"JOURNAL OF RADIOELECTRONICS" N 1, 2002
Cyclotron and Synchronous Oscillations and Waves of the Electron Beam
Vladimir A. Vanke, e-mail: firstname.lastname@example.org
Received 05 January 2002
General relations of the cyclotron and synchronous oscillations and
waves of the electron beam are discussed in a form of short review.
III. Kinematics Analysis
VIII. Selected Bibliography
Development of modern microwave communication, radio and radar facilities places additional stringent requirements upon the different microwave devices.
During last several decades mainly in Russia active and successful work is being carried on to create new electron beam microwave devices based on cyclotron and synchronous waves use [1-16] and complying with the present-day requirements.
The operation of these devices is based on the principles of transverse grouping of the electron beam in the longitudinal magnetic field. In contrast to conventional longitudinal grouping of electrons into dense bunches, this principle employs the Lorenz force as an elastic force and leads to spatial distortion of the electric beam without electron bunches being formed. In this way it is possible to considerably overcome the fundamental restrictions, which are characteristic of longitudinal grouping devices (both vacuum and solid-state ones) and are associated with non-linear influence of the space charge fields upon the process of the input signal amplification, thereby laying the basis for developing new microwave devices with essentially improved characteristics [6,9,12-16].
Author’s experience in discussions of these principles both with domestic and foreign specialists has disclosed the expediency of the preparation of this article.
Consider the motion of electrons in crossed electric and magnetic fields having the form
Let us represent the beam as an aggregate of flat discs of infinitesimally small thickness, moving along the z-axis with constant velocity . If the transverse dimensions of the discs and its displacements from the unperturbed position are small in comparison with wavelength, then the coupling between the discs, due to the longitudinal space charge, is also insignificant and can be neglected. The problem is reduced to an analysis of the transverse oscillation of the individual discs .
Electron beam cross-section
of the disc be , where is the number of electrons in the disc. Then the orbits of the individual electron relative to the mass center of the disc are described by the coordinates . The equation of the transverse motion in terms of the variables assumes, in complex notation, the forms
where are the angular cyclotron frequency and the specific charge of the electron, respectively; , are the external electric forces acting on the electrons numbered by “i” and “k”.
Principal interest attaches to the last two equations. The first of them characterized the motion of the mass center of the cross section of the beam, and consequently, describes the behavior of the signal and of the noise of the beam as a whole. While the second is important in the investigations of the internal structure of the beam (expansion of the thermal orbits, their balancing, etc).
The analytical solution of the problem with Coulomb sum in Eqs. (2.2) and (2.4), entails considerable difficulties. Appreciable simplification can be obtained for a model of a round uniformly charged cylindrical beam, if one uses the approximation
(where is the plasma frequency),
which in a number of cases makes it possible to describe correctly the actual physical processes in the system [17,18].
It is important to emphasize that in the cases when the function depends linearly on the transverse coordinates and their derivations, or else entirely independent of them, the equation of motion (2.3) for the mass center of the discs coincides with the equation of motion for a single electron without allowance for the Coulomb interaction and given by
where is the external electric force acting on an electron placed at the mass center of the disc.
III. Kinematics Analysis
Let us introduce the transit angle, and then omitting subscripts, we can write the equation of motion of one electron
In the simplest case , i.e. when electric fields are absent, the solution has the form
Now let using the constant variation method, the solution will be sought in the form
Since instead of one variable, two new variables - are introduced, it is necessary to impose an additional condition to connect these variables. A suitable condition for this purpose is 
Synchronous (a) and cyclotron (b)
radiuses of an electron (-e) motion
Thus, the change in the cyclotron orbit radius and the position of the orbit center with respect to the origin are described separately, which provides an additional illustrative representation and clarity of physical interpretation of the processes involved in the interaction of the electron with external fields.
In the general case
However, the solution of the system (3.8) can be simplified significantly in two extreme cases:
1. Short lenses, when the transit angle inside the interaction region is small (Q<<1), and consequently, in the right-hand parts of the system of equations (3.8) one can take
2. Adiabatic action of the fields on the electron motion, when relative changes in a and b are small during the oscillation period. In this case it is possible to use the method of averaging by a rapidly oscillating phase, i.e. one can write
Calculating the integrals in the right-hand parts of these equations the values of a and b should be taken constant .
The solution obtained from the smoothed (averaged) equations (3.11) takes into account only effects of first order of smallness. In those cases when this does not suffice, it is necessary to apply to Eq. (3.1) the asymptotic methods of the theory of nonlinear oscillations .
For many issues of the theory of cyclotron-wave devices, the structure of the external forces is such that Eq. (3.1) can be written in a bit more convenient form
where , is the angular frequency of the resonant harmonic or subharmonic of the external force, is the relative difference between the natural frequency of the system and , the value of the parameter is chosen such that it characterized the intensity of the external forces.
The character of the action of the field on the beam is most frequently adiabatic in practice, i.e., the values of a and b change little under the influence of this field within the period of the fundamental motion. This is mathematically reflected in the smallness of the parameter . In this of a resonance adiabatic field a solution can be sought for the n-th approximation, unlike (3.3), in the form
where the functions take into account all arbitrary harmonics in , with the exception of the zero-th and the first (the latter requirement is essentially a supplement condition similar to (3.5)).
The abbreviated equations are sought in this case also in the form of an expansion in powers of the small parameter
, , (3.15)
The first approximation of the asymptotic method does not differ from averaging method . The second and succeeding approximations refine the first and make it possible to take into account successively in the solution  the small high frequency vibrations (harmonics of the signal and pump, combination frequencies, etc.), which appear against the background of the fundamental motion of the system .
of the Electron Beam
In the simplest case when external fields have the form (2.1), the equations of motion of an electron are given by
, , . (4.2)
Now let us assume z and t to be independent variables, thereby making a transition from the description of the motion of one electron to the description of the motion of a filamentary electron beam
Then (see Fig. 4.1)
Model of a filamentary electron beam
where is the electron propagation constant.
Let us represent the external transverse electrical field in the form of two oscillations with right and left circular polarizations
where is the cyclotron propagation constant.
Note that: since the functions depend on the variable only.
Therefore (4.9) can be written as
In the undisturbed case the solution has the form
Substituting this solution in (4.4) gives
Thus the motion of a filamentary electron beam can be described by the sum of four circularly polarized waves (transverse eigenmodes of the electron beam). Two waves with the amplitudes are cyclotron waves, they have the opposite polarizations and their phase velocities depend on the cyclotron frequency
The other two waves with the amplitudes are called synchronous waves, because their phase velocities are equal (synchronous) to the electron beam velocity
The phase velocity of the wave with the amplitude may be greater than the longitudinal velocity of the electron beam and even become infinite in the case of the cyclotron resonance (Fig. 4.2) Therefore this wave was given the name fast cyclotron wave of the electron beam. This wave can be either forward or backward, depending on the phase velocity direction.
The phase velocity of the wave with the amplitude is always smaller than the longitudinal velocity of the electron beam. Therefore this wave is called a slow cyclotron wave of the electron beam. The phase velocity direction of this wave is always positive, i.e. the wave is forward.
The synchronous waves differ by the directions of their circular polarizations only, however, by analogy with cyclotron waves they are often referred to as fast () and slow () synchronous waves of the electron beam.
Dispersion characteristics of transverse waves.
FCW – Fast Cyclotron Wave, SCW- Slow Cyclotron Wave,
SW – Fast and Slow Synchronous Waves.
To clarify the structure of transverse waves it is convenient to use the system of coordinates moving at the velocity (i.e. the electron motion velocity) along the -axis. To do so, it is sufficient to assume in (4.12) that .
Thus, the amplitudes of the synchronous waves (synchronous radius) describe the shift of the electron orbit centers from the -axis. The amplitudes of the cyclotron waves determine the radius of the electron rotations with the cyclotron frequency.
If we imagine the situations when in the electron beam only one kind of waves is excited consecutively, the beam configurations in all such cases will be represented by spirals twisted spatially around the - axis (Fig. 4.3). Besides, for cyclotron waves each point (an electron) of such a spiral takes part simultaneously in two motions, namely the rotation around the z-axis at the angular frequency and the movement along the - axis at the velocity . For synchronous waves there is no rotational motion, the electrons are spatially shifted with respect to each other, thereby forming a spiral and moving along the - axis only. In all cases the beam trace (the intersection point of the beam at the plane =const) travels around a circle at the angular frequency , with the rotation direction being determined by the type of polarization.
a) - Fast forward and backward Cyclotron Waves,
b) - Slow Cyclotron Wave,
c) - Synchronous Waves with right and left polarizations.
(In all cases the internal cylinder on which the electron beam
is wound serves for the illustrative purpose only).
In the presence of an external electric field , the solution of the initial system of equations (4.1),(4.2) may also be sought in the form of the sum of four transverse waves having alternating amplitudes
Since instead of one pair of variables , two pairs of new variables are introduced, it is necessary to impose some additional conditions to connect the members of this new pair of variables. It is convenient to use the following
In the two limiting cases that of a small transit angle and that of adiabatic action of electric fields on the electron beam the solution of this system of equations can be simplified significantly.
In the first case, in the right-hand parts of the system of equations one can assume that
In the latter case, one can average the right-hand part of equations by the period of cyclotron oscillations
Besides, when averaging, the wave amplitudes should be taken to be constant.
In a more general case, when
It is also possible to represent the initial equation of motion in the form
is the angular cyclotron frequency and the longitudinal component of the magnetic flux density at the system axis, respectively.
The solution can be readily found for the cases when the longitudinal velocity of the electron beam does not acquire modulation in time and dependents on the -coordinate only, i.e. .
Let us introduce - the electron and cyclotron propagation constants, respectively.
The solution is sought in the form of the sum of two waves with right and left circular polarizations
As before, the right-hand part of equation (4.28) is represented as
Using a procedure similar to that employed above, we get
In the presence of the external fields therefore the solution will be sought in the form
Under the additional conditions
we obtain the system of differential equations describing the behavior of transverse wave amplitudes
In this case
Notes: Applicability of the filamentary electron beam model.
The equations obtained are quite well applied for the electron beam of a finite cross-section, if the external field phase changes negligibly across the beam section
where is the electron beam radius, is the propagation constant of the wave of the electrodynamics system.
In practice, however, the requirement is quite sufficient for most issues.
If a filamentary electron beam with the linear charge density is placed in the transverse electric field , the power of energy exchange between the electric field and the electron beam element of the length dz is equal to
where , the sign * denotes the complex-conjugate value.
Besides interaction with the transverse electric field, it is necessary to take into account the interaction of the electron beam with a longitudinal electric field . To determine the value of , we use the quasi-stationary approximation  , for which near the -axis
, or (5.2)
, where . (5.3)
Accordingly, the power of energy exchange with the longitudinal field has the form
The expression for the transverse velocity can be given in the form
where means the transverse velocity of the electron beam trace in the plane
Summing up the powers of the longitudinal and transverse energy exchanges, we can write
Now, we can use the representations
, . (5.10)
Integrating (5.11) over from 0 to , and taking into account that , we find
The expression between the braces under sing of integral depends only on the coordinate , therefore the following substitution was used
Now, choosing the integration limits outside (before and after) the interaction region, i.e. where the transverse electric field is equal to zero , we have
Let us first consider the simplest case
Then assuming that the electron beam has no modulation at the input into the interaction region (i.e. ) we finally get
Thus, the period-averaged power of energy exchange between the electron beam transverse waves and the external electric fields is found equal to
where are the current and the potential of the electron beam, respectively.
The positive sign of the kinetic power means that the power is added into the electron beam at the excitation of a corresponding wave, and vice versa: the negative sign implies that when a wave is excited the power is extracted from the electron beam.
It is also possible to separate the transverse and longitudinal components of the kinetic power, which are due to interactions with transverse and longitudinal electric fields, respectively. For cyclotron waves the transverse power is associated with rotation of electrons and, hence
For synchronous waves the electron beam has no transverse velocity, and hence
Let us consider now a more general case when the static magnetic field varies along the - axis. In the paraxial approximation we can write
where is the longitudinal component of the magnetic flux density at the -axis.
In this case the equations of motion of an electron have the form
Accordingly, the equations for transverse wave amplitudes are written as
of the Injecting Electron Beam
To simplify the procedure of simulation, various configurations of ‘big charged particles’ are usually used to analyze and to solve different issues of vacuum microwave electronics since the middle of 60’s.
Transverse interaction we are discussing is a 3D interaction because of the physical principles of this one and the procedure of computer simulation must be also in a 3D form. Nevertheless, in some cases an essential simplification is possible.
Let us start from the equation of motion of a single electron having number “i”
Fields are calculated separately
where and are the electromagnetic fields induced by electron beam, and are the microwave fields and is the external focusing magnetostatic field, respectively.
Let us assume and to be independent variables, thereby making a transition from the description of the motion of one electron to the description of the motion of a filamentary electron beam. Then
Such filamentary beams can be used to describe the motion of a real beam of a finite cross-section.
** As an example, let us consider the case of the interaction of circularly polarized slow-wave having
and an electron beam of a finite and round cross-section injecting along and coaxially the axis of this wave ( i.e., the - axes, see Fig. 6.1).
We can choose the form of each elementary (filamentary) beam as a helix inside the injecting electron beam (Fig. 6.1). Let us choose the step of this helix as
where is the longitudinal velocity of the injecting electron beam, is the angular frequency of the beam rotation under the inner (radial) Coulomb forces.
Helix-type discretization of the injecting electron beam
For the uniformly charged electron beam
where is the space charge density of the injecting electron beam.
In the case we are discussing, the cross-point of each elementary (filamentary) beam at the plane is rotating at the angular frequency and each such beam has stationary (concerning the fields of this circularly polarized wave) boundary (at ) conditions and as a result of it
for any beam and everywhere inside the interaction region ( is the linear charge density of this filamentary beam).
The boundary condition of the uniform charged electron beam are given by
- radius of the injecting electron beam,
- number of elementary beams on each ring,
- number of rings,
- total number of elementary beams ( 50-150 usually ),
Using continuity equation
one can write for each elementary beam
where , are the space charge linear along z-axes density and the longitudinal velocity of the elementary beam, respectively.
And secondly: the simulation may be fulfilled at any fixed moment of time. Any changing in time is completely equal to the rotation of the system around z-axis (see Fig. 6.1). In all equations
for transverse coordinates and transverse velocities of any elementary beam.
It is important to emphasize that all restrictions on helix-form of each elementary beam are essential only for the injecting beam (). Inside the interaction region (z>0) the motion of each elementary beam can be absolutely arbitrary and the electron beam interaction includes both the transverse grouping and the longitudinal one.
** It is also important to emphasize that instead of (6.6), any type of electromagnetic fields rotating around -axes at the angular frequency can be also used here.
Coulomb fields of the electron beam can be found using approximate formulas (see Appendix)
is equal to 1/2 of the min distance between partial beams at .
** As an another example, let us consider the case when the electron beam having some initial rotation around -axes is injected to the region with axially-symmetric and space-variable magnetic (or electric) static field
where is the longitudinal component of the magnetic flux density at the -axis.
Boundary conditions in such case
Where are the radius and the phase of the initial rotation of the electron beam, i.e. of each electron of the beam ( at z = 0 and t = 0 ).
** We could also consider the case when the electron beam having some initial rotation around -axes is injected into the region with electromagnetic fields rotating around -axes at the angular frequency and so on.
In the simplest case transverse waves of the electron beam can be analyzed by comparing them with the space charge waves in order to reveal the potential advantages that could be realized when using transverse grouping in electron beams.
Let us outline some of them briefly:
· In contrast to space charge waves*, phase velocities of transverse waves are independent of the reduced value of the plasma frequency (which is not constant along and across the electron beam and is also dependent nonlinearly on the signal amplitude) and, consequently, it is now possible to make the synchronism of the transverse wave interaction with external fields incomparably more stable. For the same reason, devices based on interaction with transverse waves must have phase-frequency characteristics with a considerably improved linearity.
· No restrictions were imposed on the signal amplitude value when the equations for transverse waves were derived. On the other hand, for the space charge waves the restrictions of the modulation depth are essential. Hence, one can also expect a higher linearity of the amplitude characteristics in energy exchange with transverse waves of the electron beam.
· Transverse waves of the electron beam are circularly polarized, that provides an additional possibility to select waves taking part in energy exchange, thereby making it possible to improve the quality of interaction.
· At the point of cyclotron resonance () the phase velocity of the fast cyclotron wave becomes infinite in contrast to phase velocities of other waves. Therefore just with this wave in a plane transverse and uniform electrical field of the resonator it is possible to carry out an efficient prolonged (>>1) interaction to ensure a high efficiency of signal power transmission from the resonator to the wave and vice versa.
· A slow synchronous wave has a negative kinetic power, a phase velocity equal to the electron beam velocity and is not connected with excitation of transverse velocities of the beam electrons. The interaction with this wave must be more efficient than with the space charge slow wave and provide a single velocity character of the spent electron beam, i.e. it allows an extensive recovery of the spent beam energy. In such energy exchange of the running circularly polarized wave with the field the phase-frequency characteristics must have a higher linearity.
· Some additional information [12-14] is available via Internet :
*) The phase velocities of the fast (+) and slow (-) waves of the space charge are known [1,23] to be equal to: , respectively, where - is the reduced value of the plasma frequency in the electron beam.
 V.I.Yuriev et al, “Experimental Study of the Interaction of Synchronous Waves of the Electron Beam and Travelling Wave of the Electrodynamics System”, Radiotechnique & Electronics, 1972, vol. 17, No. 4, p. 830. (In Russian).
 A.A.Zaitzev, “On the Efficiency of the Interaction of Transverse Waves of the Electron Beam and Electromagnetic Fields”, Ph.D. thesis, Faculty of Physics, Moscow State University, 1979. (in Russian).
 V.I.Gorelikov, “The Efficiency of the Excitation and Conversion of Fast Cyclotron Waves in High Power Electron Beams”, Ph.D. thesis, Faculty of Physics, Moscow State University, 1984. (in Russian).
 A.V.Konnov, “The Interaction of Circularly Polarized Electromagnetic Slow Waves and Transverse Synchronous Wave of the Electron Beam”, Ph.D. thesis, Faculty of Physics, Moscow State University, 1988. (in Russian).
 S.V.Bykovsky, “Theory of the Energy Exchange Processes of Transverse Waves of the Electron Beam and the Development Basing on this Theory of Cyclotron Wave Protectors with Brief Recovery Time of Around 10 ns”, Ph.D. thesis, The Istok Corporation, Fryazino, Moscow region, 1996.
 Vanke V.A., “Microwave Electronics Based on Electron Beam Transverse Waves Using (State-of-the-Art and Perspectives. Russian Experience)”, Journal of Radioelectronics, 1999, No. 8: http://jre.cplire.ru/jre/aug99/1/text.html.
 Vanke V.A. et al, “High Power Converter of Microwaves into DC”, Journal of Radioelectronics, 1999, No. 9: http://jre.cplire.ru/jre/sep99/1/text.html.
 Vanke V.A. et al, “Cyclotron Wave Electrostatic Amplifier”, Journal of Radioelectronics, 1999, No. 10: http://jre.cplire.ru/jre/oct99/1/text.html.
 V.A.Vanke, “Microwave Electronics Based on Electron Beam Transverse Waves Using” (Invited), Technical Report of the Institute of Electronics, Information and Communication Engineers (Japan), ED99-247, 1999, p. 1.
 I.A.Boudzinski, S.V.Bykovski, “Amplifying and Protective Devices Based on Electron Beam Fast Cyclotron Wave”, Proc of the 2nd International Vacuum Electronics Conference, April 2-4, 2001, Noordwijk, The Netherlands, p. 153.
 V.A.Vanke, V.B.Magalinsky, “On the Role of Space Charge at the Amplification of Intrinsic Noise Orbits in High-Frequency Quadrupole Field”, Higher School Reports, ser. Radiophysics, 1966, vol. 9, No. 5, p. 831. (in Russian).
In many cases
where: , are the radius of the electron beam and the radiuses of curvature of the elementary (filamentary) beams, respectively.
In such cases we can simplify the task and calculate the electric field of a charged rod placing it along the tangent of the elementary beam.
Let us consider three coordinate systems (Fig. 9.1)
The - axes is the tangent (charged rod) of the elementary beam having the cross-point coordinate at the plane .
We can write for the electric field of such charged rod at the point
Where are the linear charge density of the rod and the dielectric constant, respectively.
A small add in (6.16) is a rational way to avoid mathematical singularities