Abstract

The preceding papers of this series have examined the properties of an optical calculus which represented each of the separate elements of an optical system by means of a single matrix M. This paper is concerned with the properties of matrices, denoted by N, which refer not to the complete element, but only to a given infinitesimal path length within the element.

If M is the matrix of the optical element up to the point z, where z is measured along the light path, then the N-matrix at the point z is defined by

N(dM/dz)M-1.

Thus one may write symbolically,

N=dlogM/dz,

and

M=M0exp(Ndz).

A general introduction is contained in Part I. The definition and general properties of the N-matrices are treated in Part II. Part III contains a detailed discussion of the important special case in which the optical medium is homogeneous, so that N is independent of z; Part III contains in Eq. (3.26) the explicit relation which corresponds to the symbolic relation (C). Part IV describes a systematic method, based on the N-matrices, by which the optical properties of the system at each point may be described uniquely and quantitatively as a combination of a certain amount of linear birefringence, a certain amount of circular dichroism, etc.; the method of resolution is indicated in Table I. Part V treats the properties of the inhomogeneous crystal which is obtained by twisting a homogeneous crystal about an axis parallel to the light path.

© 1948 Optical Society of America

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References

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  1. IJ. Opt. Soc. Am. 31, 488–493 (1941); IIibid., 31, 493–499 (1941); IIIibid., 31, 500–503 (1941); IVibid., 32, 486–493 (1942); Vibid., 37, 107–110 (1947); VIibid., 37, 110–112 (1947).
    [CrossRef]
  2. H. Poincaré, Théorie Mathématique de la Lumière (Gauthier–Villars et Fils, Paris, 1892).

1941 (1)

Poincaré, H.

H. Poincaré, Théorie Mathématique de la Lumière (Gauthier–Villars et Fils, Paris, 1892).

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Tables (1)

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Table I This table shows the form of the N-matrices for eight different and independent types of crystalline behavior.

Equations (172)

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N ( d M / d z ) M - 1 .
N = d log M / d z ,
M = M 0 exp ( N d z ) .
= M ,
( X Y ) ,
M ( m 1 m 4 m 3 m 2 ) .
z = M z , z z .
N z lim z = z M z , z - 1 z - z .
z = M z 0 ,
M z , z = M z M z - 1 .
N z lim z = z M z - M z z - z M z - 1 = ( d M z / d z ) M z - 1 ,
N ( d M / d z ) M - 1 ,
d M / d z = NM ,
M = S ( 1 2 α z 2 ) ;             M - 1 = S ( - 1 2 α z 2 ) ,
d M / d z = α z S ( 1 2 α z 2 ) S ( 1 2 π )
N = ( 0 - α z α z 0 )
d / d z = ( d M / d z ) 0 .
d / d z = NM 0 .
d / d z = N .
A = A 0 exp ( i ( ω t - k z ) ) ,
d A / d z = - i k A .
M = S ( ω z ) .
N = ω S ( 1 2 π ) .
d X / d z = - ω Y , d Y / d z = ω X .
M = S ( ω ) MS ( - ω ) ,
M - 1 = S ( ω ) M - 1 S ( - ω ) ,
S ( ω ) ( cos ω - sin ω sin ω cos ω ) .
d M / d z = S ( ω ) ( d M / d z ) S ( - ω ) ,
N = ( d M / d z ) M - 1 = S ( ω ) NS ( - ω ) ,
M M - 1 = 1 ,
N = - M ( d M - 1 / d z ) .
N 2 = - ( d M / d z ) ( d M - 1 / d z ) = - M ( d M - 1 / d z ) ( d M / d z ) M - 1 = 1 2 [ M ( d 2 M - 1 / d z 2 ) + ( d 2 M / d z 2 ) M - 1 ] .
N 2 + ( d N / d z ) = ( d 2 M / d z 2 ) M - 1 , N 2 - ( d N / d z ) = M ( d 2 M - 1 / d z 2 ) .
M = exp ( N z )             M - 1 = exp ( - N z ) .
d exp ( N z ) / d z N exp ( N z ) ,
d exp ( - N z ) / d z - exp ( - N z ) N .
N 2 = - α 2 z 2 1 ,
N 2 + d N d z = - α 2 z 2 1 + α S ( 1 2 π ) = ( - α 2 z 2 - α + α - α 2 z 2 ) .
d M / d z = NM
d X / d z = n 1 X + n 4 Y , d Y / d z = n 3 X + n 2 Y .
d 2 X / d z 2 + 2 T ( d X / d z ) + D X = 0 ,
T 1 2 ( n 1 + n 2 ) ,
D n 1 n 2 - n 3 n 4 .
Q ( T 2 - D ) 1 2 = ( 1 4 ( n 1 - n 2 ) 2 + n 3 n 4 ) 1 2 .
d k M / d z k = N k M .
M = M 0 + k = 1 d k M d z k ( z - z 0 ) k k !
M = { 1 + k = 1 N k ( z - z 0 ) k k ! } M 0 ,
M = e N ( z - z 0 ) M 0 .
M = 1 + N z + 1 2 N 2 z 2 + 1 6 N 3 z 3 +
M = exp ( N z ) .
M M = λ M M ,
( d M / d z ) M = ( d λ M / d z ) M .
M = λ M M - 1 M ,
( d M / d z ) M - 1 λ M M = ( d λ M / d z ) M ,
N M = ( 1 / λ M ) ( d λ M / d z ) M .
M N ,
d log λ M / d z = λ N .
λ M = exp ( λ N z ) ,
1 = ( X 1 Y 1 )             2 = ( X 2 Y 2 ) ,
M = 1 X 1 Y 1 - X z Y 2 × ( λ 1 X 1 Y 2 - λ 2 X 2 Y 1 ( λ 2 - λ 1 ) X 1 X 2 ( λ 1 - λ 2 ) Y 1 Y 2 λ 2 X 1 Y 2 - λ 1 X 2 Y 1 ) .
N = M = ( 1 2 ( n 1 - n 2 ) ± 1 Q N n 3 ) ,
λ N = T N ± 1 Q N .
λ M = exp ( ( T N ± 1 Q N ) z ) .
M = exp ( T N z ) ( cosh Q N z + 1 2 ( n 1 - n 2 ) sinh Q N z Q N n 4 sinh Q N z Q N n 3 sinh Q N z Q N cosh Q N z - 1 2 ( n 1 - n 2 ) sinh Q N z Q N ) .
M = S ( ω z ) ,             N = ω S ( 1 2 π ) .
T M = cos ω , T N = 0 , D M = 1 , D N = ω , Q M = ± i sin ω , Q N = ± i ω ,
λ M = T M ± Q M = exp ( ± i ω ) ,
λ N = T N ± Q N = ± i ω .
n 1 = n 2 = 0 , n 3 = - n 4 = ω
D M = exp ( 2 T N z ) .
T N 1 2 ( n 1 + n 2 ) = 1 2 log ( ± 2 D M 1 2 ) ,
T M 1 2 ( m 1 + m 2 ) = exp ( T N z ) cosh Q N z ,
m 1 - m 2 = ( n 1 - n 2 ) exp ( T N z ) sinh Q N z Q N ,
m 3 = n 3 exp ( T N z ) sinh Q N z Q N ,
m 4 = n 4 exp ( T N z ) sinh Q N z Q N .
Q M = ± 1 exp ( T N z ) sinh Q N z ,
Q M = ± 1 Q N exp ( T N z ) sinh Q N z Q N .
n 1 - n 2 m 1 - m 2 = n 3 m 3 = n 4 m 4 = ± 1 Q N Q M .
Q M = ± 1 D M 1 2 sinh Q N z .
Q N Q M = ± 1 sinh - 1 [ Q M / D M 1 2 ] z Q M .
N = ( F M + 1 2 ( m 1 - m 2 ) G M m 4 G M m 3 G M F M - 1 2 ( m 1 - m 2 ) G M ) .
F M 1 z log ( ± 2 D M 1 2 ) , G M 1 z Q M sinh - 1 Q M ( ± 2 D M 1 2 ) = 1 z Q M log Q M ± 3 T M ( ± 2 D M 1 2 ) ,
M = ( i 0 0 - i ) .
T M = 0 , D M 1 2 = 1 , Q M = i ,
z F M = log 1 = 2 π i k , z G M = - i log i = 2 π l + 1 2 π ,
N z = 1 2 π i ( 4 k + 4 l + 1 0 0 4 k - 4 l - 1 ) ,
z F M = log ( - 1 ) = 2 π i k + π i , z G M = - i log ( - i ) = 2 π l - 1 2 π ,
N z = 1 2 π i ( 4 k + 4 l + 1 0 0 4 ( k + 1 ) - 4 l - 1 ) .
N z = ( π i ( 2 p + 1 2 ) 0 0 π i ( 2 q - 1 2 ) ) ,
M = ( 0 - 1 1 0 ) .
N z = 1 2 π ( 4 k i - 4 l - 1 4 l + 1 4 k i ) ,
N z = 1 2 π ( ( 4 k + 2 ) i - 4 l + 1 4 l - 1 ( 4 k + 2 ) i ) ,
N z = ( π i p - ( 2 q + p + 1 2 ) π ( 2 q + p + 1 2 ) π π i p ) ,
M = 1 2 ( 11 5 7 9 ) .
Q M = 3 , D M 1 2 = 4 , T M = 5.
F M = log 4 = 2 ϵ + 2 π i k ,
F M = log ( - 4 ) = 2 ϵ + π i ( 2 k + 1 ) ,
ϵ = 0.69315 .
3 z G M = log 2 = ϵ + 2 π i l ,
3 z G M = log ( - 1 2 ) = - ϵ + π i ( 2 l + 1 ) ,
3 z G M = log ( - 2 ) = ϵ + π i ( 2 l + 1 ) ,
3 z G M = log 1 2 = - ϵ + 2 π i l ,
6 N z = ( 13 ϵ + 2 π i ( 6 k + l ) 5 ϵ + 10 π i l 7 ϵ + 14 π i l 11 ϵ + 2 π i ( 6 k - l ) ) ,
6 N z = ( 11 ϵ + π i ( 12 k + 2 l + 1 ) - 5 ϵ + 5 π i ( 2 l + 1 ) - 7 ϵ + 7 π i ( 2 l + 1 ) 13 ϵ + π i ( 12 k - 2 l - 1 ) ) ,
6 N z = ( 13 ϵ + π i ( 12 k + 2 l + 7 ) 5 ϵ + 5 π i ( 2 l + 1 ) 7 ϵ + 7 π i ( 2 l + 1 ) 11 ϵ + π i ( 12 k - 2 l + 5 ) ) ,
6 N z = ( 11 ϵ + 2 π i ( 6 k + l + 3 ) - 5 ϵ + 10 π i l - 7 ϵ + 14 π i l 13 ϵ + 2 π i ( 6 k - l + 3 ) ) .
6 N z = ( ϵ ( 12 + ( - 1 ) p ) + π i ( 12 r + 7 q + p ) 5 ( - 1 ) p ϵ + 5 π i ( p + q ) 7 ( - 1 ) p ϵ + 7 π i ( p + q ) ϵ ( 12 - ( - 1 ) p ) + π i ( 12 r + 5 q - p ) ) ,
n i k τ k 1 8 ,
M k = 1 + N k τ k + O ( τ k 2 ) ,
M s = M 8 M 7 M 2 M 1 .
M s = 1 + k N k τ k + O ( ( k τ k ) 2 ) .
N ¯ k N k τ k k τ k .
τ k τ k .
M s = 1 + N ¯ τ + O ( τ 2 ) .
M = M s q = ( 1 + N ¯ τ + O ( τ 2 ) ) z / τ .
M s = 1 + N ¯ τ + O ( τ 2 ) ,
τ k = 1 8 τ .
Θ k 1 8 N k .
N ¯ = k Θ k .
= 0 exp ( i ω t - 2 π ( k + i n ) z / λ ) .
r = ( - i 1 ) , l = ( i 1 ) .
Θ 5 + Θ 7 = ( i g 0 i g 45 i g 45 - i g 0 ) .
( i g cos 2 α i g sin 2 α i g sin 2 α - i g cos 2 α ) ,
g 2 = g 0 2 + g 45 2 ,
tan 2 α = g 45 / g 0 .
N ¯ = ( - κ + p 0 - i η + i g 0 - ω + p 45 - i δ + i g 45 ω + p 45 + i δ + i g 45 - κ - p 0 - i η - i g 0 ) .
T N ¯ = - κ - i η , Q N ¯ 2 = ( p 0 + i g 0 ) 2 + ( p 45 + i g 45 ) 2 - ( ω + i δ ) 2 .
N ¯ = ( - i η + i g 0 - ω + i g 45 ω + i g 45 - i η - i g 0 ) ,
T N ¯ = - i η , Q N ¯ 2 = - ( g 2 + ω 2 ) = - Γ 2 ,
M ¯ = exp ( - i η z ) ( cos Γ z + i g 0 sin Γ z Γ ( - ω + i g 45 ) sin Γ z Γ ( ω + i g 45 ) sin Γ z Γ cos Γ z - i g 0 sin Γ z Γ ) .
z 0 = 2 π / ( g 2 + ω 2 ) 1 2 = 2 π / Γ .
k = 1 / z 0 = ( g 2 + ω 2 ) 1 2 / 2 π .
k g = g / 2 π ;             k ω = ω / 2 π ,
k = ( k g 2 + k ω 2 ) 1 2 .
η = - π ( p + q ) / z , g 0 = π ( p - q + 1 2 ) / z ,
η = - π p / z , ω = ( 2 q + p + 1 2 ) π / z ,
η = - π ( q + 2 r ) / z , κ = - 2 ϵ / z , ω = 1 6 ( - 1 ) p ϵ / z = p 0 , δ = 1 6 π ( p + q ) / z = g 0 , g 45 = π ( p + q ) / z , p 45 = ( - 1 ) p ϵ .
- 1 2 κ = 6 ω = 6 p 0 = p 45 = ϵ , η = δ = g 0 = g 45 = 0.
tan 2 α = p 45 / p 0 = 6 ,
α = 40.8 ° .
N = S ( k z ) N 0 S ( - k z ) ,
d d z = S ( k z ) N 0 S ( - k z ) .
S ( - k z ) .
S ( k z ) d d z + d S ( k z ) d z = S ( k z ) N 0 ,
d / d z = { N 0 - k S ( 1 2 π ) } ,
( d / d z ) S ( k z ) ω S ( k z ) S ( 1 2 π )
N N 0 - k S ( 1 2 π )
d / d z = N ,
M = exp ( N z )
= exp ( N z ) 0 ,
S ( - k z ) = exp ( N z ) 0 ,
= S ( k z ) exp ( N z ) 0 ,
M = S ( k z ) exp ( { N 0 - k S ( 1 2 π ) } z ) .
N = S ( ω ( z ) ) N 0 S ( - ω ( z ) ) ,
M = S ( ω ( z ) ) M ,
N N 0 - ( d ω ( z ) / d z ) S ( 1 2 π ) .
Θ 1 = - η ( i 0 0 i )
η = 2 π n / λ .
Θ 2 = - κ ( 1 0 0 1 )
κ = 2 π k / λ .
Θ 3 = ω ( 0 - 1 1 0 )
ω = 1 2 ( η r - η l ) ,
Θ 4 = δ ( 0 - i i 0 )
δ = 1 2 ( κ l - κ r ) .
Θ 5 = g 0 ( i 0 0 - i )
g 0 = 1 2 ( η y - η x ) ,
Θ 6 = p 0 ( 1 0 0 - 1 )
p 0 = 1 2 ( κ y - κ x ) ,
Θ 7 = g 45 ( 0 i i 0 )
g 45 = 1 2 ( η - 45 - η 45 ) ,
Θ 8 = p 45 ( 0 1 1 0 )
p 45 = 1 2 ( κ - 45 - κ 45 ) ,