ ) = T S · q * tr S ( Ω , Ω ) · q * · T S 1 = T S · q * · tr T S · tr T S 1 · tr S ( Ω , Ω ) · tr T S · tr T S 1 · q * · T S 1 = T S · q * · tr T S · tr S S ( Ω , Ω ) · tr T S 1 · q * · T S 1 ,
T S · q * · tr T S = 2 q 3 , tr T S 1 · q * · T S 1 = 1 2 q 3 ,
q 3 ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) .
S S ( Ω ˆ , Ω ˆ ) = q 3 · tr S S ( Ω , Ω ) · q 3 .
[ S S ( Ω ˆ · Ω ˆ ) ] i k = c i c k [ S S ( Ω , Ω ) ] k i , ( i , k = 1 , 2 , 3 , 4 ) ,
c i = 1 for i = 1 , 2 , 4 c i = 1 for i = 3.
S C ( n ) ( Ω ˆ , Ω ˆ ) = q 3 · tr S C ( n ) ( Ω , Ω ) · q 3 .
S CP ( Ω ˆ , Ω ˆ ) = tr S CP ( Ω , Ω ) ;
A i i ( n ˆ , n ˆ ) = A i i ( n ˆ , n ˆ ) ( i = l , r ) A l r ( n ˆ , n ˆ ) = A r l ( n ˆ , n ˆ ) A r l ( n ˆ , n ˆ ) = A l r ( n ˆ , n ˆ )
A i i ( n ˆ , n ˆ ) = A i i ( n ˆ , n ˆ ) ( i = l , r ) A l r ( n ˆ , n ˆ ) = A r l ( n ˆ , n ˆ ) A r l ( n ˆ , n ˆ ) = A l r ( n ˆ , n ˆ ) .
n ˆ · n ˆ = n ˆ · n ˆ = ( n ˆ ) · ( n ˆ ) = cos θ ,
trSS(Ω,Ω)=trTS1·trS(Ω,Ω)trTS.
SS(Ω,Ω)=TS·trTS·trTS1S(Ω,Ω)trTStrTS1·TS1=TS·trTStrSS(Ω,Ω)·trTS1·TS1,
TS·trTS=2q4,trTS1TS1=12q4,