arXiv subjects

Alexander Figotin

Publications and source records attributed to Alexander Figotin.

At least 19 recordsLinked to original sources

arXiv2026

Floquet Theory of the LC Circuit with Modulated Capacitance

Parametric resonance -- periodic variation of a system parameter driving exponential growth of oscillations -- is among the most fundamental instabilities in physics and engineering. The nondissipative LC circuit with harmonically varying capacitance is one of its simplest realizations: the modulation renders the circuit equation a Hill equation with either bounded or exponentially growing solutions. Identifying the governing equation as a special case of Ince's four-parameter Hill equation yields two main results. First, a sharp structural theorem: instability occurs only at the odd sub-harmonics of the natural frequency, while every even resonance is exactly stable at all modulation amplitudes. This selectivity, invisible to the Mathieu approximation, follows from Krein's collision theory: at odd resonances the colliding Floquet multipliers carry opposite Krein signatures, opening a tongue; at even resonances the signatures agree and the tongue collapses. Second, closed-form formulas for the widths and boundary curves of all surviving tongues, derived by a continued-fraction and Magnus--Winkler method, confirmed against Cambi's 1950 numerics and recovered via the Yakubovich--Starzhinskii series. The continued fraction also gives the Floquet exponent as an exact power series in the modulation amplitude with rational coefficients, and finite-product formulas for all Fourier coefficients of the periodic Floquet factor. The tongue boundaries consist entirely of exceptional points of degeneracy (EPD) of the monodromy matrix, enabling hypersensitive capacitance sensing: at an EPD a small perturbation splits the coincident frequencies by the square root of its size, diverging relative to linear sensing as it shrinks. A closed-form splitting formula is derived, and a work-point strategy shifting slightly into the stable zone keeps the scheme robust while preserving square-root sensitivity.

math-ph↗

arXiv2026

Factorized dispersion relations for two coupled systems

We establish that the dispersion relations of any physical system composed of two coupled subsystems, governed by a space-time homogeneous Lagrangian, admit a factorized form G_{1}G_{2}=\gamma G_{\mathrm{c}}, where G_{1} and G_{2} are the subsystem dispersion functions, G_{\mathrm{c}} is the coupling function, and \gamma is the coupling parameter. The result follows from a determinant expansion theorem applied to the block structure of the coupled system matrix, and is illustrated through three examples: the traveling wave tube, vibrations of an airplane wing, and the Mindlin-Reissner plate theory. For the Mindlin-Reissner example we carry out a complete asymptotic analysis of the coupled dispersion branches, establishing that the factorized form provides a precise quantitative measure of mode hybridization: all four branches carry the imprint of both subsystem factors for any nonzero coupling, while asymptotically recovering the identity of pure uncoupled modes at large frequencies and wavenumbers. We further analyze the universal local geometry of the coupled dispersion branches near their intersection - the cross-point model - showing it is generically hyperbolic, and present a mechanical analog in which the wavenumber is replaced by a scalar parameter, exhibiting the same factorized structure and avoided crossin

math-ph↗

arXiv2025

Super-exponential Amplification of Wavepacket Propagation in Traveling Wave Tubes

We analyze wavepacket propagation in traveling wave tubes (TWTs) analytically and numerically. TWT design in essence comprises a pencil-like electron beam in vacuum interacting with an electromagnetic wave guided by a slow-wave structure (SWS). In our study, the electron beam is represented by a one-dimensional electron flow and the SWS is represented by an equivalent transmission line model. The analytical considerations are based on the Lagrangian field theory for TWTs. Mathematical analysis of wavepacket propagation in one-dimensional space is based on the relevant Euler-Lagrange equations which are second-order differential equations in both time and space. Wavepacket propagation analysis is not simple and we develop a numerically efficient algorithm to perform the analysis efficiently. In particular, when the initial pulse has a Gaussian shape at the input port, it acquires non-Gaussian features as it propagates through the TWT. These features include: (i) super-exponential (faster than exponential) amplification, (ii) shift of the pulse frequency spectrum toward higher frequencies, and (iii) change in the shape of the pulse that becomes particularly pronounced when the pulse frequency band contains a transitional point from stability to instability.

physics.plasm-ph↗

arXiv2024

Small-Signal Model for Inhomogeneous Helix Traveling-Wave Tubes using Transfer Matrices

Abstract We introduce a practical method for modeling the small-signal behavior of frequency-dispersive and inhomogeneous helix-type traveling-wave tube (TWT) amplifiers based on a generalization of the one-dimensional Pierce model. Our model is applicable to both single-stage and multi-stage TWTs. Like the Pierce model, we assume that electrons flow linearly in one direction, parallel and in proximity to a slow-wave structure (SWS) which guides a single dominant electromagnetic mode. Realistic helix TWTs are modeled with position-dependent and frequency-dependent SWS characteristics, such as loss, phase velocity, plasma frequency reduction factor, interaction impedance, and the coupling factor that relates the SWS modal characteristic impedance to the interaction impedance. For the multi-stage helix TWT, we provide a simple lumped element circuit model for combining the stages separated by a sever, or gap, which attenuates the guided circuit mode while allowing the space-charge wave on the beam to pass freely to the next stage. The dispersive SWS characteristics are accounted for using full-wave eigenmode simulations for a realistic helix SWS supported by dielectric rods in a metal barrel, all of which contribute to the distributed circuit loss. We compare our computed gain vs frequency, computed using transfer matrices, to results found through particle-in-cell (PIC) simulations and the 1D TWT code LATTE to demonstrate the accuracy of our model. Furthermore, we demonstrate the ability of our model to reproduce gain ripple due to mismatches at the input and output ports of the TWT.

physics.plasm-ph↗

arXiv2023

Simple Reciprocal Electric Circuit Exhibiting Exceptional Point of Degeneracy

An exceptional point of degeneracy (EPD) occurs when both the eigenvalues and the corresponding eigenvectors of a square matrix coincide and the matrix has a nontrivial Jordan block structure. It is not easy to achieve an EPD exactly. In our prior studies, we synthesized simple conservative (lossless) circuits with evolution matrices featuring EPDs by using two LC loops coupled by a gyrator. In this paper, we advance even a simpler circuit with an EPD consisting of only two LC loops with one capacitor shared. Consequently, this circuit involves only four elements and it is perfectly reciprocal. The shared capacitance and parallel inductance are negative with values determined by explicit formulas which lead to EPD. This circuit can have the same Jordan canonical form as the nonreciprocal circuit we introduced before. This implies that the Jordan canonical form does not necessarily manifest systems' nonreciprocity. It is natural to ask how nonreciprocity is manifested in the system's spectral data. Our analysis of this issue shows that nonreciprocity is manifested explicitly in: (i) the circuit Lagrangian and (ii) the breakdown of certain symmetries in the set of eigenmodes. All our significant theoretical findings were thoroughly tested and confirmed by extensive numerical simulations using commercial circuit simulator software.

physics.class-ph↗

arXiv2023

Factorized form of the dispersion relations of a traveling wave tube

The traveling tube (TWT) design in a nutshell comprises of a pencil-like electron beam (e-beam) in vacuum interacting with guiding it slow-wave structure (SWS). In our prior studies the e-beam was represented by one-dimensional electron flow and SWS was represented by a transmission line (TL). We extend in this paper our previously constructed field theory for TWTs as well the celebrated Pierce theory by replacing there the standard transmission line (TL) with its generalization allowing for the low frequency cutoff. Both the standard TL and generalized transmission line (GTL) feature uniformly distributed shunt capacitance and serial inductance, but the GTL in addition to that has uniformly distributed serial capacitance. We remind the reader that the standard TL represents a waveguide operating at the so-called TEM mode with no low frequency cutoff. In contrast, the GTL represents a waveguide operating at the so-called TM mode featuring the low frequency cutoff. We develop all the details of the extended TWT field theory and using a particular choice of the TWT parameters we derive a physically appealing factorized form of the TWT dispersion relations. This form has two factors that represent exactly the dispersion functions of non-interacting GTL and the e-beam. We also find that the factorized dispersion relations comes with a number of interesting features including: (i) focus points that belong to each dispersion curve as TWT principle parameter varies; (ii) formation of 'hybrid" branches of the TWT dispersion curves parts of which can be traced to non-interacting GTL and the e-beam.

physics.acc-ph↗

arXiv2023

The Field Theory of Collective Cherenkov Radiation Associated with Electron Beams

Classical Cherenkov radiation is a celebrated physics phenomenon of electromagnetic (EM) radiation stimulated by an electric charge moving with constant velocity in a three dimensional dielectric medium. Cherenkov radiation has a wide spectrum and a particular distribution in space similar to the Mach cone created by a supersonic source. It is also characterized by the energy transfer from the charge's kinetic energy to the EM radiation. In the case of an electron beam passing through the middle of a an EM waveguide, the radiation is manifested as collective Cherenkov radiation. In this case the electron beam can be viewed as a one-dimensional non-neutral plasma whereas the waveguide can be viewed as a slow wave structure (SWS). This collective radiation occurs in particular in traveling wave tubes (TWTs), and it features the energy transfer from the electron beam to the EM radiation in the waveguide. Based on a Lagrangian field theory, we develop a convincing argument that the collective Cherenkov effect in TWTs is, in fact, a convective instability, that is, amplification. We also derive, for the first time, expressions identifying low- and high-frequency cutoffs for amplification in TWTs

physics.plasm-ph↗

arXiv2023

Parametric Modeling of Serpentine Waveguide Traveling Wave Tubes

A simple and fast model for numerically calculating small-signal gain in serpentine waveguide traveling-wave tubes (TWTs) is described. In the framework of the Pierce model, we consider one-dimensional electron flow along a dispersive single-mode slow-wave structure (SWS), accounting for the space-charge effect. The analytical model accounts for the frequency-dependent phase velocity and characteristic impedance obtained using various equivalent circuit models from the literature, validated by comparison with full-wave eigenmode simulation. The model includes a relation between the modal characteristic impedance and the interaction (Pierce) impedance of the SWS, including also an extra correction factor that accounts for the variation of the electric field distribution and hence of the interaction impedance over the beam cross section. By applying boundary conditions to our generalized Pierce model, we compute both the theoretical gain of a TWT and all the complex-valued wavenumbers of the hot modes versus frequency and compare our results with numerically intensive particle-in-cell (PIC) simulations; the good agreement in the comparison demonstrates the accuracy and simplicity of our generalized model. For various examples where we vary the average electron beam (e-beam) phase velocity, average e-beam current, number of unit cells, and input radio frequency (RF) power, we demonstrate that our model is robust in the small-signal regime.

physics.plasm-ph↗

arXiv2023

Time Modulation to Manage and Increase the Power Harvested From External Vibrations

We investigate how a single resonator with a time-modulated component extracts power from an external ambient source. However, the collected power is largely dependent on the precise choice of the modulation signal frequency. We focus on the power absorbed from external vibration using a one degree-of-freedom mechanical resonator where the damper has a time-varying component. We show that time modulation can make a significant difference in the amount of harvested power, leading to more than 10 times enhancement with respect to an analogous system without time modulation. We also find that a narrow band pair of peak and dip in the spectrum of the absorbed power occurs because of the presence of an exceptional point of degeneracy (EPD). In this narrow frequency range, the delay between the damper modulating signal and the external vibrating signal largely affects the collected power. The high frequency-selectivity of EPD-induced power management could potentially be used in sensing and spectrometer applications.

physics.app-ph↗

arXiv2022

Three-Way Serpentine Slow-Wave Structures with Stationary Inflection Point and Enhanced Interaction Impedance

We introduce two novel variants of the serpentine waveguide slow-wave structure (SWS), often utilized in millimeter-wave traveling-wave tubes (TWTs), with an enhanced interaction impedance. Using dispersion engineering in conjunction with transfer matrix methods, we tune the guided wavenumber dispersion relation to exhibit stationary inflection points (SIPs), and also non-stationary, or tilted inflection points (TIPs), within the dominant TE10 mode of a rectangular waveguide. The degeneracy is found below the first upper band-edge associated with the bandgap where neighboring spatial harmonics meet in the dispersion of the serpentine waveguide (SWG) which is threaded by a beam tunnel. The structure geometries are optimized to be able to achieve an SIP which allows for three-mode synchronism with an electron beam over a specified wavenumber interval in the desired Brillouin zone. Full-wave simulations are used to obtain and verify the existence of the SIP in the three-way coupled waveguide and fine-tune the geometry such that a beam would be in synchronism at or near the SIP. The three-way waveguide SWS exhibits a moderately high Pierce impedance in the vicinity of a nearly-stationary inflection point, making the SWS geometry potentially useful for improving the power gain and basic extraction efficiency of millimeter-wave TWTs. Additionally, the introduced SWS geometries have directional coupler-like behavior, which enables distributed power extraction at frequencies near the SIP frequency.

physics.plasm-ph↗

arXiv2022

An Accurate Analytic Model for Traveling Wave Tube Dispersion Relation

Abstract -- We construct an analytical model for the dispersion of the hot modes in a traveling wave tube (TWT) based on the Lagrangian field theory, upgrading its constants to be frequency-dependent. The frequency dependence of the parameters of the TWT slow wave structure (SWS) is recovered from full-wave simulations by standard software (e.g., CST). We applied the model to study the hot modes of a helical-based TWT and found an excellent agreement between the results from our model and those from particle in cell (PIC) simulations. Our additional studies show that the proposed approach can be applied to various SWS geometries.

physics.app-ph↗

arXiv2021

Enhanced Sensitivity of Degenerate System Made of Two Unstable Resonators Coupled by Gyrator Operating at an Exceptional Point

We demonstrate that a circuit comprising two unstable LC resonators coupled via a gyrator supports an exceptional point of degeneracy (EPD) with purely real eigenfrequency. Each of the two resonators includes either a capacitor or an inductor with a negative value, showing purely imaginary resonance frequency when not coupled to the other via the gyrator. With external perturbation imposed on the system, we show analytically that the resonance frequency response of the circuit follows the square-root dependence on perturbation, leading to possible sensor applications. Furthermore, the effect of small losses in the resonators is investigated, and we show that losses lead to instability. In addition, the EPD occurrence and sensitivity are demonstrated by showing that the relevant Puiseux fractional power series expansion describes the eigenfrequency bifurcation near the EPD. The EPD has the great potential to enhance the sensitivity of a sensing system by orders of magnitude. Making use of the EPD in the gyrator-based circuit, our results pave the way to realize a new generation of high-sensitive sensors to measure small physical or chemical perturbations.

physics.app-ph↗

arXiv2021

Analytic theory of coupled-cavity traveling wave tubes

Coupled-cavity traveling wave tube (CCTWT) is a high power microwave (HPM) vacuum electronic device used to amplify radio-frequency (RF) signals. CCTWTS have numerous applications, including radar, radio navigation, space communication, television, radio repeaters, and charged particle accelerators. The microwave-generating interactions in CCTWTs take place mostly in coupled resonant cavities positioned periodically along the electron beam axis. Operational features of a CCTWT particularly the amplification mechanism are similar to those of a multicavity klystron (MCK). We advance here a Lagrangian field theory of CCTWTs with the space being represented by one-dimensional continuum. The theory integrates into it the space-charge effects including the so-called debunching (electron-to-electron repulsion). The corresponding Euler-Lagrange equations are ODEs with coefficients varying periodically in the space. Utilizing the system periodicity we develop the instrumental features of the Floquet theory including the monodromy matrix and its Floquet multipliers. We use them to derive closed form expressions for a number of physically significant quantities. Those include in particular the dispersion relations and the frequency dependent gain foundational to the RF signal amplification. Serpentine (folded, corrugated) traveling wave tubes are very similar to CCTWTs and our theory applies to them also.

physics.acc-ph↗

arXiv2021

Exceptional Points in Gyrator-Based Circuit and Nonlinear High-Sensitivity Oscillator

We present a scheme for high-sensitive oscillators based on an exceptional point of degeneracy (EPD) in a circuit made of two LC resonators coupled by a gyrator. The frequency of oscillation is very sensitive to perturbations of a circuit element, like a capacitor. We show conditions that lead to an EPD, assuming one of the two resonators is composed of an inductor and a capacitor of negative values. The EPD occurrence and sensitivity to perturbations in the linear case are demonstrated by showing that the eigenfrequency bifurcation around the EPD is described by the relevant Puiseux (fractional power) series expansion. We also investigate the effect of small losses in the system and show that they lead to instability. We fabricate the circuit, and exploit its instability and nonlinearity, observing experimentally stable self-oscillations under the saturated regime. We measure the circuit's sensitivity to a small capacitor perturbation. A shift in frequency of oscillation after saturation is well detectable with very distinct spectral peaks with 10 Hz linewidth, clean until -70 dB from the peak value. The sensitivity is (i) higher than the one of a comparable simple LC linear resonator, (ii) comparable or better than other published EPD circuits, and (iii) applicable to both negative and positive values of the capacitance perturbation, contrary to what happens in PT-symmetric circuits. The proposed scheme can pave the way for a new generation of high-sensitive sensors to measure slight variations in physical, chemical or biological quantities.

physics.app-ph↗

arXiv2021

Analytic theory of multicavity klystron

Multicavity Klystron (MCK) is a high power microwave (HPM) vacuum electronic device used to amplify radio-frequency (RF) signals with numerous applications, including radar, radio navigation, space communication, television, radio repeaters, and charged particle accelerators. The microwave-generating interactions in klystrons take place in resonant cavities at discrete locations along the beam. Importantly, there is no electromagnetic coupling between cavities, they are coupled only by the bunched electron beam, which drifts from one cavity to the next. We advance here an analytic theory of MCKs operating in voltage amplification mode associated with the maximal gain. This theory features in particular exact formulas for the MCK instability frequencies, its dispersion relations and optimal values of the MCK parameters providing for maximal gain.

physics.acc-ph↗

arXiv2021

Exceptional Degeneracies in Traveling Wave Tubes with Dispersive Slow-Wave Structure Including Space-Charge Effect

The interaction between a linear electron beam and a guided electromagnetic wave is studied in the contest of exceptional points of degeneracy (EPD) supported by such an interactive system. The study focuses on the case of a linear beam traveling wave tube (TWT) with a realistic helix waveguide slow-wave structure (SWS). The interaction is formulated by an analytical model that is a generalization of the Pierce model, assuming a one-dimensional electron flow along a dispersive single-mode guiding SWS and taking into account space-charge effects in the system. The augmented model using phase velocity and characteristic impedance obtained via full-wave simulations is validated by calculating gain versus frequency and comparing it with that from more complex electron beam simulators. This comparison also shows the accuracy of our new model compared with respect to the non-dispersive Pierce model. EPDs are then investigated using the augmented model, observing the coalescence of complex-valued wavenumbers and the system's eigenvectors. The point in the complex dispersion diagram at which the TWT-system starts/ceases to exhibit a convection instability, i.e., a mode starts/ceases to grow exponentially along the TWT, is the EPD. We also demonstrate the EPD existence by showing that the Puiseux fractional power series expansion well approximates the bifurcation of the dispersion diagram at the EPD. This latter concept also explains the "exceptional" sensitivity of the TWT-system to changes in the beam's electron velocity when operating near an EPD.

physics.app-ph↗

arXiv2020

Exceptional points of degeneracy in traveling wave tubes

Traveling wave tube (TWT) is a powerful vacuum electronic device used to amplify radio-frequency (RF) signals with numerous applications, including radar, television and telephone satellite communications. TWT design in a nutshell comprises of a pencil-like electron beam (e-beam) in vacuum interacting with guiding it slow-wave structure (SWS). In our studies here the e-beam is represented by one-dimensional electron flow and SWS is represented by a transmission line (TL). The interaction between the e-beam and the TL is modeled by an analytic theory that generalizes the well-known Pierce model by taking into account the so-called space-charge effects particularly electron-to-electron repulsion (debunching). Many important aspects of the analytic theory of TWTs have been already analyzed in our monograph on the subject. The focus of the studies here is on degeneracies of the TWT dispersion relations particularly on exceptional points of degeneracy and their applications. The term exceptional point of degeneracy (EPD) refers to the property of the relevant matrix to have nontrivial Jordan block structure. Using special parameterization particularly suited to chosen EPD we derive exact formulas for the relevant Jordan basis including the eigenvectors and the so-called root vector associated with the Jordan block. Based on these studies we develop constructive approach to sensing of small signals.

physics.class-ph↗

arXiv2020

Perturbations of circuit evolution matrices with Jordan blocks

In our prior studies we synthesized special circuits possessing evolution matrices that involve nontrivial Jordan blocks and the corresponding degenerate eigenfrequencies. The degeneracy of this type is sometimes referred to as exceptional point of degeneracy (EPD). The simplest of these circuits are composed just of two LC-loops coupled by a gyrator and they are of our primary interest here. These simple circuits when near an EPD state can be used for enhanced sensitivity applications. With that in mind we develop here a comprehensive perturbation theory for these simple circuits near an EPD as well way to assure their stable operation. As to broader problem of numerical treatment of Jordan blocks and their perturbation we propose a few approaches allowing to detect the proximity to Jordan blocks.

physics.class-ph↗