Spatial optical solitons in nonlocal nonlinear media: from the Snyder-Mitchell model to long-range interactions in liquid-crystal and photorefractive media

Spatial optical solitons in nonlocal nonlinear media are important objects for studying the formation, stable propagation, and interaction of localized waves in nonlinear optics. Different from local nonlinear media, nonlocal response makes the refractive-index change of the medium depend on the optical-field distribution within a certain spatial range. As a result, the formation conditions, stability, and propagation dynamics of solitons can be greatly changed. This review focuses on the theoretical basis and research progress of nonlocal spatial optical solitons. It first introduces the general description of nonlocal nonlinear response, the strongly nonlocal limit, and the Snyder-Mitchell model, and analyzes their physical meanings and applicable conditions. Then, three typical systems, including liquid crystals, photorefractive crystals, and thermal nonlinear media, are discussed. The formation types, propagation features, and control methods of spatial optical solitons under different nonlocal mechanisms are summarized. Recent representative progress in long-range interactions, new localized wave structures, and controllable transport is further reviewed. The main feature of this review is that it follows the main line of strongly nonlocal theory and connects the common physical mechanisms with the specific response features of different media. It shows the important role of nonlocality in stabilizing complex localized waves, controlling soliton interactions, and realizing programmable optical-field transport. In contrast to earlier reviews that mainly emphasize an early model or one material platform, we compare liquid-crystal, photorefractive, and thermal media within the same strongly nonlocal framework, and relate the Snyder-Mitchell approximation to material anisotropy, boundaries, and competing nonlinearities. The comparison indicates that nonlocality can stabilize complex solitons and enable reconfigurable transport, although response speed, loss, boundary sensitivity, and integration remain important challenges.