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For an aperture normal to ${\mathbf{e}}_x$, let $F_y,F_z$ be its tangential electric field traces extended by zero outside the aperture. With the outgoing branch of $\kappa$, the Weyl plane-wave representation and the curl aperture integral give a spectrum proportional to $(-(\nu {\hat{F}}_y+\eta {\hat{F}}_z)/\kappa ,{\hat{F}}_y,{\hat{F}}_z)$. Each wave is transverse to $\mathbf{q}$. The longitudinal electric component is regenerated by oblique diffraction even when it vanishes at the input. For $G=e^{ikr}/(4\pi r)$ and a unit prefactor on the stated curl integral, the spectrum has an overall factor $1/2$; a doubled half-space Green kernel gives factor one. The normalization must follow the chosen Green kernel and equivalent-current convention.

Use time dependence $e^{-i\omega t}$, positive wavenumber $k$ and aperture normal $\mathbf{n}={\mathbf{e}}_x$. Write $F(y^{\prime },z^{\prime })=E_{\mathrm{inc},y}(0,y^{\prime },z^{\prime })$ on the aperture and extend $F$ by zero outside it. DefineThe outgoing branch has $\kappa \ge 0$ for propagating components and $\mathrm{Im}\kappa >0$ for evanescent components.

Fix the Green function normalization by $G_0(\mathbf{r})=e^{ik|\mathbf{r}|}/(4\pi |\mathbf{r}|)$, so $(\Delta +k^2)G_0=-\delta$. Its transverse Fourier transform satisfies $({\partial }_x^2+{\kappa }^2){\hat{G}}_0=-\delta (x)$. An outgoing solution on both sides is proportional to $e^{i\kappa |x|}$, and its derivative jump fixes ${\hat{G}}_0=i{e}^{i\kappa |x|}/(2\kappa )$. Thus the Weyl plane-wave representation for observation at $x>0$ isSince $\mathbf{n}\times {\mathbf{E}}_{\mathrm{inc}}=F{\mathbf{e}}_z$, the given aperture integral becomes a superposition of curls of ${\mathbf{e}}_z{e}^{i\mathbf{q}\cdot \mathbf{r}}$. Each curl supplies $i\mathbf{q}\times {\mathbf{e}}_z=i(\nu ,-\kappa ,0)$. Therefore, with the unit prefactor in that integral,Every vector amplitude is transverse to $\mathbf{q}$, as Maxwell's equations require. The polarizer specifies the incident aperture field; diffraction can regenerate an $x$ component outside the aperture. In contrast, its $z$ component remains zero in this representation. Evanescent plane waves are needed for the complete near field, not just the propagating disk ${\nu }^2+{\eta }^2\le k^2$.

The factor $1/2$ follows explicitly from the standard outgoing Green function for the three-dimensional Helmholtz equation and the printed unit curl prefactor. If the intended kernel is the doubled half-space kernel $2G_0$, or the aperture formula carries the usual compensating factor $2$, delete that $1/2$. The latter normalization makes the tangential output trace equal to $F$. Keeping the convention explicit avoids silently changing the supplied representation; the angular dependence and transversality of an electromagnetic plane wave are the same in either normalization. Reversing the specified normal also reverses the overall sign.

Let the two tangential aperture fields be $F_y(y,z)$ and $F_z(y,z)$, extended by zero off the aperture. The normal component does not enter $\mathbf{n}\times {\mathbf{E}}_{\mathrm{inc}}$, which equals $-F_z{\mathbf{e}}_y+F_y{\mathbf{e}}_z$. Apply the Weyl plane-wave representation and curl to both components. With the same kernel normalization as part (a),This is the electromagnetic aperture angular spectrum. The two electromagnetic polarization vectors are linearly independent and both orthogonal, bilinearly, to $\mathbf{q}$; they need not be orthogonal to each other. Arbitrary tangential electromagnetic polarization is therefore represented, with its outgoing longitudinal component fixed by $\kappa E_x+\nu E_y+\eta E_z=0$ rather than independently prescribed. Away from grazing components one could equally express the spectrum in a transverse-electric and transverse-magnetic basis. The same limiting outgoing branch handles grazing and evanescent components.

For each component the magnetic amplitude follows from Faraday's law as $\mathbf{B}=\mathbf{q}\times \mathbf{E}/\omega$. Together with ${\mathbf{q}}^2=k^2$ and transversality of an electromagnetic plane wave, this verifies the source-free Maxwell equations. If the doubled-kernel aperture convention is used, multiply the entire expression by two, exactly as in part (a).

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.

For $p^2+q^2>1$, $m=i\sqrt{p^2+q^2-1}$ and the Weyl plane-wave representation contains evanescent waves. Their amplitudes decay exponentially with the distance from the scattering region. Excluding them removes high transverse spatial frequencies, including subwavelength information. Near-field measurements can in principle detect them, but back-propagating their decay exponentially amplifies measurement errors.

For propagating data, the Fourier transfer satisfies $|{\mathbf{k}}_s-{\mathbf{k}}_i|\le 2k$. One illumination still samples only the Ewald sphere, so this frequency bound does not imply access to every mode inside that ball. The lack of a volumetric spectrum is the separate nonuniqueness described in the previous part.

The Born approximation requires the incident field to remain a good approximation inside the scatterer. A useful sufficient smallness estimate for a weak, extended inclusion is small accumulated phase $kD|n-1|\ll 1$, alongside weak reflection and multiple scattering. Small pointwise contrast alone can fail for a thick inclusion or a resonance. The scalar Helmholtz and background-medium assumptions, and the extra linearization of $n^2-1$, also belong to the model's validity conditions.

Numerically, a finite aperture windows the measured fields and limits angular and Fourier resolution. Discrete sampling must satisfy the Nyquist–Shannon sampling theorem for the retained transverse bandwidth; spacings $\Delta x,\Delta y\lesssim \pi /K_{\max }$ avoid its simplest aliasing. Finite discrete Fourier transforms also introduce periodic continuation, truncation, interpolation and window errors. Noise, unresolved evanescent modes and near-grazing modes with $m\simeq 0$ limit reliable recovery. Refining the grid cannot by itself fill the missing three-dimensional Fourier data.