Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 80 / 3 / b

Represent the thin layer by its wave phase jump at the exit $x=0$ and take the incident envelope to have amplitude one. Then $E(0,z)=e^{i\varphi (z)}$. Stationarity makes the wave phase mean a constant $\mu$; it need not be zero, since the question specifies only the variance. For a normal distribution marginal, write $\varphi =\mu +\sigma Z$ with $Z\sim N(0,1)$. If $F(t)=\mathbb{E}[e^{itZ}]$, integration by parts in the Gaussian density gives $F^{\prime }(t)=-tF(t)$ and $F(0)=1$, hence $F(t)=e^{-t^2/2}$. Thus its characteristic function givesThe Fourier transform of this constant is $2\pi M\delta (\nu )$, interpreted distributionally for the infinite plane wave. Applying the propagator from part (a),If the wave phase has zero mean, omit $e^{i\mu }$. A nonunit incident amplitude simply multiplies the expression. Only the one-point Gaussian distribution is needed for this expected value; no assumption about joint Gaussian distributions at separated points is needed.

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).