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Publication# Electromagnetic waves under nonstationary conditions, the connection with parity-time symmetry and studies on resonant systems

Abstract

In this thesis, the electromagnetic wave propagation is studied in nonstationary-medium scenarios. The electromagnetic fields under material time-modulation are shown to conserve their momentum but not their energy. The mathematical foundations and analysis to treat wave propagation in time-Floquet media are given additionally to the related parametric amplification phenomena, which are mapped to the stability analysis of the corresponding hypergeometric equations. Assuming a time-variation of permittivity, permeability and conductivity the appropriate time-domain solutions are derived, based on an observation of the fields in the past. The formulation of a time-transitioning state matrix connects the unusual energy transitions of electromagnetic fields in time-varying media with the exceptional point theory, a theory strongly connected with parity-time symmetry. Consequently, the state-matrix approach of this thesis allows the analysis of the electromagnetic waves in terms of parity and time-reversal symmetries and signify parity-time symmetric wave-states without the presence of a spatially symmetric distribution of gain and loss, or any inhomogeneities and material periodicity. The parametric amplification phenomena of time-Floquet media and more precisely those that generate a Mathieu equation at the first momentum gap are theoretically studied and numerically compared with simulations using FDTD and connected with the parity-time scattering conventional characteristics. In the last part of this thesis, studies regarding resonant acoustic and electromagnetic systems are exhibited. The theoretical foundation to treat both acoustic and electromagnetic resonant phenomena is given based on the coupled mode theory and the appropriate Hilbert space. Two examples of interest are shown leveraging the time-dynamics of a temporal resonant system. The first example is related to the design of an artificial resonant acoustic lattice with the appropriate time-modulation leading to an effective zero index of refraction. The second example is related to resonant systems with temporal coupling and the possibility to induce nonreciprocal gain by leveraging the frequency conversion occurring in parametric systems. This thesis enriches the literature and the theoretical bases for dynamical wave systems and provides an insight on the broad capabilities of time-varying systems in electromagnetics, optics and acoustics. It may be used as a guidance to realize wave devices that amplify and actively filter wave signals for many future applications in lasing, sensing, signal amplifying, energy transferring and imaging.

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Collective spin excitations can propagate in magnetically ordered materials in the form of waves. These so-called spin waves (SWs) or magnons are promising for low-power beyond-CMOS information processing, which does not rely anymore on the lossy movement of electric charges. SWs in the few GHz frequency regime possess nanoscale wavelengths about five orders of magnitude smaller than electromagnetic waves of the same frequency. This property makes SWs ideally suited for application in microwave technology, essential for on-chip processing of wireless telecommunication signals. In this thesis, three crucial challenges relevant for the technological application of SWs are addressed:
First, to functionalize SWs and exploit their small wavelengths, it is necessary to control them at the nanoscale. Here, periodically nanostructured materials, denoted magnonic crystals, are promising, as they allow to tailor the band structure of SWs. We report on SWs propagating in a prototypical one-dimensional magnonic crystal consisting of dipolarly coupled magnetic nanostripes. The remanent magnetization of individual stripes was designed to be bistable along the long axis. By magnetizing an individual stripe in opposite direction to the others, we created a magnetic defect. We measured by means of all-electrical spin wave spectroscopy and Brillouin light scattering microscopy phase and amplitude of SWs trespassing the defect. We found that SW phases and amplitudes were modified at the nanoscale, and phase shifts could be tuned by an applied bias magnetic field. Using micromagnetic simulations, we identified specific bias fields for which phase shifts of Pi are achieved without suppressing SW amplitudes. This result is highly relevant for the implementation of logic gates based on interference of phase-controlled SWs. We further measured propagation of short-waved SWs in an antiferromagnetically ordered one-dimensional magnonic crystal, where every second stripe was magnetized in opposite direction. We found a band gap closing at the Brillouin zone boundary when no magnetic bias field was applied. Our observations are promising for reprogrammable microwave filters capable of adjusting stop- and passband.
Second, we address how long-waved electromagnetic waves can be coupled efficiently to nanoscale SWs. We demonstrate by space- and time-resolved scanning X-ray transmission measurements, that excited nanogratings allow to transfer their reciprocal lattice vector and multiple of it to an underlying magnetic thin film, in which nanoscale propagating SWs are launched. Additionally, we discovered a second method for short-waved SW generation based on magnetic microwave guides. This approach is easy to fabricate and relies on the adaption of the SW wavelength to a changing effective magnetic field. Efficient coupling of electromagnetic waves to nanoscale SWs promises an unprecedented miniaturization of microwave components.
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In this paper, we study the interactions of electromagnetic waves with a non-dispersive dynamic medium that is temporally dependent. Electromagnetic fields under material time-modulation conserve their momentum but not their energy. We assume a time-variation of the permittivity, permeability and conductivity and derive the appropriate time-domain solutions based on the causality state at a past observation time. We formulate a time-transitioning state matrix and connect the unusual energy transitions of electromagnetic fields in time-varying media with the exceptional point theory. This state-matrix approach allows us to analyze further the electromagnetic waves in terms of parity and time-reversal symmetries and signify parity-time symmetric wave-states without the presence of a spatially symmetric distribution of gain and loss, or any inhomogeneities and material periodicity. This paper provides a useful arsenal to study electromagnetic wave phenomena under time-varying media and points out novel physical insights connecting the resulting energy transitions and electromagnetic modes with exceptional point physics and operator symmetries.

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We present an exact mathematical framework for electromagnetic wave propagation in periodically time-modulated media, in which the permittivity is homogenous and modulated in a step-varying fashion. By using Hill’s equation theory, we show that this problem has analytical solutions. We connect the dispersion relation, which exhibits k− gaps, with the Hill stability analysis, providing an alternative mathematical description for wave propagation in temporal crystals. Our analysis, which is exact and transposable to other kinds of waves or modulation schemes, provides general useful physical and mathematical insights, complementing the use of numerical techniques such as finite differences in the time domain, or harmonic balance schemes, with a more transparent and practical design tool. The present analytical transient mathematical analysis, in contrast with the existing frequency-domain numerical approaches, can exhibit the parametric properties of electromagnetic waves inside a time periodic medium. For this reason, it can be a useful tool for the design of active microwave and optical devices, which employ time periodic wave medium modulation to filter or parametrically amplify wave signals.

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