 Solution

Write $k=|{\mathbf{k}}_i|$ for the background wavenumber and ${\psi }_i=e^{i{\mathbf{k}}_i\cdot \mathbf{r}}$. The Helmholtz equation givesThe Born approximation for scalar wave scattering replaces $\psi$ in this source term by ${\psi }_i$. For the question's contrast $V=-k(n-1)$, also linearize $n^2-1\simeq 2(n-1)$. Since $(\Delta +k^2)e^{ikR}/R=-4\pi \delta$, the outgoing scattered field isWithout the small-contrast linearization, replace $V$ by $-k(n^2-1)/2$ in this formula.

Apply the Weyl plane-wave representation. For an observation plane wholly above or below $D$, put $\sigma =+1$ or $-1$, respectively, and choose $m=\sqrt{1-p^2-q^2}$ with nonnegative imaginary part. Define $\tilde{V}(\mathbf{\kappa })={\int }_{D}V({\mathbf{r}}^{\prime })e^{-i\mathbf{\kappa }\cdot {\mathbf{r}}^{\prime }}{d}^3{r}^{\prime }$. Substitution givesFor the lower observation plane the vertical phase has the opposite sign. This follows from the absolute separation $|z-z^{\prime }|$ in the Weyl identity; its version with $z-z^{\prime }$ alone applies only above the sources.