= Solution
Apply the first <lens>, propagate, and apply the second <lens>. Omitting only common prefactors and uniform <phases>,
$$
E_{\mathrm{out}}(x)\propto\int dx'\,
\exp\!\left[-\frac{ikx^2}{2f}+\frac{ik(x-x')^2}{2L}-\frac{ikx'^2}{2f}\right]E_{\mathrm{in}}(x').
$$
The coefficients of both $x^2$ and $x'^2$ vanish if and only if $f=L$. The remaining cross term is $-ikxx'/L$, so
$$
\boxed{f=L,\qquad E_{\mathrm{out}}(x)\propto\int dx'\,e^{-i(kx/L)x'}E_{\mathrm{in}}(x')
=\widetilde E_{\mathrm{in}}\!\left(\frac{kx}{L}\right).}
$$
The Fourier convention here is $\widetilde E(q)=\int e^{-iqx}E(x)dx$. Finite clear apertures would truncate the field and limit the spatial-frequency range; the ideal relation assumes the paraxial field fits within both <optical lens>[lenses].
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