On the Use of Distributed Feedback for Stabilization of Intense Charged Particle Beams
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This paper discusses the stabilization of intense charged particle beams by treating beam dynamics as a distributed-parameter control problem rather than as the motion of a rigid body. It reviews limitations of passive media and lumped feedback systems, then formulates a beam surrounded by a controlling medium through the Vlasov and Maxwell equations with reflected fields defined by feedback operators. The authors argue that suppressing spectra of unstable coherent oscillations generally requires a spatially distributed medium with space-time dispersion, functioning simultaneously as sensor, converter, and actuator. They suggest constructive synthesis of such media, drawing attention to anisotropic conducting shells and microwave devices such as traveling-wave tubes and distributed parametric amplifiers as possible realizations.
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On the Use of Distributed Feedback for the Stabilization of Intense Charged Particle Beams
A. A. VOROB'EV, V. A. KOCHETUROV, A. S. OGORODNIKOV
(Presented by the scientific seminar of AVTA)
In the last decade, many works have appeared on the study of various unstable types of oscillations of intense beams of charged particles. The problem of suppressing instability is a very important one, and its solution will make it possible to sharply increase beam intensity.
Let us note that this problem is one aspect of the more general problem of the stable confinement of an intense beam on an equilibrium orbit. An attempt at a compromise solution of the problem of ensuring equilibrium and the problem of stabilization by choosing fields of a special configuration and a prescribed law of their variation in time leads to unjustifiably large energy expenditures (for example, dynamic stabilization of a plasma). The point is that equilibrium configurations satisfying the requirements of economy prove to be unstable, while stable configurations are economically disadvantageous. Thus, on the one hand it is necessary to seek economical methods for ensuring equilibrium beam motion, and on the other hand methods for suppressing instabilities.
In accelerators, systems of weak and strong focusing are widely used to ensure the equilibrium motion of particles. In [1] it is proposed, for this purpose, to impose forced high-frequency oscillations on the beam. It is noted that this method may prove effective.
Theory provides methods for suppressing unstable coherent modes of oscillation based, first, on the disruption of the coherence of the oscillations and, second, on the selection of appropriate boundary conditions for the motion of the beam. In practice, the first method is implemented by virtue of the nonlinear character of betatron oscillations or by means of the spread of particles in energy. The second method is implemented as a result of surrounding the beam with an appropriate medium.
The external medium may be either passive or active. In [2] it is shown theoretically that, when a beam moves in a waveguide with walls having an inductive impedance (for example, a waveguide loaded with a dielectric), longitudinal dissipative beam instabilities are suppressed. It is shown that complete suppression of the transverse dissipative instability is impossible, since the system is passive.
In [3] the possibility of suppressing a longitudinal instability of the “negative mass” type by means of
the beam with a conducting helix. The parameters of the helix may be chosen so that the instability will be suppressed during the entire acceleration cycle.
In works [4, 5] the interaction of a bunch with various kinds of dissipative systems— a resonator, plates loaded with wave resistances—was investigated theoretically and experimentally. It was established that, with an appropriate choice of the wave resistances, intense damping of coherent betatron oscillations was observed. In [4] it was also noted that the radial instability could be suppressed by introducing feedback, the input signal of which was taken from one of the ion-clearing plates and fed through an amplifier to special radial-deflecting plates. The phase of the feedback was regulated by means of a phase shifter.
In other works the effectiveness of using feedback systems for suppressing unstable coherent oscillations of the center of gravity of a beam (bunch) is also noted [6, 7]. The systems described consist of a discrete network of sensors, converting devices, actuators, and also channels for transmitting information between them. The proposed feedback systems control the beam dynamics in one degree of freedom (the beam is regarded as a rigid body) and, naturally, in certain important cases prove ineffective. Thus, in [7] it was noted that in the case of unstable oscillations of the transverse section of a beam, when there are no appreciable oscillations of the center of gravity, the use of such a feedback system proved ineffective. The use of a feedback system with lumped parameters to ensure the stability of a beam consisting of several bunches proves ineffective, since the number of unstable modes of oscillation increases. The need arises for pulsed action on different bunches with very short pulses and a high repetition rate (respectively, \(\sim 10\) nsec and \(\sim 20\) MHz for the ADONE storage ring) [7].
It is known that the motion of particles in accelerators in the self-consistent approximation is described by the Vlasov–Maxwell system of equations; in other words, a particle beam is an object with distributed parameters. In accordance with Ashby’s principle, the controller must be characterized by no less complex properties, i.e., in order to ensure high speed and resolving power (selective action on disturbances), it must also be distributed in space. Thus the controller is represented in the form of a controlling medium, which it is expedient to regard as a material supplement to the object, imparting to it the necessary dynamic properties and, first of all, stability. The controlling medium is represented as a sheath surrounding the beam, combining the functions of a spatially distributed sensor, an operator converter, and an actuator. The viewpoint on control systems for objects with distributed parameters as a supplement to the object was discussed in [8].
Let us consider how feedback is closed in the system for controlling beam dynamics [8]. As is known, in accelerators the states of a system of particles with charges \(e\) in the space of coordinates \(r\) and momenta \(p\) are described by the distribution function \(F = F(r, p, t)\), satisfying the kinetic Vlasov equation
\[ \frac{\partial F}{\partial t} + (v \cdot \nabla_r)F + e\{(E + \mu_0 v_x H)\cdot \nabla_p\}F = 0. \tag{1} \]
The total electromagnetic field \(E, H\) in the general case is the sum of the self-consistent field \(E_0, H_0\) due to the intrinsic currents and charges, the external field \(E_e, H_e\) caused by external sources, and the field \(E_c, H_c\) created by the currents and charges of the controlling medium:
\[ E=E_0+E_e+E_c;\quad H=H_0+H_e+H_c. \tag{2} \]
With respect to the controlling medium the field
\[ E_1=E_e+E_0;\quad H_1=H_e+H_0 \tag{3} \]
is incident, while the field \(E_c, H_c\) is reflected.
The self-consistent field \(E_0, H_0\) is expressed in terms of the distribution function by means of Maxwell’s equations:
\[ rot\,E_0=-\mu_0\frac{\partial H_0}{\partial t},\quad div\,H_0=0, \tag{4} \]
\[ rot\,H_0=\varepsilon_0\frac{\partial E_0}{\partial t}+e\int vF\,dp,\quad divE_0=\frac{e}{\varepsilon_0}\int F\,dp. \]
In the presence of the controlling medium the reflected field \(E_c, H_c\) is related by certain operator relations to the field \(E_1, H_1\) acting on the medium,
\[ E_c=\hat K_E(E_1,H_1), \]
\[ H_c=\hat K_H(E_1,H_1). \tag{5} \]
The system of equations (1)—(5) is closed. To single out a definite solution it is necessary to use boundary conditions imposed on the distribution function on the basis of physical considerations, as well as boundary conditions for the electromagnetic field.
The specific form of the operators \(\hat K_E,\ \hat K_H\) is determined as a result of an analysis of the dispersion equations obtained from equations (1)—(5) for the closed system beam—controlling medium. Thus, for example, in [10] it is shown that, in order to suppress unstable coherent transverse oscillations of a beam, it is sufficient to close upon the beam, according to the feedback principle, an automatic-control system with a transfer function corresponding to the simplest inertial element, with a time constant determined by the beam parameters and the geometry of the vacuum chamber. To suppress unstable oscillations consisting of a set of many harmonics, a feedback system is correspondingly required with time constants depending on the wave vector. This can be traced in the example of the transfer function of a system for suppressing a two-beam instability in a plasma, where the transfer function corresponds to the aggregate of two inertial elements with time constants depending on the wave vector [8, 10]. Thus, in the general case there arises the need to control the spectrum of unstable oscillations. Obviously, in principle this can be accomplished only with the aid of a controlling medium possessing space-time dispersion. The theory of such media, their classification, and examples of constructive synthesis are considered in [8].
In the general case the problem of synthesizing a controlling medium for a given operator is an inverse problem of the theory of field scattering, and its solution in analytic form does not appear possible.
It is proposed to approach the solution of this question from the standpoint of constructive synthesis, i.e., by considering such a physically (and technically) realizable model of the controlling medium that would provide a broad class of space-time feedback operators. A similar approach was developed in [8] for the case of liquid conductors and plasma. It is shown that anisotropy of the controlling medium in specified directions makes it possible to realize a broad class of spatial feedback operators. The temporal dispersion of the shell is provided by the complex conductivity of the medium.
Devices possessing such properties are known in microwave engineering: for example, traveling-wave tubes (TWTs), distributed parametric amplifiers, and others. It is of interest to consider these devices from the standpoint of their implementation of the above-mentioned class of space-time operators.
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