ULTRAHIGH-FREQUENCY WAVES AND THEIR PRACTICAL APPLICATIONS1

L. Brillouin

1. SOME CHARACTERISTIC FEATURES OF ULTRAHIGH-FREQUENCY WAVES

Ultrahigh-frequency electromagnetic waves, i.e. waves with a length of several decimeters, were first used by Hertz to repeat optical experiments, such as reflection, refraction, diffraction, etc.; the complete similarity of short radio waves to optical waves was subsequently confirmed by a large number of experiments. It turned out that optical lenses, mirrors, and parabolic reflectors are very effective, provided only that their dimensions are greater than several wavelengths.

Later another analogy was discovered—between ultrahigh-frequency and acoustic waves. Since in both cases the wavelength is of one and the same order of magnitude, it is now possible to point to a whole series of electrical devices surprisingly similar to acoustic instruments: hollow tubes similar to organ pipes, hollow resonators (also called tank resonators, endovibrators) resembling Helmholtz acoustic resonators, and hollow cables, which are analogous to the tubes used in acoustics.

The idea of using hollow resonators arose chiefly because they give very little damping, i.e. have a very high \(Q\) (quality factor). A high \(Q\) means that the amplitude of free oscillations in the resonator decreases to \(1/e\) of its initial value only after the passage of a very large number of periods.

In connection with this there is another question deserving attention: the possibility first of storing energy in a hollow resonator and then radiating it into space.

If we wish to accumulate electromagnetic energy in some volume \(\Phi\), this can be done by creating strong electric \(E\) and magnetic \(H\) fields inside the given

volume; the total store of energy will be determined by the integral

\[ A=\frac{1}{8\pi}\int(\varepsilon E^2+\mu H^2)\,d\Phi, \tag{1} \]

where the electric and magnetic parts of the total store of energy are usually quantities of the same order.

But now the question arises: to what magnitude can we increase the fields? Let us suppose that the dielectric medium is air at atmospheric pressure. Its breakdown field is approximately \(30000\ \mathrm{V/cm}\), i.e. \(100\) CGSE. In practice this upper limit will never be reached, and taking an acceptable safety factor, we may assume that on average the field can reach \(15\) CGSE. This corresponds to an energy density

\[ A=\frac{1}{8\pi}15^2\simeq 10\ \mathrm{erg/cm^3} \tag{2} \]

or about \(1\ \mathrm{joule/m^3}\). Thus, for the possible order of magnitude of the total energy store we take

\[ A(\mathrm{erg})=10\Phi, \tag{3} \]

where the volume \(\Phi\) is expressed in cubic centimeters.

The storage of a large amount of energy \(A\) requires the use of a resonator of large dimensions \(\Phi\).

Let us suppose now that some quantity of energy is radiated every second. A good resonator must have a factor \(Q\) of the order of \(1000\), to which there corresponds a time constant \(\theta\) of about a thousand periods \((\tau)\):

\[ \theta=1000\tau. \tag{4} \]

The energy radiated per second, \(\frac{dA}{dt}\), may therefore be obtained from the relation

\[ \frac{dA}{dt}=\frac{A}{\theta}=\frac{A}{1000\tau}=\frac{\Phi}{100\tau}. \tag{5} \]

Let us suppose that we are designing a resonator capable of radiating a power of \(100\ W\) \((10^9\ \mathrm{erg/sec})\). Its volume will be related to the period by means of equation (5):

\[ \frac{dA}{dt}=10^9,\qquad \Phi=10^{11}\tau\ \mathrm{CGS}. \]

This result can be made more visual if we compare the dimensions of the volume \(\Phi\) with the wavelength \(\lambda\). Assuming the volume to be in the form of a cube

\[ \Phi=L^3, \]

we find

\[ L^3=10^{11}\frac{\lambda}{c}=\frac{10}{3}\lambda\ \mathrm{CGS}, \]

where \(c\) is the speed of light; the preceding equation may be written in the following form:

\[ \left(\frac{L}{\lambda}\right)^3=\frac{10}{3\lambda^2} \tag{6} \]

(\(\lambda\)—in centimeters), which leads to the following results:

a) meter radio waves

\[ \frac{L}{\lambda}\ll 1; \tag{7a} \]

to this case there correspond circuits with lumped constants;

b) decimeter radio waves

\[ \frac{L}{\lambda}\approx 1; \tag{7b} \]

c) centimeter or millimeter waves

\[ \frac{L}{\lambda}\gg 1 \quad\text{— hollow resonators.} \tag{7c} \]

For decimeter radio waves the use of ordinary capacitive-inductive circuits is impossible; one has to use two-wire or coaxial lines. These systems, however, can be rationally constructed only when the distance between the conductors is small in comparison with the wavelength. This cannot be achieved in the case of centimeter waves, and we are therefore compelled to work with hollow resonators, even (owing to the fact that \(L>\lambda\)) with resonators oscillating at higher harmonics.

Although hollow resonators are similar to acoustic ones, there are nevertheless fundamental differences here, and the complete theory of hollow resonators, hollow tubes, etc., for the case of electromagnetic oscillations must be constructed anew. The difference consists in the fact that acoustics deals with longitudinal waves, whereas in electromagnetism transverse waves are involved. An acoustic field can be defined by a single quantity (the velocity potential), whereas an electromagnetic field requires four quantities for its definition (vector and scalar potentials). Rayleigh’s acoustic calculations, brilliantly carried out, can be used only as guides or as formal examples, but they must still be completely translated into the language of transverse waves and Maxwell’s equations; this translation often requires a good command of the mathematical apparatus.

2. THE FIRST EXPERIMENTS OF CLAVIER AND DARDBER

The L. M. T. laboratory (“Laboratorie le Matériel Téléphonique”) in Paris has for a number of years been actively working in the field of ultra-high frequencies. The investigations are being conducted by Clavier, Dardber, and many other collaborators. As a result of these investigations, radio communication across the English Channel was established on a wavelength of \(18\ \mathrm{cm}\). The first experiments were carried out in 1931, and the system was at that time

in commercial use since 1934. For the generation and reception of ultrahigh-frequency waves, tubes with a retarding field were used. In Fig. 1, a gives a simplified diagram of the transmitter, and in Fig. 1, b—of the receiver. In both cases the generated (or received) oscillations are fed (or taken off) by means of a coaxial feeder to a small antenna (or from it), situated at the focus of a parabolic reflector.

The reflectors form a very slightly divergent beam, directed from the transmitting station to the receiving station. The position of the reflectors must be such that there are no obstacles in the path of the wave beam.

Fig. 1. Diagrams of the transmitter and receiver

$M$ — parabolic mirrors; $S$ — hemispherical reflectors; $D$ — dipole antennas
$F$ — coaxial feed lines; $I$ — generation indicator; $A$ — amplifier

obstacles. In communication across the English Channel, one of the stations was located on the French side, the other on the English side. The distance was approximately 56 km, and the towns were chosen so that the entire communication line was free of obstacles. The electro-optical devices were mounted on special steel towers 20 m high.

The installation was used for two-way Baudot telegraphy, as well as for duplex telephony. All details concerning the arrangement of the stations had already been published in several articles$^{1,2}$. Of all communication lines in commercial use, this line uses the shortest waves. The operating results of the line proved very good; in particular, atmospheric interference was almost entirely absent, although occasional sharp clicks were noted, the cause of which clearly could not lie in the apparatus used. Interference from nearby lightning discharges and automobiles was likewise not observed. The background noise in the receiver was very similar to normal noise

lamps. Slow fadings were sometimes observed, but they appeared very rarely, so that in telegraphing signals lasting a short time they proved immaterial. Rain and fog had no noticeable effect on radio communication; however, as a rule, stable atmospheric conditions corresponded to steady communication, whereas an unstable state of the atmosphere was usually accompanied by fadings.

3. HOLLOW CABLES; EXPERIMENTAL PART

A very important step forward in the study of ultra-high-frequency waves was made by the staff of the Bell Laboratory (USA)—Southworth, Carson, Mead, and Schelkunoff ^{3,4}, who were the first to begin investigating the propagation of ultrashort waves inside hollow conducting tubes (so-called hollow cables). Similar investigations were begun at almost the same time by Barrow at the Massachusetts Institute of Technology ^5. The first reports published by these investigators aroused the interest of many physicists. The author investigated the propagation of waves in rectangular tubes, and then in tubes of elliptical cross-section ^6. Recently many further investigations have appeared ^{7,8}. We shall dwell only on the investigations carried out by the author together with Clavier. Clavier gave brilliant experimental demonstrations before the French Physical Society in November 1938, using the equipment shown in the following figures.

Fig. 2

Very short waves were produced by a lamp with a braking field or with the aid of a magnetron.

Fig. 2 shows a magnetron generator producing waves from 1.2 to 3 cm. These waves could propagate inside a narrow copper tube 1.6 cm in diameter. For measuring the wave, a coaxial wavemeter with a movable piston was used; it is visible in the same figure near the magnetron.

Fig. 3 gives the arrangement of the experiment with hollow cables as it was demonstrated before the French Physical Society. Two different tubes were used: one—a large one, visible in the foreground of the figure, 12 cm in diameter (it served Clavier for demonstrating before the audience the distribution of the electric ...

fields inside the tube for various types of waves); the second, narrower tube, as already indicated above, had a diameter of 1.6 cm.

Fig. 3

Figure 4 shows a device that served to demonstrate the distribution of electric lines of force for various types of waves. A small dipole antenna could rotate inside the tube; its received signals, after amplification, were reproduced by a small lamp, which was oriented in the corresponding manner on the placard.

Fig. 4

Fig. 5

Figure 5 depicts a so-called wave converter, serving to change the type of wave from \(E_0\) to \(H_0\). The converter consists of small radial antennas arranged inside the tube and

having a curved part at the place where the wave of type \(H_0\) has the maximum electric field. The antenna system is located between two filters, of which one consists of radial conductors that delay the waves \(E_0\) but pass the waves \(H_0\), while the other (behind) consists of circular conductors and delays the waves \(H_0\), whereas the waves \(E_0\) can pass through it without hindrance. In Fig. 6 the antenna system of the indicated converter is shown.

Fig. 6

More detailed information concerning all this equipment can be found in the works of Clavier and Altovsky\(^{7,8}\).

4. THEORY OF HOLLOW CABLES

A hollow conducting tube for electromagnetic waves is an ordinary filter, transmitting a wave only in the case when its frequency is above some limiting frequency. This result was one of the astonishing facts obtained from the first investigations of Southworth. It can easily be understood from Fig. 7, in which the curves represent the variation of the quantity

\[ K=\frac{c}{v}=\frac{u}{c} \tag{8} \]

as a function of the ratio \(\frac{\lambda}{d}\), i.e. of the wavelength to the diameter of the tube. Here \(c\) is the velocity of light in free space, \(v\) is the phase velocity of the waves propagating in the tube, and \(u\) is the group velocity, or the speed of the signal of the waves propagating along the tube. The solid curve represents the theoretical curve for waves \(E_0\) in a tube filled with water (\(\varepsilon = 81,\ \sqrt{\varepsilon}=9\)); the dashed curve represents the same for a tube filled with air. Along the tube there can propagate only such a wave whose length is less than the limiting (critical) one.

Fig. 7

The limiting wave in this case will be determined by the condition \(\frac{\lambda}{d} \simeq 1.7\).

Fig. 8 gives an analogous curve for \(H_1\) waves in a tube filled with air, where the limiting wave will be \(\frac{\lambda}{d}=1.67\). The physical interpretation of these results will be given below. It should be noted that the theoretical curves given in Figs. 7 and 8 are ellipses.

Fig. 8

Fig. 9 is also taken from Southworth’s work; it demonstrates the attenuation of various types of waves (called \(E_0\), \(E_1\), \(H_0\), \(H_1\)) as they propagate inside a copper tube. Remarkable is the drop of the curve for \(H_0\) waves; the higher the frequency, the smaller the attenuation. This unexpected result may prove very useful for communication by hollow cables over long distances.

Fig. 9

If we consider the phenomenon in more detail, this result can be explained in a few words. Each type of wave induces electric currents in the sheath of the tube. These currents, owing to the skin effect, must flow within very thin layers, and these layers become thinner and thinner as the frequency increases. This causes an increase in losses as Joule heat within the layers for most types of waves. Thus, a higher frequency corresponds to greater attenuation. The \(H_0\) waves, however, behave somewhat differently; the intensity of the current induced by them in the wall of the tube falls very rapidly as the frequency increases, and this effect proves more than sufficient to compensate the increase in the resistance of the layer. Thus, as the frequency increases, the attenuation decreases.

Let us now return to the elliptical curves shown in Figs. 7 and 8 and explain their physical meaning. Remarkable is

the fact that the velocity of waves inside the tube differs essentially from the velocity of waves in free space. It turns out that inside a conducting tube there exists a moving interference pattern, i.e. there is formed there a certain distribution of alternating maxima and minima, and then it moves along the tube without distortion. Hence it is quite clear that the phase velocity (or propagation velocity) for such an interference pattern must be different from the usual velocity of light in free space. Some elementary examples may give a simple explanation of these properties. Fig. 10 depicts the interference pattern which takes place in front of a mirror \(M\), if a wave \(I\) is incident on it at an angle \(\theta\), and is then reflected, forming the reflected wave \(R\). The interference pattern is formed in the triangular region in front of the mirror, where the incident and reflected waves are superposed on one another.

Fig. 10

The maxima of intensity, which are denoted by dark dots, are formed at the intersections of the wave planes of the two waves. If \(\lambda\) is the wavelength in free space, then here there arises a longer wave \(\Lambda\), which represents the distance between two points measured parallel to the mirror. It should be noted that the motion of the entire interference pattern occurs parallel to the mirror, and the whole system of waves slides with velocity \(v\). This at once leads to the very simple result:

\[ \frac{c}{v}=\frac{\lambda}{\Lambda}=\sin\theta . \tag{9} \]

The dark lines, parallel to the mirror, are bands where the intensity is zero. They divide the triangle into a system of parallel layers, in which there is practically no flow of energy from one layer to another; this once again proves that the resultant velocity of the waves \(v\) is directed parallel to the mirror.

The conditions that arise on the surface of the mirror are likewise fulfilled on each of the dark bands. This means that, for example, in the case of electromagnetic waves reflected from a conducting horizontal mirror \(M\), the horizontal electric field vanishes both at the mirror and at every dark band. Thus we can, without disturbing the interference pattern, place a second mirror \(M'\) parallel to the first \(M\) on one of these dark bands, as shown in Fig. 11, which represents a system of interfering waves propagating between two mirrors with phase velocity \(v\). This corresponds to the propagation of waves inside a conducting...

…tube, if we imagine a tube with a rectangular cross-section, one of whose sides is infinitely long.

The phase velocity given by formula (9) is greater than the velocity of waves \(c\) in free space by a factor of \(\dfrac{1}{\sin \theta}\). The energy flux travels with a somewhat different velocity \(u\), smaller than the velocity of waves in free space \(c\):

\[ u = c \sin \theta; \tag{10} \]

this result becomes clear when considering the zigzag motion of rays between the mirrors \(M\) and \(M'\) (Fig. 12). On the other hand,

Fig. 11              Fig. 12

the angle \(\theta\) is related to the distance \(d\) between the mirrors and the integer \(n\), which determines the order of the overtone excited in the tube, by the following relation:

\[ d = \frac{n\lambda}{2\cos\theta} = \frac{nc}{2f\cos\theta}. \tag{11} \]

Formulas (9) and (10) lead to relation (8). Eliminating \(\theta\) from (9) and (11), we obtain

\[ \left(\frac{c}{v}\right)^2 + \left(\frac{nc}{2fd}\right)^2 = 1, \tag{12} \]

i.e. the equation of the ellipse shown in Figs. 7 and 8, where the ordinates are \(\dfrac{c}{v}\), and the abscissas are \(\dfrac{\lambda}{d} = \dfrac{c}{fd}\).

The critical frequency \(f_1\) corresponds to the limiting conditions

\[ v = \infty,\qquad u = 0,\qquad f_1 = n\frac{c}{2d},\qquad \theta = 0. \tag{13} \]

This very simple example is important for consideration, since it directly gives the most typical features of wave propagation inside hollow tubes and explains in an elementary way the relations between the velocities.

5. RECTANGULAR CROSS-SECTION

This problem can be studied without great mathematical difficulties. Inside the cross-section of the tube the interference pattern is determined by two constants \(n_1\) and \(n_2\), giving the number of nodal lines in both directions. The theory gives the following relations for the phase velocity \(v\) inside the tube:

\[ k^2 = \pi^2 \left[\left(\frac{n_1}{a}\right)^2 + \left(\frac{n_2}{b}\right)^2\right], \tag{14} \]

\[ \left(\frac{c}{v}\right)^2 + \left(\frac{kc}{2\pi f}\right)^2 = 1, \tag{15} \]

where \(a\) and \(b\) are the two sides of the rectangular cross-section, \(f\) is the frequency, and \(k\) is a quantity characterizing the distribution corresponding to the type \(n_1, n_2\). The limiting frequency \(f_1\) is obtained from (15) if \(v\) is made infinite.

\[ f_1=\frac{kc}{2\pi}. \tag{16} \]

Equations (14), (15), and (16) are very similar to equations (12), (13) of the preceding section; in particular, equation (15) is represented in exactly the same way by elliptic curves.

Fig. 13

Writing Maxwell’s equations, we find that each of the constants \(n_1, n_2\) can be used to define two different types of oscillations, of which one gives an electric wave and the other a magnetic one. These names arise from the fact that the first wave has a longitudinal component of the electric field, but has no longitudinal magnetic component. For the magnetic wave the conditions are the reverse; only one difference may be noted, namely that the electric wave disappears completely if only one of the constants \(n_1, n_2\) is made zero, whereas the magnetic wave can still be formed in this case as well.

The general result is that the quantity reciprocal to \(k\) (i.e. \(k^{-1}\)), and the critical wavelength \(\lambda_c\), for a fixed type of distribution, are proportional to one of the linear dimensions of the cross-section of the tube. This gives us the possibility of clarifying the role of the shape of the cross-section by reducing the cross-sectional area to a constant value and choosing

\[ ab=1. \]

In Fig. 13 a diagram is given showing the dependence of \(\left(\dfrac{k}{\pi}\right)^2\) on \(a\); \(a=1\) denotes a square cross-section; \(a>1\) represents a rectangle in which the horizontal side \(a\) is longer than the vertical side \(b\).

Such a diagram is very interesting for discussion. It characterizes the stability or instability of various types of waves with respect to deformations of the cross section. Some curves (for example, \(n_1 = 1\), \(n_2 = 1\), or curve \(11\)) pass through a minimum for a square cross section (\(a = 1\)); this means that a small change in the cross section will only slightly disturb the wave and the phase velocity. These types of waves are stable. But other types of waves behave quite differently. Consider, for example, the waves \(01\) and \(10\), which give two separate curves intersecting for a square cross section. A certain deviation from a square cross section toward a slightly rectangular one affects the wave by dividing it into two parts (\(01\) and \(10\)), propagating with different velocities. This case corresponds to unstable waves.

6. CIRCULAR AND ELLIPTICAL CROSS SECTIONS

Although interesting in itself, the case of tubes with a rectangular cross section may not be of great practical importance, since in all probability tubes of circular cross section will be used more in practice; consequently, the deformations in this case will consist of deviations toward an elliptical form. This explains the investigations, undertaken by various authors \(^{6,9,10}\), of wave propagation inside tubes of elliptical cross section. A complete consideration leads to the principal types of oscillations having elliptical or hyperbolic nodal lines, as shown in Fig. 14.

Fig. 14. Nodal lines for elliptical cross sections

The natural tones of the oscillations are determined by Mathieu functions, numerical calculations of which are rather tedious. The pictures obtained are in general similar to the preceding ones. Denoting by \(a\) and \(b\) the two semiaxes of the ellipse and introducing the constant area \(ab = 1\), we can draw the curves given in Figs. 15a and 15b for electric and magnetic waves. Curves having a minimum for a circular cross section (\(a = 1\)) correspond to stable waves, whereas unstable waves are represented by intersecting curves. Thus the wave \(E_0\) is stable, the wave \(E_1\) is unstable, the wave \(H_0\) is stable, and the waves \(H_1\), \(H_2\), and \(H_3\) are unstable.

A very important fact is the stability of the waves \(H_0\), since these waves can have very great practical significance owing to their small attenuation, as has already been indicated above.

One could clarify many more interesting features from consideration of the curves and diagrams shown in the preceding figures, but this would go beyond the scope of the present article.

In conclusion, it should be emphasized the enormous interest represented—

Fig. 15a. Electric waves

Fig. 15b. Magnetic waves

—by ultrahigh-frequency oscillations for both the physicist and the radio engineer. Here a new field of activity opens up for research, with highly interesting theoretical and technical problems and prospects for important practical applications.

References

1. Clavier, Electr. Communic., 12, 3, 1933.
2. Clavier and Gallant, Electr. Communic., 12, 222, 1934.
3. Southworth, Bell Syst. Techn. Journ., 15, 284, 1936.
4. Carson, Mead and Schelkunoff, Bell Syst. Techn. Journ., 15, 310, 1936.
5. Barrow, Proc. I. R. E., 24, 1298, 1936.
6. Brillouin, Rev. Gen. d’Électricité, 40, 227, 1936; Electr. Communic., 16, 350, 1938; Bull. Soc. Franç. d’Électr., 8, 899, 1938.
7. Clavier, Bull. Soc. Franç. d’Électr., 8, 385, 1938.
8. Clavier and Altovsky, Rev. Gen. d’Électr., 45, 697, 1939.
9. L. J. Chu, J. Appl. Physics, 9, 583, 1938.
10. Schelkunoff, J. Appl. Physics, 9, 484, 1938.

1. J. Frankl. Inst., 229, 709, 1940. Abridged translation by E. M. Studenkov. In addition to this article, see the article by E. M. Studenkov on p. 444 of this issue. ↩