
This subchapter can easily be turned
into a whole lecture course, so it is impossible to derive all the interesting
relations and to cover anything in depth.
This subchapter therefore just tries to give a strong flavor of the topic. 

We know,
of course, that the
index of refraction
n of a
nonmagnetic material is linked to the dielectric constant e_{r} via a simple relation, which is a rather
direct result of the
Maxwell
equations. 




But in learning about the origin of the dielectric
constant, we have progressed from a simple constant e_{r} to a complex
dielectric function with frequency
dependent real and imaginary parts. 


What happens to n then? How do we transfer the
wealth of additional information contained in the dielectric function to optical properties, which are
to a large degreee encoded in the index of
refraction? 

Well, you probably guessed it: We switch to a
complex index of
refraction! 


But before we do this, let's ask ourselves what we actually
want to find out. What are the optical properties that we like to know and that
are not contained in a simple index of
refraction? 


Lets look at the paradigmatic experiment in optics and see what we
should know, what we already know, and what we do not yet know. 




What we have is an
electromagnetic wave, an incident beam
(traveling in vacuum to keep things easy), which impinges on our dielectric
material. As a result we obtain a reflected
beam traveling in vacuum and a refracted
beam which travels through the material. What do we know about the three
beams? 


The incident beam is
characterized by its wavelength l_{i}, its frequency n_{i} and its velocity c_{0},
the direction of its polarization in some
coordinate system of our choice, and the arbitrary angle of incidence a. We know, it is hoped, the simple
dispersion
relation for vacuum. 





c_{0} is, of course, the velocity of light in
vacuum, an absoute constant of nature. 

The incident beam also has a certain amplitude of the electric field (and of the magnetic
field, of course) which we call E_{0}. The intensity
I_{i} of the light that the incident beams embodies, i.e.
the energy flow, is proportional to E_{0}^{2} 
never mix up the two! 

The reflected
beam follows one of the basic laws of optics, i.e. angle of incidence =
angle of emergence,
and its wavelength, frequency and magnitude of velocity are identical to that
of the incident beam. 


What we do not know is its
amplitude and its polarization, and these two quantities must somehow
depend on the properties of the incident beam and the properties of the dielectric. 

If we now consider the refracted beam, we know that it travels under an
angle b, has the same frequency as the
incident beam, but a wavelength l_{d}
and a velocity c that is different from l_{i} and c_{0}. 


Moreover, we must expect that it is damped or attenuated, i.e. that its amplitude decreases as a
function of penetration depth (this is indicated by decreasing thickness of the
arrow above). All parameters of the refracted beam may depend on the
polarization of the incident beam. 


Again, basic optics teaches that there are some simple
relations. We have





Snellius law 





From Maxwell equations 



c = n_{i} · l_{d} 

Always valid 





From the equations above 




A bit more involved is another basic relations coming from the
Maxwell equations. It is the equation linking c, the speed of
light in a material to the material "constants" e_{r} and the corresponding magnetic
permeability m_{0} of vacuum and
m_{r} of the material via 


c = 
1_{ }
(m_{0} · m_{r} · e_{0} · e_{r})^{1/2} 




Since most optical materials are not
magnetic, i.e. m_{r} = 1, we obtain
for the index of refraction of a dielectric material our relation
from above. 


n 
= 
c_{0}
c_{ } 
= 
(m_{0} ·
m_{r} · e_{0} · e_{r})^{1/2}
(m_{0} · e_{0})^{1/2} 
= 
e_{r}^{1/2} 




Consult the basic optics module if you have problems so
far. 