Abstract

It is shown that the usual expression for measuring the cell thickness da of an empty cell, namely, da = λ1λ2N/2(λ2 − λ1), where N is the number of fringes over the wavelength range λ1 and λ2, may only be applied under particular conditions of cell and wall thickness and spectrophotometer characteristics. This is done by first deriving the expression for the transmission coefficient of a cell taking into account all the four surfaces at which a discontinuity in the refractive index occurs (while in deriving the above formula only two such surfaces are considered). Then, by examining the spectrophotometer characteristics, we investigate what amount of the information contained in the variation of transmission coefficient with wavelength is actually recorded by the instrument. Formulas are developed for different situations, in which only cell thickness, both cell and wall thickness, and finally only wall thickness, may be measured by counting the number of fringes over a wavelength interval. The equation mentioned at the beginning is only valid for the first of these cases.

© 1967 Optical Society of America

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Figures (1)

Fig. 1
Fig. 1

The cell.

Equations (55)

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d a = λ 1 λ 2 N / 2 ( λ 2 - λ 1 ) .
E i = E 0 e i ( k a z - ω t ) ,
H i = Y a E i ;
E r = E 1 e i ( - k a z - ω t ) ,
H r = - Y a E r ;
E 2 = ( E 2 + e i k w z + E 2 - e - i k w z ) e - i ω t ,
H 2 = Y w ( E 2 + e i k w z - E 2 - e - i k w z ) e - i ω t ;
E 3 = ( E 3 + e i k a y + E 3 - e - i k a y ) e - i ω t ,
H 3 = Y a ( E 3 + e i k a y - E 3 - e - i k a y ) e - i ω t ;
E 4 = ( E 4 + e i k w x + E 4 - e - i k w x ) e - i ω t ,
H 4 = Y w ( E 4 + e i k w x - E 4 - e - i k w x ) e - i ω t ;
E t = E 5 e i ( k a u - ω t ) ,
H t = Y a E t .
E 0 + E 1 = E 2 + + E 2 - ,
Y ( E 0 - E 1 ) = E 2 + - E 2 - ;
E 2 + e i k w d w + E 2 - e - i k w d w = E 3 + + E 3 - ,
E 2 + e i k w d w - E 2 - e - i k w d w = Y ( E 3 + - E 3 - ) ;
E 3 + e i k a d a + E 3 - e - i k a d a = E 4 + + E 4 - ,
Y ( E 3 + e i k a d a - E 3 - e - i k a d a ) = E 4 + - E 4 - ;
E 4 + e i k w d w + E 4 - e - i k w d w = E 5 ,
E 4 + e i k w d w - E 4 - e - i k w d w = Y E 5 .
Y = 1 / n ,
T = E 5 2 / E 0 2 ,
[ E 2 + E 2 - ] = M [ E 0 E 1 ]
[ E 3 + E 3 - ] = N [ E 2 + E 2 - ]
[ E 4 + E 4 - ] = P [ E 3 + E 3 - ]
[ E 5 E 5 ] = Q [ E 4 + E 4 - ] ,
[ E 5 E 5 ] = A [ E 0 E 1 ] ,
A = QPNM .
A = [ a 11 a 12 a 21 a 22 ] ,
E 5 = E 0 ( det A ) / ( a 22 - a 12 ) ;
T = det A 2 / a 22 - a 12 2 .
M = ( 1 / 2 ) [ 1 + Y 1 - Y 1 - Y 1 + Y ]
N = ( 1 / 2 Y ) [ ( 1 + Y ) e i k w d w - ( 1 - Y ) e - i k w d w - ( 1 - Y ) e i k w d w ( 1 + Y ) e - i k w d w ]
P = ( 1 / 2 ) [ ( 1 + Y ) e i k a d a ( 1 - Y ) e - i k a d a ( 1 - Y ) e i k a d a ( 1 + Y ) e - i k a d a ]
Q = [ e i k w d w e - i k w d w Y - 1 e i k w d w - Y - 1 e - i k w d w ] .
det A = - 2.
a 22 - a 21 = ( n 11 m 12 + n 12 m 22 ) [ p 11 ( q 21 - q 11 ) + p 21 ( q 22 - q 12 ) ] + ( n 21 m 12 + n 22 m 22 ) [ p 12 ( q 21 - q 11 ) + p 22 ( q 22 - q 12 ) ] .
8 Y 2 ( a 22 - a 12 ) = ( 1 + Y ) 2 ( 1 - Y ) 2 × { exp [ i ( k a d a + 2 k w d w ) ] - 2 exp [ i k a d a ] + 2 exp [ - i k a d a ] + exp [ i ( k a d a - 2 k w d w ) ] } - ( 1 - Y ) 4 exp [ i ( - k a d a + 2 k w d w ) ] - ( 1 + Y ) 4 exp [ - i ( k a d a + 2 k w d w ) ] .
| i a i e i b i | 2 = i a i 2 + 2 i > j a i a j cos ( b i - b j ) .
T = 2 8 Y 4 / R ,
R = R 1 + R 2 [ 2 cos ( 2 k a d a + 2 k w d w ) - cos ( 2 k a d a + 4 k w d w ) - 2 cos ( 2 k w d w ) - cos ( 2 k a d a ) ] + R 3 [ 2 cos ( 2 k a d a + 2 k w d w ) - 4 cos ( 2 k w d w ) + cos ( 4 k w d w ) - 4 cos ( 2 k a d a ) + 2 cos ( 2 k a d a - 2 k w d w ) ] + R 4 [ 2 cos ( 2 k a d a - 2 k w d w ) - cos ( 2 k a d a ) - 2 cos ( 2 k w d w ) - cos ( 2 k a d a - 4 k w d w ) ] ,
R 1 = ( 1 + Y ) 8 + ( 1 - Y ) 8 + 10 ( 1 - Y 2 ) 4
R 2 = 2 ( 1 + Y ) 6 ( 1 - Y ) 2
R 3 = 2 ( 1 + Y ) 4 ( 1 - Y ) 4
R 4 = 2 ( 1 + Y ) 2 ( 1 - Y ) 6 .
T = 16 Y 2 ( 1 - Y ) 4 + ( 1 + Y ) 4 - 2 ( 1 - Y ) 2 ( 1 + Y ) 2 cos ( 2 k a d a ) .
F = 2 cos ( 2 k a d a + 2 k w d w ) - cos ( 2 k a d a + 4 k w d w ) - 2 cos ( 2 k w d w ) - cos ( 2 k a d a ) .
F ( λ ) = 2 cos [ 4 π ( d a + n d w ) / λ ] - cos [ 4 π ( d a + 2 n d w ) / λ ] - 2 cos ( 4 π n d w / λ ) - cos ( 4 π d a / λ ) .
F 1 ( λ ) = - 2 cos ( 4 π d a / λ ) .
F ( λ ) = 2 [ 1 - cos ( 4 π n d w / λ ) ] cos [ 4 π ( d a + n d w ) / λ ] - 2 cos ( 4 π n d w / λ ) .
n d w = λ 1 λ 2 N e / 2 ( λ 2 - λ 1 ) .
d a + n d w = λ 3 λ 4 N c / 2 ( λ 4 - λ 3 ) .
F 1 ( λ ) = - 2 cos ( 4 π n d w / λ ) ;
n d w = λ 1 λ 2 N / 2 ( λ 2 - λ 1 ) ,

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