 Solution

For the shown sequence of free space, thin lens, and free space, multiply the ray transfer matrices in traversal order:To obtain an ordinary Fourier transform of an arbitrary input field, neither an input quadratic phase nor an output quadratic phase may remain. Thus $A=D=0$, requiringUsing the physically consistent kernel derived in part b givesThe spatial Fourier coordinate is $x^{\prime }/(\lambda f)$. If the source's schematic integral normalization is retained instead, set its slowly varying denominator to $L_0$ to obtain the same Fourier transform with overall prefactor $ik{e}^{ik{L}_0}/(2\pi L_0)$. Keeping the source's incorrect quadratic sign would reverse the Fourier coordinate. These overall-normalization conventions do not alter the focal-plane placement, but an unapproximated transverse-dependent denominator would not give an exact Fourier transform.