Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 88 / 5 / c / Solution

For $p^2+q^2>1$, $m=i\sqrt{p^2+q^2-1}$ and the Weyl plane-wave representation contains evanescent waves. Their amplitudes decay exponentially with the distance from the scattering region. Excluding them removes high transverse spatial frequencies, including subwavelength information. Near-field measurements can in principle detect them, but back-propagating their decay exponentially amplifies measurement errors.

For propagating data, the Fourier transfer satisfies $|{\mathbf{k}}_s-{\mathbf{k}}_i|\le 2k$. One illumination still samples only the Ewald sphere, so this frequency bound does not imply access to every mode inside that ball. The lack of a volumetric spectrum is the separate nonuniqueness described in the previous part.

The Born approximation requires the incident field to remain a good approximation inside the scatterer. A useful sufficient smallness estimate for a weak, extended inclusion is small accumulated phase $kD|n-1|\ll 1$, alongside weak reflection and multiple scattering. Small pointwise contrast alone can fail for a thick inclusion or a resonance. The scalar Helmholtz and background-medium assumptions, and the extra linearization of $n^2-1$, also belong to the model's validity conditions.

Numerically, a finite aperture windows the measured fields and limits angular and Fourier resolution. Discrete sampling must satisfy the Nyquist–Shannon sampling theorem for the retained transverse bandwidth; spacings $\Delta x,\Delta y\lesssim \pi /K_{\max }$ avoid its simplest aliasing. Finite discrete Fourier transforms also introduce periodic continuation, truncation, interpolation and window errors. Noise, unresolved evanescent modes and near-grazing modes with $m\simeq 0$ limit reliable recovery. Refining the grid cannot by itself fill the missing three-dimensional Fourier data.