Page 142 - Modern Optical Engineering The Design of Optical Systems

Prism and Mirror Systems 125

differentiating the preceding equations with respect to the index n,
assuming that I 1 is constant, yielding,
cos I tan I′ sin I
1
2
dD 2 dn (7.7)
cos I′
2
The angular dispersion with respect to wavelength is simply dD/d
and is obtained by dividing both sides of Eq. 7.7 by d . The resulting
dn/d term on the right is the index dispersion of the prism material.

7.3 The “Thin” Prism
If all the angles involved in the prism are very small, we can, as in the
paraxial case for lenses, substitute the angle itself for its sine. This
case occurs when the prism angle A is small and when the ray is
almost at normal incidence to the prism faces. Under these conditions, we
can write
i
i′ 1
1 n
i 1
i A i′ A
2 1 n
i′ ni nA i
2 2 1
D i i′ A i nA i A
1 2 1 1
and finally
D A (n 1) (7.8a)
If the prism angle A is small but the angle of incidence I is not small,
we get the following approximate expression for D (which neglects
powers of I larger than 3).
I (n 1) .. .
2
D A (n 1) 1 (7.8b)
2n
These expressions are of great utility in evaluating the effects of a
small prismatic error in the construction of an optical system since it
allows the resultant deviation of the light beam to be determined quite
readily.
The dispersion of a “thin” prism is obtained by differentiating Eq.
7.8a with respect to n, which gives dD Adn. If we substitute A from
Eq. 7.8a, we get
dn
dD D (7.9)
(n 1)