image: Results for HeLa cells. (a) Optical path for the bright-field imaging system with a wide-field microscope. (b) Through-focus intensity distributions I(r⊥, z₀ − Δz), I(r⊥, z₀), and I(r⊥, z₀ + Δz). (c) Phase distributions reconstructed from the three intensity images. (d) Reconstructed spatial distribution of the refractive-index fluctuation Δn and (e) the attenuation coefficient μ. (f) Correlation pattern between Δn(r⊥, z₀) and μ(r⊥, z₀). The black solid line indicates the 3σ confidence level, and the green area shows the physical parameter boundaries (validity region) given by the validity-bound equation in the main text. In this sample, 65,055 events (24.82%) fall outside the green region. (g) Attenuation asymmetry Aκ as a function of optical depth log₁₀|μΔz|. The blue solid line and band denote the statistical mean and the root mean squared (RMS) value of Aκ, respectively, and the dashed line indicates Aκ = 0.
Credit: Masaki Watabe and Joe Sakamoto
Quantitative phase imaging extracts phase information from through-focus intensity images without interferometry, with the transport of intensity equation (TIE) as its mathematical foundation. The classical TIE, however, assumes purely refractive objects—an assumption that breaks down for real biological samples, where refraction and attenuation are intertwined. A research team led by Dr. Masaki Watabe (Exploratory Research Center on Life and Living Systems (ExCELLS) / (National Institute for Basic Biology (NIBB)) and Dr. Joe Sakamoto (Exploratory Research Center on Life and Living Systems (ExCELLS) / National Institute for Physiological Sciences (NIPS)) started from the paraxial wave equation with a complex optical potential and derived a coupled transport model (the coupled TIE-TPE framework) consisting of two transport equations—a generalized TIE for intensity evolution and a transport of phase equation (TPE) for phase evolution. By decomposing the refractive index into a uniform mean field and a local fluctuation field, the resulting non-divergent system simultaneously reconstructs refractive-index fluctuations (Δn) and attenuation coefficients (μ) without linearization or weak-absorption approximations.
The model also yields explicit validity bounds—derived from the paraxial condition combined with photon-counting statistics and the diffraction limit—that specify in advance the parameter range over which reconstruction is guaranteed to be consistent. Experimental validation was carried out across three systems with markedly different optical properties: precision-fabricated microlens arrays (numerical aperture NA = 0.45), cultured HeLa cells (NA = 1.20), and HeLa cell membranes imaged with an oil-immersion objective (NA = 1.49). In all cases, the reconstructed Δn and μ distributions fell within the predicted validity region, and subcellular structures including nuclei and organelles were clearly resolved through refractive-index contrast even in the transparent-limit regime where attenuation signals approach detection thresholds.
The authors further introduced an attenuation asymmetry parameter Aκ—quantifying the imbalance between forward and backward propagation—and measured it as a function of optical depth. Aκ was found to be statistically consistent with zero across three orders of magnitude in optical depth, with systematic errors from high-numerical-aperture optics remaining well below statistical uncertainties. This provides the first quantitative experimental confirmation that optical reciprocity is preserved in heterogeneous biological media at the wavelength scale.
Journal
Physical Review A
Method of Research
Computational simulation/modeling
Subject of Research
Cells
Article Title
Coupled amplitude-phase transport in heterogeneous optical media
Article Publication Date
18-May-2026