Abstract

The properties of the polarization ellipse are deduced in terms of the ratio of the Cartesian components of the complex electric vector of a beam of radiation by utilizing the Argand representation of a real two-dimensional vector as a complex number.

The two components of a beam that are accepted and rejected by a polarizer or a radio antenna are considered as orthogonal components in the directions of two complex orthonormal vectors. The intensities of the corresponding components of a polarized beam are derived and represented on the Poincare sphere. The methods are then applied to the important radio case of the Faraday effect in a uniform magneto-ionic medium. Finally, the measurable quantities characterizing a beam of partially polarized radiation are obtained from a diagonalization of the complex polarization tensor that specifies the beam.

© 1959 Optical Society of America

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References

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  1. R. Clark Jones, J. Opt. Soc. Am. 31, 488 (1941).
    [Crossref]
  2. H. Hurwitz and R. Clark Jones, J. Opt. Soc. Am. 31, 493 (1941)
    [Crossref]
  3. R. Clark Jones, J. Opt. Soc. Am. 31, 500 (1941).
    [Crossref]
  4. R. Clark Jones, J. Opt. Soc. Am. 32, 487 (1942).
  5. R. Clark Jones, J. Opt. Soc. Am. 37, 107 (1947).
    [Crossref]
  6. R. Clark Jones, J. Opt. Soc. Am. 37, 110 (1947).
    [Crossref]
  7. R. Clark Jones, J. Opt. Soc. Am. 38, 671 (1948).
    [Crossref]
  8. R. Clark Jones, J. Opt. Soc. Am. 46, 126 (1956).
    [Crossref]
  9. M. Richartz and H.-Y. Hsü, J. Opt. Soc. Am. 39, 136 (1949).
    [Crossref]
  10. M. H. Cohen, Proc. Inst. Radio Engrs. 46, 172 (1958).
  11. Hatanaka, Suzuki, and Tsuchiya, Proc. Japan. Acad. 31, 81 (1955).
  12. T. Hatanaka, Publs. Astron. Soc. Japan 8, 73 (1956).
  13. M. H. Cohen, Proc. Inst. Radio Engrs. 46, 183 (1958).
  14. H. Poincaré, Théorie Mathématique de la Lumiére II (Gauthier-Villars, Paris, 1892), Chap. XII.
  15. F. E. Wright, J. Opt. Soc. Am. 20, 529 (1930).
    [Crossref]
  16. J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), Sec. 2.20.
  17. K. C. Westfold, Am. Math. Monthly 64, 174 (1957).
    [Crossref]
  18. G. N. Ramachandran and S. Ramaseshan, J. Opt. Soc. Am. 42, 49 (1952).
    [Crossref]
  19. K. C. Westfold, J. Atmos. Terr. Phys. 1, 152 (1951).
    [Crossref]
  20. L. Landau and E. Lifshitz, The Classical Theory of Fields (Addison-Wesley Press, Inc., Cambridge, 1951), Sec. 6–7.
  21. Hatanaka, Suzuki, and Tsuchiya, Publs Astron. Soc. Japan 7, 114 (1955).
  22. S. Suzuki and A. Tsuchiya, Proc. Inst. Radio Engrs. 46, 190 (1958).

1958 (3)

M. H. Cohen, Proc. Inst. Radio Engrs. 46, 172 (1958).

M. H. Cohen, Proc. Inst. Radio Engrs. 46, 183 (1958).

S. Suzuki and A. Tsuchiya, Proc. Inst. Radio Engrs. 46, 190 (1958).

1957 (1)

K. C. Westfold, Am. Math. Monthly 64, 174 (1957).
[Crossref]

1956 (2)

T. Hatanaka, Publs. Astron. Soc. Japan 8, 73 (1956).

R. Clark Jones, J. Opt. Soc. Am. 46, 126 (1956).
[Crossref]

1955 (2)

Hatanaka, Suzuki, and Tsuchiya, Publs Astron. Soc. Japan 7, 114 (1955).

Hatanaka, Suzuki, and Tsuchiya, Proc. Japan. Acad. 31, 81 (1955).

1952 (1)

1951 (1)

K. C. Westfold, J. Atmos. Terr. Phys. 1, 152 (1951).
[Crossref]

1949 (1)

1948 (1)

1947 (2)

1942 (1)

R. Clark Jones, J. Opt. Soc. Am. 32, 487 (1942).

1941 (3)

1930 (1)

Clark Jones, R.

Cohen, M. H.

M. H. Cohen, Proc. Inst. Radio Engrs. 46, 172 (1958).

M. H. Cohen, Proc. Inst. Radio Engrs. 46, 183 (1958).

Hatanaka,

Hatanaka, Suzuki, and Tsuchiya, Proc. Japan. Acad. 31, 81 (1955).

Hatanaka, Suzuki, and Tsuchiya, Publs Astron. Soc. Japan 7, 114 (1955).

Hatanaka, T.

T. Hatanaka, Publs. Astron. Soc. Japan 8, 73 (1956).

Hsü, H.-Y.

Hurwitz, H.

Landau, L.

L. Landau and E. Lifshitz, The Classical Theory of Fields (Addison-Wesley Press, Inc., Cambridge, 1951), Sec. 6–7.

Lifshitz, E.

L. Landau and E. Lifshitz, The Classical Theory of Fields (Addison-Wesley Press, Inc., Cambridge, 1951), Sec. 6–7.

Poincaré, H.

H. Poincaré, Théorie Mathématique de la Lumiére II (Gauthier-Villars, Paris, 1892), Chap. XII.

Ramachandran, G. N.

Ramaseshan, S.

Richartz, M.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), Sec. 2.20.

Suzuki,

Hatanaka, Suzuki, and Tsuchiya, Publs Astron. Soc. Japan 7, 114 (1955).

Hatanaka, Suzuki, and Tsuchiya, Proc. Japan. Acad. 31, 81 (1955).

Suzuki, S.

S. Suzuki and A. Tsuchiya, Proc. Inst. Radio Engrs. 46, 190 (1958).

Tsuchiya,

Hatanaka, Suzuki, and Tsuchiya, Publs Astron. Soc. Japan 7, 114 (1955).

Hatanaka, Suzuki, and Tsuchiya, Proc. Japan. Acad. 31, 81 (1955).

Tsuchiya, A.

S. Suzuki and A. Tsuchiya, Proc. Inst. Radio Engrs. 46, 190 (1958).

Westfold, K. C.

K. C. Westfold, Am. Math. Monthly 64, 174 (1957).
[Crossref]

K. C. Westfold, J. Atmos. Terr. Phys. 1, 152 (1951).
[Crossref]

Wright, F. E.

Am. Math. Monthly (1)

K. C. Westfold, Am. Math. Monthly 64, 174 (1957).
[Crossref]

J. Atmos. Terr. Phys. (1)

K. C. Westfold, J. Atmos. Terr. Phys. 1, 152 (1951).
[Crossref]

J. Opt. Soc. Am. (11)

Proc. Inst. Radio Engrs. (3)

M. H. Cohen, Proc. Inst. Radio Engrs. 46, 172 (1958).

M. H. Cohen, Proc. Inst. Radio Engrs. 46, 183 (1958).

S. Suzuki and A. Tsuchiya, Proc. Inst. Radio Engrs. 46, 190 (1958).

Proc. Japan. Acad. (1)

Hatanaka, Suzuki, and Tsuchiya, Proc. Japan. Acad. 31, 81 (1955).

Publs Astron. Soc. Japan (1)

Hatanaka, Suzuki, and Tsuchiya, Publs Astron. Soc. Japan 7, 114 (1955).

Publs. Astron. Soc. Japan (1)

T. Hatanaka, Publs. Astron. Soc. Japan 8, 73 (1956).

Other (3)

H. Poincaré, Théorie Mathématique de la Lumiére II (Gauthier-Villars, Paris, 1892), Chap. XII.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), Sec. 2.20.

L. Landau and E. Lifshitz, The Classical Theory of Fields (Addison-Wesley Press, Inc., Cambridge, 1951), Sec. 6–7.

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Figures (4)

F. 1
F. 1

The polarization ellipse. Note: The angle φ is incorrectly represented. It should properly denote the position of the projection of Ɛ on to the auxiliary circle.

F. 2
F. 2

The projection from the Argand plane of Q on to the Poincaré sphere.

F. 3
F. 3

The representation of the ellipse characteristics on the Poincaré sphere.

F. 4
F. 4

The representation of Δa on the Poincaré sphere.

Equations (76)

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E = E 0 e i ϑ , B = B 0 e i ϑ ,
ϑ = ω t k n · r ,
ω B 0 = k n × E 0 ,
Ɛ = R E ξ + i R E η .
Q = E η / E ξ ,
E ξ = a ξ exp { i ( ϑ ϕ ξ ) } , E η = a η exp { i ( ϑ ϕ η ) } .
Ɛ = Ɛ + + Ɛ ,
Ɛ + = 1 2 a ξ ( 1 i Q ) * exp { i ( ϑ ϕ ξ ) } , Ɛ = 1 2 a ξ ( 1 + i Q ) exp { i ( ϑ ϕ ξ ) } ,
Ɛ e i ψ = a cos φ + i b sin φ .
( 1 i Q ) * = | 1 i Q | e i χ + , 1 + i Q = | 1 + i Q | e i χ ,
ψ = ϑ ϕ ξ + χ + = ϑ + ϕ ξ + χ ,
2 ψ = χ + + χ .
1 + i Q 1 i Q = | 1 + i Q 1 i Q | e 2 i ψ .
φ = ϑ ϕ ξ + 1 2 ( χ + χ ) ,
Ɛ = 1 2 a ξ e i ψ { ( | 1 i Q | + | 1 + i Q | ) cos φ + i ( | 1 i Q | | 1 + i Q | ) sin φ } .
cos λ = | 1 i Q | + | 1 + i Q | 2 ( 1 + | Q | 2 ) 1 2 , sin λ = | 1 i Q | | 1 + i Q | 2 ( 1 + | Q | 2 ) 1 2 ,
a ξ 2 ( 1 + | Q | 2 ) = | E | 2 , Ɛ = | E | e i ψ ( cos λ cos φ + i sin λ sin φ ) .
1 + i Q 1 i Q = 1 tan λ 1 + tan λ e 2 i ψ .
Q = tan 1 2 θ e i ϕ ,
1 + i Q 1 i Q = cos θ + i sin θ cos ϕ 1 + sin θ sin ϕ ,
sin 2 λ = sin θ sin ϕ
tan 2 ψ = tan θ cos ϕ .
Q Q * = 1 .
I = ( R E ) 2 .
I = 1 2 | E 0 | 2 .
e a · e a * = 1
e a = ( 1 , Q a , 0 ) / ( 1 + | Q a | 2 ) 1 2 ,
e a · e a * = e a * · e a = 0 ,
Q a * Q a = 1 .
E = E a e a + E a e a ,
E a = E · e a * , E a = E · e a * .
e a = ( 1 , i , 0 ) / ( 2 ) 1 2 , e a = ( 1 , i , 0 ) / ( 2 ) 1 2 ,
I = I a + I a ,
I a = 1 2 | E 0 a | 2 , I a = 1 2 | E 0 a | 2 .
E 0 = | E 0 | e ,
e = exp ( i ϕ ξ ) ( 1 , Q , 0 ) / ( 1 + | Q | 2 ) 1 2 .
I a = I | e · e a * | 2 , I a = I | e · e a * | 2 ,
| e · e a * | 2 = 1 + 2 R ( Q Q a * ) + | Q | 2 | Q a | 2 ( 1 + | Q | 2 ) ( 1 + | Q a | 2 ) ,
1 + | Q | 2 | Q a | 2 = 1 2 ( 1 + | Q | 2 ) ( 1 + | Q a | 2 ) + 1 2 ( 1 | Q | 2 ) ( 1 | Q a | 2 ) , | e · e a * | 2 = 1 2 { 1 + cos θ cos θ a + sin θ sin θ a cos ( ϕ ϕ a ) } .
cos θ cos θ a + sin θ sin θ a cos ( ϕ ϕ a ) = cos Δ a ,
| e · e a * | 2 = 1 2 ( 1 + cos Δ a ) , | e · e a * | 2 = 1 2 ( 1 cos Δ a ) ,
I a = 1 2 I ( 1 + cos Δ a ) , I a = 1 2 I ( 1 cos Δ a ) .
cos A P = cos θ = cos 2 λ cos 2 ψ , cos B P = sin θ cos ϕ = cos 2 λ sin 2 ψ , cos C P = sin θ sin ϕ = sin 2 λ ,
cos 2 A P + cos 2 B P + cos 2 C P = 1 .
Q α * Q β = Q α Q β = 1 .
μ α = ( 1 x ) 1 2 { 1 + x y | cos Θ | 2 ( 1 x ) } , μ β = ( 1 x ) 1 2 { 1 x y | cos Θ | 2 ( 1 x ) } ,
Q α = i | cos Θ | cos Θ , Q β = i | cos Θ | cos Θ .
e α = ( 1 , i , 0 ) / ( 2 ) 1 2 , e β = ( 1 , i , 0 ) / ( 2 ) 1 2 .
E = E ξ ( 1 , Q , 0 ) .
E α = E ξ ( 1 + i Q ) / ( 2 ) 1 2 , E β = E ξ ( 1 i Q ) / ( 2 ) 1 2 ,
E α 2 = E α 1 exp ( i ω μ α d / c ) , E β 2 = E β 1 exp ( i ω μ β d / c ) .
E 2 = E α 2 e α + E β 2 e β .
1 + i Q 2 1 i Q 2 = 1 + i Q 1 1 i Q 1 e i δ ,
δ = ω ( μ α μ β ) d / c .
δ = ω d c x y cos Θ ( 1 x ) 1 2 .
μ α = ( 1 x ) 1 2 , μ β = ( 1 x ) 1 2 { 1 x y 2 2 ( 1 x ) 2 }
Q α = cos Θ / i y , Q β = i y / cos Θ .
e α = ( 1 , 0 , 0 ) , e β = ( 0 , 1 , 0 ) ,
E α = E ξ , E β = E η .
Q 2 = Q 1 e i δ ,
δ = ω d 2 c x y 2 ( 1 x ) 1 2 , .
I a = 1 2 | E 0 · e a * | 2 = 1 2 E 0 · e a * E 0 * · e a .
I = 1 2 E 0 E 0 *
I a = I : e a e a * .
I = I 1 e 1 e 1 * + I 2 e 2 e 2 * .
I a = I 1 | e 1 · e a * | 2 + I 2 | e 2 · e a * | 2 .
I = I 1 + I 2 ,
I = I : U ,
I = 1 2 I U .
U = e 1 e 1 * + e 2 e 2 * ,
I = 1 2 I u U + I p ee * ,
I a = 1 2 I u + I p | e · e a * | 2 .
p = I p / I .
I a = 1 2 I + 1 2 I p cos Δ a
I a = 1 2 I 1 2 I p cos Δ a ,
cos Δ a = cos 2 λ cos 2 ( ψ ψ a ) ,