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This paper concerns the analysis of a passive, broadband approximate cloakingscheme for the Helmholtz equation in ${\mathbb R}^d$ for $d=2$ or $d=3$. Usingideas from transformation optics, we construct an approximate cloak by``blowing up" a small ball of radius $\epsilon>0$ to one of radius $1$. In theanisotropic cloaking layer resulting from the ``blow-up" change of variables,we incorporate a Drude-Lorentz-type model for the index of refraction, and weassume that the cloaked object is a soft (perfectly conducting) obstacle. Wefirst show that (for any fixed $\epsilon$) there are no real transmissioneigenvalues associated with the inhomogeneity representing the cloak, whichimplies that the cloaking devices we have created will not yield perfectcloaking at any frequency, even for a single incident time harmonic wave.Secondly, we establish estimates on the scattered field due to an arbitrarytime harmonic incident wave. These estimates show that, as $\epsilon$approaches $0$, the $L^2$-norm of the scattered field outside the cloak, andits far field pattern, approach $0$ uniformly over any bounded band offrequencies. In other words: our scheme leads to broadband approximate cloakingfor arbitrary incident time harmonic waves.

Far field broadband approximate cloaking for the Helmholtz equation with a Drude-Lorentz refractive index