Impedance matching of inverted conductors: Two-dimensional beam splitters with divergent gain

A thin conducting sheet—graphene, for example—transmits, absorbs, and reflects radiation. A sheet that is very thin, even vanishingly so, can still produce 50% absorption at normal incidence if it has conductivity corresponding to half the impedance of free space. We find that, regardless of the sheet conductivity, there exists a combination of polarization and angle of incidence that achieves this impedance-half-matching condition. If the conducting medium can be inverted, the conductivity is formally negative and the sheet amplifies the incident radiation. To the extent that a negative half-match in a thin sheet can be maintained, enormous single-pass gain in both transmission and reflection is possible. Known semiconductors (e.g., gallium nitride) have the optical properties necessary to give large amplification in a structure that is, remarkably, both thin and nonresonant.

Figure 1

Diagrams establishing sign conventions for $S$-polarized and $P$-polarized plane waves incident on a conductor sandwiched between media with refractive indices $n_1$ and $n_3$ respectively. An $\mathbf{E}$-field directed into the page is indicated with $\otimes$, and the magnetic fields $\mathbf{B}$ not shown are oriented such that $\mathbf{E}\times \mathbf{B}$ is along the wave vector $\mathbf{k}$. We consider two cases: the conductor is a sheet with thickness $d=0$ and two-dimensional conductivity ${\sigma }_{\text{2D}}$, or it is a slab with $d\ne 0$, permittivity ${\widehat{\varepsilon }}_2$, and three-dimensional conductivity ${\sigma }_{\text{3D}}$.

Figure 2

The transmission $T$, reflection $R$, and absorption $A$ by a two-dimensional (2D) conducting plane as a function of $\xi$, where $\xi ={\sigma }_{\text{2D}}{Z}_0/ncos\theta$ for $S$ polarization and $\xi ={\sigma }_{\text{2D}}{Z}_{0}cos\theta /n$ for $P$ polarization. For a given value of the conductivity ${\sigma }_{\text{2D}}$ any magnitude of $\xi$ can be arranged by judicious choice of the polarization and the angle of incidence $\theta$. Thus strong coupling $\left(\xi =\pm 2\right)$, which gives 50% absorption for positive conductivity and divergent gain with negative conductivity, is always possible.