Microscopic theory of diffraction of light from a small hole

On the basis of the Maxwell-Lorentz local-field equations and nonlocal linear response theory, a self-consistent microscopic Green function theory of diffraction of light from a single hole in a thin and plane metallic screen is established. By subtracting the scattering of identical incident fields from screens with and without a hole, a causal effective optical aperture response tensor is introduced. An approximate expression is derived for the aperture response tensor in the limit where the screen behaves like an electric-dipole absorber and radiator. In this limit the internal electron dynamics is that of a quantum well. For a screen so thin that its bound electron motion can be described by a single quantum level, a approach for a quantum mechanical calculation of the aperture response tensor is presented. When the linear dimensions of the hole become sufficiently small the so-called aperture field, defined as the difference between the prevailing electric field with and that without a hole, becomes identical to the field from an incident-field-induced electric dipole with anisotropic linear polarizability. Our theory is formulated in such a manner that preknowledge only of (i) the incident electromagnetic field and (ii) the light-unperturbed optical electron properties (the microscopic conductivity tensor) of the screen with the geometrically given hole is needed. Since the microscopic theory allows for the presence of an (oscillating) component of the sheet current density perpendicular to the plane of the screen, a generalization of (i) the standard jump conditions of the field across the sheet and (ii) the reflection symmetries of the various fields in the plane of the screen is worked out. As our theory deviates radically from the approach of all classical diffraction theories, which are based on the macroscopic Maxwell equations and some kind of pheno-menological expression for the screen conductivity $\sigma$ (often just $\sigma \to \infty )$, we give a brief review of classical diffraction theory, formulated in such a manner that a comparison to the microscopic theory is made easier. In particular, the Bethe-Bouwkamp theory of classical diffraction from a small hole in an infinitely thin and perfectly conducting $\left(\sigma \to \infty \right)$ screen is our focus. We suggest that experimental frequency and angular resolved studies of the interference of the diffracted fields from quantum wells with and without holes are undertaken to obtain detailed insight into the microscopic aperture response functions, not least in the optical near-field domain.

Figure 1

Helmholtz-Kirchhoff integral theorem applied to investigate the diffraction of an incident vector field ${\mathbf{U}}^0\left(\mathbf{r}\right)$ from an aperture in a plane screen. The surface is divided into three parts: the screen $\mathcal{S}$ (solid line) and the aperture $\mathcal{A}$ (short-dashed line) in the $z=0^{+}$ plane and the remaining hemisphere ${\mathcal{S}}_{\infty }$ (dashed line). Note that $\mathbf{r}$ is a point inside $S$ and that the normals ${\widehat{\mathbf{n}}}^{\prime }=\widehat{\mathbf{n}}\left({\mathbf{r}}^{\prime }\right)$ to the surface are inwardly directed.

Figure 2

(a) A system of source charges generates an incident field ${\mathbf{E}}^0$ which subsequently excites an infinitely extended plane screen with a hole. The screen is in the $xy$ plane and the optical electronic properties of the screen are described by the nonlocal conductivity tensor $\sigma (\mathbf{r},{\mathbf{r}}^{\prime })$. The field ${\mathbf{E}}^0\left(\mathbf{r}\right)$ induces a self-consistent current density $\mathbf{J}\left(\mathbf{r}\right)$ in the screen. (b) The same as (a), except that there is no hole in the screen.

Figure 3

The effective optical aperture in the plane of the screen—henceforth denoted $\mathcal{A}$—is the area defined by the region where ${\sigma }^{\text{cau}}-{\sigma }_{\infty }^{\text{cau}}$ is nonvanishing. Note that $\mathcal{A}$ is lager than the geometrical aperture indicated by the thick solid line.

Figure 4

Electronic $\delta$ function and (lowest order) direct electromagnetic correlation effects in the causal conductivity tensor, ${\sigma }^{\text{cau}}(\mathbf{r},{\mathbf{r}}^{\prime };\omega )$.

Figure 5

Scattering of the electron wave function from the hole in the QW screen.

Figure 6

Schematics of the asymmetries and jumps in the various electromagnetic field components (thick black lines) across a current sheet placed at $z=0$. The usual textbook result, with the assumption $J_{\perp }^S=0$, is also sketched (thin gray lines). (a) The symmetry and the jump in $B_{\perp }^S$ are the same in the two cases. (b) The jump in ${\mathbf{B}}_{\parallel }$ (only the $x$ component is shown here) is the same, but the symmetry is no longer antisymmetric with respect to $z$. (c) The jump in $E_{\perp }$ is the same, but the symmetry is modified; it is no longer antisymmetric with respect to $z$. (d) Both the jump and the symmetry of ${\mathbf{E}}_{\parallel }$ (only the $x$ component is sketched here) are modified. In the general case ${\mathbf{E}}_{\parallel }$ jumps at $z=0$ and it is not symmetric with respect to $z$. Note also that the jumps in ${\mathbf{B}}_{\parallel }$ and $E_{\perp }$ in general are not symmetric around 0 [as depicted in (b) and (c)]. For the textbook case where $J_{\perp }^S=0$ this is a condition imposed by the antisymmetry of ${\mathbf{B}}_{\parallel }$ and $E_{\perp }$. This antisymmetry is lost for $J_{\perp }^S\ne 0$. An alternative sketch of the $x$ component of ${\mathbf{B}}_{\parallel }$ is shown in Fig. 7.

Figure 7

Alternative sketch of the $x$ component of ${\mathbf{B}}_{\parallel }$. As in Fig. 6, thick black lines are the generalized case with $J_{\perp }^S\ne 0$ and thin gray lines represent the textbook result in which $J_{\perp }^S=0$.