A holographic global uniqueness in passive imaging

TL;DR

This paper proves that in passive imaging, the complex Helmholtz solution's restriction to rays and planes can be uniquely reconstructed from its imaginary part on a subset, establishing holographic global uniqueness results.

Contribution

It introduces new holographic uniqueness theorems for Helmholtz solutions, linking boundary measurements to the entire wave field in passive imaging.

Findings

Unique determination of wave solutions from imaginary parts on rays and planes.

Extension of holographic principles to passive imaging and inverse spectral problems.

Applicability to various measurement surfaces beyond planes.

Abstract

We consider a radiation solution $\psi$ for the Helmholtz equation in an exterior region in $\mathbb R^3$. We show that the restriction of $\psi$ to any ray $L$ in the exterior region is uniquely determined by its imaginary part $\Im\psi$ on an interval of this ray. As a corollary, the restriction of $\psi$ to any plane $X$ in the exterior region is uniquely determined by $\Im\psi$ on an open domain in this plane. These results have holographic prototypes in the recent work Novikov (2024, Proc. Steklov Inst. Math. 325, 218-223). In particular, these and known results imply a holographic type global uniqueness in passive imaging and for the Gelfand-Krein-Levitan inverse problem (from boundary values of the spectral measure in the whole space) in the monochromatic case. Some other surfaces for measurements instead of the planes $X$ are also considered.