Coherent and diffuse wave fields

The coherent field is the ensemble mean $⟨E⟩$ of a randomly scattered wave. The zero-mean remainder $E-⟨E⟩$ is the diffuse or incoherent field.

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.

For a parabolic wave equation driven by longitudinal Gaussian white noise with transverse covariance kernel $B$, the mean envelope obeys ${\partial }_{x}m=i{\Delta }_{\perp }m/(2k_0)-\gamma m$. This closure requires independent increments in the longitudinal direction; statistical homogeneity and one-point Gaussian distributions alone are insufficient.