Dynamic Coherent Diffractive Imaging Using Only a Support Constraint in the Complex Plane

Technical Summary: Dynamic Coherent Diffractive Imaging Using Only a Support Constraint in the Complex Plane

Problem Statement
Quantitative characterization of thin, transparent, and time-varying samples is a persistent challenge in metrology. Traditional interferometric methods require stable environments and multiple exposures, while standard Coherent Diffraction Imaging (CDI) typically assumes a stationary object. Extending CDI to dynamic samples usually necessitates complex priors, such as designed reference structures, explicit motion models, or learned data-driven constraints. This paper addresses the problem of reconstructing a time-varying phase object from a near-field diffraction movie using minimal priors, specifically avoiding the need for explicit phase unwrapping even when the accumulated phase far exceeds $2\pi$.

Methodology
The authors propose a framework based on a bounded temporal increment prior on the sample dynamics, operating under the thin-film approximation with spatially coherent illumination.

1. Multiplicative Inter-frame Parameterization:
   Instead of reconstructing each frame $\psi(x, y, t)$ independently, the field evolution is parameterized by multiplicative inter-frame update factors, $\Delta\psi(x, y, t)$. The field at time $t + \Delta t$ is expressed as $\psi(t + \Delta t) = \psi(t) \cdot \Delta\psi(t)$. This transforms the reconstruction into a feasibility problem for the sequence of factors $\Delta\psi$.

2. Circular-Sector Constraint (Object Domain):
   The core prior is a bound on the rate of change of the sample's optical path length (OPL). This physical bound translates to a constraint on the inter-frame phase increment, $|\Delta\phi| \leq \alpha/2$. Combined with a passivity condition (the sample cannot amplify the field, so $|\Delta\psi| \leq 1$), the admissible values for each pixel of $\Delta\psi$ are confined to a circular sector in the complex plane.

   • This constraint generalizes the classical non-negativity constraint (a degenerate sector on the real axis).
   • Crucially, because each increment is bounded within $(-\pi, \pi)$, the full phase trajectory is recovered by sequential integration of these small increments, automatically resolving phase wraps without explicit unwrapping algorithms.

3. Measurement-Domain Constraint:
   The second constraint enforces consistency with the recorded diffraction intensity movie. A measurement-domain projector replaces the computed amplitude of the propagated field with the measured amplitude while retaining the phase (Gerchberg–Saxton–Fienup magnitude projection).

4. Reconstruction Algorithm:
   The problem is solved as a feasibility problem using alternating projection methods with reflections (Relaxed-Reflect-Reflect and Relaxed Averaged Alternating Reflections) to handle the non-convexity of the measurement constraint.

   • Stage 1: Reconstructs the static probe field (initial condition) from the first frame.
   • Stage 2: Reconstructs the inter-frame factors $\Delta\psi$ in a forward pass (integrating from $t=0$) and a backward pass (integrating from $t=J$). The final estimate combines the forward and backward passes to balance directional error accumulation.

Key Contributions

• Minimalist Prior: The paper demonstrates that a single, physically motivated prior (bounded temporal increment) is sufficient to reconstruct dynamic phase objects without spatial regularizers (e.g., total variation, optical flow) or learned priors.
• Automatic Phase Unwrapping: By recovering the trajectory through the integration of bounded increments, the method inherently handles accumulated phases spanning many $2\pi$ cycles, eliminating the need for explicit phase unwrapping.
• Generalizability: The framework is applied without modification to both calibrated synthetic dynamics (SLM) and real-world, unknown photo-chemical processes.

Experimental Results
The approach was validated in three distinct scenarios:

1. Reaction-Diffusion Dynamics (SLM): A spatially expanding phase pattern was reconstructed. The recovered inter-frame increments and integrated phase trajectory showed good agreement with the ground truth, resolving the propagating front of the chemical species.
2. Pure-Growth Dynamics (SLM): A fixed spatial support with a phase growing monotonically to $10\pi$ was reconstructed. The method successfully recovered the phase values and spatial pattern, confirming the ability to handle large accumulated phases without unwrapping. Minor concentric ring artifacts were observed due to error accumulation over time.
3. In-situ Photo-polymer 3D Printing: The framework was applied to monitor a photo-polymerization process under patterned blue light. The phase trajectory, representing the spatiotemporal evolution of the refractive index, was recovered quantitatively. The results tracked the expansion of the polymerized region and the growth of phase magnitude, providing a direct observable for the printing process.

Significance and Claims
The authors claim that the separation of spatial coupling (handled by the Fresnel diffraction forward model) and temporal coupling (handled by the inter-frame parameterization and circular-sector constraint) allows for effective reconstruction with very few priors.

The paper posits that this approach offers a practical tool for photo-chemical system identification. The recovered phase trajectory serves as a quantitative observable of the spatiotemporal refractive-index evolution, which can be used to fit kinetic parameters of reaction-diffusion equations governing polymerization. The authors note that while error accumulation over long sequences is a limitation, the method provides a robust baseline for dynamic imaging that does not require complex training data or designed reference structures. The work suggests that bounded support constraints in the complex plane provide sufficient temporal regularization on their own for dynamic CDI.