Multipole methods have evolved to be an important class of theoretical and computational techniques in the study of photonic crystals and related problems. In this chapter, we present a systematic and unified development of the theory, and apply it to a range of scattering problems including finite sets of cylinders, two-dimensional stacks of grating and the calculation of band diagrams from the scattering matrices of grating layers. We also demonstrate its utility in studies of finite systems that involve the computation of the local density of states.

In this paper, a new computational technique is presented for the first time. In this method, physical differential equations are incorporatedin to interpolations of basic element in finite element methods. This is named physical spline finite element method (PSFEM). Theoretically, the physical spline interpolation introduces many new features. First, physical equations can be usedin the interpolations to make the interpolations problem-associated. The algorithm converges much faster than any general interpolation while keeping the simplicity of the first order Lagrange interpolation. Second, the concept of basis functions may need to be re-examined. Thirdly, basis functions could be complex without simple geometric explanations. The applications to typical one-dimensional electromagnetic problems show the great improvements of the newly developed PSFEM on accuracy, convergence andstabilit y. It can be extendedto other applications. Extension to two- andthree-d imensional problems is briefly discussed in the final section.

The usual aim with any waveguide is to operate it with only the fundamental mode propagating. With fully closed waveguides, finding the band over which this is possible turns on no more than knowledge of the cutoff frequencies of the fundamental and first higher order modes. With open waveguides, the question is not so simply answered. Such waveguides propagate at most a finite set of bound modes together with a continuous modal spectrum that has no counterpart with closed guides. In this paper, for several particular two-conductor transmission lines, we investigate the circumstances under which leaky wave modes, though not themselves members of any orthonormal set of basis functions, can be used to set bounds on the band over which it is to be expected that the transmitted field is substantially contained in the fundamental TEM mode. The method used relies on transverse resonance.

Abstract-The paper deals with the modeling, based on the Finite Difference Time Domain method, of active one- and twodimensional photonic crystals. The onset of laser oscillation is observed by simulating the active substance as having a negative frequency-dependent Lorentzian-shaped conductivity so including into Maxwell's equations an electric current density. Particular attention is devoted to the implementation of uniaxial perfectly matched layer absorbing boundary conditions for the simulation of infinitely extending structures having gain features. Laser behaviour is simulated as a function of various parameters; the threshold wavelengthand conductivity are evaluated as the wavelengthand conductivity where the transmittance diverges. Moreover, the properties of the active two-dimensional photonic band gap structures are given in terms of a Q quality factor which increases by increasing the crystal size and strongly depends on the lattice shape. For the square lattice, when the crystal size increases from N = 2 to N = 8 the Q-factor increases by about an order of magnitude (from 0.027 to 0.110) for TE polarization while for TM polarization it decreases from 0.025 to 0.022. At last the Q-factor pertaining to the chess-board lattice, to parity of other parameters, assumes greater values and already for N = 4, it reaches the values obtained for the 16×8 square lattice, for bothTE and TM polarizations.

It is shown that in hyperbolic spaces, an electromagnetic radiation experiences shifts in spectrum as a function of curvature and distance. The equation relating distance in hyperbolic space, its curvature, and spectral shift is derived by method of horospheres. The active nature of the Lobachevskian vacuum is discussed with applications to physics.

Propagation of waves in circular waveguide withth e boundary condition of hard surface is considered. The waveguide is filled with uniaxial chiral material. This study is a generalization of previously studied cases withisotropic chiral or anisotropic material filling. The eigenvalue equation is formed and the corresponding eigenmodes are presented. It is seen that the hard surface boundary condition simplifies the field analysis remarkable. While the eigenwaves in anisotropic waveguide were TE and TM fields in this more general case the eigenwaves are elliptically polarized hybrid fields. Since the eigenwaves are certain combinations of TE and TM fields and propagate withdifferen t propagation factors, uniaxial chiral medium can be used for polarization transformation. Reflection and transmission from a uniaxial chiral section of a waveguide is analyzed withn umerical examples.

An analytical model of a grid composed of small S-shaped conducting particles located on the surface of a dielectric slab is presented. This approach replaces the original one-layer structure with metallic particles printed on the interface by a multilayered structure with homogenized permittivities for each layers. This way one can homogenize the arrays of small resonant particles. The analytical model is verified by numerical simulations for the case of normal incidence of the plane wave. The homogenization is possible due to the small sizes of S-particles compared to the resonant wavelength in the substrate and due to the small thickness of the whole structure.

We consider oscillations in cylindrical slotted resonators formed by combinations of rectangular domains with several slots cut in the walls using the methods of approximate semi-inversion of integral operator-valued functions with a logarithmic singularity of the kernel. The initial boundary value problems for the Helmholtz equation are reduced to Fredholm integral equations and systems of integral equations of the first kind with a logarithmic singularity. In the case of narrow slots, the dispersion equations are obtained and evaluated using perturbations and the small-parameter method. Eigenfrequencies and eigenfields are calculated explicitly. The values of geometrical and material parameters are determined that lead to the interaction of oscillations. The results obtained are used for improving the design of filters and switches on the basis of simple model prototype structures.

A Combined Genetic Algorithm and Method of Moments design methods is presented for the design of unusual near-field antennas for use in Magnetic Resonance Imaging systems. The method is successfully applied to the design of an asymmetric coil structure for use at 190 MHz and demonstrates excellent radiofrequency field homogeneity.

This paper presents an improved variant of timedomain method for predicting crosstalk on parallel-coupled matched terminated microstrip lines. This method derives simple near-end and far-end time-domain crosstalk expressions which are applicable to lossless case with significant harmonic frequency < 1 GHz. The expressions are in polynomial form with geometrical dimensions of the structure and stimulus information as the only required entry parameters. They are simpler as compared to other methods because the difficult-to-determine distributed RLCG electrical parameters of the coupled lines are not needed. A look-up table for the polynomial coefficients is generated for easy application of this technique. The expressions are applicable for board thickness of 4-63 mils, 30-70Ω line characteristic impedance, 0.5W-4.0W (where W is the line width) inner edge to edge separation, and 3-5 dielectric constant. For significant harmonic frequency > 1 GHz, the effect of both losses and dispersion on the crosstalk levels is accounted for by investigating the gradient of the distorted driving signal. The peak crosstalk levels are then predicted by modifying the time derivative term in the lossless expressions. In addition, the far-end crosstalk is proved to saturate at half of the magnitude of the driving signal entering the active line. The saturation phenomenon is studied from the viewpoint of difference in odd-mode and even-mode propagation velocities.

The paper deals with proposition and evaluation of new and specific methods to represent vector radar data acquired by means a side-looking measurement in order to use compression process of Lind, Buzo, Gray (LBG), and Kohonen's self organizing feature maps of topology. The aim is to enable after coding, transmission, and decoding a high-resolution reconstruction image using the Synthetic Aperture Radar (SAR) methods. The approach proposed for compression uses the statistical properties of the signals to be compressed in order to perform the vector quantification in an optimal way.

A new synthesis algorithm for shaped, double-reflector antennas with complex array feed is presented. The approach presented here aims to improve the efficiency of synthesis techniques without missing the required accuracy. The algorithm is based on a convenient splitting of the original problem into two phases, each one involving a sub-problem significantly simpler than the original one. A double reflector synthesis problem involving only Fourier Transform (FT) operators is of concern during the first phase. The subre- flector surface and a first estimate of the main reflector geometry are obtained in this step. A single reflector synthesis problem is considered during the second phase wherein the final main reflector surface and the excitation coefficients of the primary feed array are obtained. While in the first phase only approximate relationships between the unknowns and the secondary radiated field are exploited, in the second phase accurate radiation operators are involved. Despite this accuracy, the second phase is still numerically effective since it involves a single reflector synthesis problem and exploits, as "good" starting point, the main reflector estimate obtained during the first phase. The effectiveness of the approach is due to the fact that the necessity of dealing simultaneously with two reflector surfaces, the key of the synthesis difficulties, is afforded only during the first phase where efficient computational tools are allowed. A numerical example shows the effectiveness of the proposed approach.

Dyadic Green's functions for inhomogeneous parallel-plate waveguides are considered. The usual residue series form of the Green's function is examined in the case of modal degeneracies, where secondorder poles are encountered. The corresponding second-order residue contributions are properly interpreted as representing "associated functions" of the structure by constructing a new dyadic root function representation of the Hertzian potential Green's dyadic. The dyadic root functions include both eigenfunctions (corresponding to first-order residues) and associated functions, analogous to the idea of Jordan chains in finite-dimensional spaces. Numerical results are presented for the case of a two-layer parallel-plate waveguide.

The field conditions inside a vibrating intrinsic reverberation chamber (VIRC) are examined. By the use of the Finite Difference Time Domain (FDTD) method,the field strength in the VIRC is calculated,and an investigation of the field uniformity and the field distribution is performed. The modes inside the cavity are excited by applying an appropriately modulated waveform on a dipoles gap. The use of this kind of waveform enables the study of the field conditions over a wide frequency range. On the contrary,an implementation of the field excitation with an unmodulated carrier would require a simulation of the FDTD method at each frequency of interest. Thus,a considerable reduction in the simulation time is achieved. The results presented,describing the field behavior inside the enclosure,agree with theory to a high degree.

In this paper, we present time-domain integral equation (TDIE) formulations for analyzing transient electromagnetic responses from three-dimensional (3-D) arbitrary shaped closed conducting bodies using the time-domain electric field integral equation (TDEFIE), the time-domain magnetic field integral equation (TD-MFIE), and the time-domain combined field integral equation (TD-CFIE). Instead of the conventional marching-on in time (MOT) technique, the solution methods in this paper are based on the Galerkin's method that involves separate spatial and temporal testing procedure. Triangular patch basis functions are used for spatial expansion and testing functions for arbitrarily shaped 3-D structures. The timedomain unknown coefficient is approximated by using an orthonormal basis function set that is derived from the Laguerre functions. These basis functions are also used as temporal testing. Using these Laguerre functions it is possible to evaluate the time derivatives in an analytic fashion. We also propose a second alternative formulation to solve the TDIE. The methods to be described result in very accurate and stable transient responses from conducting objects. Detailed mathematical steps are included and representative numerical results are presented and compared.

This paper is a theoretical study of solitons in multidimensions, also known as optical bullets, that is governed by the nonlinear Schrodinger's equation in 1 + 3 dimensions. The parameter dynamics of such multidimensional solitons has been obtained. The study is extended to obtain the adiabatic evolution of soliton parameters in presence of the perturbation terms. Furthermore, the parameter dynamics for the vector multidimensional solitons and including the presence of the perturbation terms has been obtained.

An analytic solution to the problem of plane wave scattering by an achiral multilayered sphere in a host chiral medium is obtained in this paper. By applying the radiation-to-scattering transform, the scattering problem can be considered as the specific radiation problems where the radiated source equivalent to the electromagnetic plane wave is located at infinity. The volumetric currents which generate right circular polarization (RCP) and left circular polarization (LCP) plane waves,resp ectively, are found. An integral equation consisting the volumetric current distributions and the dyadic Green's functions is formulated to obtain both the equivalent incident wave fields and the scattered fields. Two-layered lossless and lossy dielectric spheres and a conducting sphere with a dielectric coated layer buried in an infinitely extended host chiral medium are considered and the expressions for the scattered fields in far-zone are found in explicit analytic form. The characteristics of scattered fields are illustrated and discussed in terms of the circular polarization degree and linear polarization degree against different chiral admittances and sizes.

In this paper, the problem of scattering from sea surface with and without oil slicks is investigated taking the finite size of the illuminated area into account. A model of an inhomogeneous random rough surface with finite size of the scattering area is considered. To apply the results for a broad range of the random surface spectrum, an approach is developed which extends the range of validity beyond that of small perturbation theory. The general expression obtained for the scattering cross section takes into account a modulation of the rough surface by long surface waves. Analytical and numerical studies of the scattering cross section are provided to investigate the role of different mechanisms of scattering from various parts of the surface spectrum, and of diffraction caused by the finite size of the area. It is shown that the area size may affect the normalized scattering cross section in the case of the surface with a slick. Possibilities to explain the features of the suppression of the backscattering by oil slicks 238 Timchenko, Serebryannikov, and Schünemann are discussed. Furthermore, a way to distinguish between different scattering mechanisms is suggested.

The purpose of this paper is to use the analytical asymptotic extraction technique to analyze the bend discontinuity. We show that the derived analytical techniques significantly reduce the computational time while improving the accuracy compared to the conventional method. Especially, the advantage of the proposed method can eliminate the truncation error for evaluating the asymptotic part of impedance matrix. The proposed method has applied for solving the bend discontinuity, and verified with measurement results.

This paper shows the scattering cross sections of a random medium which is a simple model of moist soil by analyzing a dense medium radiative transfer equation (DMRT). The parameters in the DMRT, the extinction rate and the scattering coefficient, are calculated by a multiple scattering method called our method in this paper. Our method is valid for particles with high dielectric constant like water drops. Characteristics of the scattering cross section are made clear by changing the fractional volume of water and the incident angle, polarization of incident waves. We discuss the possibility of detection of a water content in this approach by using the characteristics of the scattering cross section.