Page 143 - Modern Optical Engineering The Design of Optical Systems

126 Chapter Seven

Now the fraction (n 1)/
n is one of the basic numbers used to charac-
terize optical materials. It is called the reciprocal relative dispersion,
Abbe V number, or V-value. Ordinarily n is taken as the index for the
helium d line (0.5876 m) and
n is the index difference between the
hydrogen F(0.4861 m) and C(0.6563 m) lines, and the V-value is
given by
n d 1
V (7.10)
n n
F C
Making the substitution of 1/V for dn/(n 1) in Eq. 7.9, we get
D
dD (7.11)
V
which allows us to immediately evaluate the chromatic dispersion pro-
duced by a thin prism.

7.4 Minimum Deviation
The deviation of a prism is a function of the initial angle of incidence I 1 .
It can be shown that the deviation is at a minimum when the ray passes
symmetrically through the prism. In this case I 1 I′ 2 ⁄2(A D) and
1
I′ 1 I 2 A/2, so that if we know the prism angle A and the minimum
deviation angle D 0 it is a simple matter to compute the index of the
prism from
1
sin ⁄2 (A D )
sin I 1
0
n (7.12)
sin I′ 1 sin ⁄2 A
1
This is a widely used method for the precise measurement of index,
since the minimum deviation position is readily determined on a spectro-
meter. This position for the prism is also approximated in most spectral
instruments because it allows the largest diameter beam to pass through
a given prism and also produces the smallest amount of loss due to
surface reflections.

7.5 The Achromatic Prism and the Direct
Vision Prism

It is occasionally useful to produce an angular deviation of a light beam
without introducing any chromatic dispersion. This can be done by com-
bining two prisms, one of high-dispersion glass and the other of low-
dispersion glass. We desire the sum of their deviations to equal D and
1,2
the sum of their dispersions to equal zero. Using the equations for “thin”
prisms (Eqs. 7.8 and 7.11), we can express these requirements as follows:

Deviation D D D A (n 1) A (n 1)
1,2 1 2 1 1 2 2