Page 151 - Modern Optical Engineering The Design of Optical Systems

134 Chapter Seven

A power series expansion yields the following expression:

tI (n 1) I ( n 3n 3)
2
2
D 1
n 6n 2
4
3
2
4
I (n 15n 15n 45n 45)
...
120n 4
For small angles, we can make the usual substitution of the angle for
its sine or tangent, or simply use the first term of the expansion to get
ti (n 1)
d
n
This lateral displacement by a tilted plate is used in high-speed cameras
(where the rotating plate displaces the image an amount approximately
equal to the travel of the continuously moving film) and in optical
micrometers. The optical micrometer is usually placed in front of a
telescope and used to displace the line of sight. The amount of displace-
ment is read off a calibrated drum connected to the mechanism which
tilts the plate.
When used in parallel light, a plane parallel plate is completely free
of aberrations (since the rays enter and leave at the same angles).
However, if the plate is inserted in a convergent or divergent beam,
it does introduce aberrations. The longitudinal image displacement
(n 1)t/n is greater for short wavelength light (higher index) than for
long, so that overcorrected chromatic aberration is introduced. The
amount of displacement is also greater for rays making large angles
with the axis; this is, of course, overcorrected spherical aberration.
When the plate is tilted, the image formed by the meridional rays is
shifted backward while the image formed by the sagittal rays (in a
plane perpendicular to the page in the figures) is shifted by a lesser
amount, so that astigmatism is introduced.
The amount of aberration introduced by a plane parallel plate can
be computed by the formulas below. Reference to Fig. 7.14 will indicate
the meanings of the symbols

U and u—slope angle of the ray to the axis
U p and u p —the tilt of the plate
t—thickness of the plate
n—index of the plate
V—Abbe V number (n d 1)/(n F n C )