Towards a mobile quantitative phase imaging microscope with smartphone phase-detection sensors

Abstract: Quantitative phase imaging (QPI) enables visualization and quantitative extraction of the optical phase information of transparent samples. However, conventional QPI techniques typically rely on multi-frame acquisition or complex interferometric optics. In this work, we introduce Quad-Pixel Phase Gradient Imaging ($QP^{2}GI$), a single-shot quantitative phase imaging method based on a commercial quad-pixel phase detection autofocus (PDAF) sensor, where each microlens on the sensor covers a $2\times2$ pixel group. The phase gradients of the sample induce focal spot displacements under each microlens, which lead to intensity imbalances among the four pixels. By deriving the phase gradients of the sample from these imbalances, $QP^{2}GI$ reconstructs quantitative phase maps of the sample from a single exposure. We establish a light-propagation model to describe this process and evaluate its performance in a customized microscopic system. Experiments demonstrate that quantitative phase maps of microbeads and biological specimens can be reconstructed from a single acquisition, while low-coherence illumination improves robustness by suppressing coherence-related noise. These results reveal the potential of quad-pixel PDAF sensors as cost-effective platforms for single-frame QPI.

• The paper introduces a novel single-shot quantitative phase imaging method that leverages quad-pixel PDAF sensors from smartphones.
• It demonstrates robust phase gradient estimation and reconstruction using a light-propagation model and Tikhonov regularization.
• Experimental results show high-speed, cost-effective imaging of biological specimens, with sensitivity challenges addressed via sensor calibration.

Single-Shot Quantitative Phase Imaging with Smartphone Quad-Pixel PDAF Sensors

Introduction

This paper presents Quad-Pixel Phase Gradient Imaging (QP$^{2}$GI), a single-shot quantitative phase imaging (QPI) technique leveraging commercial quad-pixel phase-detection autofocus (PDAF) sensors, commonly found in smartphone cameras. The method exploits the dense microlens array architecture, where each microlens covers a $2\times2$ pixel group, to infer local phase gradients from intensity imbalances induced by sample-induced wavefront tilts. This approach circumvents the need for multi-frame acquisition or complex interferometric setups, offering a compact, cost-effective, and high-speed platform for QPI.

Theoretical Framework

QP$^{2}$GI is conceptually analogous to a Shack–Hartmann wavefront sensor (SHWS), but with a much denser microlens array and only four pixels per microlens. The local phase gradient is estimated by measuring the displacement of the focal spot within each $2\times2$ pixel group. The differential signals $\gamma_x$ and $\gamma_y$ are computed as normalized intensity differences between left/right and up/down pixel pairs, respectively:

$$\gamma_x = \frac{I_{\mathrm{left}} - I_{\mathrm{right}}}{I_{\mathrm{left}} + I_{\mathrm{right}}}, \quad \gamma_y = \frac{I_{\mathrm{up}} - I_{\mathrm{low}}}{I_{\mathrm{up}} + I_{\mathrm{low}}}$$

These signals are directly related to the local phase gradient, provided that system-specific background offsets (chief ray angle, CRA mismatch) are properly subtracted. The mapping from normalized differential signals to phase gradients is established via a light-propagation model based on the Huygens–Fresnel principle, incorporating microlens geometry, CRA, and coherence properties.

Phase reconstruction is performed by solving the Poisson equation in the frequency domain, using Tikhonov regularization to stabilize low-frequency components:

$$\phi(u,v) = \mathcal{F}^{-1}\left[ \frac{-j2\pi p\,{D}_u(p,q) - j2\pi q\,{D}_v(p,q)} {(2\pi p)^2 + (2\pi q)^2 + \varepsilon} \right]$$

where $D_u$ and $D_v$ are the Fourier transforms of the measured phase gradients.

Experimental Implementation

A custom microscope system was constructed using an inverted CCTV lens and a smartphone lens to achieve finite-conjugate imaging, with a measured system magnification $M=2.3$ and NA=0.23. The quad-pixel PDAF sensor (pixel pitch $1.008~\mu$m) was integrated at the image plane. Illumination coherence was tunable via an adjustable iris, and the coherence parameter $\sigma$ was defined as the ratio of illumination NA to objective NA.

CRA mismatch between the sensor (designed for wide-FOV photography) and the microscope optics introduces a position-dependent background in the differential signals. This was characterized experimentally and mitigated by average filtering or reference subtraction. The effective microlens focal length ($f_l=0.71~\mu$m) and shift were empirically determined by fitting measured differential signals to the theoretical model.

Results

Phase Imaging of Microbeads

Single-shot phase maps of $25~\mu$m polystyrene microbeads (refractive index contrast $\Delta n=0.03$) were reconstructed under varying illumination coherence. At $\sigma=1.16$, the reconstructed phase closely matched theoretical predictions, with strong linearity between normalized differential signals and actual phase gradients. As $\sigma$ decreased (higher coherence), noise and artifacts increased, degrading phase reconstruction quality. The system maintained robust phase detection across the sensor FOV, though sensitivity decreased with distance from the center due to CRA effects.

Biological Specimens

Phase maps of Coprinus mushroom slices revealed structural features of basidiospores, demonstrating applicability to complex biological samples. Reconstruction quality deteriorated in peripheral regions, highlighting the importance of CRA matching for uniform phase sensitivity.

Numerical Performance

• Maximum measurable phase gradient: $2.73~\mathrm{rad}/\mu$m at $\lambda=530$ nm, NA=0.23.
• Calibration factor: Experimental differential signals required scaling by 0.5 relative to simulation, attributed to focal spot size and microlens geometry deviations.
• Repeatability: Phase detection remained stable as long as illumination coherence length exceeded the microlens size ($\sim2~\mu$m).

Practical and Theoretical Implications

QP$^{2}$GI enables high-speed, single-shot QPI at the full frame rate of the sensor, suitable for dynamic biological processes (e.g., cell motility, intracellular transport). The architecture is compatible with multi-camera array microscopes (MCAM), which inherently feature large CRA and low magnification, facilitating integration with quad-pixel PDAF sensors. CRA matching can be further optimized via relay optics or sensor design modifications (minimal microlens shift), improving phase measurement accuracy and FOV uniformity.

The method is robust under partially coherent illumination, which suppresses coherent artifacts and maintains measurement repeatability. However, phase sensitivity and reconstruction fidelity are limited by microlens geometry, CRA mismatch, and focal spot size. Detailed sensor characterization and calibration are essential for quantitative accuracy.

Future Directions

• 3D QPI: Rapid acquisition of phase-gradient maps at multiple focal planes enables volumetric phase imaging via computational reconstruction.
• Sensor Design: Custom PDAF sensors with minimal microlens shift and optimized CRA for microscopy could enhance compatibility and measurement uniformity.
• Integration with MCAM: Large-scale, high-throughput QPI systems for biomedical and industrial applications.
• Advanced Modeling: FDTD simulations incorporating detailed microlens and pixel stack geometry for improved calibration and sensitivity analysis.

Conclusion

QP$^{2}$GI demonstrates the feasibility of single-shot, quantitative phase imaging using commercial quad-pixel PDAF sensors, with robust performance under partially coherent illumination and practical applicability to biological microscopy. The approach offers a scalable, cost-effective platform for dynamic phase imaging, with potential for further optimization through sensor and system co-design. Theoretical and experimental analyses confirm the method's repeatability and highlight key factors influencing sensitivity and accuracy, providing a foundation for future developments in mobile and high-speed QPI instrumentation.