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Phase retrieval via media diversity

Yan Cheng, Kui Ren, Nathan Soedjak

TL;DR

The paper addresses phase retrieval for wave fields by exploiting media diversity—linear and nonlinear media—to encode phase information into intensity measurements. It proves that phase can be uniquely recovered (up to a global constant in some cases) using data from multiple media, and derives explicit reconstruction algorithms based on transport-of-intensity ideas. Importantly, quadratic nonlinearity can remove the global phase ambiguity, enabling full phase recovery under suitable diversity conditions, while GP nonlinearity does not guarantee such uniqueness. The work is validated through numerical experiments and extended to general computational retrieval frameworks, highlighting practical considerations such as noise robustness and media design for enhanced information content.

Abstract

This work studies phase retrieval for wave fields, aiming to recover the phase of an incoming wave from multi-plane intensity measurements behind different types of linear and nonlinear media. We show that unique phase retrieval can be achieved by utilizing intensity data produced by multiple media. This uniqueness does not require prescribed boundary conditions for the phase in the incidence plane, in contrast to existing phase retrieval methods based on the transport of intensity equation. Moreover, the uniqueness proofs lead to explicit phase reconstruction algorithms. Numerical simulations are presented to validate the theory.

Phase retrieval via media diversity

TL;DR

The paper addresses phase retrieval for wave fields by exploiting media diversity—linear and nonlinear media—to encode phase information into intensity measurements. It proves that phase can be uniquely recovered (up to a global constant in some cases) using data from multiple media, and derives explicit reconstruction algorithms based on transport-of-intensity ideas. Importantly, quadratic nonlinearity can remove the global phase ambiguity, enabling full phase recovery under suitable diversity conditions, while GP nonlinearity does not guarantee such uniqueness. The work is validated through numerical experiments and extended to general computational retrieval frameworks, highlighting practical considerations such as noise robustness and media design for enhanced information content.

Abstract

This work studies phase retrieval for wave fields, aiming to recover the phase of an incoming wave from multi-plane intensity measurements behind different types of linear and nonlinear media. We show that unique phase retrieval can be achieved by utilizing intensity data produced by multiple media. This uniqueness does not require prescribed boundary conditions for the phase in the incidence plane, in contrast to existing phase retrieval methods based on the transport of intensity equation. Moreover, the uniqueness proofs lead to explicit phase reconstruction algorithms. Numerical simulations are presented to validate the theory.

Paper Structure

This paper contains 22 sections, 12 theorems, 111 equations, 7 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Recovering phase gradient with linear media diversity
  3. Recovering phase with nonlinear media diversity
  4. Analytic phase reconstruction
  5. Reconstruction from diversity
  6. Reconstruction from diversity
  7. Numerical experiments
  8. Experiment I.
  9. Experiment II.
  10. Computational retrieval in general settings
  11. Single medium data
  12. Multi-media data
  13. Impact of measurement noise
  14. Concluding remarks
  15. Examples of non-uniqueness without media diversity
  16. ...and 7 more sections

Key Result

Theorem 2.1

Let $f_1, f_2\in \mathcal{F}$ be such that $\Lambda^{[0, \varepsilon]}_{f_1}(\kappa,0) = \Lambda^{[0, \varepsilon]}_{f_2}(\kappa,0)$ for all $\kappa\in \mathcal{P}$ for the linear Schrödinger model EQ:LS. Assume that there exists at least $N_m\ge d+2$ different $\kappa_1,\dots, \kappa_{N_m}\in \math is full-rank for each $\mathbf x\in\Omega$, and for each $\kappa_m$, $\left.\dfrac{\partial{|u_1|}}

Figures (7)

  • Figure 1: Phase gradient reconstruction in Experiment I by \ref{['ALGO:Phase Grad Rec']}.
  • Figure 2: Phase reconstruction in Experiment II by \ref{['ALGO:Phase Rec']}. Shown from left to right are (a) true phase, (b) phase reconstructed with data from $\beta_1\sim \beta_6$, and (c) phase reconstructed with data from $\beta_1\sim \beta_7$.
  • Figure 3: Phase reconstructions with multi-plane (top row) and single-plane (bottom row) data from a single medium. Left to right: (a) true phase, (b, e) phase reconstructed with the LSE model \ref{['EQ:LS']}, (c, f) phase reconstructed with the GP model \ref{['EQ:GP']}, and (d, g) phase reconstructed with the QNLS model \ref{['EQ:QNLS']}.
  • Figure 4: Phase reconstructions for the GP model \ref{['EQ:GP']} (top) and QNLS model \ref{['EQ:QNLS']} (bottom) using data from two different media $(\kappa_1,\beta_1)$ and $(\kappa_2,\beta_2)$ in \ref{['EQ:Kappa-Beta']}. Shown are reconstructions with multi-plane (left) and single-plane (right) data.
  • Figure 5: Same as \ref{['FIG: 2 beta kappa']} but with data from three media $(\kappa_1,\beta_1)$, $(\kappa_2,\beta_2)$, and $(\kappa_3,\beta_3)$ in \ref{['EQ:Kappa-Beta']}.
  • ...and 2 more figures

Theorems & Definitions (24)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['THM:k Unique']}
  • Corollary 2.4
  • proof
  • Remark 2.5
  • Theorem 3.1
  • ...and 14 more