Invisibility devices have captured the human imagination for many years. Recent theories have proposed schemes for cloaking devices using transformation optics and conformal mapping. Metamaterials, with spatially tailored properties, have provided the necessary medium by enabling precise control over the flow of electromagnetic waves. Using metamaterials, the first microwave cloaking has been achieved but the realization of cloaking at optical frequencies, a key step towards achieving actual invisibility, has remained elusive. Here, we report the first experimental demonstration of optical cloaking. The optical 'carpet' cloak is designed using quasi-conformal mapping to conceal an object that is placed under a curved reflecting surface by imitating the reflection of a flat surface. The cloak consists only of isotropic dielectric materials, which enables broadband and low-loss invisibility at a wavelength range of 1,400-1,800 nm.

An optical cloak made of dielectrics

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Regularization Of Inverse Problems

Transformation optics describes the capability to design the path of light waves almost at will through the use of metamaterials that control effective materials properties on a subwavelength scale. In this review, the physics and applications of transformation optics are discussed.

Transformation optics and metamaterials

Invisibility or cloaking has captured human's imagination for many years. With the recent advancement of metamaterials, several theoretical proposals show cloaking of objects is possible, however, so far there is a lack of an experimental demonstration at optical frequencies. Here, we report the first experimental realization of a dielectric optical cloak. The cloak is designed using quasi-conformal mapping to conceal an object that is placed under a curved reflecting surface which imitates the reflection of a flat surface. Our cloak consists only of isotropic dielectric materials which enables broadband and low-loss invisibility at a wavelength range of 1400-1800 nm.

Dielectric Optical Cloak

Like their optical counterparts, acoustic metamaterials are capable of manipulating sound waves in unusual ways. An acoustic hyperlens is now demonstrated that is capable of magnifying subwavelength acoustic waves, and could therefore find applications in medical imaging or underwater sonar. Acoustic metamaterials can manipulate sound waves in surprising ways, which include collimation, focusing, cloaking, sonic screening and extraordinary transmission1,2,3,4,5,6,7,8,9,10,11,12,13,14. Recent theories suggested that imaging below the diffraction limit using passive elements can be realized by acoustic superlenses or magnifying hyperlenses15,16. These could markedly enhance the capabilities in underwater sonar sensing, medical ultrasound imaging and non-destructive materials testing. However, these proposed approaches suffer narrow working frequency bands and significant resonance-induced loss, which hinders them from successful experimental realization. Here, we report the experimental demonstration of an acoustic hyperlens that magnifies subwavelength objects by gradually converting evanescent components into propagating waves. The fabricated acoustic hyperlens relies on straightforward cutoff-free propagation and achieves deep-subwavelength resolution with low loss over a broad frequency bandwidth.

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Experimental demonstration of an acoustic magnifying hyperlens.

The problem of determining a continuously distributed neuronal current inside the brain within the framework of the three-shell model was analysed in Fokas (2009 J. R. Soc. Interface 6 479–88), where it was shown that the simultaneous use of electro-encephalography and magneto-encephalography yields information about two of the three scalar functions specifying the interior current. In particular, for the spherical and ellipsoidal geometries, it is possible to determine the angular parts of these two functions, as well as to obtain certain explicit constraints satisfied by their radial parts. The complete determination of the radial parts of these two functions, as well as the determination of the third unknown function, requires some a priori assumption about the current. One such possible assumption is that the current minimizes the L2-norm. It is shown here that in the case of spherical geometry this assumption yields a unique and explicit formula for the current.

Electro-magneto-encephalography for the three-shell model: minimal L-2-norm in spherical geometry

New Analytic and Geometric Methods in Inverse Problems

We consider a Riemannian polyhedron of a special type with a piecewise smooth boundary. The associated Neumann Laplacian defines the boundary spectral data as the set of eigenvalues and restrictions to the boundary of the corresponding eigenfunctions. In this paper we prove that the boundary spectral data prescribed on an open subset of the polyhedron boundary determine this polyhedron uniquely, i.e. up to an isometry.

Inverse boundary spectral problem for Riemannian polyhedra

The problem of determining the neuronal current inside the brain from measurements of the induced magnetic field outside the head is discussed under the assumption that the space occupied by the brain is approximately spherical. By inverting the Geselowitz equation, the part of the current which can be reconstructed from the measurements is precisely determined. This actually consists of only certain moments of one of the two functions specifying the tangential part of the current. The other function specifying the tangential part of the current as well as the radial part of the current are completely arbitrary. However, it is also shown that with the assumption of energy minimization, the current can be reconstructed uniquely. A numerical implementation of this unique reconstruction is also presented.

The Unique Determination of Neuronal Currents in the Brain via Magnetoencephalography

We consider a geometric inverse problems associated with interior measurements: Assume that on a closed Riemannian manifold (M,h) we can make measurements of the point values of the heat kernel on some open subset U ⊂ M. Can these measurements be used to determine the whole manifold M and metric h on it? In this paper we analyze the stability of this reconstruction in a class of n-dimensional manifolds which may collapse to lower dimensions. In the Euclidean space, stability results for inverse problems for partial differential operators need considerations of operators with non-smooth coefficients. Indeed, operators with smoothcoefficients can approximate those with non-smooth ones. For geometric inverse problems, we can encounter a similar phenomenon: to understand stability of the solution of inverse problems for smooth manifolds, we should study the question of uniqueness for the limiting non-smooth case. Moreover, it is well-known, that a sequence of smooth n-dimensional manifolds can collapse to a non- smooth space of lower dimension. To analyze the stability of inverse problem in a class of smooth manifolds with bounded sectional curvature and diameter, we study properties of the spaces which occur as limits of smooth collapsing manifolds and study the uniqueness of the inverse problems on the class of the limit spaces. Combining these, we obtain stability results for inverse problems in the class of smooth manifolds with bounded sectional curvature and diameter.

Yaroslav Kurylev

Papers

Electro-magneto-encephalography for the three-shell model: minimal L-2-norm in spherical geometry

New Analytic and Geometric Methods in Inverse Problems

Inverse boundary spectral problem for Riemannian polyhedra

The Unique Determination of Neuronal Currents in the Brain via Magnetoencephalography

Uniqueness and Stability in Inverse Spectral Problems for Collapsing Manifolds