Nuclear Ptychoscopy: A Ptychographic Framework for Nuclear Spectroscopy

TL;DR

Nuclear Ptychoscopy presents a ptychography-inspired framework to recover the full complex nuclear response from 2D time-energy spectra by treating the Doppler-tuned analyzer as a scanning probe. The approach encompasses three reconstruction schemes—known analyzer, blind, and partial-prior—implemented through three algorithm families (geometry-based, feasible, and constrained optimization) with variants like NPRS, RAAR, and PnP methods. Experimental data and simulations demonstrate improved reconstruction accuracy and robustness, enabling phase-sensitive nuclear spectroscopy and advancing metrology, coherent control, and quantum applications in the X-ray-nuclear regime. The framework integrates priors, time-spectrum constraints, and data-driven denoisers to address non-convexity and ill-posedness, offering a versatile toolkit for high-precision nuclear spectroscopy across X-ray platforms.

Abstract

Accessing both amplitude and phase of nuclear response functions is central to fully characterizing light-matter interactions in the X-ray-nuclear regime. Recent work has demonstrated phase retrieval in two-dimensional time- and energy-resolved spectra, establishing the feasibility of phase-sensitive nuclear spectroscopy. Here, we introduce Nuclear Ptychoscopy, a ptychographic framework that adapts algorithms from coherent diffractive imaging to nuclear spectroscopy, enabling reconstruction of the complex response function by exploiting redundancy in two-dimensional spectra. We develop three complementary reconstruction schemes tailored to distinct experimental scenarios: reconstruction with a known analyzer response, blind reconstruction, and reconstruction incorporating partial prior information. In parallel, we develop geometric analysis techniques that elucidate algorithmic behavior and contribute new tools to ptychography. The framework is validated through experimental data and simulations, demonstrating its versatility across diverse nuclear spectroscopy scenarios and bridging nuclear spectroscopy with ptychography. Beyond advancing quantitative nuclear spectroscopy, our framework opens new opportunities for metrology, coherent control, and quantum applications in the X-ray-nuclear regime.

Figures (6)

• Figure 1: Experimental setup and its analogy to ptychography. Synchrotron radiation first passes through the monochromators, then through an analyzer with transmission function $T(\Delta-\Delta_D)$ mounted on a Mössbauer drive, and is subsequently scattered by a $^{57}$Fe sample characterized by the response function $R(\Delta)$. Single photons are then detected by avalanche photodiode detectors (APDs) as a function of time and Doppler detuning. Each analyzer detuning $\Delta_D^k$ probes a local region of $R(\Delta)$, and the sequence of overlapping measurements across multiple detunings provides the redundancy needed to reconstruct both amplitude and phase, in analogy to ptychographic imaging.
• Figure 2: Comparative performance of optimization algorithms in Nuclear Ptychoscopy reconstruction. (a) 2D spectrum: ground truth (labeled "Real") and reconstructions from the NPRS, NCG, LM, and L-BFGS methods. (b) Recovered intensity and (c) its corresponding phase. Insets in (b) and (c) show magnified views highlighting key structural details. (d) Measurement error (defined in Eq. \ref{['mtt21']}) and (e) relative error (defined in Eq. \ref{['mtt22']}), plotted against iteration count for each algorithm. (f) and (g) respectively record the measurement error and relative error of the four algorithms under three different loss functions: the Poisson model, the intensity-based Gaussian model, and the amplitude-based Gaussian model.
• Figure 3: Comparative performance of geometry-based optimization methods on experimental data. (a) Measured 2D spectrum alongside its theoretical prediction and reconstructions from the NPRS, NCG, LM, and L-BFGS algorithms. (b) Measurement error and (c) relative error, as defined in Eqs.\ref{['mtt21']} and \ref{['mtt22']}, plotted as functions of iteration number. (d) Reconstructed intensity and phase of the target. (e) Time spectra obtained from Fourier transforms of the reconstructed amplitude and phase for each method, compared with independently measured experimental data, demonstrating reconstruction accuracy. The corresponding measurement error is shown in each plot. Here, the spectrum (2D spectrum, intensity or phase spectrum) labeled “Theory” represents the theoretically calculated spectrum obtained using parameters extracted from fitting the independently measured time spectrum. The 2D spectrum labeled “Measured” corresponds to the experimental data acquired by the APDs.
• Figure 4: Comprehensive analysis of feasible methods. (a) Intensity reconstructed via the L-BFGS method. (b) Loss landscape of the optimization problem, which is rendered by the http://paraview.org. (c) Intensity reconstructed by the RAAR method, initialized at the L-BFGS stagnation point. (d) Iteration trajectories of L-BFGS and RAAR, with corresponding measurement errors in (e), where the curve consists of the results generated in three phases, namely the L-BFGS method, the RAAR method when $\alpha=1$, and the RAAR method when $\alpha=0.95$. (f) Recovered 2D spectrum using AP, DR and RAAR. (g) Mmeasurement error of the 2D spectrum. (h) Recovered intensity and phase of $\mathbf{R}$. (i) Relative error of $\mathbf{R}$ during iteration. (j) Recovered intensity and phase of $\mathbf{T}$. (k) Relative error of $\mathbf{T}$ during iteration.
• Figure 5: Numerical and experimental evaluation of constrained optimization methods with a smooth target prior, showing reconstruction performance under varying noise conditions. (a) Simulated and experimental input data. Top row: the true 2D spectrum and simulated measurements corrupted by noise at SNRs of 40 dB and 20 dB. Bottom row: the experimentally measured 2D spectrum and its theoretical estimation. (b) Recovered intensity, (d) recovered phase, and (c) reconstructed 2D spectrum obtained using NPRS, PnP-Prox, TV-ADMM, and TV-Prox methods for simulated data under varying noise levels and for experimental data. (e) Quantitative comparison of different methods based on recovered logarithmic relative error (LRE) and number of iterations using histograms. The heights of the histograms represent the mean values, and the corresponding error bars stand for the variance. (f) Comparison of time spectrum calculated from the recovered amplitude and phase for each method, alongside the theoretical fit and experimental data, and the measurement errors are also calculated.