Square-aperture Fresnel diffraction (source code)

= 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: $R_{\rm eff}^{-1}=p^{-1}+q^{-1}$. With aperture side $d$, put $w=(d/2)\sqrt{2/(\lambda R_{\rm eff})}$. Separation into two one-dimensional <Fresnel integrals> gives the field normalized to its unobscured value,
$$
\frac{\psi}{\psi_0}=-2i[C(w)+iS(w)]^2,\qquad
\frac{I}{I_0}=4[C(w)^2+S(w)^2]^2.
$$
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, $\psi/\psi_0=D_xD_y/(2i)$. 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>.