
How do we get to the time and frequency
dependence P(t) and P(w), respectively, of the orientation
polarization without "cutting corners" as in the backbone? 


While in principle each function is just the
Fourier transform of the other, it is not so easy to actually do the required
math. It is probably best, to start with the differential equation that
describes the system. 

Within the "relaxation time
approximation" always used for those cases we have 





S(t) is some disturbance or
signal or input  whichever term you prefer  that has some time dependence. We
need it because otherwise the system would be "dead" and not doing
anything after at most one decay if we pick suitable starting conditions. 

Whatever happens, we can always write
P(t) and S(t) as a
Fourier series, or, more general,
as Fourier integral of the correlated P(w) and S(w) "spectra" of the time functions.
Doing this we have 


P(t) = 
¥
ó
õ
–¥ 
P(w)
· 
exp(iwt )
· dt 




S(t) = 
¥
ó
õ
–¥ 
S(w)
· 
exp(iwt )
· dt 




P(w)
· exp(iwt) and
S(w) · exp(iwt) are the Fourier "coefficients"
(with values for every w, not just harmonics
as in Fourier series) for the time functions. 

We now have a linear differential equation that is
solved by some P(t) which can be expressed as a Fourier
transform and that implies that all Fourier coefficients (and any superposition
thereof) also solve the differential equation. Inserting the Fourier
coefficients directly gives 


d{P(w) · exp(iwt)}
dt 
= 
iw · P(w)exp(iwt) =
– 
P(w) · exp(iwt)
t 
+ S(w) · exp(iwt) 




From this we obtain 


P(w) · ( iw + 1/t) =

S(w) 



If we define w_{0} = 1/t (or
= A/t if we want to be more general) we now
have a simple relation between the Fourier components of input S
and output P: 


P(w) = 
1_{ }
w_{0} – iw 
· S(w) 



The dielectric function e(w) that we are trying to
calculate is simple the relation between the output P(w) and the input S(w), what we get is 


e(w)
:= 
P(w)
S(w) 
= 
1_{ }
w_{0} – iw 




Considering that the disturbance S(w) must have the dimension of an electrical field,
forces us to conclude that it actually must be an electrical field, and we
could just as well write E(w).
What we have then is essentially the dielectric function as
discussed in the
backbone. 


