= Demagnifying a distant diffraction pattern
{title2=$f=\frac{Ld}{L-d}$}
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 $kx^2(1/d-1/f)/2=kx^2/(2L)$. The cross term $-kxy/d$ then matches free propagation at the old-screen coordinate $Y=L y/d$. <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 $d\lambda/a$.
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