Macro Basis Functions Research Articles

Tailorable optical scattering properties of V-shaped plasmonic nanoantennas: a computationally efficient and fast analysis.

In this work, we introduce an efficient computational scheme, based on the macro basis function method, to analyze the scattering of a plane wave by V-shaped plasmonic optical nanoantennas. The polarization currents and scattered fields for the both symmetric and antisymmetric excitations are investigated. We investigate how the resonant frequency of the plasmonic V-shaped nanoantenna is tailored by engineering the geometrical parameters and by changing the polarization state of the incident plane wave. The computational model presented herein is faster by many orders of magnitude than commercially available finite methods, and is capable of characterizing all nanoantennas comprised of junctions and bends of nanorods.

Contour-FFT Based Spectral Domain MBF Analysis of Large Printed Antenna Arrays

A fast spectral-domain method is proposed to evaluate the reaction terms between the Macro Basis Functions in regular and non-regular arrays made of identical printed antennas. The presented technique first exploits the filtering capabilities of the Macro Basis Functions in the spectral domain. The method is then strongly accelerated with the help of a newly formulated Fast Fourier Transform-based technique, which is applicable to a contour integration in the complex plane. We name the method as Contour-FFT or C-FFT. Besides an effective homogeneous medium term treated with multipoles, a computational complexity of order N log_2 N is achieved for the the tabulation of substrate-related reaction terms for any possible relative positions. The complexity of the proposed method is independent from the complexity of the elements. Numerical results obtained with the proposed method are compared with those from a pre-validated reference solution based on the traditional Macro Basis Functions technique; an excellent agreement is observed.

Efficient integral equation-based analysis of finite periodic structures in the optical frequency range

The optical response of dense finite arrays of nanoparticles can be efficiently analyzed with the help of macro basis functions obtained by employing the array scanning method. This is demonstrated by analyzing optical collimation in arrays of silver nanorods. The accuracy of the solution obtained with the proposed method has been validated by comparison with solutions obtained employing the Krylov subspace iterative method. The relative error in the electric field distribution on an observation plane above the finite array is of the order of -25 dB, while the number of unknowns is reduced by a factor of 32.

On the Relationship Between Multiple-Scattering Macro Basis Functions and Krylov Subspace Iterative Methods

A mathematical link is provided between the Krylov subspace iterative approach based on the full orthogonalization method (FOM) and the macro basis functions (MBF) approach based on a multiple-scattering methodology. The link refers to the subspaces created by those methods as well as to the orthogonality conditions which they satisfy. Both approaches are applied to the same method-of-moments (MoM) system of equations that is preconditioned based on a closest-interaction rule, and where blocks of the MoM impedance matrix are compressed using a rank-revealing method. MBF and FOM approaches are compared numerically, with a special attention given to accuracy, for perfectly conducting objects, comprising an array of tapered-slot antennas, spheres and an aircraft. The respective advantages of both methods are briefly discussed and further prospects are given.

Accelerated Macro Basis Functions Analysis of Finite Printed Antenna Arrays Through 2D and 3D Multipole Expansions

An efficient technique is presented for the analysis of finite printed antenna arrays made of identical elements. It is based on a closed-form expression for the spatial-domain Green's function (GF) given as a finite sum of cylindrical waves (obtained through rational function fitting) plus one spherical wave. From there, a multipole expansion can be obtained for planar layered medium GFs. The macro basis function (MBF) technique is applied to the method of moments (MoM) solution of a mixed-potential integral equation, this reduces the size of the MoM impedance matrix and allows for a direct solution. However, the evaluation of the entries of this reduced matrix becomes the dominant contribution to the total computation time. The aforementioned multipole expansion is exploited to provide a fast construction of the reduced MoM matrix, whose elements are the reaction integrals between the MBFs considered to characterize the currents on the array element. The complexity of evaluating the interactions between MBFs is found to be dominated by the calculations related to the spherical wave term. Thus, taking into account the layered medium does not increase the order of the complexity with respect to a multipole-accelerated computation of reaction integrals in a homogeneous medium.

Analyzing Large-Scale Arrays Using Tangential Equivalence Principle Algorithm With Characteristic Basis Functions

In this paper, the tangential equivalence principle algorithm (T-EPA) combined with characteristic basis functions (CBFs) is presented to analyze the electromagnetic scattering of large-scale antenna arrays. The T-EPA is a kind of domain decomposition scheme for the electromagnetic scattering and radiation problems based on integral equation (IE). CBFs are macrobasis functions which are constructed by conventional local basis functions. By utilizing CBFs together with the T-EPA, the scattering analysis of large-scale arrays will be much more efficient with decreased unknowns compared with the original T-EPA. Further, the multilevel fast multipole algorithm (MLFMA) is applied to accelerate the matrix-vector multiplication in the T-EPA. Numerical results are shown to demonstrate the accuracy and efficiency of the proposed technique.

Volume Integral Equation Analysis of Thin Dielectric Sheet Using Sinusoidal Macro-Basis Functions

In this letter, we present an improvement over the conventional thin dielectric sheet (TDS) formulation for the analysis of thin dielectric sheets. Our focus is to address the problem of scattering from thin penetrable scatterers by developing a volumetric formulation based on the use of macro-basis functions. The electric fields produced by the source basis functions are derived directly, bypassing the summation of scalar and vector potentials. The latter results in a considerable time advantage in comparison to the conventional method of moments (MoM) solution of the volume electric field integral equation (V-EFIE), while the accuracy remains comparable. In contrast to the TDS formulation, both tangential and normal currents are employed so that problems with high and low permittivity, as well as those involving grazing incident angles, can be accurately analyzed. Several examples are reported that show excellent agreement with conventional techniques and demonstrate the effectiveness of the new approach.

Scattering performance of plasmonic nanorod antennas in randomly tilted disordered and Fibonacci configurations

In this letter, we formulate and investigate the optical performance of finite arrays of plasmonic nanorod antennas in randomly tilted disordered and Fibonacci configurations. To efficiently model the scattering performance of a nanorod, we take advantage of characteristic basis function method in conjunction with macro basis functions. We study how random rotations of the nanorods, as a disorder, may affect their electromagnetic response. Also, the impact of quasi-periodic order on the performance of the array in the form of two-dimensional Fibonacci quasi-lattice comprising of plasmonic nanorods with two different lengths is studied.

Interpolatory Macro Basis Functions Analysis of Non-Periodic Arrays

An efficient method-of-moments (MoM) technique for analyzing non-periodic antenna arrays of identical elements with fixed orientation is presented. The proposed method, which uses macro basis functions (MBFs), is based on a compact representation of the interactions between MBFs. These interactions are expressed via a low-order harmonic-polynomial function after an explicit pre-computation of the reaction integrals in a very limited set of relative positions. This is possible for arrays of arbitrary size thanks to three transformations- subtraction of the far-field expression for the interactions, phase correction and modified radial distance- applied successively to the pre-computed interactions. After obtaining the harmonic-polynomial expressions, the computation time for the reaction integral between two MBFs is independent from the complexity of the antenna array element.

Efficient Analysis of Large Scatterers by Physical Optics Driven Method of Moments

A new iterative procedure is presented that enables method of moment (MoM) solution of scattered field from electrically large and complex perfectly conducting bodies using significantly reduced number of unknown coefficients. In each iteration the body is excited by a plane wave and by the currents, which are obtained as an approximate solution in the previous iteration. The physical optics (PO) and modified PO techniques are used to determine the PO and the correctional PO currents, which are expressed in terms of original MoM basis functions and grouped into macro-basis functions (MBFs). Weighting coefficients of all MBFs are determined from the condition that mean square value of residuum of original MoM matrix equation is minimized. The iterative procedure finishes when the residuum decreases below the maximum allowed value. The accuracy and efficiency of the proposed method are illustrated on two examples: cube scatterer and airplane scatterer. Since the construction of MBFs by PO and modified PO techniques ensures fast convergence to the original MoM solution, the method is named PO driven MoM.

Scattering analysis of plasmonic nanorod antennas: A novel numerically efficient computational scheme utilizing macro basis functions

In this paper we introduce a versatile and numerically efficient computational technique to model the problem of scattering from plasmonic nanorod antennas. The key to achieving the numerical efficiency is to utilize macro basis functions (MBFs) that taking into account the physics of the problem to reduce the size of matrix equation we need to solve. Closed form formulations are presented for computing the fields by the transverse and longitudinal MBFs that enable us to generate the required matrix elements rapidly, while ensuring that the matrix is well-conditioned. We show that the transverse and longitudinal components of polarization current and all of the components of the scattered fields can be computed very accurately by employing only a few MBFs, i.e., by solving a relatively small-size matrix equation. The accuracy of our modeling technique has been successfully demonstrated by comparing the simulation results with those derived by using the finite difference time domain (FDTD) technique, which is considerably more time-consuming than the present approach. Interesting physical phenomena such as surface plasmon modes for polarization currents and resonance behaviors of plasmonic nanorods are illustrated.

A Model Reduction Algorithm for Solving Multiple Scattering Problems Using Iterative Methods

In this paper we propose a new method for solving high-frequency scattering problems by multiple objects. A model reduction algorithm based on the macro basis function (MBF) method is used to find an approximate solution within a subspace spanned by the solutions of several single scattering subproblems. Different iterative methods for generating the MBFs are compared. The whole process relies on a finite element approach and is applied to convex obstacle scattering.

An Open-Source Code for the Calculation of the Effects of Mutual Coupling in Arrays of Wires and for the ASM-MBF Method

A general description of mutual coupling in finite and infinite antenna arrays is provided and an open-source code is described for the analysis of mutual coupling in linear arrays of parallel dipoles. The ASM-MBF method is illustrated with the help of that code and it is shown that the method also can handle situations where eigenmodes are supported by the array: single machine precision is achieved with four Macro Basis Functions only.

Improving the macro-basis function using insertion method and iterative refinement process for planar structure with one-dimensional wave propagation

This study introduces a new method to construct the macro-basis functions that improves the accuracy of a solution for a dense planar structure with wave guided in one direction. The basic step in the approach is to divide a structure into several sub-domains. An incident wave and an artificial reflected wave are then created through a smaller problem that is made up of a few sub-domains. The remaining sub-domains are then inserted into this smaller problem progressively. Further improvement in the accuracy can be achieved by combining the new method with iterative techniques. Two iterative schemes are presented, of which one is superior to the other. Some examples are given to test the validity and efficiency of the new method in conjunction with the iterative refinement process. Numerical results from the new method coupled with the iterative refinement process show excellent agreement with those computed from the conventional method of moments.

Efficient full‐wave characterization of arrays of antennas embedded in finite dielectric volumes

This paper shows how the Macro Basis Functions (MBFs) approach can be applied to the solution of arrays of complex elements, made of metallic parts embedded in disconnected dielectric volumes. It is shown how the reduced system of equations can be obtained very efficiently with the help of a multipole approach. We also provide a subsequent formulation for the array impedance matrix and a fast method for the computation of embedded element patterns, with the help of a decomposition into a finite series of pattern multiplication problems, for which the fast Fourier transform can be exploited. Examples are provided for arrays of tapered slot antennas in separate dielectric boards. Comparisons are provided with brute‐force solutions for a 4 × 4 array and increasing number of MBFs, and with the infinite‐array solution for a 16 × 16 array. An appendix provides new insight into the problem of completeness of the MBFs bases by establishing a link between the MBF approach and Krylov subspaces.

An Interpolation Algorithm to Reduce the Reduced Matrix Fill-Time in CBFM

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This letter presents an interpolation algorithm to reduce the computation time needed to generate the reduced matrix in the characteristic basis function method (CBFM). The first step in this approach is to identify the observation points and calculate the electric fields at those points that are produced by the induced currents represented by the macro basis functions (MBFs) called the characteristic basis functions (CBFs). Next, we interpolate the above field values to obtain the same at other points on the structure. Finally, the reduced matrix is formed by testing the electric field with the CBFs. Numerical results are presented to show the accuracy and computational efficiency of the method. </para>

An Iterative Solution for Electrically Large Problems Combining the Characteristic Basis Function Method and the Multilevel Fast Multipole Algorithm

An iterative scheme for the rigorous computation of electrically large problems is presented. The approach is based on a combination of the characteristic basis function method (CBFM) and the multilevel fast multipole algorithm (MLFMA) that can deal with very large problems that require an iterative solution process, even considering that the application of the CBFM entails an important reduction of the number of unknowns when compared to Method of Moments approaches based on subdomain functions. This reduction is due to the fact that the number of macro-basis functions, called characteristic basis functions (CBFs), is lesser than the number of low-level subsectional functions used to sample the geometry. In addition, the use of the MLFMA avoids the need to calculate and store the coupling terms in the reduced matrix that are not on or close to the diagonal, thereby optimizing the CPU time and the memory storage requirements. non-uniform rational B-splines (NURBS) surfaces are employed for the representation of the geometry and the CBFs are described in terms of curved rooftops generated in the parametric space. The associated macro-testing functions are defined as aggregations of curved razor-blade functions.

A Fast Impedance and Pattern Computation Scheme for Finite Antenna Arrays

A fast numerical method, well-suited to the analysis of moderate-size arrays made of complex elements, is presented. It combines macro basis functions and multipole approaches, without an iterative procedure. This method is exploited to estimate the impedance matrix and active element patterns. For the latter, an efficient formulation is provided, as a series of pattern multiplication problems. Examples are shown for arrays of broadband dipoles. The computational gain obtained for the reduction of the original method of moments system of equations is briefly described

State of the art in computational methods for the prediction of radar cross-sections and antenna–platform interactions

In this paper, a review is presented on computational methods for the prediction of Radar Cross-Sections (RCS) and antenna–platform interactions. In a first part the techniques for RCS computations are considered. A list of frequency and time domain solvers for the Maxwell's equations are given with their performances in memory requirements and run-time. Boundary Elements Methods, Finite Difference Time Domain Methods, Finite Elements—Finite Volume Methods, Hybridization and Factorization Techniques, are reviewed. The exceptional performances of the Fast Multipole Method compared to those of the classical Moment Method are especially highlighted. We have also made mention of some recent research work on numerical techniques conducted in France. Asymptotic methods are mainly discussed in the second part of this article devoted to antenna–platform interactions. After a brief description of the historical evolution of Geometrical Theory of Diffraction tools for antenna analysis and design, the advantages and drawbacks of different techniques for generating the geometry and searching the rays are discussed. Then a list of unsolved problems and lines of future research on asymptotic techniques are presented together with an example of a computer code founded on the Uniform Theory of Diffraction. In the conclusion some new research topics such as higher order finite elements defined on surfaces represented by B-Splines and macro-basis functions containing information on the phase or derived from analytical or asymptotic solutions are briefly introduced. To cite this article: F. Molinet et al., C. R. Physique 6 (2005).

Efficient evaluation of macro‐basis‐function reaction terms in the subdomain multilevel approach

AbstractThis paper presents an improved subdomain multilevel approach (SMA), a technique efficiently used for solving large complex‐shaped antennas printed in multilayered media structure. In a large electromagnetic problem, where the memory occupation and computational time have been already significantly reduced using the SMA, the calculation of the macro basis function (MBF) reaction terms remains the most time‐consuming part of the procedure. In this paper, we present an efficient way of computing MBF reaction terms and reducing them to equivalent moments. The numerical results are compared with brute‐force computations, the SMA, and the measurements. They demonstrate conclusively the efficiency and power of the new proposed technique. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 42: 138–143, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20232