The polarization component to the
nth-order in the field is denoted as
Thus, the following equations apply,
and
where
is the
i-th
component
of the electric field strength and
is the usual '
susceptibility'
in the
absence of higher terms and
is the order of the field-induced
polarization in the material.
In an anisotropic medium,
,
and
are the medium '
hyper-susceptibilities';
they are tensors of rank 2, 3, and 4, respectively.
Linear optical responses such as absorption, light
propagation, reflection, and refraction,
involving a weak incoming field, are related to
.
Non-linear techniques are connected to the non-linear
polarization
.
Low order non-linear techniques, such as
three-wave mixing, are related to the second order optical polarization
.
For a random
isotropic medium (such as a liquid)
or for a crystal with a centrosymmetric
unit cell,
is zero by symmetry and then the lowest order non-linear techniques, as well as the
higher
order, are related to the third-order optical polarization,
,
and the corresponding hyper-susceptibility.