Page 251 - Modern Optical Engineering The Design of Optical Systems

Optical Materials 231

near-infrared (0.7 to 2.2 µm). Polaroid also produces circular (as
opposed to plane) polarizers in sheet form.
Since a plane polarizer will eliminate half the energy, it is obvious
that the maximum transmission of a “perfect” polarizer in a beam of
unpolarized light will be 50 percent. Practical values range from 25 to
40 percent for sheet polarizers, depending on the type. If two polarizers
are “crossed,” that is, oriented with their polarizing axes at 90°, the
transmission will be zero if the polarization is complete. This can be
achieved with Nicol prisms, but the sheet polarizers have a residual
4
transmission ranging from 10 6 to 5 10 , again dependent on the type.
The transmission characteristics of sheet polarizers are wavelength-
dependent as well.
When two polarizers are placed in a beam of unpolarized light, the
transmission of the pair depends on the relative orientation of their
polarization axes. If is the angle between the axes, then the trans-
mission of the pair is given by:

2
2
T K cos K sin (10.17)
0 90
where K 0 is the maximum transmission and K 90 is the minimum.
Typical value pairs for K 0 and K 90 are 42 percent and 1 or 2 percent;
32 percent and 0.005 percent; 22 percent and 0.0005 percent.
Reflection from the surface of a glass plate may also be used to produce
plane polarized light. When light is incident on a plane surface at
Brewster’s angle, one plane of polarization is completely transmitted
(if the glass is perfectly clean) and about 15 percent of the other is
reflected. This occurs when the reflected and refracted rays are at 90°
to each other; thus, Brewster’s angle is given by
n′
I arctan (10.18)
n

The reflected beam is thus completely polarized and the transmitted
beam partially so. The percentage of polarized light in the transmitted
beam can be increased by using a stack of thin plates all tilted to
Brewster’s angle. For an index of 1.52, Brewster’s angle is 56.7°. Note
that Brewster’s angle is the angle at which the tangent term in Eq. 11.1
goes to zero.
The subject of polarized light is treated at greater length in texts
devoted to physical optics, to which the reader is referred. Two addi-
tional points are worth noting: one, interference filters (Chap. 11) are
usually obliquely polarizing and are occasionally used as polarizers;
and two, opal glass and other diffusers are excellent depolarizers, as
are integrating spheres.