= Solution
Use positive carrier <wavenumber> $k$ and retain waves traveling mainly in the positive $x$ direction. Factor out their rapid <wave phase> by writing $\psi=e^{ikx}E(x,z)$. Substitution into the <Helmholtz equation> is exact and gives
$$
E_{xx}+2ikE_x+E_{zz}=0.
$$
For a transverse <Fourier mode> $e^{i\nu z}$ at a small angle, $|\nu|/k\ll1$ and its forward longitudinal <wavenumber> is
$$
k_x=\sqrt{k^2-\nu^2}=k-\frac{\nu^2}{2k}+O(\nu^4/k^3).
$$
Thus the envelope varies along $x$ on scale $k/\nu^2$, much more slowly than the carrier wavelength, and $|E_{xx}|/|kE_x|=O(\nu^2/k^2)$. Dropping this smaller term derives the <parabolic wave equation>,
$$
\boxed{2ikE_x+E_{zz}=0,\qquad E_x=\frac{i}{2k}E_{zz}}.
$$
With transform $\widehat E(x,\nu)=\int E(x,z)e^{-i\nu z}dz$, its free propagation is
$$
\widehat E(x,\nu)=\widehat E(0,\nu)e^{-i\nu^2x/(2k)},\qquad
E(x,z)=\frac1{2\pi}\int\widehat E(0,\nu)e^{i\nu z-i\nu^2x/(2k)}d\nu.
$$
This is a one-way <paraxial approximation>, not a replacement for backward waves or large-angle Helmholtz components. Its <Fourier multiplier> has unit <modulus> and therefore does not attenuate any admitted component.
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