Reciprocity Theorem Research Articles

Is it possible to consider a thin hologram as a slide?

A thin phase hologram with sinusoidal modulation of the refractive index is considered. The applicability of the approach is discussed, in which it is assumed that light passes through a hologram, as through a slide, which leads to a sinusoidal modulation of the wave phase. A consequence of this consideration are expressions for the diffraction orders amplitudes, which coincide with the expressions obtained in the Raman-Nath theory of light scattering by ultrasonic waves. The paper compares expressions for the amplitude of the first order of diffraction, obtained in the framework of several theoretical approaches, including perturbation theory in a rigorous formulation of the boundary value problem. As the thickness of the hologram tends to zero, the expression obtained in this way coincides with the expression obtained in the two-wave approximation of the coupled-wave theory. Both expressions satisfy the electrodynamic reciprocity theorem. Under the same conditions, the expression obtained as a result of the slide-like consideration does not satisfy the reciprocity theorem and differs from the indicated expressions by a factor equal to the ratio of the cosines of the diffracted and incident waves. It is concluded that considering a thin hologram as a slide can lead to noticeable errors.

Analytical solution of indentation response of an elliptical cylindrical indenter on elastic microbeam with surface tension

<p indent="0mm">In addition to the measurement of mechanical properties of bulk materials, the indentation technology is widely used to measure the mechanical properties of soft films and microscale materials. As the material scale decreases, surface effects play an important role in the force and deformation in the indentation contact problem. In this paper, the two-dimensional indentation problem of an elastic microbeam under an elliptical cylinder indenter is studied theoretically. For the microbeam with the surface tension, the differential governing equation satisfied by the surface pressure distribution and the normal deflection is solved. The analytical solutions of the surface pressure and surface displacement inside and outside the contact area are obtained, where the undetermined coefficients are determined by the contact zone half-width and the indentation depth. The deformation solution of the microbeam under uniform load and Betty’s reciprocal theorem of the indentation problem are given. Then, the contact zone half-width and indentation depth under different measurement forces are solved, and the analytical solution of the indentation problem of an elliptical cylindrical indenter on elastic microbeam is obtained. The effect of surface tension, elliptical indenter parameters and microbeam size parameters on surface pressure and deformation in the indentation problem is analyzed theoretically. This study helps to understand the surface effect on the indentation behavior of microscale materials, and provides some reference for related indentation testing techniques.

Analytical solution of acoustic emission in soft material with cracks by using reciprocity theorem

Special property in soft materials manifests itself as slight external action may occur significant changes in structure or performance followed by crack initiation and propagation. Characterizing the transition from a crack-free to a cracked soft material remains one of the most challenging problems in structural changes. New methods are needed to predict when and how these soft materials will fail due to defects. In this context, we explored the acoustic emission phenomenon of a crack generation with micro-nanometer size in rubber seal rings. We found through the reciprocity theorem and equivalent body forces that this tiny structural change is defined by the opening crack time function, crack opening body and elastic constant. We take rubber as an example and analyze the new application of acoustic emission and propose a methodology for near surface crack detection of soft materials. Additionally, we discuss the contributions of the opening crack time function, cyclic frequency and elastic constant. The present ‘acoustic emission applications’ strategy offers an effective pathway to observe soft materials' structural changes.

Theory for a non-invasive diagnostic biomarker for craniospinal diseases

Monitoring intracranial pressure (ICP) and craniospinal compliance (CC) is frequently required in the treatment of patients suffering from craniospinal diseases. However, current approaches are invasive and cannot provide continuous monitoring of CC. Dynamic exchange of blood and cerebrospinal fluid (CSF) between cranial and spinal compartments due to cardiac action transiently modulates the geometry and dielectric properties of the brain. The resulting impedance changes can be measured and might be usable as a non-invasive CC surrogate. A numerically robust and computationally efficient approach based on the reciprocity theorem was developed to compute dynamic impedance changes resulting from small geometry and material property changes. The approach was successfully verified against semi-analytical benchmarks, before being combined with experimental brain pulsation data to study the information content of the impedance variation. The results indicate that the measurable signal is dominated by the pulsatile displacement of the cortical brain surface, with minor contributions from the ventricular surfaces and from changes in brain perfusion. Different electrode setups result in complementary information. The information content from the investigated three electrode pairs was employed to successfully infer subject-specific brain pulsation and motion features. This suggests that non-invasive CC surrogates based on impedance monitoring could be established.

Pruned Inside-Out Polytopes, Combinatorial Reciprocity Theorems and Generalized Permutahedra

Generalized permutahedra are a class of polytopes with many interesting combinatorial subclasses. We introduce pruned inside-out polytopes, a generalization of inside-out polytopes introduced by Beck-Zaslavsky (2006), which have many applications such as recovering the famous reciprocity result for graph colorings by Stanley. We show (quasi-)polynomiality and reciprocity results for the integer point count of pruned inside-out polytopes by applying classical Ehrhart polynomials and Ehrhart-Macdonald reciprocity. This yields a geometric perspective on and a generalization of a combinatorial reciprocity theorem for generalized permutahedra by Aguiar-Ardila (2017), Billera-Jia-Reiner (2009), and Karaboghossian (2022). Applying this reciprocity theorem to hypergraphic polytopes allows to give a geometric proof of a combinatorial reciprocity theorem for hypergraph colorings by Aval-Karaboghossian-Tanasa (2020). This proof relies, aside from the reciprocity for generalized permutahedra, only on elementary geometric and combinatorial properties of hypergraphs and their associated polytopes.

Efficient Computation of Electromagnetic Coupling From Various Types of Antennas Based on Reciprocity Theorem

In the design of complex electronic systems, such as electric vehicles, electromagnetic coupling between a specific component and an antenna should be quantitatively estimated according to the radiated emission (RE) and radiated immunity (RI) standards. However, full-wave electromagnetic simulations of complex systems and antennas in a large space take much time and effort. The decomposition method based on reciprocity and Huygens's principle, which enables the separate analysis of source and victim parts, has been previously proposed for efficient analysis of coupling to a lumped port of the victim antenna. In this article, the decomposition method along with the extraction of an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> -parameter block is extended to handle the dominant transverse electric mode in a waveguide antenna. The proposed method removes the difficulty and uncertainty in the modeling process of a commercial waveguide antenna. Also, while applying the decomposition method, Huygens's principle is additionally applied to further reduce the full-wave simulation time of a large empty space. The proposed methods are validated with full-wave electromagnetic simulations, and also experimentally applied to RI estimation. The proposed time-efficient techniques can realize the full-wave computation of RE and RI for real complicated systems and commercial antennas.

Active Stokesian dynamics

Since its development, Stokesian dynamics has been a leading approach for the dynamic simulation of suspensions of particles at arbitrary concentrations with full hydrodynamic interactions. Although developed originally for the simulation of passive particle suspensions, the Stokesian dynamics framework is equally well suited to the analysis and dynamic simulation of suspensions of active particles, as we elucidate here. We show how the reciprocal theorem can be used to formulate the exact dynamics for a suspension of arbitrary active particles, and then show how the Stokesian dynamics method provides a rigorous way to approximate and compute the dynamics of dense active suspensions where many-body hydrodynamic interactions are important.

Reciprocal swimming at intermediate Reynolds number

In Stokes flow, Purcell's scallop theorem forbids objects with time-reversible (reciprocal) swimming strokes from moving. In the presence of inertia, this restriction is eased and reciprocally deforming bodies can swim. A number of recent works have investigated dimer models that swim reciprocally at intermediate Reynolds numbers${\textit Re} \approx 1$–1000. These show interesting results (e.g. switches of the swim direction as a function of inertia) but the results vary and seem to be case specific. Here, we introduce a general model and investigate the behaviour of an asymmetric spherical dimer of oscillating length for small-amplitude motion at intermediate${\textit {Re}}$. In our analysis we make the important distinction between particle and fluid inertia, both of which need to be considered separately. We asymptotically expand the Navier–Stokes equations in the small-amplitude limit to obtain a system of linear partial differential equations. Using a combination of numerical (finite element) and analytical (reciprocal theorem, method of reflections) methods we solve the system to obtain the dimer's swim speed and show that there are two mechanisms that give rise to motion: boundary conditions (an effective slip velocity) and Reynolds stresses. Each mechanism is driven by two classes of sphere–sphere interactions, between one sphere's motion and (1) the oscillating background flow induced by the other's motion, and (2) a geometric asymmetry induced by the other's presence. We can thus unify and explain behaviours observed in other works. Our results show how sensitive, counterintuitive and rich motility is in the parameter space of finite inertia of particles and fluid.

An algebra over the operad of posets and structural binomial identities

We study generating functions of strict and non-strict order polynomials of series–parallel posets, called order series. These order series are closely related to Ehrhart series and \(h^*\)-polynomials of the associated order polytopes. We explain how they can be understood as algebras over a certain operad of posets. Our main results are based on the fact that the order series of chains form a basis in the space of order series. This allows to reduce the search space of an algorithm that finds for a given power series f(x), if possible, a poset P such that f(x) is the generating function of the order polynomial of P. In terms of Ehrhart theory of order polytopes, the coordinates with respect to this basis describe the number of (internal) simplices in the canonical triangulation of the order polytope of P. Furthermore, we derive a new proof of the reciprocity theorem of Stanley. As an application, we find new identities for binomial coefficients and for finite partitions that allow for empty sets, and we describe properties of the negative hypergeometric distribution.

High-Selectivity FA-FA-Based Frequency Selective Surfaces Using Magnetoelectronic Dipole Antennas

A new method for frequency selective surface (FSS) with high selectivity, namely, filtenna–filtenna (FA-FA)-based technique, is proposed, which consists of two filtering antennas and impedance matching network. Based on the reciprocity theorem, the radiation null of filtenna can provide a stopband of the proposed FSS. In this work, it is validated by using a periodic array of back-to-back magnetoelectronic (ME)-dipole antennas with split-ring resonators (SRRs), and a slotted GND plane is used to connect the two ME-dipole antennas for impedance matching. Bandpass response of the proposed FSS is realized by the intrinsic high-pass characteristic of ME-dipole antennas and the bandstop characteristic of SRRs. Four transmission zeros (TZs) and three transmission poles (TPs) are generated, exhibiting a high-order filtering response. TZs and TPs are analyzed by equivalent circuit model and current distributions. The 3 dB fractional bandwidth ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {FBW}}_{\mathrm {3\,dB}}$ </tex-math></inline-formula> ) of 38.5% in the passband and the 20 dB fractional bandwidths <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$({\mathrm {FBW}}_{\mathrm {20\,dB}})$ </tex-math></inline-formula> of 8.7% in the lower rejection band and 13.8% in upper rejection band are obtained in this design. Finally, a prototype is fabricated and measured to further validate the proposed FA-FA-based technique. Simulated and measured results show that the proposed FA-FA-based FSS is with the distinct advantages, including wide passband, low profile, tunable TZs, and dual-polarized application, which verifies the proposed method of FSSs.

Application of the Reciprocity Theorem to Scattering of Surface Waves by a Surface Crack in Viscoelastic Material

In this paper, the scattered surface waves created by a surface crack in a homogeneous, isotropic, viscoelastic half space within a plane strain condition was studied. The amplitude of the scattered surface wave in the far field was determined by using the reciprocity theory and a virtual surface wave. It was shown that the amplitude of the scattered surface wave was related to the crack-opening displacement and the crack length. In the special case of low frequency and low viscosity, the tractions due to the incident surface wave applying on the crack surface can be regarded as uniform, and a finite element method (FEM) based on rubber material was performed to verify the theoretical results. It was shown that the numerical results were consistent with the theoretical solutions, which proves the reliability of the theoretical analysis. The reciprocity theorem avoids complex integral transformation and reveals the relationship between the scattered surface wave and the size of the surface crack, which is promising in the characterization of surface cracks.

Reciprocity and Representations for Wave Fields in 3D Inhomogeneous Parity-Time Symmetric Materials

Inspired by recent developments in wave propagation and scattering experiments with parity-time (PT) symmetric materials, we discuss reciprocity and representation theorems for 3D inhomogeneous PT-symmetric materials and indicate some applications. We start with a unified matrix-vector wave equation which accounts for acoustic, quantum-mechanical, electromagnetic, elastodynamic, poroelastodynamic, piezoelectric and seismoelectric waves. Based on the symmetry properties of the operator matrix in this equation, we derive unified reciprocity theorems for wave fields in 3D arbitrary inhomogeneous media and 3D inhomogeneous media with PT-symmetry. These theorems form the basis for deriving unified wave field representations and relations between reflection and transmission responses in such media. Among the potential applications are interferometric Green’s matrix retrieval and Marchenko-type Green’s matrix retrieval in PT-symmetric materials.

Transient Analysis of Transmission Lines With Mismatched Terminations Irradiated by Plane Waves Based on Reciprocity Theorem

The field-to-line coupling problem of the transmission line (TL) excited by external fields can be analyzed by adopting the electromagnetic (EM) reciprocity theorem. In this article, according to the frequency-domain expression of the test current along the line in the case of the TL is terminated by the mismatched loads, the time-domain formula, consisting of the original excitation, and multiple reflected waves caused by the loads, is derived. Furthermore, the analytical expression of the induced voltage is obtained based on the EM-reciprocity theorem. The correctness of the proposed method is verified by the numerical results generated by utilizing full-wave method.

Validating and optimizing mismatch tolerance of Doppler backscattering measurements with the beam model (invited).

We use the beam model of Doppler backscattering (DBS), which was previously derived from beam tracing and the reciprocity theorem, to shed light on mismatch attenuation. This attenuation of the backscattered signal occurs when the wavevector of the probe beam's electric field is not in the plane perpendicular to the magnetic field. Correcting for this effect is important for determining the amplitude of the actual density fluctuations. Previous preliminary comparisons between the model and Mega-Ampere Spherical Tokamak (MAST) plasmas were promising. In this work, we quantitatively account for this effect on DIII-D, a conventional tokamak. We compare the predicted and measured mismatch attenuation in various DIII-D, MAST, and MAST-U plasmas, showing that the beam model is applicable in a wide variety of situations. Finally, we performed a preliminary parameter sweep and found that the mismatch tolerance can be improved by optimizing the probe beam's width and curvature at launch. This is potentially a design consideration for new DBS systems.

Simple Derivation of Transfer Functions in Bistatic Scattering Model

We present a simple derivation of the increment of the scattering and impedance parameters among antennas caused by the scattering from a target in an inhomogeneous nonmagnetic reciprocal medium. Although these formulas have been published in the literature, we provide a straightforward and elegant derivation using the electromagnetic (EM) reciprocity theorem and the circuit theory. In this way, we provide a deeper physical insight into the formulas crucial for most microwave imaging (MWI) algorithms.

Elastodynamic response of an orthotropic layered elastic half-space to time-harmonic loading.

The elastodynamics of an orthotropic half-space coated by a thin orthotropic layer is theoretically investigated in this article. We newly propose explicit expressions of free Rayleigh waves in a layered half-space that are dependent on only one unknown constant representing amplitude. The main contribution is on deriving, in a simple manner, the theoretical predictions of far-field Rayleigh wave motion arising from time-harmonic loads using elastodynamic reciprocity theorems. These are the very first closed-form exact solutions found for the forced motion of Rayleigh waves in a layered half-space of orthotropic materials. To demonstrate the theoretical results, computation of Rayleigh wave motion in a jointed rock, including a layer of quartz-schist and a half-space of soil, is considered. We present the phase and group dispersion curves superimposed with the amplitude spectra that provide useful information on wave modes, frequencies, and displacement amplitudes. The inclusion of the amplitude spectra in the dispersion curves is a significant improvement over other dispersion curves currently available in the literature. The analytical predictions are compared with numerical results found by finite element analysis, and they show excellent agreement for the cases of a uniform distributed load and a varying distributed load both applied over a strip on the layer surface. The calculations obtained in the current study could generally be very useful for applications in seismology and materials characterization of coated structures.

On the input impedance of probe‐fed electromagnetic bandgap antennas based on lattice modes

The impedance of a coaxial probe, feeding an Electromagnetic Bandgap (EBG) structure conceived for radiation shaping, is studied from the theoretical and numerical viewpoints. The EBG medium is a square arrangement of dielectric cylinders placed in a parallel-plate waveguide, where a suitable lattice mode is excited. A semi-analytical model is developed and used to derive an approximate, closed-form expression of the probe resistance. The model is based on a modal expansion in Floquet harmonics, on which a current distribution is projected according to the Lorentz reciprocity theorem to derive the amplitude of lattice modes propagating right above the bandgap along lattice axes. The dependence of probe impedance on lattice parameters is then investigated with the numerical simulations of a finite-element method, which is also used to validate the developed model. A broad set of parametric analyses is presented, showing that the reactive part weakly depends on probe position, cylinder radius and permittivity, while the heights of probe and parallel-plate waveguide play a major role in determining the resonance condition. As to the probe resistance, it decreases with cylinder radius and permittivity and decreases with probe and waveguide heights. The derived analytical formula correctly reproduces such functional dependences and its calculation is immediate, revealing its usefulness in antenna design. Matching issues are heuristically and experimentally approached by examples, demonstrating that the proposed work can be effectively employed to improve the electrical performance of EBG antennas with an embedded source.

Flow rate-pressure drop relation for deformable channels via fluidic and elastic reciprocal theorems

Viscous flows through compliant conduits apply forces at the solid-liquid interface, leading to deformation of the cross-section, which affects the flow rate--pressure drop relation. Conventionally, calculating this relation requires solving the coupled elastohydrodynamic problem of flow and deformation. Instead, we employ reciprocal theorems for Stokes flow and linear elasticity to derive a closed-form expression for the flow rate--pressure drop relation in deformable channels, using a domain perturbation expansion that only requires the fluid flow solution and the elastic deformation due to the fluid stress distribution in an undeformed channel.

A Fast and Robust Semi-Analytical Approach for the Calculation of Coil Transient Eddy-Current Response Above Planar Specimens

The transient response of an induction coil above an infinite conducting plate is calculated in the time domain (TD) by combining a semi-analytical modal expansion approach with the Euler integration scheme. The derived state equation at a given time step is formally equivalent to the Helmholtz equation with an imaginary frequency linked to the sampling rate, which provides a significant computational advantage with respect to spectral methods based on Fourier/Laplace domain discretization. In addition, time stepping schemes prove to be more robust than the spectral approaches, where the optimal choice of the sample frequencies is an issue of major concern. The developed solution is applied for the calculation of the magnetic field in a driver–pickup coil inspection configuration involving pulsed currents, and the coil response is obtained using the TD version of the reciprocity theorem. The theoretical results are compared against reference numerical solutions demonstrating an excellent agreement.

Some results on thermopiezoelectricity of nonsimple materials

In this paper, we consider the linear theory for a model of a thermopiezoelectric nonsimple material as presentated in Passarella and Tibullo (2022), adopting the entropy production inequality proposed by Green and Laws. We establish reciprocity theorems and a variational principle for homogeneous and anisotropic thermopiezoelectric nonsimple materials with a center of symmetry. The proof of these theorems use the time convolution product and an alternative formulation of the field equations. Moreover, a uniqueness result is established without using the definiteness assumptions on internal energy.