Page 223 - Modern Optical Engineering The Design of Optical Systems

206 Chapter Ten

This relationship is often stated in the following form, where a is
called the absorption coefficient and is equal to log e t.
T e ax (10.3)

Thus, it can be seen that the total transmission through an optical ele-
ment is approximately the product of its surface transmissions and its
internal transmission. For a plane parallel plate in air, the transmis-
sion of the first surface is given (from Eq. 10.1) as
(n 1) 2 4n
T 1 R 1 (10.4)
(n 1) 2 (n 1) 2
Now the light transmitted through the first surface is partially trans-
mitted by the medium and goes on to the second surface, where it is
partly reflected and partly transmitted. The reflected portion passes
(back) through the medium and is partly reflected and partly transmitted
by the first surface, and so on. The resulting transmission can be
expressed as the infinite series
3
7
2
5
3
T T T (K K R R K (R R ) K (R R ) ... ) (10.5)
1.2 1 2 1 2 1 2 1 2
T T K
1
2

2
1 K R R
1 2
where T 1 and T 2 are the transmissions of the two surfaces, R 2 and R 1
are the reflectances of the surfaces, and K is the transmittance of the
block of material between them. (This equation can also be used to
determine the transmission of two or more elements, e.g., flat plates,
by finding first T 1,2 and R 1,2 , then using T 1,2 and T 3 together, and so on.)
2
If we set T 1 T 2 4n/(n 1) from Eq. 10.4 into Eq. 10.5, and
assume that K 1, we find that the transmission, including all inter-
nal reflections, of a completely nonabsorbing plate is given by
2n
T (10.6)
2
(n 1)
This is obviously the maximum possible transmission of an uncoated
plate of index n.
Similarly, the reflection is given by
(n 1) 2
R 1 T (10.7)
2
(n 1)
It should be emphasized that the transmission of a material, being
wavelength-dependent, may not be treated as a simple number over
any appreciable wavelength interval. For example, suppose that a filter
is found to transmit 45 percent of the incident energy between 1 and
2 m. It cannot be assumed that the transmission of two such filters