Fourier diffraction theorem

The Born approximation for scalar wave scattering relates an incident plane wave and the transverse Fourier transform of its scattered field to samples of the three-dimensional Fourier transform of a scattering potential on a shifted Ewald sphere. One incident direction at one frequency samples a surface, rather than an entire three-dimensional frequency region. Additional illuminations, frequencies or structural constraints are needed for a full reconstruction.

With $|{\mathbf{k}}_i|=k$, elastic scattering into a propagating wavevector ${\mathbf{k}}_s$ samples Fourier transfer $\mathbf{\kappa }={\mathbf{k}}_s-{\mathbf{k}}_i$. These transfers satisfy the displayed shifted sphere relation. This reciprocal-space construction appears in crystallographic diffraction and the Fourier diffraction theorem.