Transfer-function engineering in non-interferometric quantitative phase microscopy: principles, methods, and applications (<italic>invited</italic>)

Significance Quantitative phase imaging (QPI) is a label-free and non-destructive optical microscopy modality that has recently demonstrated substantial potential across the life sciences, medical diagnostics, and biophysics. In contrast to conventional microscopy methods that rely on fluorescence or staining contrast, QPI quantitatively measures the phase delay imposed by a specimen on the incident optical wavefront, enabling high-contrast, quantitative imaging of transparent or weakly absorbing biological samples. Through inversion and analysis of phase information, QPI directly reconstructs key biophysical parameters, including specimen thickness, spatial refractive-index variations, and dry-mass density, thereby providing an important tool for the quantitative investigation of live-cell growth and migration dynamics, metabolic characteristics of the tumor microenvironment, and complex three-dimensional neuronal architectures. Progress The advent of the laser advanced coherent optical imaging and led to interferometric QPI approaches represented by Michelson interferometry, Mach–Zehnder interferometry, and digital holographic microscopy. By introducing a reference beam, these techniques encode phase information into interference fringes, enabling highly sensitive phase measurements. In practice, however, interferometric QPI requires stringent optical-path stability and is susceptible to speckle noise and environmental vibrations, while also relying on highly coherent light sources. Consequently, its complex optical configuration limits performance in high-throughput imaging, wide-field observation, and deployment in realistic clinical environments. To overcome these limitations, non-interferometric “intensity-to-phase” inversion techniques have emerged as an important direction in QPI. These methods employ controlled illumination and propagation modulation to deliberately break the intrinsic symmetries of the imaging system, thereby encoding specimen phase information into measured intensity distributions in an analytically tractable manner. By establishing a mapping between intensity and phase in the form of a transfer-function model, direct inversion of phase or phase gradients from intensity measurements becomes possible. Compared with interferometric approaches, non-interferometric QPI eliminates the need for a reference arm, fringe demodulation, and phase unwrapping, while remaining compatible with conventional bright-field microscope hardware and substantially improving system stability, robustness, and imaging efficiency. In the non-interferometric QPI, transfer-function engineering has emerged as a unifying theoretical framework for systematically characterizing physical modulation mechanisms and computational reconstruction principles. It clarifies the additional engineering degrees of freedom offered by partially coherent illumination for phase retrieval, thereby establishing the theoretical basis for precise phase reconstruction. Building upon this foundation, the forward imaging model for “intensity-to-phase” inversion and its associated approximation conditions are formulated, followed by the derivation of both the phase transfer function (PTF) and the phase-gradient transfer function (PGTF). Thus, diverse non-interferometric QPI approaches can be uniformly interpreted as specific implementations and distinct modulation strategies of the PTF. Three representative approaches are discussed in detail: the transport-of-intensity equation (TIE), Fourier ptychographic microscopy (FPM), and differential phase contrast (DPC). TIE relates axial intensity variations to phase through a differential formulation, FPM synthesizes high-numerical-aperture spatial spectra via multi-angle illumination to achieve super-resolution phase recovery, and DPC constructs a phase-gradient transfer function in the frequency domain using asymmetric illumination. Through the synergistic optimization of illumination and aperture, these methods have achieved systematic improvements in illumination strategies, transfer-function characteristics, reconstruction algorithms, and overall imaging performance, thereby significantly enhancing resolution, efficiency, and throughput. Building on this foundation, non-interferometric QPI has been further extended to three-dimensional diffraction tomography. By employing multi-angle illumination or multi-plane acquisition, the two-dimensional PTF is generalized to the three-dimensional spatial-frequency domain, enabling the quantitative reconstruction of the refractive-index distribution. This offers new technical pathways for the three-dimensional structural analysis of living cells, tissue-scale microstructural characterization, and imaging within complex scattering media. Integrating representative applications in cell imaging, histopathology, and neuroscience, non-interferometric QPI demonstrates distinct advantages in high-throughput QPI and clinically translatable settings.Conclusions and Prospects Transfer-function engineering has established a unified theoretical framework for non-interferometric QPI. By actively tailoring the optical field, it enables deterministic encoding and accurate reconstruction of phase information, effectively alleviating the stringent environmental stability requirements and speckle-noise susceptibility inherent to conventional interferometric approaches. While maintaining instrumental simplicity, it realizes a fundamental transition from qualitative morphological observation to the quantitative measurement of intrinsic physical parameters. Future research will focus on developing physics-driven neural network models and ultrafast encoding strategies based on sparse sampling to address high-precision inverse problems in strongly scattering media and to challenge the physical limits imposed by the space–time bandwidth product. These technological advances are expected to accelerate the standardization and clinical translation of QPI in digital pathology, particularly for label-free virtual histological staining and intraoperative real-time diagnosis, thereby providing a critical enabling technology for next-generation precision medicine.