arXiv subjects
Publications and source records attributed to Tarek Mealy.
This erratum provides an updated fitting function for the lasing threshold of finite-length cavities operating at a stationary inflection point (SIP) or regular band edge (RBE) resonance, clarifying their asymptotic scaling with the number of unit cells of the periodic cavity.
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.
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.
The concept of lasers based on the frozen mode regime in active periodic optical waveguides with a 3rd-order exceptional point of degeneracy (EPD) is advanced. The frozen mode regime in a lossless and gainless waveguide is associated with a stationary inflection point (SIP) in the Bloch dispersion relation, where three Bloch eigenmodes coalesce forming the frozen mode. As a practical example, we consider an asymmetric serpentine optical waveguide (ASOW). An ASOW operating near the SIP frequency displays a large group delay of a non-resonant nature that scales as the cube of the waveguide length, leading to a strong gain enhancement when active material is included. Therefore, a laser operating in the close vicinity of an SIP has a gain threshold that scales as a negative cube of the waveguide length. We determine that this scaling law is maintained in the presence of small distributed losses, such as radiation associated with waveguide bends and roughness. In addition, we show that although gain causes a distortion in the modes coalescing at the SIP, the properties of the frozen mode are relatively resistant to such small perturbations and we still observe a large degree of exceptional degeneracy for gain values that bring the system above threshold. Finally, our study also reveals that lasing near an SIP is favored over lasing near a photonic band edge located in close proximity to the SIP. In particular, we observe that an SIP-induced lasing in an ASOW displays lower gain threshold compared to lasing near the photonic regular band edge (RBE), even though the SIP resonance has a lower quality factor than the RBE resonance.
We introduce a glide symmetric optical waveguide exhibiting a stationary inflection point (SIP) in the Bloch wavenumber dispersion relation. An SIP is a third order exceptional point of degeneracy (EPD) where three Bloch eigenmodes coalesce to form a so-called frozen mode with vanishing group velocity and diverging amplitude. We show that the incorporation of chirped distributed Bragg reflectors and distributed coupling between waveguides in the periodic structure facilitates the SIP formation and greatly enhances the characteristics of the frozen mode regime. We confirm the existence of an SIP in two ways: by observing the flatness of the dispersion diagram and also by using a coalescence parameter describing the separation of the three eigenvectors collapsing on each other. We find that in the absence of losses, both the quality factor and the group delay at the SIP grow with the cubic power of the cavity length. The frozen mode regime can be very attractive for light amplification and lasing, in optical delay lines, sensors, and modulators.
We design a three-way silicon optical waveguide with the Bloch dispersion relation supporting a stationary inflection point (SIP). The SIP is a third order exceptional point of degeneracy (EPD) where three Bloch modes coalesce forming the frozen mode with greatly enhanced amplitude. The proposed design consists of a coupled resonators optical waveguide (CROW) coupled to a parallel straight waveguide. At any given frequency, this structure supports three pairs of reciprocal Bloch eigenmodes, propagating and/or evanescent. In addition to full-wave simulations, we also employ a so-called ''hybrid model'' that uses transfer matrices obtained from full-wave simulations of sub-blocks of the unit cell. This allows us to account for radiation losses and enables a design procedure based on minimizing the eigenmodes' coalescence parameter. The proposed finite-length CROW displays almost unitary transfer function at the SIP frequency, implying a nearly perfect conversion of the input light into the frozen mode. The group delay and the effective quality factor at the SIP frequency show an $N^{3}$ scaling, where $N$ is the number of unit cells in the cavity. The frozen mode in the CROW can be utilized in various applications like sensors, lasers and optical delay lines.
We present an analytical method to compute the wavenumbers and electric fields of the space-charge-wave eigenmodes supported by a two-stream electron beam, consisting of a solid inner cylindrical stream and a coaxial outer annular stream, both contained within a cylindrical metallic tunnel.We extend the analytical model developed by Ramo to the case of two streams. The method accounts for the interaction between the two streams with the presence of the beam-tunnel wall; it can be used to model the complex wavenumbers associated with the two-stream instability and the plasma frequency reduction effects in vacuum electronic amplifiers and other vacuum electronic devices.
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.
We demonstrate that exceptional points of degeneracy (EPDs) are obtained in two coupled waveguides without resorting to gain and loss. We show the general concept that modes resulting from a proper coupling of forward and backward waves exhibit EPDs of order two and that there the group velocity vanishes. We verify our insight by using coupled mode theory and also by fullwave numerical simulations of light in a dielectric slab coupled to a grating, when one supports a forward wave whereas the other (the grating) supports a backward wave. We also demonstrate how to realize photonic indirect bandgaps in guiding systems supporting a backward and a forward wave, show its relations to the occurrence of EPDs, and offer a design procedure.
We propose a degenerate version of the Bragg condition, associated to a degenerate band edge (DBE). A standard Bragg condition can be implemented using a periodic grating that operates at the regular band edge. The structure we propose to realize a DBE is made of two stacked identical gratings that form two coupled periodic waveguides with broken mirror symmetry. The occurrence of the DBE is verified by using both an eigenmode solver and the calculations of scattering parameters, using full-wave simulations. The proposed structure is a good candidate to conceive a degenerate distributed feedback lasers operating at the DBE.
We propose a scheme to calculate the reduced plasma frequency of a cylindrical-shaped electron beam flowing inside of a cylindrical tunnel, based on results obtained from Particle-in-cell (PIC) simulations. In PIC simulations, we modulate the electron beam using two parallel, non-intercepting, closely-spaced grids which are electrically connected together by a single-tone sinusoidal voltage source. The electron energy and the beam current distributions along the length of the tunnel are monitored after the system is operating at steady-state. We build a system matrix describing the beam's dynamics, estimated by fitting a 2x2 matrix that best agrees with the first order differential equations that govern the physics-based system. Results are compared with the theoretical Branch and Mihran model, which is typically used to compute the plasma frequency reduction factor in such systems. Our method shows excellent agreement with the theoretical model, however, it is also general. Our method can be potentially utilized to determine the reduced plasma frequencies of electron beams propagating in differently-shaped beam tunnels, where no theoretical model yet exists, such as the case of a cylindrical or elliptical electron beam propagating inside of a metallic beam tunnel of cylindrical, square, or elliptical cross-section. It can be applied also to electron beams composed of multiple streams.
The dispersion of a three-way waveguide is engineered to exhibit exceptional modal characteristics. Two coupled waveguides with Parity-Time (PT) symmetry have been previously demonstrated to exhibit second order exceptional points of degeneracy (EPDs). In this work, we introduce and investigate a particular class of EPDs, applicable from radio frequency to optical wavelengths, whereby three coupled waveguides satisfy Glide-Time (GT) symmetry to exhibit a third order modal degeneracy with a real-valued wavenumber. GT symmetry involves glide symmetry of lossless/gainless components of the waveguide in addition to changing the sign of passive/active elements while applying a glide symmetry operation. This GT-symmetry condition allows three Floquet-Bloch eigenmodes of the structure to coalesce to a real-valued wavenumber at a single frequency, in addition of having one branch of the dispersion diagram with a purely real wavenumber. The proposed scheme may have applications including but not limited to distributed amplifiers, radiating arrays, and sensors, from radio frequency to optics.
We show the physical realization and experimental demonstration of an exceptional point of 6th order degeneracy in a triple ladder (or three-way) microwave waveguide realized using three coupled microstrips on a grounded dielectric substrate. This threeway waveguide supports six Bloch eigenmodes and all coalesce onto a degenerate single eigenmode at a given frequency. The three-way waveguide is gainless, and this exceptional point is associated to a vanishing group velocity and its multiple derivatives. Indeed the wavenumber-frequency dispersion diagram, that we call 6th order degenerate band edge (6DBE) has six coalescing branches. We provide the experimental verification of a 6th order exceptional point by evaluating the degenerate wavenumber-frequency dispersion diagram from the measurement of scattering parameters of a six-port unit cell. We also show the resonant behavior of a cavity made of the three-way waveguide with finite length. The unique properties of 6DBE can be exploited in designing innovative high-Q resonators, oscillators, filters, and pulse shaping devices.
We introduce a circuit topology based on a simple triple-ladder circuit realized with lumped reactive components that provides a sixth order degenerate band-edge (6DBE). The 6DBE is a special kind of sixth-order exceptional point of degeneracy in a lossless and gainless periodic ladder. This degeneracy provides a very flat band edge in the phase-frequency dispersion diagram. The proposed topology exhibits unique structured resonance features associated with a high loaded Q-factor. We investigate the Floquet-Bloch modes in an infinite-length periodic triple-ladder and their dispersion relation using the S parameter formalism. We also provide the approximate analytic expressions of the eigenmodes and dispersion relation around the degenerate point based on the Puiseux series expansion. We investigate the filtering characteristics of a finite-length structure terminated with loads to highlight the special properties of the 6DBE compared to ladders with regular band edge (RBE) and fourth order degenerate band edge (DBE). The circuit framework introduced here with a 6DBE can be exploited in designing novel high Q-factor oscillators, filters, sensors, and pulse shaping networks.
We demonstrate that a periodic waveguide comprising of uniform lossless segments together with discrete gain and radiating elements supports exceptional points of degeneracy (EPDs). We provide analytical expressions for all possible conditions that guarantee the occurrence of an EPD, i.e., the coalescence of eigenvalues and eigenvectors. We show that EPDs are not only achieved using symmetric gain and radiation periodic loading, but they are also obtained using asymmetric gain and radiation loss conditions. We illustrate the characteristics of the degenerate electromagnetic modes, showing the dispersion diagram and discussing the tunability of the EPD frequency. We show a special condition, we refer to it as parity-time (PT)-glide symmetry, which leads to a degeneracy that is occurring at all frequencies of operation. The class of EPDs proposed in this work is very promising for many applications that incorporate discrete-distributed coherent sources and radiation-loss elements; operating in the vicinity of such special degeneracy conditions leads to potential performance enhancement in a variety of microwave and optical resonators, antennas, and devices and can be extended to a new class of active integrated antenna arrays and radiating laser arrays.
The concept of exceptional point of degeneracy (EPD) is used to conceive a degenerate synchronization regime that is able to enhance the level of output power and power conversion efficiency for backward wave oscillators (BWOs) operating at millimeter-wave and Terahertz frequencies. Standard BWOs operating at such high frequency ranges typically generate output power not exceeding tens of watts with very poor power conversion efficiency in the order of 1%. The novel concept of degenerate synchronization for the BWO based on a folded waveguide is implemented by engineering distributed gain and power extraction along the slow-wave waveguide. The distributed power extraction along the folded waveguide is useful to satisfy the necessary conditions to have an EPD at the synchronization point. Particle-in-cell (PIC) simulation results shows that BWO operating at an EPD regime is capable of generating output power exceeding 3 kwatts with conversion efficiency of exceeding 20% at frequency of 88.5 GHz.
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.
A scheme to characterize the dynamics of the electron beam-electromagnetic power exchange along a traveling wave tube (TWT) is proposed. The method is based on defining a state vector at discrete periodic locations along the TWT and determining the transfer matrix of the unit-cell of the "hot" slow-wave structure (SWS) that takes into account the interaction between the electromagnetic guided field and the electron beam via particle-in-cell (PIC) simulations. Once the estimate of the unit-cell transfer matrix is obtained, we show how to find the hybrid, beam-electromagnetic, eigenmodes in the hot SWS, i.e., where the electromagnetic guided field interacts with an electron beam, by using Floquet theory. In particular, we show how do determine the complex-valued wavenumbers of the hybrid modes and the eigenvectors associated to them. The method is applied to find the hot modes with complex wavenumber that can be supported in a TWT amplifier with a helix SWS. We show dispersion relations of the modal complex wavenumbers of the hybrid modes when varying frequency and beam voltage; the results are in agreement with Pierce theory. The method is also applied to find the complex-wavenumber modes in a hot SWS of a millimeter wave TWT amplifier based on a serpentine waveguide. The technique is general and can be applied to any SWS geometry where electromagnetic modes interact with an electron beam.