Light coupling to a general transparent dielectric can be represented formally by an infinite perturbation expansion of the polarization vector in powers of the incident electric field,
The expansion coefficients involve higher order tensor relations reflecting the vector
nature of the electromagnetic field-material coupling. Butcher [6]
was the first to give a detailed exposition on this general expansion in
1965 and much of this earlier work has been published recently as a
textbook[7].
Crystal symmetry determines the leading order nonlinear behavior of the
perturbation expansion. For crystals lacking a center of inversion symmetry,
the 
term is nonzero and dominates the interaction. This
second order term is used practically to generate second harmonics of
the incident light field or to generate frequency up- or down-shifted
components. The article by Allan Boardman in this proceedings discusses
some novel applications of the 
nonlinearity. The third harmonic term 
is the leading order
nonlinear term for centrosymmetric crystals. For a linearly polarized
(scalar) field, this leads to the well known
optical Kerr effect (frequency degenerate case). This nonlinearity is
responsible for optical soliton formation in fibers (1D) and for critical
collapse (self-focusing) in planar waveguide (2D) and bulk (3D) media.
Linearly isotropic media can give rise to nonlinear anisotropy which
causes an intensity dependent rotation of an elliptically polarized incident
field. This effect has been exploited in optical fibers to produce all-optical
polarization switches.
The dispersive properties of the material are reflected in the nonlocal nature of the polarization expansion. The leading order linear term is simply a convolution integral and can be written in Fourier space as
For near-monochromatic optical signals (reasonably long pulses), the
linear response function 
can be expanded as a
Taylor series about a reference optical carrier frequency 
to yield,
The linear term yields the group velocity and the quadratic term the
group velocity dispersion coefficients (see Figure 1) in the NLS equation.
Recall that 
and the refractive index
. Higher order
coefficients prove to be important as the optical pulse duration decreases
(spectral bandwidth increases) and their role has been investigated
in some detail in optical fibers [2]. Relatively little is
known about the nonlinear dispersive properties of optically transparent
materials although recent progress has been made in this direction as
discussed in the article by Sheik-Bahae in this proceedings.
A particularly simple classical model which mimics a detuned quantum oscillator is the Lorentz model[8], generalized to include an instantaneous Kerr nonlinear response,
The linear susceptibility is given by
where the plasma frequency
, 
and 
are the static and infinite relative
permittivities respectively and the resonance frequency of the
Lorentz oscillator. The linear version of this simple model was proposed
prior to the discovery of Quantum Mechanics and was used extensively to model
the linear dispersive properties of light interaction with materials. A
generalized multi-oscillator model due to Sellmeir[8], is still being used to
match the measured linear absorptive and refractive properties of
a great variety of materials over large frequency bandwidths. The basic
difference between these classical models and their more modern quantum
counterparts, lies in the oscillator strength definitions which are explicitly
defined in terms of fundamental quantities in the quantum picture. The above
extended Lorentz model will be used when discussing a new class of nonlinear
solitary wave pulse solutions in transparent dielectrics in Section 3.2.