The electromagnetic scattering from a conducting cylinder coated with metamaterials, which have both negative permittivity and permeability, is derived rigorously by using the classic separation of variables technique. It is found that a conducting cylinder coated with metamaterials has anomalous scattering cross section compared to that coated with conventional dielectric materials. Numerical results are presented and discussed for the scattering cross section of a conducting cylinder coated with metamaterials.

In this paper new stability theorems for Yee's Finite- Difference Time-Domain (FDTD) formulation are derived based on the energy method. A numerical energy expression is proposed. This numerical energy is dependent on the FDTD model's E and H field components. It is shown that if the numerical energy is bounded, then all the field components will also be bounded as the simulation proceeds. The theorems in this paper are inspired by similar results in nonlinear dynamical system. The new theorems are used to prove the stability of a FDTD model containing non-homogeneous dielectrics, perfect electric conductor (PEC) boundary, nonlinear dielectric and also linear/nonlinear lumped elements. The theorems are intended to complement the well-known Courant-Friedrich-Lewy (CFL) Criterion. Finally it is shown how the theorems can be used as a test, to determine if the formulation of new lumped element in FDTD is proper or not. A proper formulation will preserve the dynamical stability of the FDTD model. The finding reported in this paper will have implications in the manner stability analysis of FDTD algorithm is carried out in the future.

Abstract-In this work, we study the reflection properties of coplanar waveguides (CPW) periodically loaded with shunt connected capacitances and periodically perturbed by varying the slot width. These structures are of interest because the low pass frequency response with spurious frequency bands, inherent to the presence of capacitors, can be improved. This is achieved through the attenuation of frequency parasitics that is obtained by the introduction of slot width modulation. Both sinusoidal and square wave patterns are considered and the effects of the relative position of reactive elements with regard to the perturbation geometry is analysed. According to coupled mode theory, the central frequencies of the rejected bands in periodic transmission media are given by the spectrum of the perturbation function. However, it is demonstrated that, due to the presence of capacitors, multiple spurious bands can be simultaneously suppressed even in the case of a singly tuned (sinusoidal) perturbation geometry. This result points out that the frequency selective behaviour associated to the presence of slot width modulation can not be interpreted in the framework of coupled mode theory, since the rejection of spurious bands in periodic loaded CPWs is not merely given by the spectrum of the perturbation geometry.

The homogenization of cubically arranged, homogeneous spherical inclusions in a background material is addressed. This is accomplished by the solution of a local problem in the unit cell. An exact series representation of the effective relative permittivity of the heterogeneous material is derived, and the functional behavior for small radii of the spheres is given. The solution is utilizing the translation properties of the solutions to the Laplace equation in spherical coordinates. A comparison with the classical mixture formulas, e.g., the Maxwell Garnett formula, the Bruggeman formula, and the Rayleigh formula, shows that all classical mixture formulas are correct to the first (dipole) order, and, moreover, that the Maxwell Garnett formula predicts several higher order terms correctly. The solution is in agreement with the Hashin-Shtrikman limits.

3-dimensional photonic bandgap structures working in the visible have been given increasing attention in recent years encouraged by the possibility to control, modify or confine electromagnetic waves in all three dimensions, since this could have considerable impact on novel passive and active optical devices and systems. Although substantial progress has been made in the fabrication of 3D Photonic crystals by means of nano-lithography and nanotechnology, it still remains a challenge to fabricate these crystals with feature sizes of the half of the wavelength in the visible. Self-assembling of colloidal particles is an alternative method to prepare 3-dimensional photonic crystals. The aim of this article is to show how to use colloidal crystals as templates for photonic crystals and how to monitor the changes of their optical properties due course of the modification.

In this work, we study a novel type of optical waveguide, whose properties derive from a periodic arrangement of fibers (not necessarily circular), and from a central structural defect along which the light is guided. We first look for propagating modes in photonic crystal fibers of high indexcore region which can be single mode at any wavelength [1-4]. Unlike the first type of photonic crystal fibers, whose properties derive from a high effective index, there exists some fundamentally different type of novel optical waveguides which consist in localizing the guided modes in air regions [4-5]. These propagating modes are localized in a low-indexstructural defect thanks to a photonic bandgap guidance for the resonant frequencies (coming from the photonic crystal cladding). We achieve numerical computations with the help of a new finite element formulation for spectral problems arising in the determination of propagating modes in dielectric waveguides and particularly in optical fibers [7]. The originality of the paper lies in the fact that we take into account both the boundness of the crystal (no Bloch wave expansion or periodicity boundary conditions) and the unboundness of the problem (no artificial boundary conditions at finite distance). We are thus led to an unbounded operator (open guide operator) and we must pay a special attention to its theoretical study before its numerical treatment. For this, we choose the magnetic field as the variable. It involves both a transverse field in the section of the guide and a longitudinal field along its axis. The section of the guide is meshed with triangles and Whitney finite elements are used, i.e., edge elements for the transverse field and node elements for the longitudinal field. To deal with the open problem, a judicious choice of coordinate transformation allows the finite element modeling of the infinite exterior domain. It is to be noticed that the discretization of the open guide operator leads to a generalized eigenvalue problem, solved thanks to the Lanczos algorithm. The code is validated by a numerical study of the classical cylindrical fiber for which the eigenmodes are known in closed form. We then apply the code to Low IndexPhotonic Crystal Fibers (LPCF) and to High IndexPhotonic Crystal Fibers (HPCF).

This paper deals with the use of Photonic Crystal (PC) structures as substrates in patch antenna configurations in order to mitigate the effect of the surface wave mode propagation. The case of a single antenna has been studied. A comparison between a conventional substrate based patch and a patch with a PC as substrate has been performed. The antennas were fabricated and measured. Improvements in all the main parameters of the antenna were obtained when usinga PC. The frequency dependence of the radiation patterns is significantly reduced when using a PC as substrate.

Photonic Band-Gapmaterials (PBG) are periodic structures composed of dielectric materials or metal. They exhibit frequency bands for which no propagation mode can propagate. Unfortunately, they are bulky and their period has to be at least a quarter wavelength. One extension of the PBG structures is called High impedance ground planes (High Z). Their period is much smaller and they exhibit frequency bands in which no surface wave can propagate. Their electromagnetic characteristics make them particularly interesting for antenna applications. On the one hand, they reduce the interaction between an antenna and its backward surroundings, with smaller size than usual ground planes. On the other hand, they can be used for planar antenna solutions, as the radiating element can be placed right on the top of the ground plane. After a presentation of the steps which lead to High Impedance ground planes, the electromagnetic characteristics of such ground planes are presented. Then, some antenna applications illustrate the interest of such structures.

This paper reviews recent advancements in the research and development of Uniplanar Compact Photonic Bandgap (UCPBG) structures for microwave and millimeter-wave applications. These planar periodic structures are particularly attractive and have been intensively investigated due to their easy fabrication, low cost, and compatibility with standard planar circuit technology. In this paper, basic properties of UC-PBG will be studied such as the slowwave effect, distinct stopband and passband, leakage suppression of surface waves, and realization of a magnetic surface. Owing to the different features of UC-PBG, these structures have been applied to microwave circuits to improve microstrip filters and patch antennas, to perform harmonic tuning in power amplifiers, to suppress leakage in conductor-backed coplanar waveguide, to realize TEM waveguides, and to implement low-profile cavity-backed slot antennas.

We have demonstrated guiding and bending of electromagnetic (EM) waves in planar and coupled-cavity waveguides built around three-dimensional layer-by-layer photonic crystals. We observed full transmission of the EM waves through these waveguide structures. The dispersion relations obtained from the experiments were in good agreement with the predictions of our waveguide models. We also reported a resonant cavity enhanced (RCE) effect by placing microwave detectors in defect structures. A power enhancement factor of 3450 was measured for planar cavity structures. Similar defects were used to achieve highly directional patterns from monopole antennas.

Metallo-dielectric electromagnetic bandgap (EBG) structures are studied in the millimeter regime with a finite difference time domain (FDTD) simulator. Several EBG waveguiding structures are considered for millimeter-wave power splitting, switching and filtering operations. It is demonstrated that triangular EBG structures lend themselves naturally to the design of Y-power splitters. Square EBG structures with circular and square rods are shown to lead naturally to straight in-line waveguide filter applications. Comparisons between EBG millimeter-wave waveguide filters formed with dielectric and metallic rods are given. It is shown that high quality broad bandwidth, millimeter-wave bandstop filters can be realized with square EBG structures with circular metallic rods. It is demonstrated that multiple bandstop performance in a single device can be obtained by cascading together multiple EBG millimeter-wave waveguide filters. It is also demonstrated that one can control the electromagnetic power flow in these millimeter-wave EBG waveguide devices by introducing additional local defects. It is shown that the Y-power splitter can be made reconfigurable by using imposed current distributions to achieve these local defects and, consequently, that a millimeter-wave EBG switch can be realized.

Abstract-Photonic Bandgap (PBG) materials have been investigated for their versatility in controlling the propagation of electromagnetic waves [1, 2]. In order to determine PBG structures responses, several analytical or numerical methods are used, such as:

• The plane wave method applied to solve Maxwell's equations [3].
• The transfer matrix method, based on the wire grating impedance developed by N. Marcuvitz [4].
• The Finite Element Method (FEM) exhibits, e.g., the frequency response of reflection and transmission coefficients of the PBG materials when they have infinite surfaces and are excited by plane wave. The FEM method can be also used in the case of finite structure fed by a dipole.
• solves the discretized Maxwell's equations in the time domain and evaluates the electromagnetic field components. These EM fields are then obtained in the frequency domain thanks to a Fourier Transform.

First of all, we present a parametrical study using a 3D Finite Element method software. This study allows to estimate the role of any parameters on the reflection and transmission coefficients and then to design a PBG structure in the X-band (8-12 GHz). Continuous and discontinuous structures are presented. Then, we present a numerical analysis of PBG structures, using the FDTD method in order to understand the propagation phenomena in these periodic materials.

Since the first demonstration of a complete photonic band gap by E. Yablonovitch in 1987 [1], photonic band gap materials have attracted a very significant interest in Electromagnetism but also in Solid State Physics. Doped photonic crystals that have a point defect (a local disturbance) have been extensively studied with the emergence of this new area of Physics. They present localized modes in the band gap and triggered many potential applications. Fewer papers have been devoted to strongly disordered photonic crystals that are periodic on the average. These structures are disturbed on the overall feature and the defect corresponding is referred to as extended. Analogue at a first glance to amorphous semiconductors, these materials could present interesting properties. Moreover, manufacture of photonic crystals is still a real challenge for the optical domain and undesirable extended defects can appear leading to a compulsory study of the tolerances of periodicity for such new materials.

Based on the eigenvalue equations of vector fields ⃗E and ⃗H by extending Bloch theorem to the vector field Maxwell equations, the characteristics of 2-D dielectric rod array with square cross-section elements arranged in square lattice is analyzed in detail. From the numerical results, empirical expressions for both the relative bandwidth of frequency band gap and the midgap frequency with respect to the average permittivity, under the optimal filling fraction of dielectric/air in cross-section for wider bandwidth, are formulated by means of data fit.

We present an original study which makes use of a convenient representation of the dispersion diagrams of Bloch modes for the design of angular selective sources. These diagrams provide us all the pertinent information about the radiative properties of the photonic crystal, and a guideline to optimize the structure in order to obtain the suitable properties. We apply these tools in two cases: when the radiated field propagates normally to the device, and also for an off-axis radiating device. Several numerical examples obtained from a rigorous numerical method show the relevance of this approach.

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.