. This enables us to link soliton characteristics
with particular features of the material nonlinearity, 
.
We have shown how the linear perspective provides an exact description of nonlinear guided waves. It can also be used in a qualitative fashion. A
nonlinear medium is characterized by how its refractive index, n,
changes with light intensity, I. Ideally, we would like to ÔreadÕ the
existence, stability and salient characteristics of solitons directly from
this graph of . This enables us to link soliton characteristics
with particular features of the material nonlinearity, .
Consider bright solitons, of both one and two-transverse
dimensions, whose field has no nodes and whose intensity falls
off monotonically from a maximum value of 
to zero at
infinity. See the insert of Fig. A.1 (upper). What classes of
these solitons propagate in a nonlinear medium whose portrait
has the hypothetical form shown in Fig. A.1 (upper)? Because
we are interested in the change of refractive index, 
, associated with the maximum soliton intensity, 
,
we label 
in Fig. A.1 (upper). It is immaterial to
us whether or not this 
graph has an analytical
representation. To underscore the simplicity of the
qualitative approach, we first show how it can be implemented
by accepting only two rules. Subsequently, we derive these
rules using high school physics.
Solitons of Circular Symmetry: To find the solitons of circular symmetry we follow two rules:
Rule
1: The qualitative approach tells us (below) that such solitons exist,
at a particular soliton power, P, if the straight line 
intersects the curve 
of the material
nonlinearity. Although the constant of proportionality has dimensions, it
is of no concern to the qualitative approach. The maximum soliton
intensity is given by the value of 
at the intersection. Soliton
power is defined as the integral of I over the infinite soliton cross
section. The entire family of bound solitons is given by the
intersections that arise by considering all values of P from zero to
infinity.
Rule 2: The qualitative approach also tells us (below) that
solitons are stable if 
. Equivalently, stability demands
that the maximum intensity, 
, of the soliton increase as the soliton
power, P, is increased.
Kerr nonlinear medium:
For the sake of comparison, we apply the qualitative approach to find the
familiar solitons of the Kerr (cubic) nonlinearity. Because the straight
line 
drawn on Fig. A.1 (upper) coincides with this
nonlinearity, there is a continuum of solitons all with the same power 
, but with an arbitrary value of maximum intensity within the
interval 
. The greater 
, the more localized
is the soliton. Because 
does not increase with
increasing power, all solitons of circular symmetry are
unstable in a Kerr medium. Even though these findings are
already directly from the graph, 
, of the
material nonlinearity, it is nonetheless convenient to
summarize them on a graph which displays the maximum soliton
intensity, 
, vs the soliton power, P. This is the
vertical curve at 
in of Fig. A.1 (lower left). We
call this graph the qualitative soliton sketch. It reveals the
important physical properties of solitons at a glance.
Hypothetical nonlinear medium:
Next, consider the bright solitons of circular symmetry that propagate in
a hypothetical medium whose material nonlinearity is given by the solid
curve in Fig. A.1 (upper). Following rule 1, we examine the
intersections of straight lines 
with the material
nonlinearity curve 
. Representative
intersections are shown, as broken curves on Fig. A.1 (upper) for
increasing values of power; 
, 
, and 
. Here
the straight lines 
are tangent to the 
curve. By considering intersections of the straight lines for
all values of power, it is easy to sketch the solid curve shown
Fig. A.1 (lower left). This soliton sketch conveys the salient
physical properties of solitons for the hypothetical
nonlinearity. Following rule 2, only those solitons associated
with a positive slope, 
, are stable.
For each value of soliton power, P, within the interval 
to 
and for 
, the soliton sketch Fig. A.1 (lower left)
shows that three solitons exist, each with the same power. The
greater the value of maximum intensity, 
, the more
localized the soliton. However, only those solitons associated
with a positive slope are stable. Such multistable solitons
exist whenever a straight line 
intersects the
material nonlinearity 
at more than one position
in Fig. A.1 (upper). For soliton powers between 
and
, only one soliton can propagate and it is stable.
As 
, the straight line 
becomes parallel to the intensity axis of the material
nonlinearity portrait of Fig. A.1 (upper). It never intersects
with negative values of 
. This demonstrates that no
solitons exist with maximum intensities, 
, within the
interval 
and 
. The soliton sketch of Fig. A.1
(lower left) shows that the maximum soliton intensity, 
,
approaches a constant value as the soliton power, P,
approaches infinity. In other words, the soliton approaches a
plane wave as 
. This `discontinuous' soliton
class is novel.
One-Dimensional Solitons:
One-dimensional solitons are found analogously from rule 1, but now by
examining the intersections of the parabolas with the material
nonlinearity, 
, instead of the straight lines.
The stability criteria for one-dimensional solitons again given
by rule 2.
By considering the intersections of parabolas 
with the
material nonlinearity 
of Fig. A.1 (upper), it
is easy to construct the soliton sketch. This sketch is
similar to that for solitons of circular symmetry but with one
important difference. Only one soliton exists below 
in Fig. A.1 (lower left). Solitons of circular symmetry are
significantly more sensitive to slight departures from a Kerr
nonlinearity than are one-dimensional solitons. A relatively
large deviation from a Kerr nonlinearity is required for the
existence of multistable, one-dimensional solitons, whereas
only small departures from a Kerr nonlinearity are required for
the existence of multistable solitons of circular symmetry.
Further details are in [68].
Comparison with Exact Solutions:
The qualitative theory has been validated for numerous examples by
comparing it with numerical solutions for bright solitons of the wave
equation. This is verified by comparing the qualitative with the exact
sketch in Fig. A.1. In presenting this comparison, the
qualitative results were fit to match the exact results in the
limit as 
.
Theory:
Optical beams have an innate tendency to spread as they propagate in a
(linear) homogeneous medium. This angular spread, 
, obeys the
proportionality 
, where 
is any
characteristic beam width. The fact that the constant of proportionality
has dimensions (being a number times wavelength) is of no concern, as we
demonstrated above. Beam diffraction can be compensated for by beam
refraction [67], if the refractive index is increased in
the region of the beam so as to form a (linear) optical
waveguide. From geometric optics [18,67], a waveguide
traps light of angular spread 
, where
is the complement of the critical angle for total
internal reflection and obeys the proportionality 
. Here 
is the maximum
change (`height') in refractive index, and we have assumed that
the maximum and minimum n are nearly equal. In a nonlinear
medium the refractive index change, 
, is created by
the intensity dependent refractive index. Accordingly, a beam
becomes self guided when diffraction and nonlinear induced refraction,
, are balanced, i.e. when 
.
It is convenient to express this balance in terms of maximum soliton
intensity, 
, and soliton power, where soliton power 
and A is the infinite cross section. Assuming
the soliton shape remains unchanged, 
for
a one-dimensional beam and 
for a beam
of circular cross section. Using these expressions for power
to eliminate 
in 
and 
, the balance
necessary for solitons (
) leads to
for beams of circular symmetry and
for one-dimensional beams. These expressions were used to
graphically construct the soliton sketch. They can also be used
analytically. Consider a power law nonlinearity, where 
with 
. We find that 
, for beams of circular symmetry, and 
, for one dimensional beams. These results are fortuitously
exact because, as assumed in the qualitative theory, all solitons of a
power law nonlinearity can be shown to have the same shape
[32].
Stability:
Intuitive physics can also be used to determine the stability of the
soliton due to infinitesimal perturbations. It is most elegant to consider perturbations that preserve both the
soliton shape and its maximum intensity, 
, but change the
soliton power, P. An increase in P implies that the beam
becomes wider so that it suffers less diffraction, (
decreases). But, because 
is unchanged, refraction is
unchanged, (
is constant). Refraction is now greater
than diffraction 
. This means that the
beam will flow in a manner to initially refract (self-focus),
resulting in an initial increase in 
. If this increase in
is in a direction to make the perturbed 
beam a soliton, then it is stable as is clear from Fig. A.2,
otherwise it is unstable. Our physics gives the condition for
stability as 
. Although derived here for one
class of perturbation only, it turns out to be exact for arbitrary
perturbations [68].