
First, lets get some easier formula.
In order to do this, we remember
that e'' was connected to the conductivity of
the material and express e'' in terms of the
(total) conductivity as 





Note that in contrast to the definition of e'' given
before in the context of the dielectric function, we have an e_{0} in the e'' part. We had, for the sake of simplicity,
made a convention that the e
in the dielectric function contain the e_{0}, but here it more convenient to write it
out, because then e' = e_{0} · e_{r} is reduced to e_{r} and directly related to the
"simple" index of refraction n 


Using that in the expression (n + ik)^{2} gives 


(n + ik)^{2} 
= 
n^{2} – k^{2} + i · 2nk 
= e'
+ i · 
s_{DK}
e_{0} · w 




We have a complex number on both sides of the
equality sign, and this demands that the real and imaginary parts must be the
same on both sides, i.e. 


n^{2} –
k^{2} 
= 
e' 



nk 
= 
s_{DK}
2e_{0}w 




Separating n and k finally gives 


n^{2} 
= 
1
2 
æ
ç
è 
e' 
+ 
æ
è 
e' ^{2}
+ 
s_{DK}^{2}
4e_{0}^{2}w^{2}_{ } 
ö
ø 
½ 
ö
÷
ø 
k^{2} 
= 
1
2 
æ
ç
è 
– e' 
+ 
æ
è 
e' ^{2}
+ 
s_{DK}^{2}
4e_{0}^{2}w^{2}_{ } 
ö
ø 
½ 
ö
÷
ø 




Similar to what we had above,
but now with basic quantities like the "dielectric constant" e' = e_{r} and the
conductivity s_{DK}^{2}.





The equations above go beyond just describing the
optical properties of (perfect) dielectrics because we can include all kinds of
conduction mechanisms into s, and all kinds
of polarization mechanisms into e'. 


We can even use these equations for things like the
reflectivity of metals, as we shall see. 

Keeping in mind that typical n's in
the visible region are somewhere between 1.5  2.5 (n
» 2.5 for diamond is one of the higher
values as your girl friend knows), we can draw a few quick conclusions: From
the simple but coupled equations for n and k follows: 


k should be rather small for
"common" optical materials, otherwise our old relation of n
= (e_{r})^{½} would be not
good. 


k should be rather small for
"common" optical materials, because optical materials are commonly
insulators, i.e. s_{DK}
» 0 applies. 


For s_{DK} = 0 (and,
as we would assume as a matter of course, e_{r} > 0) we obtain immediately
n = (e_{r})^{½}
and k = 0  the oldfashioned simple relation
between just e_{r} and
n. 


For large s_{DK}
values, both n and k will
become large. We don't know yet what k means
in physical terms, but very large n simply mean that the
intensity of the reflected beam
approaches 100 %. Light that hits a good conductor thus will get
reflected  well, that is exactly what happens between light and (polished)
metals, as we know from everyday experience. 

But now we must look at some problems that can be
solved with the complex index of refraction in order to understand what it
encodes. 