In Section 3 we showed how a nonlinear beam can be two modes of its induced linear waveguide, but that the modes must be orthogonally polarized if the waveguide is to remain axially uniform. Next, suppose the modes are identically polarized. The intensity of the beams now changes periodically with propagation and so does the waveguide it induces.
This picture of periodic beams, suggests that higher-order solitons are
described by modal beating and that such solitons result only when the
induced waveguide is multimoded. Indeed, second-order
solitons appear qualitatively like the beating of the first two even
modes,
and they curiously arise when
the fundamental 
soliton has been sufficiently scaled up
[52].
Now, the linear approach to periodic solitons is not peculiar to
Kerr material, and should hold generally. But, periodic (higher-order)
solitons are presumed to be a by-product of integrability and thus, in
optics, applicable only to one-dimensional beams, propagating in a Kerr
(cubic) nonlinear medium. Accordingly, we consider an idealized
saturating medium as an example of a nonlinearity that differs radically
from a Kerr material. This is characterized by 
for 
and 
for 
, where
. Theory shows that a beam in this medium induces a step
profile waveguide whose width changes periodically.
To give a specific example, we use numerical beam propagation to determine
what happens when a beam, initially specified by 
, is launched in a nonlinear threshold medium. Here, 
and
denote the first and second even modes of a step-profile slab
waveguide, respectively, while the relative power in each of these modes
is
specified, e.g. 
= 0.0848. The (initial) induced waveguide is
characterized by the parameter V = 4, where 
is the waveguide half-width 
,
represent the maximum and minimum refractive indices of the
threshold nonlinearity, respectively. Figure 2 shows the waveguide
induced by this beam.
This beam is clearly stable to propagation; other numerical simulations
with different launch conditions (i.e. different 
-values) were
found to yield unstable beams that did not propagate. Entirely analogous
results were found for beams of circular cross-section. For example, by
launching the first two circularly symmetric modes of a V = 5
step-profile circular waveguide such that 
= 0.1344, the
results
are qualitatively the same as those presented in Fig. 2.
We have shown numerically that second-order solitons propagate over many periods in a threshold nonlinear medium, but do they propagate indefinitely without changing their form? To gain perspective, we again appeal to the linear equivalence principle which, in the present context, is stated as follows: a periodic soliton must be a Floquet or periodic mode of the periodic (linear) optical waveguide it induces. This elementary, yet exact, concept fully describes periodic solitons. Consequently, we can borrow results freely from the literature of linear periodic waveguides [53], from which we learn that periodic modes are, in general, leaky for structures with spatial wavelength perturbations relevant to our present study. Furthermore, first-order perturbation theory shows that the radiation comes from the second even mode [54].
To test these predictions, we numerically beam propagated the above solitons for one hundred periods. Typically, we observed about 1 per cent radiation loss and also observed that it is the second even mode that loses power. Further details are given in [54].