Abstract

A unified treatment of the algebra of Stokes parameters and the coherency matrix is presented. Explicit formulas are given which relate the Jones and Mueller matrix formulations to the coherency matrix, and to themselves. The general method involves the trace of products of matrices, and the algebra is both useful and capable of generalization.

© 1969 Optical Society of America

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References

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  1. R. C. Jones, J. Opt. Soc. Am. 31, 488 (1941); J. Opt. Soc. Am. 31, 493 (1941); J. Opt. Soc. Am. 31, 500 (1941); J. Opt. Soc. Am. 32, 486 (1942); J. Opt. Soc. Am. 37, 107 (1947); J. Opt. Soc. Am. 37, 110 (1947); J. Opt. Soc. Am. 38, 671 (1948); J. Opt. Soc. Am. 46, 126 (1956).
    [CrossRef]
  2. N. G. Parke, J. Math. Phys. 28, 131 (1949).
  3. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 545.
  4. See any text on introductory quantum mechanics.
  5. G. Stokes, Trans. Cambridge Phil. Soc. 9, 399 (1852).
  6. U. Fano, Rev. Mod. Phys. 29, 74 (1957).
    [CrossRef]
  7. U. Fano, J. Opt. Soc. Am. 39, 859 (1949).
    [CrossRef]
  8. M. J. Walker, Am. J. Phys. 22, 170 (1954).
    [CrossRef]
  9. H. G. Jerrard, J. Opt. Soc. Am. 44, 634 (1954).
    [CrossRef]
  10. W. H. McMaster, Rev. Mod. Phys. 33, 8 (1961).
    [CrossRef]
  11. F. Perrin, J. Chem. Phys. 10, 415 (1942).
    [CrossRef]
  12. W. A. Shurcliff, Polarized Light (Harvard Univ. Press, Cambridge, Mass., 1962).
  13. E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley Publ. Co., Reading, Mass.1963), Ch. 9.
  14. A. S. Marathay, J. Opt. Soc. Am. 55, 969 (1965).
  15. G. N. Ramachandran and S. Ramaseshan, J. Opt. Soc. Am. 42, 49 (1952).
    [CrossRef]
  16. B. H. Billings, J. Opt. Soc. Am. 42, 12 (1952).
    [CrossRef]
  17. B. H. Billings, J. Opt. Soc. Am. 41, 966 (1951).
    [CrossRef]
  18. The symbol σ¯ is used in place of the unavailable lightface form ỏ.

1965 (1)

1961 (1)

W. H. McMaster, Rev. Mod. Phys. 33, 8 (1961).
[CrossRef]

1957 (1)

U. Fano, Rev. Mod. Phys. 29, 74 (1957).
[CrossRef]

1954 (2)

1952 (2)

1951 (1)

1949 (2)

U. Fano, J. Opt. Soc. Am. 39, 859 (1949).
[CrossRef]

N. G. Parke, J. Math. Phys. 28, 131 (1949).

1942 (1)

F. Perrin, J. Chem. Phys. 10, 415 (1942).
[CrossRef]

1941 (1)

1852 (1)

G. Stokes, Trans. Cambridge Phil. Soc. 9, 399 (1852).

Billings, B. H.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 545.

Fano, U.

U. Fano, Rev. Mod. Phys. 29, 74 (1957).
[CrossRef]

U. Fano, J. Opt. Soc. Am. 39, 859 (1949).
[CrossRef]

Jerrard, H. G.

Jones, R. C.

Marathay, A. S.

McMaster, W. H.

W. H. McMaster, Rev. Mod. Phys. 33, 8 (1961).
[CrossRef]

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley Publ. Co., Reading, Mass.1963), Ch. 9.

Parke, N. G.

N. G. Parke, J. Math. Phys. 28, 131 (1949).

Perrin, F.

F. Perrin, J. Chem. Phys. 10, 415 (1942).
[CrossRef]

Ramachandran, G. N.

Ramaseshan, S.

Shurcliff, W. A.

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

Stokes, G.

G. Stokes, Trans. Cambridge Phil. Soc. 9, 399 (1852).

Walker, M. J.

M. J. Walker, Am. J. Phys. 22, 170 (1954).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 545.

Am. J. Phys. (1)

M. J. Walker, Am. J. Phys. 22, 170 (1954).
[CrossRef]

J. Chem. Phys. (1)

F. Perrin, J. Chem. Phys. 10, 415 (1942).
[CrossRef]

J. Math. Phys. (1)

N. G. Parke, J. Math. Phys. 28, 131 (1949).

J. Opt. Soc. Am. (7)

Rev. Mod. Phys. (2)

U. Fano, Rev. Mod. Phys. 29, 74 (1957).
[CrossRef]

W. H. McMaster, Rev. Mod. Phys. 33, 8 (1961).
[CrossRef]

Trans. Cambridge Phil. Soc. (1)

G. Stokes, Trans. Cambridge Phil. Soc. 9, 399 (1852).

Other (5)

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 545.

See any text on introductory quantum mechanics.

The symbol σ¯ is used in place of the unavailable lightface form ỏ.

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

E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley Publ. Co., Reading, Mass.1963), Ch. 9.

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Figures (2)

Fig. 1
Fig. 1

The Poincaré sphere. Each point on the sphere is associated with a particular state of polarization of the light. By convention, the axis passes through the two points representing circular polarization, entering at the point representing left-circularly polarized light and exiting at the point representing right-circularly polarized light. Linear polarization at all orientations is associated with the equator, and elliptical polarization is associated with points between the axis and the equator.

Fig. 2
Fig. 2

Two coordinate systems for the Poincaré sphere. The conventional system (right) is in current use; the natural system (left) is better suited to theoretical use, and is used consistently throughout this paper.

Equations (67)

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E = ( E 1 E 2 ) .
E = ( E 1 * E 2 * ) .
I E E = E 1 * E 1 + E 2 * E 2 J E E = ( E 1 E 1 * E 1 E 2 * E 2 E 1 * E 2 E 2 * ) ;
I = Tr { J }
σ 0 = ( 1 0 0 1 ) σ 1 = ( 0 1 1 0 ) σ 2 = ( 0 - i i 0 ) σ 3 = ( 1 0 0 - 1 ) ,
Tr { σ μ σ ν } = 2 δ μ ν .
O = μ = 0 3 O μ σ μ O μ = 1 2 Tr { O σ μ } .
J = 1 2 μ Tr { J σ μ } σ μ = 1 2 μ S μ σ μ ,
S μ = Tr { J σ μ } .
σ = [ σ 0 σ ˆ ] = ( σ 0 σ 1 σ 2 σ 3 ) ,
S = Tr { J σ } = ( S 0 S 1 S 2 S 3 ) ,
J = 1 2 S · σ = 1 2 ( I σ 0 + Ŝ · σ ˆ ) .
ρ = J I = 1 2 ( J 2 + ŝ · σ ˆ ) ,
S = I Tr { ρ σ } ,
S 0 = Tr { J σ 0 } = E 1 E 1 * + E 2 E 2 * = I S 1 = Tr { J σ 1 } = E 1 E 2 * + E 2 E 1 * = U S 2 = Tr { J σ 2 } = - i ( E 1 E 2 * - E 2 E 1 * ) = V S 3 = Tr { J σ 3 } = E 1 E 1 * - E 2 E 2 * = Q .
ŝ ( Mueller ) = 1 I [ Q U V ] .
ŝ ( natural ) = 1 I [ U V Q ] .
( S 0 S 1 S 2 S 3 ) = ( 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 ) ( I Q U V ) .
Tr { K } = [ Tr ( K ) ] * ,
S = Tr { J σ } = [ Tr { ( J σ ) } ] * = S * .
E = A E .
J = E E = A J A .
J = 1 2 μ Tr { J σ μ } σ μ = 1 2 μ S μ σ μ ,
S μ = Tr { J σ μ } .
S = Tr { J σ } = Tr { J σ } ,
σ = A σ A .
S μ = 1 2 ν Tr { σ μ σ ν } Tr { σ ν J } ν M μ ν S ν ,
M μ ν = 1 2 Tr { σ μ σ ν } .
S = M S M = 1 2 Tr { σ σ } .
σ σ ( σ 0 σ 0 σ 0 σ 1 σ 0 σ 2 σ 0 σ 3 σ 1 σ 0 σ 1 σ 1 σ 1 σ 2 σ 1 σ 3 σ 2 σ 0 σ 2 σ 1 σ 2 σ 2 σ 2 σ 3 σ 3 σ 0 σ 3 σ 1 σ 3 σ 2 σ 3 σ 3 ) .
J 4 = 1 2 Tr { σ σ } = ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) .
M μ ν = 1 2 Tr { σ μ σ ν } = 1 2 Tr { A σ μ A σ ν } = 1 2 Tr { [ η 1 2 Tr { A σ η } σ η ] σ μ [ λ 1 2 Tr { A σ λ } σ λ ] σ ν } = 1 8 η λ Tr { A σ η } Tr { A σ λ } Tr { σ η σ μ σ λ σ ν } = 1 4 η λ A η * A λ G μ λ ν η ,
A λ = Tr { A σ λ }
G μ λ ν η = 1 2 Tr { σ μ σ λ σ ν σ η }
A = [ a b c d ] ,
Re ( x ) ( x + x * ) / 2 ,             and             Im ( x ) ( x - x * ) / 2. M = ( 1 2 ( a 2 + b 2 + c 2 + d 2 ) Re ( a * b + c * d ) i Im ( a * b + c * d ) 1 2 ( a 2 - b 2 + c 2 - d 2 ) Re ( a * c + b * d ) Re ( a * d + b * c ) i Im ( a * d - b * c ) Re ( a * c - b * d ) - i Im ( a * c + b * d ) - i Im ( a * d + b * c ) Re ( a * d - b * c ) - i Im ( a * c - b * d ) 1 2 ( a 2 + b 2 - c 2 - d 2 ) Re ( a * b - c * d ) i Im ( a * b - c * d ) 1 2 ( a 2 - b 2 - c 2 + d 2 ) ) .
M 00 + M 11 + M 22 + M 33 = Tr { A } Tr { A } .
A = exp [ - i n ˆ · σ ˆ ( ϕ / 2 ) ] ,
M = 1 2 Tr { exp [ i n ˆ · σ ˆ ( ϕ / 2 ) ] σ exp [ - i n ˆ · σ ˆ ( ϕ / 2 ) ] σ } ,
M μ ν = 1 2 Tr { exp [ i n ˆ · σ ˆ ( ϕ / 2 ) σ μ exp [ - i n ˆ · σ ˆ ( ϕ / 2 ) σ μ } .
exp [ ± i n ˆ · σ ˆ ( ϕ / 2 ) ] = cos ( ϕ / 2 ) ± i n ˆ · σ ˆ sin ( ϕ / 2 ) ,
M μ ν = 1 2 Tr { σ μ σ ν } + i 2 sin ϕ 2 cos ϕ 2 T r { [ σ μ , n ˆ · σ ˆ ] σ ν } + 1 2 sin 2 ϕ 2 Tr { n ˆ · σ ˆ [ σ μ , n ˆ · σ ˆ ] σ ν } M μ ν ( 0 ) + M μ ν ( 1 ) + M μ ν ( 2 ) .
[ σ μ , n ˆ · σ ˆ ] = 2 i δ μ μ ˆ ( ê μ × n ˆ ) · σ ˆ ,
δ μ μ ˆ 1 - δ μ 0
M μ ν ( 0 ) = δ μ ν M μ ν ( 1 ) = sin ϕ δ μ μ ˆ δ ν ν ˆ n ˆ · ( ê μ × ê ν ) M μ ν ( 2 ) = ( 1 - cos ϕ ) δ μ μ ˆ δ ν ν ˆ [ n ˆ · ê μ n ˆ · ê ν - ê μ · ê ν ] .
M ( i ) = ( 1 0 0 0 0 0 0 ( m ( i ) ) ) ,             i = 0 , 1 , 2
m ( 0 ) = J 3 m ( 1 ) = sin ϕ n ˆ · P m ( 2 ) = ( 1 - cos ϕ ) n ˆ · T · n ˆ ,
J 3 = ( 1 0 0 0 1 0 0 0 1 )             P = ( 0 ê z - ê y - ê z 0 ê x ê y - ê x 0 )             T = ( ( ê x ê x - 1 ) ê x ê y ê x ê z e y e x ( ê y ê y - 1 ) ê y ê z ê z ê y ê z ê y ( ê z ê z - 1 ) ) .
n ˆ = α ê x + β ê y + γ ê z
n ˆ · P = ( 0 γ - β - γ 0 α β - α 0 )             n ˆ · T · n ˆ = ( α 2 - 1 α β α γ β α β 2 - 1 β γ γ α γ β γ 2 - 1 ) .
M = ( 1 0 0 0 0 cos ϕ + 2 α 2 sin 2 ϕ 2 γ sin ϕ + 2 α β sin 2 ϕ 2 - β sin ϕ + 2 α γ sin 2 ϕ 2 0 - γ sin ϕ + 2 β α sin 2 ϕ 2 cos ϕ + 2 β 2 sin 2 ϕ 2 α sin ϕ + 2 β γ sin 2 ϕ 2 0 β sin ϕ + 2 γ α sin 2 ϕ 2 - α sin ϕ + 2 γ β sin 2 ϕ 2 cos ϕ + 2 γ 2 sin 2 ϕ 2 ) .
M = ( 1 0 0 0 0 cos ϕ 0 - sin ϕ 0 0 1 0 0 sin ϕ 0 cos ϕ ) .
exp [ - i n ˆ · σ ( ϕ / 2 ) ] = ( cos ϕ 2 + i sin ϕ 2 cos ϑ i sin ϕ 2 e i φ sin ϑ i sin ϕ 2 e - i φ sin ϑ cos ϕ 2 - i sin ϕ 2 cos ϑ )
α = sin ϑ cos φ β = sin ϑ sin φ γ = cos ϑ .
σ ¯ 0 = ( 1 0 0 0 ) σ ¯ 1 = ( 0 1 0 0 ) σ ¯ 2 = ( 0 0 1 0 ) σ ¯ 3 = ( 0 0 0 1 ) = ( σ ¯ 0 σ ¯ 1 σ ¯ 2 σ ¯ 3 ) ,
Tr { } = J 4 ,
J = λ Tr { J σ ¯ λ } σ ¯ λ = λ C λ σ ¯ λ = C · ,
C = Tr { J }
C = ( E 1 E 1 * E 1 E 2 * E 2 E 1 * E 2 E 2 * ) = 1 2 ( I + Q U + i V U - i V I - Q ) .
σ = Tr { σ } · .
Tr { J σ } = Tr { J Tr { σ } · } = Tr { σ } · Tr { J } .
T = Q C ,
T = Tr { σ } = ( 1 0 0 1 0 1 1 0 0 - i i 0 1 0 0 - 1 ) .
= 1 2 Tr { σ } · σ , Tr { J } = 1 2 Tr { σ } · Tr { J σ } , C = T - 1 S ,
T - 1 = 1 2 Tr { σ } = 1 2 ( 1 0 0 1 0 1 i 0 0 1 - i 0 1 0 0 - 1 ) .
C = N C ,
N = Tr { } .