Brillouin Flow Research Articles

Resistive wall effect on the stability of planar relativistic Brillouin flow

Previous treatments of relativistic Brillouin flow stability are extended to the case of nonzero anode resistivity, for the case of wave propagation across the equilibrium magnetic field. It is found that transverse magnetic (TM) waves are unstable at frequencies below the equilibrium cyclotron frequency, unlike the case for a perfectly conducting anode. Transverse electric (TE) waves, always oscillatory in the case of a perfectly conducting anode, are damped; an explicit expression for the damping rate, valid for large conductivities, is given.

Long Wavelength, Nonlinear Perturbations of the Brillouin Flow Equilibrium on Magnetically Insulated Lines

The Brillouin flow equilibrium in magnetically insulated lines is a state in which field-emitted electrons, confined to a sheath near the cathode by a magnetic field, drift laminarly at the ?? x ?? velocity in the self-consistent fields of the anode-cathode gap. Herein, this state is perturbed by a TM disturbance traveling along the direction of electron flow. Asymptotic expansions of all quantities of interest in some small parameter are performed in the long wavelength limit. First order quantities are seen to be governed by the Korteweg-deVries' equation, which admits soliton solutions. In the appropriate limits of either infinite wavelength or vanishingly small amplitude, the results contained herein are seen to agree with other analyses, based, respectively, upon either the nondispersive telegraphers' equations or a linearized analysis.

A generalized treatment of the electrostatic instability of rippled non-neutral magnetically focused electron beams

A generalization of the stability analysis of rippled non-neutral electron beams is presented. The treatment covers regimes ranging between those known as ‘Brillouin flow’ (ωpe = ωc/2½) and ‘immersed flow’ (ωpe≪ωc). It considers electrostatic surface waves with azimuthal symmetry (l = 0) as well as harmonic modes (l≠0). Both long-wavelength and short-wavelength domains are analysed. A general discussion of instability is given in terms of the solution of the Mathieu-Hill equation. New unstable modes are found and their growth rates are derived and compared.

Linear waves and instabilities on magnetically insulated gaps

A linear, fully relativistic and electromagnetic analysis of waves perturbing the Brillouin flow state on planar, magnetically insulated gaps is presented. Three independent classes of waves are treated individually; transverse-magnetic waves propagating normal to the insulating magnetic field, transverse-electric waves propagating in the same direction, and a set of waves propagating along the magnetic field. Dispersion relations governing discrete modes of oscillation for each class are found and solved for some cases of interest. Only the transverse-magnetic waves are found to be unstable due to the existence of a Doppler-shifted plasma reasonance layer at frequencies above the plasma frequency. The scaling of the growth rate for this instability with respect to parameters characterizing the equilibrium state is examined. In addition to these discrete modes, spectral continua associated with localized resonance phenomena in the electron sheath bounding the cathode are described and explained.

Magnetically insulated plasma sheath in coaxial transmission lines with an external magnetic field

A relativistic Brillouin flow equilibrium model is assumed for the plasma sheath and a cylindrical analytic solution including a uniform external magnetic field is obtained. A free parameter appearing in all previous magnetic insulation theories is shown to be accurately determined by maximizing the transmitted power flow. The theory is supported by particle-in-cell simulations.

Operating point of long magnetically insulated vacuum transmission lines

This theoretical investigation addresses the propagation of a power pulse in a long magnetically insulated vacuum transmission line (MITL). Relativistic Brillouin flow, with an arbitrary choice of the potential Vm at the flow boundary, explains the relationships among anode voltage, anode current, and cathode current that have been observed experimentally. The results of this investigation suggest that minimum energy considerations may be used to establish the value of Vm and hence determine the operating point of the MITL. This operating point compares very favorably with experimental data and with results from numerical simulations.

Magnetic compression of axially symmetric Brillouin-focused electron beams

A study has been made of high-current density Brillouin-focused electron beams formed with magnetic compression. Analysis of the problem was by numerical calculation of electron trajectories using a digital computer. Electrostatic field data were measured using a resistance network analog. The effect of the magnetic field was calculated as an equivalent potential. An experimental system based on the results of the analysis was designed and tested. A Müller-type electron gun was found to provide a more uniform current density than a corresponding Pierce type. Experimental results were in close agreement with design objectives. For an area magnetic compression of 10 to 1, a focused beam approaching ideal Brillouin flow was achieved.

Focusing of a Relativistic Electron Flow

The trajectory differential equation governing the motion of relativistic electrons is derived in terms of the scalar and vector potentials of the system using the principle of least action. The conservation of energy and momentum are used to develop the paraxial-ray differential equation describing the beam radius of a laminar-flow relativistic electron beam. The focusing of the electron beam in drift and accelerating regions has been examined and the conditions for perfect balancing and nonspreading of a laminar-flow drifting beam are also derived. It is shown that the equilibrium condition for Brillouin flow is that 2ωL2−ωp2[1 − (uz/c)2]=01 where ωL is the Larmor precession frequency and ωp the electron-plasma frequency. uz and c denote, respectively, the axial beam velocity and the speed of light in vacuum. The variation of the normalized ripple amplitude and the scallop wavelength of a drifting beam is discussed. The profile of a beam accelerated in a uniform longitudinal electrostatic field is also illustrated.

Direct synthesis of crossed-field electron guns—A novel gun for producing Brillouin flow

A direct approach to the synthesis of crossed-field electron guns is presented. An iterative approximation to the desired beam form is achieved by application of the paraxial ray equation alternatively solving for potential and curvature along the beam while maintaining certain end-point conditions and the desired convergence pattern. When sufficient accuracy is obtained, polynomial approximations are used to express shapes of the trajectories and the electric fields along the beam edges in analytic form. The electrodes to produce the necessary electric fields along the beam edges are calculated by Kirstein's method. A conformal transformation is used to transform the beam edge into the real axis of a complex plane and analytic continuation of a suitable complex potential function is employed to find equipotential lines. These are transformed back into the plane of real coordinates and are surfaces along which electrodes can be placed. A novel short gun which produces a Brillouin beam has been designed using the iterative approach. The availability of beam curvature as a variable has made it possible to find an electrode system which controls the transition from the cathode-region flow characteristics used in Kino's short gun to a drift Brillouin beam. Detailed experimental evaluation of the gun reveals that the beam characteristics are in good agreement with those predicted theoretically.

Some exact solutions of the equations of a stationary monoenergetic beam of charged particles

We consider the class of invariant solutions which can describe only vortex flows (curl P ≠ 0, P is the generalized momentum) and show that they contain solutions corresponding to flows from a plane or cylindrical emitter with a linear voltage drop across it (direct heating) in the temperature-limited regime*. The solution is obtained in analytic form for emission from a plane in a uniform magnetic field perpendicular to the flow plane. It also (forβ=0) defines a plane magnetron in the T-regime. The solution of the problem for a cylindrical emitter reduces to considering equations describing a cylindrical diode or magnetron in the T-regime, where the shape of the collector is given by the potential distribution curve for these cases. We can extend the results to a relativistic beam if restrictions are imposed on its relative dimensions which permit us to ignore the magnetic self-field. Brillouin type flows (including irrotational ones) are studied in which particles move without intersecting the equipotential surfaces along three-dimensional spirals on the surface of cones. An analytic solution is given for relativistic Brillouin flow in a conical diode when strict allowance is mede for the magnetic self-field.

Solutions of the Equations for Space-charge Electron Flow with a Magnetic Field at the Cathode for Beams of Non-uniform Current Density†

After a discussion of the space-charge equations, a solution is given from which the critical conditions are obtained for flows like Brillouin flow but with a magnetic field at the cathode. Results for confined flows are derived; it is shown that, compared with a beam of uniform current density, much more current can be put into a beam of given boundary voltage if the current density increases strongly with radius. Confined ribbon beams are also considered.

The effect of tolerances in conical flow Pierce guns II—The perturbed flow in the magnetic field

The first-order perturbations to a beam initially in nonrippling Brillouin flow, which results from several common manufacturing errors in conical flow Pierce guns, are calculated. The manufacturing errors are changes in cathode-anode spacing, changes in cathode curvature, tilt, and transverse displacement of the cathode-focus electrode assembly, and tilt and transverse misalignment of the whole gun to the magnetic field. Errors in cathode-to-focus electrode spacing are not discussed. The tolerances which destroy the rotational symmetry of the gun are shown to cause the smooth beam to perform a helical motion; the tolerances which preserve the symmetry are shown to produce an undulating beam. Using the results of Part I of this paper [1], the magnitudes of individual gun tolerances are related to the effects they produce in the magnetic field. A method of setting limits to the individual gun tolerances is presented and illustrated by calculating tolerances for a gun used in an X-band traveling-wave tube. The calculated tolerances are found to have values which may readily be achieved by careful engineering.

The growth of peak velocity, noise, and signals in O-type electron beams

A new phenomenon, the growth of peak velocity of an electron beam originating from a shielded gun, has been investigated. Its behavior with distance, magnetic field, and phasing between beam voltage pulse and heater current zero is presented. This phenomenon bears a general but not a detailed resemblance to the noise growth observed by previous investigators. The noise growth phenomenon has been re-examined in detail on the beam used for peak velocity growth measurements. The single most striking observation is that whereas peak velocity and 3-kmc noise growth take place at magnetic field strength primarily above the Brillouin flow value, 3-kmc signal growth occurs only at magnetic fields below the Brillouin value. A combination of linear axially symmetric surface-wave mode theory and simple intermodulation theory is unable to describe quantitatively the noise growth rate. Consideration of other axially-symmetric modes on near-Brillouin beams, namely body modes and near-synchronous modes, also leads to poor correlation with experiment. It seems likely that the correct explanation involves consideration of one or more of the following: 1) theta-varying dc beam media, 2) theta-varying RF excitation, or 3) a more accurate nonlinear representation of noise.

Velocity and Potential Distribution in a Linear, Smooth Anode Magnetron (Magnetic Diode). II

We present the results of numerical calculations of the potential, space-charge, and current distributions in linear magnetrons, taking into account the three-dimensional spread of the initial electron velocities. For most practical configurations, the potential distribution is nearly parabolic within the electron cloud, with a slight dip in front of the cathode and the space-charge distribution is remarkably constant and approximately equal to that predicted by the single stream flow, although the electron trajectories are now anything but linear. For other choices of the parameters, the space-charge distribution can be ``triangular'' or may resemble that associated with the so-called double-stream flow. In all such cases the potential, space charge, and current distributions cannot be predicted on the basis of a Brillouin flow and have to be calculated separately. We have also shown that in most practical cases the systems seem to operate in what is called the magnetic-field-limited, rather than space-charge-limited regime, so that the anode current depends on the values of the external parameters only and is independent of the amount of space charge or the potential distribution between the electrodes. In ballistic terms this means that the force due to the magnetic field acting on an electron at the peak of its trajectory is greater than the corresponding force due to the electric field. Finally, we have shown that contrary to our expectations, there seems to be a great deal of agreement between the operations of systems using finite and infinite cathodes, even when the initial electron velocities are taken into account.

Transformation of Small-Signal Energy and Momentum of Waves

The transformation of the small-signal energy and momentum of a wave between two inertial reference frames as first given by Sturrock are derived by using simple perturbation theory and the appropriate transformations of length, time, current, and electromagnetic fields. The approach allows a straightforward generalization to the case of relativistic linear transformations and to nonrelativistic transformations between two reference frames that rotate with respect to each other. These rotating and linear transformations allow one to make very general statements about the frequency and wavenumbers for which negative-energy waves are obtained in a rotating and translating medium, as, for example, an electron beam in Brillouin flow.

A synchronous-wave amplifier

The small-signal gain expression for an amplifier consisting of two resonant synchronous-wave couplers is presented and the results of an experimental investigation of such a tube are reported. Some theoretical studies of such an amplifier have been reported earlier but no experimental results have been given. The experimental tube operated at 3.00 kMc and employed a Brillouin flow beam with a dc power of 730 w. The saturation output power was 45 w. In general, the experimental results were in good qualitative agreement with the theory. The discrepancy between the theoretical gain of 13 db and the experimentally observed 8-db gain is attributed to thick-beam effects. Oscillations due to the interaction between the slow space-charge wave and the off-axis longitudinal circuit fields were observed, and it is pointed out that this effect may set some limitations on the gain that can be achieved in the sychronous-wave amplifier.

Negative Energy Fast Waves in Brillouin Flow Beams

First Page

Velocity and Potential Distribution in a Linear, Smooth Anode Magnetron (Magnetic Diode)

The paper discusses the potential, space-charge and current density distribution in a “magnetic diode,”i.e., in a linear, smooth anode magnetron. The results of the investigations show that when the initial electron velocities are taken into account, the electron cloud extends much further into the interelectrode space than could be expected on the basis of a simple Brillouin flow. Furthermore, the density of the electron cloud decreases monotonically as we move away from the cathode. At the same time, the current flowing parallel to the electrodes has a pronounced peak which is situated half way between the edge of the cloud and the cathode. These results are not only of interest in the steady-state theory of linear magnetrons, but they also throw new light on the mode of operation of some thermionic power converters.

Electron-beam defocusing due to high-intensity RF fields

The electric fields associated with a slow electro-magnetic wave propagating along an electron beam will modulate this beam; the axial component of field will produce the usual bunching, while the radial component will cause periodic perturbations in the radius of such an electron beam. This paper presents the results of analytical and experimental studies in an attempt to explain certain features of this defocusing effect of intense RF fields on the beam in a traveling-wave tube. In particular the effect of cathode flux on the RF defocusing of a beam in Brillouin flow is treated.

Multiple-stream Crossed-field Interaction†

ABSTRACT The dispersion equations are developed using admittance matching techniques for both the crossed-field double-beam forward-wave amplifier and the double-beam drift-space amplifier. The input boundary value problem is solved by matching the a.c. displacement and the a.c. charge density at the input boundary. The gain is found to be considerably greater for the double-beam device than for each of the beams acting separately. The relative ease of obtaining Brillouin flow with thin beams in practical devices makes this an attractive way of increasing the gain. A non-linear analysis is made and the results for the two-beam case are compared with those for a thick Brillouin stream and those for a non-slipping laminar stream. Graphs are presented showing the reduction in gain due to d.c. slip in crossed-field interaction.