The introduction of metamaterials and transformation optics has brought the possibilities for manipulating electromagnetic waves to an unprecedented level, suggesting applications like super-resolution imaging, cloaking, subwavelength focusing, and field localization. The refractive index of metamaterial structures in transformation optics typically has to be spatially graded. This paper presents a full analytical method for description of the field propagation through composites with gradient refractive index. The remarkable property of this approach is that it gives explicit general expressions for the field intensity and transmission and reflection coefficients, without reference to any boundary conditions. This opens a possibility for a novel fundamental theory of a number of important electromagnetic phenomena. The method enables calculation of wave propagation parameters within structures with arbitrary losses, arbitrary spectral dispersions, and arbitrary slopes of permittivity and permeability gradients, from mild to abrupt.
We propose a method to determine approximate phase-integral analytical solutions for electric and magnetic fields in flat lenses with the refractive index varying in the radial direction. In our model the gradient of refractive index is approximated by a large number of concentric annuli with step-increasing index. Here the central part contributes to a bulk of the phase transformation, while the external layers act as a graded antireflective structure, matching the impedance of the lens to that of the free space. Such lenses can be modeled as compact composites with continuous permittivity and (if needed) permeability functions which asymptotically approach unity at the boundaries of the composite cylinder. We illustrate the phase-integral approach by obtaining the approximate analytic solutions for the electric and magnetic fields for a special class of composite designs with radially graded parameters.
We present a study of exact analytic solutions for electric and magnetic fields in continuously graded flat lenses designed utilizing transformation optics. The lenses typically consist of a number of layers of graded index dielectrics in both the radial and longitudinal directions, where the central layer in the longitudinal direction primarily contributes to a bulk of the phase transformation, while other layers act as matching layers and reduce the reflections at the interfaces of the middle layer. Such lenses can be modeled as compact composites with continuous permittivity (and if needed) permeability functions which asymptotically approach unity at the boundaries of the composite cylinder. We illustrate the proposed procedures by obtaining the exact analytic solutions for the electric and magnetic fields for one simple special class of composite designs with radially graded parameters. To this purpose we utilize the equivalence between the Helmholtz equation of our graded flat lens and the quantummechanical radial Schrodinger equation with Coulomb potential, furnishing the results in the form of Kummer confluent hypergeometric functions. Our approach allows for a better physical insight into the operation of our transformation optics-based graded lenses and opens a path toward novel designs and approaches.
We investigate TE-wave propagation in a hollow waveguide with a graded dielectric barrier, using an equivalent model of the waveguide filled with a stratified medium. General formulae for the electric field components of the TE-waves, applicable to hollow waveguides with arbitrary cross sectional shapes, are presented. As an illustration, we obtain the exact analytical results for the electric field components in a rectangular waveguide, as well as the exact analytical results for reflection and transmission coefficients which are valid for waveguides of arbitrary cross sectional shapes.
A new simplified formula is derived for the absorption cross section of small dielectric ellipsoidal particles embedded in lossy media. The new expression leads directly to a closed form solution for the optimal conjugate match with respect to the surrounding medium, i.e. the optimal permittivity of the ellipsoidal particle that maximizes the absorption at any given frequency. This defines the optimal plasmonic resonance for the ellipsoid. The optimal conjugate match represents a metamaterial in the sense that the corresponding optimal permittivity function may have negative real part (inductive properties), and can not in general be implemented as a passive material over a given bandwidth. A necessary and sufficient condition is derived for the feasibility of tuning the Drude model to the optimal conjugate match at a single frequency, and it is found that all the prolate spheroids and some of the (not too flat) oblate spheroids can be tuned into optimal plasmonic resonance at any desired center frequency. Numerical examples are given to illustrate the analysis. Except for the general understanding of plasmonic resonances in lossy media, it is also anticipated that the new results can be useful for feasibility studies with e.g. the radiotherapeutic hyperthermia based methods to treat cancer based on electrophoretic heating in gold nanoparticle suspensions using microwave radiation.
An optimal plasmonic resonance and the associated Fröhlich resonance frequency are derived for a thin layer in a straight waveguide in TM mode. The layer consists of an arbitrary composite material with a Drude type of dispersion. The reflection and transmission coefficients of the layer are analyzed in detail. To gain insight into the behavior of a thin plasmonic layer, an asymptotic expansion to the first order is derived with respect to the layer permittivity.
An optimal plasmonic resonance is derived for small homogeneous and isotropic inclusions in a lossy surrounding medium. The optimal resonance is given in terms of any particular eigenmode (electrostatic resonance) associated with the double-layer potential for a smooth, but otherwise arbitrary surface.
Classical homogenization theory based on the Hashin–Shtrikman coated ellipsoids is used to model the changes in the complex valued conductivity (or admittivity) of a lung during tidal breathing. Here, the lung is modeled as a two-phase composite material where the alveolar air-filling corresponds to the inclusion phase. The theory predicts a linear relationship between the real and the imaginary parts of the change in the complex valued conductivity of a lung during tidal breathing, and where the loss cotangent of the change is approximately the same as of the effective background conductivity and hence easy to estimate. The theory is illustrated with numerical examples based on realistic parameter values and frequency ranges used with electrical impedance tomography (EIT). The theory may be potentially useful for imaging and clinical evaluations in connection with lung EIT for respiratory management and control.
We consider the inverse problem of reconstructing the shape of a deformation in one of the broad walls of a rectangular waveguide. Assuming a small deformation, resulting in weak scattering, the direct problem is solved using a first order perturbation approach. Hence, the inverse problem becomes linear and is formulated as an equation system for a set of expansion coefficients. The illposedness of the inverse problem is handled with regularization, by adding a penalty term which weight is determined by the L-curve method. The theory is tested on experimental reflection data, using the dominant mode of the waveguide. The reconstructed shape is in qualitative agreement with the true shape, but a detailed resolution cannot be obtained due to insufficient quality of the experimental data. Extensions and improvements of the method are discussed.