Because beams of a nonlinear medium create their own optical
waveguide, strong (`pump') beams can guide weak (`signal') beams of a
different wavelength [19]. Indeed, guiding light with light is a
subject of recent activity [3]
[14] and
is also the physical explanation for a class of mathematical
results about the coexistence of dark and bright solitons
[34]. In this section, we characterize the waveguide
seen by the signal.
First, we emphasize that the effective refractive index induced by a monochromatic beam is not uniquely defined. For example, consider the usual Kerr (cubic) nonlinearity which, in general, exhibits induced birefringence [16,17].
Under one interpretation, the refractive index induced by elliptically
polarized light) can be viewed as a circularly birefringent medium
[28] as discussed above in Example 2. The amount of
birefringence
depends on the material as
is reflected through the parameters 
and the relative strengths of
the two circular components. An alternative interpretation is
that of a medium with normal (or linear) birefringence, but whose optical
axes twist with propagation. The period of twist is
, where 
and 
are the
propagation constants of the two circularly polarized components that
comprise the polarization of the (dynamic) soliton discussed in
Example 2. If the pump beam is linearly polarized, the
induced birefringent waveguide is axially uniform. If it is
circularly polarized, the induced waveguide is isotropic.
The refractive index of this twisted birefringent medium is
characterized by its optical axes 
which are
aligned with the major 
, and minor, 
, components of
the polarization ellipse, where
and 
where 
and 
. When B=0, the medium is isotropic.
If the beam is linearly polarized, say 
, then
and 
.
Thus, the refractive index `seen' in the direction of polarization is
larger than in the orthogonal direction.
It turns out that the twisted birefringent interpretation of the induced refractive index is the one that is necessary for light guiding light. Consider now the waveguide `seen' by a small (signal) beam in the presence of a large (pump) self guided beam.
The wavelengths of the signal and pump differ. In this case, the induced refractive index is uniquely defined. Two extreme limits exist for the waveguide seen by signal depending on the response time of the dipoles to the difference in frequencies of the pump and signal.
The case of a `fast' material response is discussed in
[16]. For example, the signal `sees' an 
double that seen by the pump for materials whose induced
refractive index is isotropic (B=0).
If the material response is `slow', the signal `sees' only the time averaged pump intensity. Consequently, the signal `sees' the same induced waveguide as the pump, i.e. a (linear) waveguide composed of the twisted birefringent medium discussed above. The propagation characteristics of such waveguides are available in the literature [35].
Consider the special case of a pump beam that is a linearly polarized
one-dimensional soliton of a Kerr medium. The waveguide induced by this
beam is characterized by 
, where
with 
, 
the maximum
and minimum refractive index. If the weak (signal) beam is polarized in
the direction of the strong soliton (pump) beam, then it `sees' a
waveguide characterized by 
, where
and 
are respectively the wavelength of
the pump and signal.
If the signal is polarized orthogonal to the pump beam, then it sees a
waveguide characterized by
.
The special case when the signal and pump are the same wavelength, but orthogonally polarized, is discussed in Example 2 above.