A preprint dated 20 August 2026 describes a mathematical route to recovering phase-related information about a radiating wave from intensity measurements at two points. The method uses a spherical reference wave and seeks an approximate estimate of the radiation solution’s far-field pattern — its behavior in the large-distance limit — from the intensity of the combined field.
The result is a methods contribution, not a report of an imaging experiment. Its setting is the Helmholtz equation in an exterior region, where the analysis follows a reference wave, a radiation solution and the total field formed from them. The supplied document is arXiv version 1, and the evidence described is mathematical rather than empirical.
From two intensities to a directional signal
The proposed input is deliberately specific: the intensities of the total field at points x and y, together with the known reference parameters A and x0. The formulas use those quantities to estimate information about the radiation solution’s far-field pattern. In this setting, phase recovery means reconstructing the phase-related part of the wave from measurements that enter the formulas through total-field intensity.
The paper does not present the two measurements as a universal solution to every phase-recovery problem. Instead, it derives formulas for the stated spherical-reference configuration and under the asymptotic conditions used in the theorem. The result is therefore an approximate determination of a far-field quantity, not a general claim that the full wave is uniquely recoverable everywhere in the exterior region.
Why the geometry matters
The central derivation turns the two intensity relations into an approximate 2 × 2 linear system for the far-field pattern. The quantity D in the paper is the determinant of the corresponding matrix. When D is nonzero, the system can be inverted within the approximation, allowing the unknown far-field information to be expressed through the two measured intensities and the reference-wave parameters.
That determinant is controlled by the measurement geometry. At large distance, its limiting behavior is given as 2i times a sine term involving the observation and source directions. The sine factor is important because a vanishing value removes the non-degeneracy condition needed by the recovery formula.
The theorem therefore imposes geometric exclusions. It requires the relevant sine condition to be nonzero and requires either the observation direction and source direction to differ or the scaling parameter to be different from one. These conditions are part of the formula’s definition, rather than optional refinements added after the calculation.
The paper states the accuracy in asymptotic terms. As the observation distance r tends to infinity while the displacement is fixed, the theorem includes error terms of order O(r^-1) and O(1/A(αr)). In ordinary language, the result describes how the approximation behaves in a large-distance regime; it does not supply a finite-distance performance figure.
A single plane, with exceptions
The authors also apply the formulas to a different measurement arrangement: recovering the radiation solution on a sufficiently distant hyperplane from intensity data collected on that same plane. A hyperplane can be thought of as an idealized flat measurement surface, so this part of the work narrows the data requirement from separated point measurements to information distributed across one such surface.
The single-plane result still depends on direction. When the source direction lies in the opposite half-sphere, the exceptional observation directions are contained in the span of the plane normal and the source direction. In other words, the geometry identifies a restricted set of directions for which the stated recovery argument may fail or require separate treatment.
Under the assumptions of the relevant lemma, the number of exceptional directions is bounded by six. That bound does not remove the geometric qualification: the result remains conditional on the arrangement of the plane, source and observation directions.
A mathematical result with a clear boundary
The introduction describes a deeper obstacle. The general phase-recovery problem is stated to be not uniquely solvable even when intensity data are available throughout the exterior region. The two-point formulas should therefore be understood as a solution for specified large-distance configurations, not as a proof of uniqueness for the broader problem.
The evidence supplied for the preprint is entirely theoretical: definitions, asymptotic expansions, an approximate linear system, determinant analysis and theorem proofs for Helmholtz-wave configurations. There are no reported physical experiments or clinical, animal or cell results in the described study. The work consequently does not establish accurate phase recovery in a real imaging system.
Several practical questions remain outside the reported result. The asymptotic error terms do not show how the formulas perform at a particular finite distance, and the stated conditions do not establish robustness when measurements are noisy or when the assumed wave model differs from the measured field. Numerical implementation for specific geometries is also not assessed in the supplied analysis.
The authors present the work as a continuation of earlier studies using a plane-wave reference. That prior plane-wave work provides context for the choice of a spherical reference here, but the supplied analysis does not assess practical performance against those earlier methods. The contribution described in this preprint is the derivation of new asymptotic formulas and the geometric conditions attached to them.
The research was supported for V.N. Sivkin by Russian Science Foundation grant 25-71-00116. The supplied document remains an arXiv version 1 preprint dated 20 August 2026, so the results should be read as a theoretical contribution awaiting any future assessment beyond the formulas and conditions reported here.
Paper data and sources
Original title: A two-point phase recovering with spherical wave reference
Authors: R. G. Novikov, V. N. Sivkin
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text