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First Demonstration of Focusing by Plano-Concave Lens

P. Vodo, P. V. Parimi, W. T. Lu and S. Sridhar Applied Physics Letters, V.86, P.201108 (2005) We demonstrate focusing of a plane microwave by a planoconcave lens fabricated from a photonic crystal having a negative refractive index and left-handed electromagnetic properties. An inverse experiment, in which a plane wave is produced from a source […]

First Demonstration of Imaging by Flat Lens published in Nature

First Demonstration of Imaging by Flat Lens 11/27/03 P. Parimi, W. T. Lu, P. Vodo and S. Sridhar Nature, V. 426, P. 404 (2003) The positive refractive index of conventional optical lenses means that they need curved surfaces to form an image, whereas a negative index of refraction allows a flat slab of a material […]

Localization of Wavefunctions in Disordered Billiards

Experimental observation of localized wavefunctions in disordered billiards, and deviations from the Porter-Thomas distribution due to localization in disordered billiards. These were the first experiments to be performed on disordered billiards.

We show the evolution of localization with increasing scattering by measuring wavefunctions in disordered billiards.

Sorting out quantum chaos in the microwave lab

Ivars Peterson Science News, P. 264 (1995) A smooth stroke ending in a sharp tap with a billiard cue sends a ball scooting across the table’s green felt surface. The ball bounces off one cushion, then another before smacking squarely into a second ball. On a rectangular table, an expert player can return the first […]

Wavefunctions in Chaotic and Disordered Geometries

The Sinai-Stadium billiard is a prototype example of chaotic geometries. Representative eigenfunctions of this geometry are shown here. The influence of the classical dynamical structures such as periodic orbits can be seen in some of the wavefunctions. The Sinai-Stadium was designed to have no non-isolated periodic orbits, which leads to deviation from universality.

Integrability, Chaos and Disorder in Wavefunctions

A unique feature of our experiments is the ability to directly measure eigenfunctions, i.e. the spatial distribution of the waves, in essentially arbitrary geometries. The wavefunctions are obtained using a cavity perturbation technique developed by Sridhar. In this technique, a small metal bead is introduced inside the cavity. If the bead is sufficiently small compared to the wavelength, the resultant shift in frequency due to the perturbation is proportional to the square of the Electric field (hence the wavefunction), at the location of the bead. By moving the bead with a magnet, the wavefunction can be mapped out.

Not Hearing the Shape of Drums

While the theorem of isospectral domains was proved on abstract mathematical grounds, the actual resonance frequencies and wave functions of the geometries were not known. The microwave experiments were able to make an unique contribution by exploiting the the fact that below a certain frequency, the equation obeyed in thin microwave cavities is the same as the Helmholtz wave equation. The resonances were obtained by actually constructing the shapes using copper. The wavefunctions are obtained using a cavity perturbation technique. Therefore these experiments were not only able to physically realize drums which sounded alike, but also were able to show that there were no degeneracies in the first 54 resonances.

Measuring Experimental Eigenvalues and Eigenfunctions

In closed or open billiard geometries, Maxwell’s equations can be written as (Ñ2 + k2) Ψ = 0, where in general Ψ = {Εi, Βj} is a vector field. In thin 2-D geometries bounded by parallel metallic plates in the x-y plane, the wave equation for the TM modes (Βz = 0) reduces to the time independent Schrodinger equation (Ñ2 + k2) Ψ = 0, for frequencies ƒ < c/2d, where d is the plate separation. This QM-E&M mapping with Ψ = Εz allows us to study 2-D problems in Quantum Chaos by suitably constructing classically chaotic geometries. Eigenvalues, eigenfunctions, scattering resonances and widths are measured and analyzed, yielding insights towards the quantum classical correspondence.