The step profile waveguide is the most familiar. It has 
for 
and 
for 
. Under inversion
[5], the fundamental bound mode of a step profile
waveguide is a self guided beam of the idealized saturating nonlinear
medium, characterized by 
for 
and 
for 
. This
threshold nonlinearity is the simplest model possible to illustrate all
physics unique to a saturating nonlinearity (e.g. bistability
[37] change of shape with power, multimoded nature of
induced waveguide) and it does so in an elementary analytic form
for both one and two [5] transverse dimensions. (Also
see Appendix).