Abstract We formulate and validate a finite element approach to the propagation of a slowly decaying electromagnetic wave, called surface plasmon–polariton, excited along a conducting sheet, e.g., a single-layer graphene sheet, by an electric Hertzian dipole. By using a suitably rescaled form of time-harmonic Maxwell's equations, we derive a variational formulation that enables a direct numerical treatment of the associated class of boundary value problems by appropriate curl-conforming finite elements. The conducting sheet is modeled as an idealized hypersurface with an effective electric conductivity. The requisite weak discontinuity for the tangential magnetic field across the hypersurface can be incorporated naturally into the variational formulation. We carry out numerical simulations for an infinite sheet with constant isotropic conductivity embedded in two spatial dimensions; and validate our numerics against the closed-form exact solution obtained by the Fourier transform in the tangential coordinate. Numerical aspects of our treatment such as an absorbing perfectly matched layer, as well as local refinement and a posteriori error control are discussed.
In the last few decades, several groups have been studying implications of surface plasmonics; for a (definitely non-exhaustive) sample of related works, see @cite_56 @cite_27 @cite_6 @cite_53 @cite_57 @cite_29 @cite_23 @cite_0 @cite_45 . For instance, in @cite_56 the authors review macroscopic properties of the electric conductivity of graphene, derive dispersion relations for electromagnetic plane waves in inhomogeneous structures, and discuss methods for exciting SPPs; see also the integral-equation approach in @cite_45 . On the other hand, the problem of a radiating dipole source near a graphene sheet is semi-analytically addressed in @cite_27 @cite_6 . In the same vein, in @cite_29 the authors numerically study the field produced by dipoles near a graphene sheet, recognizing a region where the scattered field may be significant. Most recently, two of us derived closed-form analytical expressions for the electromagnetic field when the dipole source and observation point lie on the sheet @cite_57 .
@cite_53 @cite_29 @cite_6 @cite_56 @cite_57 @cite_0 @cite_27 @cite_45 @cite_23
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We derive and interpret solutions of time-harmonic Maxwell’s equations with a vertical and a horizontal electric dipole near a planar, thin conducting film, e.g., graphene sheet, lying between two unbounded isotropic and non-magnetic media. Exact expressions for all field components are extracted in terms of rapidly convergent series of known transcendental functions when the ambient media have equal permittivities and both the dipole and observation point lie on the plane of the film. These solutions are simplified for all distances from the source when the film surface resistivity is large in magnitude compared to the intrinsic impedance of the ambient space. The formulas reveal the analytical structure of two types of waves that can possibly be excited by the dipoles and propagate on the film. One of these waves is intimately related to the surface plasmon-polariton of transverse-magnetic polarization of plane waves. Excitation of the discrete (surface-wave plasmon propagation mode) and continuous (radiation modes) spectrum by a point current source in the vicinity of graphene is examined. The graphene is represented by an infinitesimally thin, local, and isotropic two-sided conductivity surface. The dynamic electric field due to the point source is obtained by complex-plane analysis of Sommerfeld integrals, and is decomposed into physically relevant contributions. Frequencies considered are in the GHz through mid-THz range. As expected, the TM discrete surface wave (surface plasmon) can dominate the response along the graphene layer, although this depends on the source and observation point location and frequency. In particular, the TM discrete mode can provide the strongest contribution to the total electric field in the upper GHz and low THz range, where the surface conductivity is dominated by its imaginary part and the graphene acts as a reactive (inductive) sheet. V C 2011 American Institute of Physics. [doi:10.1063 1.3662883] We discuss the properties of surface plasmons-polaritons in graphene and describe four possible ways of coupling electromagnetic radiation in the terahertz (THz) spectral range to this type of surface waves: (i) the attenuated total reflection (ATR) method using a prism in the Otto configuration, (ii) graphene micro-ribbon arrays or monolayers with modulated conductivity, (iii) a met al stripe on top of the graphene layer, and (iv) graphene-based gratings. The text provides a number of original results along with their detailed derivation and discussion. In this paper we consider the localization of a collection of small, three-dimensional bounded homogeneous inclusions via time-harmonic electromagnetic means, typically using arrays of electric or magnetic dipole transmitters and receivers with given polarization(s) at some distance from the collection, possibly also lying in the far field. The inclusions, somewhat apart or closely spaced, are buried within a homogeneous medium, and are of arbitrary contrast of permittivity, conductivity, and permeability vis-ag-vis this embedding medium. The problem is formulated as an inverse scattering problem for the full Maxwell equations and it involves a robust asymptotic modeling of the multistatic response matrix. No specific application is studied at this stage, but characterization of obstacles in subsoils, nondestructive evaluation of man-made structures, and medical imaging are primary fields of application envisaged. The proposed approach uses a MUSIC (multiple signal classification)-type algorithm, and it yields fast numbering, accurate localization, and estimates of the electromagnetic and geometric parameters (polarization tensors) of the inclusions. The mathematical machinery is detailed first, some specific attention being given to triaxial ellipsoidal inclusions and degenerate spherical shapes (for the latter known results are retrieved). Then, the viability of this algorithm—which would be easily extended to planarly layered environments by introduction of their Green’s functions—is documented by a variety of numerical results from synthetic, noiseless, and severely noisy field data.