Coherent attenuation by a Gaussian phase screen ($m=e^{-{\beta }^2/2}$)

For a unit-amplitude random phase screen $e^{i\beta W}$ with standard normal marginal distribution, the characteristic function gives coherent amplitude $m=\mathbb{E}[e^{i\beta W}]=e^{-{\beta }^2/2}$. The coherent power is $|m{|}^2=e^{-{\beta }^2}$, although the total power remains one. In homogeneous paraxial space the mean constant transverse field propagates without additional attenuation. Under decorrelation at large separation, the remaining power $1-e^{-{\beta }^2}$ is diffuse.