The right hand side of Eq. (8) accounts for form birefringence,
which is most influential for axial uniform beams with planar
symmetry. To quantify this effect, we use the familiar
perturbation result of linear weak guidance [18,30]
that the 
's of (2) are shifted by order 
for TM waves. Over comparatively long distances, this leads
to
some interesting physics, not predicted by the traditional scalar wave
approximation.
For example, the polarization state of the classical sech
spatial soliton in a homogeneous isotropic nonlinear mediuim
will rotate [59], unless it is linearly polarized along
a geometrical axis of the soliton intensity pattern. This
rotation is significant over distances 
,
where 
is the maximum change in the induced refractive
index and 
is the characteristic soliton width.