Single-shot spectral-encoded waveform reconstruction through probabilistic inversion

Abstract

Spectral encoding enables single-shot measurements of ultrafast transients by mapping temporal information onto the spectrum of a chirped probe. This encoding allows dynamics to be recorded that are beyond the response limits of conventional electronic detectors. However, because the measurements record only spectral intensity, the phase of encoded signals is lost, and dispersion in the detection process introduces waveform distortions that complicate reconstruction and quantitative interpretation of spectra. In single-shot terahertz time-domain spectroscopy (THz-TDS), these distortions manifest as a tradeoff between temporal resolution and the measurement window of signals and can produce spectral null frequencies that limit the recoverable THz bandwidth. To address this challenge, a Bayesian inversion framework is developed to recover the underlying waveform from the squared spectral observable by inferring the THz field, the modulation coefficient, and a low-dimensional empirical parameterization of the probe spectrum jointly, while a Gaussian process prior regularizes the waveform. The framework is validated using single-shot THz-TDS experiments spanning two probe spectral profiles and three chirp conditions with $α$ ranging from 14.5 to 40 ps$^{-2}$. Across all cases, the inversion reconstructs both the time-domain waveform and spectral null frequency structure within the credible interval of a delay-line reference measurement. These results establish a pathway to eliminate penalties that are associated with the detection process in spectral encoding methods without adding additional optics or alignment complexity.