Electro-Optics - Applied Optics Fundamentals and Device Applications - page 88

The problem to be addressed is to solve Maxwell's equations when there
are current and charge densities brought about by the interaction of light
with matter (or in a vacuum where the light interacts with itself). The physi-
cal origin of these interactions is the nonlinear polarizability of either the
electronic charge cloud around the nuclei or a change in the various types
of nuclear motion allowed by the degrees of freedom of the material. These
are referred to as either the electronic or nuclear contributions. Although
the charge densities can be explained as a series of monopoles, this is found
not to be applicable in the optical regime, and instead a generalized elec-
tric polarization is used. A further approximation used is the electric-dipole
approximation that essentially states that the polarization is local. This in
turn makes the problem independent of spatial coordinates. With this in
mind, Maxwell's equations take the form

1 ∂

∂

E
t

∇ XE = − (2.18)

c

1 ∂

∂

4 π

∇ XB =

t E

( + 4 π P ) + I dc

c c

∇ ⋅ ( + 4 π ) = 0

E P

∇ ⋅ = B 0

P is the local polarization and is the only time-varying source term. It is in
general a function of E and fully describes the response of the medium to the
field. At this point, the assumption is generally made that the electric field
is sufficiently weak such that the total optical polarization density can be
expanded as a power series in the electric field:

( ) 1 ( ) 2 ( ) 3 ( ) 3

= χ ⋅ + χ : + χ χ + (2.18a)

P E EE EEE

where χ ( n ) is the n th order complex optical susceptibility. This can be shown
to be valid when the optical field is less than the atomic field, that is, when
E < E at ≈ 1 × 10 9 V/cm. A full quantum mechanical derivation dealing with the
microscopic interactions in the material is necessary to exactly calculate the
susceptibilities [96, 97].
The exact form of the effect observed is dependent on the frequencies of
the optical fields and the polarization of their E-fields relative to the crystal
axes. The frequency dependency occurs because of interactions with vari-
ous resonances in the crystal. These can range from slow thermal effects