The electromagnetic fields 
of optical beams obey Maxwell's equations. For example, the
Cartesian components of 
satisfy [18,30]
where in general 
and isotropic material is
considered for simplicity of notation. Now, the relation between 
and 
is in general complicated apart from the singular
case of TE waves when the right hand side of Eq. (8) vanishes.
However, linear optical waveguides usually obey the weak
guiding condition that the maximum, 
, and minimum, 
, values of
refractive index are nearly equal, or 
. Maxwell's equations then simplify
enormously.
The resulting weak guidance approximation is the cornerstone of
linear waveguide theory [18,30]. This
theory must also apply directly to nonlinear waves, because all induced
waveguides are weakly guiding, (see [56]). Within weak guidance,
an initial TEM (
) wave remains approximately TEM
and satisfies the Helmholtz equation 
, where 
is the transverse Cartesian component
of 
and 
, with 
the minimum value of n and 
is the unit vector. The spatial variation of 
is arbitrary,
apart from our excluding `long' linear gratings that
radiate other than forward or backward [57]. If they radiate only forward, the
Helmholtz equation, within weak guidance, is identical to
the generalized vector Schrödinger equation.
To see this, we simply
substitute 
, 
, 
, and 
into the transverse part of (8) and take the limit 
, keeping 
fixed, where 
is the dimensionalized refractive index and 
is a
convienient scale length. This leads to 
with 
.
For birefringent nonlinear material, we find that
where 
is now a tensor refractive index as in Eq. (3).
In other words, the standard weak guidance theory of linear physics leads directly from Maxwell's equations to the Schrödinger equation without having to explicitly impose the usual [58] slowly varying envelope condition. Weak guidence is the physical concept from which all else follows. Conversely, weak guidance is necessary for the validity of the Schrödinger equation, even for TE waves --- a fact not well appreciated.
The weak guidance limit is analogous to the famous point dipole moment
approximation: the properties of waves at the dimensionalized positions 
, 
, and 
are unchanged in
the limit 
, provided the
dimensionalized shape 
, of the
induced waveguide and 
are individually held constant.
