= Solution
Define
$$
A=\partial_x,
\qquad
B=\left(n^2+k_0^{-2}\partial_z^2\right)^{1/2}.
$$
If $A$ and $B$ commute, the <Helmholtz equation> operator factorizes as
$$
A^2+k_0^2B^2
=(A+ik_0B)(A-ik_0B).
$$
When $n$ varies with $x$, the exact product also contains the <commutator> $[A,B]$; neglecting it assumes longitudinal changes are slow. The forward-propagating factor is
$$
(A-ik_0B)\psi=0,
$$
because it admits $e^{ik_0x}$ in a uniform medium.
Put $\psi=Ee^{ik_0x}$. The one-way equation becomes
$$
E_x=ik_0(B-1)E.
$$
For
$$
Q=n^2-1+k_0^{-2}\partial_z^2,
$$
the <Taylor expansion> $(1+Q)^{1/2}=1+Q/2+O(Q^2)$ gives
$$
E_x=\frac{i}{2k_0}E_{zz}
+\frac{ik_0}{2}(n^2-1)E.
$$
Thus the <parabolic wave equation> is
$$
\boxed{
2ik_0E_x+E_{zz}
+k_0^2(n^2-1)E=0}.
$$
The expansion is accurate under the <paraxial approximation>: transverse wavenumbers satisfy $|k_z|/k_0\ll1$, the envelope varies slowly on the carrier scale, the refractive-index contrast is weak enough for $Q^2$ to be negligible, and $n$ varies slowly in $x$ so $[A,B]$ is small. The one-way factor discards backward propagation and reflection; the square-root expansion additionally discards large-angle and higher-order diffraction, and it does not accurately represent strongly evanescent components or abrupt longitudinal interfaces.
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