Page 225 - Modern Optical Engineering The Design of Optical Systems

208 Chapter Ten

where the downward slope increases again. For optical materials we usu-
ally need concern ourselves with only one section of the curve, since most
optical materials have an absorption band in the ultraviolet and another
in the infrared and their useful spectral region lies between the two.
Many investigators have attacked the problem of devising an equation
to describe “the irrational variation of index” with wavelength. Such
expressions are of value in interpolating between, and smoothing the
data of, measured points on the dispersion curve, and also in the study
of the secondary spectrum characteristics of optical systems. Several
of these dispersion equations are listed below.
b c
Cauchy n ( ) a ... (10.8)
2 4
b d
Hartmann* n ( ) a (10.9)
(c ) (e )
b c
Conrady n ( ) a (10.10)
3.5

b d
2
Kettler-Drude n ( ) a ... (10.11)
c 2 e 2
b 2 d 2 f 2
2
Sellmeier n ( ) a ... (10.12)
c 2 e 2 g 2
e d
2
Herzberger n ( ) a b (10.13)
2
2
( 0.035) ( 0.035) 2
c d e f
2
2
Old Schott n ( ) a b (10.14)
2 4 6 8
The new Schott catalog uses the Sellmeier equation (Eq. 10.12).
The constants (a, b, c, etc.) are, of course, derived for each individual
material by substituting known index and wavelength values and
solving the resulting simultaneous equations for the constants. The
Cauchy equation obviously allows for only one absorption band at zero
wavelength. The Hartmann formula is an empirical one but does allow
absorption bands to be located at wavelengths c and e. The Herzberger
expression is an approximation of the Kettler-Drude equation and is

* After an investigation, Arthur Cox concluded that the three term Hartman equation.
n( ) a b/(c ) 1.2
“is as good as anything over 408 to 656 nm and 546 to 1014 nm.”