= Paraxial angular-spectrum propagator
{title2=$e^{-iz(k_x^2+k_y^2)/(2k)}$}
The \b[<paraxial angular-spectrum propagator>] is the quadratic-phase approximation to forward free propagation of transverse <Fourier transform> components concentrated near the optical axis. For real medium <wavenumber> $k>0$, write $q^2=k_x^2+k_y^2$. With time convention $e^{-i\omega t}$, the exact factor is $e^{iz\sqrt{k^2-q^2}}$. Expanding at $q/k\ll1$ gives
$$
k_z=k-\frac{q^2}{2k}-\frac{q^4}{8k^3}+O(q^6/k^5),\qquad
H_{\rm env}(q,z)=\exp\!\left(-\frac{izq^2}{2k}\right),
$$
after removing the common carrier $e^{ikz}$.<Interactive Optics by Sander Konijnenberg, Aurèle J.L. Adam and H. Paul Urbach>[Scalar Diffraction Optics, Angular Spectrum Method and Fresnel Approximation, equations 374–385]{ref}
The approximation preserves the magnitude of each retained spectral component and changes its <wave phase>. Recombining those components produces <diffraction>; it need not preserve the shape or intensity at an individual spatial point. In coordinate space the envelope satisfies the <parabolic wave equation>, $2ik\,\partial_zU+\nabla_\perp^2U=0$.
Small propagation angle and acceptable accumulated phase error are separate conditions. The first omitted term produces a phase discrepancy of order $zq^4/(8k^3)$. For an original component with $q/k=0.1$, the exact difference between approximate and exact phases is
$$
\Delta\phi=kz\left[1-\frac{0.1^2}{2}-\sqrt{1-0.1^2}\right].
$$
It is about 0.00789 radians at $z=100\lambda$, but about 0.789 radians at $z=10{,}000\lambda$. The direction is still only about $5.74\,{}^\circ$ off axis. Propagation distance matters when phase accuracy is the observable.
For a source-plane <Gaussian beam> $U_0=\exp[-r^2/w_0^2]$, the spectral width is of order $1/w_0$. Thus $kw_0\gg1$ concentrates the important components in the paraxial region. That provides a reason for the familiar wide-waist condition rather than treating all Gaussian formulas as exact electromagnetic solutions.
A calculation containing substantial <evanescent angular-spectrum components> or large-angle components requires the exact branch and, when appropriate, the vector <transverse polarization constraint in an angular spectrum>. The paraxial transfer factor has unit magnitude even at large $q$, so extending it blindly into the evanescent region replaces physical decay by an artificial phase rotation.
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