 Fresnel diffraction phase curvature

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 isIn 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 $\rho /z\ll 1$ and $k{\rho }^4/(8z^3)\ll 1$ 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: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 $u_{zz}+2ik{u}_z+u_{xx}+u_{yy}=0$. Neglecting $u_{zz}$ yields the parabolic wave equation. A transverse Fourier transform evolves each spatial frequency $\mathbf{\nu }$ by $\exp [-iz|\mathbf{\nu }{|}^2/(2k)]$, using the transform kernel $e^{-i\mathbf{\nu }\cdot \mathbf{x}}$. 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.