Two single-measurement uniqueness results for inverse scattering problems within polyhedral geometries

<p style='text-indent:20px;'>We consider the unique determinations of impenetrable obstacles or diffraction grating profiles in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^3 $\end{document}</tex-math></inline-formula> by a single far-field measurement within polyhedral geometries. We are particularly interested in the case that the scattering objects are of impedance type. We derive two new unique identifiability results by a single measurement for the inverse scattering problem in the aforementioned two challenging setups. The main technical idea is to exploit certain quantitative geometric properties of the Laplacian eigenfunctions which were initiated in our recent works [<xref ref-type="bibr" rid="b12">12</xref>,<xref ref-type="bibr" rid="b13">13</xref>]. In this paper, we derive novel geometric properties that generalize and extend the related results in [<xref ref-type="bibr" rid="b13">13</xref>], which further enable us to establish the new unique identifiability results. It is pointed out that in addition to the shape of the obstacle or the grating profile, we can simultaneously recover the boundary impedance parameters.</p>