Derivation of the Foldy–Lax Equations

In this article, we develop a hybrid method to calculate the propagation of microwaves in randomly distributed dielectric cylinders. The hybrid method combines off-the-shelf techniques for single object and our developed techniques of Foldy–Lax (FL) method that include extracting the T-matrix for single object, vector translation addition theorem, and solving FL multiple scattering equations. For cylindrical scatterers such as tree trunks, the T-matrix in vector three-dimensional cylindrical waves are extracted from infinite cylinder approximation (ICA). In solving FL to calculate statistical moments, we iterate one order of multiple scattering at a time, with averaging over realizations performed after each order. This physically based iterative method of calculating statistical moments converges faster than the traditional iterative method of calculating the exact solution for each realization. The main purpose is to simulate tall tree trunks at the L-band and ICA is of sufficient accuracies. Numerical results are illustrated for a large number of cylinders of up to 196 and cylinder lengths of up to 94 wavelengths, which are typical of forests at the L-band. Results of the simulations of the hybrid method show that the transmission coefficients of waves are several times larger than that of the commonly used models of the radiative transfer equation and distorted Born approximation.

The full‐wave solution of multiple scattering among cylindrical vias in planar waveguides is modeled by using Foldy–Lax equations. By using the equivalence principle, the coupling among traces with many vias is decomposed into interior and exterior problems. For the interior problem, the dyadic Green's function is expressed in terms of vector cylindrical waves and waveguide modes. The Foldy–Lax equations of multiple scattering among the cylindrical vias are calculated. The waveguide modes are decoupled in the Foldy–Lax equations. The scattering matrix of coupling among vias is calculated. Numerical simulations of the scattering matrix are illustrated for several hundred vias. © 2001 John Wiley & Sons, Inc. Microwave Opt Technol Lett 31: 201–208, 2001.

In this article, we study wave propagation in vegetation at multiple frequencies using a hybrid method. First, the multiple scattering within a single plant is captured using the high-frequency structure simulator (HFSS) for extracting the T-matrix of the plant in vector cylindrical waves (VCW). Second, the Foldy–Lax equations (FLE) are applied with the extracted T-matrices to consider the multiple scattering among different plants. The accuracy of the hybrid method is verified by comparing the scattering results from nine wheat plants with direct HFSS simulations. The hybrid method is implemented with parallel computing using a physical iterative method to perform full-wave Monte Carlo simulations of a wheat field with up to 169 plants at <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> -, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula> -, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$C$ </tex-math></inline-formula> -bands. The impact of the gaps and the plant structure on the microwave propagations are studied by assessing the scattered fields and the resulting nonuniform transmission at different regions. The results are compared with modeling results using the classical radiative transfer equations (RTEs), and two major differences are observed: 1) the transmission calculated from the hybrid method is much larger than that of the RTE and 2) the full-wave simulation results show much weaker frequency dependence than RTE with saturation as frequency increases.

This paper presents a systematic analysis of the problem of multiple scattering by a finite group of arbitrarily sized, shaped, and oriented particles embedded in an absorbing, homogeneous, isotropic, and unbounded medium. The volume integral equation is used to derive generalized Foldy-Lax equations and their order-of-scattering form. The far-field version of the Foldy-Lax equations is used to derive the transport equation for the so-called coherent field generated by a large group of sparsely, randomly, and uniformly distributed particles. The differences between the generalized equations and their counterparts describing multiple scattering by particles embedded in a non-absorbing medium are highlighted and discussed.

To calculate scattering of a vegetation field consisting of a large number of plants, an improved fast version of the hybrid method (FHM) combining fast multiple scattering theory (FMST) and a numerical electromagnetic approach has been developed. In the HM, we use a library of T-matrices and the Foldy–Lax (FL) equation to calculate the electromagnetic field interactions outside of the cylinders enclosing the individual plants. To improve CPU time and memory requirements, the FL equation is solved by a triplet of fast Fourier transforms (FFTs) consisting of two FFTs solving 2-D FFTs of spatial distribution of plants and one 1-D FFT for solving the transformation of the order of cylindrical waves in the translation addition theorem. The triple FFTs minimize the CPU time and the memory requirement for computing translation addition matrix, which has been the bottleneck of large-scale simulations in the HM. To account for nonperiodic distribution of scatterers in a corn field, the premultiplication and the postmultiplication processes were applied to the FHM. The proposed method is validated at L-band (1.4 GHz) for 100 corn plants of height 1.25 m occupying an area of 9.54 by 9.54 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{m}^{2}$ </tex-math></inline-formula> . On a standard laptop, the CPU time is less than 6 min and the memory consumption is less than 710 Mbytes. The CPU time and the memory requirements are orders of magnitude more efficient than those required for solving the problem with a commercial software or using the previous version of the HM.

Modeling of electromagnetic wave propagation and scattering in random media play an important role in geoscience and remote sensing research. Considerable efforts have been made to elucidate and understand the wave interaction processes involved in such problems, and various models have been developed for microwave active and passive remote sensing applications. With the rapid advances in computer technology and fast computational electromagnetic algorithms, numerical simulations of scattering by random media allow us to solve Maxwell's equations exactly without the limitations of approximate analytical models. The numerical models can provide a valuable means for evaluating the validity regimes of analytical scattering theories; in addition, they can potentially aid in the future development of extended analytical models. In this article, updated developments in the numerical scattering models for discrete random scatterers are presented, with emphasis on the applications of microwave remote sensing in snowcover, seafoam, and vegetation canopy.The frequency dependence of scattering by dense media at microwave frequencies is the first numerical model described, which is an important topic because multifrequency measurements are usually used in remote sensing applications. The approach used is based on the Monte Carlo simulations where the three‐dimensional solutions of Maxwell's equations are solved. The properties of absorption, scattering, and extinction are calculated for dense media consisting of sticky and nonsticky particles. Numerical solutions of Maxwell's equations indicate that the frequency dependence of densely packed sticky small particles is much weaker than that of independent scattering. Numerical results are illustrated using parameters of snow in microwave remote sensing. Comparisons are made with extinction measurements as a function of frequency.Polarimetric microwave emissions form foam‐covered ocean surfaces are studied by modeling foam as densely packed air bubbles coated with thin seawater. The absorption, scattering, and extinction coefficients are computed from the Monte Carlo solutions of Maxwell's equations for a collection of coated particles. These quantities are then applied in the dense media radiative transfer theory to calculate the polarimetric microwave emissivities of ocean surfaces with foam cover. The theoretical results of Stokes brightness temperatures with typical parameters of foam in passive remote sensing at 10.8 and 36.5 GHz are illustrated and compared with experimental measurements.It follows by describing an efficient computational model based on the sparse matrix iterative approach (SMIA) for tree scattering at VHF/UHF frequencies. The method of moments is applied to solve the volume integral equation for the tree scattering signatures. The SMIA decomposes the impedance matrix into a sparse matrix for the near interactions, and a complementary matrix for the far interactions of the tree structure. Solutions obtained from the SMIA method agree very well with the solutions obtained using extract matrix inversion and the conjugate gradient method (CGM). The key feature of the SMIA approach is that very little iteration is required to obtain convergent solutions; compared to the CGM, the SMIA approach may reduce the number of iterations by a factor of &gt;100.The UV multilevel partitioning (UV‐MLP) method is also presented for solving the general volume scattering problems. The method consists of setting up a rank table of transmitting and receiving block sizes and their separations. The table can be set up speedily using coarse–coarse sampling. For a specific scattering problem with given geometry, the scattering structure is partitioned into multilevel blocks. By looking up the rank in the pre‐determined table, the impedance matrix for a given transmitting and receiving block is expressed by a product ofUandVmatrices. We demonstrate the method for two‐dimensional volume scattering by discrete scatterers. Multiple scattering is cast into the Foldy–Lax equations of partial waves. We show that the UV decomposition can be applied directly to the impedance matrix of partial waves of higher order than the usual lowest‐order Green function. Numerical results are illustrated for randomly distributed cylinders with a diameter of 1 wavelength. For scattering by 1024 cylinders on a single PC processor with 2.6 GHz CPU and 2 GB memory, only 14 CPU minutes is needed to obtain the numerical solution and for 4096 cylinders, only 7.34 s is needed for one matrix–vector multiplication.

In this paper, we model multiple vias with irregular antipad in arbitrarily shaped 3-D integrated circuit and packaging system based on generalized Foldy-Lax equations method, boundary integral equation method, and generalized T matrix. We first obtain the impedance matrix for finite cavity, which includes the reflection features of the cavity boundaries. Then, the scattered field from a single via and a generalized T matrix, including the wall effects, are derived. The Foldy-Lax multiple scattering equations are generalized to include the wall effects using impedance matrix and the generalized T matrix. To obtain the incident field for the case of vias in the arbitrarily shaped antipad, we calculate the exciting and scattering field coefficients based on the transformation that converts surface integration of magnetic surface currents in antipad into 1-D line integration of surface charges on the vias and ground plane. The coupling among vertical vias are solved by applying Foldy-Lax multiple scattering equations. The scattering matrix of coupling among vias is calculated to make corresponding signal/power integrity analysis. Numerical results for the method are in a good agreement with a commercial full-wave numerical tool up to 50 GHz.

We present large‐scale Monte Carlo simulation results of the phase functions in multiple scattering by dense media of small 2D particles. Solution of the Foldy–Lax equations with large number of unknowns is done efficiently using the sparse‐matrix canonical‐grid (SMCG) method. The SMCG method facilitates the use of FFT and results in an N log N‐type efficiency for CPU and O(N) for memory. This dependence is demonstrated by the simulation of CPU time using up to 50000 particles that are randomly distributed through random walk in a large area of 400 square wavelengths. The bistatic phase functions for a random medium are computed. The phase function converges with the number of particles and the number of realizations. The simulation results indicate that the nonsticky particles, sticky particles, and independent scattering have similar angular distribution patterns of the phase functions. However, the dense sticky particles show stronger scattering than the independent scattering, while the dense nonsticky particles have smaller scattering than that of the independent scattering. © 2003 Wiley Periodicals, Inc. Microwave Opt Technol Lett 38: 313–317, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.11047

Modeling of electromagnetic wave propagation and scattering in random media play an important role in geoscience and remote sensing research. Considerable efforts have been made to elucidate and understand the wave interaction processes involved in such problems, and various models have been developed for microwave active and passive remote sensing applications. With the rapid advances in computer technology and fast computational electromagnetic algorithms, numerical simulations of scattering by random media allow us to solve Maxwell's equations exactly without the limitations of approximate analytical models. The numerical models can provide a valuable means for evaluating the validity regimes of analytical scattering theories; in addition, they can potentially aid in the future development of extended analytical models. In this article, updated developments in the numerical scattering models for discrete random scatterers are presented, with emphasis on the applications of microwave remote sensing in snowcover, seafoam, and vegetation canopy.The frequency dependence of scattering by dense media at microwave frequencies is the first numerical model described, which is an important topic because multifrequency measurements are usually used in remote sensing applications. The approach used is based on the Monte Carlo simulations where the three‐dimensional solutions of Maxwell's equations are solved. The properties of absorption, scattering, and extinction are calculated for dense media consisting of sticky and nonsticky particles. Numerical solutions of Maxwell's equations indicate that the frequency dependence of densely packed sticky small particles is much weaker than that of independent scattering. Numerical results are illustrated using parameters of snow in microwave remote sensing. Comparisons are made with extinction measurements as a function of frequency.Polarimetric microwave emissions form foam‐covered ocean surfaces are studied by modeling foam as densely packed air bubbles coated with thin seawater. The absorption, scattering, and extinction coefficients are computed from the Monte Carlo solutions of Maxwell's equations for a collection of coated particles. These quantities are then applied in the dense media radiative transfer theory to calculate the polarimetric microwave emissivities of ocean surfaces with foam cover. The theoretical results of Stokes brightness temperatures with typical parameters of foam in passive remote sensing at 10.8 and 36.5 GHz are illustrated and compared with experimental measurements.It follows by describing an efficient computational model based on the sparse matrix iterative approach (SMIA) for tree scattering at VHF/UHF frequencies. The method of moments is applied to solve the volume integral equation for the tree scattering signatures. The SMIA decomposes the impedance matrix into a sparse matrix for the near interactions, and a complementary matrix for the far interactions of the tree structure. Solutions obtained from the SMIA method agree very well with the solutions obtained using extract matrix inversion and the conjugate gradient method (CGM). The key feature of the SMIA approach is that very little iteration is required to obtain convergent solutions; compared to the CGM, the SMIA approach may reduce the number of iterations by a factor of &gt;100.The UV multilevel partitioning (UV‐MLP) method is also presented for solving the general volume scattering problems. The method consists of setting up a rank table of transmitting and receiving block sizes and their separations. The table can be set up speedily using coarse–coarse sampling. For a specific scattering problem with given geometry, the scattering structure is partitioned into multilevel blocks. By looking up the rank in the pre‐determined table, the impedance matrix for a given transmitting and receiving block is expressed by a product ofUandVmatrices. We demonstrate the method for two‐dimensional volume scattering by discrete scatterers. Multiple scattering is cast into the Foldy–Lax equations of partial waves. We show that the UV decomposition can be applied directly to the impedance matrix of partial waves of higher order than the usual lowest‐order Green function. Numerical results are illustrated for randomly distributed cylinders with a diameter of 1 wavelength. For scattering by 1024 cylinders on a single PC processor with 2.6 GHz CPU and 2 GB memory, only 14 CPU minutes is needed to obtain the numerical solution and for 4096 cylinders, only 7.34 s is needed for one matrix–vector multiplication.

A UV multilevel partitioning method (UV‐MLP) is developed to solve the volume‐scattering problem. The method involves creating a rank table of transmitting and receiving block sizes and their separation. The table can be set up speedily using coarse‐coarse sampling. For a specific scattering problem with given geometry, the scattering structure is partitioned into multilevel blocks. By looking up the rank in the predetermined table, the impedance matrix for a given transmitting and receiving block is expressed as a product of U and V matrices. We illustrate the method for 2D volume scattering by discrete scatterers. Multiple scattering is cast into the Foldy–Lax equations of partial waves. We show that UV decomposition can be applied directly to the impedance matrix of partial waves of higher order than the usual lowest‐order Green's function. Numerical results are illustrated for randomly distributed cylinders that are one wavelength in diameter. For scattering by 1024 cylinders on a single PC processor with a 2.6‐GHz CPU and 2‐GB memory, only 14 CPU min are required to obtain the numerical solution. If subsectional volumetric discretization with the method of moments (MoM) is applied to this problem, the equivalent number of volumetric unknowns is 80,425. Furthermore, for 4096 cylinders that have 321,700 equivalent numbers of volumetric unknowns, it takes only 7.34 sec for one matrix‐vector multiplication. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 41: 354–361, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20140

This article presents a full‐wave electromagnetic approach for analyzing the electrical performance of massively coupled through silicon vias (TSV). The TSVs are modeled with SiO2 insulation coating and are placed in the sandwiched SiO2‐Si‐SiO2 substrate. The planar guided wave is analyzed to determine the fundamental mode and high order modes in stratified media. Cylindrical wave expansions and Foldy–Lax equations for multiple scattering techniques are adapted to the TSV problems. The effect of SiO2 coating around the via is modeled by the general expression of T‐matrix coefficients. Both dispersive silicon loss and copper loss are included in this approach. Numerical simulation of a 4‐by‐4 TSV array is demonstrated to show the signal performance and crosstalk. It shows that the coupling issues among the TSVs will become significant beyond 15 GHz. The results are in excellent agreement with general purpose field solver. © 2011 Wiley Periodicals, Inc. Microwave Opt Technol Lett 53:1204–1206, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26021

This paper proposes a novel approach to enhance the computational efficiency and precision of the ultra‐wideband characteristic basis function method based on compressive sensing for calculating the wideband radar cross section (RCS) of objects, which involves the development of a new method for filling measurement matrix and constructing sparse basis. While randomly selecting rows from the original impedance matrix as the measurement matrix, the proposed method also avoids the double‐filling of equal elements in the measurement matrix. Furthermore, the Foldy–Lax equation is utilized to construct the merged ultra‐wideband characteristic basis functions (MUCBFs) at the highest frequency, and the sparse transform of the induced currents is performed using MUCBFs as the sparse basis. The numerical simulation results show that this method not only reduces the calculation effort of impedance matrix filling, but also effectively improves the calculation precision of the target wideband RCS.

Multivia interconnect structures for printed circuit boards (PCBs) have been analyzed by decomposing the geometry into exterior and interior structures. In this paper, we include the effects of dielectric substrates in the exterior structure. The exterior problem is solved by using the method of moments (MoM). The impedance-matrix elements are evaluated by using a fast method of calculating the layered-medium Green's function. The RWG basis functions are also utilized. Combined with the Foldy–Lax equation for solving the interior structure of a cylindrical vias problem, we analyze the transmission and scattering characteristics of interconnects with via structures. The numerical results of the scattering matrix are obtained for various via structures with a through-hole via and a single-layered via. The results obtained using commercial modeling tools such as HFSS and IE3D are also compared. The differential signaling of common and differential modes of coupled interconnects are illustrated. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 46: 446–452, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21013

This paper presents an analysis of stability and resolution analysis of broadband (passive or active) array imaging in the Rician fading media. The main theoretical result is the stability condition KBN ≫ M where K is the Rician factor, B is the effective number of incoherent frequencies, N is the effective number of array elements and M is the number of sufficiently separated targets. The resolution performance of various imaging functionals is analyzed for the parabolic Markovian model. The imaging method is tested numerically with randomly distributed discrete scatterers. The numerical result with the Foldy–Lax formulation can be matched to the prediction based on the effective medium theory.

An efficient algorithm for the generalized Foldy–Lax formulation