We report on the observation of Wannier-Stark ladders in the bending vibrations of elastic beams. By introducing a gradient in the length distribution of N weakly coupled beams, the elastic equivalent of the Wannier-Stark ladder is obtained in a system governed by two coupled second-degree differential equations, instead of the common wave equation, and whose oscillations are also dispersive and, above a certain critical frequency, occur with two wavelengths. We have measured for the first time, not only the spectrum of the ladders, but the wave amplitudes in this type of system, which are not directly accessible in quantum-mechanical systems, and we have found that the wavelengths are spatially localized and in good agreement with the theoretically predicted amplitudes. Due to the combination of differential equations and the boundary conditions imposed on the system, the Wannier-Stark ladder phenomenon occurs from the third band of states onwards, unlike other classical systems.

In this work, we obtain closed expressions for the transfer matrix and the transmittance of electromagnetic waves propagating in finite one-dimensional anisotropic periodic stratified media with an arbitrary number of cells. By invoking the Cayley–Hamilton theorem on the transfer matrix for the electromagnetic field in a periodic stratified media formed byNcells, we obtain a fourth-order recursive relation for the matrix coefficients that defines the so-called Tetranacci polynomials (TPs). In the symmetric case, corresponding to a unit-cell transfer matrix with a characteristic polynomial where the coefficients of the linear and cubic terms are equal, closed expressions for the solutions to the recursive relation, known as symmetric TPs, have recently been derived, allowing us to write the transfer matrix and transmittance in a closed form. We show as sufficient conditions that the 4×4 differential propagation matrix of each layer in the binary unit cell, Δ , a) has eigenvalues of the form ±p1 , ±p2 , with p1≠p2 , and b) its off-diagonal 2×2 block matrices possess the same symmetric structure in both layers. Otherwise, the recursive relations are still solvable for any 4×4 matrix and provide an algorithm to compute theNth power of the transfer matrix without carrying out explicitly the matrix multiplication ofNmatrices. We obtain analytical expressions for the dispersion relation and transmittance, in closed form, for two finite periodic systems: the first one consists of two birefringent uniaxial media with their optical axis perpendicular to thez-axis, and the second consists of two isotropic media subject to an external magnetic field oriented along thez-axis and exhibiting the Faraday effect. Our formalism applies also to lossy media, magnetic anisotropy or optical activity.

In this work, the Poincaré map numerical method was successfully developed to solve the fourth-order differential equation that describes the flexural vibrations of a beam, within the Timoshenko beam theory. The Euler-Bernoulli continuity conditions were considered, which are valid for frequencies smaller than the critical frequency. As an example, this method was used to design a complex elastic structure, characterized by a flexural frequency spectrum with a broad band gap. Such structure consists of two coupled phononic crystals, which were designed with filling factor values in such a way that in their bending frequency spectra, an allowed band of the first part, overlaps with a band gap of the second one and vice versa. The resulting composed system has a much wider effective gap than its original components, between 4 and 10.5 kHz. This system works as an elastic bending wave filter. Finally, these three structured elastic systems were constructed, and characterized by the acoustic resonance spectroscopy technique. The natural flexural frequencies as well as the corresponding wave amplitudes of each structured beam were measured. The experimental measurements show excellent agreement with the numerical simulation.

We present a simple method for customizing the optical characteristics of gold-core, silver-shell (Au@Ag) nanoparticles through controlled morphosynthesis via a seed-mediated chemical reduction approach. By systematically adjusting the concentration of cetyltrimethylammonium chloride (CTAC), we obtained precise control over both the thickness of the Ag shell and the particle shape, transitioning from spherical nanoparticles to distinctly defined nanocubes. Bright field and high-angle annular dark-field scanning transmission electron microscopy (BF-STEM and HAADF-STEM), and energy-dispersive X-ray spectroscopy (EDS) were employed to validate the structural and compositional changes. To link morphology with optical behavior, we utilized the Mie and Maxwell–Garnett theoretical models to simulate the dielectric response of the core–shell nanostructures, showing trends that align with experimental UV-visible absorption spectra. This research presents an easy and adjustable method for modifying the plasmonic properties of Ag@Au nanoparticles by varying their shape and shell, offering opportunities for advanced applications in sensing, photonics, and nanophotonics.