 Solution

Inverse Fourier transform of the expected value givesindependent of both coordinates. Its squared modulus is the coherent scalar wave intensity $e^{-{\sigma }^2}$, which must not be confused with the mean wave intensity.

For the latter use the stationary two-point statistics. Let $F_0(d\nu )$ be the uncentered spectral measure of a stationary random field of $E(0,z)$, normalized bySince $|E(0,z)|=1$, its total mass is $\int F_0(d\nu )=1$. Under free paraxial propagation each spectral component is multiplied by $e^{-i{\nu }^{2}x/(2k)}$; in its wave-field correlation this wave phase multiplies its complex conjugate and cancels. The stationary intensity under free paraxial propagation is consequentlyStationarity is essential to this pointwise statement: it makes the spectral covariance diagonal. Conservation of integrated power alone would not show that a particular spatial point keeps the same scalar wave intensity.

The same result follows locally from $I_x+J_z=0$, where $J=\mathrm{Im}(E^{\ast }{E}_z)/k$. Taking expectations makes $⟨J⟩$ independent of $z$, so ${\partial }_x⟨I⟩=0$. Thus coherent scalar wave intensity and mean wave intensity are both constant in propagation, with different values. The incoherent contribution has mean $1-e^{-{\sigma }^2}$. Individual realizations can show focusing and changing scalar wave intensity even though the expected value remains constant. All these propagation statements concern the free parabolic wave equation from part (a).