Abstract

Although there are sixteen elements of the Stokes matrix, they are constructed from basically four amplitudes and three phase differences. This of course implies that there exist nine independent relationships connecting the elements. These relationships are equalities for scattering by a single particle in a fixed orientation and in a fixed direction. When Stokes matrices from an ensemble of particles differing in size, orientation, morphology, or optical properties are added incoherently, only six equalities become one-way inequalities. These relations will prove to be very useful for providing consistency checks on experimental measurements of all sixteen elements.

© 1981 Optical Society of America

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References

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  1. H.C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 5.
  2. A. C. Holland, G. Gagne, Appl. Opt. 9, 1113 (1970).
    [CrossRef] [PubMed]
  3. R. J. Perry, A. J. Hunt, D. R. Huffman, Appl. Opt. 17, 2700 (1978).
    [CrossRef] [PubMed]
  4. R. C. Thompson, J. R. Bottiger, E. S. Fry, Appl. Opt. 19, 1323 (1980).
    [CrossRef] [PubMed]
  5. K. D. Abhyankar, A. L. Fymat, J. Math. Phys. 10, 1935 (1969).
    [CrossRef]

1980

1978

1970

1969

K. D. Abhyankar, A. L. Fymat, J. Math. Phys. 10, 1935 (1969).
[CrossRef]

Appl. Opt.

J. Math. Phys.

K. D. Abhyankar, A. L. Fymat, J. Math. Phys. 10, 1935 (1969).
[CrossRef]

Other

H.C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 5.

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Figures (2)

Fig. 1
Fig. 1

Plot of Σ i , j F ij 2; i,j = 1,2,3,4, for a single sphere as a function of scattering angle for a 3° angular average. The computations were carried out using Mie theory for a size parameter of 7.9109 and a real refractive index of 1.6146. Also shown are the normalized elements F12, F33, and F43.

Fig. 2
Fig. 2

Same as Fig. 1 except a Gaussian size distribution of spheres was used with a mean size parameter of 7.9109 and a standard deviation of the size parameter of 0.117. No angular averaging was done for this calculation.

Equations (55)

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A j = α j exp ( i β j ) , j = 1 , 2 , 3 , 4 ,
= β 1 β 2 ,
δ = β 3 β 2 ,
γ = β 1 β 4 ,
σ = β 4 β 2 ,
λ = β 1 β 3 ,
η = β 4 β 3 .
f 11 = ( α 1 2 + α 2 2 + α 3 2 + α 4 2 ) / 2 ;
f 12 = ( α 1 2 + α 2 2 α 3 2 + α 4 2 ) / 2 ;
f 13 = α 2 α 3 cos δ + α 1 α 4 cos γ ;
f 14 = α 2 α 3 sin δ α 1 α 4 sin γ ;
f 21 = ( α 1 2 + α 2 2 + α 3 2 α 4 2 ) / 2 ;
f 22 = ( α 1 2 + α 2 2 α 3 2 α 4 2 ) / 2 ;
f 23 = α 2 α 3 cos δ α 1 α 4 cos γ ;
f 24 = α 2 α 3 sin δ + α 1 α 4 sin γ ;
f 31 = α 2 α 4 cos σ + α 1 α 3 cos λ ;
f 32 = α 2 α 4 cos σ α 1 α 3 cos λ ;
f 33 = α 1 α 2 cos + α 3 α 4 cos η ;
f 34 = α 1 α 2 sin + α 3 α 4 sin η ;
f 41 = α 2 α 4 sin σ + α 1 α 3 sin λ ;
f 42 = α 2 α 4 sin σ α 1 α 3 sin λ ;
f 43 = α 1 α 2 sin + α 3 α 4 sin η ;
f 44 = α 1 α 2 cos α 3 α 4 cos η .
( f 11 + f 22 ) 2 ( f 12 + f 21 ) 2 = ( f 33 + f 44 ) 2 + ( f 43 f 34 ) 2 = 4 α 1 2 α 2 2 ;
( f 11 f 22 ) 2 ( f 21 f 12 ) 2 = ( f 33 f 44 ) 2 + ( f 43 + f 34 ) 2 = 4 α 3 2 α 4 2 ;
( f 11 + f 21 ) 2 ( f 12 + f 22 ) 2 = ( f 13 + f 23 ) 2 + ( f 14 + f 24 ) 2 = 4 α 2 2 α 3 2 ;
( f 11 f 21 ) 2 ( f 12 f 22 ) 2 = ( f 13 f 23 ) 2 + ( f 14 f 24 ) 2 = 4 α 1 2 α 4 2 ;
( f 11 + f 12 ) 2 ( f 21 + f 22 ) 2 = ( f 31 + f 32 ) 2 + ( f 41 + f 42 ) 2 = 4 α 2 2 α 4 2 ;
( f 11 f 12 ) 2 ( f 21 f 22 ) 2 = ( f 31 f 32 ) 2 + ( f 41 f 42 ) 2 = 4 α 1 2 α 3 2 ;
f 13 f 14 f 23 f 24 = f 33 f 34 + f 43 f 44 = 2 α 1 α 2 α 3 α 4 sin ( β 1 β 2 + β 3 β 4 ) ;
f 14 f 23 f 13 f 24 = f 42 f 31 f 41 f 32 = 2 α 1 α 2 α 3 α 4 sin ( β 1 + β 2 β 3 β 4 ) ;
f 31 f 41 f 32 f 42 = f 33 f 43 + f 34 f 44 = 2 α 1 α 2 α 3 α 4 sin ( β 1 β 2 β 3 + β 4 ) .
f 33 2 f 34 2 + f 43 2 f 44 2 = f 13 2 f 14 2 f 23 2 + f 24 2 = 4 α 1 α 2 α 3 α 4 cos ( β 1 β 2 + β 3 β 4 ) ,
f 33 2 f 43 2 + f 34 2 f 44 2 = f 31 2 f 41 2 f 32 2 + f 42 2 = 4 α 1 α 2 α 3 α 4 cos ( β 1 β 2 β 3 + β 4 ) ,
f 31 2 f 32 2 + f 41 2 f 42 2 = f 14 2 f 24 2 + f 13 2 f 23 2 = 4 α 1 α 2 α 3 α 4 cos ( β 1 + β 2 β 3 β 4 ) .
i , j f ij 2 = 4 f 11 2 , i , j = 1 , 2 , 3 , 4 .
f 31 = f 32 = f 13 = f 23 = f 14 = f 41 = f 42 = f 24 = 0 ,
f 11 = f 22 , f 12 = f 21 , f 33 = f 44 , f 34 = f 43 ,
f 11 2 = f 12 2 + f 33 2 + f 34 2 .
( f 11 T + f 22 T ) 2 ( f 12 T + f 21 T ) 2 = 4 i , j α 1 i α 1 i α 2 j α 2 j ,
( f 33 T + f 44 T ) 2 + ( f 43 T f 34 T ) 2 = 4 i , j α 1 i α 2 i α 1 j α 2 j cos ( i j ) .
4 i , j α 1 i α 1 i α 2 j α 2 j = 2 i , j ( α 1 i α 1 i α 2 j α 2 j + α 1 j α 1 j α 2 i α 2 i ) = 2 i , j ( α 1 i α 2 j α 1 j α 2 i ) 2 + 4 i , j α 1 i α 2 i α 1 j α 2 j 4 i , j α 1 i α 2 i α 1 j α 2 j 4 i , j α 1 i α 2 i α 1 j α 2 j cos ( i j ) ,
( f 11 T + f 22 T ) 2 ( f 12 T + f 21 T ) 2 ( f 33 T + f 44 T ) 2 + ( f 43 T f 34 T ) 2 .
( f 11 T + f 22 T ) 2 ( f 12 T + f 21 T ) 2 ( f 33 T + f 44 T ) 2 + ( f 43 T f 34 T ) 2 ,
( f 11 T f 22 T ) 2 ( f 21 T f 12 T ) 2 ( f 33 T f 44 T ) 2 + ( f 43 T f 34 T ) 2 ,
( f 11 T + f 21 T ) 2 ( f 12 T + f 22 T ) 2 ( f 13 T + f 23 T ) 2 + ( f 14 T + f 24 T ) 2 ,
( f 11 T f 21 T ) 2 ( f 12 T f 22 T ) 2 ( f 13 T f 23 T ) 2 + ( f 14 T f 24 T ) 2 ,
( f 11 T + f 12 T ) 2 ( f 21 T + f 22 T ) 2 ( f 31 T + f 32 T ) 2 + ( f 41 T + f 42 T ) 2 ,
( f 11 T f 12 T ) 2 ( f 21 T f 22 T ) 2 ( f 31 T f 32 T ) 2 + ( f 41 T f 42 T ) 2 .
i , j ( f ij T ) 2 4 ( f 11 T ) 2 i , j = 1 , 2 , 3 , 4 .
2 ij α 1 j α 2 i α 3 j α 4 i sin ( β 1 j β 2 i + β 3 j β 4 i ) ,
2 ij α 1 i α 2 i α 3 j α 4 j sin ( β 1 j β 2 j + β 3 i β 4 i ) ,
i , j ( F ij T ) 2 4 ; i , j = 1 , 2 , 3 , 4 .
i , j F ij 2
i , j F ij 2

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