DURATION OF THE FLASH IN THE VAVILOV–CHERENKOV EFFECT*)

I. M. Frank

CONTENTS

§ 1. Introduction . . . . . . . . . . . . . . . . . . . . . . 111
§ 2. On the duration of the flash . . . . . . . . . . . . . . . 113
§ 3. The Vavilov–Cherenkov effect and luminescence. A medium without dispersion . . . . . . . . . . . . . . . . . . . . . 115
§ 4. Features of light decomposed into a spectrum . . . . . . . 122
§ 5. The Vavilov–Cherenkov effect in a refracting medium . . . 132
§ 6. Counters of Vavilov–Cherenkov radiation . . . . . . . . . 136
§ 7. Achromatic counters . . . . . . . . . . . . . . . . . . . 143

§ 1. INTRODUCTION

Twenty years have passed since the time when, in the Reports of the Academy of Sciences of the USSR, there appeared the articles by S. I. Vavilov¹ and P. A. Cherenkov² containing the first data on the nature and properties of the phenomenon that later received the name of the Cherenkov effect. In these works it was shown that all, without exception, pure liquids glow under the action of the γ-rays of radium, and moreover with approximately the same brightness. A number of the principal properties of this weak blue glow were also established at once. The most important of the conclusions drawn in Vavilov’s work was that the phenomenon discovered by S. I. Vavilov and P. A. Cherenkov cannot be luminescence.

Indeed, according to S. I. Vavilov’s definition³, a necessary property of any luminescence is the presence of a finite duration of afterglow (of the order of, or greater than, \(10^{-10}\) sec). In luminescence phenomena this duration is determined by the probability of a transition from the excited state of an atom or molecule to an energetically lower state. Therefore, by acting on the excited particles it is always possible, to one degree or another, to quench—

*) Report at a colloquium of the Physical Institute of the Academy of Sciences of the USSR, dedicated to the memory of S. I. Vavilov, delivered on January 25, 1954. The text of the report has been revised and supplemented.

suppress luminescence. The shorter the duration of the afterglow, the higher must be the temperature or the concentration of quenching substances in order to cause a decrease in the brightness of the luminescence. The experiments of P. A. Cherenkov showed that, by means of the known methods of quenching, it is not possible to diminish the brightness of the universal blue glow. Hence S. I. Vavilov[^1] concluded that “the observed effect in general cannot be any kind of luminescence, for which the finite duration of excitation is a necessary characteristic feature.”

The phenomena of luminescence, as is well known, are extremely widespread in nature. In particular, Pierre and Marie Curie[^4] were well acquainted with the visible glow of various radium compounds). In the following decades the glow caused by radioactive substances—including, undoubtedly, the radiation of Vavilov—Cherenkov—was observed repeatedly). However, among the various kinds of luminescence no one noticed that what predominates in the glow of pure liquids and many solid bodies is precisely not luminescence, but a new optical phenomenon—the Vavilov—Cherenkov effect. One may therefore suppose that, had it not been for the work of S. I. Vavilov, who applied to the given case his criterion of the duration of the glow, this new phenomenon would not have been discovered for a number of years yet**).

Over the past 20 years the phenomenon has been studied in great detail. However, the development of physics in recent years compels us to return once again to the question of what the duration of the Vavilov—Cherenkov radiation is. In fact, in the works of S. I. Vavilov and P. A. Cherenkov it was shown only that the duration of the afterglow is consi—

*) In her reminiscences of P. Curie, speaking of the discovery of radium, Marie Curie[^5] wrote: “We were especially delighted when we observed that our radium-containing salts all glowed spontaneously. Pierre Curie said that this unexpected feature gave him more satisfaction than he could have dreamed of.” A vivid description of how this observation was made is contained in M. Curie’s biography[^6].

**) The glow of pure liquids under the action of γ-rays was observed, for example, by Mallet[^7].

***) That this is so is evident from consideration of certain foreign works. Thus, in several papers of 1950–1951 the glow of pure liquids under the action of γ-rays was rediscovered, and in this glow nothing was suspected except luminescence[^8]. Subsequently, in order to determine whether this glow is Cherenkov radiation or luminescence, in one of these works the observed brightness was compared with the theoretically expected brightness for the Vavilov—Cherenkov effect[^9],[^10]. No other criteria for determining the nature of the radiation were used. Thus, one may suppose that, if the Vavilov—Cherenkov effect had not been known, the authors of these works apparently would not have discovered it either.

It follows also from what has been said that the now generally accepted name “Cherenkov effect,” which the author of this article had previously used as well, is not quite exact. We shall therefore call this phenomenon the Vavilov—Cherenkov effect.

considerably less than \(10^{-10}\) sec, and went beyond what could be measured experimentally. A more detailed consideration of the question at that time was altogether pointless. Now the situation is beginning to change. Indeed, the wide use of luminescent counters has led to a significant improvement in photomultipliers and in the methods of their application. As a result it has become possible to observe the Vavilov—Cherenkov effect for individual particles and to use it for studying their properties. (For the literature on these questions see, for example, \(^{10}\).)

In solving many physical problems it is essential not only to register a particle, but also to fix the moment at which it enters a counter (various kinds of coincidence circuits). In such work, besides the amount of light energy arriving at the photocathode, the duration of the light flash is also important. As experimental technique develops, ever greater demands are placed on the resolving power of the apparatus and, consequently, on limiting the duration of the flash used\(*\). At the same time, in view of the weakness of Vavilov—Cherenkov radiation, systems are in practice used which collect on the photocathode the light emitted by the particle over a considerable path, the traversal time of which by the particle can no longer be regarded as infinitely small (for example, with a radiator length of several tens of centimeters this time is already of the order of \(10^{-9}\) sec). Therefore the question of the duration of the light flash produced by Vavilov—Cherenkov radiation from an individual particle already has practical significance. In this connection, in some works the influence of the apparatus used on the duration of the light signal is considered \(^{14}\).

In these works the role of a number of obvious factors is noted which are not fundamental, but which can in practice lengthen the duration of the flash. If, however, one disregards the imperfection of the experimental conditions, then the natural question arises: what is the minimum possible duration of the light flash in the case of the Vavilov—Cherenkov effect, and on what causes does it depend\(**\). These questions are discussed in the following sections of the article.

§ 2. On the duration of the flash

First of all it must be borne in mind that the duration of a light flash at some point, for example on the surface of the photocathode, generally speaking, does not coincide with the duration of the emission

\(*\) In the recently published work of E. K. Zavoisky, G. E. Smolkin, A. T. Plakhov, and M. M. Butslov \(^{11}\) it is indicated that the resolving time can be reduced to \(10^{-14}\) sec.

\(**\) This question was put to the author by E. K. Zavoisky. I take the opportunity to thank him for the remarks he made.

of individual elementary emitters. Indeed, the observed duration of a flash may depend on a number of causes: on the fact that the excitation of emitters located at different points of the luminous object does not occur simultaneously, that the length of the path traveled by light from these points to the photocathode is not the same, and that in some cases light may propagate from the emitters to the photocathode along different paths. The totality of all these causes, together with the duration of the emission of the elementary emitters, determines the duration of the light signal, which in what follows we shall call the duration of the flash.

In determining the duration of luminescence, for example in studying the decay of fluorescence, depending on the method employed, either the duration of the flash or the duration of the emission may be determined. Thus, by the method of fluorescence quenching one determines the duration of the molecule’s stay in the excited state, that is, in pure form, the duration of emission. When various phosphoroscopes are used, to the duration of emission there is added, to one degree or another, a time determined by the fact that the flash of the exciting light has a finite duration and that the thickness of the luminescent layer is also finite. Thus, the excitation of different molecules does not occur simultaneously, and the path of the light from individual emitters to the receiver is not quite the same; consequently, the duration being determined is the duration of the flash*). In practice, however, in such experiments the thickness of the luminous object is usually centimeters or even less, and consequently the duration of the flash exceeds the duration of emission by a time less than \(10^{-10}\) sec. Since this time is small in comparison with the duration of emission in luminescence, in most cases it makes no sense to distinguish between the duration of the flash and the duration of emission (provided only that the pulse of the exciting light is sufficiently short). An essentially different situation occurs in the Vavilov—Cherenkov effect, since the duration of emission must be, as the absence of quenching shows, extremely small.

In a general form, the question of the duration of the flash in this phenomenon has not been considered up to the present time. The relations necessary for such a consideration were obtained earlier in the work of I. E. Tamm \(^{13}\); however, not only were they not used, they are apparently very little known.

One may suppose that this is connected with the fact that the usual interpretation of the Vavilov—Cherenkov effect is based on consideration of the interfe-

*) We do not here consider the lengthening of the flash as a result of reabsorption of light, when the luminescence light is successively absorbed and emitted several times.
(See, for example, the work of M. D. Galanin \(^{12}\).)

rence of monochromatic waves. Meanwhile, in order to obtain information about the duration of the light signal it is necessary to determine the behavior of a group of waves, that is, to carry out an integration over frequency. (See the work of I. E. Tamm \(^{13}\).)

§ 3. THE VAVILOV—CHERENKOV EFFECT AND LUMINESCENCE. A MEDIUM WITHOUT DISPERSION

It is comparatively simple to clarify the question of the duration of the light pulse associated with the Vavilov—Cherenkov effect if the particle moves in a medium in which there is no dispersion of light. In a real refracting medium, the dispersion of light cannot be small over the entire spectrum. However, one may assume that in some frequency interval from \(\nu_1\) to \(\nu_2\) the refractive index of light changes so little that the dispersion of light may be neglected. Let us suppose that the conditions of the experiment are such that only radiation with frequencies lying within this interval is recorded. Then the range of wavelengths for which dispersion of light takes place is altogether excluded from consideration. As for the requirements on the smallness of the dispersion, they will be clear from what follows.

Let us turn to the usual visual interpretation of the Vavilov—Cherenkov effect. It consists in the fact that each point of the trajectory of a charged particle in a substance is regarded as a source of waves of different frequencies. If the velocity of the particle \(v\) is greater than the phase velocity of light \(u = \dfrac{c}{n(\nu)}\), then it turns out that for the propagation of light there are distinguished directions forming an angle \(\theta_\nu\) with the direction of motion of the particle. In this case

\[ \cos \theta_\nu = \frac{1}{\beta n(\nu)}, \qquad \beta = \frac{v}{c}. \tag{3.1} \]

If one constructs a cone whose vertex coincides with the instantaneous position of the moving particle (Fig. 1), and the angle between its generators and its axis is

\[ \varphi = \frac{\pi}{2} - \theta_\nu, \]

then its surface is a surface of equal phases for waves from any points of the particle trajectory. Indeed, from Fig. 1 it is seen that during the time in which the particle has time to go from some point \(C_2\), where it was at the instant \(t_2\), to the point \(O\) (the instant \(t = t_0\)), waves propagating in the direction \(\theta_\nu\) move from \(C_2\) to the surface of the cone \(AOB\). The point \(C_2\) is arbitrary; therefore the same result is obtained for waves from any other point of the particle’s path, for example \(C_1\).

Fig. 1.

To determine the field at point \(A\) at the moment \(t_1\), when the surface of the cone \(ACOB\) passes through it, one must consider the result of the superposition of waves arising on different segments of the trajectory. As is known, the field of a moving charge \(e\) can be replaced by the field of a set of stationary dipoles whose electric moment on an element of length is equal to\({}^{17}\)

\[ \mathbf{p}_v dz=\frac{e}{2\pi^2\nu}\,\mathbf{z}_1\sin\left\{2\pi\nu\left(t-\frac{z}{v}\right)\right\}dz . \tag{3,2} \]

Here it is assumed that the charge moves along the \(z\)-axis and that its position is \(z=vt\) at the moment \(t\). (In formula (3,2), \(\mathbf{z}_1\) denotes a unit vector directed along the \(z\)-axis.) In this case the Hertz vector at point \(A\) must be equal to

\[ \boldsymbol{\Pi}_v=\frac{e}{2\pi^2\nu}\mathbf{z}_1 \int \frac{ \sin\left\{2\pi\nu\left(t-\frac{z}{v}-\frac{Rn'}{c}\right)\right\} }{R}\,dz, \tag{3,3} \]

where \(R\) is the distance from the point with coordinate \(z\) to \(A\), and the integration extends over the entire trajectory of the particle.

From simple considerations based on examining Fig. 1, it follows that the superposition of waves occurs only in directions close to the angle \(\theta_v\). Thus, the main contribution to (3,3) should be given by \(z\)’s close to the point \(C_2\) (we denote the coordinate of the point \(C_2\) by \(z_2\), and its distance from the point \(A\) by \(R_0\)). It may therefore be expected that, for point \(A\), located in the wave zone, as a result of integration over \(z\) in (3,3) one obtains an expression of the form

\[ \boldsymbol{\Pi}_v=\frac{e}{2\pi^2\nu}\mathbf{z}_1 l \frac{ \sin\left\{2\pi\nu\left(t-\frac{z_2}{v}-\frac{R_0 n}{c}\right)-\varphi\right\} }{R_0}. \tag{3,4} \]

Here \(l\) is the effective length of the segment of the trajectory near the point \(C_2\) that determines the field at point \(A\), and the quantity \(\varphi\) takes into account the phase difference obtained when summing the waves coming from this segment, in comparison with the wave arriving from the point \(C_2\).

To justify this relation and determine the quantities \(l\) and \(\varphi\), we proceed as follows. We expand the quantity \(R\) in formula (3,3) in powers of \((z-z_2)\):

\[ R=\left[R_0^2+(z-z_2)^2-2R_0(z-z_2)\cos\theta_v\right]^{1/2}= \]

\[ =R_0-(z-z_2)\cos\theta_v+(z-z_2)^2\frac{\sin^2\theta_v}{2R_0}+\ldots . \tag{3,5} \]

Assuming that only \(|z-z_2|\ll R_0\) are essential for us, we restrict ourselves in the expansion (3,5) to terms up to the second degree and shall take \(R\) in the denominator of (3,3) to be equal to \(R_0\). Substituting these values of \(R\) into (3,3) and taking into account that \(\theta_v\) satisfies (3,1),

and the wavelength \(\lambda_\nu=\dfrac{c}{\nu n}\), we obtain instead of (3,3):

\[ \Pi'_\nu=\frac{e}{2\pi^2\nu R_0}\,Z_1 \int \sin\left\{2\pi\nu\left(t-\frac{z_2}{v}-\frac{nR_0}{c}\right) -\frac{\pi}{\lambda_\nu R_0}\sin^2\theta_\nu\,(z-z_2)^2\right\}\,dz . \tag{3,6} \]

Thus, the phase of the waves arriving at \(A\) from points close to \(z_2\) depends quadratically on the distance \(|z-z_2|\). When this distance increases so much that \((z-z_2)^2\sin^2\theta_\nu \gg \lambda_\nu R_0\), the sine under the integral (3,6) begins to oscillate rapidly and, consequently, the range of variation of \(z\) satisfying this inequality does not make a noticeable contribution to the value of the integral (3,6). Suppose that \(R_0\) is so large that such \(|z-z_2|\) are possible for which

\[ R_0^2 \gg (z-z_2)^2\sin^2\theta_\nu \gg \lambda_\nu R_0 . \tag{3,7} \]

In this case the approximate value of \(R\) used by us in (3,6) is correct throughout the entire range of variation of \(z\) about the point \(C_2\) that gives a noticeable contribution to \(\Pi'_\nu\), i.e. \(\Pi'_\nu\) in this region coincides with \(\Pi_\nu\). Taking the path of the particle to be sufficiently long and carrying out the integration from \(-\infty\) to \(+\infty\), from (3,6) we obtain *):

\[ \Pi_\nu=\frac{e}{2\pi^2\nu R_0}\sqrt{\lambda_\nu R_0}\, \frac{1}{\sin\theta_\nu}\,Z_1 \sin\left\{2\pi\nu\left(t-\frac{z_2}{v}-\frac{R_0 n}{c}\right)-\frac{\pi}{4}\right\}. \tag{3,8} \]

Comparing with (3,4), we have:

\[ l=\sqrt{\lambda R_0}\,\frac{1}{\sin\theta_\nu}; \qquad \varphi=-\frac{\pi}{4}. \tag{3,9} \]

From (3,9) one can immediately determine the degree of directionality of Vavilov—Cherenkov radiation.

Indeed, the direction of propagation of the waves going to \(A\) from the various portions \(l\) forms with \(C_2A\) an angle not exceeding (see Fig. 1)

\[ \delta\theta=\frac{\frac{1}{2}l\sin\theta_\nu}{R_0} =\frac{1}{2}\sqrt{\frac{\lambda}{R_0}}, \tag{3,10} \]

*) Formula (3,8), of course, coincides with that obtained earlier in the theory of the Vavilov—Cherenkov effect\(^{18,13}\), as is easily verified if one passes from the Hertz vector to the vector potential \(A_\omega=\dfrac{1}{c}\dot{\Pi}_\omega\). In the usual treatment the phase shift by \(\dfrac{\pi}{4}\) is obtained automatically when the Hankel function describing the field near the \(z\)-axis is replaced by its asymptotic value.

i.e., as was to be expected, the radiation is strictly directed*). What has been said also makes it possible at once to determine the phase of the oscillations on the surface of the wave cone. Indeed, at the moment \(t_0\), when the surface of the wave cone from (3.8) passes through the point \(A\), taking into account that \(\dfrac{z_2}{v}=t_2\) and \(t_0-t_2=\dfrac{R_0 n}{c}\), we obtain the phase \(-\dfrac{\pi}{4}\). The same holds for any point of the wave cone \(AOB\), except for the region close to the vertex, where condition (3.7) is not fulfilled. In the absence of dispersion, the position of the wave cone does not depend on the frequency and, consequently, at the moment \(t_0\) the phase at \(A\) is the same for all frequencies.

In a real medium this is true for the frequency interval \(\nu_1-\nu_2\), if the dispersion of light is so small that along the path \(C_2A=R_0\) no appreciable phase difference accumulates between waves of different frequencies, i.e.

\[ \nu\left[\left(t_0-t_2\right)-\frac{n(\nu)R_0}{c}\right]\ll 1 \tag{3.11} \]

for \(\nu\) lying in the interval from \(\nu_1\) to \(\nu_2\). Inequality (3.11) determines that restriction on the magnitude of the dispersion which is necessary for us in the present discussion, and we shall assume that it is satisfied. It is clear that this requirement is the easier to satisfy, the narrower the interval \(\Delta\nu=\nu_2-\nu_1\) and the smaller \(R_0\), i.e. the closer the point \(A\) is to the particle trajectory.

The phases of oscillations with different frequencies, which coincide at the point \(A\) at the moment \(t_\nu\), will differ at other times. Thus, at the times \(t_\nu\pm\Delta t\) the phase difference of the oscillations with frequencies \(\nu_1\) and \(\nu_2\) will become equal to \(2\pi\), if

\[ (\nu_2-\nu_1)\Delta t=1. \tag{3.12} \]

Putting \(\Delta\nu=\nu_2-\nu_1\), we obtain:

\[ \Delta t=\frac{1}{\Delta\nu}. \tag{3.13} \]

Thus, over the time interval \(\Delta t\) the phase difference will become such that waves of one half of the frequency interval \(\nu_2-\nu_1\) will be in antiphase with respect to waves of the other half. If, in addition, the amplitudes of the waves depend only weakly on the frequency (in the interval from \(\nu_1\) to \(\nu_2\)), then the light intensity at the point \(A\) will decrease from maximum to zero over the time \(\Delta t\). Thus, \(\Delta t\) from (3.13) determines the half-width of the duration of the light signal. The quantity \(\Delta t\) satisfying (3.4) is, as is known, the minimum possible duration of a light signal containing the frequency interval \(\nu_2-\nu_1\).

* The result contained in (3.9), of course, is not specific to Vavilov—Cherenkov radiation. (See the end of this paragraph.)

Let us suppose, for example, that \(\Delta t \sim 10^{-12}\) sec, and that this occurs for the frequency region of visible light \((\lambda \sim 5\cdot 10^{-5}\ \text{cm})\). Then we obtain \(\Delta \nu = 10^{12}\ \text{sec}^{-1}\), and, since \(\Delta \nu = \dfrac{c}{\lambda^{2}}\Delta \lambda\), then \(\Delta \lambda \sim 10\ \text{\AA}\). Thus, a duration exceeding \(10^{-12}\) sec proves possible only if a narrow spectral interval has been selected from the continuous spectrum, or if the emission spectrum contains narrow bands*).

As is known, the spectrum of Vavilov—Cherenkov radiation is continuous, and its intensity varies only weakly in the frequency interval \(\Delta \nu\), small in comparison with \(\nu\) (for \(n=\text{const}\) the theory gives proportionality of the intensity to \(\nu\)). Thus, in the case under consideration of a medium without dispersion, no long-lived component can arise in Vavilov—Cherenkov radiation, unless it is monochromatized or unless it should turn out that its spectrum contains narrow bands. Thus, if the phenomenon of luminescence \((\Delta t > 10^{-10}\ \text{sec})\) could play a noticeable role in Vavilov—Cherenkov radiation, this would mean the presence in the radiation spectrum of spectral lines with width \(\Delta \lambda < 0.1\ \text{\AA}\) (for \(\lambda \sim 5\cdot 10^{-5}\ \text{cm}\)). It follows from this that luminescence can manifest itself in the Vavilov—Cherenkov effect only in the case where it has a narrow line spectrum. In a dense medium, in which the Vavilov—Cherenkov effect is usually observed, such luminescence is hardly possible**).

From what has been said, of course, it does not follow that the presence of the Vavilov—Cherenkov effect to some extent excludes the possibility of luminescence. In reality, luminescence probably occurs not only in luminescent substances, but to some weak degree also in substances whose radiation under the action of fast particles is now wholly attributed to the Vavilov—Cherenkov effect. What was said above means only that this luminescence cannot give the directional radiation characteristic of the Vavilov—Cherenkov effect.

The luminescence spectrum in dense substances, as is known, is either continuous or consists of comparatively broad bands, the width of which is considerably greater than the natural width determined by damping. The actual width is determined by the fact that, as a result of various actions on the radiating system, the frequency of the emitted light or, at least, the phase of the radiation

* ) Above it was assumed that within \(\Delta \nu\) the amplitudes of the waves vary little. In the case of a line spectrum, \(\Delta \nu\), evidently, is determined by the half-width of the line.

**) For particles of very high energy the Vavilov—Cherenkov effect is possible also in a rarefied gas, in which, of course, narrow emission lines may be present. However, in this case the assumption made above about the smallness of dispersion is certainly incorrect (for the related features see §§ 4 and 5).

during the damping of the oscillations change many times. Therefore the light emitted by such radiators can interfere only during a time determined not by the duration of the damping, but by the mean interval of time before the occurrence of a perturbation. In other words, the duration of the directed radiation that can arise in this case is determined by the actual width of the radiation spectrum, i.e., satisfies relation (3.4).

The situation will not change if one assumes that the continuous luminescence spectrum in fact consists of closely spaced narrow lines, each of which, taken separately, decays slowly (of course, if only the initial phases of the oscillations are the same and are determined by the moment of excitation). In this case, at the first moment after excitation the joint action of these radiators will give a flash of directed radiation, which later, when a considerable phase difference of the oscillations for different frequencies has accumulated (see 3.12), will be extinguished. In this case the excitation energy will practically not be radiated until, as a result of collisions, the coherence of the oscillations is disturbed. After this, however, the directionality of the radiation will disappear, and the luminescence will acquire its usual properties.

The possible absence of radiation in the intermediate stage in this case—the “dark pause”—has a simple analogy with the radiation of two dipoles oscillating with the same frequency and situated at a distance much smaller than the wavelength of light. If the phases of the oscillations of these dipoles are shifted with respect to one another by \(\frac{\pi}{2}\), then their radiation is extinguished everywhere, and damping of the oscillations as a result of radiation is absent.

Analogously to this, the initial moment of radiation also has an analogy with a model of two dipoles, but oscillating in phase with one another. The wave amplitudes in this case add, and the damping of the oscillations increases. Academician L. I. Mandelstam repeatedly pointed out in his time the peculiarities of the radiation and damping of such closely spaced oscillators.*)

Thus, a nonmonochromatic long afterglow cannot give directed radiation, and consequently luminescence under conditions in which the Vavilov—Cherenkov effect is possible will have its usual properties. Certain peculiarities may be expected only for the intensity of luminescence, since both at the moment of excitation and at the initial moment of bleaching the luminescence cannot be separated from the Vavilov—Cherenkov effect.

As for the Vavilov—Cherenkov effect, as we have seen, the directionality of radiation characteristic of it, in combination with the discon—

) L. I. Mandelstam, Lectures*.

the continuity of the radiation spectrum (i.e., the absence of narrow bands even against the background of a continuous spectrum) determines the smallness of the emission time associated with this effect.

It should be noted that, in proving this assertion, use was made of the assumption, following from the theory, that the initial phases are the same for waves of different frequencies (see (3.2)). If one does not proceed from the theory of the phenomenon, then one could obtain directed radiation by taking, instead of (3.2),

\[ p_y \sim \sin\left\{2\pi \nu \left(t-\frac{z}{v}\right)+\delta(\nu)\right\}, \tag{3.14} \]

i.e., by assuming that the initial phase \(\delta(\nu)\) depends on \(\nu\). Naturally, (3.14) leads to the same equation (3.8) (of course, with the addition of \(\delta(\nu)\) under the sine sign), from which (3.9) and (3.10) are obtained. As for the quantity \(\Delta t\), in this case as well it will be determined by (3.13) if \(\delta(\nu)\) is equal to a constant quantity or varies linearly, \(\delta(\nu)=\alpha+\beta\nu\). In the latter case, as is not difficult to see, the superposition of waves of different frequencies will occur with identical phases not at the instant \(t_0\), but, depending on the sign of \(\beta\), at an earlier or later instant

\[ t' = t_0 - \frac{\beta}{2\pi}. \]

A special case can occur only if the phase \(\delta(\nu)\) changes sharply in the interval \(\Delta\nu\). In the limiting case, when \(\delta(\nu)\) repeatedly and irregularly takes all values from \(0\) to \(2\pi\), the result of the superposition of waves will in general not depend on time*).

Such behavior of \(\delta(\nu)\) would mean that the interaction of the moving charge with certain atoms or molecules contained in the medium changes sharply in a narrow frequency interval, i.e., leads to the admission of some distinguished frequencies in the interval \(\Delta\nu\). In this case one must assume that these natural frequencies are excited with initial phases which, although determined by the instant of excitation, vary from line to line. Since the radiation spectrum is continuous, we therefore return to the assumption already considered above of a fine structure of lines, but with the difference that—

\[ \text{—} \]

*) It is not difficult to construct a model corresponding to this limiting case. Suppose that, in a direction at an angle \(\theta_v\) to the \(z\)-axis, plane waves of white light propagate. Suppose that the path of the light is blocked by an opaque diaphragm having an infinitely narrow slit, the position of which is aligned with the \(z\)-axis. We shall regard this slit as a source of light. For monochromatic components of the light, obviously, equation (3.14) is applicable, and we obtain radiation directed at the angle \(\theta_v\). At the same time, the duration of the light signal at the point \(A\), of course, will not be limited. The reason for this in the present case is that \(\delta(\nu)\) varies chaotically with the change of \(\nu\). Thus, waves from different points of the slit with identical \(\nu\) are coherent with one another, but there is no coherence of waves with different \(\nu\).

that, owing to the spread of the initial phases, the dark pause will not be preceded by a flash at the initial instant. Therefore, until coherence is destroyed, i.e. until the transition to ordinary luminescence, the emission of such a system will occur slowly and, correspondingly, with low intensity. The artificiality of assumptions of this kind is more than obvious, and therefore we have no grounds to doubt the correctness of (3.13), especially in the region of the spectrum that contains no anomalies in the behavior of the refractive index. We thus arrive at the conclusion that Vavilov—Cherenkov radiation possesses not only the remarkable property of coherence of radiation arising at different points of the particle trajectory, but also the coherence of waves with different frequencies.

In this section of the paper we have neglected dispersion, as a result of which, as the waves propagate in the medium, a phase difference accumulates between waves of different frequencies. The question of the influence of light dispersion, which is always present to some extent, is considered in §§ 4 and 5.

§ 4. FEATURES OF A LIGHT BEAM RESOLVED INTO A SPECTRUM

For a real medium possessing dispersion, it is necessary to take into account that the magnitude of the group velocity of light

\[ w=\frac{c}{\,n(\nu)+\nu \dfrac{dn}{d\nu}\,} =\frac{c}{\,\dfrac{d}{d\nu}(\nu n)\,} \tag{4.1} \]

differs from the magnitude of the phase velocity \(u(\nu)=\dfrac{c}{n(\nu)}\). At the same time, as is well known, the propagation of a light pulse occurs not with the phase velocity, but with the group velocity. From this, however, it does not at all follow that in the treatment presented in § 3, from which the angle was obtained (formula 3.1), one may simply replace the phase velocity by the group velocity. There could be no doubt on this question were it not for the remark made by Sommerfeld in his book on optics \(^{14}\). Noting that the angle \(\theta_\nu\) is analogous, in acoustic phenomena, to the Mach angle for a projectile moving with supersonic velocity, Sommerfeld writes: “The velocity of the front of the radiation emitted by the molecules remaining behind is not equal to the phase velocity \(u\), but is equal to the group velocity \(w\). Therefore the Mach angle will be somewhat smaller than follows from formula (47.8) (i.e. \(\sin\varphi=\dfrac{u(\nu)}{v}\) and \(u(\nu)=\dfrac{c}{n(\nu)}\))*), since in the latter ne—

*) The angle \(\varphi=\dfrac{\pi}{2}-\theta_\nu\), i.e. this formula is identical with (3.1) (see Fig. 1).

necessary to replace \(u\) by \(w\). An exact measurement of the angle of emission of Cherenkov waves would clarify this question.”

Sommerfeld’s statement recently led American physicists\(^{15}\) to take it under consideration; as a result they arrived at the natural conclusion that Sommerfeld was mistaken. In fact, Sommerfeld’s assertion is apparently an accidental oversight, since in connection with the Vavilov—Cherenkov effect he refers to the work of I. E. Tamm, in which, in particular, the role of the group velocity of light is examined in detail.

Qualitatively, the question can be clarified by means of elementary considerations. Suppose that from the spectrum of Vavilov—Cherenkov radiation so narrow an interval \(\Delta \nu\) has been selected that, for frequencies lying within it, both the value of the phase velocity \(u(\nu)\) and the value of the group velocity \(w(\nu)\) may be regarded as constant. Then the method of finding the angle \(\theta_\nu\) (§ 3, Fig. 1 and formula (3,1)), suitable for monochromatic light, may be used in this case as well. Since \(n(\nu)\) changes little over the interval \(\Delta\nu\), the \(\theta_\nu\) determined by formula (3,1) can change only by a small amount \(\Delta\theta\). Consequently, to within the small angle \(\Delta\theta\), the direction of propagation of the waves of the entire frequency interval \(\Delta\nu\) is constant and is specified by the angle \(\theta_\nu\).

In Fig. 2 the construction of Fig. 1 is repeated for the wave cone \(AOB\). In order to determine where the light signal, defined by the group of waves with frequencies from \(\nu-\Delta\nu\) to \(\nu+\Delta\nu\), will be, we lay off from the point \(C_2\), in the direction of propagation of the waves, the magnitude \(w(t_0-t_2)\). Then, for the instantaneous position of the group of waves, we obtain the cone \(EOD\), which, for \(w<\dfrac{c}{n}\), lies inside the wave cone \(AOB\).

Fig. 2.

Such a graphical method for determining the position of the group of waves was used in the work of I. E. Tamm\(^{13}\), where it is presented as an illustration of the results obtained.

Although the group cone (for \(w<u\)), in agreement with Sommerfeld, forms a sharper angle with the axis than the wave cone, nevertheless, as is clear from Fig. 2, this angle cannot be found by simply replacing \(u\) by \(w\) in equation (3,1). A special feature of this group cone is that its generators are not perpendicular to the direction of the phase velocity. It is not difficult to see that this feature is not ...

specific to the Vavilov—Cherenkov phenomenon; it occurs in all cases where the direction of propagation of waves depends on their frequency, i.e. when light is decomposed into a spectrum. The simplest example of such decomposition into a spectrum is a parallel beam of white light that has undergone refraction at a plane boundary of a dispersive medium.

Let us suppose that plane light waves propagate in a medium, and that the direction of wave propagation depends on the frequency of oscillations in such a way that a change in the frequency of light by an amount \(\Delta \nu\) corresponds to a change in the direction of propagation of the light by \(\Delta \theta\). In what follows, for the sake of simplicity, we shall consider not a continuous spectrum, but only

Fig. 3.

two plane waves with frequencies \(\nu\) and \(\nu'=\nu+\Delta\nu\). Let the series of solid parallel lines in Fig. 3 represent the intersection with the plane of the drawing of the planes that determine, at the instant \(t=0\), the positions of the wave crests (i.e. the amplitude maxima of both signs) for a plane wave of frequency \(\nu\). Thus, the distance between the straight lines is equal to \(\lambda\nu/2\). Let the dotted straight lines represent the same thing, but for waves of frequency \(\nu'=\nu+\Delta\nu\). The dotted straight lines, in accordance with the dis...

by the directions of both waves, are inclined relative to the solid ones by a small angle \(\Delta \theta\).

In Fig. 3 the points \(A_{-2},\ A_{-1},\ A_0,\ A_{+1}\), lying at the intersections of the solid and dashed straight lines, determine the points at which the phases of the two waves are respectively equal to \(-2\pi,\ -\pi,\ 0,\ +\pi\), etc. It is not difficult to see that the straight line \(a\) passing through them is the geometric locus of the points at which the phases of the two waves are identical. Thus, this straight line (and in space, a plane) determines the points of space at which the amplitudes of the two waves add. This plane may therefore be regarded as the middle of the group formed by waves with frequencies \(\nu\) and \(\nu'\).

It is clear from Fig. 3 that this plane is inclined with respect to the wave surfaces of both waves forming the group. The planes determining the position of the intensity minima are likewise inclined. The intensity minimum is determined by the geometric locus of the points at which the phases of the two waves differ by \(\pi\). In Fig. 3, for example, the point \(A_{-3-4}\) is marked, at which the phase of the wave \(\nu\) is equal to \(-3\pi\), and that of \(\nu'\) is equal to \(-4\pi\); the point \(A_{-4-5}\), at which the phases are respectively \(-4\pi\) and \(-5\pi\), etc. Obviously, the geometric locus of the points lying on the straight line \(b\) that joins them is characterized by a phase difference of a half-wave, i.e., it determines the minimum of the group. An analogous straight line \(c\), with points \(A_{+3+4},\ A_{+4+5}\), can be drawn symmetrically, on the other side, with respect to the center of the group \(a\). The region enclosed between the straight lines \(b\) and \(c\) thus determines the geometrical dimensions of the group*).

Let us determine the extent of the group \(2l\), i.e., the distance between \(b\) and \(c\), measured along the direction of the ray velocity (the distance between the points \(A_{-5-6}\) and \(A_{+5+6}\) in Fig. 3). The distance \(l\) can be found from the condition that, for the frequencies \(\nu\) and \(\nu'\), the numbers of half-waves fitting into this length differ by unity (in the case of Fig. 3, five half-waves for the frequency \(\nu\) and six for the frequency \(\nu'\) fit into this length). Thus, \(l\) is found from the condition

\[ \frac{2l\cos\Delta\theta}{\lambda_{\nu'}}-\frac{2l}{\lambda_\nu}=1. \tag{4.2} \]

Since \(\cos\Delta\theta\) differs from unity by a quantity of second order of smallness, and \(\lambda=\dfrac{c}{\nu n(\nu)}\), we obtain:

\[ \frac{2l}{c}\,[\nu'n(\nu')-\nu n(\nu)] = \frac{2l}{c}\,\Delta\nu\,\frac{d}{d\nu}\bigl(\nu n(\nu)\bigr)=1, \]

*) If only two waves are considered, then, of course, the regions of minima and maxima repeat periodically over the entire region under consideration. In the case of a continuous spectrum, such a width of the group of waves corresponds to the frequency interval from \(\nu-\Delta\nu\) to \(\nu+\Delta\nu\), i.e., to the spectral half-width \(\Delta\nu\). The straight lines \(c\) and \(b\) then bound the region in which the principal intensity maximum is enclosed.

whence, taking (4.1) into account, we have:

\[ 2l=\frac{w}{\Delta \nu}. \tag{4.3} \]

Let us also determine the magnitude of the angle \(\chi\) between the planes \(b, c\) or \(a\) and the wave planes. For this purpose, from the point \(A_0\) let us draw the line \(A_0B\), perpendicular to the wave plane (Fig. 3). The distance from the point \(A_0\) to the next solid straight line, i.e. \(A_0B\), is by definition equal to \(\lambda_\nu/2\). The length of the segment of the line \(BD=\Delta l\) between the solid and dotted straight lines (see Fig. 3) is equal to

\[ \Delta l=\frac{\lambda_\nu}{2}-\frac{\lambda_{\nu'}}{2\cos\Delta\theta}. \tag{4.4} \]

Further, from Fig. 3 it is seen that the distance between the points \(B\) and \(A_{-1}\) is

\[ r=\frac{\Delta l}{\operatorname{tg}\Delta\theta}, \tag{4.5} \]

and the tangent of the required angle between the wave plane and the plane of the group is

\[ \operatorname{tg}\chi=\frac{\lambda}{2r}. \tag{4.6} \]

Substituting into (4.6) the value of \(r\) from (4.4) and (4.5) and replacing, in view of the smallness of \(\Delta\theta\), \(\operatorname{tg}\Delta\theta\) and \(\sin\Delta\theta\) by \(\Delta\theta\), and also putting \(-\Delta\lambda=\lambda_\nu-\lambda_{\nu'}\) and

\[ \frac{\Delta\theta}{\Delta\lambda}=\frac{d\theta}{d\lambda}, \]

we obtain:

\[ \operatorname{tg}\chi=-\lambda_\nu\frac{d\theta}{d\lambda} =-\lambda\frac{d\theta}{d\nu}\frac{d\nu}{d\lambda}. \tag{4.7} \]

Moreover,

\[ \frac{d\lambda}{d\nu} =-\frac{c}{\nu^2 n^2}\frac{d}{d\nu}(\nu n) =-\frac{\lambda^2}{w}. \tag{4.8} \]

Hence finally

\[ \operatorname{tg}\chi=\frac{w}{\lambda}\frac{d\theta}{d\nu}. \tag{4.9} \]

Thus, \(\operatorname{tg}\chi\) is proportional to the magnitude of the group velocity \(w\) and does not depend on the phase velocity \(u\).

In constructing Fig. 3 we made no assumptions about the properties of the medium in which the light propagates. The only condition that was used was that the direction of the phase velocity depends on frequency. Indeed, from (4.9) it is seen that \(\chi\) becomes zero only if \(\dfrac{d\theta}{d\nu}=0\).

In Fig. 4 a part of Fig. 3 is reproduced, but for two instants of time differing from one another by 1 second. The wave

the plane \(m\) has in this time moved in the direction of the ray velocity through a distance \(A_0E\), equal to \(u(\nu)=\dfrac{c}{n(\nu)}\). Similarly, the wave plane for the frequency \(\nu'=\nu+\Delta\nu\) has passed from the position \(n\) to \(n'\). In this case the direction of its motion forms with \(A_0E\) an angle \(\Delta\theta\), and the magnitude of the displacement is equal to \(u(\nu')\).

The intersection \(A'_0\) of the wave surfaces \(m'\) and \(n'\) determines the new position of the plane of the center of the group \(a'\). It is not difficult to verify that, if \(u(\nu)\ne u(\nu')\), i.e. if the medium possesses dispersion, then the point \(A_0\) moves in the direction \(A_0A'_0\), forming with the direction of the phase velocity \(AE\) a certain angle \(\theta_\nu\). Let us denote by \(v\) the velocity of displacement of the point \(A_0\), numerically equal to the length of the segment \(A_0A'_0\). From Fig. 4 it is seen that \(\theta\) and \(v\) are related to the magnitude of the phase velocity by the relation

Fig. 4.

\[ \cos\theta_\nu=\frac{c}{n(\nu)v}. \tag{4,10} \]

With the aid of Fig. 4 it is not difficult to determine these quantities. Indeed,

\[ FE=u(\nu)-\frac{u(\nu+\Delta\nu)}{\cos\Delta\theta} \approx -\frac{du}{d\nu}\Delta\nu =\frac{c}{n^2(\nu)}\frac{dn(\nu)}{d\nu}\Delta\nu, \]

\[ A'_0E=\frac{FE}{\operatorname{tg}\Delta\theta} \approx \frac{c}{n^2(\nu)}\frac{dn(\nu)}{d\nu}\frac{d\nu}{d\theta}, \tag{4,11} \]

\[ DE=A'_0E\operatorname{tg}\chi =\frac{v}{n(\nu)}\,w\,\frac{dn}{d\nu} =u-w. \tag{4,12} \]

In determining \(DE\), the value of \(\operatorname{tg}\chi\) from (4,7) was used, as well as the expression for the magnitude of the group velocity (4,1). In addition, we assume that \(\Delta\nu\) has been chosen so that the ratios \(\dfrac{\Delta n}{\Delta\nu}\) and \(\dfrac{\Delta\theta}{\Delta\nu}\) may be replaced by derivatives. For the quantity \(\operatorname{tg}\theta_\nu\), as a result we obtain:

\[ \operatorname{tg}\theta_\nu = \frac{A'_0E}{AE} = \frac{1}{n}\, \frac{\dfrac{dn(\nu)}{d\nu}}{\dfrac{d\theta}{d\nu}}, \tag{4,13} \]

and, finally,

\[ \upsilon=\sqrt{(A_{\upsilon}E)^2+(A'_0E)^2} =\frac{c}{n}\sqrt{\,1+\frac{1}{n^2} \left(\frac{dn}{d\nu}\middle/\frac{d\theta}{d\nu}\right)^2\,}. \tag{4,14} \]

Moreover, from (4,12) it is evident that the velocity of motion of the plane of the center of the group in the direction of propagation of the wave is indeed equal to the group velocity. In fact, this velocity is equal to (see (4,12))

\[ A_{\upsilon}D=AE-DE=w. \tag{4,15} \]

Let us turn to an examination of the results obtained. First of all, let us clarify the meaning of the velocity \(\upsilon\). Suppose that the spectrum of waves propagating in the medium is restricted to a frequency interval \(\Delta\nu\), for which \(\upsilon\) has a prescribed magnitude and direction. The electric and magnetic fields produced by these waves at an arbitrary point \(A\) depend on the distribution of amplitudes and phases of the monochromatic waves within the interval \(\Delta\nu\) and may be arbitrary. However, from the definition of the quantity \(\upsilon\) it follows that the field at the point \(B\), situated from \(A\) at a distance \(\upsilon t_1\), assumes the same values as at \(A\), but with a time delay of magnitude \(t_1\). Indeed, the construction of the points \(A_0\) and \(A'_0\) is such (see Fig. 4) that any of the waves of the frequency interval \(\Delta\nu\) has at the moment \(t\) at the point \(A_0\) the same phase as at \(A'_0\) at the moment \(t+t_1\).

In the case under consideration of the superposition of plane waves, it follows unambiguously from this that the resultant field at \(A_0\) is identical to the field at \(A'_0\) for the instant of time shifted by the amount \(t_1=1\). At the same time, for plane waves the same result in the sense of identity at comparable instants of time of the phases, and consequently also of the resultant amplitudes, will be obtained if, instead of the initial point \(A_0\), we choose any other point, i.e., under an arbitrary parallel displacement of the segment \(A_0A'_0\).

Thus, the velocity \(\upsilon\) may be called the velocity of transfer of the field pattern.

In the particular case of absence of dispersion, i.e. \(w=u\) (for the frequency interval \(\Delta\nu\)), as is seen from (4,14) and (4,15), \(\theta=0\) and \(\upsilon=u\), i.e. \(\upsilon\) coincides with the phase velocity both in direction and in magnitude.

In determining the quantity \(\upsilon\) it was assumed that the frequency interval \(\Delta\nu\) is narrow, so that the corresponding \(\Delta\theta\) is small. It is possible, however, that the frequency of oscillations and the direction of propagation of the waves are related in such a way that, for \(\Delta\nu\)’s cut out from any portions of a broad spectral interval \(\nu_1-\nu_2\), one obtains one and the same direction of the vector \(\upsilon\). Let us take this direction as

the origin for measuring the angle \(\theta_\nu\), which determines the direction of propagation of the waves. Then equation (4.13) must hold for any \(\nu\) lying within the limits \(\nu_1-\nu_2\). We write this equation in the form

\[ \tan\theta_\nu\,\frac{d\theta_\nu}{d\nu} = \frac{1}{n}\,\frac{dn(\nu)}{d\nu}. \tag{4.16} \]

Carrying out the integration and taking (4.10) into account, we obtain:

\[ n(\nu)\cos\theta_\nu = n(\nu_1)\cos\theta_{\nu_1} = \frac{c}{v}, \tag{4.17} \]

where \(\nu\) is an arbitrary frequency lying inside the interval \(\nu_1-\nu_2\). Equation (4.10) thus determines the relation between the direction of wave propagation \(\theta_\nu\) and the frequency \(\nu\) in the case when the vector \(\mathbf v\) is specified.

Let us consider an example illustrating the meaning of the quantity \(v\). Suppose a parallel beam of light from medium 1 with refractive index \(n_1=\mathrm{const}\) is incident at an angle \(\varphi\) on the interface with medium 2, which possesses dispersion. The field at the point \(C_2\) lying on the interface at the moment \(t_2\) is the same as at the point \(C_1\) at the moment \(t_1\), at which the wavefront \(C_2B\) will reach it (see Fig. 5). Hence we obtain that the velocity of transfer of the field is directed parallel to the interface and is equal to

Fig. 5.

\[ v=\frac{c}{n_1\cos\varphi}. \tag{4.18} \]

Thus, in medium 2 the waves of frequency \(\nu\) must be directed with respect to \(\mathbf v\) at an angle \(\theta_\nu\) satisfying (4.10), i.e.

\[ \cos\theta_\nu=\frac{c}{n(\nu)v}. \]

Substituting the value \(v\) from (4.18), we obtain the law of refraction of light

\[ \frac{\cos\theta_\nu}{\cos\varphi} = \frac{n_1}{n(\nu)}. \]

Another example is Vavilov—Cherenkov radiation. In this case the electromagnetic field must be constant in the coordinate system associated with the particle, provided only that the particle moves rectilinearly and uniformly for an unlimitedly long time. Hence the velocity of transfer of the field pattern coincides, both in direction and in magnitude, with the velocity of the particle. Indeed, (4.10) and condition (3.1) are identical if \(v\) is understood to mean the velocity of the particle.

Before passing to consideration of the Vavilov–Cherenkov effect, let us return once more to the question of the inclination of the plane of the group with respect to the wave plane (the angle between them satisfies (4.7)). This result also has a simple physical interpretation. To clarify it, suppose that in the path of the light beam under consideration, normally to the plane of incidence, a diaphragm with an aperture of diameter \(d\) is placed (Fig. 6). Let us determine the duration of the time interval during which a group of waves, specified by the frequency interval \(\nu \pm \Delta \nu\), will pass through this aperture.

First of all it is obvious that this interval splits into two parts. Since the group has a finite extent \(2l\), the group will pass through each point of the aperture during some time interval \(\tau_\nu\). In addition, the passage of the group of waves through different points of the aperture does not occur simultaneously, since the plane of the group forms an angle \(\chi\) with the wave plane. We shall denote this lengthening of the signal by \(\tau_1\).

Fig. 6.

The time \(\tau_\nu\) is equal to the length of the segment \(2l\) between the straight lines \(c\) and \(b\), divided by the group velocity \(w\). Since according to (4.3) the quantity

\[ 2l=\frac{w}{\Delta \nu}, \]

we have

\[ \tau_\nu=\frac{2l}{w}=\frac{1}{\Delta \nu}. \tag{4.19} \]

We again obtain, as was to be expected, the general condition (3.13), relating the duration of the pulse to the frequency interval.

For a finite aperture diameter \(d\), as already noted, an additional lengthening of the signal by the amount \(\tau_1\) occurs. From Fig. 6 it is seen that

\[ \tau_1=\frac{\operatorname{tg}\chi \cdot d}{w}. \tag{4.20} \]

Substituting the value of \(\operatorname{tg}\chi\) from (4.9), we obtain:

\[ \tau_1=\frac{d}{\lambda}\frac{d\theta}{d\nu}. \tag{4.21} \]

It is not difficult to verify that \(\tau_1\) from (4.21) is also directly connected with the general condition (3.13). For this purpose we shall use a simple relation defining the resolving power of spectroscopes. It is known that if a beam of parallel light rays containing two close

DURATION OF THE FLASH IN THE VAVILOV—CHERENKOV EFFECT

spectral lines with wavelengths \(\lambda_\nu\) and \(\lambda_{\nu'}\), has passed through the prism; and if \(\Delta\theta\) is the angle between the directions of the ray velocities and \(d\) is the width of the beam, then in the spectrum these lines can be resolved if\(^*\)

\[ d\cdot \Delta\theta=\lambda . \tag{4,22} \]

If the direction of propagation of the light is a function of the frequency, then

\[ \Delta\theta=\frac{d\theta}{d\nu}\,\Delta\nu . \tag{4,23} \]

Then from (4,22) we obtain:

\[ \frac{1}{\Delta\nu}=\frac{d}{\lambda}\frac{d\theta}{d\nu}. \tag{4,24} \]

Here \(\Delta\nu\) is the minimal frequency interval that can be resolved, i.e. separated out from the spectrum. The duration of the signal corresponding to this \(\Delta\nu\),

\[ \tau_1=\frac{1}{\Delta\nu}=\frac{d}{\lambda}\frac{d\theta}{d\nu}, \tag{4,25} \]

coincides exactly with (4,21).

Thus, the lengthening of the signal associated with the inclination of the group plane to the wave plane is determined by the fact that when \(\dfrac{d\theta}{d\nu}\) differs from zero, the wave group, without any additional devices, splits into narrower spectral groups. The duration associated with each of them is equal to \(\tau_1\), while their sequence, contained in the original signal, will have duration \(\tau_1+\tau_0\).

It also follows from what has been said that if such a diverging beam is passed through an optical system that achromatizes it, the duration of the signal will be compressed to the minimal value \(\tau_0\). Indeed, if the diverging beam is transformed into a parallel one, then the condition \(\dfrac{d\theta}{d\nu}=0\) will be satisfied, and consequently the group plane will become parallel to the wave plane, i.e. \(\tau_1=0\) (see Fig. 6).

\(^*\) This relation is not difficult to obtain. Let us consider the transverse section of a light beam, for example, in the plane of the diaphragm through which the light passes. Suppose that light of wavelength \(\lambda_\nu\) comes to some point \(O\) from all points of the diaphragm aperture in the same phase. Then for light of wavelength \(\lambda_{\nu'}\), the diameter of the aperture is equivalent to two Fresnel zones (since over the width \(d\) and at the angle \(\Delta\theta\), according to (4,22), we obtain for the extreme rays a path difference equal to one wavelength). Thus, the intensity maximum for \(\lambda_\nu\) coincides with the position of the first minimum for \(\lambda_{\nu'}\), which is, according to Rayleigh, the criterion for resolving lines.

§ 5. The Vavilov—Cherenkov Effect in a Refracting Medium

In § 4 a number of features of light beams resolved into a spectrum were clarified. Although the discussion was carried out for plane waves, it is easy to see that its results can be applied in their entirety to the consideration of the Vavilov—Cherenkov effect. The difference consists only in the fact that instead of plane wave surfaces in this case we have conical ones. However, in any plane drawn through the axis \(v\) (the direction of motion of the particle), the sections of the wave cone and of the group cone are straight lines, and the picture coincides exactly with that considered above (Figs. 3 and 4). It is not difficult, moreover, to verify that the angle \(\chi\) in Fig. 2 between the wave cone and the group cone for the radiation of Vavilov—Cherenkov indeed satisfies (4.9), i.e.

\[ \operatorname{tg}\chi=\frac{w}{\lambda}\frac{d\theta}{d\nu}. \tag{5,1} \]

In the case of the Vavilov—Cherenkov effect the direction of the phase velocity is given by (3.1), or, equivalently, by (4.10), i.e.

\[ \left. \begin{aligned} \cos\theta_{\nu}&=\frac{c}{vn(\nu)},\\ \sin\theta_{\nu}&=\frac{c}{vn(\nu)}\sqrt{\beta^{2}n^{2}-1},\\ \operatorname{tg}\theta_{\nu}&=\sqrt{\beta^{2}n^{2}-1}. \end{aligned} \right\} \tag{5,2} \]

In (5.1) the quantity \(\dfrac{d\theta}{d\nu}\) can be written in explicit form, using (4.14) or differentiating (5.2),

\[ \frac{d\theta_{\nu}}{d\nu} = \frac{1}{n}\frac{\cos\theta_{\nu}}{\sin\theta_{\nu}}\frac{dn}{d\nu} = \frac{1}{n\sqrt{\beta^{2}n^{2}-1}}\frac{dn}{d\nu}. \tag{5,3} \]

Since \(\lambda=\dfrac{c}{\nu n}\) and \(w=\dfrac{c}{\,n+\nu\dfrac{dn}{d\nu}\,}\), as a result we obtain:

\[ \operatorname{tg}\chi = \frac{\nu}{\left(n+\nu\dfrac{dn}{d\nu}\right)\sqrt{\beta^{2}n^{2}-1}} \frac{dn}{d\nu}. \tag{5,4} \]

From Fig. 2 it is seen that if \(\dfrac{dn}{d\nu}>0\), i.e. \(w<u\), then the group cone lies inside the wave cone, while if \(\dfrac{dn}{d\nu}<0\), then the group cone lies outside the wave cone. The first case corresponds to a positive \(\operatorname{tg}\chi\), and the second to a negative one.

It is not difficult to determine the angle between the generators of the cone of the group and its axis (the angle with the direction opposite to the direction of the velocity \(v\)). From Fig. 2 it is seen that this angle is

\[ \alpha=\frac{\pi}{2}-\theta_\nu-\chi, \tag{5.5} \]

that is,

\[ \operatorname{ctg}\alpha=\operatorname{tg}(\theta_\nu+\chi) =\frac{\operatorname{tg}\theta_\nu+\operatorname{tg}\chi}{1-\operatorname{tg}\theta_\nu\operatorname{tg}\chi}. \tag{5.6} \]

Using (5.2) and (5.4), we have:

\[ \operatorname{ctg}\alpha= \frac{1}{\sqrt{\beta^2 n^2-1}} \left[\beta^2 n^2-1+\beta^2 n v\,\frac{dn}{dv}\right] =g(\nu). \tag{5.7} \]

By means of a simple transformation, the quantity \(\operatorname{ctg}\alpha=g(\nu)\) can be expressed in terms of the phase and group velocities of light and the velocity of the particle in the following way (formula (5.7) of Tamm’s work\({}^{13}\)):

\[ g(\nu)=\operatorname{ctg}\alpha =\frac{v^2-uw}{w\sqrt{v^2-u^2}}. \tag{5.8} \]

Formula (5.8) is also not difficult to obtain directly with the aid of a simple trigonometric consideration, using Fig. 2.

Thus, for each frequency \(\nu\) for which \(v>u\), i.e. the Vavilov—Cherenkov effect takes place, we obtain a definite value \(g(\nu)\). If from the radiation spectrum one selects an interval of frequencies \(\Delta\nu\) so narrow that the angle \(\alpha\) within it may be regarded as constant, then the cone with angle \(\alpha\) and vertex coinciding with the position of the particle will determine the instantaneous position of the middle of the group of waves corresponding to the portion of the spectrum \(\Delta\nu\). The group of waves is then enclosed between two cones, the gap between which, measured along the direction \(\theta_\nu\), is equal to \(w\tau_0=\dfrac{w}{\Delta\nu}\).

For a broad interval of frequencies the quantity \(\alpha\) cannot be regarded as constant, and if one considers radiation with a broad frequency spectrum from \(\nu_1\) to \(\nu_2\), then we find that the radiation energy at each instant of time is concentrated between the surfaces of cones forming angles \(\alpha_1\) and \(\alpha_2\) with the axis, where

\[ \operatorname{ctg}\alpha_2=g_{\min} \quad\text{and}\quad \operatorname{ctg}\alpha_1=g_{\max}, \tag{5.9} \]

and the quantities \(g_{\min}\) and \(g_{\max}\) are the minimum and maximum values taken by the quantity \(g(\nu)\), defined by (5.7) or (5.8), when the frequency changes from \(\nu_1\) to \(\nu_2\).

Suppose that the radiation receiver has small geometrical dimensions and is located at a distance \(\rho\) from the trajectory of the particle. Since the common vertex of the cones moves with velocity \(v\), the receiver

will record radiation during the time interval

\[ \tau_2=\frac{g_{\max}-g_{\min}}{v}\rho . \tag{5,10} \]

(We neglect the quantity \(\tau_0\) here, which is possible for large \(\Delta \nu\).)

Let us consider in somewhat greater detail the question of the possible magnitude of \(\tau_2\). If the interval \(\nu_1-\nu_2\) is extended over the entire range of frequencies in which the Vavilov—Cherenkov effect is possible, then one can draw certain conclusions about the limits of variation of the quantity \(g(\nu)\), which determines \(\tau_2\). It is obvious that the Vavilov—Cherenkov radiation on the side of the higher frequencies has a boundary in the region of anomalous dispersion at the frequency at which \(n(\nu)\) decreases so much that

\[ u(\nu_2)=\frac{c}{n(\nu_2)}=v . \]

Since for \(\dfrac{dn}{d\nu}<0\), \(w>u\), in formula (5,8), at \(\nu=\nu_2\), the denominator becomes zero, while the numerator is not equal to zero and is negative. Thus,

\[ g(\nu_2)=\operatorname{ctg}\alpha_2=-\infty . \]

This means that the cone of the group is compressed toward the direction of the velocity \(\alpha=-\pi\). In fact, at this limiting frequency \(\theta_\nu=0\), i.e. the ray velocity is directed along the axis, and since \(w>u=v\), the radiation runs ahead of the particle, propagating in the same direction in which it moves.

As \(\nu\) moves away from the limiting value toward lower frequencies, the quantity \(g(\nu)\) rapidly increases. When \(uw=v\), the quantity \(\alpha=-\dfrac{\pi}{2}\). For all frequencies lying in the region of normal dispersion, \(\alpha\) is certainly less than \(\dfrac{\pi}{2}\), since \(w<u\) and, consequently, \(v^2>uw\) (see (5,8)).

If the Vavilov—Cherenkov effect has a limiting frequency also on the side of the lower frequencies, i.e. in the region of normal dispersion

\[ \left(v=u(\nu_1),\ \frac{dn}{d\nu}>0\right), \]

then, analogously, from formula (5,8) we obtain \(g(\nu_1)=\operatorname{ctg}\alpha_1=+\infty\). At this limiting frequency \(\theta=0\) and \(w<v\), i.e. the radiation lags behind the particle. Thus, if the Vavilov—Cherenkov effect is possible in a limited frequency range from \(\nu_1\) to \(\nu_2\) and the radiating particle moved for an indefinitely long time, then its radiation fills all space. If the Vavilov—Cherenkov radiation has no boundary on the side of lower frequencies, i.e. if \(\nu_1=0\), then the quantity \(g\) varies from \(-\infty\) to some finite positive value (the form of the function \(g(\nu)\) depends on the behavior of the refractive index, and in this case it need not be a monotone function of frequency). It follows from what has been said that, if the inter-

lay \(\nu_2-\nu_1\), then in formula (5.10) at least one of the limits of the quantity \(g(\nu)\) goes to infinity and, consequently, \(\tau_2=\infty\). In this case the radiation frequency at various points of space is a function of the angle of the cone of revolution whose axis coincides with the trajectory of the particle, while its vertex coincides with its instantaneous position.

Thus, the spectrum at the point at which the radiation is registered is a function of time. If, by means of a shutter of magnitude \(\tau_2\), some part is isolated, then thereby a definite frequency interval will be cut out of the radiation spectrum. Earlier, when considering the quantity \(\tau_1\), it was already noted that if \(\tau_1\) is different from zero, then by means of a diaphragm it is possible to isolate a spectral interval

\[ \Delta \nu=\frac{1}{\tau_1}\quad \text{(see (4.25)).} \]

In the present case the picture is in many respects analogous, but a narrowing of the spectral interval can be achieved not by a spatial limitation, but by a temporal limitation of the signal. Such a possibility of monochromatizing the radiation appears unexpected from the standpoint of the methods of ordinary optical spectroscopy.

Under the conditions of a real experiment the quantity \(\tau_2\) is not only not equal to infinity, but is usually very small (although it should be taken into account; see § 7). This is connected with the fact that there are always limitations both on the frequency interval that is registered and on the path over which the radiation occurs.

Probably the more essential quantity is the duration \(\tau_1\) (see § 4). Its magnitude is large in those practically important cases in which radiation produced over a considerable path of the particle enters the receiver. Suppose, for example, that the aperture of the diaphragm in Fig. 6 is the entrance aperture of a lens which focuses the Vavilov—Cherenkov radiation incident on it. The time during which a group of waves (specified by the spectral half-width \(\Delta \nu\)) reaches the surface of the lens, and hence also the duration of the pulse at the focus of the lens, is equal to \(\tau_0+\tau_1\), where \(\tau_1\) is determined by (4.25). Substituting into it the quantity \(-d\theta/d\nu\) from (5.3), we have:

\[ \tau_1=\frac{d}{\lambda n\sqrt{\beta^2 n^2-1}}\,\frac{dn}{d\nu} =\frac{d}{\lambda_0 \operatorname{tg}\theta_\nu}\,\frac{dn}{d\nu}. \tag{5.11} \]

Thus, the multiplier of \(dn/d\nu\) is the difference, expressed in fractions of \(\lambda_0=n\lambda\), between the paths from the farthest and the nearest points of the particle trajectory whose light reaches the lens.

The question of the duration of the signal as applied to real systems intended for recording Vavilov—Cherenkov radiation is considered in §§ 6 and 7.

§ 6. Counters of Vavilov–Cherenkov Radiation

As was already noted (§ 1), the question of the duration of the signal produced by Vavilov–Cherenkov radiation arises when light emitted over a considerable part of the particle’s path enters the receiver.

By the duration of a flash one always means the duration of the light signal at some definite point. Therefore, when speaking of the total duration of a flash produced by some amount of light, one must assume that this duration is determined at the point to which this light is focused, or at the point from which it is emitted. In the case of the Vavilov–Cherenkov effect, in which the emission of light takes place from different points of the particle trajectory, the question of the flash duration for all the emitted light has meaning only on the condition that this light is focused.

Under real conditions, however, focusing is never ideal. In reality the light falls on some surface, whose size may also be significant. The signal arising from this surface (for example, the flux of electrons emitted from the surface of a photocathode) is then transmitted, with the aid of one device or another, to the recording device. In this case the duration of the signal is determined not only by the duration of the light pulses at the various points of the surface and by the moments at which the light reaches them, but also by the time of the subsequent propagation of the signal from these points. Therefore, for very short flashes—for example, for the Vavilov–Cherenkov effect—the signal duration obtained in this way cannot simply be identified with the duration of the light flash.

It is obvious that the smaller the linear dimensions of the area on which the light falls, the more legitimately, down to shorter times, one may identify the time of collection of its light with the duration of the flash and regard it as the initial duration for the further propagation of the signal. Therefore, in what follows, we shall consider only counters possessing light-focusing devices. It is precisely such counters that are used in most cases, since they are based on the use of light emitted over a considerable path, which, in the absence of focusing, would not enter the receiver.

Let us consider two principal schemes of counters for recording flashes from the Vavilov–Cherenkov effect, first proposed by Getting \(^{10,19}\).

One of these systems is a radiator of conical shape (Fig. 7). We shall confine ourselves to considering the idealized case in which the particle moves exactly along the axis of the radiator and its velocity is strictly constant. At first we shall assume that the medium practically does not

possesses dispersion in the frequency interval under consideration \(\Delta \nu=\nu_2-\nu_1\). If the angle between the generatrices of the radiator cone and the axis is equal to \(\varphi=\frac{1}{2}\theta_\nu\), then all rays, after reflection from its surface, become strictly parallel to the axis. Then a collecting lens placed behind the base of the cone will completely collect in its focus all the radiation that has arisen over some segment of the particle’s path in the radiator.

Fig. 7.

As for the duration of the flash, in the case of a medium without dispersion it is, as we have seen, determined by the relation \(\tau_0=\frac{1}{\Delta \nu}\). Indeed, reflection does not change the duration of the signal, since it only rotates the direction of propagation of the waves. Nor is the duration changed when the light is collected in the focus of the lens, since the relations between the phases of the waves in the plane of the lens and in the focus remain unchanged (the lens is assumed to be achromatic\(*\)).

As for the prolonged afterglow that may arise in the radiator, it, as was shown (see § 3), should not give directed radiation and, consequently, will not be focused.

In a real refracting medium the angle \(\theta_\nu\) depends on the frequency, and therefore satisfactory focusing of the beam will be obtained only for a comparatively narrow spectral interval. Suppose that this interval \(\Delta \nu\) has been selected. Then it is necessary to take into account that the group of waves corresponding to this interval will pass through the surface of the lens not simultaneously.

\(*\) The presence of chromatic aberration will cause a lengthening of the signal. In this case the position of the focus depends on the frequency; therefore, from the focused beam it becomes possible to isolate a frequency interval \(\Delta \nu'\). This can be done by placing on the axis of the lens a diaphragm with a small aperture. If \(\Delta \nu' \ll \Delta \nu\), then the lengthening of the signal will become noticeable.

To determine the order of magnitude of \(\tau_1\), we set the quantity \(d\) in formula (5,11) equal to the radius of the lens \(R\). Then

\[ \tau_1=\frac{R}{\lambda_0}\frac{\cos\theta}{\sin\theta}\cdot\frac{dn}{d\nu}, \tag{6,1} \]

where \(\lambda_0=n\lambda\) is the wavelength of light in vacuum.

Let us estimate the magnitude of \(\tau_1\) for glass. Suppose that for \(\lambda_0=5\cdot10^{-5}\,\mathrm{cm}\) the value \(n=1.66\) and \(\dfrac{dn}{d\nu}=1.2\cdot10^{-16}\,\mathrm{sec}\), which corresponds to flint glass. Assuming that the velocity of the particles is close to the velocity of light, i.e., that one may put \(\beta=\dfrac{v}{c}\simeq 1\), we have \(\cos\theta=0.6\) and \(\sin\theta=0.8\). Substituting these values into (6,1), we obtain:

\[ \tau_1=2\cdot10^{-12}R\ \mathrm{sec}. \tag{6,2} \]

Thus, for \(R=10\,\mathrm{cm}\) we have \(\tau_1=2\cdot10^{-11}\,\mathrm{sec}\).

This means that the presence of dispersion leads to a considerable spectral decomposition of Vavilov—Cherenkov radiation, i.e., that precisely in the focus of the lens there will be collected light belonging only to a very narrow spectral interval.

At the same time it is evident that a shortening of the signal duration may also be achieved, and correspondingly an improvement in the quality of focusing, if the system is achromatized for Vavilov—Cherenkov radiation (see § 7).

In most cases the design of the “counter” differs from the scheme considered above. The difference consists in the fact that the reflecting-light cone is supplemented by a cylinder (Fig. 8). The light emitted

Fig. 8.

by a particle moving along the axis of the cylinder, after a certain number of reflections from the surface of the cylinder, enters the cone. Since, upon reflection from the surface of the cylinder, the magnitude of the angle between the direction of propagation of the light and the axis of the counter does not change, the condi-

...conditions for focusing the light entering the cone remain the same as for light produced in the cone. Thus, if the particle moves exactly along the axis of the counter, then the amount of light emerging from the base of the cone must increase in proportion to the total length of the counter.

Let us assume that the radiator has no dispersion of light. The path of the rays upon reflection from the lateral surface of the cylinder is clarified in Fig. 9. If there were no walls limiting the propagation of light in the cylinder, then during the time in which the particle traverses the path from point \(A_3\) to point \(A_0\), the light from point \(A_3\) would reach point \(C_3\). By the same instant, the light from \(A_2\) would have reached \(C_2\), and from \(A_1\) to \(D_1\). Thus, the wave front for the light emitted along the path \(A_3-A_0\) would have an instantaneous position determined by a wave cone (for \(n=\mathrm{const}\), coinciding with the group cone), one of whose generators is \(A_0C_3\). With each reflection, the component of the light velocity perpendicular to the counter axis changes sign, while the longitudinal component remains unchanged. As a result, the light emitted along the path from \(A_3\) to \(A_2\), after two reflections, will reach the surface \(D_3-D_2\). Similarly, the light emitted along the path \(A_1-A_0\) undergoes one reflection, and the instantaneous position of the wave front will be \(D_2-D_1\). Finally, the light emitted along the path \(A_1-A_0\) undergoes no reflections, and for it the wave front \(D_0A_0\) coincides with the cone \(A_0C_3\)*).

Fig. 9.

The greater the number of reflections experienced by the light, the greater the length of cylinder behind the particle in which the radiation will be contained.

*) In order not to clutter Fig. 9, it shows the path of rays emitted only to one side with respect to the trajectory of the particle. Thus, of the two generators of each cone lying in the plane of Fig. 9, only one is shown (the other is, obviously, located symmetrically with respect to the axis of the cylinder).

Radiation emerging from the cylinder is thereby split into a series of successive light pulses filling a time interval that increases with increasing cylinder length.

In the idealized case under consideration, the reflected waves remain mutually coherent. If one considers the propagation of a monochromatic wave with frequency \(\nu\), then the total amplitude will be obtained as a result of superposing the sequence of reflected waves. In this case, waves only of those frequencies for which the path difference acquired in successive reflections is equal to an integral number of wavelengths will be amplified. Thus, the radiation spectrum acquires a line character. The resulting lengthening of the signal, as was to be expected, is connected with the monochromatization of the radiation.

Thus, in this case the increase in the duration of the signal occurs as the result of a mechanism analogous to the action of an interference spectral apparatus*).

Let us determine the length of the path traversed by light in a counter consisting of a cylinder and a cone. Suppose that light emitted from a point \(A_k\), lying at the beginning of a cylinder of length \(L\) (Fig. 8), undergoes \(k\) reflections from its side walls. Here the \(k\)-th reflection is the last, so that after it the light at the point \(A_0\) crosses the axis of the counter and then falls on the lateral surface of the cone. Thus, the further path of the light from the point \(A_0\) is the same as for light emitted from this point at the moment when the particle passes through it.

The path traversed by light from the point \(A_k\) to \(A_0\) is equal to

\[ s_k=\frac{2kr}{\sin\theta}, \tag{6.3} \]

where \(r\) is the radius of the cylinder. The distance between these points along the straight line (which the particle traverses) is equal to

\[ c_k=2kr\operatorname{ctg}\theta. \tag{6.4} \]

Light emitted from the point \(A_k\) will reach the point \(A_0\) with a delay in time

\[ t_k=\frac{n}{c}s_k-\frac{1}{v}c_k =2kr\frac{n}{c}\sin\theta. \tag{6.5} \]

In formula (6.5) the relation between the quantity \(\theta\) and \(v\) has been taken into account (see (5.2)).

It is obvious that waves arriving at \(A_0\) after \(k\) reflections may differ in phase from the waves emitted from this point. Using

* In paper 16 there is an incorrect statement that a particle moving for an indefinitely long time along the axis of an infinite cylinder, light from which cannot emerge because of total internal reflection, does not lose energy to radiation. This question is considered in detail in the dissertation of B. M. Bolotovsky.

formula (6.5), one can determine the path difference of the rays that accumulates as a result of each reflection. It proves to be equal to

\[ \vartheta = 2\omega r\,\frac{n}{c}\,\sin\theta-\gamma+\frac{3}{4}\pi . \tag{6.6} \]

The first term in \(\vartheta\) is obtained from (6.5) by multiplying \(t_k\) at \(k=1\) by the light frequency \(\omega=2\pi\nu\), while the quantity \(\gamma\) is the phase change of the wave under total internal reflection. In addition, two phase shifts must be taken into account. First, upon reflection the direction of propagation of the ray is rotated about the straight line perpendicular to the plane of incidence, and together with it the direction of the electric vector is rotated (\(E_a\) changes into \(E_r\), see Fig. 9). The projection of the vector \(E\) onto the direction of the velocity changes sign, and when interference is considered this is equivalent to a phase shift by \(\pi\). Secondly, as was shown in § 3 (see formulas (3.8) and (3.9)), the phase obtained when waves from a segment of the trajectory surrounding the given point (here \(A_k\)) are added differs by \(\frac{\pi}{4}\) from the phase of the wave from this point. The additional phase shift by \(\frac{3}{4}\pi\) in (6.6) takes both of these causes into account. As for the quantity \(\gamma\), when the electric vector lies in the plane of incidence (which is the case in the Vavilov—Cherenkov effect), then

\[ \gamma = 2\operatorname{arctg}\frac{n\sqrt{\,n^{2}\cos\theta-1\,}}{\sin\theta}. \tag{6.7} \]

The quantity \(\vartheta\) may also, with the aid of (5.2), be written as

\[ \gamma = 2\operatorname{arctg}\frac{n^{2}\sqrt{\,1-\beta^{2}\,}}{\sqrt{\,\beta^{2}n^{3}-1\,}} . \tag{6.8} \]

If \(\vartheta\) is set equal to a multiple of \(2\pi\), then (6.6) determines the eigenfrequencies of Vavilov—Cherenkov radiation for a particle moving along the axis of the cylinder*). Equation (6.6) is evidently also valid for a medium possessing dispersion (since in that case \(n\), as well as \(\sin\theta\) and \(\gamma\), are functions of frequency).

However, such exact coincidence of the particle trajectory with the cylinder axis is hardly possible that, after several reflections, the light waves would remain coherent with one another. Under real conditions the signal obtained from a counter consisting of a cylinder and a cone will appear as a sequence of independent light pulses. At the maximum number of reflections \(k\), the last signal is delayed by a time \(t_k\) relative to the first (see (6.5)).

*) This is a special case of a more general condition for eigenfrequencies obtained in the dissertation of B. M. Bolotovskii.

Suppose that the length of the cylinder is \(L=20\) cm; then, since

\[ L \simeq 2kr\operatorname{ctg}\vartheta, \tag{6.9} \]

putting, as in the case considered earlier,

\[ n=1.66,\qquad \cos\vartheta=0.6 \quad \text{and} \quad \sin\vartheta=0.8, \]

we obtain:

\[ \Delta t_k \sim 1\cdot 10^{-9}\ \text{sec}. \]

Thus, the increase in the amount of light achieved by attaching a cylinder to the counter entails a proportional and quite noticeable increase in the total duration of the signal.

In connection with this it may be noted that measurement of the duration \(t_k\) may prove useful for determining the velocity of the particle. Suppose that the velocity of the particle varies within small limits in such a way that the number of reflections \(k\) does not change. Then the interval \(t_k\) is proportional to \(\sin\vartheta\), and hence measurement of \(t_k\) is equivalent to determination of the quantity \(\vartheta\). At the same time it is known that measurement of \(\vartheta\) can be used to find the velocity of the particle \(v\). Obviously, for an individual particle it is more convenient to determine the duration of a time interval than an angle.

When the velocity \(v\) decreases, the quantity \(\sin\vartheta\), and consequently \(t_k\), decrease. Differentiating \(t_k\) with respect to \(v\), we obtain:

\[ \frac{d}{dv}t_k=\frac{2kr}{v^2}\operatorname{ctg}\vartheta \simeq \frac{L}{v^2}. \tag{6.10} \]

Thus, the magnitude of the derivative of \(t_k\) with respect to \(v\) is the same as for the time of flight of a particle over the path \(L\), i.e. for

\[ t_k=\frac{L}{v}, \]

but it has the opposite sign. In fact the dependence of \(t_k\) on the velocity is still more significant, since for large changes of \(v\) there is a change not only in \(\sin\vartheta\), but also in the number of reflections \(k\),

\[ k\sim \frac{L}{2r}\operatorname{tg}\vartheta. \tag{6.11} \]

In principle, determination of the number of light pulses contained in the light signal can also serve as a method for measuring the velocity of the particle.

For a cylinder of small diameter, in which the pulses determined by individual reflections from the walls merge with one another, for the value \(t_k\) from (6.5) and (6.11) we obtain:

\[ t_k=L\,\frac{n}{c}\,\frac{\sin^2\vartheta}{\cos\vartheta}. \tag{6.12} \]

In the case of a real medium possessing dispersion, certain corrections must be introduced into the formulas obtained above. First of all it should be borne in mind that the velocity of propagation of the light pulse is equal to the group velocity. Thus, in formula (6.5) the quantity \(s_k\) should be divided not by \(u=\dfrac{c}{n}\), but by the group velocity \(w\). Further, it should be kept in mind that the phase jump upon reflection from the wall of the cylinder depends on the frequency (since both \(\theta_\nu\) and \(n(\nu)\) in this case are functions of frequency). This leads to the fact that, although the group cone will undergo specular reflection, in the calculation the reflecting surface should be regarded as somewhat displaced relative to the geometrical boundary of the cylinder. The additional time spent in this process on reflection is, as can be shown,

\[ \Delta t=-\frac{1}{2\pi}\frac{d\gamma}{d\nu}, \tag{6.13} \]

where \(\gamma\) is the phase jump upon reflection*).

Thus, instead of (6.5) we obtain:

\[ t_k=\frac{1}{w}s_k-\frac{1}{v}c_k+k\Delta t. \tag{6.14} \]

The quantity \(\Delta t\) is found by differentiating (6.8) and is of the order of

\[ \frac{1}{2\pi}(1-\beta^2)\frac{dn}{d\nu}. \]

This quantity, for example, for the case of glass considered by us, with \(\dfrac{dn}{d\nu}=1.2\cdot10^{-16}\), is negligibly small. Therefore from (6.14), instead of (6.5), we obtain:

\[ t_k=\frac{n+\nu\dfrac{dn}{d\nu}}{c}s_k-\frac{1}{v}c_k =2kr\frac{n}{c}\left[\sin\theta_\nu+\frac{1}{n\sin\theta_\nu}\nu\frac{dn}{d\nu}\right]. \tag{6.15} \]

Thus, qualitatively the picture is preserved, but a correction must be introduced into the quantity \(t\).

§ 7. ACHROMATIC COUNTERS

From what has been said in the preceding sections it follows that the shortest light pulse can be obtained with the aid of a radiator in which the light is not split into a series of beams, and provided that the radiator possesses no dispersion.

This can be achieved in various ways, for example with the aid of a conical radiator, if it is made achromatic,

*) I take this opportunity to thank S. M. Rytov for clarifying a number of questions connected with reflection of a group of waves.

i.e., to arrange that, for the beam emerging from the radiator,

\[ \frac{d\theta}{d\nu}=0. \]

Possible schemes of this kind are shown in Figs. 10a and 10b. Suppose that a particle moves along the axis \(ob\) in radiator \(A\), whose substance has normal dispersion. The light emitted by the particle

a)

b)

Fig. 10.

at an angle \(\theta_\nu\) to the direction \(ob\) can be transformed into a beam parallel to this direction not by means of reflection, but by means of refraction at the boundary between medium \(A\) and medium \(B\), which has a conical shape (the vertex of the cone is at \(b\), and its angle with the axis is \(\varepsilon < \frac{\pi}{2}\); see Fig. 10a). From the law of refraction we obtain the requirement

\[ \frac{n_B}{n_A} = \frac{\cos(\varepsilon-\theta_\nu)}{\cos\varepsilon} = \cos\theta_\nu+\sin\theta_\nu \tan\varepsilon, \tag{7,1} \]

where \(n_B\) and \(n_A\) are the refractive indices of medium \(B\) and medium \(A\).

Since \(\varepsilon > \theta_v\), in order that (7.1) can be satisfied it is necessary that

\[ \frac{n_B}{n_A}>\frac{1}{\cos\theta_v}=\beta n_A,\quad \text{i.e.}\quad n_B>\beta n_A^2 . \tag{7.2} \]

((7.2) is obtained from (7.1) by replacing the quantity \(\operatorname{tg}\varepsilon\) by its minimum value, equal to \(\operatorname{tg}\theta_v\).) If inequality (7.2) is fulfilled, then one can find such an angle \(\varepsilon>\theta_v\), and less than \(\dfrac{\pi}{2}\), that the radiation traveling at the angle \(\theta_v\), after refraction, will proceed parallel to the axis.

The same result can also be obtained for a cone with an obtuse angle \(\varepsilon\) (see Fig. 10, б). Equality (7.1) is valid in this case as well, but \(\cos(\varepsilon-\theta)\) and \(\cos\varepsilon\) are negative. Since \(\operatorname{tg}\varepsilon\) in this case is less than zero, and its maximum value is equal to zero, equality (7.1) for an obtuse angle can be satisfied if

\[ \frac{n_B}{n_A}<\cos\theta_v=\frac{1}{\beta n_A},\quad \text{i.e.}\quad n_B<\frac{1}{\beta}. \tag{7.3} \]

Using (7.1) and taking into account that \(\cos\theta_v=\dfrac{1}{\beta n_A}\), we obtain

\[ \operatorname{tg}\varepsilon=\frac{\beta n_B-1}{\operatorname{tg}\theta_v}; \tag{7.4} \]

in this case inequality (7.2) must be fulfilled for \(\varepsilon<\dfrac{\pi}{2}\), or (7.3) for \(\varepsilon>\dfrac{\pi}{2}\). Thus, the refraction of light by a conical surface can be used, similarly to a conical mirror, to transform a beam of rays traveling at an angle \(\theta\) to the axis into a beam parallel to the axis.

The difference from a mirror consists, however, in the fact that in the present case it is possible to transform into a parallel beam not only rays traveling at a definite angle \(\theta_v\) (i.e. at a given velocity \(v\), only for one \(\nu\)). In the case of a refracting cone this can be achieved for some interval of angles \(\Delta\theta\), corresponding to an interval of frequencies \(\Delta\nu\). For this it is necessary to require that the quantity \(\varepsilon\) in (7.4) not change when the frequency varies within the limits \(\Delta\nu=\nu_1-\nu_2\).

Differentiating the right-hand side of (7.4) with respect to frequency and setting the derivative equal to zero, we obtain the following condition for achromatization of the radiation:

\[ \frac{dn_B}{dn_A} = \frac{\beta n_B-1}{\beta n_A\sin^2\theta_v}\,\frac{dn_A}{d\nu} = \frac{\beta n_A(\beta n_B-1)}{\beta^2 n_A^2-1}\,\frac{dn_A}{d\nu}. \tag{7.5} \]

(In the differentiation, formula (5.3) has been taken into account; in it the subscript \(A\) should be assigned to the refractive index.)

In the case of an acute angle \(\varepsilon<\dfrac{\pi}{2}\), both media may have normal dispersion. Indeed, since in this case, according to (7.2),

\[ n_B>\beta n_A \]

and, moreover, \(\beta n_A>1\), then

\[ \left|\frac{dn_B}{d\nu}\right|>\left|\frac{dn_A}{d\nu}\right| \]

and both derivatives have the same sign.

It is in principle possible that condition (7.5) will be satisfied in a comparatively broad frequency interval from \(\nu_2\) to \(\nu_1\), and then for the refracted rays in this interval

\[ \frac{d\theta'}{d\nu}=0 \]

(\(\theta'\) is the direction of propagation of the refracted rays; \(\theta'=0\)). As a result, the duration \(\tau\) must vanish, and, moreover,

\[ \tau_0=\frac{1}{\nu_1-\nu_2} \]

for large \(\Delta\nu=\nu_1-\nu_2\) will also be very small. Therefore one may expect that at the focus of a lens collecting a parallel beam of light emerging from the base of the cone, a very short flash will be obtained.

However, this is valid only so long as \(\Delta\nu\) is still so small that the values of the group velocity for \(\nu_1\) and \(\nu_2\) may be regarded as identical. As \(\Delta\nu\) increases, the duration connected with \(\tau_2\) (see § 5, formula (5.10)) becomes significant. Indeed, the angle \(\alpha\) between the generators of the group cone and the axis of the counter (Fig. 2 and formula (5.7)) depends on frequency and is equal, for \(\nu_1\) and \(\nu_2\), respectively to \(\alpha_1\) and \(\alpha_2\). In Fig. 10,a the position of the generators of these cones 1 and 2 is shown at the moment when the particle emitting the light approaches the interface of the two media at point \(b\). When the particle reaches point \(d\), the generators of these group cones will occupy, in medium \(A\), positions 3 and 4 (it is obvious that the propagation of light in medium \(A\) does not depend on the presence of medium \(B\)). In medium \(B\), the light propagates in the form of a parallel beam of plane waves in the direction of the counter axis, and therefore instead of the group cone we obtain planes \(a_1\) and \(a_2\), perpendicular to the axis and intersecting cones 3 and 4 at the interface. Indeed, upon refraction of light there is no change in the phase of the waves and, consequently, the group of waves must pass continuously through the interface.

From Fig. 10,a it is seen that as the particle advances inside medium \(B\), the planes \(a_1\) and \(a_2\) move farther and farther apart. The velocities of their displacement in the medium are obviously equal to the magnitude of the group velocity for the frequencies \(\nu_1\) and \(\nu_2\). From the condition that the planes \(a_1\) and \(a_2\) intersect at the interface with the group cones, it is not difficult to obtain, using Fig. 10,a, that the group velocity in medium \(B\) must satisfy the relation

\[ \frac{1}{w_B}=\frac{1}{v}\left(\frac{\operatorname{tg}\varepsilon}{\operatorname{tg}\alpha}+1\right). \tag{7.6} \]

The existence of such a relation is explained by the fact that the properties of medium \(A\)

and \(B\) are related to each other by condition (7.5). Indeed, if one takes into account the relations (5.7) and (7.4), which determine the quantities \(\tg \alpha\) and \(\tg \varepsilon\), then (7.6) follows from (7.5).

If the height of cone \(B\) is equal to \(L_B\), then the wave packets with frequencies \(\nu_1\) and \(\nu_2\) will traverse a path equal to the height of the cone in times differing by the amount

\[ \begin{aligned} \tau_2 &= L_B \left|\frac{1}{w_B(\nu_2)}-\frac{1}{w_B(\nu_1)}\right| = \\ &= \frac{L_B}{c}\left| r_B(\nu_2)-r_B(\nu_1)+(\nu_2-\nu_1)\frac{dn_B}{d\nu}\right| = \frac{2L_B}{c}\left|(\nu_2-\nu_1)\frac{dn_B}{d\nu}\right|. \end{aligned} \tag{7.7} \]

(Here it is assumed that the change of \(n_B\) in the frequency interval from \(\nu_1\) to \(\nu_2\) may be regarded as linear, i.e. \(\frac{dn_B}{d\nu}=\mathrm{const}\).) Suppose, for example, that medium \(B\) is made of dense flint. Then for the quantities appearing in the right-hand side of (7.7) one may take the following values: for \(\lambda_C=6563\,\text{\AA}\), \(n_B=1.715\), and for \(\lambda_F=4861\,\text{\AA}\), \(n=1.735\), \(\frac{dn}{d\nu}=1.43\cdot 10^{-16}\,\text{sec}\), and \(\nu_F-\nu_C=1.68\cdot 10^{14}\,\text{sec}^{-1}\). Substituting these values into (7.7), we obtain:

\[ \tau_2=1.67\cdot 10^{-12} L_B \,\text{sec}. \]

Taking, for example, \(L_B=10\,\text{cm}\), we find that the magnitude \(\tau_2\) really cannot be neglected if a wide interval of the radiation spectrum is used.

Thus, from what has been said above it follows that, with the aid of an achromatic counter, it is indeed possible to obtain a very short light flash in the case where the frequency interval \(\Delta \nu\) is suitably restricted. This, however, leads to a decrease in the amount of light used. It is therefore essential that a reduction of the quantity \(\tau_2\) can be achieved in another way as well.

For this purpose let us place, after cone \(B\), a plane-parallel plate \(C\), possessing, in contrast to \(A\) and \(B\), anomalous dispersion (Fig. 10, a). In such a plate \(w(\nu_2)>w(\nu_1)\), and consequently the group of waves with frequency \(\nu_2\) will overtake in it the group of waves with frequency \(\nu_1\). Obviously, by selecting the thickness of the compensator \(C\), it is in principle possible to reduce \(\tau_2\) to zero. In the case of a counter with an obtuse angle \(\varepsilon\) (Fig. 10, b), the role of such a compensator is played by medium \(B\) itself. Indeed, repeating the same reasoning that was carried out above, we find that the planes \(a_1\) and \(a_2\) will approach each other as the particle approaches the vertex of the cone—the point \(b\) (see Fig. 10, b). On passing through the point \(b\) they coincide, i.e. \(\tau_2\) becomes zero. Such compression of the wave packet is explained by the fact that, as was already indicated, for \(\varepsilon>\frac{\pi}{2}\) medium \(B\) must necessarily possess anomalous dispersion, if the dispersion in medium \(A\) is normal.

Thus, unlike ordinary optical problems, in the case under consideration it is not sufficient to make the system achromatic. To obtain a short-duration flash, it is required that the propagation time of the group of waves, within known limits, not depend on the frequency \(\nu\)*).

The principal schemes of Figs. 10,a and 10,b are, of course, not the only possible types of achromatic counters.

The Vavilov—Cherenkov radiation can be achromatized by other methods as well, for example by means of successive reflection and refraction by conical surfaces. One such possible scheme is shown in Fig. 11. The trajectory of the particle \(ab\) is assumed to coincide with the axis of the cones, and its direction is indicated by an arrow. The emitted light, after reflection from the surface of the cone \(cbd\), propagates in the direction opposite to the direction of the trajectory, at an angle \(ab\) to the axis. By choosing a suitable angle of the reflecting cone one can transform the angle \(\theta_r\) into an arbitrary angle \(\theta_y\). The angle of the refracting cone \(\varepsilon\), for which the refracted ray goes parallel to the axis, is determined by the same condition (7,1), if in it \(\theta\) is replaced by \(\theta_r\). Then

Fig. 11.

\[ \operatorname{tg}\varepsilon = \frac{n_B-n_A\cos\theta_r}{n_A\sin\theta_r}. \tag{7,8} \]

For an acute angle \(\varepsilon\), from the requirement (7,1) and \(\varepsilon>\theta_r\), we again obtain condition (7,2)

\[ \frac{n_B}{n_A}>\frac{1}{\cos\theta_r}. \tag{7,9} \]

The condition for the independence of \(\operatorname{tg}\varepsilon\) from the frequency is obtained by setting equal to zero the derivative with respect to \(\nu\) of the right-hand side of (7,8), moreover \(\dfrac{d\theta_r}{d\nu}=\dfrac{d\theta}{d\nu}\), and, consequently, equal to (5,3)

\[ \frac{dn_B}{dn_A} = \left( \frac{n_B}{n_A} + \frac{\dfrac{n_B}{n_A}\cos\theta_r-1} {\sin\theta_r\sqrt{\beta^2 n_A^2-1}} \right) \frac{dn}{d\nu}. \tag{7,10} \]

It is obvious that condition (7,10), since it contains the angle \(\theta_r\),

*) In an exact calculation of the thickness of the compensator \(C\), one must also take into account the propagation of light in the achromatic lens that focuses the light emerging from the counter.

the magnitude of which within known limits can be chosen arbitrarily, is easier to satisfy than (7.5).

It should also be borne in mind that the use of a refracting cone in combination with an achromatic lens is by no means obligatory. Achromatization of the radiation may, for example, also be achieved by means of a lens possessing a suitably chosen chromatic aberration, so that the light emerging from the counter is focused at a single point.

Above we considered arrangements corresponding to ideal conditions. In reality these conditions are never fulfilled strictly. In fact, the trajectory of a particle may be displaced relative to the axis of the counter and inclined with respect to it. Multiple scattering, as well as loss of velocity as a result of ionization losses, introduce additional deviations of the true trajectory from the ideal one. It is evident that all these causes produce a spread in the length of the path traversed by the light, i.e., they cause an increase in the duration of the signal.

An increase in the duration of the signal in these cases at the same time also leads to a deterioration of the focusing of the radiation emerging from the counter. A displacement of the trajectory by a distance from the axis of the order of a millimeter will cause a lengthening of the signal by \(10^{-12}\)--\(10^{-11}\) sec. Thus, the deviation from the ideal experimental conditions is a substantial and, probably, the principal cause limiting the possibility of shortening the duration of the signal.

What has been said in the preceding sections of this article makes it possible to draw a certain conclusion.

Already in the first works of S. I. Vavilov and P. A. Cherenkov it was shown that the glow they discovered possesses a decay time that is, in any case, small in comparison with the duration of luminescence. Indeed, the directionality of the radiation characteristic of the Vavilov--Cherenkov effect, in combination with its continuous spectrum, proves to be incompatible with the presence of a long afterglow (§ 3).

The directionality of the radiation is very important for the possibility of its registration, since it makes it possible to focus the light emitted by a particle over a considerable path, i.e., over a comparatively long interval of time. The features of the radiation in this case are such that, under certain conditions, the duration of the flash obtained in the focus of the lens can nevertheless remain very short.

In considering this question one must take into account a number of circumstances. Thus, for example, a significant increase in the duration of the signal occurs in counters in which different rays undergo a different number of reflections (a cylindrical counter). This feature can be used for measuring

particle’s velocity (§ 6); however, such counters should not be used for obtaining extremely short flashes.

It should further be borne in mind that the presence of light dispersion in the radiator substance causes the radiation to be decomposed into a spectrum and leads to a lengthening of the signal (§ 5). This cause of lengthening of the flash, however, can be eliminated if the counter is made achromatic (§ 7). In addition, there is a lengthening of the flash associated with the spreading of the wave packet as it passes through a dispersive medium. This cause, too, can be eliminated with the aid of a compensator possessing anomalous dispersion (§ 7).

Thus, Vavilov—Cherenkov radiation can serve as a source of radiation whose duration is theoretically minimal \(\left(\tau_0 = \frac{1}{\Delta \nu}\right)\). The actual duration of the flash observed experimentally must be greater and is determined by practical causes (imperfection of the optics and deviation of the particle trajectory from the ideal one, etc.). Nevertheless, in recording particles of high energy, which almost do not lose their velocity in the radiator and are only weakly scattered, it is possible to obtain very short flashes.

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