  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.
