Maxwell's equations for a nonmagnetic dielectric material with no free charges are given, in MKS units, by
Here 
is the electric field, 
the magnetic field,
the magnetic induction and 
the electric displacement vector.
is the magnetic permeability and 
the vacuum dielectric
permeability. The relation for as written, separates the
electric displacement vector into a vacuum contribution ( proportional
to the vacuum permeability ) and a material induced polarization
( 
). In general will be a complicated nonlinear functional of the
electric field .
Once a relation between and is given, the full problem
is specified. This is a far from trivial step as we shall see in the discussion
of the constitutive relations below.
The envelope approximation to the electric field provides a robust description of light interaction with materials. We assume a linearly polarized incident field (scalar assumption) propagating in the z-direction and write the electric field vector in the form,
The envelope A(x,y,z,t) is a slowly varying function in space on the
scale of the optical wavelength ( 
) and its time variation
is much slower than the inverse of the optical frequency 
. The envelope assumption
may break down in circumstances where typical space scales approach the
wavelength of light and time scales approach the inverse of . This
happens in the vicinity of the focus in critical collapse or when the incident
pulse contains just a few optical cycles.
The scalar field obeys the nonlinear wave equation
where the speed of light 
.
Finally, the envelope function A(x,y,z,t) can be shown to obey
a parabolic type pde of Nonlinear Schrödinger type[3],
where the nonlinear term derives from the third order induced polarization
described below and a coordinate transformation has been made to a reference
frame moving with the group velocity (discussed below). The linear term,
proportional to 
, arises as a consequence of the dispersive
property of materials to an incident light field and this coefficient
can be of either sign. Figure 1 shows an experimentally measured graph
of the group velocity dispersion (GVD) coefficient versus optical wavelength
for water. As water is the dominant constituent of the human body and in
particular, of the vitreous humor making up the human eye, the absorptive and
refractive properties to light of water prove to be of great importance.
The sign of changes from positive
to negative just beyond 
.
Here 
denotes the transverse Laplacian
operator and measures the diffractive spreading of an initially transversely
localized beam. A reasonably general initial data for the above envelope
equation or variants thereof, is a optical pulse with a transverse Gaussian
spatial and a longitudinal hyperbolic secant profile in time,
where 
is the minimum beam waist (1/e value of the transverse
profile) and 
is the pulse duration. The phase curvature term with
radius R, allows
for an initial linear focusing or defocusing of the light pulse and mimics
the action of a positive or negative lens, respectively. The Gaussian shape
is a good approximation to the fundamental 
(transverse
electromagnetic) mode of a laser and the sech shape is typical of a
mode-locked pulsed laser output.
The diffraction and dispersion terms combined provide a potentially rich
scenario for the study of ultrashort pulse self-focusing in transparent
optical materials. When 
the overall dispersive
term is positive definite and an initial data of the type above is expected
to undergo 3D supercritical collapse[4]. For 
( mixed
sign of the dispersion), relatively little is known quantitatively about
the initial value problem but it has been shown that the 2D critical
collapse singularity development is arrested when the initial GVD is
sufficiently weak[5].