Quantum-mechanical diffraction theory of light from a small hole: Extinction-theorem approach

In a recent paper [Phys. Rev. A 90, 043830 (2014)] it was shown that the so-called aperture response tensor is the central concept in the microscopic quantum theory of light diffraction from a small hole in a flat screen. It was further shown that the quantum mechanical theory of diffraction only requires a preknowledge of the incident field plus the electronic properties of identical screens with and without a hole. Starting from the quantum mechanical expression for the linear conductivity tensor, we study the related causal conductivity tensor paying particular attention to diamagnetic electron dynamics. Using a nonlocal-potential separation assumption, we present a calculation of the diamagnetic causal surface conductivity for a jellium quantum-well screen using a two-dimensional Hartree-Fock model. In the diamagnetic case the difference between the light-unperturbed electron densities for screens with $\left(n^0\right)$ and without $\left(n_{\infty }^0\right)$ holes are the primary quantities for the diffraction theory. In a central part (Sec. IV) of this article we determine $n^0$ via a quantum-mechanical two-dimensional extinction-theorem approach related to elastic electron scattering from a hole with an electronic selvedge. For heuristic purposes we illustrate aspects of the extinction-theorem theory by applying the approach for an infinitely high potential barrier to the vacuum hole. Finally, we calculate and discuss the aperture response tensor in the small hole limit and in the zeroth-order Born approximation. Our final result for the aperture response tensor establishes the bridge to the anisotropic electric dipole polarizability tensor of the hole. It turns out that the effective optical aperture (hole) size relates closely to the extension of the relevant electronic wave functions scattered from the hole.

Figure 1

Zero-, first-, and second-order contributions to the local-field tensor in the diamagnetic case.

Figure 2

Schematic illustration of the electronic and electromagnetic couplings in the kernel $\mathbf{K}(\mathbf{r},{\mathbf{r}}^{\prime };\omega )$ entering the integral equation for the local-field tensor $\mathbf{\Gamma }(\mathbf{r},{\mathbf{r}}^{\prime };\omega )$.

Figure 3

(a) Vacuum hole in a 2D QW screen. The hole area denoted by $S_0$ is surrounded by the selvedge area $S_E$. Outside the selvedge the bulk area $S_B$ of the screen extends to infinity. (b) Schematic illustration of the potential profile from the bulk through the selvedge and into the hole.

Figure 4

Plane area $S$ with outward-directed normal vectors ${\widehat{\mathbf{n}}}_0$ and ${\widehat{\mathbf{n}}}_1$ along the curves $C_0$ and $C_1$ that bound the area $S$.

Figure 5

The 2D hole geometry. The hole area $S_0$ is bounded by the curve $C_0$, which has an outward-directed normal vector denoted by ${\widehat{\mathbf{n}}}_0$. The selvedge area $S_E$ surrounds the hole area $S_0$ and is bounded by the curve $C_0$ at the inner edge and $C_E$ at the outer edge. The outward-directed normal vector to $C_E$ is denoted ${\widehat{\mathbf{n}}}_E$. The bulk area $S_B$ surrounds the selvedge area $S_E$ and is bounded by the curves $C_E$ and $C_{\infty }$ at the inner and outer [remote $\left(\infty \right)]$ edges, respectively. The outward-directed normal vector to $C_{\infty }$ is denoted by ${\widehat{\mathbf{n}}}_{\infty }$. ${\mathbf{r}}_0,{\mathbf{r}}_E$, and ${\mathbf{r}}_B$ denote ${\mathbf{r}}_{\parallel }$ points inside the hole, the selvedge, and the bulk, respectively.

Figure 6

Schematic illustration of the contributions to ${\psi }^i\left({\mathbf{r}}_B\right)$.

Figure 7

Schematic illustration of the extinction theorems in Eqs. (62) and (63), (a) and (b), respectively.

Figure 8

Magnification of the boundary between the hole and the selvedge. $+$ is in the hole region and $-$ is in the selvedge region.

Figure 9

Feynman diagrams related to the Hartree (top figure) and exchange (bottom figure) potential.