Abstract

In order to analyze an elliptical light wave, we must find its state of polarization which is completely determined by three parameters, i.e., the direction (azimuth), the form (ellipticity), and the sense of the ellipse described by the light vector. In this paper a survey is given of the direct and indirect methods used for the measurement of the magnitudes of the three parameters. First, a mathematical treatment, which uses the calculus of quaternions, of the state of polarization for special cases will give the background to the theories of the methods discussed.

© 1949 Optical Society of America

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References

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  1. H. G. Jerrard, J. Opt. Soc. Am. 38, 35 (1948).
    [Crossref]
  2. R. Clark Jones, J. Opt. Soc. Am. 31, 488 (1941).
    [Crossref]
  3. See reference 2, p. 490.
  4. H. Y. Hsü, M. Richartz, and Y. K. Liang, J. Opt. Soc. Am. 37, 99 (1947).
    [Crossref]
  5. H. Hurwitz and R. Clark Jones, J. Opt Soc. Am. 31, 493 (1941).
    [Crossref]
  6. Max Born, Optik (Verlagsbuchhandlung Julius Springer, Berlin, 1933), p. 23.
  7. O. Schönrock, Handbuch der Physik 19, 749 (1928).
  8. F. Lippich, Wien. Ber. 91, 1059 (1885).
  9. G. Szivessy, Handbuch der Physik 19, 955 (footnote 2) (1928); T. M. Lowry, Opt. Rotatory Power (Longmans, Green and Company, London, 1935), p. 189.
  10. See reference 7, p. 750.
  11. See reference 9, p. 932.
  12. E. Bertrand, Bull. Soc. Mineral. 1, 26 (1878).
  13. S. Nakamura, Centralblatt f. Min., 267 (1905).
  14. Such rotator was proposed by Hans Mueller at M.I.T. in a private conversation (July1946).
  15. J. Strong, Rev. Sci. Inst. 6, 243 (1935).
    [Crossref]
  16. See reference 9, p. 955.
  17. Chauvin, Ann. de Toulouse 3 (J), 30 (1889).
  18. L. Chaumont, Ann. de physique (9),  4, 175 (1915).
  19. G. Szivessy and W. Herzog, Zeits. f. Instrumentenk. 58, 229, 345 (1938).
  20. F. Gabler and P. Sokob, Zeits. f. Instrumentenk 58, 301 (1938).
  21. M. Richartz, Zeits. f. Instrumentenkunde 60, 360 (1940).
  22. G. Szivessy and A. Dierkesmann, Ann. d. Physik 11, 909 (1931).
  23. M. Richartz, “Measurement of phase differences of half-shadow plates,” Zeits. f. Instrumentenk. 60, 358 (1940).
  24. See reference 19, p. 345.
  25. See reference 24, p. 229.
  26. See reference 9, p. 955.
  27. M. Richartz, Zeits. f. Instrumentenkunde 61, 148 (1941); J. Opt. Soc. Am. 31, 292 (1941).
    [Crossref]
  28. See reference 9, p. 956.
  29. A. Q. Tool, Phys. Rev. 31, 1 (1910).
  30. T. M. Lowry, Optical Rotatory Power (Longmans, Green and Company, London, 1935), p. 186.
  31. L. B. Tuckerman, Univ. Stud. of the University of Nebraska,  9, 194 (1909).
  32. C. A. Skinner, J. Opt. Soc. Am. 10, 491 (1925).
    [Crossref]
  33. G. Szivessy, Zeits. f. Instrumentenk. 47, 148 (1925).
  34. C. Bergholm, Physik. Zeits. 21, 137 (1920).
  35. Reference 9, p. 961.
  36. G. Szivessy, Zeits. f. Instrumentenk. 46, 454 (1926).
  37. W. Voigt, Physik. Zeits. 2, 303 (1901); see reference 9, p. 963.
  38. R. S. Minor, Ann. di. Physik 10, 581 (1903).
    [Crossref]

1948 (1)

1947 (1)

1941 (3)

H. Hurwitz and R. Clark Jones, J. Opt Soc. Am. 31, 493 (1941).
[Crossref]

R. Clark Jones, J. Opt. Soc. Am. 31, 488 (1941).
[Crossref]

M. Richartz, Zeits. f. Instrumentenkunde 61, 148 (1941); J. Opt. Soc. Am. 31, 292 (1941).
[Crossref]

1940 (2)

M. Richartz, “Measurement of phase differences of half-shadow plates,” Zeits. f. Instrumentenk. 60, 358 (1940).

M. Richartz, Zeits. f. Instrumentenkunde 60, 360 (1940).

1938 (2)

G. Szivessy and W. Herzog, Zeits. f. Instrumentenk. 58, 229, 345 (1938).

F. Gabler and P. Sokob, Zeits. f. Instrumentenk 58, 301 (1938).

1935 (1)

J. Strong, Rev. Sci. Inst. 6, 243 (1935).
[Crossref]

1931 (1)

G. Szivessy and A. Dierkesmann, Ann. d. Physik 11, 909 (1931).

1928 (2)

G. Szivessy, Handbuch der Physik 19, 955 (footnote 2) (1928); T. M. Lowry, Opt. Rotatory Power (Longmans, Green and Company, London, 1935), p. 189.

O. Schönrock, Handbuch der Physik 19, 749 (1928).

1926 (1)

G. Szivessy, Zeits. f. Instrumentenk. 46, 454 (1926).

1925 (2)

C. A. Skinner, J. Opt. Soc. Am. 10, 491 (1925).
[Crossref]

G. Szivessy, Zeits. f. Instrumentenk. 47, 148 (1925).

1920 (1)

C. Bergholm, Physik. Zeits. 21, 137 (1920).

1915 (1)

L. Chaumont, Ann. de physique (9),  4, 175 (1915).

1910 (1)

A. Q. Tool, Phys. Rev. 31, 1 (1910).

1909 (1)

L. B. Tuckerman, Univ. Stud. of the University of Nebraska,  9, 194 (1909).

1905 (1)

S. Nakamura, Centralblatt f. Min., 267 (1905).

1903 (1)

R. S. Minor, Ann. di. Physik 10, 581 (1903).
[Crossref]

1901 (1)

W. Voigt, Physik. Zeits. 2, 303 (1901); see reference 9, p. 963.

1889 (1)

Chauvin, Ann. de Toulouse 3 (J), 30 (1889).

1885 (1)

F. Lippich, Wien. Ber. 91, 1059 (1885).

1878 (1)

E. Bertrand, Bull. Soc. Mineral. 1, 26 (1878).

Bergholm, C.

C. Bergholm, Physik. Zeits. 21, 137 (1920).

Bertrand, E.

E. Bertrand, Bull. Soc. Mineral. 1, 26 (1878).

Born, Max

Max Born, Optik (Verlagsbuchhandlung Julius Springer, Berlin, 1933), p. 23.

Chaumont, L.

L. Chaumont, Ann. de physique (9),  4, 175 (1915).

Chauvin,

Chauvin, Ann. de Toulouse 3 (J), 30 (1889).

Clark Jones, R.

H. Hurwitz and R. Clark Jones, J. Opt Soc. Am. 31, 493 (1941).
[Crossref]

R. Clark Jones, J. Opt. Soc. Am. 31, 488 (1941).
[Crossref]

Dierkesmann, A.

G. Szivessy and A. Dierkesmann, Ann. d. Physik 11, 909 (1931).

Gabler, F.

F. Gabler and P. Sokob, Zeits. f. Instrumentenk 58, 301 (1938).

Herzog, W.

G. Szivessy and W. Herzog, Zeits. f. Instrumentenk. 58, 229, 345 (1938).

Hsü, H. Y.

Hurwitz, H.

H. Hurwitz and R. Clark Jones, J. Opt Soc. Am. 31, 493 (1941).
[Crossref]

Jerrard, H. G.

Liang, Y. K.

Lippich, F.

F. Lippich, Wien. Ber. 91, 1059 (1885).

Lowry, T. M.

T. M. Lowry, Optical Rotatory Power (Longmans, Green and Company, London, 1935), p. 186.

Minor, R. S.

R. S. Minor, Ann. di. Physik 10, 581 (1903).
[Crossref]

Mueller, Hans

Such rotator was proposed by Hans Mueller at M.I.T. in a private conversation (July1946).

Nakamura, S.

S. Nakamura, Centralblatt f. Min., 267 (1905).

Richartz, M.

H. Y. Hsü, M. Richartz, and Y. K. Liang, J. Opt. Soc. Am. 37, 99 (1947).
[Crossref]

M. Richartz, Zeits. f. Instrumentenkunde 61, 148 (1941); J. Opt. Soc. Am. 31, 292 (1941).
[Crossref]

M. Richartz, “Measurement of phase differences of half-shadow plates,” Zeits. f. Instrumentenk. 60, 358 (1940).

M. Richartz, Zeits. f. Instrumentenkunde 60, 360 (1940).

Schönrock, O.

O. Schönrock, Handbuch der Physik 19, 749 (1928).

Skinner, C. A.

Sokob, P.

F. Gabler and P. Sokob, Zeits. f. Instrumentenk 58, 301 (1938).

Strong, J.

J. Strong, Rev. Sci. Inst. 6, 243 (1935).
[Crossref]

Szivessy, G.

G. Szivessy and W. Herzog, Zeits. f. Instrumentenk. 58, 229, 345 (1938).

G. Szivessy and A. Dierkesmann, Ann. d. Physik 11, 909 (1931).

G. Szivessy, Handbuch der Physik 19, 955 (footnote 2) (1928); T. M. Lowry, Opt. Rotatory Power (Longmans, Green and Company, London, 1935), p. 189.

G. Szivessy, Zeits. f. Instrumentenk. 46, 454 (1926).

G. Szivessy, Zeits. f. Instrumentenk. 47, 148 (1925).

Tool, A. Q.

A. Q. Tool, Phys. Rev. 31, 1 (1910).

Tuckerman, L. B.

L. B. Tuckerman, Univ. Stud. of the University of Nebraska,  9, 194 (1909).

Voigt, W.

W. Voigt, Physik. Zeits. 2, 303 (1901); see reference 9, p. 963.

Ann. d. Physik (1)

G. Szivessy and A. Dierkesmann, Ann. d. Physik 11, 909 (1931).

Ann. de physique (9) (1)

L. Chaumont, Ann. de physique (9),  4, 175 (1915).

Ann. de Toulouse (1)

Chauvin, Ann. de Toulouse 3 (J), 30 (1889).

Ann. di. Physik (1)

R. S. Minor, Ann. di. Physik 10, 581 (1903).
[Crossref]

Bull. Soc. Mineral. (1)

E. Bertrand, Bull. Soc. Mineral. 1, 26 (1878).

Centralblatt f. Min. (1)

S. Nakamura, Centralblatt f. Min., 267 (1905).

Handbuch der Physik (2)

O. Schönrock, Handbuch der Physik 19, 749 (1928).

G. Szivessy, Handbuch der Physik 19, 955 (footnote 2) (1928); T. M. Lowry, Opt. Rotatory Power (Longmans, Green and Company, London, 1935), p. 189.

J. Opt Soc. Am. (1)

H. Hurwitz and R. Clark Jones, J. Opt Soc. Am. 31, 493 (1941).
[Crossref]

J. Opt. Soc. Am. (4)

Phys. Rev. (1)

A. Q. Tool, Phys. Rev. 31, 1 (1910).

Physik. Zeits. (2)

C. Bergholm, Physik. Zeits. 21, 137 (1920).

W. Voigt, Physik. Zeits. 2, 303 (1901); see reference 9, p. 963.

Rev. Sci. Inst. (1)

J. Strong, Rev. Sci. Inst. 6, 243 (1935).
[Crossref]

Univ. Stud. of the University of Nebraska (1)

L. B. Tuckerman, Univ. Stud. of the University of Nebraska,  9, 194 (1909).

Wien. Ber. (1)

F. Lippich, Wien. Ber. 91, 1059 (1885).

Zeits. f. Instrumentenk (1)

F. Gabler and P. Sokob, Zeits. f. Instrumentenk 58, 301 (1938).

Zeits. f. Instrumentenk. (4)

M. Richartz, “Measurement of phase differences of half-shadow plates,” Zeits. f. Instrumentenk. 60, 358 (1940).

G. Szivessy, Zeits. f. Instrumentenk. 46, 454 (1926).

G. Szivessy, Zeits. f. Instrumentenk. 47, 148 (1925).

G. Szivessy and W. Herzog, Zeits. f. Instrumentenk. 58, 229, 345 (1938).

Zeits. f. Instrumentenkunde (2)

M. Richartz, Zeits. f. Instrumentenkunde 60, 360 (1940).

M. Richartz, Zeits. f. Instrumentenkunde 61, 148 (1941); J. Opt. Soc. Am. 31, 292 (1941).
[Crossref]

Other (12)

See reference 9, p. 956.

See reference 19, p. 345.

See reference 24, p. 229.

See reference 9, p. 955.

T. M. Lowry, Optical Rotatory Power (Longmans, Green and Company, London, 1935), p. 186.

Reference 9, p. 961.

See reference 9, p. 955.

Such rotator was proposed by Hans Mueller at M.I.T. in a private conversation (July1946).

See reference 7, p. 750.

See reference 9, p. 932.

See reference 2, p. 490.

Max Born, Optik (Verlagsbuchhandlung Julius Springer, Berlin, 1933), p. 23.

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Figures (11)

Fig. 1
Fig. 1

Ellipse described by the light vector E; OF, fixed direction; OA, vibration direction of analyzer; α, its azimuth; OR1, OR2, direction of axes of the ellipse; r1, r2, magnitude of major and minor semiaxes; ϑ, azimuth of the major axis; r2/r1=tanχ, ellipticity.

Fig. 2
Fig. 2

Optical system and azimuths of the directions of vibration for a system of three doubly refracting plates. A, analyzer; P, polarizer; P1, P2, P3, “fast” directions of vibration of the three plates; α, γ1, γ2, and γ3, azimuths of the analyzer and the plates 1, 2, and 3, respectively, measured from OP.

Fig. 3
Fig. 3

Directions of the major axis, ϑ′, for different γ′(ae); numbers 0, 1, 2, 3, 4, indicate the values of ρ=0, <π/4, π/4, >π/4 and π/2, respectively.

Fig. 4
Fig. 4

Ellipses 1 which are *(ρ, γ′) together with the rotator S(γ′) as fore multiplier. The values of γ and ϑ correspond to the values of γ′ and ϑ′ in Fig. 3.

Fig. 5
Fig. 5

The Schönrock analyzer in the half-shadow position.

Fig. 6
Fig. 6

The effect of the Nakamura plate upon the elliptical vibration.

Fig. 7
Fig. 7

The effect of the half-wave plate when its axes are parallel to that of the incident, elliptical wave.

Fig. 8
Fig. 8

Chaumont’s half-shadow device. In Fig. 8a the half-wave plate covers the first and fourth quadrants with phase differences π and −π/2, the quarter-wave plate the third and fourth quadrants with phase differences π/2 and −π/2. Figure 8b shows the phase differences in the four quadrants when the half-shadow device is combined with the compensating quarter-wave plate.

Fig. 9
Fig. 9

Vibration directions of the combination: doubly refracting biplate and analyzer; B1, B2, fast axes of the half-plates 1 and 2, respectively.

Fig. 10
Fig. 10

Richartz’ elliptical analyzer. (a) Combination of half-shadow quarter-wave plate, H, and rotating biplate, N. (b) Change of the elliptical vibration by the combination when the axes of plate H are parallel to that of the ellipse. (c) Half-shadow azimuths of the analyzer, α0 and and α1, for the lower and upper halves of the field of view, respectively.

Fig. 11
Fig. 11

Voigt’s arrangement, and system of zero points.

Tables (1)

Tables Icon

Table I Azimuth ϑ′ (and ϑ), electric vector *(ρ, γ′), and ellipticity F*(ρ, γ′) for given values of ρ and γ′ (or γ).

Equations (130)

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( E x E y ) = ( cos γ - sin γ sin γ cos γ ) ( e i ρ 0 0 e - i ρ ) × ( cos γ sin γ - sin γ cos γ ) ( E x 0 E y 0 ) ,
= S ( γ ) G ( ρ ) S ( - γ ) 0 ,
M S ( γ ) G ( ρ ) S ( - γ )
I = ( 1 0 0 1 ) ;         i = ( i 0 0 - i ) ; J = ( 0 1 - 1 0 ) ;             k = ( 0 i i 0 ) ,
i 2 = j 2 = k 2 = - 1 , jk = i = - kj ;             ki = j = - ik ;             ij = k = - ji .
q = t + x i + y j + z k ,
q ¯ = t - x i - y j - z k ,
q q ¯ = t 2 + x 2 + y 2 + z 2 .
S ( γ ) = cos γ - sin γ j .
M = cos ρ + sin ρ cos 2 γ i + sin ρ sin 2 γ k .
0 = ( E x 0 E y 0 ) = ( 1 0 ) A e i ω t ,
1 = M 1 0 = ( cos ρ 1 + i sin ρ 1 cos 2 γ 1 i sin ρ 1 sin 2 γ 1 ) A e i ω t .
1 ( a ) = S ( - α ) 1 ,
a = ( E x , 1 ( a ) 0 ) ,
E x , 1 ( a ) = ( 1 0 ) S ( - α ) 1 = { cos ρ 1 cos α + i sin ρ 1 cos ( α - 2 γ 1 ) } A e i ω t .
J 1 = K E x , 1 ( a ) E x , 1 ( a ) ¯ = cos 2 α - sin 2 ρ 1 ( cos 2 α - cos 2 ( α - 2 γ 1 ) ) = cos 2 α + sin 2 δ 1 / 2 sin 2 γ 1 sin 2 ( α - γ 1 ) ,
S ( γ 1 ) S ( γ 2 ) = S ( γ 1 + γ 2 ) ,
M ( 2 ) = M 2 M 1 = MS ( - γ ) ,
cos ρ cos γ = cos ρ 2 cos ρ 1 - sin ρ 2 sin ρ 1 × cos ( 2 γ 2 - 2 γ 1 ) ,
sin ρ cos ( 2 γ + γ ) = cos ρ 2 sin ρ 1 cos 2 γ 1 + sin ρ 2 cos ρ 1 cos 2 γ 2 ,
cos ρ sin γ = sin ρ 2 sin ρ 1 sin ( 2 γ 2 - 2 γ 1 ) ,
sin ρ sin ( 2 γ + γ ) = cos ρ 2 sin ρ 1 sin 2 γ 1 + sin ρ 2 cos ρ 1 sin 2 γ 2 .
tan γ = x 2 x 1 sin ( 2 γ 2 - 2 γ 1 ) 1 - x 2 x 1 cos ( 2 γ 2 - 2 γ 1 ) ,
tan ( 2 γ + γ ) = x 1 sin 2 γ 1 + x 2 sin 2 γ 2 x 1 cos 2 γ 1 + x 2 cos 2 γ 2 ,
x 2 = tan 2 ρ = x 1 2 + 2 x 1 x 2 cos ( 2 γ 2 - 2 γ 1 ) + x 2 2 1 - 2 x 1 x 2 cos ( 2 γ 2 - 2 γ 1 ) + x 1 2 x 2 2 .
cos ρ 0 cos γ 0 = cos ρ cos γ , sin ρ 0 cos ( 2 γ 0 + γ 0 ) = sin ρ cos ( 2 γ + γ ) , cos ρ 0 sin γ 0 = cos ρ sin γ , sin ρ 0 sin ( 2 γ 0 + γ 0 ) = sin ρ sin ( 2 γ + γ ) .
tan γ 0 = tan γ ,             tan ( 2 γ 0 + γ 0 ) = tan ( 2 γ + γ ) .
γ 0 = γ ;             γ 0 = γ             or             γ + π / 2.
ρ 0 = ρ             when             γ 0 = γ ,
ρ 0 = - ρ             when             γ 0 = γ + π / 2.
x = a cos ω t ,             y = b cos ( ω t - ψ ) .
x 2 a 2 sin 2 ψ - 2 x y cos ψ a b sin 2 ψ + y 2 b 2 sin 2 ψ = 1.
tan 2 ϑ = 2 ab cos ψ a 2 - b 2 ,             sin 2 χ = 2 a b sin ψ a 2 + b 2 .
1 = M 1 0 = S ( γ ) S ( π / 4 ) G ( ρ ) S ( - π / 4 ) × S ( - γ ) ( 1 0 ) A e i ω t = S ( γ ) ( cos ρ cos γ - i sin ρ sin γ - cos ρ sin γ + i sin ρ cos γ ) A e i ω t S ( γ ) * ( ρ , γ ) A e i ω t ,
a 2 = cos 2 ρ cos 2 γ + sin 2 ρ sin 2 γ , b = cos 2 ρ sin 2 γ + sin 2 ρ cos 2 γ .
tan 2 ϑ = - sec 2 ρ tan 2 γ , sin 2 χ = sin 2 ρ cos 2 γ ,
r 2 / r 1 = tan χ = [ 1 - ( 1 - sin 2 2 ρ cos 2 2 γ ) 1 2 1 + ( 1 - sin 2 2 ρ cos 2 2 γ ) 1 2 ] 1 2 F * ( ρ , γ ) ,
r 1 2 + r 2 2 = a 2 + b 2 = 1.
tan 2 γ = ± ( cos 2 ρ ) 1 2 .
* ( ρ , γ ) = ( cos ρ i sin ρ ) ,
E x = A cos ρ cos ω t ,             E y = A sin ρ cos ( ω t + π / 2 ) .
2 = M ( 2 ) 0 = M 2 M 1 ( 1 0 ) A e i ω t , 1 = M 1 0 = M 1 ( 1 0 ) A e i ω t ,
M 2 = S ( h ) G ( η 2 ) S ( - h ) = cos η 2 + sin η 2 cos 2 h i + sin η 2 sin 2 h k
M 1 = S ( ± π 4 ) G ( δ 2 ) S ( π 4 ) = cos δ 2 ± sin δ 2 k .
2 = ( cos η 2 cos δ 2 sin η 2 sin δ 2 sin 2 h + i sin η 2 cos δ 2 cos 2 h ± sin η 2 sin δ 2 cos 2 h + i { sin η 2 cos δ 2 sin 2 h ± cos η 2 sin δ 2 } ) A e i ω t ,
1 = ( cos ( δ / 2 ) ± i sin ( δ / 2 ) ) A e i ω t .
E 2 = E x , 2 ( a ) = { ± sin η 2 sin δ 2 cos 2 h + i ( sin η 2 cos δ 2 sin 2 h ± cos η 2 sin δ 2 ) } A e i ω t ,
E 1 E x , 1 ( a ) = ± i sin ( δ / 2 ) A e i ω t ;
J 2 = sin 2 ( δ / 2 ) + sin 2 ( η / 2 ) cos δ sin 2 2 h ± 1 2 sin δ sin η sin 2 h ,
J 1 = sin 2 ( δ / 2 ) .
sin 2 h 0 = 0 ,             or             tan δ = tan η 2 sin 2 h 0 .
s = 1 J 0 | h ( J - J ) | h = h 0
s = 1 E 1 E ¯ 2 | h ( E 2 E ¯ 2 - E 1 E ¯ 1 ) | h = h 0 = 2 J 1 | R ( E 2 ) h R ( E 2 ) + I ( E 2 ) h I ( E 2 ) | = 2 | sin η tan δ / 2 | | ( 1 - tan 2 δ tan 2 η / 2 ) 1 2 | ,
3 ( a ) = S ( - β ) M ( 3 ) 0 = S ( β ) M 3 M ( 2 ) 0 = S ( β ) S ( k ) G ( κ / 2 ) S ( - k ) M ( 2 ) 0 = S ( k ) G ( κ / 2 ) S ( - k ) S ( β ) M ( 2 ) 0 = M ( 3 ) S ( β ) 0 .
3 ( a ) = M ( 3 ) ( cos η + δ 2 ± sin η + δ 2 k ) ( cos β sin β ) A e i ω t .
a = ( E x , 3 ( a ) 0 ) ,
E x , 3 ( a ) = { cos β ( cos κ 2 cos δ + η 2 sin κ 2 × sin δ + η 2 sin 2 k + i sin κ 2 cos δ + η 2 cos 2 k ) + sin β ( sin κ 2 sin δ + η 2 cos 2 k ± i cos κ 2 sin δ + η 2 + i sin κ 2 cos δ + η 2 sin 2 k ) } A e i ω t .
J 3 = cos 2 β - cos 2 β ( sin 2 ( δ + η 2 ) + sin 2 κ 2 cos ( δ + η ) sin 2 2 k ± cos κ 2 sin κ 2 × sin ( δ + η ) sin 2 k ) + sin 2 β sin 2 κ 2 sin 2 k cos 2 k .
± ( 1 - 2 sin 2 k ¯ sin 2 κ 2 ) sin ( δ + η 2 ) + sin 2 k ¯ sin κ cos ( δ + η 2 ) = 0.
δ = sin κ ( sin 2 k ¯ - sin 2 k ¯ 0 ) ,
3 ( a ) = S ( - α ) M ( 3 ) 0 .
3 ( a ) = S ( - α ) S ( ± π 4 ) G ( η 2 ) S ( π 4 ) × S ( k ) G ( κ 2 ) S ( - k ) × S ( ± π 4 ) G ( δ 2 ) S ( π 4 ) 0 .
a = ( E x , 3 ( a ) 0 ) ,
E x , 3 ( a ) = [ cos α ( cos κ 2 cos δ + η 2 sin κ 2 sin δ + η 2 sin 2 k + i sin κ 2 cos δ - η 2 cos 2 k ) + sin α ( ± sin κ 2 sin δ - η 2 cos 2 k ± i cos κ 2 sin δ + η 2 + i sin κ 2 cos δ + η 2 sin 2 k ) ] A e i ω t .
J 3 = cos 2 α - cos 2 α ( sin 2 δ + η 2 + sin 2 κ 2 cos ( δ + η ) - sin 2 κ 2 cos 2 2 k cos δ cos η ± sin κ 2 cos κ 2 sin ( δ + η ) sin 2 k ) ) + sin 2 α ( sin κ 2 cos 2 k { sin κ 2 sin 2 k cos δ ± cos κ 2 sin δ } ) .
± sin δ ( cos κ cos η 3 sin 2 k ¯ sin κ sin η 2 ) + cos δ { sin κ sin 2 k ¯ cos η 2 ± sin η 2 [ 1 - 2 sin 2 2 k ¯ sin 2 κ 2 ] } = 0.
1 = S ( ϑ ) ( cos ρ ± i sin ρ ) A e i ω t .
1 ( a ) = S ( - α ) 1             and             a = ( E x , 1 ( a ) 0 ) ,
E x , 1 ( a ) = { cos ρ cos ( α - ϑ ) ± sin ρ sin ( α - ϑ ) } A e i ω t ;
2 ( a ) = S ( ) ( a ) *             and             a = ( E x , 2 ( a ) 0 ) ,
E x , 2 ( a ) = cos { cos ρ cos ( α - ϑ + ) ± sin ρ sin ( α - ϑ + ) } A e i ω t .
J 1 = cos 2 ρ cos 2 ( α - ϑ ) + sin 2 ρ sin 2 ( α - ϑ ) , J 2 = cos 2 { cos 2 ρ cos 2 ( α - ϑ + ) + sin 2 ρ sin 2 ( α - ϑ + ) } .
sin { cos 2 ρ cos 2 ( α - ϑ + ) + sin 2 ρ sin 2 ( α - ϑ + ) = - sin ( 2 α - 2 ϑ + ) cos 2 ρ .
cos 2 ρ sin ( 2 α - 2 ϑ ) sin 2 = 0.
1 ( a ) = S ( - α ) S ( ϑ + θ ) ( cos ρ ± i sin ρ ) A e i ω t
a = ( E x , 1 ( a ) 0 ) ,
E x , 1 ( a ) = { cos ( α - ϑ - θ ) cos ρ ± i sin ( α - ϑ - θ ) sin ρ } A e i ω t .
J 1 = cos 2 ρ cos 2 ( α - ϑ - θ ) + sin 2 ρ sin 2 ( α - ϑ - θ ) ,
M 2 M 1 = { cos 2 [ γ + ( π / 2 ) ± ] i + sin 2 [ γ + ( π / 2 ) ± ] k } × { cos 2 γ i + sin 2 γ k } = cos ( ± 2 ) - sin ( ± 2 ) j = S ( ± 2 ) ,
2 = { cos 2 ( α + ) i + sin 2 ( α + ) k } S ( ϑ ) ( cos ρ ± i sin ρ ) A e i ω t = ( ± sin ρ sin ( ϑ - 2 α - 2 ) + i cos ρ cos ( ϑ - 2 α - 2 ) ± sin ρ cos ( ϑ - 2 α - 2 ) - i cos ρ sin ( ϑ - 2 α - 2 ) ) A e i ω t ,
2 ( a ) = S ( - α ) 2 ,             a = ( E x , 2 ( a ) 0 ) ,
E x , 2 ( a ) = { ± sin ρ sin ( ϑ - α - 2 ) + i cos ρ cos ( ϑ - α - 2 ) } A e i ω t .
J 2 = sin 2 ρ sin 2 ( ϑ - α - 2 ) + cos 2 ρ cos 2 ( ϑ - α - 2 ) .
1 = S ( ϑ ) ( cos ρ ± i sin ρ ) A e i ω t ,             1 ( a ) = S ( - α ) 1
a = ( E x , 1 ( a ) 0 ) ,
E x , 1 ( a ) = { cos ρ cos ( α - ϑ ) ± i sin ρ sin ( α - ϑ ) } A e i ω t .
J 1 = cos 2 ρ cos 2 ( α - ϑ ) + sin 2 ρ sin 2 ( α - ϑ ) .
cos 2 ρ sin ( 2 α - 2 ϑ + 2 ) sin 2 = 0.
2 = ( ± sin ρ sin ϑ - i cos ρ cos ϑ sin ρ cos ϑ - i cos ρ sin ϑ ) A e i ω t = - i S ( ϑ ) ( cos ρ i sin ρ ) A e i ω t ,
2 ( a ) = S ( - α ) S ( k ) G ( Δ 2 ) S ( - k ) S ( ϑ ) × S ( ± π 4 ) G ( δ 2 ) S ( π 4 ) 0 = S ( - α + ϑ ) S ( k - ϑ ) G ( Δ 2 ) S ( - k + ϑ ) × S ( ± π 4 ) G ( δ 2 ) S ( π 4 ) 0 .
J = cos 2 ( α - ϑ ) - cos 2 ( α - ϑ ) { sin 2 δ 2 + sin 2 Δ 2 cos δ × sin 2 2 ( k - ϑ ) ± 1 2 sin Δ sin δ sin 2 ( k - ϑ ) } + sin 2 ( α - ϑ ) { sin 2 Δ 2 cos 2 ( k - ϑ ) × sin 2 ( k - ϑ ) cos δ ± 1 2 sin Δ sin δ cos 2 ( k - ϑ ) } .
sin δ = 2 e 1 + e 2 ,             cos δ = 1 - e 2 1 + e 2 ,
J = 1 1 + e 2 { cos 2 ( α - ϑ ) + e 2 sin 2 ( α - ϑ ) - ( 1 - e 2 ) sin 2 Δ 2 sin 2 ( k - ϑ ) sin 2 ( k - α ) e sin Δ sin 2 ( k - α ) } .
δ J = J - J = 1 1 + e 2 | 2 sin Δ 2 sin ( β 1 - β 2 ) × { ( 1 - e 2 ) sin Δ 2 sin 2 ( - α - ϑ + β 1 + β 2 ) × cos ( β 1 - β 2 ) 2 e cos Δ 2 cos ( β 1 + β 2 - 2 α ) } | ,
β 1 + β 2 = 2 α             or             β 1 + β 2 = 2 α π .
sin 2 ( α ¯ - ϑ ) = ± ( tan 2 χ / cos 2 β ) cot ( Δ / 2 ) .
γ = 1 2 arc sin [ ( tan 2 χ / cos 2 β ) cot ( Δ / 2 ) ] , ( - π / 4 γ + π / 4 ) ;
α ¯ 1 = ± γ + ϑ ;             α ¯ 2 = π / 2 γ + ϑ ; α ¯ 3 = π ± γ + ϑ ;             α ¯ 4 = 3 π / 2 γ + ϑ .
ϑ = ( t 2 + t 2 ) / 2 - ( t 0 + 90 ° ) = ( t 4 + t 4 ) / 2 - ( t 0 + 90 ° ) .
sin 2 χ = sin 2 γ sin δ .
sin 2 χ = sin δ ,             and             χ = δ / 2.
= M 2 M 1 ( 1 0 ) = ( cos ρ 2 cos ρ 1 - sin ρ 2 sin ρ 1 cos 2 ( γ 2 - γ 1 ) + i [ cos ρ 2 sin ρ 1 cos 2 γ 1 + sin ρ 2 cos ρ 1 cos 2 γ 2 ] - sin ρ 2 sin ρ 1 sin 2 ( γ 2 - γ 1 ) + i [ cos ρ 2 sin ρ 1 sin 2 γ 1 + sin ρ 2 cos ρ 1 sin 2 γ 2 ] ) .
cos ρ 2 sin ρ 1 cos 2 γ 1 + sin ρ 2 cos ρ 1 cos 2 γ 2 cos ρ 2 cos ρ 1 - sin ρ 2 sin ρ 1 cos 2 ( γ 2 - γ 1 ) = cos ρ 2 sin ρ 1 sin 2 γ 1 + sin ρ 2 cos ρ 1 sin 2 γ 2 - sin ρ 2 sin ρ 1 sin 2 ( γ 2 - γ 1 ) ,
= ( 1 2 ) 1 2 ( 1 + i ) ( cos ρ ± sin ρ )             or         ( 1 2 ) 1 2 ( 1 - i ) ( cos ρ sin ρ ) .
2 h 0 = ( θ 1 + θ 2 + θ 3 ) / 3 ,
tan 2 χ = cos 2 β tan Δ 2 sin 2 γ ,
χ = 1 2 arctan ( cos 2 β tan Δ 2 sin t 2 - t 2 ) = 1 2 arctan ( cos 2 β tan Δ 2 sin t 4 - t 4 ) .
tan 2 χ = tan 2 ρ 1 = tan 2 ρ 2 sin 2 γ 2 = tan δ sin 2 ϑ .
ϑ ¯ = ( p 2 - p 0 ) / 2 + ( p 1 - p 0 ) / 2 + π / 4.
sin 2 χ = sin δ cos 2 ( γ - π / 4 ) = sin δ sin 2 γ .
tan 2 ϑ = tan 2 γ cos δ ,             cos 2 γ = cos 2 χ cos 2 ϑ .
cos 2 χ = cos 2 γ 1 cos 2 ϑ 1 = cos 2 γ 2 cos 2 ϑ 2 = sin ( γ 2 - γ 1 ) sin ( ϑ 2 - ϑ 1 )
cos 2 χ = sin ( q 2 - q 1 ) sin ( p 2 - p 1 ) .
ϑ = ( α 1 + α 2 ) / 2 + π / 2 ,
cos 2 χ = sin [ ( k 2 - k 1 ) - ( α 2 - α 1 ) ] sin ( k 2 - k 1 ) · sin ( k 2 - k 1 ) sin [ ( k 2 - k 1 ) - ( α 2 - α 1 ) ] ,
δ = sin δ ( sin 2 k - sin 2 k 0 ) ,
δ = sin δ ( cos 2 k ¯ 0 - cos 2 k ¯ 0 ) .
sin 2 γ 2 cos 2 ρ 1 = 0.
tan 2 χ = sin 2 ϑ tan δ
d f / d t = 1 2 ν a b sin δ .
δ = 2 π ( d / λ ) ( μ Y - μ X ) ,
sin 2 χ = sin 2 γ sin δ .
δ 01 = ± 2 n π - η / 2 ,             ( n = 0 , 1 , 2 , ) .
δ 02 = ± 2 n π + η / 2.
( δ 01 + δ 02 ) / 2 = ± 2 n π .
δ 1 = - δ - η / 2 ± 2 n π ) .
δ 2 = - δ + η / 2 ± 2 n π ) .
δ = p 01 + p 02 2 - p 1 + p 2 2 .
1 = ( 1 exp ( - ( δ + δ ) i ) ) A e i ω t .
2 = S ( θ ) 1 ,