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1. Snapshot phase-shifting diffraction phase microscope

   https://doi.org/10.1364/OL.394033  

   Tian, Xiaobo ; Liang, Rongguang ( June 2020 , Optics Letters)

   We propose a novel and simple snapshot phase-shifting diffraction phase microscope with a polarization grating and spatial phase-shifting technology. Polarization grating separates the incident beam into left and right circular polarization beams, one of which is used as the reference beam after passing through a pinhole. Four phase-shifted interferograms can be captured simultaneously from the polarization camera to reconstruct the high spatial resolution phase map. The principle is presented in this Letter, and the performance of the proposed system is demonstrated experimentally. Due to the near-common-path configuration and snapshot feature, the proposed system provides a feasible way for real-time quantitative phase measurement with minimal sensitivity to vibration and thermal disturbance.

    

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2. General phase regularized reconstruction using phase cycling

   https://doi.org/10.1002/mrm.27011  

   Ong, Frank ; Cheng, Joseph Y. ; Lustig, Michael ( November 2017 , Magnetic Resonance in Medicine)

   To develop a general phase regularized image reconstruction method, with applications to partial Fourier imaging, water–fat imaging and flow imaging.

   The problem of enforcing phase constraints in reconstruction was studied under a regularized inverse problem framework. A general phase regularized reconstruction algorithm was proposed to enable various joint reconstruction of partial Fourier imaging, water–fat imaging and flow imaging, along with parallel imaging (PI) and compressed sensing (CS). Since phase regularized reconstruction is inherently non‐convex and sensitive to phase wraps in the initial solution, a reconstruction technique, named phase cycling, was proposed to render the overall algorithm invariant to phase wraps. The proposed method was applied to retrospectively under‐sampled in vivo datasets and compared with state of the art reconstruction methods.

   Phase cycling reconstructions showed reduction of artifacts compared to reconstructions without phase cycling and achieved similar performances as state of the art results in partial Fourier, water–fat and divergence‐free regularized flow reconstruction. Joint reconstruction of partial Fourier + water–fat imaging + PI + CS, and partial Fourier + divergence‐free regularized flow imaging + PI + CS were demonstrated.

   The proposed phase cycling reconstruction provides an alternative way to perform phase regularized reconstruction, without the need to perform phase unwrapping. It is robust to the choice of initial solutions and encourages the joint reconstruction of phase imaging applications. Magn Reson Med 80:112–125, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

    

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3. Phase-factor spectra of turbulent phase screens

   https://doi.org/10.1364/JOSAA.429928  

   Muschinski, Andreas ( August 2021 , Journal of the Optical Society of America A)

   The optical phase$\mathrm{\varphi <\#comment/>}$is a key quantity in the physics of light propagating through a turbulent medium. In certain respects, however, the statistics of the phasefactor,$\mathrm{\psi <\#comment/>}=\exp <\#comment/>(i\mathrm{\varphi <\#comment/>})$, are more relevant than the statistics of the phase itself. Here, we present a theoretical analysis of the 2D phase-factor spectrum$F_{\mathrm{\psi <\#comment/>}}(\mathit{\kappa <\#comment/>})$of a random phase screen. We apply the theory to four types of phase screens, each characterized by a power-law phase structure function,$D_{\mathrm{\varphi <\#comment/>}}(r)=(r/r_c{)}^{\mathrm{\gamma <\#comment/>}}$(where$r_c$is the phase coherence length defined by$D_{\mathrm{\varphi <\#comment/>}}(r_c)=1{\mathrm{r}\mathrm{a}\mathrm{d}}^2$), and a probability density function$p_{\mathrm{\alpha <\#comment/>}}(\mathrm{\alpha <\#comment/>})$of the phase increments for a given spatial lag. We analyze phase screens with turbulent ($\mathrm{\gamma <\#comment/>}=5/3$) and quadratic ($\mathrm{\gamma <\#comment/>}=2$) phase structure functions and with normally distributed (i.e., Gaussian) versus Laplacian phase increments. We find that there is a pronounced bump in each of the four phase-factor spectra$F_{\mathrm{\psi <\#comment/>}}(\mathrm{\kappa <\#comment/>})$. The precise location and shape of the bump are different for the four phase-screen types, but in each case it occurs at$\mathrm{\kappa <\#comment/>}\sim <\#comment/>1/r_c$. The bump is unrelated to the well-known “Hill bump” and is not caused by diffraction effects. It is solely a characteristic of the refractive-index statistics represented by the respective phase screen. We show that the second-order$\mathrm{\psi <\#comment/>}$statistics (covariance function, structure function, and spectrum) characterize a random phase screen more completely than the second-order$\mathrm{\varphi <\#comment/>}$counterparts.

    

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4. Frame Phase-Retrievability and Exact Phase-Retrievable Frames

   https://doi.org/10.1007/s00041-019-09696-8  

   Han, Deguang ; Juste, Ted ; Li, Youfa ; Sun, Wenchang ( December 2019 , Journal of Fourier Analysis and Applications)
5. Low-coherence snapshot phase-shifting diffraction phase microscope

   https://doi.org/10.1016/j.optlaseng.2023.107618  

   Sun, Yuanyuan ; Wang, Yihan ; Wang, Daodang ; Pau, Stanley ; Liang, Rongguang ( August 2023 , Optics and Lasers in Engineering)

   We proposed a Wollaston-prism-based snapshot phase-shifting diffraction phase microscope (WP-SPDPM) for low-coherence snapshot quantitative phase imaging and videography. Wollaston prism separates two orthogonally linearly polarized beams with high degrees of polarization at a sufficiently small separation angle; one of the beams passing through a pinhole serves as the reference beam. Four phase-shifted interferograms are simultaneously acquired with a polarization camera to accurately retrieve a high spatial resolution phase map. The system is nearly common-path in configuration and can achieve a large slope range and high accuracy. In addition to the ability to resist environmental noise, the WP-SPDPM is suitable for phase measurement using low-coherence light. The accuracy and large measurable slope range of the proposed system is validated and compared experimentally with a commercial profilometer. We believe WP-SPDPM is a powerful tool for the accurate and robust quantitative phase measurement and has a significant potential of the real-time phase imaging. 

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