Page 146 - Modern Optical Engineering The Design of Optical Systems

Prism and Mirror Systems 129

When the angle of incidence reaches a value such that sin I n′/n,
then sin I′ 1.0 and I′ 90°. At this point none of the light is trans-
mitted through the surface; the ray is totally reflected back into the
denser medium, as is any ray which makes a greater angle to the
normal. The angle
n′
I arcsin (7.15)
c
n
is called the critical angle and for an ordinary air-glass surface has a
value of about 42° if the index of the glass is 1.5; for an index of 1.7,
the critical angle is near 36°; for an index of 2.0, 30°; for an index of
4.0, 14.5°.
For practical purposes, if the boundary surface is smooth and clean,
100 percent of the energy is redirected along the totally reflected ray.
However, it should be noted that the electromagnetic field associated
with the light actually does penetrate the surface for a relatively short
(to the order of a wavelength) distance. If there is anything near the
other side of the boundary surface, the total internal reflection can be
“frustrated” to some extent and a portion of the energy will be trans-
mitted. Since the distance of effective penetration is only to the order
of the wavelength of the light involved, this phenomenon has been
used as the basis of a light valve, or modulator. In the German “Licht-
Sprecher,” an external piece of glass was placed in contact with the
reflecting face of a prism to frustrate the reflection, and then moved an
extremely short distance away (e.g., a few micrometers) to reinstate
the reflection.
It should also be noted that the reflection of a totally reflecting
surface is decreased by aluminizing or silvering the surface. When
this is done, the reflectance drops from 100 percent to the reflectance
of the coating applied to the surface.

7.7 Reflection from a Plane Surface
Since the prism systems which are discussed in the balance of this
chapter are primarily reflecting prisms (the majority of which can
be replaced by a system of plane mirrors), we shall first discuss the
imaging properties of a plane reflecting surface. Rays originating at an
object are reflected according to the law of reflection, which states that
both the incident and reflected rays lie in the plane of incidence and
that both rays make equal angles with the normal to the surface. The
normal to the surface is the perpendicular at the point where the ray
strikes the surface, and the plane of incidence is that plane containing
the incident ray and the normal.