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.