Abstract

Reusch and Sohncke have examined the properties of a system containing a large number n of identical retardation plates, each of which is rotated with respect to the one preceding it through the angle ω. The product of n and ω must be equal to μπ, where μ is an integer. Under certain conditions this system is optically equivalent to a simple rotator. This system, which would be very difficult to examine by ordinary methods, is given a treatment which is completely rigorous and which is more general than any given heretofore.

© 1941 Optical Society of America

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References

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  1. E. Reusch, Berliner Monatsber. (Gesammtsitzung, July8, 1869) 530 (1869); Pogg. Ann. 138, 628 (1869).
  2. L. Sohncke, Math. Ann. 9, 504 (1876). See also Pogg. Ann. Ergänzungsband 8, 16 (1878), and Zeits. f. Krist. 13, 214 (1888).
    [Crossref]
  3. F. Pockels, Lehrbuch der Kristalloptik (Teubner, Leipzig, 1906), pp. 289–290.
  4. L. Sohncke, Entwickelung einer Theorie der Krystallstructur (Teubner, Leipzig, 1879). See particularly pp. 238–246.
  5. T. M. Lowry, Optical Rotatory Power (Longmans Green, New York, 1935), p. 194, Fig. 79(a).
  6. M. F. Billet, Traité D’optique Physique (Paris, 1859), Vol. 2, Chapter 20, Art. 1, pp. 328–351.

1876 (1)

L. Sohncke, Math. Ann. 9, 504 (1876). See also Pogg. Ann. Ergänzungsband 8, 16 (1878), and Zeits. f. Krist. 13, 214 (1888).
[Crossref]

Billet, M. F.

M. F. Billet, Traité D’optique Physique (Paris, 1859), Vol. 2, Chapter 20, Art. 1, pp. 328–351.

Lowry, T. M.

T. M. Lowry, Optical Rotatory Power (Longmans Green, New York, 1935), p. 194, Fig. 79(a).

Pockels, F.

F. Pockels, Lehrbuch der Kristalloptik (Teubner, Leipzig, 1906), pp. 289–290.

Reusch, E.

E. Reusch, Berliner Monatsber. (Gesammtsitzung, July8, 1869) 530 (1869); Pogg. Ann. 138, 628 (1869).

Sohncke, L.

L. Sohncke, Math. Ann. 9, 504 (1876). See also Pogg. Ann. Ergänzungsband 8, 16 (1878), and Zeits. f. Krist. 13, 214 (1888).
[Crossref]

L. Sohncke, Entwickelung einer Theorie der Krystallstructur (Teubner, Leipzig, 1879). See particularly pp. 238–246.

Math. Ann. (1)

L. Sohncke, Math. Ann. 9, 504 (1876). See also Pogg. Ann. Ergänzungsband 8, 16 (1878), and Zeits. f. Krist. 13, 214 (1888).
[Crossref]

Other (5)

F. Pockels, Lehrbuch der Kristalloptik (Teubner, Leipzig, 1906), pp. 289–290.

L. Sohncke, Entwickelung einer Theorie der Krystallstructur (Teubner, Leipzig, 1879). See particularly pp. 238–246.

T. M. Lowry, Optical Rotatory Power (Longmans Green, New York, 1935), p. 194, Fig. 79(a).

M. F. Billet, Traité D’optique Physique (Paris, 1859), Vol. 2, Chapter 20, Art. 1, pp. 328–351.

E. Reusch, Berliner Monatsber. (Gesammtsitzung, July8, 1869) 530 (1869); Pogg. Ann. 138, 628 (1869).

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Equations (31)

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ω ˜ = 1 8 n Δ 2 cot ( π / n ) .
ω k = - ( k - 1 ) π μ / n .
M ( n ) = S ( π μ ) ( S ( ω ) G ) n = ( - 1 ) μ ( S ( ω ) G ) n ,
ω π μ / n .
D = TAT - 1 , A = T - 1 DT ,
T - 1 T = T T - 1 = 1.
A n = T - 1 DTT - 1 DT T - 1 DT = T - 1 D n T .
a 1 = d 1 - d 2 d 1 - d 2 a 1 + d 2 d 1 - d 1 d 2 d 1 - d 2 , a 2 = d 1 - d 2 d 1 - d 2 a 2 + d 2 d 1 - d 1 d 2 d 1 - d 2 , a 3 = d 1 - d 2 d 1 - d 2 a 3 . a 4 = d 1 - d 2 d 1 - d 2 a 4 .
A = S ( ω ) G = ( e i γ cos ω - e - i γ sin ω e i γ sin ω e - i γ cos ω ) .
( e i γ cos ω - d ) ( e - i γ cos ω - d ) + sin 2 ω = 0
d 1 = e i χ ,             d 2 = e - i χ ,
cos χ = cos ω cos γ .
d 1 = e i n χ ,             d 2 = e - i n χ ,
M ( n ) = ( - 1 ) μ A n = ( - 1 ) μ A ,
a 1 = sin n χ sin χ cos ω e i γ - sin ( n - 1 ) χ sin χ , a 2 = α 1 * , a 3 = sin n χ sin χ sin ω e i γ , a 4 = - a 3 * .
m 1 = cos n ( χ - ω ) + i [ sin n ( χ - ω ) ] cos ω sin γ sin χ , m 2 = m 1 * , m 3 = [ sin n ( χ - ω ) ] sin ω sin χ e i γ , m 4 = - m 3 * .
M ( n ) = S ( ω ˜ ) [ S ( ω ˜ ) G ¯ S ( - ω ˜ ) ] .
tan ω ˜ = tan ( A + B ) = m 3 - m 4 m 1 + m 2 = [ tan n ( χ - ω ) ] tan ω tan χ ,
tan ( ω ˜ + 2 ω ˜ ) = tan ( A - B ) = m 3 + m 4 m 1 - m 2 = tan ω ,
tan γ ¯ = sin ( A + B ) sin ( A - B ) tan γ = sin ω ˜ tan γ sin ω .
| tan γ sin ω | 1.
γ ω = π μ / n .
sin ( χ - ω ) = sin ω cos ω cos γ × [ ( 1 + tan 2 γ sin 2 ω ) 1 2 - 1 ] ,
tan 2 χ tan 2 ω = 1 + tan 2 γ sin 2 ω .
ω ˜ n ( χ - ω ) n sin ( χ - ω ) 1 2 n cot ω sin γ tan γ 1 2 n γ 2 cot ω
ω ˜ = O ( n 2 γ 2 / 2 π μ ) .
ω ˜ 1 2 π μ .
S ( a b γ ) G ( a π / 4 ) [ S ( b π / 4 ) G ( γ ) × S ( - b π / 4 ) ] G ( - a π / 4 ) ,
a = ± 1 ,             b = ± 1.
G ( a b ω ˜ ) S ( a π / 4 ) [ G ( b π / 4 ) S ( ω ˜ ) × G ( - b π / 4 ) ] S ( - a π / 4 ) ,
a = ± 1 ,             b = ± 1.