Our thesis is that both the physical understanding and the mathematical description of nonlinear self-guided waves follows naturally from linear guided waves. Conceptually speaking, nonlinear beams interact with matter to create their own (linear) optical waveguides of arbitrary shape and form. These beams then propagate along their induced waveguides according to the familiar physics of linear optics. This elementary concept of linear physics can be used for a rigorous and systematic approach to nonlinear guided wave optics.
To set the stage, recall the physics of an optical waveguide. Optical beams have an innate tendency to spread as they propagate in a homogeneous medium. However, this beam diffraction can be compensated for by beam refraction if the refractive index is increased in the region of the beam. The resulting optical waveguide can provide an exact balance between diffraction and refraction if the medium is uniform in the direction of propagation. But, waveguides are not generally axially uniform, whereupon the balance between beam refraction and beam diffraction is not necessarily maintained. This gives rise to radiation `leakage'.
The propagation of optical beams is described by Maxwell's equations.
For waves that are monochromatic, the refractive index n of a
nonlinear medium depends on the time-averaged intensity
of the electric field for the usual
reasons [15]
[17]. Accordingly,
as a beam propagates in a nonlinear medium, it creates its
own (linear) optical waveguide (or modifies existing ones)
and then travels along the induced waveguide, according to
the familiar physics of linear optics.
This induced optical waveguide is characterized by the
refractive index profile 
. In
general, the waveguide is of rather arbitrary shape in x, y,
z. Of course, the unique aspect of nonlinear propagation is
that the particular induced (linear) waveguide depends upon
the initial condition of the beam.
We have implicitly assumed that the induced waveguide is isotropic. In general, it is anisotropic [16,17] and n is a tensor whose elements depend upon the dyadic
product 
. While such nonlinear material is considered
here, it is easier to introduce the linear perspective via
isotropic nonlinearities.
Self-Consistency. In theory, the linear perspective can
be used to solve beam propagation in any specified nonlinear
medium. To see this, suppose we know how an initial beam
propagates along every possible inhomogeneous linear optical
waveguide, 
. Call this solution 
.
Now, out of this infinity of linear waveguides, one is
exactly that induced by the beam propagating in a specified
nonlinear medium. This equivalent waveguide is identified
by the self-consistency relation
where we have suppressed the explicit spatial dependence 
which accounts for inhomogeneous material. More generally,
n is a tensor to describe anisotropy.
While exact, the above procedure is often impractical. We do not have the solutions of all possible linear problems and, if we did, it would be time consuming to find the one which is self-consistent. Nevertheless, the fact that every nonlinear problem has a linear equivalent provides a powerful conceptual tool, one that guides us in a physical manner to the fundamental equations and to their general solutions.
We begin by showing how the linear perspective anticipates classes of nonlinear waves and their characteristics, rather than the traditional approach of solving for a particular beam propagating in a specified medium. This provides physical insight and motivates novel phenomena.
