Abstract

We describe the historical and mathematical development of the polarization ellipse and the Poincaré sphere. We point out the limitations of the Poincaré sphere in its present use. To overcome these limitations we describe a new polarization sphere that we call the hybrid polarization sphere. This name is used because phase shifting and rotation of polarization components are described by small circles. Furthermore, longitudinal and latitudinal great circles are introduced so that the coordinates of a point on the sphere can be read. The hybrid polarization sphere is described and applied to polarizers, wave plates, and rotators. As a result, the hybrid polarization sphere can be used for both visualization and calculation and enables the difficulties associated with the Poincaré sphere to be overcome.

© 2008 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  2. E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, 1992).
  3. H. Poincaré, Théorie Mathématique de la Lumier (Gauthiers-Villars, 1892).
  4. H. G. Jerrard, “Transmission of light through birefringent and optically active media: the Poincaré sphere,” J. Opt. Soc. Am. 44, 634-640 (1954).
    [CrossRef]
  5. M. J. Walker, “Matrix calculus and the Stokes parameters of polarized radiation,” Am. J. Phys. 22, 170-174 (1954).
    [CrossRef]
  6. F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415-427 (1942).
    [CrossRef]
  7. E. Collett, “The description of polarization in classical physics,” Am. J. Phys. 36, 713-725 (1968).
    [CrossRef]
  8. E. Collett and B. Schaefer, “Visualization and calculation of polarized light. II. Applications of the hybrid polarization sphere,” Appl. Opt. 47, 4017-4024 (2008).
    [PubMed]

1999 (1)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

1992 (1)

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, 1992).

1968 (1)

E. Collett, “The description of polarization in classical physics,” Am. J. Phys. 36, 713-725 (1968).
[CrossRef]

1954 (2)

H. G. Jerrard, “Transmission of light through birefringent and optically active media: the Poincaré sphere,” J. Opt. Soc. Am. 44, 634-640 (1954).
[CrossRef]

M. J. Walker, “Matrix calculus and the Stokes parameters of polarized radiation,” Am. J. Phys. 22, 170-174 (1954).
[CrossRef]

1942 (1)

F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415-427 (1942).
[CrossRef]

1892 (1)

H. Poincaré, Théorie Mathématique de la Lumier (Gauthiers-Villars, 1892).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Collett, E.

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, 1992).

E. Collett, “The description of polarization in classical physics,” Am. J. Phys. 36, 713-725 (1968).
[CrossRef]

E. Collett and B. Schaefer, “Visualization and calculation of polarized light. II. Applications of the hybrid polarization sphere,” Appl. Opt. 47, 4017-4024 (2008).
[PubMed]

Jerrard, H. G.

Perrin, F.

F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415-427 (1942).
[CrossRef]

Poincaré, H.

H. Poincaré, Théorie Mathématique de la Lumier (Gauthiers-Villars, 1892).

Schaefer, B.

Walker, M. J.

M. J. Walker, “Matrix calculus and the Stokes parameters of polarized radiation,” Am. J. Phys. 22, 170-174 (1954).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Am. J. Phys. (2)

M. J. Walker, “Matrix calculus and the Stokes parameters of polarized radiation,” Am. J. Phys. 22, 170-174 (1954).
[CrossRef]

E. Collett, “The description of polarization in classical physics,” Am. J. Phys. 36, 713-725 (1968).
[CrossRef]

Appl. Opt. (1)

J. Chem. Phys. (1)

F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415-427 (1942).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (3)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, 1992).

H. Poincaré, Théorie Mathématique de la Lumier (Gauthiers-Villars, 1892).

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Figures (11)

Fig. 1
Fig. 1

Plot of the equation for the polarization ellipse, Eq. (2). In general, the ellipse is not in its standard form, where E x ( z , t ) and E y ( z , t ) are directed along the x and y axes, but along an axis rotated through an angle ψ.

Fig. 2
Fig. 2

Poincaré sphere of unit radius represented in a Cartesian coordinate system. The spherical coordinates of point P are P ( ψ , χ ) .

Fig. 3
Fig. 3

Linearly and circularly polarized light points plotted on the Poincaré sphere. The points LHP, L+45P , and RCP are linearly horizontal polarized light, linear + 45 polarized light, and right circularly polarized light, respectively, and appear on the X, Y, and Z axes.

Fig. 4
Fig. 4

Polarization sphere using the angular coordinates represented by the traditional latitude and longitude sphere. We note that the axes representing the Stokes polarization parameters are labeled differently than the Poincaré sphere.

Fig. 5
Fig. 5

Plot of Eq. (15) showing a nonaxial view on the polarization sphere.

Fig. 6
Fig. 6

Rotation of a polarized beam described by Eq. (25) on the polarization sphere.

Fig. 7
Fig. 7

Plot of the polarization states emerging from a rotating polarizer. Regardless of the polarization state of the input beam, the polarization state of the output beam emerging from a linear polarizer is always restricted to the prime meridian.

Fig. 8
Fig. 8

The small-circle polarization sphere. The equator is in the S 2 S 1 plane. Both the latitude and the longitude lines are separated by 15 ° increments. The number of small circles that are shown has been kept to 12 to avoid cluttering the figures.

Fig. 9
Fig. 9

(a) Sphere consisting only of great circles. (b) With rotation points, which follow small circles. The vertical great circles are the longitudinal latitude lines, and the horizontal great circles are the latitudinal lines.

Fig. 10
Fig. 10

The HPS consisting of small circles and great circles. The longitudinal great-circle lines are used to read the angle δ, and the latitudinal great-circle lines are used to read the angle 2 α . The combination leads to a polarization sphere that we call the HPS.

Fig. 11
Fig. 11

Determining the final polarization state of an input L+45PL beam that propagates through a wave plate with a phase shift of + 45 ° and is rotated through an angle of + 22.5 ° on the HPS.

Equations (33)

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E x ( z , t ) = E 0 x cos ( ω t κ z + δ x ) , E y ( z , t ) = E 0 y cos ( ω t κ z + δ y ) ,
E x ( z , t ) 2 E 0 x 2 + E y ( z , t ) 2 E 0 y 2 2 E x ( z , t ) E y ( z , t ) E 0 x E 0 y cos δ = sin 2 δ ,
E ξ = a cos ( τ + δ 0 ) , E η = ± b sin ( τ + δ 0 ) .
a 2 + b 2 = E 0 x 2 + E 0 y 2 ,
tan 2 ψ = ( tan 2 α ) cos δ , 0 ψ < π , sin 2 χ = ( sin 2 α ) sin δ , π / 4 < χ π / 4 ,
tan α = E 0 y E 0 x , 0 α π / 2 , tan χ = ± b a , π / 4 < χ π / 4.
2 x = cos 2 χ cos 2 ψ , 2 y = cos 2 χ sin 2 ψ , 2 z = sin 2 χ ,
S 0 = 1 , S 1 = cos 2 χ cos 2 ψ , S 2 = cos 2 χ sin 2 ψ , S 3 = sin 2 χ ,
S 1 2 + S 2 2 + S 3 2 = 1.
S 0 2 S 1 2 S 2 2 S 3 2 = 0 ,
S 0 = E 0 x 2 + E 0 y 2 , S 1 = E 0 x 2 E 0 y 2 , S 2 = 2 E 0 x E 0 y cos δ , S 3 = 2 E 0 x E 0 y sin δ , δ = δ y δ x .
tan α = E 0 y E 0 x , 0 α π / 2.
S 0 = 1 , S 1 = cos 2 α , S 2 = sin 2 α cos δ , S 3 = sin 2 α sin δ .
S = ( S 0 S 1 S 2 S 3 ) = ( 1 cos 2 α sin 2 α cos δ sin 2 α sin δ ) , 0 α π / 2 , 0 δ < 2 π .
S = ( 1 cos ( 5 π / 12 ) sin ( π / 12 ) cos δ sin ( 5 π / 12 ) cos δ ) .
S 1 = cos 2 α = cos 2 χ cos 2 ψ , S 2 = sin 2 α δ = cos 2 χ sin 2 ψ , S 3 = sin 2 α sin δ = sin 2 χ .
tan 2 ψ = tan 2 α cos δ , sin 2 χ = sin 2 α sin δ .
cos 2 α = cos 2 χ cos 2 ψ , cot δ = cot 2 χ sin 2 ψ .
S = M WP ( ϕ ) · S ,
M WP ( ϕ ) = ( 1 0 0 0 0 1 0 0 0 0 cos ϕ sin ϕ 0 0 sin ϕ cos ϕ ) ,
S = ( S 0 S 1 S 2 S 3 ) = M WP ( ϕ ) · S = ( 1 cos 2 α sin 2 α cos ( δ + ϕ ) sin 2 α sin ( δ + ϕ ) ) , 0 α π / 2 , 0 δ < 2 π .
S = M ROT ( θ ) · S ,
M ROT ( θ ) = ( 1 0 0 0 0 cos 2 θ sin 2 θ 0 0 sin 2 θ cos 2 θ 0 0 0 0 0 ) .
S = M ROT ( θ ) = ( 1 cos 2 θ cos 2 α + sin 2 θ sin 2 α cos δ sin 2 θ cos 2 α + cos 2 θ sin 2 α cos δ sin 2 α sin δ ) .
S = ( 1 ( 3 / 2 ) sin 2 θ ( 3 / 2 ) cos 2 θ 1 / 2 ) .
S = 1 cos ( 2 α 2 θ ) sin ( 2 α 2 θ ) 0 .
tan 2 ψ = S 2 S 1 = sin ( 2 α 2 θ ) cos ( 2 α 2 θ ) = tan ( 2 α 2 θ ) .
M POL ( p x , p y ) = 1 2 ( p x 2 + p y 2 p x 2 p y 2 0 0 p x 2 p y 2 p x 2 + p y 2 0 0 0 0 2 p x p y 0 0 0 0 2 p x p y ) , 0 p x , p y 1.
S = M POL ( p x , p y ) · S .
M POL ( θ ) = 1 2 ( 1 cos 2 θ sin 2 θ 0 cos 2 θ cos 2 2 θ cos 2 θ sin 2 θ 0 sin 2 θ cos 2 θ sin 2 θ sin 2 2 θ 0 0 0 0 0 ) .
S = M POL ( θ ) · S = 1 2 ( S 0 + S 1 cos 2 θ + S 2 sin 2 θ ) ( 1 cos 2 θ sin 2 θ 0 ) .
M WPROT = M ROT ( θ ) · M WP · M ROT ( θ ) .
M WROT = ( 1 0 0 0 0 cos ( 2 × 22.5 ° ) sin ( 2 × 22.5 ° ) 0 0 sin ( 2 × 22.5 ° ) cos ( 2 × 22.5 ° ) 0 0 0 0 1 ) · ( 1 0 0 0 0 1 0 0 0 0 cos ( 45.0 ° ) sin ( 45.0 ° ) 0 0 sin ( 45.0 ° ) cos ( 45.0 ° ) ) · ( 1 0 0 0 0 cos ( 2 × 22.5 ° ) sin ( 2 × 22.5 ° ) 0 0 sin ( 2 × 22.5 ° ) cos ( 2 × 22.5 ° ) 0 0 0 0 1 ) = ( 1 0 0 0 0 0.854 0.146 0.5 0 0.146 0.854 0.5 0 0.5 0.5 0.707 ) .

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