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For a slowly varying envelope $E=\psi e^{-ikx}$, the paraxial approximation to the Helmholtz equation isBecause $n=1+\mu W$ and ${\mu }^2\ll 1$, passage through a sufficiently thin phase screen producesIts modulus is one at the screen exit. Beyond the screen, $n=1$, so the parabolic wave equation is $E_x=i{E}_{zz}/(2k)$. A Taylor expansion in propagation distance givesSincewe findIt follows thator, equivalently, $|E{|}^2=1-x{\varphi }^{\prime \prime }/k+O(x^2)$. Thus random phase curvature produces local focusing and defocusing: free-space diffraction converts phase fluctuations into amplitude fluctuations immediately after the screen.

The phase screen adds the phase accumulated across its thickness, so the reduced field just after the screen isFor a zero-mean random variable $w$ with a normal distribution and variance ${\sigma }^2$, its characteristic function givesThe same expression is approximately valid for a non-Gaussian weak fluctuation: the cumulant expansion begins with $-k^2{\xi }^2{\sigma }^2/2$, while higher cumulants give higher-order corrections.

For $x>0$, every realization obeys the parabolic wave equationLinearity permits ensemble averaging, soThe initial mean is independent of $z$, hence diffraction does not change it and $⟨E(x,z)⟩=\exp (-k^2{\xi }^2{\sigma }^2/2)$ for every $x\ge 0$.

For a scalar complex wave amplitude $E$, the squared-amplitude scalar wave intensity convention is $I=E{E}^{\ast }$. It is nonnegative and unchanged by multiplying $E$ by a unit-modulus wave phase factor. This convention is useful in paraxial propagation and phase screen models. Conversion to physical energy flux also depends on the field variable and medium; an arbitrary superposition need not have its full flux determined by this scalar alone. For a random field, $⟨I⟩$ is the mean wave intensity, whereas $|⟨E⟩{|}^2$ is the squared coherent field; these generally differ.