Square-aperture Fresnel diffraction

For an axial point source a distance $p$ before a square aperture and an observer a distance $q$ beyond it, source and observation wave phase curvatures add: $R_{\mathrm{eff}}^{-1}=p^{-1}+q^{-1}$. With aperture side $d$, put $w=(d/2)\sqrt{2/(\lambda R_{\mathrm{eff}})}$. Separation into two one-dimensional Fresnel integrals gives the field normalized to its unobscured value,At an off-axis observer $(X,Y)$, centre the dimensionless coordinates on $x_0=pX/(p+q)$ and $y_0=pY/(p+q)$. If $D_x,D_y$ are the directed Cornu spiral chords between the corresponding aperture-edge parameters, $\psi /{\psi }_0=D_x{D}_y/(2i)$. This is a product of complex interval factors, rather than a sum of two intensities. The unobscured field is evaluated at the same observation point and includes its common propagation wave phase.