= Solution
The nullspace in the <Fourier diffraction theorem> already prevents uniqueness, and the rapid decay of evanescent high-frequency information prevents stable recovery. Both make this an <ill-posed inverse problem>.
Let $A_\pm$ be the linear <Born approximation> operators mapping a real contrast supported in a known region $D$ to the measured fields on the two observation planes. For data $d_\pm$, a suitable <Tikhonov reconstruction from two-plane scattering data> minimizes
$$
\boxed{J_\alpha(V)=\|A_+V-d_+\|_{L^2(S_+)}^2
+\|A_-V-d_-\|_{L^2(S_-)}^2+\alpha\|V\|_{L^2(D)}^2,
\qquad \alpha>0.}
$$
A <Sobolev norm> penalty can instead express a smoothness preference. Reality, known support and $n=1-V/k>0$ can be imposed as constraints. For the unconstrained complex Hilbert-space problem with the displayed quadratic penalty, write $A=(A_+,A_-)$ and $d=(d_+,d_-)$. Its <Tikhonov normal equation> is
$$
(A^*A+\alpha I)V_\alpha=A^*d,
\qquad V_\alpha=(A^*A+\alpha I)^{-1}A^*d.
$$
<Tikhonov regularization> selects a stable preferred estimate; it does not establish that the original two-plane data uniquely determine the true contrast.
Back to article page
