Abstract

It is pointed out that the concept of the Poincaré sphere appreciably simplifies the mathematical treatment of phenomena accompanying the passage of polarized light through a medium which exhibits birefringence, optical activity or both simultaneously. This is exemplified by using the Poincaré sphere to evolve techniques which could be used for determining the true Faraday rotation in the presence of birefringence. When birefringence is present, measurements made with the half-shade at the polarizer and analyzer ends are not equivalent. In either arrangement, the errors introduced as a result of birefringence are largely reduced by taking the mean of two measurements for opposite directions of the field. Formulae are also derived by which the magnitudes of the error can be calculated for the particular experimental set up, knowing the value of the birefringence. In certain cases, even this need not be known, and the true rotation can be determined purely from measurements of the apparent rotations for two different azimuths of the incident plane of polarization.

© 1952 Optical Society of America

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References

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  1. H. Poincaré, Traite de la lumiere, Paris 2, 165 (1892).
  2. F. A. Wright, J. Opt. Soc. Am. 20, 529 (1930).
    [Crossref]
  3. C. A. Skinner, J. Opt. Soc. Am. 10, 490 (1925).
  4. J. Becquerel, Communs. Phys., Lab. Univ. Leiden No. 91C (1928); 211a (1930).
  5. G. Bruhat and P. Grivet, J. phys. et radium 6, 12 (1935).
    [Crossref]
  6. G. Szivessy and C. Schweers, Ann. Physik 1, 891 (1929).
    [Crossref]
  7. S. Ramaseshan and V. Chandrasekharan, Current (India) Sci. 20, 150 (1951). S. Ramaseshan, Proc. Indian Acad. Sci. 34A, 32 (1951).
  8. G. N. Ramachandran and V. Chandrasekharan, Proc. Indian Acad. Sci. 33A, 199 (1951).
  9. G. Szivessy, Handbuch der Physik (Julius Springer, Berlin, Germany), Vol. 20, Ch. 11.
  10. H. Schutz, Handbuch Exp. Phys. 26, 48 (1936).
  11. M. Chauvin, Compt. rend. 102, 972 (1886).

1951 (2)

S. Ramaseshan and V. Chandrasekharan, Current (India) Sci. 20, 150 (1951). S. Ramaseshan, Proc. Indian Acad. Sci. 34A, 32 (1951).

G. N. Ramachandran and V. Chandrasekharan, Proc. Indian Acad. Sci. 33A, 199 (1951).

1936 (1)

H. Schutz, Handbuch Exp. Phys. 26, 48 (1936).

1935 (1)

G. Bruhat and P. Grivet, J. phys. et radium 6, 12 (1935).
[Crossref]

1930 (1)

1929 (1)

G. Szivessy and C. Schweers, Ann. Physik 1, 891 (1929).
[Crossref]

1928 (1)

J. Becquerel, Communs. Phys., Lab. Univ. Leiden No. 91C (1928); 211a (1930).

1925 (1)

C. A. Skinner, J. Opt. Soc. Am. 10, 490 (1925).

1892 (1)

H. Poincaré, Traite de la lumiere, Paris 2, 165 (1892).

1886 (1)

M. Chauvin, Compt. rend. 102, 972 (1886).

Becquerel, J.

J. Becquerel, Communs. Phys., Lab. Univ. Leiden No. 91C (1928); 211a (1930).

Bruhat, G.

G. Bruhat and P. Grivet, J. phys. et radium 6, 12 (1935).
[Crossref]

Chandrasekharan, V.

S. Ramaseshan and V. Chandrasekharan, Current (India) Sci. 20, 150 (1951). S. Ramaseshan, Proc. Indian Acad. Sci. 34A, 32 (1951).

G. N. Ramachandran and V. Chandrasekharan, Proc. Indian Acad. Sci. 33A, 199 (1951).

Chauvin, M.

M. Chauvin, Compt. rend. 102, 972 (1886).

Grivet, P.

G. Bruhat and P. Grivet, J. phys. et radium 6, 12 (1935).
[Crossref]

Poincaré, H.

H. Poincaré, Traite de la lumiere, Paris 2, 165 (1892).

Ramachandran, G. N.

G. N. Ramachandran and V. Chandrasekharan, Proc. Indian Acad. Sci. 33A, 199 (1951).

Ramaseshan, S.

S. Ramaseshan and V. Chandrasekharan, Current (India) Sci. 20, 150 (1951). S. Ramaseshan, Proc. Indian Acad. Sci. 34A, 32 (1951).

Schutz, H.

H. Schutz, Handbuch Exp. Phys. 26, 48 (1936).

Schweers, C.

G. Szivessy and C. Schweers, Ann. Physik 1, 891 (1929).
[Crossref]

Skinner, C. A.

C. A. Skinner, J. Opt. Soc. Am. 10, 490 (1925).

Szivessy, G.

G. Szivessy and C. Schweers, Ann. Physik 1, 891 (1929).
[Crossref]

G. Szivessy, Handbuch der Physik (Julius Springer, Berlin, Germany), Vol. 20, Ch. 11.

Wright, F. A.

Ann. Physik (1)

G. Szivessy and C. Schweers, Ann. Physik 1, 891 (1929).
[Crossref]

Communs. Phys., Lab. Univ. Leiden No. 91C (1)

J. Becquerel, Communs. Phys., Lab. Univ. Leiden No. 91C (1928); 211a (1930).

Compt. rend. (1)

M. Chauvin, Compt. rend. 102, 972 (1886).

Current (India) Sci. (1)

S. Ramaseshan and V. Chandrasekharan, Current (India) Sci. 20, 150 (1951). S. Ramaseshan, Proc. Indian Acad. Sci. 34A, 32 (1951).

Handbuch Exp. Phys. (1)

H. Schutz, Handbuch Exp. Phys. 26, 48 (1936).

J. Opt. Soc. Am. (2)

F. A. Wright, J. Opt. Soc. Am. 20, 529 (1930).
[Crossref]

C. A. Skinner, J. Opt. Soc. Am. 10, 490 (1925).

J. phys. et radium (1)

G. Bruhat and P. Grivet, J. phys. et radium 6, 12 (1935).
[Crossref]

Proc. Indian Acad. Sci. (1)

G. N. Ramachandran and V. Chandrasekharan, Proc. Indian Acad. Sci. 33A, 199 (1951).

Traite de la lumiere, Paris (1)

H. Poincaré, Traite de la lumiere, Paris 2, 165 (1892).

Other (1)

G. Szivessy, Handbuch der Physik (Julius Springer, Berlin, Germany), Vol. 20, Ch. 11.

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Figures (6)

Fig. 1
Fig. 1

Method of representing the state of polarization of light on the Poincaré sphere.

Fig. 2
Fig. 2

Top half—Construction for determining the intensity of light transmitted by an elliptic analyzer. Bottom half—Locus of points representing the emergent light in Chauvin’s experiment.

Fig. 3
Fig. 3

The difference between putting the half-shade at the polarizer and the analyzer ends, illustrated on the Poincaré sphere.

Fig. 4
Fig. 4

Graph showing the variation of the intensity I1 and I2 of the two halves when the half-shade is put at the polarizer end and the linear analyzer is rotated.

Fig. 5
Fig. 5

Graph showing the variation of ΔI/I with the setting of the analyzer.

Fig. 6
Fig. 6

Construction for proving the result in Sec. 6.

Tables (3)

Tables Icon

Table I Effect of the position of the half-shade on measurements of apparent rotation.

Tables Icon

Table II Effect of an error in setting on the apparent rotation, when the azimuth is near 0° (α=2°52′).

Tables Icon

Table III Effect of an error in setting on the apparent rotation, when the azimuth is near 45° (α=42°8′).

Equations (28)

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b / a = tan ω .
tan 2 γ = 2 ρ 0 / δ 0 .
Δ 0 = ( δ 0 2 + 4 ρ 0 2 ) 1 2 .
2 ω = 2 π s / f .
cos 2 ( ω 1 - ω 2 ) cos 2 ( λ 1 - λ 2 ) + sin 2 ( ω 1 + ω 2 ) sin 2 ( λ 1 - λ 2 ) .
1 2 [ 1 + sin 2 ω 1 sin 2 ω 2 + cos 2 ω 1 cos 2 ω 2 cos 2 ( λ 1 - λ 2 ) ]
= 1 2 ( 1 + cos P S ) = cos 2 P S / 2 ,
α =             tan 2 ψ = sin 2 γ sin Δ / ( cos 2 2 γ + sin 2 2 γ cos Δ ) ,
α = 45°             tan 2 ψ = sin 2 γ tan Δ .
α =             tan 2 ψ = sin 2 γ tan Δ ,
α = 45°             tan 2 ψ = sin 2 γ sin Δ / ( cos 2 2 γ + sin 2 2 γ cos Δ . )
arc Q Q = π / 2 ,             arc Q C l = π / 2.
Q C Q = π / 2             or             2 λ = 2 λ + π / 2 ,
λ = λ + π / 4.
α = α + π / 4 ,
ψ = λ - α = λ - α = ψ .
cot 2 λ 1 = cos 2 2 γ + sin 2 2 γ cos Δ - sin 2 γ sin Δ tan 2 α tan 2 α cos Δ + sin 2 γ sin Δ .
cot 2 λ 2 = cos 2 2 γ + sin 2 2 γ cos Δ + sin 2 γ sin Δ tan 2 α tan 2 α cos Δ - sin 2 γ sin Δ .
cot 2 ( λ 1 - λ 2 ) = [ ( cos 2 2 γ + sin 2 2 γ cos Δ ) 2 - sin 2 2 γ sin 2 Δ + tan 2 2 α ( cos 2 Δ - sin 2 2 γ sin 2 Δ ) ] ÷ 2 sin 2 γ sin Δ [ cos 2 2 γ + sin 2 2 γ cos Δ + tan 2 2 α cos Δ ] .
tan 2 γ = [ cos 2 α - cos 2 ω cos 2 ( α + ψ ) ] / sin 2 ω cos Δ = 1 - ( 1 - cos 2 ω cos 2 ψ ) / ( 1 - cos 2 2 γ cos 2 2 α ) δ = Δ cos 2 γ ;             2 ρ = Δ sin 2 γ . } .
2 ψ 0 = 2 ρ ( 1 - δ 2 / 6 ) 7 ,
2 ψ 45 = 2 ρ ( 1 + δ 2 / 3 ) .
tan 2 ψ = sin 2 2 γ sin Δ - ( 1 - cos Δ ) cos 2 γ tan 2 α + sin 2 γ sin Δ tan 2 2 α cos 2 2 γ + sin 2 2 γ cos Δ + cos Δ tan 2 2 α .
( tan 2 ψ ) m = [ cot Δ ( cot Δ + cosec Δ cot 2 2 γ ) ] - 1 2 .
2 ψ m = 2 ρ ( 1 + δ 2 / 12 ) .
( 2 ψ 0 + ψ 45 ) / 3 = ρ ,
2 ψ = 2 ρ sin δ / δ ,
tan 2 ψ = 2 ρ tan δ / δ ,