On the fundamental limitations of imaging with evanescent waves

There has been significant interest in imaging and focusing schemes that use evanescent waves to beat the diffraction limit, such as those employing negative refractive index materials or hyperbolic metamaterials. The fundamental issue with all such schemes is that the evanescent waves quickly decay between the imaging system and sample, leading to extremely weak field strengths. Using an entropic definition of spot size which remains well defined for arbitrary beam profiles, we derive rigorous bounds on this evanescent decay. In particular, we show that the decay length is only $w / \pi e \approx 0.12 w$, where $w$ is the spot width in the focal plane, or $\sqrt{A} / 2 e \sqrt{\pi} \approx 0.10 \sqrt{A}$, where $A$ is the spot area. Practical evanescent imaging schemes will thus most likely be limited to focal distances less than or equal to the spot width.