  We show that cloaking of isolated objects is subject to a diameter-bandwidth
product limitation: as the size of the object increases, the bandwidth of good
(small cross-section) cloaking decreases inversely with the diameter, as a
consequence of causality constraints even for perfect fabrication and materials
with negligible absorption. This generalizes a previous result that perfect
cloaking of isolated objects over a nonzero bandwidth violates causality.
Furthermore, we demonstrate broader causality-based scaling limitations on any
bandwidth-averaged cloaking cross-section, using complex analysis and the
optical theorem to transform the frequency-averaged problem into a single
scattering problem with transformed materials.
