Abstract

The parameters necessary to define an elliptically polarized vibration, namely the azimuth, the shape, and the sense of rotation of the ellipse described by the light vector can be represented geometrically by a point on a sphere. The method was suggested by Poincaré in 1892. The theory of the Poincaré sphere is presented in detail, and its application to tracing the passage of light through doubly refracting and optically active media fully illustrated. A simple model, designed on the principles involved, is described: it is suitable for instruction and demonstration.

© 1954 Optical Society of America

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References

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  1. L. B. Tuckermann, University of Nebraska Studies9, 157 (1909).
  2. H. Poincaré, Théorie Malhématique de la Lumiere (Gauthiers-Villars, Paris, 1892), Vol. 2, Chap. 12.
  3. J. Becquerel, Communs. Phys. Lab. Univ. Leiden. No.  91C (1928);Communs. Phys. Lab. Univ. Leiden.211a (1930).L. Chaumont, Compt. rend. 156, 1604 (1913);Ann. chim. et phys. (9th series)  4, 101 (1915).C. A. Skinner, J. Opt. Soc. Am. 10, 490 (1925).F. E. Wright, ibid. 20, 529 (1930).G. Bruhat and P. Grivet, J. phys. radium 6, 12 (1935).Y. Björnstahl, Physik. Z. 42, 437 (1939);Z. Instrumentenk. 59, 425 (1939).O. Snellman and Y. Björnstahl, Kolloid-Beih. 52, 403 (1941).M. F. Bokstein, J. Tech. Phys. (U.S.S.R.) 18, (5) 673 (1948).G. N. Ramachandran and S. Ramaseshan, J. Opt. Soc. Am. 42, 49 (1952).
    [Crossref]
  4. C. A. Skinner, J. Opt. Soc. Am. 10, 503 (1925).
    [Crossref]
  5. A. Johannsen, Manual of Petrographic Methods (McGraw-Hill Book Company, Inc., New York, 1918), second edition, Chapter 2.
  6. H. G. Jerrard, J. Opt. Soc. Am. 38, 35 (1948).
    [Crossref]
  7. G. N. Ramachandran and V. Chandrasekharan, Proc. Indian Acad. Sci. 33A, 199 (1951).
  8. S. Ramaseshan and V. Chandrasekharan, Current Sci. (India) 20, 150 (1951);S. Ramaseshan, Proc. Indian Acad. Sci. 34A, 32 (1951).
  9. G. N. Ramachandran and S. Ramaseshan, J. Opt. Soc. Am. 42, 49 (1952).
    [Crossref]

1952 (1)

1951 (2)

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

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

1948 (1)

1928 (1)

J. Becquerel, Communs. Phys. Lab. Univ. Leiden. No.  91C (1928);Communs. Phys. Lab. Univ. Leiden.211a (1930).L. Chaumont, Compt. rend. 156, 1604 (1913);Ann. chim. et phys. (9th series)  4, 101 (1915).C. A. Skinner, J. Opt. Soc. Am. 10, 490 (1925).F. E. Wright, ibid. 20, 529 (1930).G. Bruhat and P. Grivet, J. phys. radium 6, 12 (1935).Y. Björnstahl, Physik. Z. 42, 437 (1939);Z. Instrumentenk. 59, 425 (1939).O. Snellman and Y. Björnstahl, Kolloid-Beih. 52, 403 (1941).M. F. Bokstein, J. Tech. Phys. (U.S.S.R.) 18, (5) 673 (1948).G. N. Ramachandran and S. Ramaseshan, J. Opt. Soc. Am. 42, 49 (1952).
[Crossref]

1925 (1)

C. A. Skinner, J. Opt. Soc. Am. 10, 503 (1925).
[Crossref]

Becquerel, J.

J. Becquerel, Communs. Phys. Lab. Univ. Leiden. No.  91C (1928);Communs. Phys. Lab. Univ. Leiden.211a (1930).L. Chaumont, Compt. rend. 156, 1604 (1913);Ann. chim. et phys. (9th series)  4, 101 (1915).C. A. Skinner, J. Opt. Soc. Am. 10, 490 (1925).F. E. Wright, ibid. 20, 529 (1930).G. Bruhat and P. Grivet, J. phys. radium 6, 12 (1935).Y. Björnstahl, Physik. Z. 42, 437 (1939);Z. Instrumentenk. 59, 425 (1939).O. Snellman and Y. Björnstahl, Kolloid-Beih. 52, 403 (1941).M. F. Bokstein, J. Tech. Phys. (U.S.S.R.) 18, (5) 673 (1948).G. N. Ramachandran and S. Ramaseshan, J. Opt. Soc. Am. 42, 49 (1952).
[Crossref]

Chandrasekharan, V.

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

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

Jerrard, H. G.

Johannsen, A.

A. Johannsen, Manual of Petrographic Methods (McGraw-Hill Book Company, Inc., New York, 1918), second edition, Chapter 2.

Poincaré, H.

H. Poincaré, Théorie Malhématique de la Lumiere (Gauthiers-Villars, Paris, 1892), Vol. 2, Chap. 12.

Ramachandran, G. N.

G. N. Ramachandran and S. Ramaseshan, J. Opt. Soc. Am. 42, 49 (1952).
[Crossref]

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

Ramaseshan, S.

G. N. Ramachandran and S. Ramaseshan, J. Opt. Soc. Am. 42, 49 (1952).
[Crossref]

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

Skinner, C. A.

C. A. Skinner, J. Opt. Soc. Am. 10, 503 (1925).
[Crossref]

Tuckermann, L. B.

L. B. Tuckermann, University of Nebraska Studies9, 157 (1909).

Communs. Phys. Lab. Univ. Leiden. (1)

J. Becquerel, Communs. Phys. Lab. Univ. Leiden. No.  91C (1928);Communs. Phys. Lab. Univ. Leiden.211a (1930).L. Chaumont, Compt. rend. 156, 1604 (1913);Ann. chim. et phys. (9th series)  4, 101 (1915).C. A. Skinner, J. Opt. Soc. Am. 10, 490 (1925).F. E. Wright, ibid. 20, 529 (1930).G. Bruhat and P. Grivet, J. phys. radium 6, 12 (1935).Y. Björnstahl, Physik. Z. 42, 437 (1939);Z. Instrumentenk. 59, 425 (1939).O. Snellman and Y. Björnstahl, Kolloid-Beih. 52, 403 (1941).M. F. Bokstein, J. Tech. Phys. (U.S.S.R.) 18, (5) 673 (1948).G. N. Ramachandran and S. Ramaseshan, J. Opt. Soc. Am. 42, 49 (1952).
[Crossref]

Current Sci. (India) (1)

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

J. Opt. Soc. Am. (3)

Proc. Indian Acad. Sci. (1)

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

Other (3)

A. Johannsen, Manual of Petrographic Methods (McGraw-Hill Book Company, Inc., New York, 1918), second edition, Chapter 2.

L. B. Tuckermann, University of Nebraska Studies9, 157 (1909).

H. Poincaré, Théorie Malhématique de la Lumiere (Gauthiers-Villars, Paris, 1892), Vol. 2, Chap. 12.

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Figures (9)

F. 1
F. 1

Parameters of an elliptical vibration having components of amplitudes A and B along the OX and OY axes, respectively. tanν = B/A. The major and minor axes of the ellipse are 2a and 2b; the ellipticity e = b/a = tan; the azimuth is θ with respect to the axis OX.

F. 2
F. 2

Representation of elliptically polarized light by a point m on a plane. Δ is the phase difference between the components of the ellipse.

F. 3
F. 3

Stereographic projection of a plane on a sphere. Elliptically polarized light is represented by the points m and M on the plane and sphere, respectively. The vector Om projects into the arc OM of length 2ν; the angle Δ projects into the spherical angle MOT. The latitude and longitude of M are l and k, respectively.

F. 4
F. 4

The Poincaré sphere showing the representation of an elliptically polarized vibration by a point M. From the spherical triangle OMT, the parameters of the vibration can be found. Points on the equator OO′ represent linearly polarized light. The sense of rotation of the ellipse is left and right in the upper and lower hemispheres, respectively. The poles P1 and P2 represent left and right circularly polarized light, respectively.

F. 5
F. 5

Representation of passage of linearly polarized light through a doubly refracting plate of phase angle Δ1 in azimuth i1. In (a), OP and Ox are the vibration direction of the polarizer and fast direction of the plate, respectively; Oe is the major axis of the emergent elliptical vibration. θ1 and φ1 are azimuths. In (b) and (c), M and M3 represent the incident (linear) and emergent (elliptical) light respectively referred to OP. In (b), M1 and M2 are the same factors referred to Ox. In (b) the passage from M to M3 is by a shift (1) of M to M1 by rotation about P1P2 through 2i1 to change the axis of reference from OP to Ox, (2) of M1 to M2 by rotation about OO′ through Δ1, and (3) of M3 to M3 by rotation about P1P2 through −2i1 to return to the axis OP. In (c) the passage from M to M3 is by a single rotation about the axis OO′ set at longitude −2i1. The lower figures are traces on the equatorial plane.

F. 6
F. 6

Representation of passage of polarized light through 2 doubly refracting plates of phase angles Δ1 and Δ2 in azimuths i1 and i2, respectively, measured from an arbitrary axis of reference represented by OO′. Δ1 is a phase lead, Δ2 a phase lag. M, M1, and M2 represent the incident light, and the light leaving the first and second plate, respectively. The shift M to M1 is by a counter-clockwise rotation of Δ1 about an axis H1H1 at longitude 2i1; the shift M1 to M2 is by a clockwise rotation of Δ2 about H2H2 at longitude 2i2. θ, θ1, and θ2 are the azimuths of the light with respect to original direction of reference. The lower figure is a trace on the equatorial plane.

F. 7
F. 7

Intensity of light after passage through a polarizer, doubly refracting plate and analyzer. In (a), OP, OA, and Ox represent the vibration directions of the polarizer, analyzer, and plate, respectively. σ, i1, and θ1 are azimuth angles. In (b), M represents the elliptical vibration leaving the plate referred to Ox. This is referred to OA by a rotation about P1P2 through (σ+i1) and M shifts to M1. The intensity is cos2ρ, where the arc OM1 is 2ρ.

F. 8
F. 8

(a) Intensity of light after passage through a polarizer M and an analyzer M1 at an azimuth σ with respect to the vibration direction of the polarizer. The small circle, center M1, radius MM1, represents points for which the intensity of the light leaving M1 is the same. (b) Representation of the action of a λ/4 compensator on an elliptical vibration. M represents linearly polarized light incident on a plate of phase lag Δ1, in azimuth 45°. M1 represents the elliptical vibration leaving this plate and M2 is the linear vibration leaving the λ/4 plate in azimuth zero.

F. 9
F. 9

Simple model of a Poincaré sphere.

Equations (27)

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X = Ā e i ω t ,
Y = B ¯ e i ω t ,
B ¯ Ā = B A e i Δ = ( B A ) cos Δ + i ( B A ) sin Δ = u + i υ .
X 2 A 2 + Y 2 B 2 2 X Y A B cos Δ = sin 2 Δ .
tan = b / a ( 0 π / 2 ) ,
tan ν = B / A ( 0 ν π / 2 ) .
A 2 + B 2 = a 2 + b 2 ,
A 2 B 2 = ( a 2 b 2 ) cos 2 θ ,
A B sin Δ = ± a b ,
2 A B cos Δ = ( a 2 b 2 ) sin 2 θ .
cos 2 = cos 2 ν cos 2 θ + sin 2 ν sin 2 θ cos Δ ,
± sin 2 = sin 2 ν sin Δ ,
tan 2 θ = tan 2 ν cos Δ ,
cos 2 ν = cos 2 cos 2 θ ,
± tan 2 = sin 2 θ tan Δ .
u 2 + υ 2 + 2 cot 2 θ u 1 = 0 ,
u 2 + υ 2 2 cosec 2 υ + 1 = 0 .
x + 1 2 1 = y tan ν cos Δ tan ν cos Δ = z tan ν sin Δ tan ν sin Δ .
x 2 + y 2 + z 2 = 1 4 ,
2 x = cos 2 cos 2 θ ,
2 y = cos 2 sin 2 θ ,
2 z = sin 2 .
sin l = 2 z = sin 2 ,
tan k = y / x = tan 2 θ .
I = 1 2 [ 1 + cos 2 i 1 cos 2 ( σ + i 1 ) + sin 2 i 1 sin 2 ( σ + i 1 ) cos Δ 1 ] .
I = 1 2 [ 1 + cos 2 1 cos 2 ( θ 1 + σ + i 1 ) ] .
cos 2 1 cos 2 ( θ 1 + σ + i 1 ) = cos 2 ρ ,