The perfect lens on a finite bandwidth
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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 $H^2$ of the upper half-plane, i.e., a $H^2$ function satisfying $χ(-ω)=\bar{χ(ω)}$. An additional requirement is that the imaginary part of $χ$ be nonnegative for nonnegative arguments. Given an interval $I$ on the positive half-axis, we compute the distance in $L^\infty(I)$ 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.