Lorentz Medium Research Articles

SANDWICH-STRUCTURE WAVEGUIDES FOR VERY HIGH-POWER GENERATION AND TRANSMISSION USING LEFT-HANDED MATERIALS

The high power generation and transmission for TM modes in a parallel-plate waveguide filled with three-layered medium called sandwich structure are investigated. The transmission power could never exceed the input power of the source in a conventional parallel-plate waveguide filled with homogeneous or inhomogeneous right-handed material (RHM) or homogenous left-handed material (LHM). Based on the stratified medium theory, extremely high power can be generated and transmitted if the waveguide is composed of RHM-LHM-RHM or LHM-RHM-LHM sandwich structure with the medium parameters and layer thickness being properly chosen. Different from the TE case, the dominant mode exists in such a structure is TM0 mode which can be always supported despite the size of the waveguide. From our numerical results, we find that the high power generation and transmission can be easily realized even in the more realistic case where the LHM is described by the Lorentz medium model.

Brillouin precursor propagation in the THz region in Lorentz media

We numerically calculated the formation of Brillouin precursor in a single resonance Lorentz medium. The contribution from medium dispersion and absorption in the formation of Brillouin precursor is described. Numerical calculations show that Brillouin precursor can be observed for pulses in the THz region propagating in ZnTe.

SIMULATIONS OF THE LEFT-HANDED MEDIUM USING DISCONTINUOUS GALERKIN METHOD BASED ON THE HYBRID DOMAINS

Abstract—The increasing interest in electromagnetic effects in the Left-Handed medium (LHM) requires the formulae capable of the full analysis of wave propagation in such materials. First, we develop a novel technique for discretization of the Lorentz medium model. In order to overcome the instability inherent in the standard perfectly matched layer (PML) absorbing boundary condition (ABC), we derive the modified PML ABC which can be extended to truncate the boundary of LHM. Then a convergent high-order accurate scheme based on triangle domains, discontinuous Galerkin method (DGM), is extended to the new DGM based on hybrid domains, triangle domains and quadrilateral domains. Finally, we adopt the new DGM and modified PML formulations to analysis the electromagnetic phenomena in the LHM. The simulation results show accuracy and stability of the proposed scheme.

Time-domain electromagnetic energy in a frequency-dispersive left-handed medium

From Maxwell's equations and the Poynting theorem, the time-domain electric and magnetic energy densities are generally defined in the frequency-dispersive media based on the conservation of energy. As a consequence, a general definition of electric and magnetic energy is proposed. Comparing with existing formulations of electric and magnetic energy in frequency-dispersive media, the new definition is more reasonable and is valid in any case. Using the new definition and staring from the equation of motion, we have shown rigorously that the total energy density and the individual electric and magnetic energy densities are always positive in a realistic artificial left-handed medium (LHM) [R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001)], which obeys actually the Lorentz medium model, although such a LHM has negative permittivity and negative permeability simultaneously in a certain frequency range. We have also shown that the conservation of energy is not violated in LHM. The earlier conclusions can be easily extended to the Drude medium model and the cold plasma medium model. Through an exact analysis of a one-dimensional transient current source radiating in LHM, numerical results are given to demonstrate that the work done by source, the power flowing outwards a surface, and the electric and magnetic energy stored in a volume are all positive in the time domain.

High-order compact schemes for dispersive media

Staggered high-order compact (HOC) finite difference schemes are developed for modelling electromagnetic wave propagation in dispersive media. The main advantage of HOC schemes is their very low dispersion error, which is dominant in low-order methods. The high accuracy of HOC schemes is demonstrated by examples of wave propagation through first-order Debye and Lorentz media in one dimension.

The time-frequency interpretation for transient evolution of wave propagation through dispersive medium

The propagation of electromagnetic waves in a linear, dispersive Lorentz medium is calculated using the finite-difference time-domain (FDTD) method; their time-frequency (TF) characteristics are studied using the Gabor extension. Numerical results show that the TF spectrum gives a clear interpretation for transient evolution of ultra wideband pulse propagation through a Lorentz medium.

Comparison between Staggered and Unstaggered Finite-Difference Time-Domain Grids for Few-Cycle Temporal Optical Soliton Propagation

The auxiliary-differential-equation formulation of the finite-difference time-domain method has become a powerful tool for modeling electromagnetic wave propagation in linear and nonlinear dispersive media. In the first part of this paper, we compare the stability and accuracy of second- and fourth-order-accurate spatial central discretizations on staggered grids with a third-order-accurate spatial discretization on an unstaggered grid, combined with a second-order leapfrog time integration scheme for modeling linear dispersive phenomena in a one-dimensional single-resonance Lorentz medium. We use on the unstaggered grid the NS2 scheme introduced by Y. Liu, which combines forward and backward differencing for the spatial derivatives. Phase and attenuation errors are determined analytically across the entire frequency band of the low-loss single-resonance Lorentz model. In the second part of the paper, we compare the use of one-dimensional staggered and unstaggered grids for modeling the transient evolution of few-cycle optical localized pulses in a dispersive Lorentz medium with a delayed third-order nonlinearity. The focus of the study is on the decay of higher-order solitons under the combined action of dispersion, self-phase modulation, self-steepening, and stimulated Raman scattering. Numerical results from the simulations on the unstaggered and staggered grids are in excellent agreement. This study demonstrates the accuracy of the unstaggered scheme augmented with a linear/nonlinear dispersive media formulation for temporal soliton propagation.

An FDTD algorithm with perfectly matched layers for general dispersive media

A three-dimensional (3-D) finite difference time domain (FDTD) algorithm with perfectly matched layer (PML) absorbing boundary condition (ABC) is presented for general inhomogeneous, dispersive, conductive media. The modified time-domain Maxwell's equations for dispersive media are expressed in terms of coordinate-stretching variables. We extend the recursive convolution (RC) and piecewise linear recursive convolution (PLRC) approaches to arbitrary dispersive media in a more general form. The algorithm is tested for homogeneous and inhomogeneous media with three typical kinds of dispersive media, i.e., Lorentz medium, unmagnetized plasma, and Debye medium. Excellent agreement between the FDTD results and analytical solutions is obtained for all testing cases with both RC and PLRC approaches. We demonstrate the applications of the algorithm with several examples in subsurface radar detection of mine-like objects, cylinders, and spheres buried in a dispersive half-space and the mapping of a curved interface. Because of their generality, the algorithm and computer program can be used to model biological materials, artificial dielectrics, optical materials, and other dispersive media.

Hybrid analytical–numerical approach for modeling transient wave propagation in Lorentz media

The advent of ultrawideband, short-pulse sources has recently generated renewed interest in the problem of pulse propagation in dispersive media. When one is trying to numerically simulate transient plane-wave propagation through a dispersive medium, the main difficulties are encountered in trying to transform from the frequency domain to the time domain. We develop an asymptotic-extraction technique wherein asymptotic methods are used in conjunction with a fast Fourier transform (FFT) to overcome the limitations of each method. The basic idea behind the asymptotic-extraction technique is to extract and analytically invert as much of the signal as possible. The remainder of the signal is then transformed by use of a numerical FFT. We demonstrate that it is possible to construct functions that possess analytical inverse transforms and also asymptotically model the low- and high-frequency behavior of a field propagating in a single-resonance Lorentz medium. Extraction of the low- and high-frequency responses from the spectral representation for the waveform in essence preconditions the waveform for application of a FFT, thereby significantly reducing the number of sample points required by the FFT. It is also demonstrated that these two extracted functions, whose inverse Fourier transforms are evaluated analytically in terms of known special functions, provide a good approximation for the transient signal for the case of a square-wave-pulse-modulated source, provided that the carrier frequency resides far enough from the absorption band.

Electromagnetic waves in a suddenly created magnetized Lorentz medium (transverse propagation)

The interaction of electromagnetic waves in a suddenly created Lorentz medium with impressed steady uniform magnetic field is considered. It was shown that for the case of the incident electric field normal to the static magnetic field, the source wave splits into six new extraordinary waves whose frequencies are different from the incident waves. Three of those six new waves are transmitted waves, and the other three are reflected waves. The dispersion relation, power transmission, and reflection coefficients are derived and discussed.

Precursor propagation in dispersive media from short-rise-time pulses at oblique incidence

The derivation and the application of a numerical method designed to study the propagation of pulsed electromagnetic fields into dispersive dielectric materials are reported. A Fourier series based methodology, appropriate for a useful class of pulse train incident signals, is presented and utilized to study the dynamics of dispersive pulse propagation in a half-space constructed of a Debye or a Lorentz model medium. Computed solutions are presented showing precursor propagation excited by pulses obliquely incident upon a planar boundary to the medium. The results show that oblique incident pulses upon a Lorentz medium can excite different types of precursor, propagating in unique directions within the half-space.

Transient electromagnetic wave propagation in waveguides

This paper focuses on propagation of transient electromagnetic waves in waveguides of general cross section with perfectly conducting walls. The solution of the transient wave propagation problem relies on a wave splitting technique, which has been frequently used in direct and inverse scattering problems during the last decade. The field in the waveguide is represented as a time convolution of a Green function and the excitation. Some numerical computations illustrate the method. A new way of calculating the first precursor in a Lorentz medium is presented. This method, which is not based upon the classical asymptotic methods, gives an expression of the first precursor at all depths in the medium. The excitation of the waveguide modes for time dependent sources is also addressed.

Stability and phase error analysis of FD-TD in dispersive dielectrics

Four FD-TD extensions for the modeling of pulse propagation in Debye or Lorentz dispersive media are analyzed through studying the stability and phase error properties of the coupled difference equations corresponding to Maxwell's equations and to the equations for the dispersion. For good overall accuracy the author shows that all schemes should be run at their Courant stability limit, and that the timestep should finely resolve the medium timescales. Particularly, for Debye schemes it should be at least /spl Delta/t=10/sup /spl minus/3//spl tau/, while for Lorentz schemes it should be /spl Delta/t=10/sup /spl minus/2//spl tau/, where /spl tau/ is a typical medium relaxation time. A numerical experiment with a Debye medium confirms this. The author has determined that two of the discretizations for Debye media are totally equivalent. In the Lorentz medium case the author establishes that the method that uses the polarization differential equation to model dispersion is stable for all wavenumbers, and that the method using the local-in-time constitutive relation is weakly unstable for modes with wavenumber k such that k/spl Delta/x>/spl pi//2.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Transformation of a time-harmonic electromagnetic field in a suddenly created two-resonance Lorentz medium

The dispersion relation for the six waves generated in the suddenly created two-resonance lossy Lorentz medium has been derived and analyzed by the use of a Brillouin-like dispersion diagram. In comparison to the single-resonance Lorentz medium, another two new waves were excited. The amplitude, power transmission, and reflection coefficients for these new modes have been derived and analyzed for several values of particle density. It was shown that less than 4% of the pump wave power is transformed into the reflected modes.

Uniform asymptotic description of ultrashort Gaussian-pulse propagation in a causal, dispersive dielectric.

The complete asymptotic description of ultrashort Gaussian-pulse propagation in a single-resonance Lorentz medium in the mature-dispersion regime is presented and compared with the results of two independent numerical experiments of the propagated-field evolution. The nonuniform asymptotic method of Olver [Stud. Appl. Math. Rev. 12, 228 (1970)] (an extension of the method of steepest descents) is first applied to obtain the standard asymptotic description of the propagated field that is due to the given input Gaussian-modulated field. The description afforded by this asymptotic method, although nonuniform in certain space-time regions, is found to be in excellent agreement with purely numerical results, especially when exact, numerically determined saddle-point locations, and exact expressions for the derivatives of the phase function are used in the implementation of the theoretical approach. Modern uniform asymptotic techniques, which generalize Olver's nonuniform description, are then employed to obtain a rigorous description of ultrashort Gaussian-pulse propagation that is uniformly valid for all space-time points in the mature-dispersion regime. This asymptotic description clearly shows that the propagated field can be expressed solely in terms of a generalized Sommerfeld and a generalized Brillouin precursor field, the first of which dominates the total propagated field when the carrier frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$ is well above resonance while the second generalized precursor field dominates when ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$ is below or near resonance. It is further shown that the pulse distortion is due solely to the manner in which the precursor field amplitude is modified by the initial Gaussian-pulse envelope spectrum. Finally, the frequency dependence of the signal velocity of the input ultrashort, Gaussian-modulated harmonic field in a single-resonance Lorentz medium is discussed.

On the propagation of short optical pulses in a linear dispersive medium with absorption

A very accurate numerical algorithm based on the Laplace transform is used to describe the field evolution of ultrashort Gaussian-modulated pulses in a single resonance Lorentz medium. The results are compared with previously published numerical results by I. Christov (see ibid., vol.24, p.1548-53, 1988) in which a simple Taylor series expansion of the input pulse spectrum about the carrier frequency is used to explain the observed pulse dynamics. It is shown, by numerical Laplace transform analysis, that these previously published numerical results are inaccurate and that the first few terms of the Taylor series expansion technique about the carrier frequency do not properly explain the physically correct dispersive pulse evolution properties for rapid rise time pulses. In response to the comments the original author recalculated the dispersive propagation of femtosecond optical pulses by using different numerical methods. >

Direct time integration of Maxwell's equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses

The initial results for femtosecond pulse propagation and scattering interactions for a Lorentz medium obtained by a direct time integration of Maxwell's equations are reported. The computational approach provides reflection coefficients accurate to better than 6 parts in 10,000 over the frequency range of dc to 3 x 10 to the 16th Hz for a single 0.2-fs Gaussian pulse incident upon a Lorentz-medium half-space. New results for Sommerfeld and Brillouin precursors are shown and compared with previous analyses. The present approach is robust and permits 2D and 3D electromagnetic pulse propagation directly from the full-vector Maxwell's equations.

Electromagnetic waves in a suddenly created Lorentz medium and plasma

The dispersion relations for the new waves excited by a suddenly created Lorentz media are derived and presented by the use of a Brillouin-like diagram. The amplitude and power transmission and reflection coefficients for these new modes are derived and discussed. It is shown that less than 4% of the pump wave power is transferred to the reflected modes.

Pulse reflection from a lossy Lorentz medium half-space (TM polarization)

A closed form solution for transient reflection from a lossy Lorentz medium half-space has been expressed as an infinite series of Bessel functions of the first kind and fractional order. The delta pulse plane wave excitation in the case of TM polarization has been considered. The variation of a reflected waveform with the angle of incidence, normalized collision frequency and normalized characteristic binding frequency of a lossy Lorentz medium are presented in several computer plots. The time domain Brewster angle is noted in the early time response. It is shown how the obtained transient response might be used for diagnosing the main characteristic parameters of the Lorentz medium.

Pulse propagation in a linear, causally dispersive medium

The uniform asymptotic description of optical pulse propagation in a linear, causally dispersive, absorptive dielectric, as described by the classical Lorentz model, is presented. The resultant pulse distortion in a single resonance Lorentz medium is shown to be primarily due to the Sommerfeld and Brillouin precursor fields. The mathematical approach does not rely upon any quasimonochromatic or slowly varying envelope approximation and consequently provides a canonical description of linear pulse dispersion dynamics that is completely valid for rapid risetime pulses of arbitrary time duration. The results do not depend upon any nth-order dispersion approximation so that the causality relations that are critical to the physically proper analysis of linear dispersive pulse propagation phenomena are maintained. >