In the enormously cited paper, Frank Stern [1] gave the fomula of dielectric function of two dimensional material. But the process remains very obscure to me.
The longitudinal electric field $\vec{E}(\vec{q},\omega)=\vec{E_0}(\vec{q},\omega)exp(i\vec{q}\cdot\vec{r}-i\omega t)$ is applied on the two dimensional system, then the induced polarization is $$\vec{P}(\vec{q},\omega)=\chi(\vec{q},\omega)\vec{E}(\vec{q},\omega)\delta(z)$$.
The field is localized near the surface, so it's easy to assume that the polarization is proportional to $exp(i\vec{q}\cdot\vec{r}-i\omega t-\beta|z|)$, where $\beta=\sqrt{q^2-k_b \times \omega^2/c^2}$, here $k_b $ is the dielectric constant of the medium which the two dimensional system is embedded in.
Everything seems logical to me so far, until all of sudden the author came to the conclusion that the induced electric field in the plane $z=0$ is found to be $$\vec{E_{ind}}(\vec{q},\omega)=-2\pi\chi(\vec{q},\omega)\vec{E}(\vec{q},\omega)/k_b$$.
So can someone help me find where is the link to the final step of derivation. I am sure maxwell eqation is utilized here, but have no clue how it is used.
Ref: [1] Stern, Frank. "Polarizability of a two-dimensional electron gas." Physical Review Letters 18.14 (1967): 546.
