Approximate Cloaking Research Articles

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

This paper concerns the analysis of a passive, broadband approximate cloaking scheme for the Helmholtz equation in Rd for d=2 or d=3. Using ideas from transformation optics, we construct an approximate cloak by “blowing up” a small ball of radius ϵ>0 to one of radius 1. In the anisotropic cloaking layer resulting from the “blow-up” change of variables, we incorporate a Drude-Lorentz-type model for the index of refraction, and we assume that the cloaked object is a soft (perfectly conducting) obstacle. We first show that (for any fixed ϵ) there are no real transmission eigenvalues associated with the inhomogeneity representing the cloak, which implies that the cloaking devices we have created will not yield perfect cloaking at any frequency, even for a single incident time harmonic wave. Secondly, we establish estimates on the scattered field due to an arbitrary time harmonic incident wave. These estimates show that, as ϵ approaches 0, the L2-norm of the scattered field outside the cloak, and its far field pattern, approach 0 uniformly over any bounded band of frequencies. In other words: our scheme leads to broadband approximate cloaking for arbitrary incident time harmonic waves.

Optimized lattice-based metamaterials for elastostatic cloaking

An optimization-based approach is proposed to design elastostatic cloaking devices in two-dimensional (2D) lattices. Given an elastic lattice with a defect, i.e. a circular or elliptical hole, a small region (cloak) around the hole is designed to hide the effect of the hole on the elastostatic response of the lattice. Inspired by the direct lattice transformation approach to elastostatic cloaking in 2D lattices, the lattice nodal positions in the design region are obtained using a coordinate transformation of the reference (undisturbed) lattice nodes. Subsequently, additional connectivity (i.e. a ground structure) is defined in the design region and the stiffness properties of these elements are optimized to mimic the global stiffness characteristics of the reference lattice. A weighted least-squares objective function is proposed, where the weights have a physical interpretation—they are the design-dependent coefficients of the design lattice stiffness matrix. The formulation leads to a convex objective function that does not require a solution to an additional adjoint system. Optimization-based cloaks are designed considering uniaxial tension in multiple directions and are shown to exhibit approximate elastostatic cloaking, not only when subjected to the boundary conditions they were designed for but also for uniaxial tension in directions not used in design and for shear loading.

On corners scattering stably and stable shape determination by a single far-field pattern

In this paper, we establish two sharp quantitative results for the direct and inverse time-harmonic acoustic wave scattering. The first one is concerned with the recovery of the support of an inhomogeneous medium, independent of its contents, by a single far-field measurement. For this challenging inverse scattering problem, we establish a sharp stability estimate of logarithmic type when the medium support is a polyhedral domain in $\mathbb{R}^n$, $n=2,3$. The second one is concerned with the stability for corner scattering. More precisely if an inhomogeneous scatterer, whose support has a corner, is probed by an incident plane-wave, we show that the energy of the scattered far-field possesses a positive lower bound depending only on the geometry of the corner and bounds on the refractive index of the medium there. This implies the impossibility of approximate invisibility cloaking by a device containing a corner and made of isotropic material. Our results sharply quantify the qualitative corner scattering results in the literature, and the corresponding proofs involve much more subtle analysis and technical arguments. As a significant byproduct of this study, we establish a quantitative Rellich's theorem that continues smallness of the wave field from the far-field up to the interior of the inhomogeneity. The result is of significant mathematical interest for its own sake and is surprisingly not yet known in the literature.

Approximate cloaking for electromagnetic waves via transformation optics: Cloaking versus infinite energy

We study the approximate cloaking via transformation optics for electromagnetic waves in the time harmonic regime in which the cloaking device only consists of a layer constructed by the mapping technique. Due to the fact that no-lossy layer is required, resonance might appear and the analysis is delicate. We analyze both non-resonant and resonant cases. In particular, we show that the energy can blow up inside the cloaked region in the resonant case and/whereas cloaking is achieved in both cases. Moreover, the degree of visibility depends on the compatibility of the source inside the cloaked region and the system. These facts are new and distinct from known mathematical results in the literature.

Approximate Cloaking Using Transformation Optics for Acoustic and Electromagnetic Waves

This is a survey on approximate cloaking using transformation optics on acoustic and electromagnetic waves. Both the time-harmonic and the time regimes are discussed.

Approximate Cloaking for Time-dependent Maxwell Equations via Transformation Optics

We study approximate cloaking using transformation optics for electromagnetic waves in the time domain. Our approach is based on estimates of the degree of visibility in the frequency domain for all frequencies in which the frequency dependence is explicit. The difficulty and the novelty analysis parts are in the low and high frequency regimes. To this end, we implement a variational technique in the low frequency domain and multiplier and duality techniques in the high frequency domain. Our approach is inspired by the work of Nguyen and Vogelius on the wave equation.

Approximate cloaking for the heat equation via transformation optics

In this paper, we establish approximate cloaking for the heat equation via transformation optics. We show that the degree of visibility is of the order $\epsilon$ in three dimensions and $|\ln\epsilon|^{-1}$ in two dimensions, where $\epsilon$ is the regularization parameter.

Solving approximate cloaking problems using finite element methods

Motivated by the approximate cloaking problem, we consider a variable coefficient Helmholtz equation with a fixed wave number. We use finite element methods to discretize the equation. Numerical results are shown to exhibit cloaking behaviour. References A. Alu and N. Engheta. Achieving transparency with plasmonic and metamaterial coatings. Phys. Rev. E 72 :016623, 2005. doi:10.1103/PhysRevE.72.016623 W. Cai, U. K. Chettiar, A. V. Kildishev and V. M. Shalaev. Optical cloaking with metamaterials. Nature Photonics 1 :224–227, 2007. doi:10.1038/nphoton.2007.28 A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann. Cloaking devices, electromagnetic wormholes and transformation optics. SIAM Rev. 51 :3–33, 2009. doi:10.1137/080716827 R. V. Kohn, D. Onofrei, M. S. Vogelius and M. I. Weinstein. Cloaking via change of variables for the Helmholtz equation. Comm. Pure Appl. Math. 63 :973–1016, 2010. doi:10.1002/cpa.20341 R. V. Kohn, H. Shen, M. S. Vogelius and M. I. Weinstein. Cloaking via change of variables in electric impedance tomography. Inverse Problems 24 :015016, 2008. doi:10.1088/0266-5611/24/1/015016 U. Leonhardt. Optical conformal mapping. Science 312 :1777–1780, 2006. doi:10.1126/science.1126493 M. Maischak. Book of Numerical Experiments (B.O.N.E) . http://www.ifam.uni-hannover.de/ maiprogs/ J. B. Pendry, D. Schurig and D. R. Smith. Controlling electromagnetic fields. Science 312 :1780–1782, 2006. doi:10.1126/science.1125907 D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr and D. Smith. Metamaterial electromagnetic cloak at microwave frequencies. Science 314 :977–980, 2006. doi:10.1126/science.1133628

Cloaking a vertical cylinder via homogenization in the mild-slope equation

We describe a method to construct devices which allows a vertical rigid cylinder to be cloaked for any far-field observer in the case of linear water waves. An adaptation of parameters given by a geometric transform performed in the mild-slope equation is achieved via homogenization. The final device, which respects the physical constraints of the problem, is obtained with a conformal mapping. The result of this algorithm is a structure surrounding the vertical cylinder, composed of an annular region with varying bathymetry and with rigid vertical objects piercing the free surface. An approximate cloaking is achieved, which implies a reduction of the mean drift force acting on the cylinder.

Approximate cloaking for the full wave equation via change of variables: The Drude–Lorentz model

This paper concerns approximate cloaking by mapping for a full, but scalar wave equation, when one allows for physically relevant frequency dependence of the material properties of the cloak. The paper is a natural continuation of [20], but here we employ the Drude–Lorentz model in the cloaking layer, that is otherwise constructed by an approximate blow up transformation of the type introduced in [10]. The central mathematical problem is to analyze the effect of a small inhomogeneity in the context of this non-local full wave equation.

The Blow-up of Electromagnetic Fields in 3-Dimensional Invisibility Cloaking for Maxwell's Equations

Transformation optics constructions have allowed the design of cloaking devices that steer electromagnetic, acoustic, and quantum waves around a region without penetrating it, so that this region is hidden from external observations. The proposed material parameters are anisotropic, and singular at the interface between the cloaked region and the cloaking device. The presence of these singularities causes various mathematical problems and physical effects on the interface surface. In this paper, we analyze the 3-dimensional cloaking for Maxwell's equations when there are sources or sinks present inside the cloaked region. In particular, we consider nonsingular approximate invisibility cloaks based on the truncation of the singular transformations. We analyze the limit of solutions when the approximate cloaking approaches the ideal cloaking in the sense of distributions. We show that the solutions in the approximate cloaks converge to a distribution that contains Dirac's delta distribution supported on the interface surface. In particular, this implies that the limit of solutions are not measurable functions, making them outside of those classes of functions that have earlier been used in the models of the ideal invisibility cloaks. Also, we give a rigorous meaning for the “extraordinary surface voltage effect" considered in physical literature of invisibility cloaks.

Nearly cloaking the elastic wave fields

In this work, we develop a general mathematical framework on regularized approximate cloaking of elastic waves governed by the Lamé system via the approach of transformation elastodynamics. Our study is rather comprehensive. We first provide a rigorous justification of the transformation elastodynamics. Based on the blow-up-a-point construction, elastic material tensors for a perfect cloak are derived and shown to possess singularities. In order to avoid the singular structure, we propose to regularize the blow-up-a-point construction to be the blow-up-a-small-region construction. However, it is shown that without incorporating a suitable lossy layer, the regularized construction would fail due to resonant inclusions. In order to defeat the failure of the lossless construction, a properly designed lossy layer is introduced into the regularized cloaking construction. We derive sharp asymptotic estimates in assessing the cloaking performance. The proposed cloaking scheme is capable of nearly cloaking an arbitrary content with a high accuracy.

Spectral effectiveness of engineered thermal cloaks in the frequency regime.

We analyse basic thermal cloaks designed via different geometric transforms applied to thermal cloaking. We evaluate quantitatively the effectiveness of these heterogeneous anisotropic thermal cloaks through the calculation of the standard deviation of the isotherms. The study addresses the frequency regime and we point out the cloak's spectral effectiveness. We find that all these cloaks have comparable effectiveness irrespective of whether or not they have singular conductivity at their inner boundary. However, approximate cloaking with multi-layered cloak critically depends upon the homogenization algorithm and it is shown that the standard deviation varies linearly with the inverse of the number of layers.

G-convergence, Dirichlet to Neumann maps and invisibility

We establish optimal conditions under which the G-convergence of linear elliptic operators implies the convergence of the corresponding Dirichlet to Neumann maps. As an application we show that the approximate cloaking isotropic materials from [19] are independent of the source.

Electromagnetic scattering by approximately cloaked cylindrical bodies with nonhomogeneous anisotropic cloaking material

In cloaking, a body is hidden from detection by surrounding it by a coating consisting of an unusual anisotropic nonhomogeneous material. The permittivity and permeability of such a cloak are determined by the coordinate transformation of compressing a hidden body into a point (3D or spherical configuration) or a line (2D or cylindrical configuration). Some components of the electrical parameters of the cloaking material are required to have infinite or zero value at the boundary of the hidden object. Approximate cloaking can be achieved by transforming the cylindrical bodies (dielectric and conducting) virtually into a small cylinder rather than a line, which eliminates the zero or infinite values of the electrical parameters but produces scattering. The solution is obtained by rigorously solving Maxwell equations using angular harmonics expansion. In this work, the scattering pattern and the back-scattering cross-section against the frequency for cloaked conducting and dielectric cylinders are studied for both transverse magnetic (TMz) and transverse electric (TEz) polarizations of the incident plane wave for different transformed body radii.

Suppression of the Resonant Scattering in Imperfect Acoustic Cloaking with a Lossy Medium in ℝ3

It has been realized that resonance frequencies of imperfect acoustic cloaking based on a small perturbation of the transformation acoustics in ℝ2 are located near Dirichlet eigenvalues of the cloaked region [Chin. Phys. Lett. 26 (2009) 014301; 29 (2012) 124301]. In this work, we study the performance of the three-dimensional approximate cloaking system based on the transformation acoustics and show that the cloaking effect may be deteriorated at zeroth order Neumann eigenvalues of the concealed region. In particular, transmitted fields into the concealed region can be extremely resonated at frequencies corresponding to the zeroth-order Neumann eigenvalues while scattered fields are suppressed well for any frequency. To enhance the cloaking effect at resonance frequencies, we introduce a lossy medium inside the cloaked region and show that the new proposal can reduce the intensity of transmitted fields significantly due to the lossy medium.

Approximate electromagnetic cloaking of a conducting cylinder using homogeneous isotropic multi-layered materials

Cloaking refers to hiding a body from detection by surrounding it with a coating consisting of an unusual anisotropic nonhomogeneous material. Its function is to deflect the rays that would have struck the object, guide them around the object, and return them to their original trajectory, thus no waves are scattered from the body. The permittivity and permeability of such a cloak are determined by the coordinate transformation of compressing a hidden body into a point or a line. Some components of the electrical parameters of the cloaking material (ɛ, μ) are required to have infinite or zero value at the boundary of the hidden object. Approximate cloaking can be achieved by transforming the cylindrical body (dielectric and conducting) virtually into a small cylinder rather than a line, which eliminates the zero or infinite values of the electrical parameters. The radially-dependent cylindrical cloaking shell can be approximately discretized into many homogeneous anisotropic layers; each anisotropic layer can be replaced by a pair of equivalent isotropic sub-layers, where the effective medium approximation is used to find the parameters of these two equivalent sub-layers. In this work, the scattering properties of cloaked perfectly conducting cylinder is investigated using a combination of approximate cloaking, together with discretizing the cloaking material using pairs of homogeneous isotropic sub-layers. The solution is obtained by rigorously solving Maxwell equations using angular harmonics expansion. The scattering pattern, and the back scattering cross section against the frequency are studied for both transverse magnetic (TMz) and transverse electric (TEz) polarizations of the incident plane wave for different transformed body radii.

Enhanced approximate cloaking by optimal change of variables

The aim of (passive) cloaking with respect to electromagnetic (or acoustic) sensing is to surround a region of space with a material layer—the cloak—that renders its contents and even the existence of the layer undetectable by such measurements. At least theoretically this can be achieved using the coordinate invariance of the underlying wave equation, through so-called cloaking by mapping. However, a practical realization of the cloaking by mapping schemes discussed in the literature frequently requires the design of highly anisotropic materials with extreme dielectric properties. In this work we consider, in the electrostatic case, a regularized, approximate cloaking by mapping scheme and discuss the problem of optimal choice of radial maps, that determine the conductivity distribution of the cloak. We consider two different optimality criteria: minimal maximal anisotropy and minimal mean anisotropy of this conductivity distribution. Using both criteria we show that it is possible to achieve significantly lower anisotropy (for a prescribed level of invisibility) or significantly lower visibility (for a prescribed level of anisotropy). For example, in two dimensions one may achieve exponentially small visibility with a cloak, that in terms of anisotropy (and lowest and highest conductivity) is no worse than the traditional affine map cloak, which only yields quadratically small visibility.

Analysis of an approximate cloaking for acoustic scattering problems in [formula omitted

In this paper, we consider the acoustic cloaking based on singular transformations and its approximation in the context of scattering problems in R3. The cloaking based on singular transformations can not avoid a singularity of the resulting Laplace–Beltrami operator because singular transformations blow up one point to a sphere. In order to treat the singularity properly, we use the variational method proposed in Greenleaf et al. (2007) [1] and prove the possibility of the perfect cloaking. To design an approximation scheme that avoids the singularity of the perfect cloaking, we regularize the singular transformation and show that transmitted fields into the cloaked region in H1-norm and scattered fields on any compact set outside of the cloaked region in L∞-norm converge to zero as a regularization (approximation) parameter approaches zero provided that wave number is not a Neumann eigenvalue of the Helmholtz equation in the cloaked region.

Nearly Cloaking the Electromagnetic Fields

The approximate cloaking is investigated for time-harmonic Maxwell's equations via the approach of transformation optics. The problem is reduced to certain boundary effect estimates due to an inhomogeneous electromagnetic inclusion with an asymptotically small support but an arbitrary content enclosed by a thin high-conducting layer. Sharp estimates are established in terms of the asymptotic parameter, which are independent of the material tensors of the small electromagnetic inclusion. The result implies that the “blow-up-a-small-region” construction via the transformation optics approach yields a near-cloak for the electromagnetic waves. A novelty lies in the fact that the geometry of the cloaking construction of this work can be very general. Moreover, by incorporating the conducting layer developed in the present paper right between the cloaked region and the cloaking region, arbitrary electromagnetic contents can be nearly cloaked. Our mathematical technique extends the general one developed in [H. Y. Liu and H. Sun, J. Math. Pures Appl., 99 (2013), pp. 17--42] for nearly cloaking scalar optics. In order to investigate the approximate electromagnetic cloaking for general geometries with arbitrary cloaked contents in the present study, new techniques and analysis tools are developed for this more challenging vector optics case.