arXiv subjects

Eric Michielssen

Publications and source records attributed to Eric Michielssen.

17 recordsLinked to original sources

arXiv2022

Wigner-Smith Time Delay Matrix for Electromagnetics: Guiding and Periodic Systems with Evanescent Modes

The Wigner-Smith (WS) time delay matrix relates an electromagnetic system's scattering matrix and its frequency derivative. Previous work showed that the entries of WS time delay matrices of systems excited by propagating waves consist of volume integrals of energy-like field quantities. This paper introduces a generalized WS relationship that applies to systems excited by mixtures of propagating and evanescent fields. Just like its predecessor, the generalized WS relationship allows for the identification of so-called WS modes that interact with the system with well-defined time delays. Furthermore, a technique is developed to compute the WS time delay matrix of a composite system from the WS time delay matrices of its subsystems. Numerical examples demonstrate the usefulness of the generalized WS method when characterizing time delays experienced by fields interacting with guiding and periodic structures that have ports supporting evanescent modes.

eess.SY↗

arXiv2022

Wigner-Smith Time Delay Matrix for Electromagnetics: Systems with Material Dispersion and Losses

The Wigner-Smith (WS) time delay matrix relates a system's scattering matrix to its frequency derivative and gives rise to so-called WS modes that experience well-defined group delays when interacting with the system. For systems composed of nondispersive and lossless materials, the WS time delay matrix previously was shown to consist of volume integrals of energy-like densities plus correction terms that account for the guiding, scattering, or radiating characteristics of the system. This study extends the use of the WS time delay matrix to systems composed of dispersive and lossy materials. Specifically, it shows that such systems' WS time delay matrix can be expressed by augmenting the previously derived expressions with terms that account for the dispersive and lossy nature of the system, followed by a transformation that disentangles effects of losses from time delays. Analytical and numerical examples demonstrate the new formulation once again allows for the construction of frequency stable WS modes that experience well-defined group delays upon interacting with a system.

physics.comp-ph↗

arXiv2022

Wigner-Smith Time Delay Matrix for Acoustic Scattering: Computational Aspects

The Wigner-Smith (WS) time delay matrix relates an acoustic system's scattering matrix to its wavenumber derivative. The entries of the WS time delay matrix can be expressed in terms of energy density-like volume integrals, which cannot be efficiently evaluated in a boundary element method framework. This paper presents two schemes for efficiently populating the WS time delay matrix. The direct formulation casts the energy density-like volume integrals into integrals of the incident field and the field and/or its normal derivative over the scatterer surface. The indirect formulation computes the system's scattering matrix and its wavenumber derivative, again via surface integration, and then invokes the WS relationship to compute the WS time delay matrix. Both the direct and the indirect formulations yield equivalent results and can be easily integrated into standard boundary element codes.

cs.CE↗

arXiv2022

Wigner-Smith Time Delay Matrix for Acoustic Scattering: Theory and Phenomenology

The Wigner-Smith (WS) time delay matrix relates a lossless system's scattering matrix to its frequency derivative. First proposed in the realm of quantum mechanics to characterize time delays experienced by particles during a collision, this article extends the use of WS time delay techniques to acoustic scattering problems governed by the Helmholtz equation. Expression for the entries of the WS time delay matrix involving renormalized volume integrals of energy densities are derived, and shown to hold true independent of the scatterer's geometry, boundary condition (sound-soft or sound-hard), and excitation. Numerical examples show that the eigenmodes of the WS time delay matrix describe distinct scattering phenomena characterized by well-defined time delays.

cs.CE↗

arXiv2021

The Design of Dual Band Stacked Metasurfaces Using Integral Equations

An integral equation-based approach for the design of dual band stacked metasurfaces which are invariant in one-dimension is presented. The stacked metasurface will generate collimated beams at desired angles in each band upon reflection. The conductor-backed stacked metasurface consists of two metasurfaces (a patterned metallic cladding supported by a dielectric spacer) stacked one upon the other. The stacked metasurface is designed in three phases. First the patterned metallic cladding of each metasurface is homogenized and modeled as an inhomogeneous impedance sheet. An Electric Field Integral Equation (EFIE) is written to model the mutual coupling between the homogenized elements within each metasurface, and from metasurface to metasurface. The EFIE is transformed into matrix equations by the method of moments. The nonlinear matrix equations are solved at both bands iteratively resulting in dual band complex-valued impedance sheets. In the second phase, optimization is applied to transform these complex-valued impedance sheets into purely reactive sheets suitable for printed circuit board fabrication by introducing surface waves. In the third phase, the metallic claddings of each metasurface are patterned for full-wave simulation of the dual band stacked metasurface. Using this approach, two dual band stacked metasurfaces are designed

physics.app-ph↗

arXiv2021

Wigner-Smith Time Delay Matrix for Electromagnetics: Theory and Phenomenology

Wigner-Smith (WS) time delay concepts have been used extensively in quantum mechanics to characterize delays experienced by particles interacting with a potential well. This paper formally extends WS time delay theory to Maxwell's equations and explores its potential applications in electromagnetics. The WS time delay matrix relates a lossless and reciprocal system's scattering matrix to its frequency derivative and allows for the construction of modes that experience well-defined group delays when interacting with the system. The matrix' entries for guiding, scattering, and radiating systems are energy-like overlap integrals of the electric and/or magnetic fields that arise upon excitation of the system via its ports. The WS time delay matrix has numerous applications in electromagnetics, including the characterization of group delays in multiport systems, the description of electromagnetic fields in terms of elementary scattering processes, and the characterization of frequency sensitivities of fields and multiport antenna impedance matrices.

physics.comp-ph↗

arXiv2020

Wigner-Smith Time Delay Matrix for Electromagnetics: Computational Aspects for Radiation and Scattering Analysis

The WS time delay matrix relates a lossless and reciprocal system's scattering matrix to its frequency derivative, and enables the synthesis of modes that experience well-defined group delays when interacting with the system. The elements of the WS time delay matrix for surface scatterers and antennas comprise renormalized energy-like volume integrals involving electric and magnetic fields that arise when exciting the system via its ports. Here, direct and indirect methods for computing the WS time delay matrix are presented. The direct method evaluates the energy-like volume integrals using surface integral operators that act on the incident electric fields and current densities for all excitations characterizing the scattering matrix. The indirect method accomplishes the same task by computing scattering parameters and their frequency derivatives. Both methods are computationally efficient and readily integrated into existing surface integral equation codes. The proposed techniques facilitate the evaluation of frequency derivatives of antenna impedances, antenna patterns, and scatterer radar cross sections in terms of renormalized field energies derived from a single frequency characterization of the system.

physics.comp-ph↗

arXiv2020

Butterfly factorization via randomized matrix-vector multiplications

This paper presents an adaptive randomized algorithm for computing the butterfly factorization of a $m\times n$ matrix with $m\approx n$ provided that both the matrix and its transpose can be rapidly applied to arbitrary vectors. The resulting factorization is composed of $O(\log n)$ sparse factors, each containing $O(n)$ nonzero entries. The factorization can be attained using $O(n^{3/2}\log n)$ computation and $O(n\log n)$ memory resources. The proposed algorithm applies to matrices with strong and weak admissibility conditions arising from surface integral equation solvers with a rigorous error bound, and is implemented in parallel.

math.NA↗

arXiv2019

Magnetic and Combined Field Integral Equations Based on the Quasi-Helmholtz Projectors

Boundary integral equation methods for analyzing electromagnetic scattering phenomena typically suffer from several of the following problems: (i) ill-conditioning when the frequency is low; (ii) ill-conditioning when the discretization density is high; (iii) ill-conditioning when the structure contains global loops (which are computationally expensive to detect); (iv) incorrect solution at low frequencies due to current cancellations; (v) presence of spurious resonances. In this paper, quasi-Helmholtz projectors are leveraged to obtain a magnetic field integral equation (MFIE) formulation that is immune to drawbacks (i)-(iv). Moreover, when this new MFIE is combined with a regularized electric field integral equation, a new quasi-Helmholtz projector combined field integral equation is obtained that also is immune to (v). Numerical results corroborate the theory and show the practical impact of the newly proposed formulations.

physics.comp-ph↗

arXiv2018

Computation of Electromagnetic Fields Scattered From Objects With Uncertain Shapes Using Multilevel Monte Carlo Method

Computational tools for characterizing electromagnetic scattering from objects with uncertain shapes are needed in various applications ranging from remote sensing at microwave frequencies to Raman spectroscopy at optical frequencies. Often, such computational tools use the Monte Carlo (MC) method to sample a parametric space describing geometric uncertainties. For each sample, which corresponds to a realization of the geometry, a deterministic electromagnetic solver computes the scattered fields. However, for an accurate statistical characterization the number of MC samples has to be large. In this work, to address this challenge, the continuation multilevel Monte Carlo (CMLMC) method is used together with a surface integral equation solver. The CMLMC method optimally balances statistical errors due to sampling of the parametric space, and numerical errors due to the discretization of the geometry using a hierarchy of discretizations, from coarse to fine. The number of realizations of finer discretizations can be kept low, with most samples computed on coarser discretizations to minimize computational cost. Consequently, the total execution time is significantly reduced, in comparison to the standard MC scheme.

physics.comp-ph↗

arXiv2017

Mode Control in a Multimode Fiber Through Acquiring its Transmission Matrix from a Reference-less Optical System

A simple imaging system together with complex semidefinite programming is used to generate the transmission matrix of a multimode fiber. Once the transmission matrix is acquired, we can modulate the phase of the input signal to induce strong mode interference at the fiber output. The optical design does not contain a reference arm and no interferometric measurements are required. We use a phase-only spatial light modulator to shape the profile of the propagating modes and the output intensity at an individual pixel is monitored. The semidefinite program uses a convex optimization algorithm to generate the transmission matrix of the optical system using intensity only measurements. This simple yet powerful method can be used to compensate for modal dispersion in multimode fiber communication systems. It also yields great promises for the next generation biomedical imaging, quantum communication, and cryptography.

physics.optics↗

arXiv2017

Controlling Light Transmission Through Highly Scattering Media Using Semi-Definite Programming as a Phase Retrieval Computation Method

Complex Semi-Definite Programming (SDP) is introduced as a novel approach to phase retrieval enabled control of monochromatic light transmission through highly scattering media. In a simple optical setup, a spatial light modulator is used to generate a random sequence of phase-modulated wavefronts, and the resulting intensity speckle patterns in the transmitted light are acquired on a camera. The SDP algorithm allows computation of the complex transmission matrix of the system from this sequence of intensity-only measurements, without need for a reference beam. Once the transmission matrix is determined, optimal wavefronts are computed that focus the incident beam to any position or sequence of positions on the far side of the scattering medium, without the need for any subsequent measurements or wavefront shaping iterations. The number of measurements required and the degree of enhancement of the intensity at focus is determined by the number of pixels controlled by the spatial light modulator.

physics.optics↗

arXiv2016

A Butterfly-Based Direct Integral Equation Solver Using Hierarchical LU Factorization for Analyzing Scattering from Electrically Large Conducting Objects

A butterfly-based direct combined-field integral equation (CFIE) solver for analyzing scattering from electrically large, perfect electrically conducting objects is presented. The proposed solver leverages the butterfly scheme to compress blocks of the hierarchical LU-factorized discretized CFIE operator and uses randomized butterfly reconstruction schemes to expedite the factorization. The memory requirements and computational cost of the direct butterfly-CFIE solver scale as $O(N\mathrm{log}^2N)$ and $O(N^{1.5}\mathrm{log}N)$, respectively. These scaling estimates permit significant memory and CPU savings when compared to those realized by low-rank (LR) decomposition-based solvers. The efficacy and accuracy of the proposed solver are demonstrated through its application to the analysis of scattering from canonical and realistic objects involving up to 14 million unknowns.

math.NA↗

arXiv2016

A HSS Matrix-Inspired Butterfly-Based Direct Solver for Analyzing Scattering from Two-dimensional Objects

A butterfly-based fast direct integral equation solver for analyzing high-frequency scattering from two-dimensional objects is presented. The solver leverages a randomized butterfly scheme to compress blocks corresponding to near- and far-field interactions in the discretized forward and inverse electric field integral operators. The observed memory requirements and computational cost of the proposed solver scale as O(Nlog^2N) and O(N^1.5 logN), respectively. The solver is applied to the analysis of scattering from electrically large objects spanning over ten thousand of wavelengths and modeled in terms of five million unknowns.

math.NA↗

arXiv2015

The transmission coefficient distribution of highly scattering sparse random media

We consider the distribution of the transmission coefficients, i.e. the singular values of the modal transmission matrix, for 2D random media with periodic boundary conditions composed of a large number of point-like non-absorbing scatterers. The scatterers are placed at random locations in the medium and have random refractive indices that are drawn from an arbitrary, known distribution. We construct a randomized model for the scattering matrix that retains scatterer dependent properties essential to reproduce the transmission coefficient distribution and analytically characterize the distribution of this matrix as a function of the refractive index distribution, the number of modes, and the number of scatterers. We show that the derived distribution agrees remarkably well with results obtained using a numerically rigorous spectrally accurate simulation. Analysis of the derived distribution provides the strongest principled justification yet of why we should expect perfect transmission in such random media regardless of the refractive index distribution of the constituent scatterers. The analysis suggests a sparsity condition under which random media will exhibit a perfect transmission-supporting universal transmission coefficient distribution in the deep medium limit.

cond-mat.dis-nn↗

arXiv2014

Iterative, backscatter-analysis algorithms for increasing transmission and focusing light through highly-scattering random media

Scattering hinders the passage of light through random media and consequently limits the usefulness of optical techniques for sensing and imaging. Thus, methods for increasing the transmission of light through such random media are of interest. Against this backdrop, recent theoretical and experimental advances have suggested the existence of a few highly transmitting eigen-wavefronts with transmission coefficients close to one in strongly backscattering random media. Here, we numerically analyze this phenomenon in 2-D with fully spectrally accurate simulators and provide rigorous numerical evidence confirming the existence of these highly transmitting eigen-wavefronts in random media with periodic boundary conditions that is composed of hundreds of thousands of non-absorbing scatterers. Motivated by bio-imaging applications where it is not possible to measure the transmitted fields, we develop physically realizable algorithms for increasing the transmission through such random media using backscatter analysis. We show via numerical simulations that the algorithms converge rapidly, yielding a near-optimum wavefront in just a few iterations. We also develop an algorithm that combines the knowledge of these highly transmitting eigen-wavefronts obtained from backscatter analysis, with intensity measurements at a point to produce a near-optimal focus with significantly fewer measurements than a method that does not utilize this information.

physics.optics↗

arXiv2014

Backscatter analysis based algorithms for increasing transmission through highly-scattering random media using phase-only modulated wavefronts

Recent theoretical and experimental advances have shed light on the existence of so-called `perfectly transmitting' wavefronts with transmission coefficients close to 1 in strongly backscattering random media. These perfectly transmitting eigen-wavefronts can be synthesized by spatial amplitude and phase modulation. Here, we consider the problem of transmission enhancement using phase-only modulated wavefronts. We develop physically realizable iterative and non-iterative algorithms for increasing the transmission through such random media using backscatter analysis. We theoretically show that, despite the phase-only modulation constraint, the non-iterative algorithms will achieve at least about 25$π$% or about 78.5% transmission assuming there is at least one perfectly transmitting eigen-wavefront and that the singular vectors of the transmission matrix obey a maximum entropy principle so that they are isotropically random. We numerically analyze the limits of phase-only modulated transmission in 2-D with fully spectrally accurate simulators and provide rigorous numerical evidence confirming our theoretical prediction in random media with periodic boundary conditions that is composed of hundreds of thousands of non-absorbing scatterers. We show via numerical simulations that the iterative algorithms we have developed converge rapidly, yielding highly transmitting wavefronts using relatively few measurements of the backscatter field. Specifically, the best performing iterative algorithm yields approx 70% transmission using just 15-20 measurements in the regime where the non-iterative algorithms yield approximately 78.5% transmission but require measuring the entire modal reflection matrix.

physics.optics↗