Optical turbulence retrieval of heterogeneous media

TL;DR

The paper addresses the limitation of the transport of intensity equation (TIE) for absorbing and scattering media by deriving a non-divergent, coupled TIE–TPE framework from the paraxial wave equation with a complex optical potential. It decomposes the refractive-index field as $n(\mathbf{r}_\perp,z)=n_0[1+\Delta n(\mathbf{r}_\perp,z)]$ to enable simultaneous reconstruction of $\Delta n$ and attenuation $\mu$ from three through-focus intensities via a four-step pipeline, culminating in the conversion of intermediate quantities to physical maps. The study establishes explicit reconstruction validity bounds that delineate the measurable parameter space and demonstrates robust experimental validation with microlens arrays and living HeLa cells, including the first experimental verification of attenuation symmetry $\mathcal{A}_\kappa$. By revealing attenuation symmetry and providing a rigorous NA-correction framework, the work enables quantitative phase imaging in thick, heterogeneous samples while preserving reciprocity, which has significant implications for imaging deep or complex biological tissues.

Abstract

The transport of intensity equation (TIE) has revolutionized phase retrieval in optical microscopy, yet its application to complex media with absorption/scattering remains challenging. Here, we present a coupled TIE-TPE (transport of phase equation) framework derived directly from the paraxial wave equation with complex optical potential. By decomposing the refractive index field into a spatially uniform mean field and local fluctuation field, our approach enables simultaneous reconstruction of refractive-index fluctuations and attenuation coefficients without linearization assumptions. We establish reconstruction validity bounds that define the measurable parameter region where reconstruction remains physically consistent. Experimental demonstration with microlens arrays and HeLa cells shows robust recovery of optical properties even in the transparent-limit regime where attenuation signals approach detection thresholds. Furthermore, we provide the first experimental verification of attenuation symmetry -- a fundamental property of wave propagation that characterizes reciprocity in light-matter interactions.

Figures (12)

• Figure 1: Phase retrieval via TIE imaging typically involves a through-focus series of image intensities at three different z-positions, $I(z_0 - \Delta z)$, $I(z_0)$ and $I(z_0 + \Delta z)$. These images are then used to compute the axial intensity derivatives, which, when combined with the intensity distributions at the in-focus and half of the defocus z-positions, solves the TIE and reconstructs the phase distributions $\phi(z_0 - \Delta z/2)$, $\phi(z_0)$ and $\phi(z_0 + \Delta z/2)$.
• Figure 2: The computational inputs of HeLa cells: (a) the image intensities ($I_{-}$, $I_{0}$ and $I_{+}$), and (b) the phases ($\phi_{-}$, $\phi_0$ and $\phi_{+}$) reconstructed by using the GMGM.
• Figure 3: Results for HeLa cells: (a) Spatial distribution of refractive-index fluctuations $\Delta n(r_{\bot},z_0)$ and (b) attenuation coefficients $\mu(r_{\bot},z_0)$. The sign of the $\mu(r_{\bot},z_0)$ indicates forward and backward wave propagations along the z-axis. (c) Correlation patterns between $\Delta n(r_{\bot},z_0)$ and $\mu(r_{\bot},z_0)$. The black solid line is the $3\sigma$ confidence level. The green area represents the theoretically measurable region defined by Eq. (\ref{['eqn;limits']}), where reconstruction remains physically consistent despite operating in the transparent-limit regime. (d) Measured value of the asymmetry $\mathcal{A}_{\kappa}$ in bins of the optical depth $\log_{10}|\mu \Delta z|$. The blue line and bands are the statistical average and the root mean squared (RMS) value, respectively. Dashed line represents $\mathcal{A}_{\kappa} = 0$.
• Figure S1: Optical path for the bright-field imaging systems: (a) laser-scanning confocal microscope and (b) wide-field microscope.
• Figure S2: Parameter allowed regions in the $\Delta n(r_{\bot},z_0)$-$\mu(r_{\bot},z_0)$ spaces: (a) the MLA sample, (b) HeLa cells and (c) HeLa cell membrane. Red contours represent the statistically-allowed regions of the $1\sigma\ (68.3\%)$, $2\sigma\ (95\%)$ and $3\sigma\ (99.7\%)$ confidence intervals. Green area exhibits the physically-allowed region given by the Eq. (10) in the main text.