The book is entirely dedicated to the exploration of time-domain electromagnetic fields in the presence of thin, high contrast sheets, with an emphasis on metasurfaces combining magnetic and dielectric properties.
Since the number of problems that are amenable to exact solutions in terms of analytic functions is limited, the book's analysis is not restricted to analytical methods only, attention is paid to the development of computational and approximate techniques too. All the solution methodologies presented in the book heavily rely on the Cagniard-DeHoop technique. Regrettably, perhaps because of its origins in seismology, this powerful mathematical tool is still not fully appreciated in the electromagnetics and antenna community. It is hoped that this book will demonstrate the truly broad applicability of the Cagniard-DeHoop technique to achieving both analytical and numerical time-domain solutions. The book is for advanced researchers in computational electromagnetics and those looking for new approaches to modelling of metasurfaces.
The book also includes a foreword from Professor Adrianus T. de Hoop, the originator of the Cagniard-DeHoop technique, who ingeniously simplified a joint transform initially put forward by the French geophysicist Cagniard.
Inspec keywords: method of moments; electromagnetic wave propagation; electromagnetic wave scattering; time-domain analysis; electromagnetic metamaterials
Other keywords: thin films; electromagnetic wave propagation; Maxwell equations; method of moments; electromagnetic wave scattering; conducting bodies; frequency-domain analysis; electromagnetic waves; electromagnetic metamaterials; time-domain analysis
Subjects: Electromagnetic wave propagation; Single antennas; General electrical engineering topics; Education and training; Textbooks; Handbooks and dictionaries; Artificial electromagnetic wave materials and structures; Metamaterials and structures; Electromagnetic waves: theory; Monographs, and collections
The controlled synthesis of thin-film structures has given rise to major technological breakthroughs in electronics and semiconductor industry with diverse applications including micro- and nanoelectronic devices, data and energy storage technologies, integrated optical devices and optical coatings. Design methodologies of such structures are essentially based on computational tools for solving, in an approximate way, Maxwell's equations. Owing to the multiscale nature of thin-film problems, the standard, general-purpose numerical techniques lead to unacceptably high computational efforts and very limited insights into the physics of thin layers. Despite the still increasing number of applications relying entirely upon the pulsed electromagnetic (EM)-field operation, dedicated time-domain (TD) modeling techniques for analyzing the pulsed EM-field interaction with thin-layer structures are very rare in the relevant literature. With this gap in EM theory in mind, this book aims at developing new sophisticated modeling methods and closed-form solutions, thereby providing enablers for future developments of thin-layer-based technologies.
The Cagniard-DeHoop (CdH) technique is a sophisticated mathematical tool for solving initial value problems in wave field physics that can be viewed as an ingenious simplification of the intricate joint transform procedure due to Cagniard [1, 2]. The technique has been initially developed for solving seismic pulse problems [3 4], but thanks to its versatility, it has become popular in various research fields including acoustics [5], electromagnetics [6, 7, 8] and elastodynamics [9]. More recently, the CdH technique has been successfully applied to tackle several fundamental problems in antenna theory [10, 11] or electromagnetic (EM) compatibility [12, 13], for instance.
The widespread use of composite materials in the aircraft, automotive and building industries raised the need for efficient computational methodologies capable of providing their electromagnetic (EM) scattering characteristics. Owing to the multiscale nature of structures involving composites, however, both standard direct-discretization techniques (e.g. the finite difference time-domain (TD) method) and volumetric integral-equation-based approaches (e.g. the method of moments) suffer from the heavy computational burden or/and meshing issues. A way out of the difficulty has been offered by thin-sheet transition conditions (e.g. [1, Section 2.4]) that considerably simplify calculations by reducing the actual volumetric solution domain to the corresponding surface one. Accordingly, thin-sheet saltus-type conditions suitable for analyzing pulsed EM fields in the presence of thin-sheet layers are the main subject of this chapter. Their TD formulation concerning thin-film structures with dielectric and conductive properties has been introduced [2], while the idea behind the thin-sheet cross-layer conditions can be traced back to the works of Levi--Civita from the beginning of the twentieth century [3, p. 19].
In this chapter, we shall provide illustrative applications of the thin-sheet, high-contrast boundary conditions introduced in Chapter 3. In particular, we shall explore reflection and transmission properties of a metasurface excited by impulsive electromagnetic (EM) sources. Starting from a simple, plane-wave-excited two-dimensional (2-D) configuration, the problem is gradually extended by analyzing the time-domain (TD) response to a localized line-source excitation and the loop-to-loop pulsed EM-field signal transfer across the sheet. In all the analyzed problems, a discussion on achieving the EM transparency is given.
A considerable success achieved in the beginning of the twentieth century in the experimental wireless transmission of electromagnetic (EM) signals over long distances has triggered the intensive research into the EM wave propagation mechanisms. Besides the explanation based on the EM-field reflection against the Kenelly-Heaviside ionized layer [1], a large amount of attention has been paid, under the influence of the discovery of surface wave phenomena traveling along a free surface of a semi-infinite elastic solid [2, 3], to the existence of EM surface waves propagating along the air-earth interface. These efforts have yielded useful mathematical tools for solving a class of EM boundary-value problems [4], but their interpretation has given rise to long-standing controversies (see [5, 6], for example) and terminology confusion [7]. While such disputes are still not fully settled in the realm of a frequency-domain (FD) analysis, the situation is truly transparent in the time domain (TD), where any wave phenomena do manifest themselves. Indeed, under the causality-preserving Cagniard-DeHoop (CdH)-like representation, any singularity in the complex slowness plane can be directly associated with a physical phenomenon occurring in the resulting wave motion [8]. Hence, the presence of a pole singularity in the pertaining slowness complex plane implies the existence of a true (causal) surface wave (e.g. the Rayleigh wave at the traction-free boundary of a solid or the Scholte wave along a fluid-solid interface [9]). Therefore, the CdH-technique is also employed throughout this chapter, where TD EM surface effects excited along the surface of a thin, highly contrasting layer are analyzed analytically.
Electromagnetic (EM) diffraction by a perfectly electrically conducting (PEC) semi-infinite screen is a canonical problem in EM wave theory whose first exact solution is attributed to Sommerfeld [1]. The need for physical insights into EM diffraction problems that would enable the evaluation of antenna performance and EM propagation in a complex environment called for a general solution methodology capable of handling more involved problem configurations. Significant progress in these efforts has been accomplished via integral-equation formulations [2, 3] that are amenable to the Wiener-Hopf (WH) method [4]. Through this methodology, the original Sommerfeld's half-plane problem has been generalized by incorporating finite conductivity of the diffracting screen [5, 6] or by placing the semi-infinite PEC screen on the planar interface of two media [7]. A detailed summary of major achievements in this field of research is given in Ref. [8].
Despite a myriad of applications relying entirely on pulsed EM fields, the majority of available analytical solutions have been obtained under the assumption of the sinusoidal time dependence only. In these closed-form frequency-domain (FD) solutions, the (real-valued) frequency parameter generally occurs in intricate functional dependencies, which does not allow achieving their time-domain (TD) counterparts analytically without the use of an inverse (fast) Fourier transform. To provide closed-form analytical solutions directly in the TD, we shall next combine the Cagniard-DeHoop (CdH) technique (see Chapter 1) with the WH method. For truly seminal works pioneering the presented methodology, we refer the reader to [9, 10].
Electromagnetic (EM) scattering by an infinitely long, narrow perfectly electrically conducting (PEC) strip has been approximately analyzed in the real frequency domain (FD) by Schelkunoff using the concept of external (per-unit-length) impedance [1, Section 8.5]. This efficient approach has been later generalized in Ref. [2] with the aid of the time-domain (TD) EM reciprocity theorem (see [3]), thus formulating the scattering problem in the form of an integral equation of the time-convolution type. This line of reasoning is also pursued in this chapter, where an efficient computational methodology capable of achieving the transient electric- and magnetic-current distributions induced along a narrow metastrip is introduced. The combined effect of (magneto-)dielectric and conductive properties of EM-penetrable metastrips is incorporated using the concept of thin-sheet, high-contrast transition conditions as introduced in Chapter 3.
Closed-form solutions of pulsed electromagnetic (EM) scattering problems in terms of analytic functions are attainable for a limited number of configurations only. Several examples from this category including infinite, semi-infinite and narrow thin screens are presented in Chapters 4, 6 and 7, respectively. Whenever the problem under consideration cannot be plausibly approximated by one of the limiting cases, one has to resort to a numerical technique. An efficient computational method capable of handling pulsed EM scattering by bounded planar screens is termed as the Cagniard-DeHoop method of moments (CdH-MoM) (see [1, Chapters 2 and 14]). Its application to analyzing the pulsed EM scattering by two-dimensional (2-D) planar structures located on a homogeneous background is hence the main subject matter of this chapter.
The need for efficient and controlled synthesis of thin-film structures that are encountered in various applications of electronics and semiconductor industries (e.g. microelectronic circuits, integrated optical devices) puts severe demands on electromagnetic (EM) solvers for achieving their EM scattering characteristics. Such structures typically consist of highly conductive layers located in a piecewise homogeneous embedding, which cannot be directly analyzed with the aid of the Cagniard-DeHoop method of moments (CdH-MoM) formulation introduced in Chapter 8. Therefore, it is shown in this chapter how the CdH-MoM can be generalized, thus providing an efficient computational tool for analyzing the pulsed EM scattering by bounded, highly contrasting thin sheets on a layered background. For the sake of clarity, the methodology is illustrated based on the analysis of EM-penetrable sheets supporting electric currents only. The incorporation of combined effects of both electric and magnetic currents flowing on metasurfaces can be handled along similar lines.
In order to evaluate electromagnetic (EM) phenomena such as cross talk or substrate noise coupling in thin-film multilayered structures, the pulsed EM field interaction between conductive strips has to be properly incorporated in a computational model. This is exactly the main purpose of this chapter, where it is demonstrated how mutual-coupling effects can be included in the Cagniard-DeHoop method of moments (CdH-MoM). Again, the main mathematical tool for the analysis is the EM reciprocity theorem of the time-convolution type. The reciprocity-based formulation yields a reciprocity relation that is upon discretizing the (space-time) solution domain cast into its matrix form, elements of which represent local and remote interactions involved. A few illustrative examples, including the EM coupling between two plane screens on the dielectric half-space and on the grounded dielectric slab, are discussed and analyzed in detail. More complex problems can be analyzed along the same lines.
We have demonstrated in Chapter 10 that the Cagniard-DeHoop method of moments (CdH-MoM) can serve for evaluating the electromagnetic (EM) coupling between planar strips supporting electric currents. In this chapter, the EM reciprocity theorem of the time-convolution type is applied, again, to put forward an efficient computational methodology that is capable of calculating the pulsed EM scattering by multiple metasurfaces exhibiting both magnetic and dielectric constitutive properties. In particular, it is shown that under the assumption of piecewise linear space-time distribution of unknown currents, the pertaining EM reciprocity relation can be cast into a system of time-convolution type that is amenable to its solution via the marching-on-in-time technique. In addition, a special case concerning the EM coupling between relatively narrow metastrips is analyzed in detail.
If the analyzed structure and its excitation do not vary along the y-direction, the electromagnetic (EM) scattering problem can be solved through a set of y-independent Maxwell's equations supplemented with the pertaining boundary conditions. A variety of such two-dimensional (2-D) EM scattering problems have been analyzed throughout the previous chapters. If the problem dimensionality reduction is no longer appropriate, one has to resort to a general, fully three-dimensional (3-D) methodology. Such a computational approach is the subject of this chapter, where the pulsed EM scattering by thin, highly contrasting layers is analyzed with the aid of the Cagniard-DeHoop method of moments (CdH-MoM) [1]. Special attention is paid to the incorporation of combined magneto-dielectric EM constitutive properties of metasurfaces.
Electromagnetic (EM)-field penetration through a slot in a planar sheet is a fundamental problem of EM theory that has been the subject of in-depth research in the real-frequency domain (FD) (see [1, 2] and the references therein). In this chapter, it is demonstrated how the problem of EM scattering by apertures in a perfectly electrically conducting (PEC) planar sheet can be solved in the time domain (TD). To that end, we shall first employ the Cagniard-DeHoop method of moments (CdH-MoM) to tackle the two-dimensional (2-D) problem consisting of a slot in a PEC screen. Second, under the assumption that the slot is relatively narrow, we shall pursue the approach previously applied to narrow strips (see Chapter 7) and describe its EM scattering properties in terms of Kirchoff-circuit equivalent parameters such as the "aperture admittance" [3]. With an emphasis on the proper problem formulation, the analysis is further generalized to incorporate the presence of a dielectric half-space, a rectangular filled groove and a homogeneous slot's filling. Finally, it is shown that the CdH-MoM computational methodology can be readily extended to analyzing the pulsed EM scattering by a bounded aperture of arbitrary shape.
Throughout the chapter, the EM reciprocity theorem of the time-convolution type is taken as the point of departure (see [4, Section 28.4] and [5, Section 1.4.1]). For details regarding the EM scattering by slots in the real-FD, the reader is referred to thorough texts on the subject (see [6, Section 5.2] and [7, Section 14.9]).
Space-time metasurfaces hold great promise for their virtually unlimited potentialities for the development of fundamentally new material-based devices [1, 2]. Transient electromagnetic (EM) scattering by structures with time-varying constitutive properties is, however, a little understood phenomenon that is analyzed either numerically via finite-difference approaches [3, 4], or analytically under the restriction of a periodically modulated medium [5, 6], or the constant wave impedance [7, 8].
In this chapter, we introduce a straightforward analytical procedure for analyzing the pulsed plane-wave EM scattering by simple time-varying metasurfaces. In particular, it is demonstrated that the approach is applicable to EM-impenetrable perfectly electrically/magnetically conducting (PEC/PMC) metasurfaces [9] and to thin highly conducting sheets with the time-dependent electric conductivity.
Whenever the transverse dimension of a waveguide structure is relatively small such that its characteristic electromagnetic (EM) properties can be described via the concept of Kirchhoff's circuit lumped elements, the EM propagation along the structure's axis is governed by the transmission-line (TL) equations [1, Chapter VII]. To evaluate the pulse propagation along nonideal practical lines (e.g. an overhead wire over a lossy ground [2, Chapter 8]), one has to rely on approximate or/and numerical approaches. This strategy is also pursued in this chapter, where the Cagniard-DeHoop method of moments (CdH-MoM) is applied to analyze the pulsed EM scattering by a straight transmission line that is located above a thin highly contrasting layer. Accordingly, the computational approach introduced below bridges the gap between the EM reciprocity-based, EM-field-to-line coupling model [3, Chapters 11-13] and the CdH-MoM analysis of a narrow planar strip above the perfectly electrically conducting (PEC) ground [3, Chapter 17]. For its applications to the transient analysis of a transmission line in the presence of a highly conducting thin layer, the reader is referred to [4].
In this chapter, we shall analyze selected illustrative problem configurations to demonstrate the use of some Cagniard-DeHoop (CdH)-based analytical tools for achieving closed-form time-domain (TD) solutions. First, to demonstrate the principle of a version of the CdH technique that is suitable for describing the short-range electromagnetic (EM) signal wireless transfer between antennas, we shall study the loop-to-loop signal transmission via a thin highly conductive layer. The introduced analytical methodology is directly applicable to the estimation of the (low-frequency) shielding effectiveness in standardized experimental scenarios (see [1, Appendix C], for instance), for instance, and can be further extended to more complex problem configurations consisting of fundamental EM radiators integrated in a layered structure. Consequently, we shall demonstrate how the analytical methodology described in Section 4.3.1 can be extended to incorporate plasmonic properties of the layer lying in between the transmitting and receiving loop antennas. Finally, a modified Kirchhoff approximation is applied to analyze the pulsed EM radiation from a two-dimensional (2-D) slot-excited Fabry-Pérot antenna. Again, the approximate analysis yields TD expressions that give some insights into the pulsed EM phenomenon.
The edge-diffracted wave on a semi-infinite perfectly electrically conducting (PEC) sheet is expressed in the form of a two-dimensional cylindrical wave propagating away from the edge (see (6.12)). To reveal the link between Sommerfeld's analytical solution [1] and the CdH-based TD solution represented by (6.12), we shall derive an explicit expression for the diffracted wave amplitude.
The elements of the impeditivity array that occurs in the Cagniard-DeHoop method of moments (CdH-MoM) solution of electromagnetic (EM) scattering by a bounded perfectly electrically conducting (PEC) screen can be expressed through the following wave-slowness representation (cf. (E.1)).
The elements of the excitation array that occurs in the Cagniard-DeHoop method of moments (CdH-MoM) solution of electromagnetic (EM) scattering by a bounded screen can be expressed through the following wave-slowness representation.
The elements of the impeditivity array that occurs in the Cagniard-DeHoop method of moments (CdH-MoM) solution of electromagnetic (EM) scattering by a bounded screen with conductive and dielectric properties can be expressed through the following wave-slowness representation.
The elements of the admittivity array that occurs in the Cagniard-DeHoop method of moments (CdH-MoM) solution of electromagnetic (EM) scattering by a bounded perfectly magnetically conducting (PMC) screen can be expressed through the following wave-slowness representation (cf. (B.1)).
The impeditivity elements that serve for analyzing the electromagnetic (EM) scattering by a grounded screen (see Section 9.1.2) and the mutual EM coupling between two electrically conducting screens (see (10.13)) can be expressed as follows (cf. (B.1)).
The elements of the impeditivity array that occurs in the Cagniard-DeHoop method of moments (CdH-MoM) solution of electromagnetic (EM) scattering by a bounded screen located on a dielectric half-space can be expressed as follows (cf. (B.1)).
The elements of the impeditivity array that occurs in the Cagniard-DeHoop method of moments (CdH-MoM) solution of electromagnetic (EM) scattering by a screen above a grounded slab can be expressed as follows (cf. (F.1) and (G.1)).
The admittivity elements that serve for analyzing the mutual electromagnetic (EM) coupling between two magnetically conducting screens can be expressed as follows (cf. (11.24) and (11.25)).
The electromagnetic (EM) coupling between two noncoplanar metasurfaces can be analyzed through arrays, the inverse transformation of which is demonstrated on the elements defined by (11.26a) and (11.27b).
The elements of the impedance array that occurs in the Cagniard-DeHoop method of moments (CdH-MoM) solution of time-domain (TD) electromagnetic (EM) scattering by a 3-D bounded screen with dielectric and conductive properties can be expressed through (12.12a)-(12.13b).
The problem solution based on the Cagniard-DeHoop method of moments (CdH-MoM) leads to a system of equations of the convolution type (see (8.10)), the (time-dependent) coefficients of which are frequently expressed in the form of convolution-type integrals (e.g. (F.7)). To facilitate an efficient calculation of such integrals, a dedicated technique based on a recursive scheme (see [1]) was introduced in [2, Appendix H]. In this appendix, the recursive-convolution technique is briefly described.