A Numerical Method to Solve a Phaseless Coefficient Inverse Problem from a Single Measurement of Experimental Data

Abstract

We propose in this paper a globally numerical method to solve a phaseless coefficient inverse problem: how to reconstruct the spatially distributed refractive index of scatterers from the intensity (modulus square) of the full complex valued wave field at an array of light detectors located on a measurement board. The propagation of the wave field is governed by the 3D Helmholtz equation. Our method consists of two stages. On the first stage, we use asymptotic analysis to obtain an upper estimate for the modulus of the scattered wave field. This estimate allows us to approximately reconstruct the wave field at the measurement board using an inversion formula. This reduces the phaseless inverse scattering problem to the phased one. At the second stage, we apply a recently developed globally convergent numerical method to reconstruct the desired refractive index from the total wave obtained at the first stage. Unlike the optimization approach, the two-stage method described above is global in the sense that it does not require a good initial guess of the true solution. We test our numerical method on both computationally simulated and experimental data. Although experimental data are noisy, our method produces quite accurate numerical results.

Phaseless Parametric Inversion for System Calibration and Obtaining Prior Information

“…The difficulty in measuring phase is even more applicable at optical frequencies, where the phase often cannot be measured at all, and as such optical-frequency phaseless problems have received significant attention, e.g. [35]. The phaseless inversion problem is widely recognized as being more difficult to solve than its full-data counterpart.…”

Single‐Particle Characterization by Elastic Light Scattering

“…This is also known as a coefficient inverse problem, if differential Maxwell's equations are considered as a forward model. [ 231 ] The necessary prior information is reflected mostly in the approximation used or in the knowledge of the RI value (then only the boundary is reconstructed). The main drawback of these methods is the enormous amount of required experimental data, severely limiting high‐throughput implementations.…”

Approximate Lipschitz stability for phaseless inverse scattering with background information

“…In addition to Problem 1, there are also other phaseless inverse problems for equation (1.1) and for related equations; see, for example, [3], [4], [7], [8], [10], [11], [12], [13], [14], [15], [21], [22], [23], [24], and references therein.…”