Brillouin Diagram Research Articles

Finite-Difference Time-Domain Modeling of Periodic Structures: A review of constant wave vector techniques

This article reviews state-of-the-art periodic boundary conditions (PBCs) in finite difference time-domain (FDTD) simulations that operate by enforcing constant wave vector components. The mathematical principles and 3D FDTD implementation details are systematically outlined. Techniques for extracting scattering parameters, Brillouin diagrams, and attenuation constants are presented along with the array scanning method (ASM) used to model the interaction of nonperiodic sources with periodic structures. Through these techniques, the robustness, utility, and efficiency of PBCs are demonstrated and a unified view of the various approaches to the FDTD implementation of PBCs is presented.

Higher-Order Space Harmonics in Substrate Integrated Waveguide Leaky-Wave Antennas

The concept and feasibility of deploying higher-order space harmonics (HSHs) in 1-D periodic leaky-wave antennas (1-D periodic LWAs) is proposed, studied, analyzed, and demonstrated in this article. This can lead to the creation of multiband multibeam antennas (MBMBAs) with the capability of wide scanning ranges. To show the proposed concept, the excitation and flexibility of odd (n ε [-1, -3]) and even (n ε [-2, -4]) HSHs based on the analysis of a Brillouin diagram along with the effect of unit-cell geometrical structures of substrate integrated waveguide (SIW) for different propagating media are investigated. The proposed analysis is experimentally validated by prototyping and measuring a 1-D periodic LWA in SIW technology in which two HSHs (n ε [-1, -2]) are utilized for achieving a dual-band multibeam LWA with wide scanning range capability. In addition, a triple-band multibeam LWA design based on this concept is also analyzed and demonstrated where we take advantage of the characteristics of four HSHs (n ε [-1, -2, -3, -4]). Furthermore, the band edge issue of different stop-bands of HSH is also examined, and a method is proposed and used to address this issue where we use a quarter wavelength technique to overcome the matching problem at different frequencies related to different stop-bands of various HSHs (n ε [-1, -2, -3, -4]). Another design of a SIW-based 1-D periodic LWA is also discussed to extend and generalize the proposed concept in solving the mismatch issues of HSHs focusing on broadside radiation of the n $= -2$ space harmonic. The measured results are found to agree very well with their simulation and analysis counterparts.

Investigation of Instabilities in a Folded-Waveguide Sheet-Beam TWT

Particle-in-cell (PIC) simulation was used for analyzing the instabilities due to zero-drive oscillations in a 220-GHz folded-waveguide (FW) traveling-wave tube. The equivalent circuit analysis based on the transmission line approach was used for the parametric design of the FW structures (FWSs) with circular and rectangular (horizontal and vertical) beam holes. Experimental validations with reference to the dispersion and interaction impedance characteristics were carried out on a scaled model of the slow wave structure in the Ku band. The oscillations observed in various configurations of the FWS were subsequently analyzed using the Brillouin diagram. The FWS with horizontal-rectangular beam hole was found to have a better immunity to instability compared to the other two configurations.

Higher-order sensitivity analysis of periodic 3-D eigenvalue problems for electromagnetic field calculations

Abstract. An algorithm to perform a higher-order sensitivity analysis for electromagnetic eigenvalue problems is presented. By computing the eigenvalue and eigenvector derivatives, the Brillouin Diagram for periodic structures can be calculated. The discrete model is described using the Finite Integration Technique (FIT) with periodic boundaries, and the sensitivity analysis is performed with respect to the phase shift φ between the periodic boundaries. For validation, a reference solution is calculated by solving multiple eigenvalue problems (EVP). Furthermore, the eigenvalue derivatives are compared to reference values using finite difference (FD) formulas.

Wideband analytical equivalent circuit for one-dimensional periodic stacked arrays.

A wideband equivalent circuit is proposed for the accurate analysis of scattering from a set of stacked slit gratings illuminated by a plane wave with transverse magnetic or electric polarization that impinges normally or obliquely along one of the principal planes of the structure. The slit gratings are printed on dielectric slabs of arbitrary thickness, including the case of closely spaced gratings that interact by higher-order modes. A Π-circuit topology is obtained for a pair of coupled arrays, with fully analytical expressions for all the circuit elements. This equivalent Π circuit is employed as the basis to derive the equivalent circuit of finite stacks with any given number of gratings. Analytical expressions for the Brillouin diagram and the Bloch impedance are also obtained for infinite periodic stacks.

Accurate and Efficient Computation of Layered Medium Doubly Periodic Green's Function in Matrix-Friendly Formulation

This communication proposes a novel approach to evaluate the doubly periodic Green's function for layered medium in matrix-friendly formulation. Several techniques, including factorization of the Green's function, generalized pencil of function method and high order Taylor expansion, are employed in a delicate manner to derive the high order asymptotic expressions, which are then evaluated by new fast convergent series derived from Ewald transformation. This approach delivers fast and highly accurate results as well as high order convergence, and also allows fast frequency sweep for calculating Brillouin diagram in eigenvalue problem and for normal incidence in scattering problem.

On the Suppression Band and Bandgap of Planar Electromagnetic Bandgap Structures

Electromagnetic bandgap structures are considered a viable solution for the problem of switching noise in printed circuit boards and packages. Less attention, however, has been given to whether or not the introduction of EBGs affects the EMI potential of the circuit to couple unwanted energy to neighboring layers or interconnects. In this paper, we show that the bandgap of EBG structures, as generated using the Brillouin diagram, does not necessarily correspond to the suppression bandwidth typically generated usingS-parameters. We show that the reactive near fields radiating from openings within the EBG layers can be substantial and are present in the entire frequency band including propagating and nonpropagating mode regions. These fields decay fast with distance; however, they can couple significant energy to adjacent layers and to signal lines. The findings are validated using full-wave three-dimensional numerical simulation. Based on this work, design guidelines for EBG structures can be drawn to insure not only suppression of switching noise but also minimization of EMI and insuring signal integrity.

WAVE PROPAGATION IN A WAVEGUIDE SECTION WITH SHARP CORRUGATIONS

The rigorous excitation equations are used to analyze wave propagation in a rectangular waveguide section with sharp periodic corrugations whose height is much greater than the period. It has been shown that the Brillouin diagram of the corrugated and regular waveguides in the case of strong gradient scattering is characterized by qualitatively the same frequency dependence of the longitudinal wavenumber. The only difference is that the cutoff frequencies of the corrugated waveguide modes increase with decreasing the corrugation period, and it is always possible to implement the single-mode regime within an arbitrary frequency range lying higher than the first cutoff frequency. The effect can be used to suppress the mode competition in microwave amplifiers, as well as in narrow-band oscillators provided that the latter involve the Bragg reflectors as a feed-back.

Full-Space Scanning Periodic Phase-Reversal Leaky-Wave Antenna

A novel full-space scanning periodic phase-reversal leaky-wave antenna array is proposed, designed in offset parallel stripline technology, and demonstrated experimentally. This antenna radiates from its small phase-reversing cross-overs, which leads to a small leakage factor and subsequently a large directivity. The operation principle of the antenna is explained from the Brillouin diagram, which shows how single-beam scanning, using the m = -1 space harmonic, is achieved as a result of the π lateral shift of the dispersion curves due to phase reversal. One of the benefits of phase reversal is to permit this radiation performance with relatively small permittivity substrates (ee,min = 4 compared to ee,min = 9 for antennas without phase reversal). An unitcell matching technique is applied to avoid reflections, and thereby prevent the presence of an open stopband so as to permit continuous space scanning with efficient broadside radiation. An efficient array synthesis procedure, based on a transmission line modeling of the structure, is utilized for the design of the antenna following specifications in terms of frequency, scanning, directivity, radiation efficiency, and sidelobe level. A uniform-aperture antenna prototype, including a balun-transformer input transition, is presented, featuring experimental beamwidth and gain at the broadside frequency (25 GHz) of 4° and 15.7 dBi, respectively.

Guided modes in a rectangular waveguide with semiconductor metamaterial

The dispersion equations of bulk modes and surface modes in a rectangular waveguide of semiconductor metamaterial are derived by a modified “Marcatili's method". The cutoff frequencies of the lowest TM bulk mode are discussed, and the Brillouin diagrams of different bulk modes are drawn. They demonstrate that different heights correspond to different guidance frequency ranges which have no superposition with each other and a waveguide with a larger height possesses a wider passband of light. In addition, tendencies of degeneracy for different modes are observed. Finally, the existence of surface modes is verified by a graphical method.

Generalized Brillouin diagrams for evanescent waves in metamaterials with interelement coupling

A graphical representation for two-dimensional periodic structures is introduced that provides the full information contained in the dispersion equation: not only for the pass band but for the neighboring stop bands as well. It is a seamless natural extension of Brillouin diagrams obtained by adding eight new zones which incorporate the properties of evanescent waves. The accuracy of the infinite lattice approximation is tested both by simulations and experiments using magnetoinductive waves. For waves propagating in one direction and evanescent in the other direction, the results of the three different methods are shown to be in good agreement. The versatility of the new representation and its applicability to the design of near-field manipulating metamaterials is discussed.

Efficient Modal Analysis of Periodic Structures Loaded With Arbitrarily Shaped Waveguides

In this paper, a novel full-wave method for the modal characterization of closed metallic periodic structures based on arbitrarily shaped waveguides is presented. This method relies on an integral equation formulation solved via the method of moments, which finally leads to the solution of a standard eigenvalue problem. The required modal spectrum of waveguides with arbitrary cross section is determined through the boundary integral-resonant mode expansion technique. For validation purposes, the proposed analysis method is first successfully applied to standard waveguide periodic geometries already considered in the technical literature. Our new algorithm is then used to identify the higher order Floquet modes, as well as to compute the related Brillouin diagrams, of complex closed metallic periodic structures loaded with arbitrarily shaped waveguides.

Full-Wave Modal Analysis of Slow-Wave Periodic Structures Loaded With Elliptical Waveguides

In this paper, a novel full-wave method for the modal characterization of periodic structures loaded with elliptical waveguides is presented. This method relies on an integral equation formulation solved via the method of moments, which finally leads to the solution of a standard eigenvalue problem. The required modal spectrum of elliptical waveguides is determined through the boundary integral-resonant mode expansion technique. For validation purposes, the proposed analysis method is first successfully applied to periodic waveguide structures already considered in the technical literature. Then, our new algorithm is used to compute the related Brillouin diagrams and the interaction impedance of new periodic structures loaded with elliptical waveguides. Not only the main interacting mode (such as the TM 01 mode) is studied, but higher-order Floquet modes are also considered. These results have potential applications as slow-wave structures for high-power microwave devices and possibly filtering structures at millimeter-wave frequencies.

Finite Element Electromagnetic Analysis of TWT Slow-Wave Structures in Grid Environment

In this paper, some new finite element procedures are presented to analyze TWT helix slow-wave structures in a distributed computing environment. In particular, very computing time expensive analyses required in TWT SWS simulations are parallelized in order to obtain a remarkable reduction of computing time. The example presented regards the construction of the Brillouin diagram of a helix SWS. The obtained results confirmed us the good performance of the grid environment for the numerical analysis and optimization of TWTs.

Intra-Brillouin-zone bandgaps due to periodic misalignment in one-dimensional magnetophotonic crystals

One-dimensional (1D) magnetophotonic crystals (MPCs) can incorporate optical gyrotropy induced by a bias magnetic field, crystalline misalignment, and differential linear birefringence in a single photonic-crystal structure. A 1D MPC whose unit cell contains two layers—one magnetophotonic, the other not—displays intra-Brillouin-zone photonic bandgaps (PBGs) in the Brillouin diagram. While the optical gyrotropy makes the PBG bandwidths tunable by a bias magnetic field, the bicrystalline misalignment modifies and can even trump this magnetic tunability. Magnetic tunability is greatly affected by a proper selection of the two materials; e.g., a large birefringence ratio between the two layers can dramatically enhance the magnetic tunability of the MPC. We also expect our 1D MPCs to be useful for detecting magnetic fields.

A New Brillouin Dispersion Diagram for 1-D Periodic Printed Structures

Dispersion and radiation properties for bound and leaky modes supported by 1-D printed periodic structures are investigated. A new type of Brillouin diagram is presented that accounts for different types of physical leakage, namely, leakage into one or more surface waves or also simultaneously into space. This new Brillouin diagram not only provides a physical insight into the dispersive behavior of such periodic structures, but it also provides a simple and convenient way to correctly choose the integration paths that arise from a spectral-domain moment-method analysis. Numerical results illustrate the usefulness of this new Brillouin diagram in explaining the leakage and stopband behavior for these types of periodic structures.

Dispersion properties of circular dielectric waveguides loaded with periodic metal disks

AbstractThe dispersion properties of a circular dielectric waveguides loaded at the center with periodic center‐protruded metal disks were investigated using a mode‐matching technique in conjunction with the Floquet theorem. The metal disks with rectangular and crisscrossed cross‐sectional shapes in the longitudinal direction were considered. The effects of the cylinder radius, disk size, and disk spacing on the dispersion curves plotted in the form of the Brillouin diagram were studied. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 2057–2061, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22645

Study on relationships of electromagnetic band gap structures and left/right handed structures

Two types of dual periodic circuits are introduced. The distributions of passbands and stopbands are generated from their dispersion relationships. Based on the study, Brillouin diagrams of three representative special cases are drawn; S parameters of these three cases are simulated by Aglient ADS; the S parameters of one of the three cases are verified by an experiment. The phase characteristics are compared with those generated from the dispersion relationship. The theoretical analysis and the experimental verification show that both types of the periodic structures can behave as electromagnetic band gap (EBG) structures, right-handed structures (RHS), and left-handed structures (LHS), when they operate at different frequency ranges. Thus, the possibility of a physical structure showing these three different characteristics at different frequency ranges is proven.

Periodic FDTD analysis of leaky-wave structures and applications to the analysis of negative-refractive-index leaky-wave antennas

The determination of the attenuation constants of periodic leaky-wave structures via the finite-difference time-domain (FDTD) method has been pursued so far via the simulation of a number of unit cells that is large enough to guarantee the convergence of the computed value. On the other hand, Brillouin diagrams of periodic structures can be readily extracted via the simulation of a single unit cell, terminated in periodic boundary conditions. This paper demonstrates a methodology that enables the concurrent extraction of leaky-wave attenuation constants and Brillouin diagrams of periodic structures, through the FDTD simulation of a unit cell. The proposed methodology is first validated and then employed to model leaky-wave radiation from a two-dimensional negative-refractive-index transmission-line (NRI-TL) medium. Apart from evaluating the characteristics of forward and backward leaky-wave radiation from such a medium, a lumped-element macro-model, with element values determined from the FDTD simulation, is extracted. The FDTD analysis, combined with this equivalent circuit, is used to investigate theoretically the possibility of the NRI-TL medium, as a leaky-wave antenna, to achieve continuous scanning from backward to forward end-fire and broadside radiation.

Numerical investigations of guided plasma waves beyond the quasistatic limits

Departing from general expressions for a guided waves along the interface between a classical and a nonconventional dielectric with negative permittivity, one observes that the zeroes of the dispersion function might be sought in a natural way by a numerical procedure known as the fixed point method. The paper relates this suggestion to existing software solutions for iteration procedures and the numerical idea has been tested and verified on the general problem of wave propagation along a helix in a plasma waveguide. The Brillouin diagrams and the diagrams of characteristic impedance are compared to existing results obtained previously by the application of the quasi-static approximation.