Page 144 - Modern Optical Engineering The Design of Optical Systems

Prism and Mirror Systems 127

D 1 D 2
Dispersion dD dD dD 0
1,2 1 2 V V
1 2
A (n 1) A (n 1)
2
2
1
1

V V
1 2
A simultaneous solution for the angles of the two prisms gives
D V
1,2
1
A
1
(n 1) (V V )
1 1 2
(7.13)
D V
1,2
2
A
2 (n 1) (V V )
2 2 1
It is apparent that the prism angles will have opposite signs and
that the prism with the larger V-value (smaller relative dispersion)
will have the larger angle. A sketch of an achromatic prism is shown
in Fig. 7.3. Note that the emerging rays are not coincident but are
parallel, indicating an identical angular deviation.
In the direct vision prism it is desired to produce a dispersion without
deviating the ray. By setting the deviation D 1,2 equal to zero and pre-
serving the dispersion term dD 1,2 in the preceding equations we can
solve for the angles of two prisms which will produce the desired
result. The solution is
dD V V
1,2
2
1
A
1
(n 1) (V V )
1 2 1
(7.14)
dD 1,2 V 1 V 2
A
2
(n 1) (V V )
2 1 2
A two-element direct vision prism is shown in Fig. 7.4a. In order to
obtain a large enough dispersion for practical purposes it is often
necessary to use more than two prisms. Figure 7.4b shows the appli-
cation of such a prism to a hand spectroscope.
Since Eqs. 7.13 and 7.14 were derived using the equations for thin
prisms, it is obvious that the values of the component prism angles

Figure 7.3 An achromatic prism.
The red and blue rays emerge
parallel to each other; no chro-
matic dispersion is introduced
by the deviation.