The perfect lens on a finite bandwidth

Abstract

The resolution associated with the so-called perfect lens of thickness d is −2πd/ln(|χ+2|/2). Here the susceptibility χ is a Hermitian function in H2 of the upper half-plane, i.e., a H2 function satisfying χ(−ω)=χ(ω)¯. An additional requirement is that the imaginary part of χ be non-negative for non-negative arguments. Given an interval I on the positive half-axis, we compute the distance in L∞(I) from a negative constant to this class of functions. This result gives a surprisingly simple and explicit formula for the optimal resolution of the perfect lens on a finite bandwidth.