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For real $h\in C_c^{\infty }(D)$, the displayed real compactly supported contrast has Fourier transform $\hat{V}(\mathbf{\kappa })=[|\mathbf{\kappa }{|}^4-4({\mathbf{k}}_i\cdot \mathbf{\kappa }{)}^2]\hat{h}(\mathbf{\kappa })$. This vanishes on the shifted Ewald sphere because $|\mathbf{\kappa }{|}^2+2{\mathbf{k}}_i\cdot \mathbf{\kappa }=0$ there. Thus nonzero contrasts can have zero fixed-frequency, fixed-incidence Born approximation data. A small amplitude preserves weak contrast and a positive refractive index.

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.

Let ${\hat{\psi }}_s(\mathbf{K},z)=\int {\psi }_s(x,y,z)e^{-i(K_{x}x+K_{y}y)}dxdy$, and use the same two-dimensional Fourier transform for $V$ at a fixed depth. For propagating modes, $|\mathbf{K}|<k$ and $m=\sqrt{1-|\mathbf{K}{|}^2/k^2}$. Fourier transforming the angular spectrum in the previous part givesThe factor $k^2$ cancels against the change of transverse spectral variables $\mathbf{K}=k(p,q)$. Hence the recoverable samples satisfyEquivalently, in terms of the requested two-dimensional transform of $V$,This is the Fourier diffraction theorem. It recovers Fourier samples on the two hemispheres of a shifted Ewald sphere, since the outgoing wavevector ${\mathbf{k}}_s=(\mathbf{K},\sigma km)$ has length $k$ and the Fourier transfer is $\mathbf{\kappa }={\mathbf{k}}_s-{\mathbf{k}}_i$.

For a general three-dimensional $V$, this is not enough to recover $\hat{V}(\mathbf{K},z^{\prime })$ at every depth or reconstruct $V$ uniquely. At a fixed transverse frequency, the two planes supply only two longitudinal Fourier samples. Additional incident directions, frequencies or prior constraints are required. Thus the inverse problem as worded yields the boxed sampling relation, rather than a unique unconstrained volumetric solution.

An explicit nullspace makes this obstruction stronger than a dimension count. For any real smooth $h$ with compact support in $D$, take the Born non-scattering contrasts at one incident directionIts Fourier transform is $[|\mathbf{\kappa }{|}^4-4({\mathbf{k}}_i\cdot \mathbf{\kappa }{)}^2]\tilde{h}$. On the sampled sphere, $|\mathbf{\kappa }{|}^2+2{\mathbf{k}}_i\cdot \mathbf{\kappa }=0$, so this polynomial factor vanishes. Nonzero real compactly supported contrasts can therefore give identical propagating data to zero contrast in this linear approximation. Their amplitude can be made small enough to keep $n>0$ and remain in the weak-scattering regime. In fact the associated Born field is ${\psi }_s=2k{e}^{i{\mathbf{k}}_i\cdot \mathbf{r}}(\Delta -2i{\mathbf{k}}_i\cdot \nabla )h$. Applying $\Delta +k^2$ gives the source $2kV{e}^{i{\mathbf{k}}_i\cdot \mathbf{r}}$, while compact support makes this field zero on both exterior measurement planes. Thus this example also defeats recovery from their evanescent components.

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.