Periodic Media Research Articles

Bragg resonances in left-handed bigyrotropic media based on periodic ferromagnetic semiconductor

In this work, a theoretical model for a periodic bigyrotropic left-handed medium, which is a ferromagnetic semiconductor, is developed. The presence of periodicity in such a structure leads to the formation of Bragg band gaps in the spectrum of propagating waves, both in the microwave and terahertz ranges. Band gaps in the microwave range are formed in the frequency region with double negative effective parameters of the medium. The band gaps density decreases with increasing frequency in the microwave range and increases in the terahertz range.

Pulsating waves in a multidimensional reaction-diffusion system of epidemic type

In this work we investigate the existence of front propagation for a two-component reaction -diffusion system of epidemic type posed in a multi-dimensional periodic medium. This system is a spatially heterogeneous version of the well-known Kermack–McKendrick epidemic model with Fickian diffusion. We derive sufficient conditions for the existence of pulsating travelling waves propagating in an arbitrarily given unit direction. Specifically, we prove that the set of admissible wave speeds contains a semi-infinite interval. Then, for each direction of propagation and each admissible speed, there exists a pulsating travelling wave solution of the system which is globally bounded.

Reflection and transmission characteristic waves on the periodic rough interface of fluid-porous medium

Abstract Nondestructive testing (NDT) is a major method in practical applications such as material exploration and safety assessment. The interaction of ultrasonic waves with roughness material is crucial in NDT, thus the related ultrasonic diffraction theory at the rough interface is widely investigated and analyzed, especially for the periodic elastic medium interface. However, the actual surface is usually porous due to the long-term moisture infiltration. In this paper, the reflection and transmission characteristics of ultrasonic waves on the fluid-porous medium periodic rough surfaces are studied in details.

Effects of dynamic diffraction in coherent X-ray radiation from a beam of relativistic electrons in a periodic layered medium with three layers in a period

The possibilities of manifestation of dynamic diffraction effects in coherent X-ray radiation (CXR) of relativistic electrons in a periodic layered medium (PLM) with three different layers in a period are being investigated. A dynamic theory of CXR of relativistic electrons has been developed for the case of such a target. The corresponding formulas for the spectral-angular characteristics of parametric X-ray radiation (PXR) and diffracted transition radiation (DTR) are obtained and studied. The influence of asymmetric diffraction on the spectral-angular densities of PXR and DTR is shown. A striking effect of anomalous photoabsorption in PXR in a layered target, similar to the Bormann effect in a single crystal, is predicted.

Dynamic homogenization of inhomogeneous elastic media based on energy equivalence

This paper presents a new method of dynamic homogenization of periodic elastic media under harmonic antiplane deformation based on energy equivalence. The dynamic homogenization is based on two main steps, (1) determining the dispersion relation and the detailed local response in a representative volume element (RVE) by analyzing the propagation of Bloch waves and (2) using energy equivalence between the periodic and effective media to determine the effective properties. The method is first applied to a one-dimensional periodic medium, from which the analytical solutions of the effective material properties are obtained. The results are compared with that from the original periodic medium and an excellent agreement is observed. The dynamic homogenization method is then applied to general two-dimensional periodic media to determine the effective properties and to predict wave fields under typical loading conditions. Illustrative examples are presented and compared with the results from the multiple scattering model. The method has also been applied to multiscale modeling of complicated inhomogeneous media containing multiple groups of periodic inhomogeneities. By treating each group as a homogeneous material with effective properties determined by the current homogenization method, the wave field is obtained using the boundary element method. The resulting wave fields from the current method show an excellent agreement with that from the multiple scattering model with matching local response details.

Coherent X-ray radiation excited by a relativistic electron in a periodic layered medium

The work is devoted to the development of the theory of coherent X-ray radiation near the direction of Bragg scattering of a beam of relativistic electrons crossing a target with a periodic layered structure with three different amorphous layers per period. The coherent X-ray radiation is considered within the two-wave approximation of the dynamic theory of diffraction as a sum of parametric X-ray radiation and diffracted transition radiation excited in the target by a relativistic electron. Expressions describing the spectral-angular and angular densities of PXR and DTR have been obtained and studied.

Towards Sustainable Local Development: Comparative Analysis of Periodic Plans in Nepalese Municipalities

The Local Government Operation Act of 2017 mandates that rural municipalities and municipalities in Nepal develop periodic, annual, and strategic medium- and long-term plans for local-level development. Guided by directives from the National Planning Commission, these plans outline visions, goals, and strategies across social, economic, and physical domains, crucial for sustainable development amidst political stability. These plans integrate governance, environmental considerations, and social inclusion by emphasizing alignment with sustainable development goals and national policies. As primary governance agents, local governments are tasked with formulating these plans to enhance prosperity and socio-economic development within constitutional frameworks. This study examines the objectives, formulation processes, and comparative analysis of periodic plans across Jiri Municipality, Mahankal Rural Municipality, Gauri Shankar Rural Municipality, and Mahalaxmi Municipality in Nepal. It compares their long-term visions, SWOT analyses, sector-wise budgetary allocations, strengths, weaknesses, opportunities, and threats. The findings highlight distinct priorities and challenges each municipality faces, offering insights into tailored strategies for balanced and sustainable development.

Rapid Establishment of a Regeneration System for Astragalus membranaceus Mongholicus

Background: Astragalus membranaceus Mongholicus (Astragalus) is one of the most widely used medicinal plants in the world. It is a traditional Chinese medicine plant that has been used to treat various human diseases and conditions, such as inflammation, gastrointestinal bacterial infections and malignant tumors. With the expansion of artificial planting areas, the demand for high-quality Astragalus seedlings is also increasing. The traditional seedling-raising methods of direct seeding and induced callus have problems such as a long growth period, unstable genetic traits, strict sampling and rooting difficulties. Therefore, it is very important to multiply the excellent germplasm by optimizing the tissue culture regeneration system. Methods: In this study, Astragalus field cultivated plants, which are easily available, were used as materials and the method of tissue culture cuttings was used to select the direct inoculation method of stem segments as explant materials to establish a plant regeneration system of Astragalus and the best sampling period, sampling site and optimal medium for regeneration of field cultivated plants were screened. Result: The best sampling period for plant regeneration of Astragalus in the field was about 40 d after the emergence of the 3-year-old Astragalus and the rooting rate of the regenerated plants was 100.00%. The best sampling site is the middle section of the stem and the regeneration system using the middle section of the stem as the explant is better than using the top and bottom of the stem as the explant and the rooting rate of the regenerated plants is 100.00%; the best medium for the regenerated plants is N22: 1/2MS+1 mg/L NAA and T28: 1/2MS+1 mg/L ABT, the rooting rates were 76.67% and 96.67%, respectively. The regenerated plants established with the stems of the field cultivated plants of Astragalus as explants were all able to induce seedlings after 1 week and the regenerated seedlings start to take root after 3 weeks of growth and can be transplanted and propagated after 6 weeks of culture. The regeneration system established in this study by direct cuttings from cultivated Astragalus stems is more rapid and efficient, breaking through the difficulties of limited access to material and long growth period of regenerated seedlings in the traditional regeneration system and providing technical support for the conservation and utilization of wild resources of Astragalus and rapid expansion of new varieties and germplasm conservation.

Propagation and dispersion of Bloch waves in periodic media with soft inclusions

We investigate the behavior of waves in a periodic medium containing small soft inclusions or cavities of arbitrary shape, such that the homogeneous Dirichlet conditions are satisfied at the boundary. The leading terms of Bloch waves, their dispersion relations, and low frequency cutoff are rigorously derived. Our approach reveals the existence of exceptional wave vectors for which Bloch waves are comprised of clusters of perturbed plane waves that propagate in different directions. We demonstrate that for these exceptional wave vectors, no Bloch waves propagate in any one specific direction.

Brillouin Klein space and half-turn space in three-dimensional acoustic crystals

The Bloch band theory and Brillouin zone (BZ) that characterize wave-like behaviors in periodic mediums are two cornerstones of contemporary physics, ranging from condensed matter to topological physics. Recent theoretical breakthrough revealed that, under the projective symmetry algebra enforced by artificial gauge fields, the usual two-dimensional (2D) BZ (orientable Brillouin two-torus) can be fundamentally modified to a non-orientable Brillouin Klein bottle with radically distinct manifold topology. However, the physical consequence of artificial gauge fields on the more general three-dimensional (3D) BZ (orientable Brillouin three-torus) was so far missing. Here, we theoretically discovered and experimentally observed that the fundamental domain and topology of the usual 3D BZ can be reduced to a non-orientable Brillouin Klein space or an orientable Brillouin half-turn space in a 3D acoustic crystal with artificial gauge fields. We experimentally identify peculiar 3D momentum-space non-symmorphic screw rotation and glide reflection symmetries in the measured band structures. Moreover, we experimentally demonstrate a novel stacked weak Klein bottle insulator featuring a nonzero Z2 topological invariant and self-collimated topological surface states at two opposite surfaces related by a nonlocal twist, radically distinct from all previous 3D topological insulators. Our discovery not only fundamentally modifies the fundamental domain and topology of 3D BZ, but also opens the door towards a wealth of previously overlooked momentum-space multidimensional manifold topologies and novel gauge-symmetry-enriched topological physics and robust acoustic wave manipulations beyond the existing paradigms.

An optimal control problem for diffusion–precipitation model

We present an optimal control problem associated to a chemical transportation phenomena in a periodic porous medium. Posing controls on the porous part of the medium (distributed control), we set up a convex minimization problem. The main objective of this article is to characterize an arbitrary control to be an optimal control. We establish a relation between the optimal control and the corresponding adjoint state. At first, we analyse the microscopic description of the controlled system, then we homogenised the system by rigorous two-scale convergence method and periodic unfolding method.

Willis coupling in one-dimensional poroelastic laminates

We employ the Baker–Campbell–Hausdorff formula to derive closed-form expressions for the effective properties, including emergent Willis coupling, of a one-dimensional heterogeneous poroelastic medium consisting of a periodically repeating two-layer unit-cell. In contrast to the elastic and fluidic analogs, the Willis coupling of this periodic poroelastic medium does not vanish in the low-frequency limit. However, the effective wavenumber and impedance of this asymmetric lamellar material demonstrate symmetric reflection and absorption behavior that is indicative of symmetric structures in the low-frequency limit due to the effect of Darcy’s law on the dynamic effective density, which is inversely proportional to frequency. These closed-form expressions are validated against results obtained by direct numerical evaluation. The scattering coefficients, particularly the two reflection coefficients for incidence from either side of a finite length asymmetric structure, are different at non-zero frequencies but still in the metamaterial limit and are correct when the Willis coupling is included. The results show that asymmetry in poroelastic layers results in direction-dependent absorption coefficients, one of which could be optimized based on layer properties and asymmetry factors. As a consequence, the frequency range of validity of these scattering coefficients is wider when the Willis coupling matrix is accounted for than in its absence. This work paves the way for better control of elastic and acoustic waves in multiphase materials by considering poroelastic behavior.

The time‐harmonic electromagnetic wave scattering by a biperiodic elastic body

This paper concentrates on an interaction scattering problem between the time‐harmonic electromagnetic waves and an unbounded periodic elastic medium. The uniqueness results of the interaction problem are established for small frequencies or all frequencies except a discrete set in both the absorbing and nonabsorbing medium, and then the existence of solutions is derived by the classical Fredholm alternative. The perfectly matched layer (PML) method is proposed to truncate the unbounded scattering domain to a bounded computational domain. We prove the well‐posedness of the solution for the truncated PML problem, where a homogeneous boundary condition is imposed on the outer boundary of the PML. The exponential convergence of the PML method is established in terms of the thickness and parameters of the PML. The proof is based on the PML extension and the exponential decay properties of the modified fundamental solution.

Bragg scattering of stochastic beams in PT-symmetric photonic lattices.

The propagation of a Gaussian-Schell beam through a PT-symmetric optical lattice, whose index of refraction is represented by a sinusoidal type of function, is theoretically investigated. Within the framework of standard coherence theory, one is able to access and elucidate unexpected consequences of the interplay between the spatial coherence properties of the beam and the non-Hermitian nature of the photonic lattice. We describe how one may use a non-Hermitian periodic medium to enhance the spatial coherence properties of a partially coherent beam.

Dispersion diagram reconstruction of effectively bianisotropic composite periodic media

A dispersion diagram reconstruction technique is proposed for arbitrarily bianisotropic composite periodic media, which utilizes a previously introduced parameter retrieval technique based on eigenvalue analysis and field averaging. We initially retrieve the effective electromagnetic parameters of a composite periodic medium consisting of Edge-Coupled Split-Ring Resonators (EC-SRRs) via this homogenization technique using alternative integration approaches for the averaging of the field components. Subsequently, we derive the analytical framework for the wave propagation in a homogeneous medium of arbitrary bianisotropy and extract the appropriate equations which solve for the complex propagation constants. We then involve the retrieved effective parameters in these equations and reconstruct the dispersion diagrams for all three orthogonal directions, thereby spanning the whole irreducible Brillouin zone. An excellent agreement is found between the original dispersion diagrams and the reconstructed ones; a result which further validates the utilized parameter retrieval technique. The reconstruction technique moreover allows one to interpret the slope differences observed in the dispersion diagrams for in-plane and normal incidence modes of the examined composite medium. It may also be used as a tool for the confirmation of the accuracy of other formerly proposed homogenization techniques existing in the literature.

Reflective properties of active layered media at second optical harmonic generation

We studied the reflective properties of an active periodic medium with quadratic nonlinearity using numerical modeling based on the description of the generation of the second optical harmonic in the quasi-optical approximation. The model under consideration appears to be a system of four coupled equations for slowly varying amplitudes of forward and backward waves. Based on numerical simulation, the influence of the asymmetric Bragg coupling of counterpropagating waves at the fundamental frequency on the absorbing/generating and reflecting properties of the active medium is analyzed.

Computation of the dynamic scalar response of large two-dimensional periodic and symmetric structures by the wave finite element method

In the past, the study of periodic media mainly focused on one-dimensional periodic structures (meaning periodic along one direction), on the one hand to determine the dispersion curves linking the frequencies to the wavenumbers and on the other hand to obtain the response of a structure to an external excitation, both for bounded or unbounded structures. In the latter case, effective approaches have been obtained, based on methods such as the Wave Finite Element (WFE). Two-dimensional periodic media are more complex to analyse but dispersion curves can be obtained rather easily as in the one-dimensional case. Obtaining the steady state response of two-dimensional periodic structures to time-harmonic excitations is much more difficult than for one-dimensional media and the results mainly concern infinite media. This work is about this last case of the steady state response of finite two-dimensional periodic structures to time-harmonic excitations by limiting oneself to structures described by a scalar variable (acoustic, thermal, membrane behaviour) and having symmetries compared to two orthogonal planes parallel to the edges of a substructure. Using the WFE for a rectangular substructure and imposing the wavenumber in one direction, we can numerically calculate the wavenumbers and mode shapes associated with propagation in the perpendicular direction. By building solutions with null forces on parallel boundaries, we can decouple the waves in the two directions parallel to the sides of the rectangle. The solution of each of these two problems is obtained by a fast Fourier transformation giving the amplitudes associated with the waves. By summing the contributions of all these waves we obtain the global solution for a two-dimensional periodic medium with a large number of substructures and a low computing time. Examples are given for the case of a two-dimensional membrane with many substructures and different types of heterogeneities.

Multiscale Topology Optimization of modulated fluid microchannels based on asymptotic homogenization

Dehomogenization techniques are becoming increasingly popular for enhancing lattice designs of compliant mechanical systems with ultra-large resolutions. Their effectiveness hinges on computing a deformed periodic grid that enable to reconstruct fine-scale designs with modulated and oriented patterns. In this paper, we propose an approach for extending dehomogenization methods to laminar fluid systems. We initiate our methodology by asymptotically deriving Darcy’s law on a periodically porous medium deformed by a diffeomorphism. Unlike the mechanical context, we reveal that the homogenized permeability matrix depends not solely on local the orientation but also on the local dilation of the deformed periodic medium. This distinction presents one of the several challenges to be tackled when adapting dehomogenization-based topology optimization techniques to porous media. To accommodate existing methodologies, we formulate a simplified “poor man’s” homogenized model, which streamlines various aspects, yet still leans on periodic cell problems to estimate the spatially varying permeability matrix. Specifically, we overlook boundary layer effects, we presume periodic grid deformations, and we neglect local dilation, solely considering the relationship with local cell orientations. Subsequently, we present a numerical approach for designing a system that redistributes an input flow across numerous regularly spaced outlets at an output interface. Leveraging the homogenized model, we deduce optimized geometric arrangements of local channel spacing parameters and orientations. We then use established methods to reconstruct grid deformations and fine-scale designs. The fidelity of these reconstructions is then validated through fine-scale simulations. Our observations indicate that while the proposed designs yield satisfactory performance when subjected to the full-scale model, discernible deviations from the homogenized model persist, appealing to future improvements.

Modal analysis of waveguide for the study of frequency bandgaps of a bounded periodic medium

A novel method based on waveguide modal analysis is presented to evaluate the effectiveness in terms of frequency bandgaps of bounded periodic metamaterials. The effect of the size of the bounded medium, i.e., the number of periodic unit cells, on the frequency bandgaps predicted by Floquet–Bloch theory in the infinite case is investigated and quantified. The waveguides considered in this work have a bounded cross-section and an infinite longitudinal axis simulated by Perfectly Matched Layers. Its useful area is made of a bounded 1D or 2D-periodic matrix-inclusion metamaterial. In the 2D-periodic case, an approach combining the waveguide modal analysis with the Floquet–Bloch transform is further proposed. Applying the proposed method, the SH-waves propagating in a waveguide are studied by using the finite element method. The discrete spectrum of the waveguide and the associated attenuated or trapped wave modes are calculated and analyzed, which provides, within a semi-analytic framework, an exact characterization regarding the attenuation coefficient and the frequency bandgaps of the studied bounded periodic medium. More particularly, effectiveness indicators are defined to compare the width and the position of frequency bandgaps between the bounded and infinite cases as a function of the cell number. Last but not least, the proposed method is also successfully applied to bounded periodic media with local resonators to characterize their filtering and attenuation effectiveness, which shows its interest in such a case of great practical interest.

Treating disorder in introductory solid state physics

Introductory textbooks in solid state physics present solvable models for illustrating the occurrence of allowed bands and forbidden gaps in the energy spectrum of Bloch electrons. However, the quantum mechanical description of electrons in non-periodic solids, such as amorphous materials, is beyond the scope of introductory courses because of its intrinsic complexity. The tight-binding approximation can account for such a scenario by letting the atomic levels vary at random from lattice site to site. We theoretically tackle the study of the average properties of the energy spectrum by introducing a transfer matrix method that allows us to obtain closed expressions for the so-called coherent potential. The coherent potential is energy-dependent and constant in space. It replaces the actual atomic random potential, thus generating a periodic effective medium with the same average properties as the non-periodic solid. We demonstrate that the average density of states can be calculated within this framework without relying on heavy mathematical machinery. Thus, our approach is suitable for introductory courses in solid state physics and materials science.