Demagnifying a distant diffraction pattern ($f=\frac{Ld}{L-d}$)

A thin lens placed immediately after an aperture can reproduce its free-propagated pattern from distance L on a nearer screen d if $1/f=1/d-1/L$. In the paraxial Fresnel integral, the lens changes the aperture's quadratic propagation phase from $k{x}^2/(2d)$ to $k{x}^2(1/d-1/f)/2=k{x}^2/(2L)$. The cross term $-kxy/d$ then matches free propagation at the old-screen coordinate $Y=Ly/d$. Intensities have the same shape under transverse scale d/L, with an overall brightness factor in a lossless lens of sufficient aperture. For a narrow uniformly illuminated slit, the far-field first-zero estimate on the new screen is $d\lambda /a$.