INTERFERENCE, DIFFRACTION, SPECTRAL DECOMPOSITION IN OPTICS AND RADIO

G. S. Gorelik

This article is didactic in character. The desire to write an article of this kind for the anniversary issue of Uspekhi Fizicheskikh Nauk may, I hope, be justified by the fact that many readers of Uspekhi teach physics and often make use in their teaching of the reviews published in it.

1. DOES THE PLACE OF RADIOPHYSICS IN TEACHING CORRESPOND TO ITS PLACE IN SCIENCE?

One of the striking features of the development of physics over the last 30 years is the place that radiophysical lines of inquiry and methods of research have won in it. The study of electrical fluctuations, of the frequency dependence of the electrical, magnetic, and mechanical properties of matter, radiospectroscopy, radioastrophysics; the abundance of tube amplifiers, electronic oscilloscopes, high-frequency generators in acoustic, optical, nuclear, and all sorts of other laboratories; the creation of cyclotrons and synchrotrons—this is far from a complete list of examples. The editor of this journal recently quite rightly noted¹ that radio engineering has placed at the disposal of physicists “entirely new means that have revolutionized the whole technique of physical experiment.” The language of radiophysics (for example, the term “modulation”) is penetrating general physical terminology.

But here, as in many other respects, teaching (not only in secondary school, but often in higher school as well) lags behind the development of living science by at least several decades.

It is considered inadmissible for a student taking an examination in physics not to know the construction of a prismatic binocular, various kinds of photometers, or an organ pipe. Yet few are troubled by whether he is familiar with the construction of an electronic oscilloscope, the principle of operation of directional antennas, or the idea of the interference method for measuring the velocity of propagation of radio waves, created by Mandelstam and Papaleksi. It is hardly possible to prove that the first group

questions has greater practical or general-educational significance than the second.

Is there a more visual and more convenient way of studying the addition of mutually perpendicular oscillations than by means of an electronic oscilloscope? Should one, in setting forth the basic molecular-kinetic ideas, remain silent about the fact that noise in radio apparatus has a fluctuation origin and depends on the temperature of the input circuit; that it is precisely fluctuations which limit the range of radar stations; that the classical expression $kT$ enters into the engineering formula by which this range is calculated?

It seems to me that it is high time substantially to increase the place occupied by radiophysics in curricula and textbooks, in lectures and laboratory work in general physics. I should also like to draw attention to another side of the matter.

What L. I. Mandelstam called the “international language of the theory of oscillations”—a unified approach to all oscillatory phenomena: mechanical, acoustical, radiophysical, optical—is acquiring ever greater importance in science. Such an approach is not only legitimate, but also productive in the highest degree: oscillatory intuition acquired in one field, when necessary, is immediately and naturally used in another. This approach has already entered courses on the theory of oscillations taught at some of our universities. The experience of teaching at Gorky University shows that such an approach is possible and also yields positive results in the general physics course. It not only brings teaching closer to scientific research, but to a great extent saves time and eases the work of both lecturer and student. The effectiveness of a lecture devoted, say, to the addition of harmonic oscillations is at least tripled when it is illustrated by both acoustical and electrical experiments, and when, over the course of several weeks, the theorems derived in it are applied successively to alternating currents, to acoustics, and to optics. With such a unified approach, radiophysical facts acquire a significance going beyond what was spoken of at the beginning. Radiophysical facts often appear as those on which it is easiest to develop oscillatory intuition, as a key to understanding, say, the most difficult points of optics. This is connected above all with the fact that in radiophysics we deal with spatial scales (wavelength) and time scales (duration of the cycle) far closer to us than in optics, and can see (oscillograph) each individual oscillation, which is fundamentally impossible for light.

The purpose of the following paragraphs is to illustrate this idea with several examples.

2. ANTENNAS AND THE DIFFRACTION GRATING

A very elementary, elegant, and eloquent application of the theorems on interference (i.e. superposition) of sinusoidal waves is the theory of the directional action of the simplest antenna systems.

What could be simpler and at the same time more instructive for the beginner than the calculation of the directivity (in the equatorial plane) of two parallel antennas \(A_1, A_2\), separated from one another by a quarter wavelength (Fig. 1), with the current in \(A_1\) leading in phase by \(\pi/2\) the current in \(A_2\), and, in particular, the proof that in the direction indicated by the arrow there is a maximum of radiation intensity, while in the opposite direction there is no radiation?

Fig. 1.

Fig. 2.

The field radiated in the direction \(\theta\) by a grating consisting of \(N\) in-phase antennas, having period \(d\) (Fig. 2), we find as the superposition of \(N\) oscillations of equal amplitude (denote it by \(A\)), whose phases form an arithmetic progression with difference \(kd\sin\theta\) (\(k=2\pi/\lambda\) is the wave number). The amplitude of the resulting field is therefore equal to

\[ A \frac{\sin\left(N\frac{kd\sin\theta}{2}\right)} {\sin\left(\frac{kd\sin\theta}{2}\right)} . \]

This formula coincides with the formula for an optical diffraction grating. But to begin with a grating made of antennas is much more natural. Here the field of each element of the grating is given to us. The case of the optical grating, however, reduces to the one just considered—approximately and not without logical strain—by means of the Huygens–Fresnel principle, whose fruitfulness consists precisely in the fact that it replaces diffraction problems by incomparably simpler problems of the type illustrated by the antenna grating.

The presence of principal and secondary maxima, the disappearance of principal maxima of all orders except the zero order when \(d<\lambda\)—the latter being especially important for understanding the diffraction theory of the microscope—all this is very vividly illustrated by the form of the lobes

customary to a radio engineer, the radiation pattern of a grating. This pattern is so expressive (see, for example, Fig. 3) that it ought finally to be admitted even onto the pages of general physics courses.

3. THE STRUGGLE FOR THE SHORT WAVE IN RADIOLOCATION AND MICROSCOPY

The grating formula written above indicates that the angular width of the central beam (the only principal maximum when \(d<\lambda\)) is equal to the ratio \(\lambda/D\), where \(D=Nd\) characterizes the overall size of the grating; that is, in order to obtain sharp directivity of the central beam it is necessary to have a small value of this ratio. This criterion is also applicable in the limiting case when \(d/\lambda \to 0\), \(N \to \infty\), and in such a way that \(Nd\) and \(N\lambda\) remain constant. From this it is easy to pass to a discussion of the sharpness of directivity of the ultrasonic beam produced by a piezoquartz plate.

Fig. 3.

The criterion \(\lambda/D\) has fundamental significance for understanding the main tendency in the development of radiolocation technology. For \(\lambda\) of the order of 1 meter, obtaining an angular beam width of the order of \(1^\circ\) would require \(D\) of the order of 100 meters. Great angular directivity with not very bulky apparatus can be achieved only by shortening the wave. The “struggle for the short wave,” caused to a large extent precisely by this circumstance, led to the appearance of magnetrons, klystrons, waveguides—everything that makes the newest radio devices so unlike, even in their outward appearance, what could be seen in radio laboratories not only in Popov’s time, but even quite recently.

The Huygens–Fresnel principle makes it possible to transfer (approximately) the criterion \(\lambda/D\) also to the case of a parabolic mirror, in whose focus a microwave antenna is located (Fig. 4). The transition to the theory of the resolving power of optical instruments is obvious. An analogy suggests itself between the struggle for the short wave in radiolocation and microscopy, where it led to the appearance of the electron microscope, in which light is replaced by electrons with their de Broglie wavelength, which is in approximately the same ratio to the wavelength of visible light as the wavelengths used in radiolocation are to the wavelengths in the ordinary broadcasting range.

4. THE OPTICAL PROJECTOR AND THE RADIO PROJECTOR

The radio projector (Fig. 4) is analogous to an inverted telescope, but not to an optical projector. The fact that both in the optical projector and in the radio projector \(\lambda/D \ll 1\) still does not give

grounds to regard them as similar systems. Indeed, the beam of a light searchlight, as everyone knows, is the thicker the larger \(D\) is. The directivity of a radio searchlight is the sharper the larger \(D\) is. What is the reason for this apparent contradiction? This question inevitably arises for any thoughtful student.

The answer to it is simple and, it seems to me, instructive. The criterion for comparing an optical searchlight and a radio searchlight is not the ratio \(\lambda/D\), but the ratio \(\sqrt{\lambda r}/D\), where \(r\) is the distance from the searchlight to the observation point, i.e. the ratio to the size of the apparatus not of the wavelength, but of the size of the central Fresnel zone. It is wrong to think that geometrical optics, on the basis of which the beam of an optical searchlight is the thicker the larger \(D\) is, is applicable to it because for it \(\lambda/D \ll 1\). It would then be applicable in this case also to a radio searchlight. Geometrical optics is applicable to an optical searchlight because \(\sqrt{\lambda r}/D\) remains small for it up to \(r\) of the order of tens of kilometers. For a radio searchlight, on the contrary, even for \(D = 1\ \text{m}\), \(\lambda = 1\ \text{cm}\), we have \(\sqrt{\lambda r}/D \gg 1\) already for \(r\) of the order of one kilometer.

Fig. 4.

Analogous considerations show (contrary to what is sometimes thought) that geometrical optics cannot be applied to the reflection of microwaves from one object or another solely on the grounds that its characteristic dimensions are large compared with the wavelength. Suppose, for example, that a wave with \(\lambda = 1\ \text{cm}\) is incident on a plane metallic mirror of diameter \(D = 1\ \text{m}\). Already at a distance \(r = 1\ \text{km} \gg D^2/\lambda\), what will be observed is not a “spot” reflected according to the law “the angle of incidence is equal to the angle of reflection,” but a Fraunhofer diffraction pattern.

5. QUESTIONS OF COHERENCE*)

The nonmonochromaticity of light explains the fact that, in the superposition of light emitted by two independent (incoherent) sources, the resulting intensity is always the sum of their “partial” intensities, whereas if the oscillations were strictly monochromatic, the resulting intensity could be both greater and less than this sum. But it is wrong to think—this was repeatedly explained by L. I. Mandelstam \(^{2,3}\)—that only opti-

*) The lecture demonstrations described in §§ 5, 6 were set up by V. S. Troitskii at Gorky State University.

cal waves fundamentally cannot be absolutely monochromatic. The same is true of those oscillations that arise in any acoustic or radiophysical device. And if we can easily observe, in the superposition of two acoustic or radio oscillations generated by independent sources, the difference between the resulting intensity and the sum of the partial intensities, then here the matter is not a fundamental difference from light, but a difference in time scales. If, in the case of such acoustic or radio oscillations, the experiment is made to last sufficiently long, the average intensity will also be equal to the sum of the average partial intensities.

Fig. 5.

It is easy to carry out radiophysical experiments that vividly illustrate this circle of ideas*).

If we apply to the input of a radio receiver a voltage proportional to the anode current of a tube operating in the saturation regime, we shall hear in the loudspeaker, owing to the shot effect of this tube, a powerful fluctuating noise. If, at the output of the receiver, we replace the loudspeaker by a weakly damped low-frequency circuit (Fig. 5), we shall obtain in it nonmonochromatic oscillations, to be sure, but very weakly nonmonochromatic ones, which resemble thereby a narrow spectral line. Roughly speaking, one obtains successive wave trains of oscillations with a frequency equal to the natural frequency of the circuit (the carrier frequency). Each wave train contains a limited, but large, number of individual oscillations; the phase and amplitude change irregularly from train to train: a chaotic modulation of amplitude and phase takes place (Fig. 6). If this modulation is sufficiently slow, it can be followed directly by eye by applying the oscillation from the circuit to the vertical-deflection plates of an electronic oscilloscope and choosing the sweep period so that it is close to an integral multiple of the natural period of the circuit.

Fig. 6.

In practice it is especially convenient to use, not an \(LC\)-circuit, but a weakly damped (narrow-band) \(RC\)-filter, for example a three-

*) See also \(^{4}\).

chain filter, the principal circuit of which is shown in Fig. 7. Such a device, for \(Sr>29\), where \(S\) is the slope of the tube (in amperes per volt, if \(r\) is in ohms), is a self-oscillating system (an \(RC\) generator). If \(r\) is somewhat smaller than is required by this condition, then the system is equivalent to a weakly damped circuit whose natural period can be varied over wide limits by selecting \(R\) or \(C\), and whose damping (the width of the pass band) can be varied by selecting \(r\). The closer to the threshold of self-excitation \(Sr=29\), the smaller the damping and the greater the duration of the wave trains. With the aid of an \(RC\) filter it is easy to obtain, at a carrier frequency of several tens of hertz, a duration of the wave trains of the order of a second.

Fig. 7.

Observing simultaneously two oscillations \(u_1, u_2\), produced by two independent devices of the type just described, with the aid of a two-beam oscilloscope (or two ordinary single-beam ones), we obtain a model of two incoherent light oscillations. Observing, moreover, with the aid of still another oscilloscope, the sum \(u=u_1+u_2\) of these oscillations, we shall see that at some moments the amplitudes of the oscillations add, at others they subtract.

Let us turn to the observation of the mean intensity with the aid of a device possessing large inertia. Let us take, for example, a tube (diode) operating in the mode of square-law rectification, and connect to its output an electrical measuring instrument with a time constant far exceeding the duration of the wave train, for example of the order of 10 sec., if the duration of a train is of the order of 1 sec. (This is easy to do artificially by connecting an ordinary microammeter according to the circuit shown in Fig. 8 and selecting \(R_0, C_0\).) We shall apply to the input of the tube in succession \(u_1, u_2, u\). Each time, after waiting until the pointer has settled, we take the reading of the instrument: respectively \(\alpha_1, \alpha_2, \alpha\). We shall be convinced that \(\alpha=\alpha_1+\alpha_2\); the intensity values add over a long time.

Fig. 8.

It is not difficult to demonstrate also the case of the addition of coherent oscillations, when \(\alpha \ne \alpha_1+\alpha_2\). For this it is sufficient to take the voltages \(u_1\) and \(u_2\) from one and the same filter and to create a phase difference between them with the aid of a phase shifter (Fig. 9).

Let us apply the oscillations \(u_1, u_2\) from independent sources (with the sweep switched off) to both pairs of deflecting plates of the oscilloscope. We shall see, if the carrier frequencies are the same or close, an alternation of ellipses of every possible form and orientation and of different sizes. By reducing the duration of the trains, the flickering of the ellipses may become so rapid that the eye will see only a certain averaged pattern—a bright spot possessing central symmetry. These experiments explain well the nature of natural (unpolarized) quasimonochromatic light. We shall obtain a model of polarized light if we repeat the same operations, taking \(u_1, u_2\) from a single source. By feeding one of these oscillations through a phase shifter, we can obtain a model not only of linearly, but also of elliptically polarized light: the size of the ellipses will change chaotically, but their form and orientation will remain unchanged.

Fig. 9.

6. TIME PICTURE OF THE PROCESS OF SPECTRAL DECOMPOSITION

White light or noise can be represented mathematically in the form of a Fourier integral; a spectral apparatus physically carries out this decomposition, and does so the more perfectly, the greater its resolving power. Has everything thereby been said? Anyone who strives to learn to think physically will wish to form for himself a visual picture of how the spectral apparatus “works,” how it transforms the chaotic action applied to it into a quasimonochromatic oscillation.

The first experiment of § 5 is directly related to this question. Let us simultaneously observe, with the aid of two oscilloscopes, the fluctuations (noise) at the input of an \(RC\)-filter and the oscillations at its output. The transformation of noise into a chaotic, but slowly modulated, oscillation is precisely what, in mathematical language, is called the extraction of a narrow portion of the Fourier decomposition. The oscillation arising at the output of the \(RC\)-filter may be regarded as a superposition of damped trains produced by each of the random impulses, the succession of which constitutes the fluctuation noise at its input. The pulsations of the amplitude and phase of the oscillation thus arising occur the more slowly, the longer the oscillation born of each separate impulse lasts, i.e. the smaller the damping of the filter is—in other words: the narrower its pass-

scanning. Here there is a complete analogy with the fact that the grating, the more perfectly it reworks white light into monochromatic light, the greater is its resolving power—the longer the time of “prolongation” \(NmT\) (\(N\) is the number of rulings, \(m\) the order of the spectrum, \(T\) the period of the light oscillation) of the action of an individual light pulse (see, for example, \(^{5}\)).

The duration of the change of patterns can be brought, by narrowing the pass band of the \(RC\)-filter, to several seconds. The fact that fluctuation impulses, following one another on the average, say, every \(10^{-4}\) sec (a time of the order of the reciprocal of the width of the receiver’s pass band), can change the amplitude of the oscillation at the filter output only after several seconds have elapsed, produces a strong impression. We “feel” that the time needed for chaotic impulses to appreciably change the amplitude and phase of the oscillation maintained by them in the spectral apparatus is determined by the properties of this apparatus and increases as the sharpness of its tuning is increased. And this is precisely the main thing that must be brought to awareness in revealing the “mechanism” of spectral decomposition.

CITED LITERATURE

1. E. V. Shpolsky, Uspekhi Fizicheskikh Nauk, 30, 1 (1946).
2. L. I. Mandelstam, Izv. Ak. N., physical series, No. 4, 525 (1938).
3. L. I. Mandelstam and N. D. Papaleksi, collection Recent Studies on the Propagation of Radio Waves along the Earth’s Surface, Gostekhizdat, 1945 (Complete Works, 2, 232).
4. G. S. Gorelik, Uspekhi Fizicheskikh Nauk, 34, 321 (1948).
5. Schuster, Introduction to Theoretical Optics, Gostekhizdat, 1935 (p. 119).