General scaling limitations of ground-plane and isolated-object cloaks

Abstract

We prove that, for arbitrary three-dimensional transformation-based invisibility cloaking of an object above a ground plane or of isolated objects, there are practical constraints that increase with the object size. In particular, we show that the cloak thickness must scale proportional to the thickness of the object being cloaked, assuming bounded refractive indices, and that absorption discrepancies and other imperfections must scale inversely with the object thickness. For isolated objects, we also show that bounded refractive indices imply a lower bound on the effective crosssection.

Diameter-bandwidth product limitation of isolated-object cloaking

“…The only assumptions are that the ambient medium ε a and µ a is approximately lossless and dispersionless over the bandwidth of interest (e.g. for cloaking in air) and that the attainable refractive index contrast (eigenvalues of εµ/ε a µ a ) is ≤ B for some finite bound B (similar to our previous work [3]). Our analysis in the following subsections constitutes the following steps.…”

“…If the ambient materials are ε a and µ a , then an ideal cloak is obtained by constructing the materials ε = J ε a J ′ / det J and µ = J µ a J ′ / det J in the cloak, where J is the Jacobian matrix of the transformation. In our previous work [3] we showed that the cloaking problem at one frequency becomes increasingly difficult as the size of the object being cloaked increases. In particular, we proved that the imperfections due to absorption losses and random fabrication disorder must decrease asymptotically with the object diameter in order to maintain "good" cloaking performance: for the total (scattering + absorption) cross-section to be less than a given fraction f of a geometric cross section s g .…”

“…Fourth, in Sec. II D we use analysis similar to our proof in [3] to show the "losses" introduced by these effective complex materials must scale with diameter, and hence mean σ tot must scale with diameter. Finally, for the special case of a bandwidth-limited cloak (a cloak that is very good at one frequency but spoiled at other frequencies by material dispersion), we show that the bandwidth must narrow inversely with diameter in Sec.…”

“…Second, we map the complex-frequency problem to an equivalent realfrequency problem with transformed materials. We can then use the fact that Im ε and Im µ are > 0 for any physical causal material at Re ω and Im ω > 0 [4] to analyze an "effective" absorption loss in a manner similar to our previous work [3]. Finally, in Sec.…”

“…However, as pointed out by Pendry using a speed-of-light argument [1] and by Miller with a more formal approach [2], perfect cloaking of an isolated object in vacuum over a non-zero bandwidth is impossible due to causality. This severe limitation helped inspire the idea of ground-plane cloaking [33][34][35][36][37] and its experimental demonstrations [38][39][40][41][42][43][44][45][46][47], which circumvents such causality limitations (although it is still subject to other practical scaling difficulties [3,48]). However, the Pendry and Miller proofs say little about imperfect cloaking (a nonzero but small scattering cross-section).…”