Infrared (IR) glass-ceramics (GCs) hold the potential to dramatically expand the range of optical material solutions available for use in bulk and planar optical systems in the IR. Current material solutions are limited to single-or polycrystalline materials and traditional IR-transparent optical glasses. GCs that can be processed with spatial control and extent of induced crystallization present the opportunity to realize an effective refractive index variation, enabling arbitrary gradient refractive index elements with tailored optical function. This work discusses the role of the parent glass composition and morphology on nanocrystal phase formation in a multicomponent chalcogenide glass. Through a two-step heat treatment protocol, a Ge-As-Pb-Se glass is converted to an optical nanocomposite where the type, volume fraction, and refractive index of the precipitated crystalline phase(s) define the resulting nanocomposite's optical properties. This modification results in a giant variation in infrared Abbe number, the magnitude of which can be tuned with control of crystal phase formation. The impact of these attributes on the GCs' refractive index, transmission, dispersion, and thermo-optic coefficient is discussed. A systematic protocol for engineering homogeneous or gradient changes in optical function is presented and validated through experimental demonstration employing this understanding.

Abstract. We consider the inverse problem of recovering an isotropic electrical conductivity from interior knowledge of the magnitude of one current density field generated by applying current on a set of electrodes. The required interior data can be obtained by means of MRI measurements. On the boundary we only require knowledge of the electrodes, their impedances, and the corresponding average input currents. From the mathematical point of view, this practical question leads us to consider a new weighted minimum gradient problem for functions satisfying the boundary conditions coming from the Complete Electrode Model (CEM) of Somersalo, Cheney and Isaacson. We show that this variational problem has non-unique solutions. The surprising discovery is that the physical data is still sufficient to determine the geometry of (the connected components of) the level sets of the minimizers. We thus obtain an interesting phase retrieval result: knowledge of the input current at the boundary allows determination of the full current vector field from its magnitude. We characterize locally the non-uniqueness in the variational problem. In two and three dimensions we also show that additional measurements of the voltage potential along a curve joining the electrodes yield unique determination of the conductivity. The proofs involve a maximum principle and a new regularity up to the boundary result for the CEM boundary conditions. A nonlinear algorithm is proposed and implemented to illustrate the theoretical results.

We consider the problem of reconstruction of a sufficiently smooth planar conductivity from the knowledge of the magnitude |J| of one current density field inside the domain, and the corresponding voltage and current on a part of the boundary. Mathematically, we are lead to the Cauchy problem for the the 1-Laplacian with partial data. Different from existing works, we show that the equipotential lines are characteristics in a first order quasilinear partial differential equation. The conductivity can be recovered in the region flown by the characteristics originating at parts of the boundary where the data is available. Numerical experiments show the feasibility of this alternative method.

We consider the numerical solvability of the Dirichlet problem for the 1-Laplacian in a planar domain endowed with a metric conformal with the Euclidean one. Provided that a regular solution exists, we present a globally convergent method to find it. The global convergence allows to show a local stability in the Dirichlet problem for the 1-Laplacian nearby regular solutions. Such problems occur in conductivity imaging, when knowledge of the magnitude of the current density field (generated by an imposed boundary voltage) is available inside. Numerical experiments illustrate the feasibility of the convergent algorithm in the context of the conductivity imaging problem.

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