Abstract

Part IV is divided into two sections. The first is devoted to some additions to the general theory developed in Part I, and the second section to the derivation of the matrices representing two optical elements which were not treated in Parts II and III: (1) plates possessing circular dichroism, and (2) plates cut from crystals of such low symmetry that the principal axes of absorption and refraction are not parallel. In case (2), the discussion is limited to monoclinic and triclinic crystals which do not possess optical activity.

© 1942 Optical Society of America

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References

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  1. R. Clark Jones, J. Opt. Soc. Am. 31, 488 (1941);Henry Hurwitz and R. Clark Jones, J. Opt. Soc. Am. 31, 493 (1941);R. Clark Jones, J. Opt. Soc. Am. 31, 500 (1941).
    [Crossref]
  2. M. Plancherel, Rend. di Palermo 30, 289 (1910).
    [Crossref]
  3. For the definition of M†, see the footnote on p. 492 in Part I.
  4. L. R. Ingersoll, P. Rudnick, F. G. Slack, and M. Underwood, Phys. Rev. 57, 1145 (1940).
    [Crossref]
  5. M. Berek, Fortschr., Mineral. Krist. Petrog. 22, 1–104 (1937).

1941 (1)

1940 (1)

L. R. Ingersoll, P. Rudnick, F. G. Slack, and M. Underwood, Phys. Rev. 57, 1145 (1940).
[Crossref]

1937 (1)

M. Berek, Fortschr., Mineral. Krist. Petrog. 22, 1–104 (1937).

1910 (1)

M. Plancherel, Rend. di Palermo 30, 289 (1910).
[Crossref]

Berek, M.

M. Berek, Fortschr., Mineral. Krist. Petrog. 22, 1–104 (1937).

Clark Jones, R.

Ingersoll, L. R.

L. R. Ingersoll, P. Rudnick, F. G. Slack, and M. Underwood, Phys. Rev. 57, 1145 (1940).
[Crossref]

Plancherel, M.

M. Plancherel, Rend. di Palermo 30, 289 (1910).
[Crossref]

Rudnick, P.

L. R. Ingersoll, P. Rudnick, F. G. Slack, and M. Underwood, Phys. Rev. 57, 1145 (1940).
[Crossref]

Slack, F. G.

L. R. Ingersoll, P. Rudnick, F. G. Slack, and M. Underwood, Phys. Rev. 57, 1145 (1940).
[Crossref]

Underwood, M.

L. R. Ingersoll, P. Rudnick, F. G. Slack, and M. Underwood, Phys. Rev. 57, 1145 (1940).
[Crossref]

Fortschr., Mineral. Krist. Petrog. (1)

M. Berek, Fortschr., Mineral. Krist. Petrog. 22, 1–104 (1937).

J. Opt. Soc. Am. (1)

Phys. Rev. (1)

L. R. Ingersoll, P. Rudnick, F. G. Slack, and M. Underwood, Phys. Rev. 57, 1145 (1940).
[Crossref]

Rend. di Palermo (1)

M. Plancherel, Rend. di Palermo 30, 289 (1910).
[Crossref]

Other (1)

For the definition of M†, see the footnote on p. 492 in Part I.

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Equations (60)

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M ( n ) Ɛ ( i ) = d i Ɛ ( i ) ,
det [ M ( n ) d i ] = 0
( m 1 d i ) ( m 2 d i ) = m 3 m 4 .
Ɛ ( 1 ) ( a 1 b 1 ) ( m 1 d 2 m 3 ) ( m 4 m 2 d 2 ) ( d 1 m 2 m 3 ) ( m 4 d 1 m 1 )
Ɛ ( 2 ) ( a 2 b 2 ) ( m 1 d 1 m 3 ) ( m 4 m 2 d 1 ) ( d 2 m 2 m 3 ) ( m 4 d 2 m 1 ) ,
T 1 M ( n ) T = D ( d 1 0 0 d 2 ) .
Δ a 1 b 2 a 2 b 1
T ( a 1 a 2 b 1 b 2 ) ; T 1 Δ 1 ( b 1 a 2 b 1 a 1 )
Ɛ ( 1 ) ( a 1 b 1 ) ; Ɛ ( 2 ) ( a 2 b 2 )
M ( n ) = TD T 1 Δ 1 ( d 1 a 1 b 2 d 2 a 2 b 1 ( d 1 d 2 ) a 1 a 2 ( d 1 d 2 ) b 1 b 2 d 2 a 1 b 2 d 1 a 2 b 1 ) .
Ɛ ( 1 ) ( 1 0 ) ; Ɛ ( 2 ) ( 0 1 )
Ɛ ( E x Ɛ y ) E x Ɛ ( 1 ) + E y Ɛ ( 2 ) .
F ( F 1 F 2 ) F 1 Ɛ ( 1 ) + F 2 Ɛ ( 2 ) .
Ɛ = T F ; F = T 1 Ɛ ,
Ɛ n = M ( n ) Ɛ 0 ,
F n = T 1 M ( n ) T F 0 .
Ɛ ¯ ( 1 ) Ɛ ( 2 ) * a a * + b b * = 1
Ɛ ¯ ( 1 ) Ɛ ( 2 ) * a 1 a 2 * + b 1 b 2 * = 0 .
M M = M M .
Ɛ ¯ Ɛ * = F ¯ F * .
Ɛ ¯ j Ɛ k * = F ¯ j F k * .
Ɛ ( 1 ) ( a 1 b 1 ) ; Ɛ ( 2 ) ( b 1 * a 1 * )
a 1 a 1 * + b 1 b 1 * = 1 ,
Ɛ n 1 = M ( n ) Ɛ 01
Ɛ n 2 = M ( n ) Ɛ 02 .
( | M ( n ) Ɛ 01 | 2 + | M ( n ) Ɛ 02 | 2 ) = 1 2 ( | m 1 | 2 + | m 2 | 2 + | m 3 | 2 + | m 4 | 2 ) .
T | M Ɛ | 2 / | Ɛ | 2 .
Ɛ = ( e i γ sin θ e i γ cos θ ) .
T | M Ɛ | 2 = ( | m 1 | 2 + | m 3 | 2 ) sin 2 θ + ( | m 2 | 2 + | m 4 | 2 ) cos 2 θ + [ ( m 1 m 4 * + m 2 * m 3 ) e 2 i γ + ( m 1 * m 4 + m 2 m 3 * ) e 2 i γ ] sin θ cos θ .
γ = 1 2 arg ( m 1 * m 4 + m 2 m 3 * ) ,
tan 2 θ = 2 | m 1 * m 4 + m 2 m 3 * | | m 2 | 2 + | m 4 | 2 | m 1 | 2 | m 3 | 2 ,
T stat = 1 2 ( | m 1 | 2 + | m 2 | 2 + | m 3 | 2 + | m 4 | 2 + 1 2 ( | m 2 | 2 + | m 4 | 2 | m 1 | 2 | m 2 | 2 ) sec 2 θ ,
T stat = 1 2 ( | m 1 | 2 + | m 2 | 2 + | m 3 | 2 + | m 4 | 2 ) ± 1 2 [ ( | m 2 | 2 + | m 4 | 2 | m 1 | 2 | m 3 | 2 ) 2 + 4 | m 1 m 4 * + m 2 * m 3 | 2 ] 1 2
Ɛ ( 1 ) = 2 1 2 ( 1 i ) , Ɛ ( 2 ) = 2 1 2 ( 1 i )
d 1 = e i ω , d 2 = e i ω .
( cos ω sin ω sin ω cos ω ) S ( ω ) .
( 1 2 ( d 1 + d 2 ) 1 2 i ( d 1 d 2 ) 1 2 i ( d 1 d 2 ) 1 2 ( d 1 + d 2 ) ) .
S ( ω G ) GS ( ω G ) ,
G ( e i γ 0 0 e i γ ) .
S ( ω P ) PS ( ω P ) ,
P ( p 1 0 0 p 2 ) = ( κ e ξ 0 0 κ e ξ )
0 p 1 1 , 0 p 2 1 , κ = ( p 1 p 2 ) 1 2 , ξ = 1 2 log ( p 1 / p 2 ) . }
B S ( ω G ) GS ( ω G ) S ( ω P ) PS ( ω P )
A S ( ω G ) BS ( ω G ) = GS ( ω ) PS ( ω ) ,
ω ω P ω G
B n = S ( ω G ) A n S ( ω G ) .
a 1 = κ e i γ ( cosh ξ + cos 2 ω sinh ξ ) , a 2 = κ e i γ ( cosh ξ cos 2 ω sinh ξ ) , a 3 = κ e i γ sin 2 ω sinh ξ , a 4 = κ e i γ sin 2 ω sinh ξ . }
d 1 , d 2 = α ± ( α 2 1 ) 1 2 = ( 1 + β 2 ) 1 2 ± β = α ± β , }
α cosh ξ cos γ + i cos 2 ω sinh ξ sin γ , β ( α 2 1 ) 1 2 , }
d 1 = ( α + β ) n , d 2 = ( α β ) n
γ 0 = n γ , ξ 0 = n ξ , κ 0 = κ n .
lim a 1 = lim ( κ n / 2 β ) [ ( a 1 1 + β ) ( 1 + β ) n ( a 1 1 β ) ( 1 β ) n = κ 0 ( cosh z + Ψ 1 sinh z ) , lim a 2 = κ 0 ( cosh z Ψ 1 sinh z ) , lim a 3 = lim a 4 = κ 0 Ψ 2 sinh z , }
z 2 ξ 0 2 γ 0 2 + 2 i x 0 γ 0 cos 2 ω = lim n 2 β , Ψ 1 ( ξ 0 cos 2 ω + i γ 0 ) / z , Ψ 2 ( ξ 0 sin 2 ω ) / z . }
lim B n = S ( ω G ) ( lim A ) S ( ω G ) = Θ ( κ 0 ( cosh z + Γ 1 sinh z ) κ 0 Γ 2 sinh z κ 0 Γ 2 sinh z κ 0 ( cosh z Γ 1 sinh z ) )
z 2 ξ 0 2 γ 0 2 + 2 i ξ 0 γ 0 cos 2 ( ω P ω G ) , Γ 1 ( ξ 0 cos 2 ω P + i γ 0 cos 2 ω G ) / z , Γ 2 ( ξ 0 sin 2 ω P + i γ 0 sin 2 ω G ) / z . }
n y n x = λ γ 0 / π t
k x = λ ( log κ 0 + ξ 0 ) / 2 π t , k g = λ ( log κ 0 ξ 0 ) / 2 π t .
Ɛ ( 1 ) ( 1 + Γ 1 Γ 2 ) ( Γ 2 1 Γ 1 )
Ɛ ( 2 ) ( Γ 1 1 Γ 2 ) ( Γ 2 1 + Γ 1 ) .
Γ 1 2 + Γ 2 2 1