A New Type of Microwave Amplifier

V. M. Lopukhin

1. Introduction

The amplification and generation of centimeter and decimeter radio waves are important problems of modern microwave radio engineering. The new type of microwave amplifier and generator described in the present article*), based on the interaction of parallel electron streams possessing different average velocities, is a known step forward in this important field.

In considering the operation of the device, we direct the reader’s attention not to the technical details of the tube design, but to the fundamental physical aspect of the problem.

In order to understand correctly the place of the new device among other devices serving the same purpose, let us briefly recall the difficulties that arise when one attempts to use ordinary circuits and radio tubes (for example, triodes) for the amplification of microwaves (for more detail see review \(^{1}\)). Ordinary oscillatory circuits containing lumped inductance \(L\), capacitance \(C\), and resistance \(R\) have, at microwaves, a poor quality factor, which is explained by the small value of \(\dfrac{L}{C}\), and also by the radiation of electromagnetic waves by the circuit.

An ordinary radio tube in a circuit with a grounded grid also begins to operate poorly at microwaves. Its operation begins to be adversely affected by the finite time of flight of the electrons between the electrodes—so-called transit-time effects arise. They lead to the appearance of conductivity in the cathode–grid space. This conductivity shunts the oscillatory circuit, thereby reducing the amplification coefficient.

The reasons indicated above force one to pass to circuits of a special “closed” type (so-called cavity resonators), which have a good quality factor, and to small-sized tubes in which the influence of transit-time effects is slight.

*) In what follows we shall call decimeter and centimeter radio waves microwaves.

Another direction in solving the problem of a microwave amplifier and generator is the traveling-wave tube, described in detail earlier[^1].

In this tube, the interaction is effected between an electron stream and a slowed electromagnetic wave of the electric type*), having a velocity close to the mean velocity of the electron stream. As the retarder of the electromagnetic wave, either a conducting helix[^2] or a system of endovibrators or conducting planes (“comb”)[^3] may be used (Figs. 1, 2, 3).

In the presence of an electron stream, the electromagnetic wave propagating along the system is split into three waves, one of which is close to the unperturbed wave of the retarding system without an electron stream, while the other two have a phase velocity somewhat smaller than the electron stream. One of these waves decays exponentially with the coordinate, and the other grows.

Fig. 1. A conducting helix and electron stream used in an ordinary traveling-wave tube; \(v_e\) is the electron velocity

The growing wave leads to amplification of the signal. Speaking in the language of analogies, one may say that the “electron wind,” moving faster than the wave, leads to an increase in its amplitude. The physical reason for the amplification lies in the fact that, when the electron beam moves with a velocity greater than the velocity of the exponentially growing wave, the electrons bunch in the decelerating portions of the field-strength wave (for more detail, see the review[^1]).

Fig. 2. A system of endovibrators of the “slot–aperture” type, slowing the electromagnetic wave, and an electron stream of velocity \(v_e\)

The appearance of growing solutions in the interaction of an electron stream and slowed electromagnetic waves of the electric type, close in velocity to the electron stream, is a general property of retarding systems of various types.

A traveling-wave tube used as a microwave amplifier has the following positive features[^1],[^4]:

*) A wave of the electric type, propagating along the \(x\)-axis, has \(E_x \ne 0,\ H_x = 0\).

V. M. LOPUKHIN

1. The duration of interaction of the electron beam with the wave, which leads to the effectiveness of such interaction and to a good value of the gain (in the tube described in ⁴, the power gain is 23 decibels).

2. The wide pass band of the device using a spiral (in the tube ⁴ the pass band is 800 MHz at a carrier frequency of 3600 MHz).

The disadvantages of electron-beam tubes with slow-wave systems are the following:

1. The most intense alternating electromagnetic field interacting with the electron beam is concentrated near the conducting surfaces of the slow-wave system. Therefore deposition of the beam electrons on these conductors is inevitable, which leads to a reduction in the efficiency of the device.

Fig. 3. System of conducting planes (“comb”) slowing an electromagnetic wave, and an electron beam with velocity \(v_e\).

Fig. 4. Negative resistance used in a new amplifier of centimeter waves. Two parallel electron beams possessing different velocities \(\xi_1\) and \(\xi_2\).

1. When, for example, a system of the type of Fig. 2 is used in an oscillation-generation regime, the frequency is determined by the resonant frequency of the endovibrators located at the periphery of the system. The \(Q\)-factor of these endovibrators, and consequently the \(Q\)-factor of the system as a whole, decreases in passing to shorter (millimeter) waves, since in this case the linear dimensions of the endovibrators decrease.

Consequently, with the aid of the systems shown in Figs. 2 and 3, it is hardly possible to generate millimeter radio waves with good efficiency.

A new type of electron-beam amplifier—a two-beam, or in the general case an \(n\)-beam, electron tube—does not have the indicated disadvantages. The idea of the design of a two-beam tube is extremely simple. Let two electron beams move in one and the same direction, possessing somewhat different velocities \(\xi_1\) and \(\xi_2\) (in the general case one should consider \(n\) beams or even a continuous distribution with respect to velocities) (Fig. 4). It turns out,

that within a known interval of velocity ratios \(\xi_1\) and \(\xi_2\) the system of two beams behaves like a negative resistance, so that if, for example, the slower electron stream is modulated in density or in velocity, this modulation will increase with the coordinate. A two-beam tube constructed on the basis of the use of this negative resistance\(^{12}\) gave a gain of 80 decibels in power at a frequency of 3000 Mc/s \((\lambda = 10\ \mathrm{cm})\) with a passband of 900 Mc/s.

The physical foundations of the operation of the tube, the theory of its operation, and its comparison with the traveling-wave tube will be given below. It should already be pointed out, however, that there is a certain analogy between a two-beam tube and an ordinary traveling-wave tube. The slower electron beam is velocity-modulated by the signal at the input to the system and is transformed into a beam modulated in charge density. This modulated beam creates along the tube an approximately periodic structure, performing the role of a slow-wave structure for the electromagnetic wave. The role of this periodic structure is analogous to the role of the helix or the “comb” in a traveling-wave tube. The faster electron stream thus moves in a periodic field, and physically the problem reduces to the interaction of this beam with a slowed electromagnetic wave, i.e., to a traveling-wave tube. Below another, more complete theory of the two-beam tube will be given.

2. THEORY OF THE OPERATION OF THE DEVICE

The explanation of the principle of operation and of the features of the operation of a two-beam (and, in the general case, an \(n\)-beam) tube is connected with the consideration of a very important and interesting problem concerning the interaction of many-electron systems. The problem of the interaction of electron streams, each of which consists of \(N\) electrons having in each stream a distribution both in velocities and in density, should be solved on the basis not only of electrodynamics but also of statistics. We shall regard the constant component of the electron charge \(\rho_e^-\) as compensated by the charge of the immobile ions \(\rho_e^+\). Then the problem of the two-beam amplifier is close to the problem of a plasma (an external electron stream, neutralized on the average by ions) excited by an electron stream (an internal, faster electron stream). The problem of the interaction of many-electron systems was considered in the works of A. A. Vlasov\(^{5,6,7}\).

In contrast to the methods of the gas-kinetic approach, based on the concept of pair “collisions” between electrons, A. A. Vlasov established that many-electron systems should be considered as a single ensemble of all \(N\) particles present, in which all \(N-1\) other particles act on a given particle, and the physical picture of the behavior of the system is in an essential way

is determined by interactions with remote electrons of the system (long-range interactions). The long-range interactions and the statistical character of the ensemble were taken into account by him in the joint solution of the continuity equation in the phase space of coordinates and velocities*) and the equations of electrodynamics.

Assuming that the deviations of all quantities from their mean values are small (linear approximation), the system of equations describing the distribution of plasma electrons is reduced to the form

\[ \frac{\partial f}{\partial t}+\xi \frac{\partial f}{\partial x}+\frac{e}{m}E\frac{\partial f_0}{\partial \xi}=0, \tag{1} \]

\[ \frac{\partial E}{\partial x}=4\pi e\int_{-\infty}^{+\infty} f(x,\xi)\,d\xi, \tag{2} \]

where \(f\) is the deviation of the distribution function of the plasma electrons from the equilibrium function \(f_0\), \(\xi\) is the projection of the velocity of the plasma electrons on the \(x\)-axis, in the direction of which the beam electrons move, \(E\) is the electric-field strength, and \(e\) and \(m\) are the charge and mass of the electron.

The deviations of all quantities from their equilibrium values are assumed to be small in comparison with their equilibrium values.

Equations (1) and (2), as well as all subsequent equations, are written under the assumption that the problem is one-dimensional.

In the linear approximation, the equations for \(f\) and \(E\) split into two independent systems containing vortical and potential fields. The joint solution of the kinetic equation and the equations of electrodynamics leads to essentially new results only for potential fields. Everywhere below, by the field \(E\) we shall mean the potential component of the total field.

We shall also regard the ions of the positive background, which compensates the mean electronic charge, as immobile.

To solve the system of equations (1) and (2), put:

\[ f=T(t)f(x,\xi), \tag{3} \]

\[ E=T(t)\cdot E(x). \tag{4} \]

Substituting (3) and (4) into (1) and (2), we obtain:

\[ \frac{dT/dt}{T} = -\frac{\xi\dfrac{\partial f}{\partial x}+\dfrac{e}{m}E\dfrac{\partial f_0}{\partial \xi}}{f} =\mathrm{const.}=i\omega, \tag{5} \]

where \(i\omega\) is the separation constant. The dependence on time has the form

\[ T(t)=T(0)\cdot e^{i\omega t}. \]

*) Below we shall conventionally call this equation the kinetic equation, although it does not contain a term taking collisions into account.

To determine \(f(x,\xi)\), we obtain the system of equations:

\[ i\omega f+\xi \frac{\partial f}{\partial x}+\frac{e}{m}E\cdot\frac{\partial f_0}{\partial \xi}=0, \tag{6} \]

\[ \frac{\partial E}{\partial x}=4\pi e \int_{-\infty}^{+\infty} f(x,\xi)\,d\xi . \tag{7} \]

We shall seek solutions of (6) and (7) in the form of a superposition of plane waves of the form

\[ f_k(\xi)e^{-ikx}\quad \text{and}\quad E_k\cdot e^{-ikx}, \tag{8,9} \]

where \(f_k(\xi)\) and \(E_k\) are the amplitudes corresponding to the wave number \(k\). Substituting (8) and (9) into (6) and (7), we have:

\[ i\omega f_k(\xi)-ik\xi f_k(\xi)+\frac{e}{m}E_k\frac{\partial f_0}{\partial \xi}=0, \tag{10} \]

\[ -ikE_k=4\pi e \int_{-\infty}^{+\infty} f_k(\xi)\,d\xi . \tag{11} \]

Eliminating from (10) and (11) the coefficients \(f_k\) and \(E_k\), we arrive at the dispersion equation, first obtained by A. A. Vlasov in 1938:

\[ 1=\frac{4\pi e^2}{mk^2}\int_{-\infty}^{+\infty} \frac{\dfrac{\partial f_0}{\partial \xi}\,d\xi}{\left(\xi-\dfrac{\omega}{k}\right)} . \tag{12} \]

Performing in (12) one integration by parts and using the condition

\[ f_0(\xi)\Big|_{-\infty}^{\infty}=0, \]

we rewrite (12) in the form

\[ 1=\frac{4\pi e^2}{mk^2}\int_{-\infty}^{+\infty} \frac{f_0(\xi)\,d\xi}{\left(\xi-\dfrac{\omega}{k}\right)^2}. \tag{13} \]

\(f_0(\xi)d\xi\) has the physical meaning of the number of electrons per unit volume \(N(\xi)\) whose velocities lie in the interval \((\xi,\xi+d\xi)\), so that

\[ f_0(\xi)d\xi=N(\xi)\quad \text{and}\quad \frac{4\pi e^2}{m}N(\xi)=\omega_0^2(\xi), \tag{14} \]

where \(\omega_0\) is the frequency of the natural oscillations of the plasma.

We shall not discuss here questions of the existence and meaning of the roots of the dispersion equation (13) in the case of arbitrary \(k\)

and \(f_0\), and restrict ourselves to considering such \(f_0\) and \(k\) for which the integrals in (13) and (12) are certainly convergent\({}^{9}\) and the legitimacy of using formulas (13) and (12) is not in doubt\({}^{8,9,10,11}\).*)

Let us turn to the problem of the motion of \(n\) parallel interacting electron beams having average drift velocities \(\xi_1,\xi_2,\ldots,\xi_n\) and, respectively, electron densities \(N_1,N_2,\ldots,N_n\). The number of electrons \(f\,d\xi\), whose velocities lie in the interval \((\xi,\xi+d\xi)\), may be approximated by the function\({}^{25,26}\)

\[ f_0 d\xi=\sum_{i=1}^{n} N_i \left(\frac{m}{2\pi kT_{ei}}\right)^{\frac12} e^{-\frac{m}{2kT_{ei}}(\xi-\xi_i)^2}\,d\xi, \tag{15} \]

which is a superposition of Maxwellian distributions with allowance for translational motions with velocities \(\xi_i\). Here \(T_{ei}\) is the temperature of the electrons in beam number \(i\). This temperature is close to the temperature of the filament emitting the electrons\({}^{26}\) of beam number \(i\), and is usually less than the electron temperature of the plasma. It is easy to see that \(f\,d\xi\) has a maximum whenever \(\xi\) approaches \(\xi_i\).

We shall make one more approximation: we shall neglect the distribution of electrons by velocities within the individual beams. In this case, instead of (15) one should put:

\[ f_0(\xi)\,d\xi=\sum_{i=1}^{n} N_i\,\delta(\xi-\xi_i)\,d\xi. \tag{16} \]

Using (13), (16) and the properties of the delta-function, we obtain:

\[ 1=\sum_{i=1}^{n}\frac{\omega_{0i}^{2}}{(k\xi_i-\omega)^2}. \tag{17} \]

Let us emphasize that (17) assumes the absence of velocity dispersion within each individual beam.

*) \(f_0(\xi)\) must decrease sufficiently rapidly as \(\xi\) increases (as \(f_0\) one may, for example, take the Maxwell distribution function or, which is a cruder approximation, the \(\delta\)-function—see below), and \(k\) must be sufficiently small. We shall assume that

\[ kD\ll 1, \]

where \(D\) is the Debye radius

\[ D=\sqrt{\frac{kT_e}{4\pi Ne^2}} \]

\((k=1.38\cdot10^{-16}\ \mathrm{erg/grad},\ N\) is the number of electrons per unit volume, \(T_e\) is the electron temperature). Under real plasma conditions \(T_e\) depends chiefly on the kind of gas, with other things being equal, on its pressure and on the discharge-current strength; \(T_e\) usually lies in the range from \(5\cdot10^3\) to \(10^5\ \mathrm{grad\ abs.}\) \(D\) is of the order of the mean distance between plasma electrons. The inequality \(kD\ll1\) is well satisfied in the problems that will be considered below.

Formula (17) was obtained by Gaev\(^ {12}\) in 1949 (i.e., 11 years after Vlasov) from more elementary, but less precise, considerations, without using the kinetic equation.

By passing from (17) to an integral, Gaev obtained for a continuous distribution of electron beams over velocities the following dispersion equation:

\[ 1=\int_{-\infty}^{+\infty}\frac{\dfrac{\partial \omega_0^2}{\partial \xi}\,d\xi}{(k\xi-\omega)^2}. \tag{18} \]

However, the exact hydrodynamic equation has the form:

\[ \frac{du}{dt}+(\mathbf{u}\,\operatorname{grad})\,\mathbf{u}+\operatorname{div} T=\frac{e}{m}\mathbf{E}, \]

where \(\mathbf{u}\) is the electron velocity, \(\mathbf{E}\) is the electric-field strength, and \(T\) is the stress tensor, which includes allowance for the thermal spread of electrons over velocities, i.e. a qualitatively new effect. In the equations used by Gaev, the term \(\operatorname{div} T\) is omitted; i.e., the influence of the thermal spread of electron velocities in (17) is not taken into account. The same also applies to (18), if the limiting transition from (17) to (18) is carried out correctly. (This transition was made by Gaev without any justification.) Therefore formula (18) should at best be regarded as unjustified, and when considering cases with a continuous distribution of electrons over velocities it is necessary to proceed from (13). This nontrivial question is considered in the forthcoming book by A. A. Vlasov, Theory of Many Particles.\(^*\)

For the purpose of applying the theory to a two-beam tube, let us consider the case \(n=2\). Then (17) takes the form

\[ 1=\frac{\omega_1^2}{(k\xi_1-\omega)^2}+\frac{\omega_2^2}{(k\xi_2-\omega)^2}. \tag{19} \]

(19) is the dispersion equation of a two-beam system, which determines \(k(\omega)\). Let the roots of this equation be complex, \(k(\omega)=\operatorname{Re} k+i\,\operatorname{Im} k\). Then the solution for which \(\operatorname{Im} k>0\) will grow with the coordinate, since all quantities are proportional to \(e^{-ikx}\). In the case of a homogeneous electron beam, when \(\omega_2=0\), (19) assumes the form

\[ (\omega-k\xi_1)^2=\omega_1^2. \tag{20} \]

The solution of (20) gives the well-known expression for the propagation constants of waves in a single-beam system

\[ k_{1}=\frac{\omega}{\xi_1}\pm\frac{\omega_1}{\xi_1}. \tag{21} \]

\(^*\) The author of the present article was kindly informed by A. A. Vlasov of the unjustified nature of Gaev’s integral formula.

Expression (21) shows that two electromagnetic waves can propagate along the beam; the velocity of one of them is somewhat greater, and the velocity of the other is less than the velocity of the electron stream $\xi_1$. We note that for $\omega_1 < \omega$ both of these waves will propagate in one and the same direction, while for $\omega_1 > \omega$ one of the waves will propagate in the negative direction (toward the electrons of the beam). These waves have constant amplitudes; growing solutions are absent. In the presence of two beams, the dispersion equation (17) will be of the fourth degree. Generally speaking, it has four distinct roots for $k(\omega)$. This means that four waves will propagate in a system of two beams, as indeed should be the case in a system of two distributed interacting systems.

The situation here is analogous to that which occurs in a waveguide or in some slow-wave system with an electron stream. The electron stream causes the forward wave to split into three waves; the backward wave does not split.

Let us note that (19) can also be regarded as an equation with respect to $\omega$. In this case complex values of $\omega$ with $\operatorname{Im}\omega < 0$ would mean growth of the process in time according to an exponential law (all quantities are proportional to $e^{i\omega t}$). The solution of (19) with respect to $\omega$ is meaningful only in the case when $k$ is specified, i.e. when a spatial inhomogeneity of the field is specified. This can be achieved, for example, by imposing some periodic boundary conditions on the system under study. An analogous problem arises in the study of the natural frequencies of a multisegment magnetron,^14 for which the field at the anode has a definite periodicity connected with the presence of slots. In the presence of an electron stream, the dispersion equation of the magnetron will also be of the fourth degree with respect to $\omega$. The regions where $\operatorname{Im}\omega < 0$ correspond to the excitation regions of the magnetron, i.e. to the regions where generation begins. Thus we see that the problem of amplification is connected with the problem of generation. In both problems the solutions have the form $e^{i(\omega t-kx)}$, and in the amplification problem $\operatorname{Im}\omega = 0$ and $\operatorname{Im}k > 0$, while in the generation problem $\operatorname{Im}k = 0$ and $\operatorname{Im}\omega < 0$. Both amplification and generation constitute a certain excitation of the system, in which the kinetic energy of the electron beam is transformed into the energy of the electromagnetic field. From signal amplification one may pass to generation by introducing a connection between the input and output of the amplifying system (feedback). To solve (19) in the case $\xi_1 \ne \xi_2$ we shall assume:

\[ \xi_1=\xi+\delta,\qquad \xi_2=\xi-\delta, \]

where $\xi$ is the mean velocity of the electrons of the beam; $2\delta$ is the difference of the velocities of the two beams, i.e.

\[ 2\delta=\xi_1-\xi_2. \]

Putting

$$ k=\frac{\omega}{\xi}+\gamma, $$

we obtain

$$ \left. \begin{aligned} \omega-k\xi_1&=-\frac{\delta}{\xi}\,\omega-\gamma\xi,\\ \omega-k\xi_2&=\frac{\delta}{\xi}\,\omega-\gamma\xi. \end{aligned} \right\} \tag{22} $$

With the aid of (22), we write (19) in the form

$$ 1=\frac{1}{\left(\dfrac{\delta\omega}{\xi\omega_1}+\dfrac{\gamma\xi}{\omega_1}\right)^2} +\frac{1}{\left(\dfrac{\delta\omega}{\xi\omega_1}-\dfrac{\gamma\xi}{\omega_1}\right)^2}. \tag{23} $$

It is easy to show that (23) has the following solutions:

$$ \frac{\xi}{\omega_1}\cdot\gamma = \pm\sqrt{\chi^2+1\pm\sqrt{4\chi^2+1}}. \tag{24} $$

The dimensionless factor \(\chi=\dfrac{\delta\omega}{\xi\omega_1}\) is connected with the “degree of inhomogeneity” of the system of two electron streams.

Fig. 5. Real and imaginary components of the quantity \(\dfrac{\xi}{\omega_1}\gamma\).

Let \(\gamma=\operatorname{Re}\gamma+i\operatorname{Im}\gamma\), where \(\operatorname{Re}\gamma\) and \(\operatorname{Im}\gamma\) are, respectively, the real and imaginary components of \(\gamma\). The quantity \(\gamma\dfrac{\xi}{\omega_1}\) as a function of the parameter \(\chi\) is presented in Fig. 5.

The imaginary component \(\operatorname{Im}\gamma\) differs from zero only in a limited interval of values of the “inhomogeneity factor” \(\chi\), namely, for \(0<\chi<\sqrt{2}\). Only for values of \(\chi\) lying in the indicated interval are growing solutions possible (for \(\operatorname{Im}\gamma>0\))

and, consequently, amplification of the signal is possible. The maximum value of \(\operatorname{Im}\gamma \cdot \dfrac{\xi}{\omega_1}\) occurs at \(\varkappa=\dfrac{\sqrt{3}}{2}\) and is equal to 0.5.

The real components of the propagation constant \(\operatorname{Re}\gamma\) increase together with \(\varkappa\) for \(0<\varkappa<\sqrt{2}\). When \(\varkappa>\sqrt{2}\), the amplification effect is absent.

Let us note that the curves in Fig. 5 give the variation of the real and imaginary components of \(\gamma\) as functions of the frequency \(\omega\), since \(\varkappa\) is proportional to \(\omega\). The ratio of the square of the signal amplitude in the cross section \(x\) to the square of the signal amplitude in the cross section \(x=0\) gives the signal power gain over the length \(x\).

For large values of \(\operatorname{Im}\gamma \cdot x\), the gain increases exponentially with the length \(x\).

3. DESCRIPTION OF THE EXPERIMENTAL TUBES. EXPERIMENTS\(^{12,15}\). COMPARISON OF THEORY AND EXPERIMENT

Figure 6 shows a two-beam tube which uses an electron stream nonuniform in velocities for the amplification of microwaves. The sources of electrons are the cathodes \(K_1\) and \(K_2\), placed near the end of the tube which we shall conventionally call its “input.” Behind the cathode are placed focusing and accelerating electrodes; behind them is placed a long metallic cylinder, usually called the drift cylinder. At the opposite end of the tube there is an electron collector. The dis-

Fig. 6. Arrangement of the two-beam tube: \(K_1\) and \(K_2\)—cathodes of the tube, \(A\)—input of the tube, \(B\)—drift cylinder, \(C\)—focusing solenoid, \(D\)—electron streams, \(E\)—output of the tube, \(F\)—collector, \(V_{K1}, V_{K2}, V_B, V_F\)—potentials of the cathodes, drift cylinder, and collector.

A New Type of Microwave Amplifier

The scattering of the electron stream is partially reduced by a constant magnetic field directed along the axis of the beam. In Fig. 6 are shown the coils producing this magnetic field. Below we shall apply the theory set forth in § 2 to a two-beam tube. In doing so the following should be kept in mind:

1. The constant magnetic field focusing the electron stream is not fully equivalent to a positive ion background; therefore it cannot be asserted that the plasma equations used in the theory describe exactly the processes in multibeam tubes.

2. The equation investigated above, (19), assumes monochromaticity of the electron streams with respect to velocities, which is only a very crude approximation to reality.

3. The whole theory is constructed in the linear approximation, i.e. it is applicable, strictly speaking, only for sufficiently small values of the alternating components of the current densities and of the potentials induced by them. The theory should correctly give the ranges of electron velocities where amplification begins, but it becomes inapplicable for large amplitudes of the amplified signals.

4. The theory is applicable only to the one-dimensional problem, when

\[ \frac{\partial}{\partial y}=\frac{\partial}{\partial z}=0 \quad\text{and only}\quad \frac{\partial}{\partial x}\ne 0. \]

1. The theory is constructed under the assumption of mixed streams, whereas in reality the fast and slow electron streams are spatially separated.

Taking the above into account, one should hardly demand of the theory that it give good quantitative agreement with experiment. The signal is fed to the input of the tube by means of a cable ending in a short helix, along the axis of which the electron stream passes. This signal produces the initial perturbations in the electron stream, which subsequently grow exponentially with the length of the tube and then induce an emf at the output of the tube, constructed analogously to its input.

In order to obtain maximum amplification, the potentials on the electrodes and the frequency ratio \(\frac{\omega}{\omega_1}\) were chosen so that the inhomogeneity factor \(\varkappa=\frac{\sqrt{3}}{2}\). Since this factor includes the resonant plasma frequency \(\omega_1\), the optimum conditions for amplification depend on the current density in the beam. Taking into account that \(\max \operatorname{Im}\gamma \frac{\xi}{\omega_1}=0.5\), we can write the maximum amplification in amplitude, given by a two-beam tube, in the form

\[ G_{\text{decibels}} = 20\lg e^{\operatorname{Im}\gamma\cdot x} = 1330\,x \sqrt{ \frac{I}{V^{3/2}} \frac{cm\, (a/cm^2)^{1/2}}{b^{5/4}} }. \tag{25} \]

Fig. 7 presents the power gain of the device, in decibels, as a function of the potential of the drift tube. This graph was obtained theoretically with the aid of the graph in Fig. 5. The effect of interference of the component waves shows itself in the change of gain as the potential of the drift tube is increased.

Fig. 8 gives the experimental curve of the output voltage as a function of the potential of the drift tube. It is easy to see that the experimental curve very closely resembles the theoretical one.

Fig. 7. Theoretical gain curve of the two-beam tube \(G\) as a function of the potential of the drift tube \(V_B\), measured in units \(\left(\dfrac{\delta\omega}{\xi\omega_1}\right)^{-4}\); \(A\)—noise curve, \(B\)—noise + signal.

In exactly the same way, the experimentally obtained dependence of the gain on the potential difference of the electrodes (Fig. 9) very closely resembles the curve
\(\dfrac{\xi}{\omega_1}|\operatorname{Im} k|\), plotted as a function of \(\chi\) (see Fig. 5).

The gain was measured as follows. The signal was fed to the voltmeter through an isolating device*) and an amplifier, first with the two-beam tube present in the circuit and then with it absent. The ratio of the signals in the first and second experiments gave the amplitude gain coefficient.

As can be seen from Fig. 9, for some values of the electrode potential difference the gain coefficient of the system is less

*) An isolating device is a device that separates the generator from the load and reduces the reaction of the load on the generator. At centimeter wavelengths the role of an isolating device can be performed by a section of cable containing a dielectric.

Fig. 8. Experimental curve of the output voltage (in microvolts) of a two-beam tube as a function of the drift-tube potential \(V_B\). Beam currents \(I_1\) and \(I_2\) are respectively \(4.5\ \mathrm{mA}\) and \(9.5\ \mathrm{mA}\); beam potential difference \(V_1 - V_2 = 50\ \mathrm{V}\); \(A\)—noise curve, \(B\)—noise + signal.

Fig. 9. Experimental curve of the power gain \(G\) of a two-beam tube as a function of the difference of the cathode potentials \(V_1 - V_2\). Total beam current \(I_1 + I_2 = 15\ \mathrm{mA}\), frequency \(f = 3000\ \mathrm{MHz}\) (which corresponds to \(\lambda = 10\ \mathrm{cm}\)).

units. The reason for this is the absence of matching at the input and output of the system.

Figure 10 shows the relative bandwidth of the tube passband*) as a function of the gain in decibels.

It is clear from the figure that, up to gains of the order of 100 decibels, the relative bandwidth of the passband is still greater than 30%.

Thus, we see that the two-beam tube gives good amplification (\(\sim 80\) decibels) and has a very large passband width. The latter circumstance is connected with the absence in the tube of any resonant systems limiting the range of amplified frequencies.

Fig. 10. Relative width of the passband \(\dfrac{\Delta \omega}{\omega}\) as a function of the gain \(G\) in decibels.

In the same article\(^{12}\) a “single-beam” tube is described (Fig. 11), the action of which is based on the interaction of parallel electron streams having slightly different velocities. This difference in electron velocities appears owing to the dependence of the potential \(V(r)\) of the electron beam on \(r\), where \(r\) is the distance from the axis of the system. The input and output of the amplifier are made in the form of cavity resonators (end vibrators). The coupling of these resonators with external circuits, necessary for supplying and, correspondingly, withdrawing power, is carried out by a loop (magnetic coupling).

*) The passband of the tube is determined in the usual way as the difference of the frequency values at two points at which the gain is equal to 3 decibels (i.e. the amplitude is two times smaller than the maximum corresponding to resonance).

This tube has approximately the same characteristics (with respect to amplification factor and passband width) as the two-beam tube described above.

We shall not dwell in detail on a number of works by other authors on multibeam tubes15–18, 27, the results of which are approximately the same. Let us only note that in works17, 18 an attempt was made to take into account the influence of the radius of the electron flow. The work is carried out by the method of matching the complex impedances at the boundary of beams having different velocities. This is a known step forward in the development of the theory.

Fig. 11. Single-beam tube. \(A\)—tube input, \(K\)—cathode, \(B\)—drift tube, \(C\)—focusing coil, \(D\)—output device, \(E\)—anode, \(F\)—collector, \(V_E, V_B, V_F\)—potentials of the anode, drift tube, and collector.

4. EXCITATION OF PLASMA BY AN ELECTRON FLOW

Let us now consider the theory of a two-beam tube, based on the approximation of such a tube by a plasma penetrated by an electron flow. The role of the plasma is played by the slower electron flow, whose mean charge is compensated by the charge of ions; the role of the exciting electron flow is played by the faster electron flow. For studying the excitation of waves in a plasma by an electron flow, it is convenient to start from the hydrodynamic approximation of the plasma, which was also proposed by A. A. Vlasov1.

Vlasov showed that, in a certain approximation,* the plasma may be described hydrodynamically, regarding it as a certain charged fluid characterized by elasticity under compression.

* The wavelength \(\Delta\) of the inhomogeneity in the plasma must be much greater than

\[ D=\sqrt{\frac{kT}{4\pi N e^2}}, \]

i.e. \(kD \ll 1\), where \(k=\frac{2\pi}{\Delta}\), which is the case under the real conditions of a two-beam tube.

The equations describing a plasma penetrated by an electron stream have, in the linear approximation, the form:

\[ \frac{\partial^2 \varphi}{\partial x^2}=-\frac{1}{\eta_0}\frac{\partial \sigma}{\partial t}, \tag{26} \]

\[ \frac{\partial^2 \Phi}{\partial x^2}=-4\pi \rho_1-4\pi \rho_2, \tag{27} \]

\[ \frac{\partial \sigma}{\partial t}+\frac{e}{m}\eta_1+\frac{v_0^2}{\eta_0}\sigma=0, \tag{28} \]

\[ \rho_e\frac{\partial v_2}{\partial x}+v_e\frac{\partial \rho_2}{\partial x} =-\frac{\partial}{\partial t}\rho_2, \tag{29} \]

\[ \frac{d}{dt}v_2=\frac{e}{m}E=-\frac{e}{m}\frac{\partial \Phi}{\partial x}, \tag{30} \]

where \(\varphi\) is the velocity potential of the plasma charges, so that the velocity of motion of the plasma charges is \(\mathbf{v}_1=\operatorname{grad}\varphi\) \((v_{1x}=\frac{\partial \varphi}{\partial x}=v_1)\); \(\Phi\) is the electric potential, so that the potential component of the field is \(\mathbf{E}=-\operatorname{grad}\Phi\); \(\eta\) is the material density of the medium (plasma); \(\eta_0\) is the zero approximation of the density of the medium; \(\sigma=\eta-\eta_0\); \(v_0\) is the propagation velocity of acoustic oscillations in the plasma, related to the temperature \(T_e\) of the plasma electrons by the relation

\[ v_0^2=\frac{3kT_e}{m} \]

\((k=1.36\cdot10^{-16}\ \text{erg/degree})\)—thus, \(v_0\) coincides with the mean thermal velocity of motion of the plasma electrons; \(\rho_1\) is the charge density associated with the inhomogeneities of the plasma; \(\rho_e+\rho_2\) is the charge density of the electron stream penetrating the plasma; \(\rho_e\) is the constant component of the density of the electron stream penetrating the plasma; \(v_e+v_2\) is the velocity of the stream electrons; \(v_e\) is the constant component of the velocity of the stream electrons.

Equations (26), (27), (28) are, respectively, the equations of continuity, Poisson, and Bernoulli–Euler for the plasma. Equations (29) and (30) are the equations of continuity and motion for the beam electrons.

In equations (26), (27), (28) terms of second order of smallness \((\sim \sigma^2,\ (\operatorname{grad}\varphi)^2,\ \operatorname{grad}\varphi\cdot\operatorname{grad}\sigma)\) have been discarded; in equation (29) the term \(\rho_2 v_2\) has been discarded.

In equations (26)—(30) it is assumed that the constant components of the charge density of the electron stream \(\rho_e\) and of the plasma electrons \(\rho_0\) are compensated by a positive background, while the variable components \(\rho_1\ll\rho_0\), \(\rho_2\ll\rho_e\), \(v_2\ll v_e\) (linear approximation).

Noting that \(\rho_1=\dfrac{e}{m}\sigma;\ \rho_0=\dfrac{e}{m}\eta_0\), we rewrite (26)—(30) in the following form:

\[ \frac{\partial^2\varphi}{\partial x^2}+\frac{1}{\rho_0}\frac{\partial}{\partial t}\rho_1=0, \tag{31} \]

\[ \frac{\partial^2\Phi}{\partial x^2}+4\pi\rho_1+4\pi\rho_2=0, \tag{32} \]

\[ \frac{\partial\varphi}{\partial t}+\frac{e}{m}\Phi+v_0^2\frac{\rho_1}{\rho_0}=0, \tag{33} \]

\[ \rho_e\frac{\partial v_2}{\partial x}+v_e\frac{\partial\rho_2}{\partial x}=-\frac{\partial}{\partial t}\rho_2, \tag{34} \]

\[ \frac{d}{dt}v_e=-\frac{e}{m}\frac{\partial\Phi}{\partial x}. \tag{35} \]

Assuming solutions of (31)—(35) in the form \(\sim e^{i(\omega t-kx)}\) and noting that

\[ \frac{d}{dt}=\frac{\partial}{\partial t}+\frac{dx}{dt}\cdot\frac{\partial}{\partial x}\sim \frac{\partial}{\partial t}+v_e\frac{\partial}{\partial x}, \tag{36} \]

we have:

\[ -k^2\varphi+\frac{i\omega}{\rho_0}\rho_1=0, \tag{37} \]

\[ -k^2\Phi+4\pi\rho_1+4\pi\rho_2=0, \tag{38} \]

\[ i\omega\varphi+\frac{e}{m}\Phi+v_0^2\frac{\rho_1}{\rho_0}=0, \tag{39} \]

\[ -ik\rho_e v_2-ikv_e\rho_2+i\omega\rho_2=0, \tag{40} \]

\[ i\omega v_2-ikv_2\cdot v_e-\frac{e}{m}ik\Phi=0. \tag{41} \]

(37)—(41) constitute a system of five homogeneous equations with respect to the variables \(\varphi,\Phi,\rho_1,\rho_2,v_2\).

Equating the determinant of this system to zero, we obtain the dispersion equation\({}^{19}\), relating \(\omega\) and \(k\):

\[ \left(1-\frac{\Omega_0^2}{(\omega-kv_e)^2}\right)\left(\omega^2-k^2v_0^2\right)=\omega_0^2. \tag{42} \]

Here

\[ \Omega_0^2=\frac{4\pi e\rho_e}{m};\qquad \omega_0^2=\frac{4\pi e\rho_0}{m}; \]

\(\Omega_0\) is the resonant frequency of a plasma having the same constant density \(\rho_e\) as the electron beam; \(\omega_0\) is the resonant frequency of a plasma; \(\rho_0\) is the constant component of the density of the electron charge of the plasma.

V. M. Lopukhin

Let \(\Omega_0=0\); then (42) takes the form

\[ v_0^2 k^2=\omega^2-\omega_0^2, \tag{43} \]

which corresponds to two waves that can propagate in the plasma in the absence of an electron current:

\[ k=\pm \frac{1}{v_0}\sqrt{\omega^2-\omega_0^2}; \tag{44} \]

the phase velocity of these waves is

\[ v_{\phi}=\frac{\omega}{|k|}=\frac{v_0}{\sqrt{1-\omega_0^2/\omega^2}}>v_0 . \tag{45} \]

In the presence of an electron current \((\Omega_0\ne 0)\), the dispersion equation (42) will, generally speaking, be of the fourth degree with respect to \(k\), and thus in the general case will give four waves.

Fig. 12. Theoretical curve of \(\dfrac{v_e}{\omega_1}\operatorname{Im} k\) as a function of \(x=\dfrac{v_e}{v_0}\). \(v_e\) is the mean velocity of the electron current, \(v_0\) is the velocity of the acoustic wave in the plasma.

Solutions for \(k\) characterized by \(\operatorname{Im} k>0\) lead to solutions increasing with the coordinate.

Let us consider the following specific problem:

\[ \omega_0=\Omega_0=\frac{\omega}{\sqrt{2}} \]

(conditions close to those indicated occurred in the two-beam tube described above\({}^{12}\)). Then the complex roots of equation (42) occur for \(1.3v_0<v_e<4.2v_0\), i.e., only in a certain interval of values of the electron velocity \(v_e\).

The quantity $\left|\operatorname{Im} k\right| \dfrac{v_e}{\omega_0}$ is shown in Fig. 12 as a function of $\chi = \dfrac{v_e}{v_0}$. The character of the variation of this quantity, as well as its maximum value, is very reminiscent of $\left|\operatorname{Im} k\right| \dfrac{\xi}{\varphi_1}$ in Fig. 5.

In the case of a two-beam tube, $v_e$ should be understood as the difference between the velocities of the electron streams. The maximum value that $v_e$ assumes in a two-beam tube is of the order of 100 V (in units of potential).

The weak point of this calculation is the well-known uncertainty in the thermal velocity $v_0$ of the electrons (equal to the velocity of propagation of acoustic waves in the plasma).

If this velocity is calculated from the condition $\chi = 2.5 = \dfrac{v_e}{v_0}$, at which $\dfrac{v_e}{\omega_0} \left|\operatorname{Im} k\right|$ (see Fig. 12) reaches a maximum, and $v_e$ is calculated from the condition $\dfrac{\delta \omega}{v_e \omega_0} = 0.7$, at which $\left|\omega_0 \operatorname{Im} k\right|$ (see Fig. 5) reaches a maximum, then for $v_0$ one obtains extremely large values corresponding to $T_e \sim 10^5$ degrees.

These difficulties indicate that the description of a two-beam tube as a plasma penetrated by an electron stream is very approximate and can convey only the general character of the processes.

Independently of the application to the theory of the two-beam tube, equation (42) shows that, with the aid of a plasma penetrated by an electron stream, microwaves can be generated, contrary to the conclusion sometimes drawn that it is impossible to generate centimeter and millimeter radio waves by means of a plasma.^20 In making this conclusion it is assumed that the period of plasma oscillations $T \sim N_0^{-1/2}$ ($N_0$ is the density of the plasma electrons), while the time between collisions of electrons with positive ions is $T \sim N_0^{-1}$; therefore, with an increase of $N_0$, necessary for the transition to centimeter and millimeter radio waves, the role of collisions of electrons with positive ions increases strongly. These collisions remove the electrons from the oscillatory process. As Akhiezer and Feinberg^19 note, considerations of this kind are inapplicable to the case of excitation of a plasma by an electron stream, since in this problem the physical conditions are different from those in the ordinary oscillations of electrons in the plasma of a gas discharge, which were discussed above.^20 In the problem of excitation of a plasma by an electron stream, besides the plasma, there is an electron stream penetrating it and moving with a velocity greater than the thermal velocity of the plasma electrons. In this case the generated frequency is determined not only

by the density of the plasma electrons, but also by the ratio of the velocities \(v_e\) and \(v_0\).

For other ratios of the parameters \(\omega_0\) and \(\Phi_0\), the range of values of \(v_e\) in which excitation is possible will change; however, the appearance of increasing solutions (plasma excitation) always occurs for \(v_e > v_0\). This reveals the analogy between plasma excitation and the Cherenkov effect, as well as excitation by means of electron flows of slowing-down systems of the helix type (Fig. 1), “comb” systems (Fig. 3), etc.

In the Cherenkov effect\({}^{24}\), radiation also begins when the electron velocity \(v_e\) is greater than the phase velocity of light in the medium \(v_0\).

However, in contrast to the plasma case considered above, this radiation takes place up to electron velocities \(v_e \sim c\) \((c = 3 \cdot 10^{10}\ \text{cm/sec})\).

In the excitation of helices, “combs,” and other slowing-down systems, the optimum conditions for excitation occur for \(v_e \simeq v_0 < c\) (where \(v_0\) is the phase velocity of the slowed wave of electrical type propagating along the system in the absence of an electron flow).

The boundaries of the excitation region usually have the form \(v_1 < v_e < v_2\), where \(v_1\) and \(v_2\) are determined by the specific conditions of the problem\({}^{1–4, 21, 22}\).

Let us note that, with the aid of the theory of multibeam tubes, one can explain the emission of microradio waves by the Sun. In this connection two points of view are possible. First, one may assume that the atmosphere of the Sun, representing a plasma, is permeated and excited by radial electron streams. This point of view was adopted by Shklovskii\({}^{23}\) in his 1946 theory. This theory, based on Vlasov’s work, leads to satisfactory agreement with experiment. Second, one may regard the electron streams of the upper parts of the Sun’s atmosphere as a multibeam tube, since these electron streams possess a velocity dispersion. A similar point of view was held by Gaev\({}^{13}\) in 1949. However, many of the calculations presented by him are not justified, since he proceeded from the dispersion equation (18) (see above).

CONCLUSION

Thus, the multibeam tube is a new amplifier and generator of microradio waves. Existing tubes have a power gain of \(\sim 80\) decibels and an extremely large pass band \(\left(\dfrac{\Delta \omega}{\omega} = 0.3\right.\) for \(\omega = 2\pi \cdot 3 \cdot 10^9\), which corresponds to \(\left.\Delta \omega = 900 \cdot 2\pi\ \text{MHz}\right)\). These characteristics of the multibeam tube are close to the operating characteristics of ordinary traveling-wave tubes,

which are based on the interaction of a slowed electromagnetic wave and an electron beam.

A disadvantage of the latter type of tubes is the loss of electrons when they settle on the conductors of the slowing devices (helix, chain of end vibrators) and the consequent heating of the conductors. These losses should increase especially strongly at millimeter radio waves. The multibeam tube does not have this type of loss; therefore it may be assumed that this tube will prove effective in amplifying and generating millimeter radio waves.

On the other hand, the weak point of multibeam tubes should be noise. With an increase in the number of interacting electron streams, the role of density fluctuations in these streams should increase.

From the brief survey of existing theories of multibeam tubes given above it is clear that there is at present no complete and finished theory of these tubes; all existing theories are only certain approximations to reality. At the same time it is clear that the multibeam tube is a very interesting device; for a correct explanation of the physical aspect of the operation of this device one should use the ideas developed by A. A. Vlasov in his theory of plasma, connected with the interaction of many-electron systems.

The phenomena observed in the multibeam tube have a certain analogy with the Cherenkov effect, and also with the excitation by an electron stream of slowing systems (helices, “comb,” plasma, etc.).

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