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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.

The two Born approximation measurement operators $A_{\pm }$ map a supported scattering potential to fields on planes above and below it. The displayed Tikhonov regularization functional balances their residuals against a positive quadratic penalty. With an $L^2$ penalty, the unconstrained minimizer is $(A^{\ast }A+\alpha I{)}^{-1}{A}^{\ast }d$, where $A=(A_{+},A_{-})$. Regularization stabilizes estimation but cannot remove the nonuniqueness of Born non-scattering contrasts at one incident direction without extra information.