Nonlinear Photonic Crystals: IV. Nonlinear Schrodinger Equation Regime

We study here the nonlinear Schrodinger Equation (NLS) as the first term in a sequence of approximations for an electromagnetic (EM) wave propagating according to the nonlinear Maxwell equations (NLM). The dielectric medium is assumed to be periodic, with a cubic nonlinearity, and with its linear background possessing inversion symmetric dispersion relations. The medium is excited by a current $\mathbf{J}$ producing an EM wave. The wave nonlinear evolution is analyzed based on the modal decomposition and an expansion of the exact solution to the NLM into an asymptotic series with respect to some three small parameters $\alpha$, $\beta$ and $\varrho$. These parameters are introduced through the excitation current $\mathbf{J}$ to scale respectively (i) its amplitude and consequently the magnitude of the nonlinearity; (ii) the range of wavevectors involved in its modal composition, with $\beta ^{-1}$ scaling its spatial extension; (iii) its frequency bandwidth, with $\varrho ^{-1}$ scaling its time extension. We develop a consistent theory of approximations of increasing accuracy for the NLM with its first term governed by the NLS. We show that such NLS regime is the medium response to an almost monochromatic excitation current $\mathbf{J}$. The developed approach not only provides rigorous estimates of the approximation accuracy of the NLM with the NLS in terms of powers of $\alpha$, $\beta$ and $\varrho$, but it also produces new extended NLS (ENLS) equations providing better approximations. Remarkably, quantitative estimates show that properly tailored ENLS can significantly improve the approximation accuracy of the NLM compare with the classical NLS

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

Novel concept for pulse compression via structured spatial energy distribution

We present a novel concept for pulse compression scheme applicable at RF, microwave and possibly to optical frequencies based on structured energy distribution in cavities supporting degenerate band-edge (DBE) modes. For such modes a significant fraction of energy resides in a small fraction of the cavity length. Such energy concentration provides a basis for superior performance for applications in microwave pulse compression devices (MPC) when compared to conventional cavities. The novel design features: larger loaded quality factor of the cavity and stored energy compared to conventional designs, robustness to variations of cavity loading, energy feeding and extraction at the cavity center, substantial reduction of the cavity size by use of equivalent lumped circuits for low energy sections of the cavity, controlled pulse shaping via engineered extraction techniques. The presented concepts are general, in terms of equivalent transmission lines, and can be applied to a variety of realistic guiding structures.Comment: 18 pages, 10 figure

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 TWT

Wave-Corpuscle Mechanics for Electric Charges

It is well known that the concept of a point charge interacting with the electromagnetic (EM) field has a problem. To address that problem we introduce the concept of wave-corpuscle to describe spinless elementary charges interacting with the classical EM field. Every charge interacts only with the EM field and is described by a complex valued wave function over the 4-dimensional space time continuum. A system of many charges interacting with the EM field is defined by a local, gauge and Lorentz invariant Lagrangian with a key ingredient—a nonlinear self-interaction term providing for a cohesive force assigned to every charge. An ideal wave-corpuscle is an exact solution to the Euler-Lagrange equations describing both free and accelerated motions. It carries explicitly features of a point charge and the de Broglie wave. Our analysis shows that a system of well separated charges moving with nonrelativistic velocities are represented accurately as wave-corpuscles governed by the Newton equations of motion for point charges interacting with the Lorentz forces. In this regime the nonlinearities are “stealthy” and don’t show explicitly anywhere, but they provide for the binding forces that keep localized every individual charge. The theory can also be applied to closely interacting charges as in hydrogen atom where it produces discrete energy spectrum