When laser beams are passed through dielectric materials, the high intensity
field can induce a nonlinear relation between 
and . In practice, devices exploiting such nonlinear behavior also involve
anisotropic behavior. For simplicity, we assume the material is isotropic,
and concentrate on the nonlinearities.
We first illustrate a model for examining the nonlinearity, the Lorentz
equation. We examine the microscopic effect of an electric field
on an atom in the material. Let r denote the displacement of the electron
cloud due to the electric field; that is, the induced dipole moment is p=-er where e is the charge of the electron. Now, is a
macroscopic quantity, and it is not valid to use this value on an atomic
scale (the neighboring atoms, for example, influence the local behavior);
let us represent the ``local'' electric field by 
, where g is
a constant ``fudge factor''.
Now, if we perturb the electron cloud, there will be a tendency for it to
return to equilibrium (the Coulombic attraction to the nucleus, for example,
induces a restoring force); let us quantify this restoring force by 
. That is, if we think of a simple harmonic oscillator, is
the frequency of oscillation. There is a damping constant 
, which
represents resistance to perturbation. Then the differential equation
relating r to 
is:
where 
is the charge-to-mass ratio of the electron.
The nonlinear behavior of the system is modeled by the introduction of the
term 
in (4.1). The parameter 
is a constant, and
is very small in most materials. We can obtain a power series solution:
and hence the total displacement can be represented as the sum of
displacements 
proportional to the 
power of E. The
corresponding polarization density is 
where N is the volume
density of electrons. Hence:
where:
For the isotropic case, the coefficients 
are
scalars. In most materials, the nonlinear terms ( 
and higher order terms) are anisotropic, represented by tensors (matrices).
In many applications, the electric field is comprised of fields at a few,
discrete frequencies, 
. For example, there may be a bias field ( 
) and a laser
beam 
present in the material. In this case, 
will be comprised of frequencies at and its harmonics ( 
, etc.), as well as a bias term.
The second order susceptibility coefficients are called Pockels
coefficients, and the corresponding effect is called the Pockels or
electro-optic effect. The third order coefficients are called Kerr coefficients, and the corresponding effect is called the Kerr
effect. The Pockels coefficients are very small, several orders of magnitude
smaller than the linear dielectric constant 
, and the Kerr
coefficients are even smaller. In particular, the Kerr effect is generally
relevant or noticeable only in materials which exhibit special symmetries
(called inversion symmetry) so the Pockels effect is identically 0.
We will examine the Pockels effect, though.
