Page 265 - Modern Optical Engineering The Design of Optical Systems

Optical Coatings 245

where the symbol |z| indicates the modulus of a complex number z, so
that

|z| |x iy| x y 2
2
and
x iy 1 2 x y 2
2
1
1
2
2
1
2
R |z| x y 2
x iy
2
2 2 x y 2
2
If the computation has been for normal incidence through nonabsorb-
ing materials, the transmission is given by
T 1 R (11.18)
Otherwise, the transmission is given by
2
n 0 cos 0 E 0
T (11.19a)

n cos E
p p p
or
n cos E 2
0
0
T p (11.19b)

n cos E p
p 0
where Eq. 11.19a is used for light polarized with the electric vector
perpendicular to [s] and Eq. 11.19b for the electric vector parallel to [p]
the plane of incidence.
A discussion of the design of multilayer coatings is beyond the scope
of this volume; the interested reader should pursue the subject in the
references listed at the end of this chapter. By suitable combinations of
thin films of different indices and thicknesses a tremendous number
of transmission and reflection effects can be created. Among the types
of interference coatings which are readily available are long- or short-
pass transmission filters, bandpass filters, narrow bandpass (spike fil-
ters), achromatic extra-low-reflection coatings as well as the reflection
coatings described in the next section. An extremely valuable property
of thin-film coatings is their spectral versatility. Once a combination of
films has been designed to produce a desired characteristic, the wave-
length region can be shifted at will by simply increasing or decreasing
all the film thicknesses in proportion. For example, a spike filter
designed to transmit a very narrow spectral band at 1 m can be
shifted to 2 m by doubling the thickness of each film in the coating.
This, of course, is limited by the absorption characteristics of the sub-
strate and the film materials.
The characteristics of a number of typical interference coatings are
shown in Fig. 11.5. Note that the wavelength scale is plotted in arbitrary
units, with a central wavelength of 1, since (within quite broad limits)