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Place the input plane one focal length before a single positive lens and the output plane one focal length after it. The lens maps each incident transverse spatial frequency to its corresponding point in the back focal plane. Referencing the input to the front focal plane removes the otherwise present output phase curvature, giving an exact scaled Fourier transform within the same paraxial model.

A short ray-matrix check makes the geometry explicit. With free propagation $P(f)$ and lens matrix $L(f)$,The vanishing diagonal entries are the Fourier-plane condition; the output is proportional to ${\tilde{E}}_{\mathrm{in}}(kx/f)$. The final panel of the original diagram in part a shows the two focal distances. A lens with input immediately on its surface and output at the back focal plane would generally leave a quadratic output phase; the front-focal-plane choice avoids it.

For a slowly varying envelope $E=\psi e^{-ikx}$, the paraxial approximation to the Helmholtz equation isBecause $n=1+\mu W$ and ${\mu }^2\ll 1$, passage through a sufficiently thin phase screen producesIts modulus is one at the screen exit. Beyond the screen, $n=1$, so the parabolic wave equation is $E_x=i{E}_{zz}/(2k)$. A Taylor expansion in propagation distance givesSincewe findIt follows thator, equivalently, $|E{|}^2=1-x{\varphi }^{\prime \prime }/k+O(x^2)$. Thus random phase curvature produces local focusing and defocusing: free-space diffraction converts phase fluctuations into amplitude fluctuations immediately after the screen.