Photon-Counting Underwater Optical Links with Temporal Pseudo-Random Noise Signaling and Spatio-Temporal Dimensional Signaling: A Regime-Aware Rate–Range Study

We study underwater optical communication under photon-counting (Poisson) detection with realistic attenuation, background radiance, directionality, and pointing uncertainty. Information is embedded in (i) a temporal dictionary of pseudo-random noise (PRN) intensity sequences and (ii) an optional spatio-temporal extension, denoted SIM–TS (spatial-index modulation with temporal signaling), that combines temporal coding with spatial indexing across multiple transmit/receive apertures. For a fixed optical energy-per-symbol (photon budget), these structured waveforms increase observation dimensionality and improve maximum-likelihood separability under Poisson statistics. We present a layered modeling framework, derive the corresponding Poisson detection metrics, and use Monte Carlo evaluation to extract maximum range at a target symbol error rate. The results show that dimensional signaling provides a modest but repeatable gain in clear-water photon-limited regimes: at 100 kbps, SIM–TS increases the clear-water range from 593.8 m to 617.2 m at 450 nm (3.95%) and from 457.8 m to 473.4 m at 420 nm (3.41%) under fixed total power. In coastal water the gain falls below 1%, while in the 1 Gbps benchmark SIM–TS under fixed total power remains within about 2% of on–off keying (OOK) and the larger improvement under power combining is attributable primarily to increased photon budget. These rate–range trade-offs clarify when dimensional signaling yields practical gains and when attenuation, background, and misalignment dominate the link budget.