HOLLOW SYSTEMS IN MICROWAVE ENGINEERING

E. M. Studenkov, Moscow

Judging by the great attention that Brillouin devotes in his review1 to hollow systems, one may infer the enormous importance of these systems in microwave engineering.

It is now already quite clear that in all branches of microwave engineering—from generation and radiation to reception, amplification, and the channeling of electromagnetic energy over considerable distances—hollow systems will everywhere have to play a leading role. This explains the tremendous interest in hollow systems that has appeared recently, stimulated by the general interest in microwaves.

The study of hollow systems is being pursued mainly in three directions, corresponding to their prospective principal applications: hollow oscillatory systems, or resonators (endovibrators), hollow radiating and receiving devices (horns, diffraction antennas), and, finally, hollow systems for channeling microwave electromagnetic energy (hollow cables).

The theoretical foundations and some experimental data of American investigators concerning the transmission of microwave energy through tubes and dielectric rods were given in the review by H. H. Malov. Brillouin’s article provides information on the work of French authors in this direction. But this does not encompass all the extensive material that has by now been obtained in the study of hollow systems both abroad and in the USSR. The aim of the present review is to give at least brief information on the data obtained in recent times.

A. HOLLOW OSCILLATORY SYSTEMS

A whole series of questions concerning the physical aspect of the phenomena occurring during oscillations in hollow metallic resonators of the simplest forms was clarified theoretically already comparatively long ago in connection with certain other problems of physics[^2][^3][^4][^5]. Among such questions are the establishment of the possible types of oscillations of a resonator, the elucidation of the character of the distribution of the electric and magnetic fields for a given type of oscillation, and, finally,

determination of the natural frequencies of the resonator. In the case of resonators of simple shape, such as, for example, a sphere, a circular cylinder, or a rectangular parallelepiped, this problem reduces to solving Maxwell’s equations in the corresponding coordinate system and using boundary conditions requiring that the tangential component of the electric field vanish on the surface of the metallic shell. As a result, it becomes possible to determine completely which types of oscillations are possible in a resonator of the given shape, what the distribution of the electric and magnetic fields will be for the given type of oscillation, and what the corresponding natural frequencies will be.

The development of microwave technology and the consequent necessity of the real use of hollow oscillatory systems called forth another, more practical approach to the study of the properties of hollow resonators. In addition to knowing the natural wavelengths of resonators, \(\lambda\), it became necessary also to calculate certain quantities that are often used in describing the properties of ordinary oscillatory circuits with lumped constants.

To estimate the quality of a resonator in the sense of the duration of its free oscillations, it is necessary to know the quantity \(Q\), which may be different for resonators of different shape. In addition, it is often desirable to know the quantity \(\rho=\sqrt{\dfrac{L}{C}}\), called the characteristic of the oscillatory circuit (or wave resistance), as well as the resonance resistance \(Z_r=\dfrac{L}{RC}\).

The quantity \(Q\) can always be determined exactly as the product of \(\omega\) by the ratio of the total energy of the system \(A\) to the energy \(W\) expended per period:

\[ Q=\frac{\omega A}{W}. \]

As for the quantities \(\rho\) and \(Z_r\), it would seem that the simplest way to calculate them consists in first finding the quantities corresponding to the capacitance \(C\), inductance \(L\), and resistance \(R\) of an ordinary circuit, which could then be used to determine \(\rho\) and \(Z_r\). This means that the hollow resonator must be reduced to an equivalent circuit with lumped constants, the parameters of which can then serve to determine the above-mentioned quantities.

The first attempt at such an interpretation of hollow resonators was undertaken by Hansen\(^6\). His calculations showed that for hollow resonators it is impossible to determine exactly the quantities corresponding to \(L\), \(C\), and \(R\) of ordinary circuits. This is understandable, since for an exact description of an oscillatory system comparable with the wavelength one cannot make use of concepts inherent in systems with lumped constants. However, as Hansen showed, there are several different, but, generally speaking, quite equivalent ways of determining quantities closely corresponding to \(L\), \(C\), and \(R\). In this case the quantities obtained, irrespective of—

depending on the manner of their definition, must of course satisfy the conditions

\[ \left. \begin{gathered} \sqrt{LC}=\frac{\lambda}{2\pi c},\\ \frac{L\omega}{R}=Q, \end{gathered} \right\} \tag{1} \]

i.e., the equivalent circuit obtained must have the same wavelength and the same attenuation as the given cavity resonator; for the final choice of the values \(L\), \(C\), and \(R\), and consequently also for the determination of \(\rho=\sqrt{\frac{L}{C}}\) and \(Z_r=\frac{L}{RC}\), certain additional conditions are needed, depending on the choice of which these quantities take one or another value. The quantity \(Z_r\), for example, can be determined in three ways and, correspondingly, three values can be obtained, which, however, differ very little from one another, so that in practice any of them may be used.

In his work\(^{6}\) Hansen gives not only the values of the natural wavelengths for resonators of the simplest forms (cylinder, parallelepiped, sphere), but also the quantities \(L\), \(C\), \(R\), \(\rho\), \(Z_r\), \(Q\), and a number of others calculated by him.

For cavity resonators \(Q\) is usually obtained of the order of \(\frac{\lambda}{\delta}\), where \(\lambda\) is the wavelength, and \(\delta=\sqrt{\frac{\rho}{\pi\omega}}\) is the surface attenuation of the wall material. The quantity corresponding to the resonance resistance, in absolute electromagnetic units, is obtained of the order \(Qc\), where \(c\) is the speed of light.

For practical purposes, the greatest interest is presented only by the quantities \(\lambda\) and \(Q\), which sufficiently characterize the properties of the resonator. Proceeding from this, it may perhaps be unnecessary each time, for a given resonator, to find the equivalent lumped-constant circuit and calculate the values \(L\), \(C\), \(R\), etc. In later works Born\(^{7}\), Bunimovich\(^{8}\), and Neumann\(^{9}\) devoted their principal attention precisely to the calculation only of the quantities \(\lambda\) and \(Q\) for resonators of the same shapes as Hansen’s.

Hollow resonators in the form of a parallelepiped or cylinder are finding ever wider application in high-frequency technology because of their ease of manufacture, rigidity and compactness of construction, and high frequency stability\(^{10,11}\). However, in some cases resonators of such simple shape prove not entirely suitable, and then it becomes necessary to choose other, more complicated forms of resonator surface.

In particular, the need to choose a more complicated form of resonator arises in the construction of ultrahigh-frequency generators and amplifiers of the newest type, operating on the principle of velocity modulation of an electron beam\(^{12,13,14,15}\). The point is that a resonator operating in any electron-beam device, in addition to the general necessary qualities, such as sufficient simplicity

construction, frequency stability, high \(Q\), etc., must also satisfy several other requirements, the chief of which are the following:

a) The presence of a sufficiently high resonance resistance \(Z_r\), which should be of the order of several megohms. This requirement follows from the following: the amplitude of the alternating voltage of high frequency arising in the oscillatory system is determined by the equality

\[ U_1 = i_1 \frac{L}{RC}. \]

If one takes into account that in electron-beam generators usually \(U_1 \simeq U_0 \simeq 1000\ \mathrm{V}\) and \(i_1 \simeq i_0 \simeq 10\ \mathrm{mA}\) (where \(U_0\) is the constant accelerating voltage, and \(i_0\) is the magnitude of the constant beam current), then for

\[ Z_r = \frac{L}{CR} \]

one obtains a value of the order of \(10^5 \Omega\).

b) The distance between the walls of the resonator in the region of the greatest electric field must be such that the beam electrons can traverse this region in a sufficiently small interval of time, in any case not greater than a half-period, for only then will there be an effective extraction of oscillatory energy from the electron beam by the cavity resonator. This means that in the region of passage of the electron ray the distance between the boundaries of the resonator must be of the order of \(\frac{v}{c}\cdot\frac{\lambda}{2}\), where \(v\) is the electron velocity and \(\lambda\) is the natural wavelength of the resonator, i.e. certainly less than \(\lambda\).

As for the first condition, resonators of ordinary shape fully satisfy it. Calculation of \(Z_r\) for them gives precisely a value of the required order. The second condition, however, can be satisfied only by somehow modifying the shape of ordinary resonators. For example, a sphere having a natural wave equal to \(1.1d\) plainly does not satisfy this condition; as for a cylinder or a parallelepiped, they could be used for electron-beam devices in the case of the so-called flat type of oscillations, examined in detail by Bunimovich\(^{8}\). In this case the electromagnetic field in the cavity does not depend on the coordinate \(z\), if the \(Z\)-axis is perpendicular to the bases. However, for this it would be necessary greatly to reduce the height \(h\) of the resonator relative to the dimensions of the bases, and since the quality factor in this case is approximately proportional to \(h\), this would considerably lower \(Q\), which, of course, is undesirable.

All this made it necessary to investigate resonators whose general form is shown in Fig. 1. The study of resonators of this type was undertaken by Hansen and Richtmyer\(^{16}\). As before, the task was to determine for these resonators the quantities \(\lambda\), \(Q\), \(Z_r\), etc.

Of all the forms presented in Fig. 1, the easiest to calculate is resonator \(a\), which has the form of a sphere into which enter two cones with vertex angles \(\Theta = \Theta_0\). Of course, in practice such a

form cannot be strictly realized; complete cones, for example, must necessarily be replaced by slightly truncated ones (the dotted line in Fig. 1, a), so that the electron beam can be passed through their upper bases, made in the form of grids. However, the results obtained for the idealized form can in practice be quite acceptable, provided that the area of the grids is sufficiently small.

The solution of the vector wave equation and the use of boundary conditions requiring the disappearance of \(E_\Theta\) on the walls of the resonator lead to an expression for the natural wavelength \(\lambda\) of the resonator of form a, which is found to be equal to \(4r\), independently of the angle of the cones entering the sphere. Further, analogously to how this was done for the sphere or cylinder, in the present case one can in exactly the same way calculate the values of \(Q\) and \(Z_r\). These quantities turn out to depend on \(\Theta_0\), and at certain angles they attain their greatest values. The maximum values of \(Q\) and \(Z_r\) are obtained somewhat reduced in comparison with the same quantities for a spherical resonator; however, this reduction is not very considerable. For example, \(Q\) changes from \(0.45 \dfrac{\lambda}{\delta}\) for the sphere to \(0.155 \dfrac{\lambda}{\delta}\) for a resonator of form a.

Fig. 1

Correspondingly, \(Z_r\) decreases from \(4.92 \dfrac{\lambda}{\delta} c\) to \(1.31 \dfrac{\lambda}{\delta} c\). Substituting into the latter expression the value \(\delta\) for copper and \(\lambda = 10\) cm, one obtains for \(Z_r\) \(\sim 2.66 \cdot 10^6 \Omega\), i.e. a value of the order of several megohms.

Thus, by modifying a spherical resonator by including two cones in it, one can obtain a resonator quite suitable for electron-beam devices both with respect to its form and with respect to the high value of \(Q\) and \(Z_r\).

A calculation similar to that given for the resonator of form a is also carried out by Hansen and Richtmyer for other modifications of the sphere; in all these cases the problem can be solved sufficiently rigorously and exactly. However, the form of the resonator obtained by modifying the sphere by including in it either two cones or, for example, two paraboloids nevertheless proves in practice to be of little use because of the small area that can be assigned to the grids. More valuable in this respect are the resonator forms b and c, one of which is as it were a segment of a coaxial line, while the other resembles a toroid. An exact calculation of resonators of such form proves to be very difficult from the mathematical point of view; however, there exist approximate calculation methods based on the fact that the regions with the smallest distance between the walls of the resonator, denoted in the figure by \(K\), are pri-

are taken as the capacitance of the system, while the remaining part is regarded as possessing only inductance. Proceeding from this assumption, one can calculate the natural wavelength, the \(Q\), and all other quantities characterizing the resonator. For a toroidal resonator, for example, a similar calculation is given by Neyman\(^9\).

In those cases where the distance between the walls of the resonator in the region \(K\) is comparable with its overall dimensions, the approximate calculation indicated above may prove not entirely sufficient. For a more exact calculation of resonators of a type similar to \(b\), Hansen developed a special method\(^ {17}\), the essence of which reduces to the following: the entire cavity of the resonator is divided into several regions, in the present case (form \(b\)) into two regions, denoted in the figure by \(I\) and \(II\). Next, vector functions corresponding to the electromagnetic field in each region are constructed, and then these functions are matched in such a way that they have the same value or, at least, pass more or less smoothly into one another at the boundary of contact of regions \(I\) and \(II\). From the matching conditions one can obtain an expression for \(\lambda\). Calculations carried out in this way, although they appear cumbersome, nevertheless give more accurate data regarding the natural waves of the resonator and its other characteristics than all other known methods.

Fig. 2

The difficulty of the mathematical treatment of resonators of more or less complicated form is further increased by the fact that, for one and the same prescribed form, a resonator can oscillate in several different ways. This difficulty can in many cases be successfully avoided by experimentally studying the properties of resonators of the forms that interest us.

In this respect a good example is the work of Barrow and Mieher\(^ {18}\), who subjected to careful experimental investigation a resonator of special construction which could, at will, be transformed from a section of a closed coaxial line into a completely hollow cylinder, passing through all intermediate stages. This was achieved by moving the inner rod in the closed coaxial resonator, as shown in Fig. 2.

The natural frequencies of the resonator in the case where it is a hollow cylinder (\(VI\)), and in the case where it represents a section of a closed coaxial line (\(I\)), can be calculated exactly. The intermediate forms of the resonator, which in practice are precisely of the greatest interest, are much more difficult to calculate. Barrow and Mieher calculated and experimentally verified the first ten natural frequencies for each limiting case. Then, by moving the inner rod, various intermediate forms of the resonator were obtained and meas-

the corresponding frequencies measured. The data obtained give a complete picture of the transition of the resonator from one form to another for various types of oscillations and can be used in designing resonators whose form is close to any one of the intermediate forms considered.

The value of the results obtained in this way is further increased by the fact that, for resonators of this kind, the principle of similarity holds, according to which a change in the linear dimensions of the resonator causes a proportional change in the resonant frequency. In practice this can very often be used. However, it should be borne in mind here that, for certain types of oscillations, the resonant frequency depends not on all the dimensions of the resonator, but only on some one of them. In such a case the similarity must be referred only to this dimension.

As we see, most authors who until recently have studied the properties of hollow resonators approached this problem chiefly from the point of view of using resonators of this kind in ultrahigh-frequency radio engineering. In accordance with this, one or another quality of the resonators was determined, the forms under study were selected, etc. However, it is quite clear that hollow resonators, possessing very small decrements, may prove very valuable in solving a number of physical problems, such as, for example, the investigation of the electrical properties of substances in the ultrahigh-frequency region.

There is as yet almost no data reporting the results of the use of hollow resonators in this field, although information has already appeared^19,20 on work beginning in this direction. It may be hoped that, besides their wide application in ultrahigh-frequency radio engineering, hollow resonators will soon be used also in the study of the electrical properties of substances, where they will likewise be of great benefit.

B. HOLLOW RADIATORS AND RECEIVERS OF ELECTROMAGNETIC WAVES

Already in his first work concerning the question of the propagation of electromagnetic waves inside metallic pipes, Barrow^21 noted that pipes open at one end can serve as very convenient radiators or receivers of electromagnetic waves whose frequency is above the critical frequency for the given pipe dimensions.

The action of radiators and receivers of electromagnetic waves of this kind very closely resembles the action of analogous acoustic devices. Electromagnetic energy propagating along a pipe in the form of waves of one type or another, on reaching the open end of the pipe, will be partially reflected back and create standing waves in the pipe, but a considerable part of it will also be radiated into the surrounding space. Thus, a pipe open at one end can be used as a transmitting antenna of a definite type.

In exactly the same way, electromagnetic waves propagating in space, on reaching the open end of a tube, will partly propagate inside it and, upon reaching the other end, can easily be received and demodulated. Consequently, a tube open at one end can be used as a receiving antenna.

It is quite understandable that the efficiency of radiation depends on the conditions under which electromagnetic energy emerges from the end of the tube. In order to reduce the reflection coefficient at the end of the tube and to achieve the maximum output of energy into the external space, it was natural to try, by analogy with acoustics, to make the outlet in the form of a horn. Thus arose the idea of electromagnetic horns, which were also proposed by Barrow as radiators and receivers of ultrahigh-frequency waves.

The entirely obvious ease and simplicity of construction of such hollow radiators and receivers, as well as the hope of obtaining, with their aid, sharply directional transmission and reception of signals, led to intensive study of their properties; and at the present time one can already note a whole series of very valuable results achieved in this field.

Fig. 3

The study naturally began with the very simplest forms of hollow radiators and receivers. Thus, at the end of 1938 Barrow and Green \(^{22}\) published the results of a theoretical and experimental investigation of the radiating properties of a section of tube of rectangular cross-section open on one side. The rectangular form of the tube was chosen on the basis of the consideration that, when waves of type \(H_{0n}\) are excited in it, the electromagnetic field both inside and outside the tube is obtained with a strictly expressed polarization, which in certain practical cases is very important.

In a rectangular tube having dimensions \(a\) along the \(Y\) axis and \(b\) along the \(Z\) axis, waves of type \(H_{0n}\) are characterized by the fact that in any transverse section of the tube the electric field \(E\) is directed parallel to one of the coordinate axes, say the \(Y\) axis. It has a uniform distribution along this axis (index zero) and a sinusoidal distribution in the perpendicular direction, i.e. along the \(Z\) axis; moreover, the number of half-sinusoids fitting across the width of the tube \(b\) (index \(n\)) determines the order of the wave.

The investigation was carried out under the condition that the tube contained a wave of the simplest type, i.e. \(H_{01}\). In accordance with this, the tubular radiator had the construction schematically shown in Fig. 3. Excitation of the wave of type \(H_{01}\) was produced by means of a vertical

located inside the tube, a rod \(R\), connected to the generator by a coaxial feeder \(F\). Behind the rod \(R\) there was a sleeve \(P\), movable along the tube, which served for tuning.

The problem was to try to calculate theoretically the character of the radiation of such a section of tube, i.e., to find the shapes of the radiation patterns both in the vertical and in the horizontal planes, and then to check this calculation experimentally.

Barrow and Green calculated the radiation patterns on the assumption that the field distribution in the plane of the tube aperture is the same as in any other of its transverse sections. This assumption is, of course, not entirely exact; however, it was found experimentally that the field distribution in the aperture of the tube differs very little from the distribution in any other section, so that a calculation based on such an assumption may in practice be considered quite sufficient.

Starting from the given field distribution in the plane of the tube aperture and using Huygens’ principle, Barrow and Green calculated the field outside the tube at a distance very large in comparison with the wavelength \(\lambda\) and the transverse dimensions of the tube \(a\) and \(b\). A dependence was obtained for the absolute value of the electric-field strength on the angles \(\vartheta\) and \(\varphi\), by means of which it was possible to plot radiation patterns in the vertical and horizontal planes for various wavelengths and tube dimensions.

Of the greatest practical interest is the clarification of the question: what determines the sharpness of the directionality of the radiation, and by what means can it be increased? If, as a measure of the width of the radiated beam in the vertical plane, one takes the quantity \(\Theta_v\), and in the horizontal plane the quantity \(\Theta_h\) (\(\Theta_v\) and \(\Theta_h\) are the angles enclosed between the first nulls from the center in the vertical and horizontal radiation patterns), then for these quantities the following expressions are obtained:

\[ \begin{aligned} \Theta_v &= 2 \arc \sin \left( \frac{1}{W_v} \right),\\ \Theta_h &= 2 \arc \sin \left( \frac{3}{2W_h} \right), \end{aligned} \tag{2} \]

where \(W_v\) and \(W_h\) are the dimensions of the tube expressed in wavelengths of the radiated waves, i.e.,

\[ W_v = \frac{a}{\lambda}, \]

\[ W_h = \frac{b}{\lambda}, \]

called the vertical and horizontal apertures of the tube.

From the expressions for \(\Theta_v\) and \(\Theta_h\) two main conclusions may be drawn: first, we see that for a square cross section of the tube \((a=b)\) the beam in the vertical plane will be sharper than in the horizontal one, and in order to obtain a beam of equal width in the horizontal and vertical planes it is necessary to make the tube dimensions greater in the \(Z\) direction than in the \(Y\) direction in the ratio-

for the ratio \(b/a = 3/2\); secondly, the sharpness of the beam in both planes should increase with an increase in the horizontal and vertical apertures of the tube \(W_v\) and \(W_h\). Thus, in order to obtain the greatest possible directivity of radiation, it is necessary either to increase the transverse dimensions of the tubes \(a\) and \(b\), or to decrease the length of the emitted wave \(\lambda\).

Experimentally these conclusions were confirmed with sufficient persuasiveness. The experiments were carried out with a tubular radiator of dimensions \(a = 15\ \text{cm}\), \(b = 50\ \text{cm}\), and length \(4.78\ \text{m}\), at wavelengths from 100 to 50 cm. The radiation diagrams obtained experimentally proved to agree quite closely with the diagrams calculated theoretically; moreover, they clearly revealed an increase in the sharpness of the beam with decreasing wavelength, in complete agreement with expressions (2). Thus, by increasing the transverse dimensions of the tube or, conversely, by decreasing the length of the emitted wave, it is possible, with such a rather simple design of a hollow radiator, to obtain a sharply directed beam of electromagnetic waves.

Fig. 4

Nevertheless, the necessity of constructing a tubular radiator of fairly large dimensions in comparison with the wavelength, and the associated difficulty of exciting in it only waves \(H_{01}\), excluding any other types, are in practice somewhat inconvenient.

With the aim of finding more perfect radiator designs, a study was soon undertaken of the radiating properties of tubes whose open end has the form of a horn. At the beginning of 1939 there appeared a paper by Barrow and Lewis \(^{23}\), in which were described the results of an experimental study of the so-called sectoral horn, the scheme of which is shown in Fig. 4. As we see, this horn is a slight modification of the tubular radiator considered above, consisting in the fact that two side walls of the rectangular tube are flared out by a certain angle \(\Phi_0\).

A theoretical consideration of a horn of this type was carried out by Barrow and Chu \(^{24}\) and appeared in print simultaneously with the description of the experimental data. The task of the theoretical analysis in this case again consisted in calculating the character of the horn radiation and in clarifying on what the directivity of the radiation depends. In the first part of their theory the authors find solutions of Maxwell’s equations for the electromagnetic field inside the horn, taking into account the existing boundary conditions, and on this basis determine the possible types of waves in the horn and the corresponding distributions of the electric and magnetic fields. In the second part, using

by Huygens’ principle and, starting from the distribution of the field over the surface of the horn aperture, which is approximately taken to be the same as it would be in the given cross-section with infinitely long horn walls, the authors find the distribution of the field at a large distance from the aperture and, on this basis, draw the radiation patterns.

All the calculations in the second part are carried out under the assumption that a wave \(H_{01}\) propagates in the horn, which in its character closely resembles the same wave in a rectangular pipe; namely, it has a uniform field distribution along the \(Y\) axis and a sinusoidal one in the direction \(\Phi\), i.e. along an arc of a circle whose center is at the vertex of the horn.

The radiation patterns obtained from these calculations showed that the directivity of the horn radiation must depend on the magnitude of its flare angle \(\Phi_0\), on the horn length \(\rho_0\), and on the frequency of the radiated waves.

If, for generality, the ratio

\[ \frac{\rho_0}{\lambda} \]

is taken as the measure of the horn length, then this dependence may be formulated as follows:

1. At a constant flare angle \(\Phi_0\), the sharpness of the radiated beam increases with increasing horn length \(\frac{\rho_0}{\lambda}\); however, for certain sufficiently large values of \(\frac{\rho_0}{\lambda}\), a further increase in the horn length no longer has an appreciable effect on the sharpness of the beam.

2. At a constant horn length \(\frac{\rho_0}{\lambda}\), there is an optimum flare angle of the horn \(\Phi_{\mathrm{opt}}\), at which the radiated beam is sharpest. For example, for \(\frac{\rho_0}{\lambda}=8\), \(\Phi_{\mathrm{opt}}\) is about \(40^\circ\).

3. The optimum flare angle decreases as the horn length increases.

4. At a constant aperture

\[ \left(\frac{b'}{\lambda}\right) \]

of the horn, the beam narrows as the flare angle decreases and becomes sharpest at \(\Phi_0=0^\circ\), i.e. when the horn passes into an ordinary pipe.

All these conclusions can easily be explained by using the regularities that were found earlier for the tubular radiator. There we saw that the sharpness of the beam in the \(ZX\) plane depends on the magnitude of the horizontal aperture \(W_h=\frac{b}{\lambda}\) and increases with its increase. Correspondingly, in a horn the sharpness of the beam must increase with increasing \(\frac{b'}{\lambda}\), where the arc length \(b'=\Phi_0\rho_0\) plays the role of the width of the pipe aperture. An increase in \(\frac{b'}{\lambda}\) may occur either by lengthening the horn walls or by increasing the flare angle \(\Phi_0\) (with \(\lambda\) regarded as constant); however, it is easy to see that, in contrast to the tubular radiator, an increase in the horn aperture achieved by increasing the flare angle \(\Phi_0\) will have a narrowing effect on the beam only up to certain limits. As soon as the flare angle becomes too large, ...

guided electromagnetic waves, already inside the horn, must propagate within a very wide angle and, upon emerging into the surrounding space, will give an ever less and less sharply defined beam as \(\Phi_0\) is further increased. This explains the existence of an optimum aperture angle \(\Phi_{\mathrm{opt}}\), at which the emitted beam has the smallest width.

In the experimental investigation of the properties of sectoral horns, carried out by Barrow and Lewis\(^{23}\), all the theoretical conclusions indicated above were fully confirmed.

The authors constructed two horns: one of them (horn I) was made so that its side walls could be moved apart, thereby making it possible to obtain different aperture angles \(\Phi_0\). This horn was intended for investigating the directivity of radiation as a function of the magnitude of the horn aperture angle. The other horn (II) was made with a constant aperture angle \(\Phi_0 = 40^\circ\) and was used to study the character of the radiation directivity when the wavelength \(\lambda\) was varied (this is equivalent to changing the horn length at constant \(\lambda\)).

Excitation in the horns of waves of type \(H_{01}\) was produced, as in the tubular radiator, by means of a coaxial feeder \(F\) and a rod \(R\), which was placed either at the beginning of the tube \(T\) joined to the horn (Fig. 4), or directly in the throat of the horn. Both methods of feeding the horn are almost equivalent with respect to the character of the radiation, but in some cases, in the sense of saving space and material, the second method may prove more expedient.

Fig. 5. \(\Phi_0\) equals: for \(A\)—\(0^\circ\), for \(B\)—\(10^\circ\), for \(C\)—\(20^\circ\), for \(D\)—\(30^\circ\), for \(E\)—\(40^\circ\), for \(F\)—\(60^\circ\), for \(G\)—\(70^\circ\), for \(H\)—\(90^\circ\).

With horn I a series of radiation diagrams was taken in the horizontal plane at aperture angles \(\Phi_0\) varying from zero (which corresponds simply to a tube with an open end) to \(90^\circ\). Some of these diagrams, obtained in operation at a wavelength of \(50\ \mathrm{cm}\), are shown in Fig. 5. We see that the directivity of the radiation at first increases with increasing angle \(\Phi_0\). For \(\Phi_0\) lying between \(40^\circ\) and \(60^\circ\), the sharpness of the radiation becomes greatest, while at still larger aperture angles the sharpness of the beam is disturbed by the appearance of secondary loops, which make it less directional. Thus, in order to obtain the sharpest beam under the given conditions, an aperture angle of \(40\)–\(50^\circ\) proves optimal.

The investigation of the dependence of the character of the radiation on the wavelength was carried out at wavelengths from \(98\) to \(44\ \mathrm{cm}\). For this purpose horn II with the optimum aperture angle of \(40^\circ\) was used. The obtained radiation diagrams, in complete agreement with the theoretical predictions, exhibit at first a very rapid and then a gradual increase-

the sharpness of the beam with decreasing wavelength. At a wavelength of 50 cm the sharpest beam was obtained, although the sharpness of the radiation could be considered almost the same both for waves of 67 cm and for waves of 44 cm, i.e., the directivity of the radiation changed essentially little when the wavelength was varied by almost 50%. This feature of the horn radiator may prove very useful in those cases where transmission over a wide frequency band is required.

Hollow systems having the form of a section of pipe or the form of a horn have also been investigated from the point of view of using them as receivers of electromagnetic waves. The first work in this direction was carried out by Southworth and King^25. They investigated the operating features of hollow receivers having various forms, beginning with sections of pipes of different cross sections and sizes and ending with various horn shapes. Thus far only the results of the study of hollow receivers having the form of a section of pipe of circular cross section and the form of a conical horn have been published.

Fig. 6

The general arrangement of the hollow receivers studied is given in Fig. 6. The receiver is a section of pipe of circular cross section, one end of which is open and can be connected either to similar pipes but of different diameter, or to conical horns of different sizes and with different aperture angles. The other end of the pipe is closed by a tuning piston \(P\), in front of which is located a diametral conductor \(R\), provided with a detector \(D\) and a tuning device \(T\). The conductor \(R\) must be situated in the plane in which the electric vector of the received waves lies; it is assumed that the received electromagnetic energy propagates along the pipe in the form of waves of type \(H_1\).

The experiments carried out with these receivers were mainly of two kinds: first, reception directivity diagrams were taken as a function of the length of the horn and of its aperture angle; second, the relative gain in received power with a horn present and without it was measured. The work was carried out at wavelengths from 10 to 15 cm, obtained from a Barkhausen generator located at a distance of 30–100 wavelengths from the receiver.

The results of the experiments concerning the directivity of reception show that hollow receivers of electromagnetic waves behave in all respects almost the same as hollow radiators of the same shapes.

In Fig. 7 several diagrams are given which characterize the directivity of reception as a function of the aperture angle of a conical horn. We see that the sharpness of reception increases with an increase of the aperture angle approximately up to 50°, after which it decreases again. However, this optimum angle applies only to the horn dimensions shown in the figure; in general it was found that for more

for long horns smaller optimum angles are obtained, as was also the case for hollow radiators. The same analogy with radiators is obtained also in the case when we vary the length of the horn while keeping its aperture angle constant (for example, at \(40^\circ\)). Here too the directivity of reception at first increases rather rapidly with increasing horn length, but then, at a certain length, it becomes maximal, and any further increase in the horn length no longer gives any advantages.

Measurement of the relative gain in receiving power in the case of a horn, as compared with reception without a horn under optimum conditions, gives a value of the order of several hundreds. This indicates the very great practical value of receivers of this kind.

Fig. 7

The results obtained up to the present in the study of hollow radiators and receivers are, of course, not exhaustive, since only their very simplest forms have so far been investigated, and there is as yet no reason to regard these forms as the most advantageous. However, even from these preliminary experiments it is clearly evident that hollow radiators and receivers possess a whole series of advantages in comparison with other types of ultrahigh-frequency antennas, such, for example, as parabolic reflectors or complex antenna systems, and that in the region of centimeter waves they will unquestionably win for themselves a firm place.

The devices known up to now for directed radiation (reflectors, antenna systems), although they can give radiation with very high directivity, nevertheless have many shortcomings that somewhat hinder their use. As an example one may point to such shortcomings as the presence in their radiation diagrams of very large side lobes, the strong critical dependence of the nature of the radiation on the length of the radiated wave, the difficulty of tuning the antenna system, the presence of various kinds of insulating supports, instability of the construction, and a number of others. Hollow radiators are to a considerable extent free of these shortcomings,

They give radiation in the form of a single sharply defined central beam without noticeable side lobes, are simple in construction and convenient for operation under any conditions, do not require complicated adjustment, and, most importantly, make it possible to transmit rather broad frequency bands with the same directivity. Owing to these advantages, hollow radiators, and also receivers, can be successfully used for transmitting and receiving television and for radiotelephone communication from point to point over great distances, for marine and aerial navigation, for determining distances, altitudes, directions, and in other very valuable fields.

In the literature there are already data^26 reporting the first experiments in the practical application of hollow radiators having the form of the sectoral horn considered above. Such a horn, when placed in a vertical plane, will give a sharp beam only in this plane; in the perpendicular direction the beam will be diffuse. This beam shape proves very convenient for determining the gliding angle of a descending airplane and can be used successfully to carry out blind landings of aircraft. The first experiments in this direction, despite the fact that they were conducted at rather long wavelengths (from 40 to 100 cm), gave good results, giving hope that, upon going over to shorter waves (5–10 cm), it will be possible to realize a very accurate blind-landing system, which will ensure an accuracy of aircraft touchdown not exceeding the limits of a circle 15 m in diameter.

At the present time the study of hollow radiators is continuing in the direction of investigating various horn shapes, as well as studying the combined action of systems consisting of several coupled horns^27. It turns out that a system of several horns not only gives an advantage over a single horn in the sense of attaining greater directivity with relatively smaller horn dimensions, but also makes it possible to control the radiated beam freely by simple adjustment of the phases between the coupled horns, which in practice may sometimes prove very useful.

Alongside the study of hollow radiators and receivers having the form of tubes, horns, etc., investigation is also being carried out of such radiating systems which are simply hollow resonators having a small aperture in their shell^28,29. Radiators and receivers of this kind, in contrast to horns, can operate only at strictly definite frequencies determined by the dimensions of the resonator. The power radiated from the aperture depends strongly on its dimensions. Usually, in order not to reduce too greatly the quality factor of the hollow resonator, the aperture in its shell is made small (of the order of several tenths of a fraction of \(\lambda\)), as a result of which the radiated power also proves small. The aperture likewise does not possess sharp radiation directivity. To obtain radiation in the form of a directed beam, Reber^28 proposed a combination of a parabolic reflector and a hollow resonator, the radiating aperture of which is located precisely at the focus of the reflector. On the other hand, Neumann^29 proposed…

so-called multi-hole or diffraction antennas, in which the desired radiation pattern is obtained as a result of the joint action of several apertures, the dimensions and arrangement of which have been selected in a definite way. Such radiators and receivers, while possessing certain features that may be very useful in practice, nevertheless, in simplicity of construction and ease of obtaining sharply directional action, are undoubtedly inferior to electromagnetic horns.

Radiation from an aperture may be successfully used in a number of other fields—those in which a high directivity of radiation is not essential, while the radiated power can be made considerable even with small aperture dimensions, owing to the fact that the aperture is located at the point where the electromagnetic field is most concentrated. Such systems concentrating the electromagnetic field will be discussed below.

B. HOLLOW CHANNELIZING SYSTEMS

At the present time it may be said that almost all phenomena connected with the propagation of electromagnetic waves in hollow conductors of various cross sections have, from the mathematical point of view, already been elucidated in considerable detail. The experimental verification of the theoretical conclusions, although it cannot yet be considered complete, has also already been carried out to a considerable extent, especially for the simplest types of waves and the simplest pipe cross sections.

However, most of the work carried out up to the present time has had as its object of study only pipes with a constant transverse cross section and perfectly straight pipes, having no bends or branches. In practice, however, when, for example, implementing a system for feeding antennas by means of hollow feeders or transmitting electromagnetic energy through pipes over large distances, as is readily seen, only in rare cases can one confine oneself to the use of pipes of the simplest kind, i.e. straight pipes with constant cross section. It may often happen that in a channelizing system it will be necessary, for example, to connect two pipes having different cross sections; in the simplest case this can be done by means of a section of conical, pyramidal, or wedge-shaped pipe, depending on the shape of the cross section of the pipes being connected. It may also prove necessary to change the direction in which the electromagnetic energy is transmitted, or to transmit it simultaneously in several directions. This can be accomplished by means of sections of pipes bent at one angle or another, or of pipes having, for example, the form of a tee or some other branch. All this makes necessary the theoretical and experimental study of the properties of such more complex forms of pipes.

Indications of the use, in experimental practice, of pipes of conical form are found in Southworth,³⁰ who used them to connect circular pipes of different diameters. Barrow³¹ reports

There is evidence of the use, in certain cases, of tubes bent at a right angle, serving to change the direction of energy transmission; similarly, bent tubes are used by him in horn systems[^27] for coupling between separate horns. However, the phenomena that occur in the propagation of waves through tubes of variable cross-section or in branched tubes have as yet almost nowhere been examined in detail. The study of these phenomena proves to be extremely interesting not only from the point of view of using such tubes for various kinds of connections in channeling systems or for changing the direction of energy transmission, but also from several other aspects. It turns out that tubes of variable cross-section can be successfully used for the concentration of electromagnetic energy, accomplished by placing the feeding device in the wide part of the tube and the radiating aperture, playing the role of an antenna, in its narrow part; this is undoubtedly of great practical interest. Branched tubes, in some

Fig. 8

cases, make it possible to transform one type of wave into another, which is also interesting from the standpoint of finding the best methods of exciting a wave of the desired type.

The idea of using tubes of variable cross-section as transmitting systems that simultaneously concentrate electromagnetic energy was first put forward by N. N. Malov[^32],[^33]. In order to obtain quantitative data on the possible degree of such concentration of the electromagnetic field as a function of the shape of the tube, its dimensions, etc., N. N. Malov examined in detail two tubes whose shapes are most convenient for practical application (Fig. 8, a, b). The first of them (Fig. 8, a) has a variable rectangular cross-section and, by its form, may be called a wedge-shaped tube; the second (Fig. 8, b) has the form of a truncated cone. For a mathematical description of the character of the waves arising in such tubes, and in order to determine the conditions for their occurrence, coordinate systems corresponding to the shapes of the tubes are chosen: cylindrical for the wedge-shaped tube and spherical for the conical tube.

All subsequent calculations are carried out under the assumption that the tubes are closed on both sides by reflecting walls \(A\) and \(B\), having respectively the form of cylindrical surfaces for the first tube and the form of spherical surfaces for the second. The radiating

the apertures located in the walls \(B\) of both tubes are assumed to be small, so that the internal electromagnetic field can be represented approximately in the form of standing waves of one type or another.

The solution of Maxwell’s equations for the first tube, taking into account the boundary conditions on its walls, leads to the conclusion that in a wedge-shaped tube the simplest possible wave types are \(H_{10}\) and \(H_{01}\). The first of these (\(H_{10}\)) in many respects resembles the corresponding wave in a tube of constant rectangular cross-section; namely, it has a sinusoidal distribution of the fields along the coordinate \(y\) and a uniform distribution along the coordinate \(\varphi\). The dimension \(b\) of the tube along the \(Y\)-axis determines the critical wavelength of this type, which, as in an ordinary rectangular tube, is given by the relation \(\lambda_{\mathrm{cr}} = 2b\). The aperture angle of the tube \(\varphi_0\) has no effect on the field distribution.

The wave of the second type (\(H_{01}\)), considered by Barrow and Chu \(^{24}\), represents, as it were, the same picture of field distribution, but rotated by \(90^\circ\) with respect to the side walls of the tube. Here we have a sinusoidal distribution of the field along the coordinate \(\varphi\) and a uniform one along the coordinate \(y\). The dimensions of the tube along the \(Y\)-axis may be arbitrary, while the magnitude of the aperture angle of the tube \(\varphi_0\), if it varies within small limits, affects only the distribution of the field along the radius \(\rho\); thus, in this case there is no critical wavelength.

Unlike tubes of constant cross-section (where the distribution of the magnetic and electric fields along the axis of the tube, in the presence of standing waves in it, always has a harmonic character), in a tube of the type under consideration the distribution of the fields along the radius \(\rho\) is governed by Bessel functions. This means that the wavelength in the tube is not constant, but changes together with the change in the dimensions of the cross-section and becomes longest in the narrowest part of the tube. At the same time the density of the electromagnetic field also changes; in the narrow part of the tube it may turn out to be several times greater than in its wide part. Comparing the field strength obtained near the radiating aperture in the narrow part of the tube with the field that would be obtained at the same point in an ordinary tube of constant cross-section having the same dimensions and operating under the same conditions as the wedge-shaped tube considered, one can determine the magnitude of the resulting field amplification. For a certain specified aperture of the tube \(\varphi_0\), the magnitude of the field amplification proves to depend on the number \(n\), denoting the number of antinodes (or nodes) in the standing waves formed in the tube between the reflecting walls \(A\) and \(B\). As \(n\) is increased, which can be achieved either by lengthening the tube or by using as short waves as possible, the magnitude of the field amplification also increases. Simple calculations show that even with a comparatively short wedge-shaped tube (\(n\) of the order of \(20\)—\(30\)) a five- or sixfold increase in field strength is possible, which, if desired, can be made considerably larger.

Preliminary experiments have confirmed the possibility of concentrating energy by means of a wedge-shaped tube.

Consideration of a conical tube leads to almost analogous results both with respect to the magnitude of the field amplification obtained and with respect to the dependence of the amplification on the parameters of the tube and on the length of the wave used. However, an essential difference between conical tubes and wedge-shaped ones consists in the fact that excitation in them of waves both of type \(H\) and of type \(E\), possessing circular symmetry (i.e. having a uniform distribution of the fields with respect to the coordinate \(\varphi\)), proves possible not for all aperture angles of the tube, but only for certain quite definite ones. These “critical” angles are different for \(E\) and \(H\) waves, owing to which conical tubes with definite aperture angles, in addition to being used as concentrating systems, apparently may also find application in channeling systems for filtering waves.

Thus, even the first steps undertaken in the study of tubes with variable cross section reveal many extremely important phenomena, interesting both from the physical standpoint and from the standpoint of their practical use.

The study of tubes having bends or branches will in all probability also reveal a large number of new facts, but at the present time there are still very few data relating to tubes of this kind.

Brillouin \(^{34}\) considered qualitatively the question of the influence of small bends of a round tube on the stability of waves of type \(H_0\) propagating in it. Waves \(H_0\) attract attention because of their small attenuation, and therefore the investigation of their stability with respect to small deviations from straightness of the tube is of great practical interest. Brillouin showed that at each bend of a round tube the waves \(H_0\) will generate waves \(E_1\) propagating with the same velocity, which have considerably greater attenuation and will gradually reduce the effectiveness of the waves \(H_0\).

To eliminate this undesirable phenomenon it is proposed to use tubes with an elliptical cross section, in which these two types of waves can easily be separated from one another.

In Buchholz’s work \(^{35}\) the question is considered in detail of what influence the bend of a rectangular tube exerts on waves of type \(H\) or \(E\) propagating in it. The radius of curvature of the bend is here assumed to be very large in comparison with the transverse dimensions of the tube, so that the bent tube differs little from a rectilinear one. Phenomena of wave reflection, which may take place at the bend, are not taken into account. The conclusion obtained is analogous to that previously indicated by Brillouin, i.e. that after passing through the bent part of the tube a wave of some definite type \(E\) or \(H\) is transformed into a wave of mixed type.

All these data suggest that the phenomena of partial transformation of waves from one type into another, predicted by the theory of bent tubes, should increase as the angle of bend increases.

pipe and that they will manifest themselves most distinctly in a pipe bent at a right angle, or in a pipe having a branch perpendicular to it.

The author of the present article2 undertook an experimental study of the phenomena of transformation of waves from one type into another, occurring in pipes with perpendicular branches. The experiments confirmed the assumption stated above and showed that, under known conditions, such a transformation can be carried out in an almost pure form for a whole series of waves.

Thus, for example, by exciting a standing wave of type \(E_0\) in a round pipe \(A\) (Fig. 9), having a branch \(B\) located precisely at the place in the pipe where there is a bundle of the axial electric field of the wave \(E_0\), one can obtain in the branch a wave \(H_1\), polarized perpendicular to the plane passing through the axes of the two pipes. The pipe \(B\) in this case must be chosen of such a diameter that the wavelength of \(H_1\) in it is equal to twice the diameter of pipe \(A\).

Fig. 9

By attaching the feeding device to branch \(B\) and exciting in it the wave \(H_1\), polarized perpendicular to the plane parallel to the axis of pipe \(A\), one can carry out the inverse transformation, i.e., obtain from the wave \(H_1\) in the branch the wave \(E_0\) in the main pipe \(A\).

In an analogous way a number of other transformations can also be carried out, among which the transformation of waves from type \(H_1\) into \(H_0\) is of the greatest practical interest. The point is that waves of type \(H_1\) or \(E_0\) can be excited comparatively easily and by means of rather simple devices. For this purpose an exciting rod is simply introduced into the pipe, arranged either along the axis of the pipe (for waves \(E_0\)) or along its diameter (for waves \(H_1\)); but waves \(H_0\), which in practice are the most interesting, cannot be excited by such simple means. It is therefore natural to suggest first exciting waves of type \(E_0\) or \(H_1\), and then transforming them into waves \(H_0\). A wave transformer of type \(E_0 \to H_0\) was constructed by Clavier and Altovsky3, who successfully used it for obtaining \(H_0\)-waves in all their experiments with these waves.

Using the wave-transformation phenomena described above in a branched pipe, the author succeeded in constructing an analogous transformer, but of type \(H_1 \to H_0\). Its construction is shown in Fig. 10. In a transverse section of pipe \(A\), passing through the axis of the lateral

tubes, an annular conductor \(C\) is placed, the radius of which is chosen so that the conductor is located along the circumference corresponding to the maximum of the electric field of the wave \(H_0\) \((r=0.48R)\). The wave \(H_1\), excited in the side tube \(B\), must be polarized in a plane parallel to the axis of the main tube \(A\). Falling on the annular conductor, it excites in it circular currents, which serve as the source of the waves \(H_0\) in tube \(A\). To suppress waves of type \(E\) that may arise in this process, at a certain distance from the annular conductor there is placed a volumetric filter \(V\) (analogous to that of Klav'e and Altovsky),

Fig. 10

which delays the waves \(E_0\), but freely passes \(H_0\). The piston \(P\) serves for tuning the system; the same tuning piston is also located at the beginning of tube \(A\) (not shown in the figure).

Tests of such a converter showed that, with properly chosen tube dimensions and a suitable generator frequency, it can serve as a quite satisfactory source for obtaining waves of type \(H_0\).

LITERATURE

1. N. N. Malov, Uspekhi fizich. nauk, 23, 403, 1940.
2. G. Mie, Ann. Physik, 25, 377, 1908.
3. Debye, Ann. Physik, 30, 57, 1909.
4. Wolfsohn, Handb. d. Physik, 20, 291, 307.
5. Hansen and Backerley, Proc. I. R. E., 24, 1594, 1936.
6. Hansen, J. Appl. Physics, 9, 654, 1938.
7. Borgnis, Ann. Physik, 35 (5), 359, 1939.
8. Bunimovich, Zhurnal tekhnich. fiziki, 9, 984, 1939.
9. Neĭman, Izv. el.-prom. sl. toka, No. 9—11, 1939.
10. Barrow, Rev. Sci. Instr., 9, 170, 1938.

1. Goodman, QST dev. to amat. Radio, 24, 33, 1940.
2. E. M. Studenkov, Uspekhi fizich. nauk, 23, 417, 1940.
3. Kovalenko, Izv. Akad. nauk SSSR, 4, 449, 1940.
4. V. I. Kalinin, Izv. Akad. nauk SSSR, 4, 532, 1940.
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6. Hansen and Richtmyer, J. Appl. Physics, 10, 189, 1939.
7. Hansen, J. Appl. Physics, 10, 38, 1939.
8. Barrow and Mieher, Proc. I. R. E., 28, 184, 1940.
9. Patrushev, Izv. Akad. nauk SSSR, 4, 571, 1940.
10. Lamont, Phil. Mag., 30, 1, 1940.
11. Barrow, Proc. I. R. E., 24, 1298, 1936.
12. Barrow and Greene, Proc. I. R. E., 26, 1498, 1938.
13. Barrow and Lewis, Proc. I. R. E., 27, 41, 1939.
14. Barrow and Chu, Proc. I. R. E., 27, 51, 1939.
15. Southworth and King, Proc. I. R. E., 27, 95, 1939.
16. Electronics, No. 11, 12, 1939.
17. Barrow and Schulman, Proc. I. R. E., 28, 130, 1940.
18. Reber, Communications, No. 12, 5, 1938.
19. Neiman, Izv. el.-prom. sl. toka, No. 6, 1, 1940.
20. Southworth, Bell. Syst. Techn. Journ., 15, 284, 1936.
21. Chu and Barrow, Proc. I. R. E., 26, 1520, 1938.
22. N. N. Malov, Elektrosvyaz (in press).
23. N. N. Malov, ZhETF (in press).
24. Brillouin, Electr. Communic., 16, 350, 1938.
25. Buchholz, Elektr. Nachr. Techn., 16, 73, 1939.
26. E. M. Studenkov, Dissertation, Moscow, 1941.
27. Clavier and Altovsky, Electr. Communic., 18, 81, 1939.

1. See p. 430 of this issue. ↩