= 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 $|\mathbf k_s-\mathbf k_i|\leq2k$. 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 $kD|n-1|\ll1$, 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 $m\simeq0$ limit reliable recovery. Refining the grid cannot by itself fill the missing three-dimensional Fourier data.
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