Snell's law describes refraction at a flat interface. Real interfaces are not flat. A scratched lens, a choppy sea surface, a biological membrane — all have roughness at some scale. The question is what happens to the refracted beam.
Gomez and Sølna (arXiv:2603.17488) derive generalized Snell's laws for rough interfaces using asymptotic analysis. The answer depends on the ratio of the interface's correlation length to the beam width.
When the roughness correlates on the same scale as the beam, the reflected and transmitted waves stay in specular cones — they follow Snell's law directions but with random fluctuations. No speckle forms. The wave sees the roughness as a single distortion, not a scattering medium.
When the roughness is finer than the beam, two things happen simultaneously. The rough interface homogenizes into an effective flat interface with deterministic specular reflection — an averaged Snell's law. But around the specular cone, a broader speckle cone appears, carrying energy scattered by the fine-scale roughness. The specular reflection is clean; the scattered light surrounds it as a diffuse halo.
The transition between regimes is not gradual. It is a consequence of scale separation: once the roughness correlation length drops below the beam width by enough, the homogenization kicks in sharply. The interface is either rough (regime 1) or effectively flat with speckle (regime 2). The generalized Snell's law is not a correction to the original — it is a replacement that reduces to the original in the appropriate limit.