Optical waveguides also have unbounded modes which comprise the
continuous mode spectrum [18]. These correspond to plane wave
scattering from the waveguide and, in general, consist of an incident
and scattered wave, except at special angles of incidence [36].
However, the 
waveguide (with 
) is unusual, in that it is reflectionless to every
angle of plane wave incidence [18] where V is defined
in the above example. Consequently, arbitrary beams will pass
through this linear waveguide without reflection. Upon
inverting the unbound mode, we find [36] that the fields
of the unbound (reflectionless) modes of the linear 
waveguide with 
are the fields of the
dark solitons propagating in a (negative) Kerr medium
with 
. In sum, the familiar 
profile of linear optical waveguides (with 
)
gives us the family of bright and dark solitons of a
Kerr medium, as well as the knowledge that their induced
waveguides are reflectionless.
The unbounded reflectionless modes of the linear slab waveguide occur only for
discrete values of plane wave incidence. Consequently, there is generally some
reflected power for beams incident on a step profile waveguide. Under
inversion [36], the unbounded reflectionless modes of the slab are
the
self guided dark beams of a (negative) threshold nonlinearity (
for 
).