GSCP-Net: Gram-Structured Coded-Probe Recovery of Complex Optical Fields Through Atmospheric Turbulence

Recovering complex optical fields through atmospheric turbulence is a challenging inverse problem. Our key idea is to cast the complex optical field recovery as parameter estimation, rather than to learn an end-to-end mapping from received to source fields. To make this estimation tractable, we compress the high-dimensional complex field into a low-dimensional latent representation with a variational autoencoder. Inspired by unitary space–time modulation in communications, we exploit the unitary structure of lossless turbulent propagation and propose a latent-variable estimation method that combines multiple unitary coded probes ${\mathbf{Q}}_k$ with Gram geometry. Under lossless reception, the Gram geometry is independent of the instantaneous turbulence channel, whereas a finite receiver aperture truncates the field. This truncation breaks the ideal invariance, and the joint effect of propagation and truncation reduces to an effective metric $\mathbf{R}$. Since $\mathbf{R}$ can be modeled explicitly, we keep the autoencoder fixed and train only ${\mathbf{Q}}_k$. Based on this formulation, we propose the Gram-Structured Coded-Probe Network (GSCP-Net) and evaluate it on a 2-km distributed-turbulence channel spanning 42 turbulence–aperture conditions. Compared with end-to-end baselines, GSCP-Net is substantially more accurate under strong turbulence and small apertures, while using orders of magnitude fewer channel-dependent parameters and only a fraction of their training time. Even under $C_n^2=3\times 10^{-13}{\mathrm{m}}^{-2/3}$ with a 140-mm receiver aperture, GSCP-Net reaches a recovery fidelity of 0.660 using 15 probes. Without retraining or channel tracking, the learned probes and fixed receiver keep the fidelity nearly unchanged over 20 coherence times of evolving turbulence. These results establish Gram-structured coded probing as a lightweight, physics-guided approach to robust complex-field recovery across a broad range of turbulence strengths and receiver apertures. Code is available at https://anonymous.4open.science/r/GSCP-Net/.