Abstract

A detailed survey is given of the present-day knowledge of optical compensators. The compensators discussed are those of Babinet, Soleil, Rayleigh, De Forest Palmer, Brace, Szivessy, Senármont, and Richartz. Each instrument is described, the theory developed, the method of use for the measurement of small phase differences given, and reference made to the sensitivity and accuracy.

© 1948 Optical Society of America

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References

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  1. For a full account of the influence of reflections on a doubly refracting plate see F. Gabler and P. Sokob, Zeits. f. Physik 116, 47 (1940). The effect of absorption which is negligible is discussed by M. Berek, Ann. d. Physik 58, 165 (1919).
    [Crossref]
  2. G. Szivessy and W. Herzog, Zeits. f. Instrumentenk. 57, 53 (1937). For alternative forms see F. Gabler and P. Sokob, Zeits. f. Instrumentenk 58, 302 (1938). For a generalized intensity formula for a system of n plates see Hsien-Yü Hsü, M. Richartz, and Yüng-K’ang Liang, J. Opt. Soc. Am. 37, 99 (1947).
    [Crossref]
  3. G. Szivessy, Handbuch der Physik 19, 926 (1928).
  4. A. Bravais, Comptes rendus 32, 115 (1851); G. Szivessy, Zeits. f. Physik 29, 372 (1924).
    [Crossref]
  5. G. Szivessy, Handbuch der Physik 19, 941 (1928).
  6. Soleil, Comptes rendus,  21, 426 (1845).
  7. M. E. Mascart, Traité d’Optique (Gauthier-Villars, Paris, 1891), Vol. 2, p. 61.
  8. A composite plate consisting of two equally thick plates, usually of quartz cut at π/4 to the optical axis, lying on top of each other with their principal planes at right angles. See G. Szivessy, Handbuch der Physik 19, 942 (1928).
  9. Bravais, Zehnder, Biernacki, Zahrzewski, and Koenigsberger, Handbuch der Physik 19, 943 (1928).
  10. G. Szivessy, Zeits. f. Physik 29, 372 (1924).
    [Crossref]
  11. Rayleigh, Phil Mag 4, 680 (1902); Sci. Pap. 5, 64 (1905).
  12. By using this compensator in conjunction with a Soleil, the fringe displacement of the former can be compensated whence the drum reading will give δ directly.
  13. P. Zeeman and C. M. Hoogenboom, Physik. Zeits. 13, 914 (1912).
  14. A. De Forest Palmer, Phys. Rev. 17, 409 (1921); H. A. Boorse, Phys. Rev. 46, 187 (1934).
    [Crossref]
  15. In the original the half-shade and compensator springs produced tensions of 0.285 g and 2.04 g per micrometer division, respectively. If PH and Pc are the tension expressed in micrometer divisionsη=2.00PH×10-6,         κ=6.112Pc×10-6;possible maxima being η=3×54×10−4·2π and 1.77×10−4·2π whence the maximum value of the phase difference which can be measured is κ0=1.77×10−4·2π.
  16. D. B. Brace, Phil. Mag. 7, 320 (1904); Phys. Rev. 18, 70 (1904); ibid. 19, 218 (1904); G. Szivessy, Zeits. f. Instrumentenk. 57, 49 and 89 (1937).
    [Crossref]
  17. The effect of the sequence of the plates has been discussed in the introduction. The possible arrangements for the Brace compensator are shown in Fig. 10, where it can be seen that the essential difference between arrangements A and B is that in one case K never lies between H and D, in the other it always does.
  18. If ϕ0 is negative, 2ϕ=±2nπ−2ϕ0 or ±(2n+1)π+2ϕ0, (n=0,1,2⋯).
  19. For a detailed discussion see G. Szivessy, Zeits. f. Instrumentenk. 57, 92 (1937).
  20. To obtain a half-shadow plate of very small difference which can also be varied, a plate of phase difference η can be used in azimuth h(−π/2⋜h⋜+π/2) differing from ±π/4 so that the effective phase difference in this azimuth is η=±η sin2h, the positive value holding if 0⋜h⋜+π/2, the negative if −π/2⋜h⋜0. Such a plate was devised by Wedeneewa (Zeits. f. Instrumentenk. 43, 17 (1923)), but its application is limited to cases where small phase differences are involved and, if arrangement A is used, the polarizer and analyzer must be crossed, thus precluding the use of the most favorable polarizer azimuth.
  21. G. Szivessy, Zeits. f. Physik 54, 594 (1929).
    [Crossref]
  22. G. Szivessy and A. Dierkesmann, Ann. d. Physik 11, 949 (1931).
    [Crossref]
  23. G. Szivessy and A. Dierkesmann, Ann. d. Physik 11, 956 (1931).
  24. G. Szivessy and W. Herzog, Zeits. f. Instrumentenk. 57, 305 (1937).
  25. Note that a value for η can be found without recourse to an independent measurement, fortan2η/2=4-tan22ϵ1(cot2α¯01+cot2α¯02)2sin22ϵ1(cot2α¯01-cot2α¯02)2;W. Hlerzog, Zeits. f. Physik 97, 225 (1935).
    [Crossref]
  26. If, in arrangement B, Eq. (82) is to be used for the calculation of δ, then it is immaterial whether OD and OH are parallel or perpendicular whatever the sign of sin2δ cosκ tanη/2.
  27. H. de Sénarmont, Ann. Chim. Phys. 73, 337 (1840); F. Gabler and P. Sokob, Zeits. f. Instrumentenk. 58, 301 (1938); F. Gabler and P. Sokob, Zeits. f. Instrumentenk. 61, 298 (1941); F. Gabler and P. Sokob, Physik. Zeits. 42, 319 (1941).
  28. A symmetrical prism with right end faces, consisting of two Glan-Thomson prisms cemented together so that the vibration directions make a small angle 2ϵ with each other. O. Schönrock, Handbuch der Physik 19, 750 (1928).
  29. A polarizing prism combined with a λ/2 plate covering half the field of view. The vibration directions make an angle ϵ with each other; this angle can be varied. M. Chauvin, J. de Phys. 9, 21 (1890); Ann. de Toulouse 3(J) 28, (1889).
  30. For a full discussion see F. Gabler and P. Sokob, Zeits f. Instrumentenk. 58, 301 (1938) (see reference 27).
  31. F. Gabler and P. Sokob, Zeits. f. Instrumentenk. 58, 301 (1938).
  32. F. Gabler and P. Sokob, Zeits. f. Physik 116, 47 (1940).
    [Crossref]
  33. Reference 32, Eq. 22.
  34. Since s is dependent on the angle ϵ, which may be easily changed, this gives the Chauvin analyzer the decided advantage of variable sensitivity.
    [Crossref]
  35. M. Richartz, Zeits. f. Instrumentenk. 60, 357 (1940); J. Opt. Soc. Am. 31, 292 (1941).

1940 (3)

For a full account of the influence of reflections on a doubly refracting plate see F. Gabler and P. Sokob, Zeits. f. Physik 116, 47 (1940). The effect of absorption which is negligible is discussed by M. Berek, Ann. d. Physik 58, 165 (1919).
[Crossref]

F. Gabler and P. Sokob, Zeits. f. Physik 116, 47 (1940).
[Crossref]

M. Richartz, Zeits. f. Instrumentenk. 60, 357 (1940); J. Opt. Soc. Am. 31, 292 (1941).

1938 (2)

For a full discussion see F. Gabler and P. Sokob, Zeits f. Instrumentenk. 58, 301 (1938) (see reference 27).

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

1937 (3)

G. Szivessy and W. Herzog, Zeits. f. Instrumentenk. 57, 305 (1937).

G. Szivessy and W. Herzog, Zeits. f. Instrumentenk. 57, 53 (1937). For alternative forms see F. Gabler and P. Sokob, Zeits. f. Instrumentenk 58, 302 (1938). For a generalized intensity formula for a system of n plates see Hsien-Yü Hsü, M. Richartz, and Yüng-K’ang Liang, J. Opt. Soc. Am. 37, 99 (1947).
[Crossref]

For a detailed discussion see G. Szivessy, Zeits. f. Instrumentenk. 57, 92 (1937).

1935 (1)

Note that a value for η can be found without recourse to an independent measurement, fortan2η/2=4-tan22ϵ1(cot2α¯01+cot2α¯02)2sin22ϵ1(cot2α¯01-cot2α¯02)2;W. Hlerzog, Zeits. f. Physik 97, 225 (1935).
[Crossref]

1931 (2)

G. Szivessy and A. Dierkesmann, Ann. d. Physik 11, 949 (1931).
[Crossref]

G. Szivessy and A. Dierkesmann, Ann. d. Physik 11, 956 (1931).

1929 (1)

G. Szivessy, Zeits. f. Physik 54, 594 (1929).
[Crossref]

1928 (5)

G. Szivessy, Handbuch der Physik 19, 941 (1928).

G. Szivessy, Handbuch der Physik 19, 926 (1928).

A composite plate consisting of two equally thick plates, usually of quartz cut at π/4 to the optical axis, lying on top of each other with their principal planes at right angles. See G. Szivessy, Handbuch der Physik 19, 942 (1928).

Bravais, Zehnder, Biernacki, Zahrzewski, and Koenigsberger, Handbuch der Physik 19, 943 (1928).

A symmetrical prism with right end faces, consisting of two Glan-Thomson prisms cemented together so that the vibration directions make a small angle 2ϵ with each other. O. Schönrock, Handbuch der Physik 19, 750 (1928).

1924 (1)

G. Szivessy, Zeits. f. Physik 29, 372 (1924).
[Crossref]

1923 (1)

To obtain a half-shadow plate of very small difference which can also be varied, a plate of phase difference η can be used in azimuth h(−π/2⋜h⋜+π/2) differing from ±π/4 so that the effective phase difference in this azimuth is η=±η sin2h, the positive value holding if 0⋜h⋜+π/2, the negative if −π/2⋜h⋜0. Such a plate was devised by Wedeneewa (Zeits. f. Instrumentenk. 43, 17 (1923)), but its application is limited to cases where small phase differences are involved and, if arrangement A is used, the polarizer and analyzer must be crossed, thus precluding the use of the most favorable polarizer azimuth.

1921 (1)

A. De Forest Palmer, Phys. Rev. 17, 409 (1921); H. A. Boorse, Phys. Rev. 46, 187 (1934).
[Crossref]

1912 (1)

P. Zeeman and C. M. Hoogenboom, Physik. Zeits. 13, 914 (1912).

1904 (1)

D. B. Brace, Phil. Mag. 7, 320 (1904); Phys. Rev. 18, 70 (1904); ibid. 19, 218 (1904); G. Szivessy, Zeits. f. Instrumentenk. 57, 49 and 89 (1937).
[Crossref]

1902 (1)

Rayleigh, Phil Mag 4, 680 (1902); Sci. Pap. 5, 64 (1905).

1890 (1)

A polarizing prism combined with a λ/2 plate covering half the field of view. The vibration directions make an angle ϵ with each other; this angle can be varied. M. Chauvin, J. de Phys. 9, 21 (1890); Ann. de Toulouse 3(J) 28, (1889).

1851 (1)

A. Bravais, Comptes rendus 32, 115 (1851); G. Szivessy, Zeits. f. Physik 29, 372 (1924).
[Crossref]

1845 (1)

Soleil, Comptes rendus,  21, 426 (1845).

1840 (1)

H. de Sénarmont, Ann. Chim. Phys. 73, 337 (1840); F. Gabler and P. Sokob, Zeits. f. Instrumentenk. 58, 301 (1938); F. Gabler and P. Sokob, Zeits. f. Instrumentenk. 61, 298 (1941); F. Gabler and P. Sokob, Physik. Zeits. 42, 319 (1941).

Biernacki,

Bravais, Zehnder, Biernacki, Zahrzewski, and Koenigsberger, Handbuch der Physik 19, 943 (1928).

Brace, D. B.

D. B. Brace, Phil. Mag. 7, 320 (1904); Phys. Rev. 18, 70 (1904); ibid. 19, 218 (1904); G. Szivessy, Zeits. f. Instrumentenk. 57, 49 and 89 (1937).
[Crossref]

Bravais,

Bravais, Zehnder, Biernacki, Zahrzewski, and Koenigsberger, Handbuch der Physik 19, 943 (1928).

Bravais, A.

A. Bravais, Comptes rendus 32, 115 (1851); G. Szivessy, Zeits. f. Physik 29, 372 (1924).
[Crossref]

Chauvin, M.

A polarizing prism combined with a λ/2 plate covering half the field of view. The vibration directions make an angle ϵ with each other; this angle can be varied. M. Chauvin, J. de Phys. 9, 21 (1890); Ann. de Toulouse 3(J) 28, (1889).

De Forest Palmer, A.

A. De Forest Palmer, Phys. Rev. 17, 409 (1921); H. A. Boorse, Phys. Rev. 46, 187 (1934).
[Crossref]

de Sénarmont, H.

H. de Sénarmont, Ann. Chim. Phys. 73, 337 (1840); F. Gabler and P. Sokob, Zeits. f. Instrumentenk. 58, 301 (1938); F. Gabler and P. Sokob, Zeits. f. Instrumentenk. 61, 298 (1941); F. Gabler and P. Sokob, Physik. Zeits. 42, 319 (1941).

Dierkesmann, A.

G. Szivessy and A. Dierkesmann, Ann. d. Physik 11, 956 (1931).

G. Szivessy and A. Dierkesmann, Ann. d. Physik 11, 949 (1931).
[Crossref]

Gabler, F.

For a full account of the influence of reflections on a doubly refracting plate see F. Gabler and P. Sokob, Zeits. f. Physik 116, 47 (1940). The effect of absorption which is negligible is discussed by M. Berek, Ann. d. Physik 58, 165 (1919).
[Crossref]

F. Gabler and P. Sokob, Zeits. f. Physik 116, 47 (1940).
[Crossref]

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

For a full discussion see F. Gabler and P. Sokob, Zeits f. Instrumentenk. 58, 301 (1938) (see reference 27).

Herzog, W.

G. Szivessy and W. Herzog, Zeits. f. Instrumentenk. 57, 305 (1937).

G. Szivessy and W. Herzog, Zeits. f. Instrumentenk. 57, 53 (1937). For alternative forms see F. Gabler and P. Sokob, Zeits. f. Instrumentenk 58, 302 (1938). For a generalized intensity formula for a system of n plates see Hsien-Yü Hsü, M. Richartz, and Yüng-K’ang Liang, J. Opt. Soc. Am. 37, 99 (1947).
[Crossref]

Hlerzog, W.

Note that a value for η can be found without recourse to an independent measurement, fortan2η/2=4-tan22ϵ1(cot2α¯01+cot2α¯02)2sin22ϵ1(cot2α¯01-cot2α¯02)2;W. Hlerzog, Zeits. f. Physik 97, 225 (1935).
[Crossref]

Hoogenboom, C. M.

P. Zeeman and C. M. Hoogenboom, Physik. Zeits. 13, 914 (1912).

Koenigsberger,

Bravais, Zehnder, Biernacki, Zahrzewski, and Koenigsberger, Handbuch der Physik 19, 943 (1928).

Mascart, M. E.

M. E. Mascart, Traité d’Optique (Gauthier-Villars, Paris, 1891), Vol. 2, p. 61.

Rayleigh,

Rayleigh, Phil Mag 4, 680 (1902); Sci. Pap. 5, 64 (1905).

Richartz, M.

M. Richartz, Zeits. f. Instrumentenk. 60, 357 (1940); J. Opt. Soc. Am. 31, 292 (1941).

Schönrock, O.

A symmetrical prism with right end faces, consisting of two Glan-Thomson prisms cemented together so that the vibration directions make a small angle 2ϵ with each other. O. Schönrock, Handbuch der Physik 19, 750 (1928).

Sokob, P.

For a full account of the influence of reflections on a doubly refracting plate see F. Gabler and P. Sokob, Zeits. f. Physik 116, 47 (1940). The effect of absorption which is negligible is discussed by M. Berek, Ann. d. Physik 58, 165 (1919).
[Crossref]

F. Gabler and P. Sokob, Zeits. f. Physik 116, 47 (1940).
[Crossref]

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

For a full discussion see F. Gabler and P. Sokob, Zeits f. Instrumentenk. 58, 301 (1938) (see reference 27).

Soleil,

Soleil, Comptes rendus,  21, 426 (1845).

Szivessy, G.

For a detailed discussion see G. Szivessy, Zeits. f. Instrumentenk. 57, 92 (1937).

G. Szivessy and W. Herzog, Zeits. f. Instrumentenk. 57, 305 (1937).

G. Szivessy and W. Herzog, Zeits. f. Instrumentenk. 57, 53 (1937). For alternative forms see F. Gabler and P. Sokob, Zeits. f. Instrumentenk 58, 302 (1938). For a generalized intensity formula for a system of n plates see Hsien-Yü Hsü, M. Richartz, and Yüng-K’ang Liang, J. Opt. Soc. Am. 37, 99 (1947).
[Crossref]

G. Szivessy and A. Dierkesmann, Ann. d. Physik 11, 956 (1931).

G. Szivessy and A. Dierkesmann, Ann. d. Physik 11, 949 (1931).
[Crossref]

G. Szivessy, Zeits. f. Physik 54, 594 (1929).
[Crossref]

G. Szivessy, Handbuch der Physik 19, 926 (1928).

A composite plate consisting of two equally thick plates, usually of quartz cut at π/4 to the optical axis, lying on top of each other with their principal planes at right angles. See G. Szivessy, Handbuch der Physik 19, 942 (1928).

G. Szivessy, Handbuch der Physik 19, 941 (1928).

G. Szivessy, Zeits. f. Physik 29, 372 (1924).
[Crossref]

Wedeneewa,

To obtain a half-shadow plate of very small difference which can also be varied, a plate of phase difference η can be used in azimuth h(−π/2⋜h⋜+π/2) differing from ±π/4 so that the effective phase difference in this azimuth is η=±η sin2h, the positive value holding if 0⋜h⋜+π/2, the negative if −π/2⋜h⋜0. Such a plate was devised by Wedeneewa (Zeits. f. Instrumentenk. 43, 17 (1923)), but its application is limited to cases where small phase differences are involved and, if arrangement A is used, the polarizer and analyzer must be crossed, thus precluding the use of the most favorable polarizer azimuth.

Zahrzewski,

Bravais, Zehnder, Biernacki, Zahrzewski, and Koenigsberger, Handbuch der Physik 19, 943 (1928).

Zeeman, P.

P. Zeeman and C. M. Hoogenboom, Physik. Zeits. 13, 914 (1912).

Zehnder,

Bravais, Zehnder, Biernacki, Zahrzewski, and Koenigsberger, Handbuch der Physik 19, 943 (1928).

Ann. Chim. Phys. (1)

H. de Sénarmont, Ann. Chim. Phys. 73, 337 (1840); F. Gabler and P. Sokob, Zeits. f. Instrumentenk. 58, 301 (1938); F. Gabler and P. Sokob, Zeits. f. Instrumentenk. 61, 298 (1941); F. Gabler and P. Sokob, Physik. Zeits. 42, 319 (1941).

Ann. d. Physik (2)

G. Szivessy and A. Dierkesmann, Ann. d. Physik 11, 949 (1931).
[Crossref]

G. Szivessy and A. Dierkesmann, Ann. d. Physik 11, 956 (1931).

Comptes rendus (2)

A. Bravais, Comptes rendus 32, 115 (1851); G. Szivessy, Zeits. f. Physik 29, 372 (1924).
[Crossref]

Soleil, Comptes rendus,  21, 426 (1845).

Handbuch der Physik (5)

A composite plate consisting of two equally thick plates, usually of quartz cut at π/4 to the optical axis, lying on top of each other with their principal planes at right angles. See G. Szivessy, Handbuch der Physik 19, 942 (1928).

Bravais, Zehnder, Biernacki, Zahrzewski, and Koenigsberger, Handbuch der Physik 19, 943 (1928).

G. Szivessy, Handbuch der Physik 19, 941 (1928).

G. Szivessy, Handbuch der Physik 19, 926 (1928).

A symmetrical prism with right end faces, consisting of two Glan-Thomson prisms cemented together so that the vibration directions make a small angle 2ϵ with each other. O. Schönrock, Handbuch der Physik 19, 750 (1928).

J. de Phys. (1)

A polarizing prism combined with a λ/2 plate covering half the field of view. The vibration directions make an angle ϵ with each other; this angle can be varied. M. Chauvin, J. de Phys. 9, 21 (1890); Ann. de Toulouse 3(J) 28, (1889).

Phil Mag (1)

Rayleigh, Phil Mag 4, 680 (1902); Sci. Pap. 5, 64 (1905).

Phil. Mag. (1)

D. B. Brace, Phil. Mag. 7, 320 (1904); Phys. Rev. 18, 70 (1904); ibid. 19, 218 (1904); G. Szivessy, Zeits. f. Instrumentenk. 57, 49 and 89 (1937).
[Crossref]

Phys. Rev. (1)

A. De Forest Palmer, Phys. Rev. 17, 409 (1921); H. A. Boorse, Phys. Rev. 46, 187 (1934).
[Crossref]

Physik. Zeits. (1)

P. Zeeman and C. M. Hoogenboom, Physik. Zeits. 13, 914 (1912).

Zeits f. Instrumentenk. (1)

For a full discussion see F. Gabler and P. Sokob, Zeits f. Instrumentenk. 58, 301 (1938) (see reference 27).

Zeits. f. Instrumentenk. (6)

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

G. Szivessy and W. Herzog, Zeits. f. Instrumentenk. 57, 305 (1937).

For a detailed discussion see G. Szivessy, Zeits. f. Instrumentenk. 57, 92 (1937).

To obtain a half-shadow plate of very small difference which can also be varied, a plate of phase difference η can be used in azimuth h(−π/2⋜h⋜+π/2) differing from ±π/4 so that the effective phase difference in this azimuth is η=±η sin2h, the positive value holding if 0⋜h⋜+π/2, the negative if −π/2⋜h⋜0. Such a plate was devised by Wedeneewa (Zeits. f. Instrumentenk. 43, 17 (1923)), but its application is limited to cases where small phase differences are involved and, if arrangement A is used, the polarizer and analyzer must be crossed, thus precluding the use of the most favorable polarizer azimuth.

M. Richartz, Zeits. f. Instrumentenk. 60, 357 (1940); J. Opt. Soc. Am. 31, 292 (1941).

G. Szivessy and W. Herzog, Zeits. f. Instrumentenk. 57, 53 (1937). For alternative forms see F. Gabler and P. Sokob, Zeits. f. Instrumentenk 58, 302 (1938). For a generalized intensity formula for a system of n plates see Hsien-Yü Hsü, M. Richartz, and Yüng-K’ang Liang, J. Opt. Soc. Am. 37, 99 (1947).
[Crossref]

Zeits. f. Physik (5)

For a full account of the influence of reflections on a doubly refracting plate see F. Gabler and P. Sokob, Zeits. f. Physik 116, 47 (1940). The effect of absorption which is negligible is discussed by M. Berek, Ann. d. Physik 58, 165 (1919).
[Crossref]

G. Szivessy, Zeits. f. Physik 29, 372 (1924).
[Crossref]

G. Szivessy, Zeits. f. Physik 54, 594 (1929).
[Crossref]

Note that a value for η can be found without recourse to an independent measurement, fortan2η/2=4-tan22ϵ1(cot2α¯01+cot2α¯02)2sin22ϵ1(cot2α¯01-cot2α¯02)2;W. Hlerzog, Zeits. f. Physik 97, 225 (1935).
[Crossref]

F. Gabler and P. Sokob, Zeits. f. Physik 116, 47 (1940).
[Crossref]

Other (8)

Reference 32, Eq. 22.

Since s is dependent on the angle ϵ, which may be easily changed, this gives the Chauvin analyzer the decided advantage of variable sensitivity.
[Crossref]

If, in arrangement B, Eq. (82) is to be used for the calculation of δ, then it is immaterial whether OD and OH are parallel or perpendicular whatever the sign of sin2δ cosκ tanη/2.

M. E. Mascart, Traité d’Optique (Gauthier-Villars, Paris, 1891), Vol. 2, p. 61.

By using this compensator in conjunction with a Soleil, the fringe displacement of the former can be compensated whence the drum reading will give δ directly.

In the original the half-shade and compensator springs produced tensions of 0.285 g and 2.04 g per micrometer division, respectively. If PH and Pc are the tension expressed in micrometer divisionsη=2.00PH×10-6,         κ=6.112Pc×10-6;possible maxima being η=3×54×10−4·2π and 1.77×10−4·2π whence the maximum value of the phase difference which can be measured is κ0=1.77×10−4·2π.

The effect of the sequence of the plates has been discussed in the introduction. The possible arrangements for the Brace compensator are shown in Fig. 10, where it can be seen that the essential difference between arrangements A and B is that in one case K never lies between H and D, in the other it always does.

If ϕ0 is negative, 2ϕ=±2nπ−2ϕ0 or ±(2n+1)π+2ϕ0, (n=0,1,2⋯).

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Figures (17)

Fig. 1
Fig. 1

OX and OY are the main vibration directions in the plate; OP, is the vibration direction of the incident wave.

Fig. 2
Fig. 2

The parameters. P, polarizer; A, analyzer; OP and OA are the vibration directions of the polarizer and analyzer, respectively; OX1, OX2, and OX3 are the main vibration directions (fast) of the plates 1, 2, and 3; OY1, OY2, and OY3 are the corresponding directions for the slow rays. δ1, δ2, and δ3 are the phase differences introduced by the plates 1, 2, and 3. φ, γ1, γ2, and γ3 are the azimuths of the analyzer and the plates 1, 2, and 3, respectively, measured from OP. J3 is the intensity.

Fig. 3
Fig. 3

The Babinet compensator.

Fig. 4
Fig. 4

(a). The Bravais compensator. (b). The Soleil compensator.

Fig. 5
Fig. 5

Vibration directions for (a) Babinet and Soleil compensators, (b) Soleil with half-shadow plate. OP and OA are the vibration directions of the polarizer and analyzer, respectively. OD, OH, and OK are the vibration directions for the plates D, H, and K, respectively.

Fig. 6
Fig. 6

The Rayleigh compensator. OP and OA are the vibration directions of the polarizer and analyzer, respectively. G is the neutral layer in azimuth π/4.

Fig. 7
Fig. 7

Schematic arrangement of the De Forest Palmer compensator. H, half-shadow plate; K, compensator plate.

Fig. 8
Fig. 8

The optical system and azimuths of the directions of vibration for the De Forest Palmer compensator. P, polarizer; A, analyzer (azimuth ϕ); D, plate under test; H, half-shadow plate; K, compensator plate (azimuth k); OP, OA, OD, OH, and OK, their respective vibration directions. δ, η, and κ phase differences produced by D, H, and K, respectively.

Fig. 9
Fig. 9

The optical arrangements A and B and azimuths of the directions of vibration for the Brace compensator. P, polarizer; A, analyzer; D, plate under test; H, half-shadow plate; K, compensator plate. OP, OA, OD, OH, and OK, their respective vibration directions. δ, η, and κ are the phase differences produced by D, H, and K, respectively. In arrangement A the azimuths of D, H, K, and P are d′, h′, k′, and β, respectively, measured from OA. In arrangement B the corresponding azimuths are d, h, k, and α, respectively, measured from OP. In the text d=h=d′=h′=±π/4.

Fig. 10
Fig. 10

Possible arrangements for the Brace compensator. D, plate under test; H, half-shadow plate; K, compensator plate. Arrangements A(1), A(2), and B(2) are independent of the polarizer azimuth at the half-shadow position; arrangements A(3), A(4), and B(1) are independent of the analyzer azimuth.

Fig. 11
Fig. 11

Solutions of the equation sin2ϕ=X where X is a positive number.

Fig. 12
Fig. 12

The optical arrangements A and B and azimuths of the directions of vibration for the Szivessy and modified Szivessy compensators. P, polarizer; A, analyzer; D, plate under test; H, half-shadow and compensator plate; OP, OA, OD, OH their respective vibration directions. δ and η are the phase differences produced by D and H, respectively. In (a) (Szivessy) the azimuths are measured from OP. In (b) (modified Szivessy) the azimuths β, α, and ϵ of P, A, and H respectively, are measured from OD.

Fig. 13
Fig. 13

The optical arrangements A and B and azimuths of the directions of vibration for the modified Brace compensator. P, polarizer; A, analyzer; D, plate under test; H, half-shadow plate; K, compensator plate; OP, OA, OD, OH, OK, their respective vibration directions. δ, η, κ, are the phase differences produced by D, H, and K, respectively. The azimuths δ, β, σ, and ρ of OA, OP, OD, and OH, respectively, are measured from OK.

Fig. 14
Fig. 14

Initial orientation for the modified Brace compensator. (a). The extinction azimuths for arrangements A and B. (b). Initial azimuths before seeking half-shadow condition.

Fig. 15
Fig. 15

The optical system and azimuths of the directions of vibration for the Sénarmont compensator with half-shadow. P, polarizer; A, analyzer; D, plate under test; H, half-shadow plate; K, compensator plate; OP, OA, OD, OH, and OK, their respective vibration directions. δ, η, and κ are the phase differences produced by D, H, and K, respectively. σ is the azimuth of the dividing line, ϵ the angle between the vibration directions.

Fig. 16
Fig. 16

The optical system and azimuths of the directions of vibration for the Richartz compensator; P, polarizer; A, analyzer; D, plate under test; H, half-shadow plate; K, compensator plate; OP, OA, OD, OH, and OK their respective vibration directions. δ, η, and κ are the phase differences produced by D, H, and K, respectively.

Fig. 17
Fig. 17

Diagrammatic summary. P, polarizer; A, analyzer; D, plate under test; H, half-shadow plate; K, compensator plate. δ, η, and κ are the phase differences.

Equations (157)

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D 1 x = a cos ( ν t - α ) , D 1 y = 0 , D 1 z = 0 , D 2 x = 0 , D 2 y = b cos ( ν t - β ) , D 2 z = 0.
D X = D 1 x + D 2 x = a cos ( ν t - α ) , D Y = D 1 y + D 2 y = b cos ( ν t = β ) , D Z = D 1 z + D 2 z = 0 ; }
D X = a cos γ cos ( ν t - α ) , D Y = a sin γ cos ( ν t - β ) ,
δ = β - α = 2 π ( d / λ ) ( μ y - μ x ) ,
J 3 = cos 2 ϕ + 4 sin 2 γ 1 sin 2 ( φ - γ 3 ) cos 2 ( γ 2 - γ 1 ) cos 2 ( γ 3 - γ 2 ) sin 2 δ 1 / 2 sin 2 δ 2 / 2 sin 2 δ 3 / 2 + sin 2 γ 1 sin 2 ( φ - γ 1 ) sin 2 δ 1 / 2 + sin 2 γ 2 sin 2 ( φ - γ 2 ) sin 2 δ 2 / 2 + sin 2 γ 3 sin 2 ( φ - γ 3 ) sin 2 δ 3 / 2 - sin 2 γ 1 sin 2 ( φ - γ 3 ) [ cos 2 ( γ 2 - γ 1 ) sin 2 δ 1 / 2 sin δ 2 sin δ 3 + sin 2 δ 2 / 2 sin δ 3 sin δ 1 + cos 2 ( γ 3 - γ 2 ) sin 2 δ 3 / 2 sin δ 1 sin δ 2 ] + 2 [ sin 2 γ 1 sin 2 ( φ - γ 2 ) sin δ 1 / 2 sin δ 2 / 2 { cos δ 1 / 2 cos δ 2 / 2 - cos 2 ( γ 2 - γ 1 ) sin δ 1 / 2 sin δ 2 / 2 } + sin 2 γ 2 sin 2 ( φ - γ 3 ) sin δ 2 / 2 sin δ 3 / 2 { cos δ 2 / 2 cos δ 3 / 2 - cos 2 ( γ 3 - γ 2 ) sin δ 2 / 2 sin δ 3 / 2 } + sin 2 γ 1 sin 2 ( φ - γ 3 ) sin δ 3 / 2 sin δ 1 / 2 { cos δ 3 / 2 cos δ 1 / 2 - cos 2 ( γ 1 - γ 3 ) sin δ 3 / 2 sin δ 1 / 2 } ] .
J 2 - cos 2 φ + sin 2 γ 1 sin 2 ( φ - γ 1 ) sin 2 δ 1 / 2 + sin 2 γ 2 sin 2 ( φ - γ 2 ) sin 2 δ 2 / 2 + 2 sin 2 γ 1 sin 2 ( φ - γ 2 ) sin δ 1 / 2 sin δ 2 / 2 { cos δ 1 / 2 cos δ 2 / 2 - cos 2 ( γ 2 - γ 1 ) sin δ 1 / 2 sin δ 2 / 2 } ,
J 2 = cos 2 φ + sin 2 γ 1 cos 2 ( φ - γ 2 ) sin 2 ( γ 2 - γ 1 ) sin 2 δ 1 / 2 + cos 2 γ 1 sin 2 ( φ - γ 2 ) sin 2 ( γ 2 - γ 1 ) sin 2 δ 2 / 2 + sin 2 γ 1 sin 2 ( φ - γ 2 ) cos 2 ( γ 2 - γ 1 ) sin 2 [ ( δ 1 + δ 2 ) / 2 ] - sin 2 γ 1 sin 2 ( φ - γ 2 ) sin 2 ( γ 2 - γ 1 ) sin 2 [ ( δ 1 - δ 2 ) / 2 ] .
J 1 = cos 2 φ + sin 2 γ 1 sin 2 ( ϕ - γ 1 ) sin 2 δ 1 / 2.
J 3 x = 4 sin 2 γ 1 sin 2 γ 3 cos 2 ( γ 2 - γ 1 ) cos 2 ( γ 3 - γ 2 ) sin 2 δ 1 / 2 sin 2 δ 2 / 2 sin 2 δ 3 / 2 + sin 2 2 γ 1 sin 2 δ 1 / 2 + sin 2 2 γ 2 sin 2 δ 2 / 2 + sin 2 2 γ 3 sin 2 δ 3 / 2 - sin 2 γ 1 sin 2 γ 3 [ cos 2 ( γ 2 - γ 1 ) sin 2 δ 1 / 2 sin δ 2 sin δ 3 + sin 2 δ 2 / 2 sin δ 3 sin δ 1 + cos 2 ( γ 3 - γ 2 ) sin 2 δ 3 / 2 sin δ 1 sin δ 2 ] + 2 [ sin 2 γ 1 sin 2 γ 2 sin δ 1 / 2 sin δ 2 / 2 { cos δ 1 / 2 cos δ 2 / 2 - cos 2 ( γ 2 - γ 1 ) sin δ 1 / 2 sin δ 2 / 2 } + sin 2 γ 2 sin 2 γ 3 sin δ 2 / 2 sin δ 3 / 2 { cos δ 2 / 2 cos δ 3 / 2 - cos 2 ( γ 3 - γ 2 ) sin δ 2 / 2 sin δ 3 / 2 } + sin 2 γ 1 sin 2 γ 3 sin δ 3 / 2 sin δ 1 / 2 { cos δ 3 / 2 cos δ 1 / 2 - cos 2 ( γ 1 - γ 3 ) sin δ 3 / 2 sin δ 1 / 2 } ] .
J 2 x = sin 2 2 γ 1 sin 2 δ 1 / 2 + sin 2 2 γ 2 sin 2 δ 2 / 2 + 2 sin 2 γ 1 sin 2 γ 2 sin δ 1 / 2 sin δ 2 / 2 { cos δ 1 / 2 cos δ 2 / 2 - cos 2 ( γ 2 - γ 1 ) sin δ 1 / 2 sin δ 2 / 2 } ,
J 2 x = - sin 2 γ 1 cos 2 γ 2 sin 2 ( γ 2 - γ 1 ) sin 2 δ 1 / 2 + cos 2 γ 1 sin 2 γ 2 sin 2 ( γ 2 - γ 1 ) sin 2 δ 2 / 2 + sin 2 γ 1 sin 2 γ 2 cos 2 ( γ 2 - γ 1 ) sin 2 [ ( δ 1 + δ 2 ) / 2 ] - sin 2 γ 1 sin 2 γ 2 sin 2 ( γ 2 - γ 1 ) sin 2 [ ( δ 1 - δ 2 ) / 2 ] ,
J 1 x = sin 2 2 γ 1 sin 2 δ 1 / 2.
sin 2 γ 1 sin δ 1 / 2 = sin 2 γ 2 sin δ 2 / 2 = sin 2 γ 3 sin δ 3 / 2 = 0.
J 3 = cos 2 φ + sin 2 γ 1 sin 2 ( φ - γ 1 ) sin 2 [ ( δ 1 + δ 2 ) / 2 ] + sin 2 γ 3 sin 2 ( φ - γ 3 ) sin 2 δ 3 / 2 + 2 sin 2 γ 1 sin 2 ( φ - γ 3 ) sin [ ( δ 1 + δ 2 ) / 2 ] sin δ 3 / 2 × { cos [ ( δ 1 + δ 2 ) / 2 ] cos δ 3 / 2 - cos 2 ( γ 3 - γ 1 ) sin [ ( δ 1 + δ 2 ) / 2 ] sin δ 3 / 2 } ,
S = ( J - J ) / ( J + J ) .
S = ( / p ) ( J - J ) p = p 0 2 J 0 . Δ p = 1 2 s Δ p , where s = ( / p ) ( J - J ) p = p 0 J 0 .
Δ 1 = ( 2 π / λ ) d 1 ( μ ϵ - μ 0 ) , Δ 2 = ( 2 π / λ ) d 2 ( μ 0 - μ ϵ ) ,
κ = Δ 1 + Δ 2 = ( 2 π / λ ) ( μ ϵ - μ 0 ) ( d 1 - d 2 ) .
d 1 - d 2 = ± n λ / ( μ ϵ - μ 0 ) ,
d 1 - d 2 = ± ( 2 n + 1 ) μ ϵ - μ 0 λ
d 1 - d 2 = ( 2 t - l ) tan α = ± n λ / ( μ ϵ - μ 0 ) .
2 d t · tan α = ± λ ( μ ϵ - μ 0 ) d n , i . e . ,             d t = ± ( λ / 2 tan α ( μ ϵ - μ 0 ) )             for             d n = 1 ,
m = ± 2 π / ( p 1 - p 2 ) .
κ = 2 π x tan α ( μ ϵ - μ 0 ) / λ .
κ = ( 2 π / λ ) ( μ ϵ - μ 0 ) ( d 1 - d 2 ) ,
d 1 - d 2 = ± n λ / ( μ ϵ - μ 0 ) ,             ( n = 0 , 1 , 2 , ) ,
d 1 - d 2 = ± ( 2 n + 1 ) μ ϵ - μ 0 · λ             ( n = 0 , 1 , 2 , ) .
J 1 x = sin 2 κ / 2 ,
J 2 x = sin 2 κ / 2 + sin 2 2 h sin 2 η / 2 ± 2 sin 2 h sin κ / 2 sin η / 2 × { cos κ / 2 cos η / 2 sin κ / 2 sin η / 2 sin 2 h } .
sin 2 h 0 = 0 ,
tan κ 0 = sin 2 h tan η / 2.
tan η ¯ / 2 = tan κ 0 = sin 2 h tan η / 2 ,
κ 0 = ± 2 n π η ¯ / 2 ,             ( n = 0 , 1 , 2 , ) .
            δ = - p Y .
δ = - p ( y 2 - y 1 ) ,
J 2 = cos 2 φ + sin 2 k sin 2 ( φ - k ) sin 2 [ ( η - κ ) / 2 ] ,
J 1 = cos 2 φ + sin 2 k sin 2 ( φ - k ) sin 2 κ / 2.
η = 2 κ 0 .
η = 2 ( κ 1 ± δ ) .
± δ = κ 0 - κ 1 .
s = sin 2 k sin 2 ( φ - k ) sin κ cos 2 φ + sin 2 k sin 2 ( φ - k ) sin 2 · κ / 2 .
s / φ = 0 ,             s / k = 0 ,             and             s / κ = 0 ,
φ = π / 2 ,             k = ± π / 4 ,             and             κ = 0 ,
s max ( excluding κ = 0 ) = cot κ / 2.
± δ = p 0 - p 1 .
J 2 x - J 1 x = sin η / 2 { sin 2 k sin κ cos η / 2 ± ( 1 - 2 sin 2 2 k sin 2 κ / 2 ) sin η / 2 } ,
sin 2 k 0 sin κ cos η / 2 ± ( 1 - 2 sin 2 2 k 0 sin 2 κ / 2 ) sin η / 2 = 0.
φ = - β , γ 1 = - ( β - d ) , γ 2 = - ( β - h ) , γ 3 = - ( β - k ) ; δ 1 = δ ,             δ 2 = η ,             δ 3 = κ ,
φ = α ,             γ 1 = d ,             γ 2 = k ,             γ 3 = h ;             δ 1 = δ ,             δ 2 = κ ,             δ 3 = η .
J 3 - J 2 = cos 2 β sin η / 2 × { ± ( 1 - 2 sin 2 2 k sin 2 κ / 2 ) sin ( δ + η / 2 ) + sin 2 k sin κ cos ( δ + η / 2 ) } .
( a )             cos 2 β = 0 ,             ( b )             sin η / 2 = 0 ,
( c )             ± ( 1 - 2 sin 2 2 k ¯ sin 2 κ / 2 ) sin ( δ + η / 2 ) + sin 2 k ¯ sin κ cos ( δ + η / 2 ) = 0.
J 3 - J 2 = cos 2 α sin η / 2 [ ± sin δ ( cos κ cos η / 2 sin 2 k sin κ sin η / 2 ) + cos δ { sin 2 k sin κ cos η / 2 ± ( 1 - 2 sin 2 2 k sin 2 κ / 2 ) sin η / 2 } ] ,
( a )             cos 2 α = 0 ,             ( b )             sin η / 2 = 0 ,
( c )             ± sin δ ( cos κ cos η / 2 sin 2 k ¯ sin κ sin η / 2 ) + cos δ { sin 2 k ¯ sin κ cos η / 2 ± ( 1 - 2 sin 2 2 k ¯ sin 2 κ / 2 ) sin η / 2 } = 0.
± ( 1 - 2 sin 2 2 k ¯ 0 sin 2 κ / 2 ) sin η / 2 + sin 2 k ¯ 0 sin κ cos η / 2 = 0 ,
tan ( δ + η / 2 ) tan κ .
tan δ = sin κ ( sin 2 k ¯ - sin 2 k ¯ 0 ) ( 1 + 2 sin 2 k ¯ sin 2 k ¯ 0 sin 2 κ / 2 ( 1 - 2 sin 2 2 k ¯ sin 2 κ / 2 ) ( 1 - 2 sin 2 2 k ¯ 0 sin 2 κ / 2 ) · ) .             0 δ 2 π .
± ( 1 - 2 sin 2 2 k ¯ 0 sin 2 κ / 2 ) tan η / 2 + sin 2 k ¯ 0 sin κ = 0 ,
tan δ = sin κ ( sin 2 k ¯ - sin 2 k ¯ 0 ) ( 1 + 2 sin 2 k ¯ sin 2 k ¯ 0 sin 2 κ / 2 cos κ ( 1 - 2 sin 2 2 k ¯ 0 sin 2 κ / 2 ) + sin 2 k ¯ sin 2 k ¯ 0 sin 2 κ ) .             0 δ 2 π .
δ = sin κ ( sin 2 k ¯ - sin 2 k ¯ 0 )
δ = tan κ ( sin 2 k ¯ - sin 2 k ¯ 0 ) ,
s A = p cos 2 β cos 2 β - F cos 2 β + G sin 2 β ,
s B = p cos 2 α cos 2 α - F cos 2 α + G sin 2 α ,
sin 2 β ¯ = - sin 4 k ¯ sin 2 κ / 2 ;
sin 2 α ¯ = - sin 4 k ¯ sin 2 κ / 2 cos δ cos 2 k ¯ sin κ sin δ .
sin 2 β ¯ 0 = - sin 4 k ¯ 0 sin 2 κ / 2 ,
sin 2 α ¯ 0 = - sin 4 k ¯ 0 sin 2 κ / 2.
2 φ = ± 2 n π + 2 φ 0             or             ± ( 2 n + 1 ) π - 2 φ 0 , ( n = 0 , 1 , 2 , ) ,
φ 01 = φ 0 ,             φ 02 = π / 2 - φ 0 ,             φ 03 = π + φ 0 ,             φ 04 = 3 π / 2 - φ 0 .
k 1 = k ¯ ,             k 2 = π / 2 - k ¯ ,             k 3 = π + k ¯ ,             k 4 = 3 π / 2 - k ,
2 k ¯ = ( θ 1 + θ 2 + θ 3 ) / 3 ,
θ 1 = ( π / 2 ) - ( p 2 - p 1 ) , θ 2 = ( p 3 - p 2 ) - ( π / 2 ) , θ 3 = ( π / 2 ) - ( p 4 - p 3 ) ,
p 3 - p 1 = p 4 - p 2 = π ,
J 2 x = sin 2 δ / 2 + sin 2 2 h sin 2 · η / 2 ± 2 sin 2 h sin η / 2 sin δ / 2 × { cos δ / 2 cos η / 2 sin δ / 2 sin η / 2 sin 2 h } ,
J 1 x = sin 2 δ / 2.
sin 2 h 0 = 0 ,
tan δ = ± sin 2 h 0 tan η / 2 ,             ( 0 δ 2 π ) ,
tan η / 2 tan δ .
s = 2 | sin η tan δ / 2 | × | ( 1 - tan 2 δ tan 2 η / 2 ) 1 2 | ,
h 01 = h 0 ,             h 02 = π / 2 - h 0 ,             h 03 = π + h 0 ,             h 04 = 3 π / 2 - h 0 ,
2 h 0 = ( θ 1 + θ 2 + θ 3 ) / 3 ,
tan ( δ + η / 2 ) tan κ ,
tan δ tan η / 2
φ = ( α - β ) ,             γ 1 = ( ϵ - β ) ,             γ 2 = - β ;             δ 1 = η ,             δ 2 = δ ,
J 2 - J 1 = - sin 2 ( β - ϵ ) × sin η / 2 { sin 2 α ( cos 2 ϵ sin η / 2 cos δ + cos η / 2 sin δ ) - cos 2 α sin 2 ϵ sin η / 2 } ,
( a )             sin 2 ( β - ϵ ) = 0 , satisfied by             β = ϵ             or             ϵ π / 2 ,
( b )             sin η / 2 = 0 ,             one solution being η = 2 π ,
cos 2 ϵ cos δ + sin δ cot η / 2 - sin 2 ϵ cot 2 α ¯ = 0.
α ¯ 1 = α ¯ 0 ( α ¯ 1 > 0 )             and             α ¯ 2 = α ¯ 0 - π / 2 ( α ¯ 2 < 0 ) .
α ¯ 3 = - α ¯ 2 = π / 2 - α ¯ 0 ( α ¯ 3 > 0 )
α ¯ 4 = - α ¯ 1 = - α ¯ 0 ( α ¯ 4 < 0 ) .
2 α ¯ 0 = p 1 - p 4 = π - ( p 3 - p 2 ) .
cos 2 ϵ 1 cos δ + sin δ cot η / 2 - sin 2 ϵ 1 cot 2 α ¯ 01 = 0 ,
- cos 2 ϵ 1 cos δ + sin δ cot η / 2 + sin 2 ϵ 1 cot 2 α ¯ 02 = 0 ,
2 sin δ = sin 2 ϵ 1 ( cot 2 α ¯ 01 - cot 2 α ¯ 02 ) tan η / 2 ,
2 cos δ = tan 2 ϵ 1 ( cot 2 α ¯ 01 + cot 2 α ¯ 02 ) .
sin δ = ± cot 2 α ¯ tan η / 2             ( 0 δ 2 π ) ,
s = 4 cos 2 β sin η / 2 ( sin 2 α ¯ sin η / 2 ± cos 2 α ¯ cos η / 2 sin δ ) 1 + cos 2 α ¯ cos 2 β + sin 2 α ¯ sin 2 β cos δ ,
sin 2 β m = - sin 2 α ¯ cos δ .
s max = 4 tan η / 2 sin δ ( sin 2 η / 2 - sin 2 δ cos 2 η / 2 ) ,
sin δ = cot [ + ( p 1 - p 4 ) ] tan η / 2 = - cot [ + ( p 3 - p 2 ) ] tan η / 2             ( 0 δ 2 π ) ,
+ ( p 1 - p 0 ) = + ( p 2 - p 0 ) + π / 2 , and + ( p 3 - p 0 ) = + ( p 4 - p 0 ) + π / 2.
J 3 = cos 2 ( α - β ) - cos 2 α cos 2 β sin 2 [ ( δ ± η ) / 2 ] - sin 2 α sin 2 β sin 2 κ / 2 ± 1 2 sin 2 α cos 2 β sin ( δ ± η ) sin κ ,
J 2 = cos 2 ( α - β ) - cos 2 α cos 2 β sin 2 δ / 2 - sin 2 α sin 2 β sin 2 κ / 2 ± 1 2 sin 2 α cos 2 β sin δ sin κ .
J 3 - J 2 = cos 2 β sin η / 2 { cos 2 α sin ( δ ± η / 2 ) sin 2 α sin κ cos ( δ ± η / 2 ) } ,
( a )             cos 2 β = 0 , satisfied by β = ± π / 4 ,
( b )             sin η / 2 = 0 , one solution then being η = 2 π ,
cot 2 α ¯ = ± sin κ cot ( δ ± η / 2 ) .
ϕ = α - β ,             γ 1 = γ 3 = ± π / 4 - β
J 3 - J 2 = cos 2 β sin η / 2 × { cos 2 α ( sin δ cos κ cos η / 2 ± cos δ sin η / 2 ) sin 2 α sin κ cos η / 2 } ,
cot 2 α ¯ = ± sin κ / ( sin δ cos κ ± cos δ tan η / 2 ) .
tan δ = ± sin κ tan 2 α ¯ - tan η / 2 1 ± sin κ tan 2 α ¯ tan η / 2             ( 0 δ 2 π ) ,
tan δ = - cos κ sin η / 2 cos 2 2 α ¯ ± sin κ sin 2 α ¯ ( cos 2 2 α ¯ - sin 2 κ cos 2 η / 2 ) 1 2 cos η / 2 ( cos 2 2 α ¯ - sin 2 κ ) ,             ( 0 δ 2 π ) ,
cot 2 α ¯ 0 = ± sin κ cot ± η / 2 ,
tan δ = ± sin κ tan 2 α ¯ - tan 2 α ¯ 0 1 + tan 2 α ¯ 0 tan 2 α ¯ sin 2 κ             ( 0 δ 2 π ) ,
cos κ sin δ ± tan η / 2 cos δ sin κ tan 2 α ¯ 01 = 0 cos κ sin δ tan η / 2 cos δ sin κ tan 2 α ¯ 02 = 0 } γ 1 = γ 3 , γ 1 = γ 3 ± π / 2 ,
2 cos δ = ± sin κ cot η / 2 ( tan 2 α ¯ 01 - tan 2 α ¯ 02 ) ,
2 sin δ = ± tan κ ( tan 2 α ¯ 01 + tan 2 α ¯ 02 ) .
sec 2 ( δ ± η / 2 ) d δ = ± 2 sin κ sec 2 2 α ¯ · d α ¯ ,
( cos κ cos δ tan η / 2 sin δ ) d δ = ± 2 sin κ sec 2 2 α ¯ · d α ¯ ,
s A = M A N A ,
s B = M B N B ,
sin 2 β m = - sin 2 α ¯ cos κ ,
s A = 4 ( tan η / 2 / sin κ ) × { sin 2 ( δ ± η / 2 ) + sin 2 κ cos 2 ( δ ± η / 2 ) } ,
s B = 4 sin η / 2 sin κ · ( sin δ cos κ cos η / 2 ± cos δ sin η / 2 ) 2 + sin 2 κ cos 2 η / 2 ( cos δ ± sin δ sin κ ) cos η / 2 + [ cos 2 η / 2 + ( sin δ cos κ cos η / 2 ± cos δ sin η / 2 ) 2 ] 1 2 .
ϕ 0 = ± π / 2
γ 1 = + π / 4 ,             γ 2 = 0     or     π / 2 ;             δ 1 = δ ,             δ 2 = π / 2 ,
J 2 = cos 2 φ - cos 2 φ sin 2 δ / 2 ± sin 2 φ sin δ / 2 cos δ / 2 ,
J 2 = cos 2 ( φ δ / 2 ) ,
φ 1 = ± π / 2 ± δ / 2.
φ 1 - φ 0 = ± δ / 2 ,
J 2 = cos 2 ( σ + ϵ ) - cos 2 ( σ + ϵ ) sin 2 δ / 2 ± sin 2 ( σ + ϵ ) sin δ / 2 cos δ / 2 ,
J 2 = cos 2 ( σ - ϵ ) - cos 2 ( σ - ϵ ) sin 2 δ / 2 ± sin 2 ( σ - ϵ ) sin δ / 2 cos δ / 2.
tan 2 σ ¯ = ± tan δ ,
σ ¯ = ± π / 2 ± δ / 2 ,
            σ ¯ - σ 0 = ± δ / 2.
J 3 = cos 2 σ - cos 2 σ sin 2 δ / 2 - sin 2 σ sin δ / 2 cos δ / 2 - sin 2 ( σ + ϵ ) sin 2 ϵ cos δ - cos 2 ( σ + ϵ ) sin 2 ϵ sin δ .
( a )             sin 2 ϵ = 0 , i . e . , ϵ = 0 or π / 2. ( b )             tan 2 ( σ ¯ + ϵ ) = ± tan δ ,
σ ¯ + ϵ = ± π / 2 ± δ / 2 ,
σ ¯ - σ 0 = ± δ / 2.
γ 1 = π / 4 + Δ γ 1 ,             γ 2 = π / 2 + Δ γ 2 ; δ 1 = δ ,             δ 2 = π / 2 + Δ δ 2 ,             δ 3 = π + Δ δ 3 ,
tan 2 σ = ± cos Δ δ 2 tan δ .
tan 2 σ 1 = cos Δ χ 2 sin δ x + y cos δ ,
tan 2 σ 2 = cos Δ χ 2 sin δ x - y cos δ ,
tan ( σ 1 - σ 2 ) cos Δ χ 2 tan δ
tan ( σ 1 - σ 2 ) = tan δ
s = 2 sin 2 ϵ 1 ± cos 2 ϵ             ( 0 < ϵ < π / 4 ) ,
J 3 = cos 2 ( α ± δ ± η 2 ) ,
J 2 = cos 2 ( α ± δ / 2 ) .
2 α ¯ = π ( δ ± η / 2 ) .
2 α ¯ 0 = π .
2 α ¯ 1 = π 2 δ ,
α ¯ 1 - α ¯ 0 = δ .
p 1 - p 0 = δ .
η = 2.00 P H × 10 - 6 ,             κ = 6.112 P c × 10 - 6 ;
tan 2 η / 2 = 4 - tan 2 2 ϵ 1 ( cot 2 α ¯ 01 + cot 2 α ¯ 02 ) 2 sin 2 2 ϵ 1 ( cot 2 α ¯ 01 - cot 2 α ¯ 02 ) 2 ;