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1. BLOCH WAVES IN HIGH CONTRAST ELECTROMAGNETIC CRYSTALS

   Lipton, Robert; Viator, Robert; Jimenez Bolanos, Silvia; Adli, Abiti (October 2021, ArXivorg)

   Analytic representation formulas and power series are developed describing the band structure inside non-magnetic periodic photonic three-dimensional crystals made from high dielectric contrast inclusions. Central to this approach is the identification and utilization of a resonance spectrum for quasiperiodic source-free modes. These modes are used to represent solution operators associated with electromagnetic and acoustic waves inside periodic high contrast media. A convergent power series for the Bloch wave spectrum is recovered from the representation formulas. Explicit conditions on the contrast are found that provide lower bounds on the convergence radius. These conditions are sufficient for the separation of spectral branches of the dispersion relation for any fixed quasi-momentum. 

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2. Bloch Spectra for High Contrast Elastic Media

   Lipton, Robert; Perera, Ruchira (October 2021, ArXivorg)

   Analytic representation formulas and power series are developed to describe the band struc- ture inside periodic elastic crystals made from high contrast inclusions. We use source free modes associated with structural spectra to represent the solution operator of the Lam ́e system inside phononic crystals. Convergent power series for the Bloch wave spectrum are obtained using the representation formulas. An explicit bound on the convergence radius is given through the structural spectra of the inclusion array and the Dirichlet spectra of the inclusions. Suffi- cient conditions for the separation of spectral branches of the dispersion relation for any fixed quasi-momentum are identified. A condition is found that is sufficient for the emergence of band gaps. 

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3. Waves Generated by Electron Beam in a Crater-Shaped Flux Rope

   https://doi.org/10.3389/fphy.2021.734437  

   Dokgo, Kyunghwan; Hwang, Kyoung-Joo; Burch, James L.; Yoon, Peter H. (September 2021, Frontiers in Physics)

   Understanding the nature and characteristics of high-frequency waves inside a flux rope may be important as the wave-particle interaction is important for charged-particle energization and the ensuing dissipation process. We analyze waves generated by an electron beam in a crater-shaped magnetic flux rope observed by MMS spacecraft on the dawnside tailward magnetopause. In this MMS observation, a depression of magnetic field, or a crater, of ∼100 km is located at the center of the magnetic flux rope of ∼650 km. There exist parallel and perpendicular electrostatic wave modes inside the depression of the magnetic field at the center of the flux rope, and they are distinguished by their locations and frequencies. The parallel mode exists at the center of the magnetic depression and its power spectrum peaks below F ce (electron cyclotron frequency). In contrast, the perpendicular mode exists in the outer region associated with the magnetic depression, and its power spectrum peaks near F ce . The linear analysis of kinetic instability using a generalized dispersion solver shows that the parallel mode can be generated by the electron beam of 5,000 km/s. They can thermalize electrons ≲100 eV effectively. However, the generation mechanism of the perpendicular mode is not clear yet, which requires further study. 

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4. General Conditions to Realize Exceptional Points of Degeneracy in Two Uniform Coupled Transmission Lines

   https://doi.org/10.1109/TMTT.2020.2999498  

   Mealy, Tarek; Capolino, Filippo (June 2020, IEEE Transactions on Microwave Theory and Techniques)

   We present the general conditions to realize a fourth-order exceptional point of degeneracy (EPD) in two uniform (i.e., invariant along z) lossless and gainless coupled transmission lines (CTLs), namely, a degenerate band edge (DBE). Until now the DBE has been shown only in periodic structures. In contrast, the CTLs considered here are uniform and subdivided into four cases where the two TLs support combinations of forward propagation, backward propagation, and evanescent modes (when neglecting the mutual coupling). We demonstrate, for the first time, that a DBE is supported in uniform CTLs when there is proper coupling between: 1) propagating modes and evanescent modes, 2) forward and backward propagating modes, or 3) four evanescent modes (two in each direction). We also show that the loaded quality factor of uniform CTLs exhibiting a fourth-order EPD at k=0 is robust to series losses due to the fact that the degenerate modes do not advance in phase. We also provide a microstrip possible implementation of a uniform CTL exhibiting a DBE using periodic series capacitors with very subwavelength unit-cell length. Finally, we show an experimental verification of the existence DBE for a microstrip implementation of a CTL supporting coupled propagating and evanescent modes. 

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5. Asymptotic and Spectral Analysis of a Model of the Piezoelectric Energy Harvester with the Timoshenko Beam as a Substructure

   https://doi.org/10.3390/app8091434  

   Shubov, Marianna (September 2018, Applied Sciences)

   Mathematical analysis of the well known model of a piezoelectric energy harvester is presented. The harvester is designed as a cantilever Timoshenko beam with piezoelectric layers attached to its top and bottom faces. Thin, perfectly conductive electrodes are covering the top and bottom faces of the piezoelectric layers. These electrodes are connected to a resistive load. The model is governed by a system of three partial differential equations. The first two of them are the equations of the Timoshenko beam model and the third one represents Kirchhoff’s law for the electric circuit. All equations are coupled due to the piezoelectric effect. We represent the system as a single operator evolution equation in the Hilbert state space of the system. The dynamics generator of this evolution equation is a non-selfadjoint matrix differential operator with compact resolvent. The paper has two main results. Both results are explicit asymptotic formulas for eigenvalues of this operator, i.e., the modal analysis for the electrically loaded system is performed. The first set of the asymptotic formulas has remainder terms of the order O ( 1 n ) , where n is the number of an eigenvalue. These formulas are derived for the model with variable physical parameters. The second set of the asymptotic formulas has remainder terms of the order O ( 1 n 2 ) , and is derived for a less general model with constant parameters. This second set of formulas contains extra term depending on the electromechanical parameters of the model. It is shown that the spectrum asymptotically splits into two disjoint subsets, which we call the α -branch eigenvalues and the θ -branch eigenvalues. These eigenvalues being multiplied by “i” produce the set of the vibrational modes of the system. The α -branch vibrational modes are asymptotically located on certain vertical line in the left half of the complex plane and the θ -branch is asymptotically close to the imaginary axis. By having such spectral and asymptotic results, one can derive the asymptotic representation for the mode shapes and for voltage output. Asymptotics of vibrational modes and mode shapes is instrumental in the analysis of control problems for the harvester. 

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