NEW INSTRUMENTS AND METHODS OF MEASUREMENT

ELECTRONIC METHODS FOR GENERATING ULTRASHORT PULSES

A. S. Koporskii, A. V. Chernetskii, N. V. Korotkikh,
V. I. Voznesenskii

INTRODUCTION

Pulse technology is finding wide application in various fields of science and engineering. Radar, radio astronomy, counting technology, nuclear physics—this is far from a complete list of the main directions in the development of modern science and technology whose progress is determined to a considerable extent by advances in pulse technology.

The principal task in the development of pulse technology is to obtain pulses of very short duration with a high repetition rate. Until now, radio-engineering generators have been used almost exclusively as pulse generators. However, at present radio circuits do not make it possible to obtain and transform pulses whose duration is less than \(10^{-9}\) sec. Even for pulse durations of the order of \(10^{-7}\) sec and shorter, the parasitic capacitances of the assembly, the interelectrode capacitances of electron tubes, and their input dynamic conductance begin to have a strong effect. No less complex are the transforming devices. To amplify pulses of such duration, wide-band amplifiers with distributed parameters are used. These amplifiers, as is known, operate on the principle of addition, not multiplication, of the signal. Therefore they require a significant increase in the number of tubes. In addition, their manufacture is quite complicated. Circuits with storage devices (Fig. 1) have now become widely used as pulse generators.

Fig. 1. Schematic diagram of a pulse generator with a storage device: 1—charging resistance, 2—line (coaxial cable), 3—switch, 4—load resistance.

Thyratrons, mercury or mechanical relays may serve as the switch in these generators. The pulse repetition rate in such generators is low and lies within the range of hundreds of kHz. In thyratron generators it is limited by the deionization time. In generators with a mechanical

the repetition frequency is determined by the inertia of the mechanical system; the same applies to generators with mercury interrupters. An important advantage of these generators is the possibility of obtaining pulses of large amplitude, which eliminates the need for amplification.

In connection with the above, attempts were made to construct pulse generators free of the listed shortcomings. A number of electronic methods for generating very short pulses were proposed.

The principle of operation of electronic pulse generators consists in obtaining electron bunches of the required duration. As a rule, cathode-ray devices are used for this purpose. The principle of operation of electronic pulse generators is as follows. A continuous, focused electron beam is, by one method or another, converted into discrete groups of electrons—bunches. Since the duration of the output pulse is determined by the duration of the bunch, the bunches themselves may be subjected to further transformation, i.e., to a reduction of their duration.

The electron bunches then excite an output device, in which the kinetic energy of the electron bunches is converted into the electromagnetic energy of pulses. A distinctive feature of such devices is the possibility of obtaining extremely short pulses with a high repetition frequency*).

Before proceeding to a consideration of existing pulsed electronic generators, let us briefly dwell on the principal problems characteristic of all generators of this type.

§ 1. PROBLEMS ARISING IN THE GENERATION OF PULSES BY ELECTRONIC METHODS

Spreading of the bunch. One of the principal problems is to preserve, as far as possible unchanged, the “external” dimensions of electron bunches obtained by some method during the time of their flight from the forming device to the output device. In some designs this time reaches relatively large values; therefore the so-called spreading of the bunch is observed. It occurs for two reasons. The first is the stretching of elementary charged particles under the action of Coulomb forces. The second is spreading of the bunch as a result of the spread of electron velocities.

Table I

The magnitude of the spread of electron velocities in the bunch is determined by the type of generator. In some types it reaches a minimum value—the magnitude of the spread of the initial velocities of the electrons. Table I illustrates the values of the “path lengths” of bunches \(S\) at various current densities \(J\) and various accelerating voltages for a disk-shaped bunch with uniform velocities, where by “path length” is meant the length by which the extent of the bunch increases by 10%.

*) At the present time pulses of duration on the order of \(10^{-12}\) sec have been obtained, with repetition frequencies in the hundreds of MHz.

Obviously, when increasing the current density in the bunch, which is desirable from the point of view of increasing the power of pulse generators, it is necessary to increase the accelerating voltages and to reduce the length of the bunch path, i.e., to reduce its flight time to the output device.

The output devices presently used are most often made in the form of a coaxial cable or a two-wire line. In the following section an example is given of calculating the excitation of such an output device by a specified bunch current.

Excitation of the output device. Suppose that a convection current with linear density \(j(x,t)\) is incident on one of the conductors of an ideal infinite two-wire line or coaxial cable, while the other conductor is at a constant potential. Let us consider, in general form, using the apparatus of discontinuous functions¹, the wave processes occurring in such a line under the action of a current pulse of arbitrary shape. Without loss of generality in the formulation of the problem, suppose that the density of the convection current \(j(x,t)\) is equal to the constant quantity \(J_0/l\) along a certain segment of the line of length \(l\), and in time represents a rectangular function of duration \(\tau_0\). In other words:

\[ j(x,t)=J_0/l\cdot[\sigma_0(t)-\sigma_0(t-\tau_0)][\sigma_0(x)-\sigma_0(x-l)], \tag{1} \]

where \(J_0\) is the total current strength, and \(\sigma_0\) is the primitive discontinuous function of zero order.

If necessary, one can easily pass from this case, by means of the Duhamel integral, to a convection-current density of arbitrary form.

The wave equation describing the distribution of the potential in the line \(V(x,t)\) has the following form:

\[ \frac{\partial^2 V}{\partial t^2}-a^2\frac{\partial^2 V}{\partial x^2} =\frac{1}{C}\frac{\partial j}{\partial t}=f(x,t), \tag{2} \]

where \(a=\dfrac{1}{\sqrt{LC}}\), \(L\) is the inductance per unit length, and \(C\) is the capacitance per unit length.

It is known that the solution of the inhomogeneous wave equation for an infinite line with zero initial conditions has the form

\[ V(x,t)=\frac{1}{2a}\int_0^t d\tau \int_{x-a(t-\tau)}^{x+a(t-\tau)} f(\xi,\tau)\,d\xi . \tag{3} \]

Substituting into (3) the expression for \(j(x,t)\), we obtain:

\[ V(x,t)=\frac{1}{2}\frac{J_0}{l}\sqrt{\frac{L}{C}} \int_0^t [\delta(\tau)-\delta(\tau-\tau_0)] \{\sigma_1[x-a(t-\tau)]- \]

\[ -\sigma_1[x-l-a(t-\tau)]\}\,d\tau =\frac{1}{2}\frac{J_0}{l}\sqrt{\frac{L}{C}} \{\sigma_1(x-at)-\sigma_1(x-l-at)- \]

\[ -\sigma_1[x-a(t-\tau_0)]+\sigma_1[x-l-a(t-\tau_0)]\}. \tag{4} \]

Here \(\sigma_1\) denotes the primitive discontinuous function of the first order (Fig. 2).

The expression in brackets is easily deciphered: it is an isosceles trapezoid moving to the right with velocity \(a\). The length of the lower base of the trapezoid is \(l+at_0\), and the length of the upper one is \(at_0-l\); its height is equal to \(l\).

The amplitude of the voltage pulse is equal to

\[ V_0=\frac{1}{2} I_0 \sqrt{\frac{L}{C}} \tag{5} \]

for \(I_0=10^{-2}\,a,\ \sqrt{\frac{L}{C}}=100\ \text{ohms},\ V_0=0.5\ \text{V}.\)

Thus, when an electron bunch strikes one of the conductors of the line, an electromagnetic pulse is excited in the line and begins to propagate along it (in the general case, two). This pulse may be regarded as a certain sum of harmonic components. It should be remembered that the purpose of the present article is to consider electronic methods of generating pulses shorter than \(10^{-9}\) sec. Consequently, the harmonic components of these pulses will have very small wavelengths (centimeter and millimeter lengths). As is known, at these frequencies the question of matching the output device to the transmission line is very important. Matching is made considerably more difficult by the fact that electronic generators are vacuum devices.

Fig. 2. Primitive discontinuous function of the first order.

Matching remains to this day the bottleneck of many pulse generators.

We shall proceed to set forth the existing methods of pulse generation and to describe generator designs.

§ 2. KLYSTRON-TYPE PULSE GENERATOR

The principle of operation of the generator is based on the long-known and widely used effect of klystron bunching\(^{2,3}\), which consists in velocity modulation of an electron beam. As a result of this, the beam current becomes nonconstant in time, i.e., the electrons are bunched into packets, which then excite the output device.

The design of a klystron pulse generator differs somewhat from that of a klystron generator of sinusoidal oscillations. In a pulse generator, first, there is no feedback. The alternating high-frequency voltage is applied to the modulator (grid) from an external (“foreign”) generator. Second, its output system is not a resonator, but a coaxial cable or a long line.

The literature\(^{4}\) describes a pulse generator operating on this principle. Figure 3 shows the generator together with the measuring device. High-frequency energy from a 240 Mc/s generator is fed to a resonator made in the form of a section of coaxial line, located at one end and terminating at the other end in grids spaced 10 mm apart, across which a potential difference of 400 V is produced. The electron beam, being accelerated by high-

frequency voltage, is modulated in velocity. In the drift space the electron beam is focused by means of a longitudinal magnetic field. The length of the drift space is 400 mm. A 70-ohm coaxial cable is used as the output device. The outer conductor of the cable ends at a grid through which the electron packets pass before they reach the inner conductor of the cable (the collector). The distance between the inner conductor and the grid is 4 mm. When a packet of electrons strikes the inner conductor of the cable, an electromagnetic pulse begins to propagate along the line. This generator makes it possible to obtain pulses with a duration of the order of \(2 \cdot 10^{-10}\) sec.

For studying the pulses obtained, the circuit shown in Fig. 3 was used.

Fig. 3. Pulse generator of the klystron type: 1 — generator of sinusoidal oscillations with frequency 210 Mc/s, 2 — indicator, 3 — 10-cm waveguide, 4 — mixer, 5 — 10-cm generator, 6 — intermediate-frequency amplifier (60 Mc/s), 7 — focusing coils.

Electromagnetic energy from the line was directed into the 10-cm waveguide. At the end of the line there was a milliammeter. The waveguide ended in a mixer, to which energy from a special 10-cm generator was also supplied. The signal of the difference frequency was amplified with the aid of an intermediate-frequency amplifier. A diode current meter served as the indicator. This device was necessary for selecting pulse harmonics.

In the study of the pulses it turned out that the maximum energy of the 14th harmonic is about \(10^{-4}\) erg and is reached under the following conditions: 1) \(V_1 = 250\text{–}270\) V, \(V_0 = 2000\) V and 2) \(V_1 = 140\) V, \(V_0 = 1350\) V, where \(V_0\) is the accelerating dc voltage, \(V_1\) the alternating voltage.

Klystron-type generators find application in the study of wideband coincidence circuits, for counting the number of particles, and also for other purposes. The advantages of generators of this type are the possibility of obtaining relatively powerful pulses, as well as the high utilization coefficient of the beam current.

Among the disadvantages should be included: the need to fix the collector at a definite distance from the resonator, which is inconvenient if further transformations of the electron bunch are required; the presence of tails (blurring of the fronts) in the pulses obtained.

To obtain shorter pulses by the method of klystron compression and to get rid of the “tails,” it is necessary to use several stages of compression, similar to a multiresonator klystron, which complicates the design.

§ 3. TUBE WITH TRANSVERSE DEFLECTION OF THE BEAM AS A GENERATOR OF VERY SHORT PULSES

At present, a generator of the type described in \(^{5,6}\) has found wide application. The principle of operation of the generator is as follows. A flat electron beam (Fig. 4) passes between deflecting pla-

walls, to which a high-frequency voltage is applied, creating a transverse electric field. This alternating field causes the electron beam to pass across electrode 3 twice during a period of the high-frequency voltage. Thus, through the slit in electrode 3, groups of electrons will pass twice per period in the direction of collector 1. The collector may be the inner conductor of a coaxial cable or of a two-wire line (Fig. 5).

Fig. 4. Cross section of a pulse generator with transverse deflection:
1 — collector (anode), 2 — antidynatron grid, 3 — screen with a slit, 4 — deflecting plates, 5 — accelerating electrodes, 6, 7 — focusing electrodes, 8 — cathode, 9 — screen.

Fig. 5. Cross section of a generator with an output device in the form of a two-wire line:
1 — anode, 2 — antidynatron grid, 3 — line (or coaxial cable), 4 — deflecting plates, 5 — accelerating electrodes, 6, 7 — focusing electrodes, 8 — cathode, 9 — screen.

Let us consider the design of generators of this type. The device uses a Pierce gun with a rectangular cathode, which creates a flat ribbon beam. Electrode 7, forming an angle of 67.5° with the plane perpendicular to the emitting surface of the cathode, along with the focusing control, also controls the magnitude of the beam current. Electrodes 6, made in the form of rods, serve for additional focusing of the beam. Plates 5 are accelerating electrodes. The potentials of the electrodes are chosen so as to focus the beam on electrode 3. Plates 4 serve to deflect the beam. By applying an alternating voltage to them, we force the beam to oscillate in the plane of the drawing. In this case only a definite part of the beam electrons, as already stated, reaches the collector. Electrode 2, made in the form of a grid, plays the role of an antidynatron grid.

The output current has the form of either a trapezoidal or a triangular pulse, depending on the ratio of the slit width to the beam thickness. Thus, within certain limits it is possible to vary the pulse shape. The main parameters of this generator are: the slope of the output characteristic, which to a considerable degree determines the steepness of the leading edge of the output pulse, and the total voltage between the deflecting plates required for the complete sweep of the beam from the right plate of electrode 3 to the left, or vice versa, i.e., the voltage required to obtain a packet of electrons.

The slope \(S\) of the output characteristic of a tube with transverse control (the increment of the collector current \(i_k\) for a finite increment of the voltage on plates 4) is determined by the relation \(S=\dfrac{\partial i_k}{\partial u}\).

The slope \(S\) can also be determined from the geometry of the structure and its other parameters: \(S \simeq i_k \cdot B \cdot q\), where \(q\) is the deflection sensitivity, \(B\) is the beam width, and \(i_k\) is the current density in front of the collector in the plane of the slit.

Expressing the current density at the collector \(i_k\) and the deflection sensitivity \(q\) through the structural dimensions of the device, we obtain

\[ S \simeq j_0 B \frac{l}{\sqrt{u_1}}, \]

where \(j_0\) is the current density at the cathode, \(l\) is the length of the deflecting plates, and \(u_1\) is the average voltage of the deflecting plates relative to the cathode.

It should be noted that increasing the beam width \(B\) is limited by the difficulty of focusing it. The average direct voltage \(u_1\) is determined from the condition of optimum beam focusing, since the constant electrostatic field between the accelerating plates 5 and the deflecting plates is used as an additional lens for focusing the beam.

The length of the deflecting plates \(l\) is limited, at high frequency, as is known, by the transit time of the electrons between the plates. The maximum frequency of the control voltage must be less than or, in the limiting case, equal to:

\[ f \leq \frac{1}{2\tau_{\mathrm{tr}}}, \]

where \(\tau_{\mathrm{tr}}\) is the transit time of the electrons between the plates.

In deriving the dependence of the slope, it was assumed that the electron beam touches the deflecting plates, although in practice it occupies about \(1/3\) of the space between the deflecting plates. Therefore the maximum slope of the output characteristic will be approximately \(1/3\) of the calculated value, amounting to \(1.9\ \mathrm{ma/v}\) (for \(B = 3\ \mathrm{cm}\), \(j_0 = 10\ \mathrm{ma/cm^2}\), \(l = 0.8\ \mathrm{cm}\), and \(U_1 = 150\ \mathrm{v}\)), i.e. \(0.5 \div 0.6\ \mathrm{ma/v}\), which is confirmed quite well by experimental results. The total deflecting voltage \(U_4\), necessary for shifting the beam, can be determined if the deflection sensitivity \(q\), the beam thickness in the plane of the slit \(\delta\), the slit width in the screen \(a\), and \(d\), the distance between the plates, are known. The deflection sensitivity is calculated from the structural dimensions of the device, the average voltage of the deflecting plates, and the voltage of the screen with the slit, as follows:

\[ q = \frac{l}{2dU_4}\left(\frac{U_{q1}}{U_1}\right)^{1/2} a = \frac{0.8\cdot 1.5}{2\cdot 0.4(200)^{1/2}\cdot (150)^{1/2}} = 0.1\ \mathrm{mm/v}. \]

Then the total deflecting voltage \(U_4\) is expressed as follows:

\[ U_4 = \frac{\delta + h}{S} = 30\ \mathrm{v} \]

for \(\delta = 1\ \mathrm{mm}\), \(h = 2\ \mathrm{mm}\), \(S = 0.1\ \mathrm{mm/v}\), \(d = 0.4\ \mathrm{cm}\).

The pulse duration at the generator output is easily calculated from the obvious relation:

\[ \tau_U = \frac{\arcsin U_4/2U_m}{\pi f} \tag{4} \]

for \(f = 10\ \mathrm{Mc/s}\), \(U_m = 500\ \mathrm{v}\), \(U_4 = 30\ \mathrm{v}\), the pulse duration is found to be of the order of \(10^{-9}\ \mathrm{sec}\).

It is interesting that the duration of the pulses obtained in the described generator is measured in the same bulb in which the generator itself is assembled. Indication is carried out by means of circular scanning of the electron packet. For this purpose, two additional mutually perpendicular pairs of deflecting plates are inserted inside the generator, and a small hole is cut in the collector. Voltages shifted relative to one another by \(90^\circ\) are applied to these plates.

In the absence of deflecting voltage, the electron beam draws a circle on the phosphor-coated screen. When the pulse is cut off, instead of a circle two arcs appear on the screen, the lengths of which correspond to the duration of the output pulses. If the period of the scanning voltage is \(T\), the diameter of the circle on the screen is \(D\), and the length

of luminous arcs \(l'\), then the pulse duration is determined as \(\tau_U=\dfrac{Tl'}{\pi D}\). Two luminous arcs mean that during one period of the voltage variation at the plate cutouts two pulses are formed.

Experimental verification showed that with such a generator it is possible to obtain, comparatively easily, pulses of duration of the order of \(10^{-9}\) sec. The generator is relatively simple to adjust and to manufacture. However, only a small part of the beam current in it is used usefully.

In addition, it is natural that in attempts to obtain pulses of shorter duration the amplitude of the output pulse will decrease. This type of generator makes it possible, with a 100-ohm load (a coaxial cable, for example), to obtain pulses with an amplitude of the order of \(0.3\)—\(0.5\) V, which to a considerable extent limits the possibility of applying such generators, although in principle, undoubtedly, there is the possibility of increasing the power of the output pulses.

Fig. 6. Cross section of a combined pulse generator:
1 — fluorescent screen, 2 — circular-sweep system, 3 — resonator tuned to a frequency of 3000 Mc/s, 4 — screen with a slit, 5 — deflection system (Lecher line), 6 — lens, 7 — electron gun.

§ 4. COMBINED GENERATOR

A pulse generator\(^{7}\) is a combination of the two devices already described by us. Its principle of operation is as follows. Electron packets are cut out by the method described in the preceding paragraph. These packets are then compacted by klystron compression (Fig. 6). It should be noted that in this instrument a Lecher line tuned to resonance was used as the deflecting system. The beam passed between the conductors of this line. The deflecting device was excited at a frequency of 300 Mc/s. The electron packets obtained by cutting then passed through a compressing chamber tuned to resonance at a frequency of 3000 Mc/s, in which their velocity modulation was carried out. The duration of the resulting electron packets was measured in the same way as in the preceding case, that is, by means of circular sweep. As a result of combining the two methods, it proved possible to obtain packets with a duration of approximately \(2\cdot10^{-12}\) sec. This result is, undoubtedly, a major achievement in pulse technique.

§ 5. GENERATOR OF PULSES WITH MAGNETIC BRAKING

This type of generator\(^{8,9}\) is so far the only known device that makes it possible to obtain powerful short pulses. The principle of operation of the generator described differs from the generators described above. The device is comparatively simple and highly efficient. The construction of the device is not complicated (Fig. 7). A vacuum grounded device \(T\) with longitudinal symmetry contains a high-efficiency cathode \(K\), which is shielded from magnetic fields, and a metal grid \(G\). Between the grid \(G\) and the cathode \(K\) a pulse generator is connected. At the moment when an external pulse is applied to the grid, its potential reaches the value \(V_0\) with respect to the cathode. The duration of the external pulse is greater than the transit time of the electrons in the device \(t_0\). The length of the first section, as well as of the third section, is small in comparison with the length of the second section, which at

several orders of magnitude greater than the radius \(R\) of the tube. The radius of the third section \(R'\) is somewhat greater than the radius \(R\) of the second section. The length of the third section is several times greater than its radius.

Inside the second section an axial magnetic field \(B_z\) is produced. This focusing magnetic field, produced by a coil and usually assumed negligible in the region of the cathode, must be uniform in the second section and rise rapidly between \(z_2-z_1=\Delta z\) to the value \(B_1\). A sharp rise in the field strength can be obtained by making the first two sections of magnetic material, and the last (third) section of nonmagnetic material. The use of such a magnetic field solves the problem of focusing a cylindrical beam with a high charge density over a large distance without a significant change in the magnitude of the beam cross-sectional area, or spreading of the current and potential of the beam. Moreover, it makes it possible to change the axial velocity of the beam easily.

The principle of operation of the device is as follows. A powerful electron stream is emitted by the cathode and forms a cylindrical beam in the second section under the action of a constant focusing magnetic field \(B_0\). All electrons of the beam rotate around the longitudinal axis with the same angular velocity \(d\Phi/dt\) and approach the third section with longitudinal velocity \(dz/dt\). During the flight time \(t_0\) a constant current \(i_0\) passes through the system, except for a small part deposited on the grid \(G\). Entering the region in which the magnetic-field strength increases, the beam undergoes radial compression of the front due to the Lorentz force until this force is again compensated by the repulsive force and the centrifugal force. At the same time the axial velocity of the electrons decreases and becomes zero in the region where \(B(r,z)=B_1\). Thus, the space-charge density increases; for this reason the potential falls, and the potential barrier rises and prevents the passage of electrons. The gradient of the axial potential becomes negative in the second section, zero at the point \(z=z^*\), and positive in the third section.

Fig. 7. Section of a generator with magnetic braking of a beam with a magnetic system:
\(1\) — shell, \(2\) — electromagnet windings, \(3\) — compensating coil or ring of ferromagnet, \(4\) — insulator, \(T\) — tube, \(K\) — cathode, \(G\) — grid, \(R'\) — radius of the wide part of the tube, \(R\) — radius of the second section, not indicated in the figure.

Since the virtual cathode grows, the beam current falls because the magnetic field falls, inducing an axial electric field. Thus, an exchange of magnetic and electric energy takes place. The induced magnetic field arises only in the third section. In the second section it is compensated by the electric field owing to the nonuniform distribution of space charge along the beam during the period. Exactly the same phenomenon is observed, for example, in a series resonant circuit consisting of an inductance and a capacitance, which under certain conditions becomes an ideal conductor. During the conversion of magnetic energy into electric energy, the electric field along the beam, which creates the magnetic field, becomes equal to zero, since there is superposition of the induced...

of the electric field and of the electric field caused by a change in the concentration of conducting electrons.

The induced lines of force begin on the inner wall of the tube \(T\) and end on the frontal electrons of the beam. Because of the high density of the space charge of the beam, they do not penetrate deep into it. This shielding effect by the surface electrons is what causes the formation of the potential barrier. The frontal electrons, on which the induced lines of force terminate, form a charge behind the potential barrier in the region of the positive potential gradient. The focusing magnetic field there, although larger than \(B_1\), reducing the longitudinal velocity of the approaching electrons in the initially field-free space, nevertheless does not exceed the limiting value for the electrons in the induced electric field behind the virtual cathode. The electrons are accelerated; consequently they absorb the energy of the electric field, which is converted into magnetic energy, since the electrons increase their velocity while moving in the third section.

Thus, in the third section, initially free of field, the electromagnetic field arises discontinuously and disappears when the electrons reach the tube walls. Consequently, this discharge process may be regarded as a very short electromagnetic pulse, which the tube will radiate if it is made as a reflector and if its end is made of a dielectric, for example glass.

During the discharge, the space-charge density in the segment \(\Delta z\) decreases and the virtual cathode moves in the negative direction of the \(z\)-axis, gradually dissipating. At the same time the axial magnetic field in the second section decreases.

The fact that the magnetic energy of the space charge is proportional to its electromagnetic moment suggests that the beam in the second section loses its total electromagnetic moment when the magnetic field decreases.

The same can be explained from the nature of the internal electromagnetic forces, which increase under compression and which have antiparallel components with respect to the electron velocity. Since the Larmor force disappears when the electron moment becomes zero, the radial field of the space charge pushes the electrons in the direction of the tube walls, where they finally settle.

The second and third sections may be regarded as waveguides with very different wave impedances. The mismatch of the waveguides causes reflection of the wave associated with the beam at the boundary between the two sections. In the reflection process, higher harmonics are generated in the third section, in accordance with the electrons that have been accelerated in the tube.

The mathematical treatment of the phenomenon of braking of the electron beam by the magnetic field reduces to the solution of the Lagrange equation describing the motion of the electron flow in electric and magnetic fields.

It follows from the solution that

\[ (dz/dt)^2 = 2kV_0 - \left(\frac{1}{2}kbB_0\right)^2 \cdot (1 + 2\ln R/b), \]

where \(b\) is the beam radius, \(V\) is the scalar potential, and \(A\) is the vector potential. Moreover,

\[ d\Phi/dt = -kA\dot{\Phi}/r = -\frac{1}{2}kB_0. \]

These equalities show that the energy of the beam is partly transformed into rotational energy. The longitudinal velocity decreases with increasing \(B_0\) and vanishes at \(B_0=B_l\), where \(B_l\) is determined by the limiting field given by the equality

\[ B_l=\left(\frac{8V_0}{kb^2(1+2\log R/b)}\right)^{1/2}\quad(\text{gauss}) \]

\(b\) is in centimeters, \(V\) in volts. Consequently, no current can pass through the tube if \(B_0>B_l\).

The expression for the current has the form

\[ i_0=\frac{4\pi\varepsilon_0(2k)^{1/2}a^3(1-a^2)^{1/2}}{1+2\log R/b}\,V_0^{3/2}, \]

where

\[ a=B_0/B_l<1. \]

To determine the energy of the generated pulse and its duration, it is necessary to turn to Poynting’s theorem:

\[ \operatorname{div}\mathbf S=-\rho\mathbf v\mathbf E-d\omega/dt, \]

where \(\mathbf S\) is the Poynting vector, \(\omega\) is the density of electromagnetic energy. This expression means that the flow of energy per unit time into a unit volume goes partly to the acceleration of electrons and partly to an increase in electromagnetic energy.

The magnetic energy is determined by the expression

\[ W_m(t_0)=\frac{4\pi\varepsilon_0a^2V_0^2z_1}{1+2\log R/b} \left(1-\frac{a^2(1+4\log R/b)}{(1+2\log R/b)}\right) \]

and

\[ \Delta t=\frac{1}{2}\pi\sqrt{LC}, \]

where \(L\) is the inductance and \(C\) is the capacitance, determined from the energy relations:

\[ W_m(t_0)=\frac{1}{2}i_0^2L, \]

\[ W_C(t_0)=i_0V_0t'-\frac{1}{2}V^{*2}C \]

\(V^*\) is the potential of space at the point \(z^*\).

Numerical calculations give the following results:

1. For \(V_0=3\cdot10^4\ \text{V}\), \(z_1=100\ \text{cm}\), \(R=R'=2.5\ \text{cm}\), \(b=0.34\ \text{cm}\), \(a=0.9\) and \(B_l=1510\ \text{gauss}\), and \(B_0=1360\ \text{gauss}\), \((dz/dt)_0=4.5\cdot10^9\ \text{cm/sec}\), \(t_0=2.22\times10^{-8}\ \text{sec}\), \(W_m(t_0)=1.03\cdot10^{-2}\ \text{W}\cdot\text{sec}\), \(C=1.1\cdot10^{-14}\ \text{F}\), \(L=3.55\cdot10^{-5}\ \text{H}\),

\[ \Delta t=9.8\cdot10^{-10}\ \text{sec}. \]

1. \(R=b=0.34\ \text{cm}\), \(R'=5\ \text{cm}\), \(a=0.99\), gives:

\[ B_l=3400\ \text{gauss},\quad B_0=3350\ \text{gauss},\quad (dz/dt)_0=1.45\cdot10^9\ \text{cm/sec}, \]

\[ t_0=6.9\cdot10^{-8}\ \text{sec},\quad W_m(t_0)=7.35\cdot10^{-2}\ \text{Wsec},\quad L=6.6\cdot10^{-5}\ \text{H}, \]

\[ C=6\cdot10^{-15}\ \text{F},\quad \Delta t=10^{-9}\ \text{sec}. \]

Thus, the calculations show that a system with magnetic braking of the beam makes it possible to obtain sufficiently powerful and short electromagnetic pulses.

Experimental verification of the operating principle and design of the device at low frequency (60 cps) confirmed the possibility of generating pulses by the method of magnetic braking.

The disadvantages of the generator are the dependence of pulse power on the repetition frequency, as well as the limited scope of its application.

Conclusion

Electronic generators have a great future. Their principal advantages are simplicity, stability in operation, and the possibility of obtaining very short pulses over a wide range of repetition frequencies. The fact that at present they have limited application is explained above all by the novelty of the methods themselves for electronic generation of pulses. They are little known to a broad circle of specialists; work in this field is only just beginning to develop. In addition, existing generators are for the most part of low power and have a limited range of application. However, undoubtedly, the development of the methods described, or of methods that may be proposed in the future, will open new possibilities for pulse technology.

References

1. A. A. Kharkevich, Nonstationary Wave Phenomena, Moscow, 1947.
2. Collection “Reflective Klystrons,” part I, Sov. Radio Publishing House, 1954.
3. Collection “Klystrons,” part II, Sov. Radio Publishing House, 1954.
4. D. B. Hastid, Proceeding of Phys. Soc. 60, 340 (1948).
5. E. W. Grant, H. Von-Foerster, J. Ap. Phys. 25, 674 (1954).
6. L. R. Bloom, H. Von-Foerster, Rev. Sci. Inst. 25, 649 (1954).
7. E. W. Ernst, H. Von-Foerster, J. Ap. Phys. 25, 674 (1954).
8. Randorf, Wireless Eng. 28, 215 (1951).
9. Billington, Randorf, Wireless Eng. 31, No. 11 (1954).