(a) (b)Figure 74: a) Phonon dispersion and photon dispersion; b) Dispersion relationship of polaritons.The photon dispersion can also be written as
k^2 =μ 0 ε 0 ε(ω)ω^2. (127)Now we need to describe the motion of the atoms. We think of the atoms as masses connected by
springs but the atoms also have a charge that reacts to an electric field so Newton’s law for the atoms
is
m
d^2 x
dt^2
=−eE−CX. (128)Around the crossing point the dispersion of the optical phonons is very flat and we can assume that
the transverse optical phonons have the frequency
ωT=√
C
m. (129)
We solved the differential equation for the position of the atoms but the position is related to the
polarization like this:
P=−Nex (130)If we solve this equation forxand plug it into eqn. (128) we get
ω^2 mP
Ne
=−eE+
mωT^2 P
Ne(131)
−ω^2 P+ωTP=
Ne^2 E
m(132)
With eqn. (126) and eqn. (132) we have two equations for the relation between the electric field and
the polarization. (The electric field of the light wave is coupled to the polarization of the atoms.) We