Analytical solution of the almost-perfect-lens problem

TLDR

These results also extend to the electrostatic limit with minor alterations. The field behavior is governed by singularities from surface‑polariton excitation at large wave vectors. An analytic solution for |σ|≪1 shows the near‑field symmetry depends on σ’s sign and that resolution scales logarithmically with |σ|.

Abstract

The problem of imaging for a slab of a lossless left-handed material with refractive index n=−(1−σ)1/2 is solved analytically for |σ|≪1. The electromagnetic field behavior is determined largely by singularities arising from the excitation of surface polaritons with wave vector q→±∞. Depending on the sign of σ, the near-field is either odd or even with respect to the lens middle plane. Consistent with other nonanalytical studies, the resolution depends logarithmically on |σ|. With minor alterations, these results apply as well to the electrostatic limit.