Abstract

The constants of an elliptical vibration are determined by transforming it either to a rectilinear vibration or to one of small, known ellipticity and then measuring the azimuth of the transformed vibration. From these measurements the constants of the elliptical vibration can be computed or ascertained by graphical methods. For this transformation, crystal plates of known thickness and birefringence are used. In each one of a series of transformations the incoming vibration is referred to the axes (α′, γ′) of the crystal plate along which the vibrations take place on passage through the plate; the phase lag or lead, introduced by the plate, is then added and the form of the vibration emerging from the plate thus ascertained. This procedure is repeated for each crystal plate and for the analyzer. Poincaré showed in 1892 how the relations of any given elliptical vibration can be represented by a point on a sphere or in a stereographic projection. His graphical method of representation is now in general use. In this connection a specially prepared angle or stereographic projection chart, as a base, greatly facilitates the graphical solutions. A brief description of this chart and of the methods developed for its use, as applied to the derivation of the equations and to the solution of problems in elliptical vibrations, is presented.

Throughout the paper the effect, on a given elliptical vibration represented by a point in the projection, of changing from one set of reference coordinate axes to a second set is obtained in the projection by a rotation about the vertical, NS, axis through twice the angle between the sets. Similarly the effect of the introduction of a phase angle, ϕ, by a crystal plate on a given elliptical vibration, referred to the vibration axes (α′, γ′) of the crystal plate and represented by a point in the projection, is obtained by a rotation of the projection about the horizontal, EW, axis through the angle, ϕ.

© 1930 Optical Society of America

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References

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  1. H. Poincaré, Théorie mathématique de la lumière. II, Chap, xii, 1892; see also J. Walker, Analytical theory of light, 303–318; 1904; F. Pockels, Lehrbuch der Kristalloptik, 11–13, 267–283; 1906; L. B. Tuckerman, Doubly refracting plates and elliptic analyzers. Univ. of Nebraska Studies IX, 157–219; 1909; C. A. Skinner, A universal polarimeter. J. Opt. Soc. & Rev. Sci. Inst. 10, 490–520; 1925.
  2. F. E. Wright, Am. J. Sci. (4) 26, 377–378; 1908.
  3. C. G. Stokes, Phil. Mag. (4),  2, 420; 1851.
  4. D. B. BraceHalf shade elliptical polarizer and compensator. Phys. Review (I) 18, 701901; Phys. Review (I) 19, 218, 1902.
  5. A. Q. Tool, Phys. Rev.,  31, 1; 1910.

1910 (1)

A. Q. Tool, Phys. Rev.,  31, 1; 1910.

1908 (1)

F. E. Wright, Am. J. Sci. (4) 26, 377–378; 1908.

1901 (1)

D. B. BraceHalf shade elliptical polarizer and compensator. Phys. Review (I) 18, 701901; Phys. Review (I) 19, 218, 1902.

1851 (1)

C. G. Stokes, Phil. Mag. (4),  2, 420; 1851.

Brace, D. B.

D. B. BraceHalf shade elliptical polarizer and compensator. Phys. Review (I) 18, 701901; Phys. Review (I) 19, 218, 1902.

Poincaré, H.

H. Poincaré, Théorie mathématique de la lumière. II, Chap, xii, 1892; see also J. Walker, Analytical theory of light, 303–318; 1904; F. Pockels, Lehrbuch der Kristalloptik, 11–13, 267–283; 1906; L. B. Tuckerman, Doubly refracting plates and elliptic analyzers. Univ. of Nebraska Studies IX, 157–219; 1909; C. A. Skinner, A universal polarimeter. J. Opt. Soc. & Rev. Sci. Inst. 10, 490–520; 1925.

Stokes, C. G.

C. G. Stokes, Phil. Mag. (4),  2, 420; 1851.

Tool, A. Q.

A. Q. Tool, Phys. Rev.,  31, 1; 1910.

Wright, F. E.

F. E. Wright, Am. J. Sci. (4) 26, 377–378; 1908.

Am. J. Sci. (4) (1)

F. E. Wright, Am. J. Sci. (4) 26, 377–378; 1908.

Phil. Mag. (4) (1)

C. G. Stokes, Phil. Mag. (4),  2, 420; 1851.

Phys. Rev. (1)

A. Q. Tool, Phys. Rev.,  31, 1; 1910.

Phys. Review (I) (1)

D. B. BraceHalf shade elliptical polarizer and compensator. Phys. Review (I) 18, 701901; Phys. Review (I) 19, 218, 1902.

Other (1)

H. Poincaré, Théorie mathématique de la lumière. II, Chap, xii, 1892; see also J. Walker, Analytical theory of light, 303–318; 1904; F. Pockels, Lehrbuch der Kristalloptik, 11–13, 267–283; 1906; L. B. Tuckerman, Doubly refracting plates and elliptic analyzers. Univ. of Nebraska Studies IX, 157–219; 1909; C. A. Skinner, A universal polarimeter. J. Opt. Soc. & Rev. Sci. Inst. 10, 490–520; 1925.

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Figures (19)

Fig. 1
Fig. 1

Elliptically polarized light may be resolved into two harmonic components of the same period but of different amplitudes and vibrating along the x and y axes, respectively; the y component lags behind the x component by the phase angle, ϕ. In the figure the two harmonic components are the x component of the vector OA and the y component of the vector OB, respectively, both vectors rotating at the same constant speed about O; the angular phase lag of OB behind OA is the angle BOC=ϕ.

Fig. 2
Fig. 2

In this figure the angle included between the x-axis and the principal axis, a, of the ellipse is XOa=θ. The harmonic components of any point, F, on the ellipse along the OX and OY axes, and also along the OX0 and OY0 axes are indicated by the intercepts of the dotted lines on the several axes.

Figs. 3a, 3b
Figs. 3a, 3b

illustrate the dependence of the direction of movement of the elliptic vector, whether counter-clockwise or clockwise, on the phase difference between the two harmonic x and y components.

Fig. 4
Fig. 4

illustrates the Poincaré construction on the sphere for the representation of elliptically polarized light. From the spherical triangle CDE, the relations given in equations 13 to 18 can be read off directly.

Fig. 5
Fig. 5

Angle projection chart for use in the study of elliptically polarized light. The great and also the small circles are 2° apart.

Fig. 6
Fig. 6

indicates the change in phase angle ϕ and in the amplitude ratio tan i=B/A, when the elliptical vibration represented by C is referred to another set of coordinate axes which include an angle θ′ with the original set.

Fig. 7
Fig. 7

On changing the phase lag from ϕ to ϕ′, but maintaining the amplitude ratio constant, the point C in the projection is shifted to C′; the ellipticity is changed from tan =b/a to tan ′=b′/a′ and the azimuth angle, from θ to θ′.

Figs. 8a to 8f
Figs. 8a to 8f

illustrate the shapes and positions of the vibration ellipse emerging from a crystal plate of phase angle ϕ=60°; its vibration direction, α′, includes the angle i=0°, 10°, 20°, 30°, 40°, and 45°, respectively, with the incident plane polarized light.

Fig. 9
Fig. 9

The curves in this figure indicate the change in azimuth angle, θ (ordinates), with change in position angle, i (abscissae), for different phase angles, ϕ (curves).

Fig. 10
Fig. 10

Diagram to illustrate the positions of the axes of the two superimposed crystal plates. PO represents the direction of the vibration of the plane polarized light entering the first crystal plate with axes X1O and Y1O; the position angle X1OP=i1. After emerging from this plate the light enters the second crystal plate with axes, X2O and Y2O; the position angle X2OP=i2. The azimuth angle X2OX2″=θ2″ is the angle included between the major axis a2 of the emergent elliptical vibration with the X2O axis. The dotted lines indicate at their intersections with the several axes the vector components of the successive elliptical vibrations.

Fig. 11
Fig. 11

In this figure C1 represents the elliptical vibration emerging from the first crystal plate. This vibration is referred to the vibration axes of the second plate by rotation of the projection about the vertical, NS, axis through the angle 2(i2i1) and C1 is shifted to C2′. The phase angle ϕ2 is introduced by rotation of the projection about the horizontal, EW, axis through ϕ2; whereby C2′ is shifted to C2″. To refer the vibration ellipse represented by C2″ to the original axes, x1, y1, the projection is rotated about the vertical axis through the angle 2(i1i2) and C2″ is shifted to C2‴.

Fig. 12
Fig. 12

In this figure C1 represents the elliptical vibration emerging from the crystal plate. Refer this vibration to coordinate axes of the analyzer by rotating the projection about the vertical, NS, axis through the angle 2(i′−i1); i′ is the angle between the polarizer and the analyzer. The point C1 is shifted to C″. The angle 2i″ of the triangle DEC″ meets the requirements of equations 34 and 35 by means of which the intensity can be computed directly.

Fig. 13
Fig. 13

The point C1 represents the incoming elliptic vibration. Refer this to the axes of the greater wave plate by rotating the projection about the vertical axis through angle 2(i2i1)=−2θ1; C1 is shifted to C2′; rotate projection about horizontal, EW, axis through phase angle±π/2 of quarter-wave plate and shift C2′ to −C2″ or to +C2″, depending on the position of the quarter-wave plate. C2″ is on the equator and represents a rectilinear vibration.

Fig. 14
Fig. 14

To refer the vibration from the quarter-wave plate (C1) to the axes of the crystal plate rotate the projection about the vertical axis through 2(i2i1)=90° and shift C1 to C2′. Rotate projection about the horizontal, EW axis through the phase angle, −ϕ2, and shift C2′ to C2″ on the equator. C2″ is then shifted to E by rotation of the projection about the vertical axis through he angle 2(i1i2).

Fig. 15
Fig. 15

The points C2, C2″, and C1 represent, respectively, the elliptical vibration from the compensator, the restored rectilinear vibration and the unknown elliptic vibration. To find the position of C1 the projection is rotated about the horizontal, EW, axis through the phase angle, ϕ2, and C2″ is shifted to C2′. The point C1 lies on the small circle C2C1. The azimuth of C2′ is 2θ2″=2(θ1+i2i1); therefore 2(i2θ2″)=2(i1θ1). On the small circle C2C1 locate the point C1 for which 2(i1θ1)=2(i2θ2″) and with it determine 2i1 and ϕ1.

Fig. 16
Fig. 16

The point C1 represents the vibration from the crystal plate; C2′ this vibration referred to the axes of the quarter-wave plate; C2″, the vibration after the phase angle, −ϕ2, of the quarter-wave plate has been introduced. The angle EC2″ is the phase angle, ϕ1, of the crystal plate.

Fig. 17
Fig. 17

The point C1 represents the circular vibration emerging from the first quarter-wave plate; it can be referred to any set of coordinate axes without change in position. Introduce phase angle, −ϕ2, of any point on great circle EabW by shifting C1 to C2′. To refer this vibration to the original x1, y1 axes rotate the projection about the vertical axis through angle 2(i1i2) and shift C2′ to a′ or some other point on the small circle C2G. If the axes of the second quarter-wave plate coincide with x1, y1, the phase angle, ±π/2, is introduced by rotating the projection about the horizontal, EW, axis through ±π/2. If the phase angle is −π/2, the pole N is shifted to M at the midpoint of the projection and the small circle C2G to the small circle C3G′ concentric about M. To refer any point on the small circle C3G′ to the coordinate axes of the analyzer, distant 45° from the x1, y1 axes, rotate the projection about the vertical axis through the angle, 2×45°=90°, and shift the small circle C3G′ to the small circle C4ab‴ concentric about E. All points on this small circle have the same intensity of illumination.

Fig. 18
Fig. 18

The points C1, C2 and C3 represent the elliptical vibrations of the first quarter-wave plate, the crystal plate, and the second quarter-wave plate respectively. Refer the elliptical vibration C1 to the C2 axes by shifting C1 to C2′; introduce the phase angle, ϕ2, by shifting C2′ to C2″. Refer the vibration C2″ to the axes of the second quarter-wave plate by shifting C2″ to C3‴ and introduce the phase angle, −π/2, by shifting C3‴ to C3iv on the equator. The angle EC3iv is equal to ϕ2 and is twice the angle between the polarizer and analyzer.

Fig. 19
Fig. 19

The point C1 represents the incoming elliptical vibration; C2, the elliptical vibration of the compensator and C3 (or C3′) that of the thin mica strip attached at a definite angle to the analyzer and covering only a portion of the field. The point C3′ (or C3′) represents the elliptical vibration emerging from the uncovered portion of the field; C3″ (or C3′) the elliptical vibration from the covered part. The angle between C3′ and the equator is ϕ3/2; that between C3″ and the equator is −ϕ3/2. To find the position of C2, shift C1 to the point C2′ for which the phase angle is ϕ2+ϕ3/2. The angle of rotation 2(i2i1) on this shift can be read off directly in the projection and the position angle 2i2 thus ascertained and with it the position of C2 on the great circle of phase angle −ϕ2. Shift C2′ to C2″ and introduce phase angle −ϕ2. Refer the vibration C2″ to the axes of the mica strip by shifting C2″ to C3′ (or C3′ if the angle between α′ of the mica strip and the analyzer is different from 45°). The point C3′ represents the vibration from the uncovered portion of the field. To find the vibration from the covered portion shift C3′ to C3″ through the phase angle −ϕ3. Points C3′ and C3″ are on the same meridian, on opposite sides of, and equidistant from, the equator; the two fields are therefore equally bright.

Tables (1)

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Table 1 Character of elliptical vibration (+ or −)

Equations (97)

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u = A · cos ω t
v = B · cos ( ω t - ϕ )
u 2 A 2 + v 2 B 2 - 2 u v A B · cos ϕ = sin 2 ϕ .
u A - v B = 0             or             u A + v B = 0
tan i = B A             and             tan ( π - i ) = - B A .
u 2 A 2 + v 2 B 2 = 1
u 0 = u · cos θ + v · sin θ = a · cos ( ω t - ϕ )
v 0 = - u · sin θ + v · cos θ = b · cos ( ω t - ϕ ) .
ϕ - ϕ = ( 4 n + 1 ) π 2             or
ϕ - ϕ = ( 4 n - 1 ) π 2
ϕ - ϕ = π 2             or
ϕ - ϕ = - π 2 .
A · cos θ + B · sin θ · cos ϕ = a · cos ϕ
B · sin θ · sin ϕ = a · sin ϕ
- A · sin θ + B · cos θ · cos ϕ = b · cos ϕ = b · sin ϕ
B · cos θ · sin ϕ = b · sin ϕ = ± b · cos ϕ
a 2 + b 2 = A 2 + B 2
a 2 - b 2 = ( A 2 - B 2 ) · cos 2 θ + 2 A B · sin 2 θ · cos ϕ
± a b = A B · sin ϕ
tan 2 θ = 2 A B A 2 - B 2 · cos ϕ .
tan i = B A , or sin 2 i = 2 A B A 2 + B 2 , cos 2 i = A 2 - B 2 A 2 + B 2 , tan 2 i = 2 A B A 2 - B 2
tan = b a , or sin 2 = 2 a b a 2 + b 2 , cos 2 = a 2 - b 2 a 2 + b 2 , tan 2 = 2 a b a 2 - b 2 .
cos 2 = cos 2 i · cos 2 θ + sin 2 i · sin 2 θ · cos ϕ
± sin 2 = sin 2 i · sin ϕ
tan 2 θ = tan 2 i · cos ϕ
cos 2 i = cos 2 · cos 2 θ
± tan 2 = sin 2 θ · tan ϕ .
u = A · cos ω t
v = B · cos ( ω t - ϕ )
K · cos ω t
u = cos i · cos ω t v = sin i · cos ω t .
Δ = d ( γ - α ) .
ϕ = 2 π Δ λ
u = cos i · cos ω t
v = sin i · cos ( ω t - ϕ ) .
u 1 = cos i · cos ω t
v 1 = sin i · cos ( ω t - ϕ 1 ) .
u 2 = u 1 · cos ( i 2 - i 1 ) - v 1 · sin ( i 2 - i 1 )
v 2 = u 1 · sin ( i 2 - i 1 ) + v 1 · cos ( i 2 - i 1 )
u 2 = cos i 1 · cos ( i 2 - i 1 ) · cos ω t - sin i 1 · sin ( i 2 - i 1 ) · cos ( ω t - ϕ 1 )
v 2 = cos i 1 · sin ( i 2 - i 1 ) · cos ( ω t - ϕ 2 ) + sin i 1 · cos ( i 2 - i 1 ) cos ( ω t - ϕ 1 - ϕ 2 ) .
u 0 = u 2 · cos θ 2 + v 2 · sin θ 2 = a 2 · cos ( ω t - ϕ )
v 0 = - u 2 · sin θ 2 + v 2 · cos θ 2 = b 2 · cos ( ω t - ϕ )
± sin 2 2 = cos 2 1 · sin 2 ( θ 1 + i 2 - i 1 ) · sin ϕ 2 ± sin 2 1 · cos ϕ 2
tan 2 θ 2 = sin 2 ( θ 1 + i 2 - i 1 ) · cos ϕ 2 tan 2 1 · sin ϕ 2 cos 2 ( θ 1 + i 2 - i 1 ) .
2 = 1
θ 2 = θ 1 + i 2 - i 1 ,     or     θ 2 - θ 1 = i 2 - i 1 .
cos 2 i 2 = cos 2 1 · cos 2 ( θ 1 + i 2 - i 1 )
cot ϕ 2 = cot 2 1 · sin 2 ( θ 1 + i 2 - i 1 )
cot 2 i 2 = cos ϕ 2 · cot 2 ( θ 1 + i 2 - i 1 ) .
sin 2 2 = sin 2 i 2 · sin ( ϕ 2 + ϕ 2 )
tan 2 θ 2 = tan 2 i 2 · cos ( ϕ 2 + ϕ 2 )
cos 2 i 2 = cos 2 2 · cos 2 ( θ 2 + i 1 - i 2 )
cot ϕ 2 = cot 2 2 · sin 2 ( θ 2 + i 1 - i 2 ) .
u 2 = cos i 1 · cos ( i - i 1 ) · cos ω t - sin i 1 · sin ( i - i 1 ) · cos ( ω t - ϕ 1 ) = A · cos ( ω t - ϕ ) .
cos i 1 · cos ( i - i 1 ) - sin i 1 · sin ( i - i 1 ) · cos ϕ 1 = A · cos ϕ
- sin i 1 · sin ( i - i 1 ) · sin ϕ 1 = A · sin ϕ .
cos 2 i 1 · cos 2 ( i - i 1 ) + sin 2 i 1 · sin 2 ( i - i 1 ) - 1 2 · sin 2 i 1 · sin 2 ( i - i 1 ) · cos ϕ 1 = A 2 = I
I = 1 2 · ( 1 + cos 2 i 1 · cos 2 ( i - i 1 ) - sin 2 i 1 · sin 2 ( i - i 1 ) · cos ϕ 1 ) or
I = cos 2 i + sin 2 i 1 · sin 2 ( i - i 1 ) · sin 2 ϕ 1 2 .
I = 1 2 · [ 1 + cos 2 1 · cos 2 ( θ 1 + i - i 1 ) ] .
I = 1 2 · ( 1 + cos 2 i ) = cos 2 i .
I = sin 2 2 i 1 2 ( 1 - cos ϕ 1 )
I = 1 2 · ( 1 + cos 2 i 2 · cos 2 ( i - i 2 ) - sin i 2 · sin 2 ( i - i 2 ) · cos ϕ 2 )
I = 1 2 · [ 1 + cos 2 2 · cos 2 ( θ 2 + i - i 2 ) ] = 1 2 · ( 1 + cos 2 i ) = cos 2 i .
I = sin 2 i 2 2 · ( 1 - cos ϕ 2 ) .
d I d i = cos 2 i 1 · sin 2 ( i 1 - i ) - sin 2 i 1 · cos 2 ( i 1 - i ) · cos ϕ = 0
tan 2 ( i 1 - i ) = tan 2 i 1 · cos ϕ .
tan 2 θ 1 = tan 2 i 1 · cos ϕ .
2 ( i 1 - i ) = 2 θ 1 ; i = i 1 - θ 1 or i = π 2 + ( i 1 - θ 1 )
I = 1 2 · ( 1 ± cos 2 i 1 · cos 2 θ 1 + sin 2 i 1 · sin 2 θ 1 · cos ϕ 1 ) = 1 2 · ( 1 ± cos 2 1 ) = cos 2 1 or sin 2 1 .
I = cos 2 ( i + ρ ) - sin 2 ( i 1 + ρ ) · sin 2 ( i 1 - i ) · sin 2 ϕ 2
I = cos 2 ( i - ρ ) - sin 2 ( i 1 - ρ ) · sin 2 ( i 1 - i ) · sin 2 ϕ 2
tan 2 i 1 = tan 2 ( i 1 - i ) · cos ϕ 1
sin 2 2 = cos 2 1 · sin 2 ( θ 1 + i 2 - i 1 )
tan θ 2 = tan 2 1 cos 2 ( θ 1 + i 2 - i 1 ) .
sin 2 2 = 0.
2 ( θ 1 + i 2 - i 1 ) = 0 or i 2 = i 1 - θ 1 or i 2 - i 1 = - θ 1
θ 2 = - 1 .
sin 2 ( θ 1 + i 2 - i 1 ) = tan 2 1 · cot ϕ 2
cot 2 θ 2 = cot 2 ( θ 1 + i 2 - i 1 ) · cos ϕ 2 .
sin 2 2 = sin 2 i 2 · sin ϕ 2 tan 2 θ 2 = tan 2 i 2 · cos ϕ 2
2 i 2 - 2 θ 2 = 2 ( i 1 - θ 1 ) = S .
cos 2 i 1 = cos 2 1 · cos 2 θ 1
tan 2 θ 1 = cos S - cos 2 2 sin S
sin 2 i = sin 2 sin ϕ
sin 2 θ = tan 2 · cot ϕ
i 1 + i 2 = π 2     and     θ 1 + θ 2 = π 2 .
cos ϕ = tan 2 θ 1 tan 2 i 1 = tan 2 θ 2 tan 2 i 2 = cot ( θ 2 - θ 1 ) cot ( i 2 - i 1 )
cos 2 = cos 2 i 1 cos 2 θ 1 = cos 2 i 2 cos 2 θ 2 = sin ( i 2 - i 1 ) sin ( θ 2 - θ 1 )
x = cos ϕ 2 · cos ω t y = sin ϕ 2 · sin ω t .
cot 2 i 2 = cot 2 θ 2 · cos ϕ 3 2 .
sin 2 2 = sin 2 i 2 · sin ( ϕ 2 + ϕ 3 2 ) tan 2 θ 2 = tan 2 i 2 · cos ( ϕ 2 + ϕ 3 2 )
2 i 2 - 2 θ 2 - 2 ( i 1 - θ 1 ) = S
cos 2 i 1 = cos 2 1 · cos 2 θ 1
cos ( S + 2 θ 1 ) = cos 2 2 · cos 2 θ 1             or tan 2 θ 1 = cos S - cos 2 2 sin S .
I = cos 2 i + sin 2 i 1 · sin 2 ( i 2 - i 1 ) · cos 2 ( i - i 2 ) · sin 2 ϕ 1 2 + cos 2 i 1 · sin 2 ( i 2 - i 1 ) · sin 2 ( i - i 2 ) · sin 2 ϕ 2 2 - sin 2 i 1 · sin 2 ( i 2 - i 1 ) · sin 2 ( i - i 2 ) · sin 2 ϕ 1 - ϕ 2 2 + sin 2 i 1 · cos 2 ( i 2 - i 1 ) · sin 2 ( i - i 2 ) · sin 2 ϕ 1 + ϕ 2 2