= Fresnel diffraction phase curvature
{c}
{title2=$\phi_{\rm aperture}=\pi(\xi^2+\eta^2)/(\lambda z)$}
The quadratic <wave phase> follows from an ordinary distance calculation. Write $\rho^2=(x-\xi)^2+(y-\eta)^2$. The <Taylor expansion> of the distance to a screen <point> is
$$
r=\sqrt{z^2+\rho^2}=z+\frac{\rho^2}{2z}-\frac{\rho^4}{8z^3}+O\!\left(\frac{\rho^6}{z^5}\right).
$$
In the slowly varying propagation prefactor, $r\simeq z$ suffices. In the <complex exponential> $e^{ikr}$, however, a small distance error is multiplied by the large <wavenumber> $k$. Retaining the quadratic term while dropping the quartic term requires both $\rho/z\ll1$ and $k\rho^4/(8z^3)\ll1$ over the contributing <aperture> and the screen region being considered. The second condition is a <wave phase> accuracy requirement; the first is the geometric <paraxial approximation>.
Expanding the retained quadratic term separates three effects:
$$
U(x,y,z)\simeq\frac{e^{ikz}e^{i\pi(x^2+y^2)/(\lambda z)}}{i\lambda z}\iint_{\mathbb R^2}\underbrace{U_0(\xi,\eta)e^{i\pi(\xi^2+\eta^2)/(\lambda z)}}_{\text{field with aperture phase curvature}}e^{-2\pi i(x\xi+y\eta)/(\lambda z)}d\xi\,d\eta.
$$
The outside quadratic factor changes the <wave phase> at each screen <point> but has unit <complex modulus>, so it leaves that screen's <light intensity> unchanged. The quadratic factor inside the <integral> changes relative <wave phases> between contributions from different aperture <points>, so it changes their <wave interference>. The remaining linear factor supplies a <Fourier transform>. <Fraunhofer diffraction> drops the aperture quadratic factor when it is nearly constant; it does not require dropping the screen quadratic factor when predicting a later coherent field.
There is also a useful proof from the <Helmholtz equation>. Substituting $U=e^{ikz}u$ gives $u_{zz}+2iku_z+u_{xx}+u_{yy}=0$. Neglecting $u_{zz}$ yields the <parabolic wave equation>. A transverse <Fourier transform> evolves each <spatial frequency> $\boldsymbol\nu$ by $\exp[-iz|\boldsymbol\nu|^2/(2k)]$, using the transform kernel $e^{-i\boldsymbol\nu\cdot\mathbf x}$. Its inverse <Fourier transform> is the quadratic <Fresnel propagator> kernel above. Thus quadratic <wave phase> in space and quadratic <wave phase> in <spatial frequency> are two views of the same propagation.
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