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On Phase Unwrapping via Digital Wavefront Sensors

Simon Hubmer, Victoria Laidlaw, Ronny Ramlau, Ekaterina Sherina, Bernadett Stadler

TL;DR

The paper addresses the 2D phase unwrapping problem by reframing wrapped phase data as optical wavefront aberrations and propagating them through digital wavefront sensors. By exploiting the fact that sensor measurements depend on $e^{i\phi}$ and cannot distinguish wrapped from unwrapped phases, the authors apply standard WFS reconstructors to obtain smooth, unwrapped phase estimates $\phi_r$. They develop a general digital-WFS framework with concrete SH-WFS and Fourier-type WFS implementations, leveraging CuReD, PCuReD, and NOPE reconstructions. Numerical experiments in free-space optical communications demonstrate that the proposed digital-WFS unwrapping approaches achieve competitive or superior accuracy compared to state-of-the-art methods, with favorable computational efficiency and applicability to AO and FSOC contexts.

Abstract

In this paper, we derive a new class of methods for the classic 2D phase unwrapping problem of recovering a phase function from its wrapped form. For this, we consider the wrapped phase as a wavefront aberration in an optical system, and use reconstruction methods for (digital) wavefront sensors for its recovery. The key idea is that mathematically, common wavefront sensors are insensitive to whether an incoming wavefront is wrapped or not. However, typical reconstructors for these sensors are optimized to compute smooth wavefronts. Thus, digitally "propagating" a wrapped phase through such a sensor and then applying one of these reconstructors results in a smooth unwrapped phase. First, we show how this principle can be applied to derive phase unwrapping algorithms based on digital Shack-Hartmann and Fourier-type wavefront sensors. Then, we numerically test our approach on an unwrapping problem appearing in a free-space optical communications project currently under development, and compare the results to those obtained with other state-of-the-art algorithms.

On Phase Unwrapping via Digital Wavefront Sensors

TL;DR

The paper addresses the 2D phase unwrapping problem by reframing wrapped phase data as optical wavefront aberrations and propagating them through digital wavefront sensors. By exploiting the fact that sensor measurements depend on and cannot distinguish wrapped from unwrapped phases, the authors apply standard WFS reconstructors to obtain smooth, unwrapped phase estimates . They develop a general digital-WFS framework with concrete SH-WFS and Fourier-type WFS implementations, leveraging CuReD, PCuReD, and NOPE reconstructions. Numerical experiments in free-space optical communications demonstrate that the proposed digital-WFS unwrapping approaches achieve competitive or superior accuracy compared to state-of-the-art methods, with favorable computational efficiency and applicability to AO and FSOC contexts.

Abstract

In this paper, we derive a new class of methods for the classic 2D phase unwrapping problem of recovering a phase function from its wrapped form. For this, we consider the wrapped phase as a wavefront aberration in an optical system, and use reconstruction methods for (digital) wavefront sensors for its recovery. The key idea is that mathematically, common wavefront sensors are insensitive to whether an incoming wavefront is wrapped or not. However, typical reconstructors for these sensors are optimized to compute smooth wavefronts. Thus, digitally "propagating" a wrapped phase through such a sensor and then applying one of these reconstructors results in a smooth unwrapped phase. First, we show how this principle can be applied to derive phase unwrapping algorithms based on digital Shack-Hartmann and Fourier-type wavefront sensors. Then, we numerically test our approach on an unwrapping problem appearing in a free-space optical communications project currently under development, and compare the results to those obtained with other state-of-the-art algorithms.
Paper Structure (14 sections, 1 theorem, 25 equations, 13 figures, 2 tables)

This paper contains 14 sections, 1 theorem, 25 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Physical background on wavefront sensors
  3. The Shack-Hartmann wavefront sensor
  4. Fourier-type wavefront sensors
  5. Digital wavefront sensors for phase unwrapping
  6. Fourier transforms and Fourier optics
  7. Key ideas and generic unwrapping algorithm
  8. The digital Shack-Hartmann wavefront sensor
  9. Digital Fourier-type wavefront sensors
  10. Numerical experiments
  11. Comparison with existing methods
  12. Numerical results
  13. Conclusion
  14. Support

Key Result

Proposition 3.1

Let $E^m_{j,k}$ be defined as in def_Emjk, let $E^m$ satisfy model_aperture and assume that $\mathcal{F}(E) \equiv 1$. Furthermore, assume that on each subdomain $\Omega_{j,k}$ the wavefront aberration $\phi$ can be reasonably well approximated by a linear function, i.e., there holds Then for all indices $j,k$ and all $\boldsymbol{x} \in {\mathbb{R}^2}$ there holds

Figures (13)

  • Figure 1.1: Example of a non-wrapped phase $\phi$ and its wrapped form $\phi_{w}$.
  • Figure 2.1: Schematic depiction of the physical principle of a SH-WFS. Left and center: Cross cuts parallel to the optical axis for unperturbed and perturbed wavefronts. Right: Front view onto the sensor array and blurred/shifted images of the distant point source. See Section \ref{['subsect_SHWFS']} for details. Image taken from Hubmer_Sherina_Ramlau_Pircher_Leitgeb_2023.
  • Figure 2.2: Schematic depiction of a Fourier-type WFS measurement system; cf. HuNeuSha_2023.
  • Figure 2.3: Optical transfer function (top) and corresponding intensity image (bottom) of several Fourier-type WFSs with the phase $\phi$ from Figure \ref{['fig_example_wrapping']} as the incoming wavefront aberration. The first two columns refer to PWFSs with different apex angles.
  • Figure 3.1: Schematic depiction of two-lens imaging system corresponding to \ref{['model_aperture']}.
  • ...and 8 more figures

Theorems & Definitions (5)

  • Remark
  • Proposition 3.1
  • proof
  • Remark
  • Remark