Lensing effects in a random inhomogeneous medium

Curvature of the refractive index profile can cause the focusing or defocusing of a propagating beam. Such lensing can have detrimental e ects on, for example, imaging through the atmosphere, where it can cause features to appear stretched or regions to appear artificially bright. We consider the problem of describing propagation in the presence of both curvature and stochastic fluctuations in the refractive index profile. For di erent scenarios motivated by atmospheric propagation data, we derive concise, robust formulas for the mean ray paths that include the e ects of turbulence. In addition, for the case of a local maximum of the mean refractive index, we show that the focal length associated with the mean trajectories in the random medium is smaller than in the deterministic setting.