Slow Electromagnetic Waves

A. I. Akhiezer and Ya. B. Fainberg

Contents

I. Slow waves in waveguides partially filled with dielectric . . . 322
II. A chain of endovibrators . . . 331
III. Waveguides loaded with disks . . . 339
IV. Helical waveguides . . . 348
V. Interaction of a beam of charged particles with slow waves . . . 355
VI. Interaction of a beam of charged particles with an electron plasma . . . 360

Recently, in solving a number of physical and radio-engineering problems, slow electromagnetic waves have been widely used, i.e., waves whose phase velocity of propagation is less than the velocity of light in vacuum. The important significance of slow waves is due to the possibility of effective interaction with them of charged particles. This interaction is used in a number of devices (such as, for example, the klystron, the traveling-wave tube, the multicavity magnetron, the two-beam amplifier, the magnetron amplifier, etc.) for the excitation and amplification of electromagnetic oscillations of ultrahigh frequencies. Despite the great variety of devices of this kind, the physical processes occurring in them have much in common and can be explained by considering the general problem of the interaction of charged particles with a slow wave. An important role in this interaction is played by the Cherenkov effect, which consists in the fact that a charged particle, moving uniformly in a system in which slow waves can propagate, is itself capable of radiating such waves, provided only that its velocity exceeds a certain critical value. In this case radiation takes place regardless of whether the particle moves in a dielectric or in vacuum; in the latter case it is only necessary that artificially, for example by a special arrangement of metallic

partitions with apertures in the path of propagation of the wave, conditions were created under which the phase velocity of the wave is less than the velocity of light in vacuum.

If it is not a single particle that interacts with slow waves, but an unmodulated beam of charged particles of sufficiently low density, and if the velocity of the beam exceeds the critical value of the velocity in the Cherenkov effect, then the state of the beam becomes unstable and the density fluctuations present in it propagate in the form of waves of charge density with increasing amplitude. The slow electromagnetic waves excited in this process have the same character. The instability of the beam and the appearance of waves with increasing amplitude play an essential role in the mechanism of excitation and amplification of ultrahigh-frequency oscillations in all devices in which the interaction of an electron beam with slow waves is used (for example, in such typical devices as the traveling-wave tube, the multicavity magnetron, and the magnetron amplifier).

In the present article we shall review the known methods for obtaining slow waves, using waveguides partially filled with a dielectric, as well as periodic structures not containing dielectrics (of the endovibrator-chain type) and helical waveguides; in addition, we shall set forth the general laws governing the processes of interaction of charged particles with slow waves, which play an essential role in the generation and amplification of ultrahigh-frequency oscillations.

I. SLOW WAVES IN WAVEGUIDES PARTIALLY FILLED WITH A DIELECTRIC

1. It is known that in ordinary homogeneous waveguides that do not contain dielectrics, the phase velocity of wave propagation (along the axis of the waveguide) exceeds the velocity of propagation of light in vacuum. If the waveguide is completely filled with a dielectric, then in the frequency range

\[ \omega > \frac{\omega_0}{\sqrt{\varepsilon - 1}} \]

(\(\varepsilon\) is the dielectric constant, \(\omega_0\) is the cutoff frequency of the empty waveguide) the phase velocity will be less than the velocity of light in vacuum, \(c\). However, in order to reduce the phase velocity, it is not necessary to fill the waveguide completely with dielectric. It turns out that if the walls of the waveguide are coated with a layer of dielectric, then in the region not occupied by the dielectric, beginning with a certain critical frequency, the phase velocity of propagation \(v\) will be less than \(c\).

In this section we shall briefly describe the features of wave propagation in waveguides partially filled with dielectric\(^{1,2}\). For simplicity we shall restrict ourselves to the consideration of a cylindrical-

waveguide whose walls are covered with a layer of dielectric. We shall regard the dielectric as ideal and isotropic.

Let the radius of the waveguide be denoted by \(b\), and the thickness of the dielectric layer by \(d=b-a\) (Fig. 1). We shall consider monochromatic waves propagating along the axis of the waveguide (the \(z\)-axis), and assume that the field components are proportional to \(e^{i(\omega t-k_3 z)}\) (\(k_3\) is the projection of the wave vector on the \(z\)-axis). In the case of an \(E\)-wave (\(E_z \ne 0,\ H_z=0\)) the field components in a cylindrical

Fig. 1.

coordinate system are conveniently expressed by the following formulas\(^3\):

\[ \begin{aligned} E_z&=\frac{\partial^2 U}{\partial z^2}+\varepsilon\mu k^2 U, & E_r&=\frac{\partial^2 U}{\partial r\,\partial z}, & E_\varphi&=\frac{1}{r}\frac{\partial^2 U}{\partial\varphi\,\partial z}, \\ H_z&=0, & H_r&=i\,\frac{\varepsilon k}{r}\frac{\partial U}{\partial\varphi}, & H_\varphi&=-i\varepsilon k\,\frac{\partial U}{\partial r}, \end{aligned} \tag{1,1} \]

where the function \(U\) satisfies the equation

\[ \frac{\partial^2 U}{\partial z^2} +\frac{\partial^2 U}{\partial r^2} +\frac{1}{r}\frac{\partial U}{\partial r} +\frac{1}{r^2}\frac{\partial^2 U}{\partial\varphi^2} +\varepsilon\mu k^2U=0 \tag{1,2} \]

\[ \left(k=\frac{\omega}{c},\ \varepsilon \text{ is the dielectric constant and } \mu \text{ is the magnetic permeability}\right). \]

For \(H\)-waves (\(E_z=0,\ H_z\ne0\)), instead of (1,1) the following relations hold:

\[ \begin{aligned} H_z&=-\frac{\partial^2 U}{\partial z^2}-\varepsilon\mu k^2U, & H_r&=-\frac{\partial^2 U}{\partial r\,\partial z}, \\ H_\varphi&=-\frac{1}{r}\frac{\partial^2 U}{\partial\varphi\,\partial z}, \\ E_z&=0, & E_r&=i\,\frac{\mu k}{r}\frac{\partial U}{\partial\varphi}, & E_\varphi&=-i\mu k\,\frac{\partial U}{\partial r}. \end{aligned} \tag{1,3} \]

The particular solutions of equation (1.2) have the following form:

\[ U= \begin{cases} U_1J_n(k_1r)\cos n\varphi\, e^{i(\omega t-k_3z)}, & \text{if } r\leq a,\\ \left[U'_2J_n(k_2r)+U''_2N_n(k_2r)\right]\cos n\varphi\, e^{i(\omega t-k_3z)}, & \\ \qquad\text{if } a\leq r\leq b, \end{cases} \tag{1.2′} \]

where \(J_n\) and \(N_n\) are the Bessel and Neumann functions,

\[ k_1=\sqrt{k^2-k_3^2},\qquad k_2=\sqrt{\varepsilon\mu k^2-k_3^2},\qquad k=\frac{\omega}{c}, \]

\(n\) is an integer, and \(U_1, U'_2, U''_2\) are constants.

We shall assume that the walls of the waveguide are perfectly conducting. Therefore the tangential component of the electric field at \(r=b\) must vanish. In addition, at \(r=a\) the tangential components of the electric and magnetic fields must be continuous. Thus, the following boundary conditions hold:

\[ \begin{aligned} E_{z\,r=a-0}&=E_{z\,r=a+0},& E_{\varphi\,r=a-0}&=E_{\varphi\,r=a+0},\\ H_{z\,r=a-0}&=H_{z\,r=a+0},& H_{\varphi\,r=a-0}&=H_{\varphi\,r=a+0}. \end{aligned} \tag{1.4} \]

It is easy to see that separate \(E\)- and \(H\)-waves corresponding to \(n\ne0\) cannot satisfy all these boundary conditions. The only exception is provided by axially symmetric \(E\)- and \(H\)-waves, whose fields separately can satisfy the boundary conditions (1.4). For \(n\ne0\), the boundary conditions can be satisfied only by a linear combination of \(E\)- and \(H\)-waves. In other words, the separate existence of \(E\)- and \(H\)-waves is possible only in the case of an axially symmetric field; in all other cases, only superpositions of \(E\)- and \(H\)-waves can propagate in the waveguide.

In what follows we shall be chiefly interested in the interaction of fields with charged particles moving in the paraxial region of the waveguide. From this point of view, the axially symmetric \(E\)-wave corresponding to \(n=0\), for which the component \(E_z\) on the axis of the waveguide is different from zero, is of greatest interest. (For waves with \(n\ne0\), \(E_z\), being proportional to \(J_n(k_1r)\), vanishes on the axis of the waveguide.) We shall therefore restrict ourselves to considering the axially symmetric \(E\)-wave (the \(E_0\)-wave).

The nonzero components of the electric and magnetic fields for the \(E_0\)-wave are expressed by the following formulas: in the region \(r\leq a\),

\[ \begin{aligned} E_z&=E_0J_0(k_1r)e^{i(\omega t-k_3z)},\\ E_r&=i\,\frac{k_3}{k_1}\,E_0J_1(k_1r)e^{i(\omega t-k_3z)},\\ H_\varphi&=i\,\frac{k}{k_1}\,E_0J_1(k_1r)e^{i(\omega t-k_3z)}; \end{aligned} \tag{1.5} \]

in the region \(a \leq r \leq b\)

\[ \left. \begin{aligned} E_z&=\bigl[A_2J_0(k_2r)+B_2N_0(k_2r)\bigr]e^{i(\omega t-k_3z)},\\ E_r&=i\,\frac{k_3}{k_2}\bigl[A_2J_1(k_2r)+B_2N_1(k_2r)\bigr]e^{i(\omega t-k_3z)},\\ H_\varphi&=i\varepsilon\,\frac{k}{k_2}\bigl[A_2J_1(k_2r)+B_2N_1(k_2r)\bigr]e^{i(\omega t-k_3z)}. \end{aligned} \right\} \tag{1,6} \]

The boundary conditions (1,4) lead to the relations:

\[ \left. \begin{aligned} E_0J_0(k_1a)&=A_2J_0(k_2a)+B_2N_0(k_2a),\\ \frac{1}{k_1}E_0J_1(k_1a)&=\frac{\varepsilon}{k_2}\bigl[A_2J_1(k_2a)+B_2N_1(k_2a)\bigr],\\ A_2J_0(k_2b)+B_2N_0(k_2b)&=0. \end{aligned} \right\} \tag{1,7} \]

Eliminating from these the constants \(E_0, A_2, B_2\), we obtain an equation relating \(k_3\) to the frequency \(\omega\) and the waveguide parameters \(^{1,2}\),

\[ \frac{J_0(k_1a)}{J_1(k_1a)}\,\frac{k_1\varepsilon}{k_2} = \frac{J_0(k_2a)N_0(k_2b)-N_0(k_2a)J_0(k_2b)} {J_1(k_2a)N_0(k_2b)-N_1(k_2a)J_0(k_2b)}. \tag{1,8} \]

This relation makes it possible to determine the phase velocity of the wave

\[ v_\varphi=\frac{\omega}{k_3} \]

as a function of the frequency. In what follows we shall call the dependence of the phase velocity on the frequency the dispersion relation, and equation (1,8) the dispersion equation*).

The phase velocity of the waves under consideration has one and the same value in the region not occupied by the dielectric and in the dielectric layer. Therefore, for sufficiently large values of the dielectric constant \(\varepsilon\) and of the thickness of the dielectric layer \(b-a\), a substantial reduction of the phase velocity to values much smaller than \(c\) can be achieved.

If the dielectric almost completely fills the waveguide, then the phase velocity \(v_\varphi\) differs little from the value which it has in a waveguide completely filled with dielectric.

*) We have derived the dispersion equation for a waveguide whose walls are covered with a dielectric layer. In an analogous way one may consider another case of partial filling of a waveguide with dielectric, when a dielectric rod is introduced into the waveguide. In this case the dispersion equation has the form \(^{3}\)

\[ \frac{J_0(k_1a)}{J_1(k_1a)}\,\frac{k_1}{k_2\varepsilon} = \frac{J_0(k_2a)N_0(k_2b)-J_0(k_2b)N_0(k_2a)} {J_1(k_2a)N_0(k_2b)-J_0(k_2b)N_1(k_2a)}, \]

where

\[ k_1=\sqrt{\varepsilon\mu k^2-k_3^2},\qquad k_3=\sqrt{k^2-k_3^2}. \]

In this latter case the phase velocity of wave propagation is equal to

\[ v_\varphi=\frac{c}{\sqrt{\varepsilon-\left(\frac{\omega_0}{\omega}\right)^2}},\qquad a=0, \]

where \(\omega_0=\frac{2.405}{b}c\) is the critical frequency of a waveguide of radius \(b\) containing no dielectric. If \(\omega>\omega_0(\varepsilon-1)^{-1/2}\), then \(v_\varphi<c\).

In the case where the dielectric almost completely fills the waveguide \((a\ll b)\), it is easy to investigate the dispersion equation (1.8). Assuming that \(k_2\) is close to the unperturbed value \(\frac{2.405}{b}\), and using the known asymptotic representations of the Bessel functions, valid for small values of the argument \((x\ll 1)\),

\[ J_0(x)\simeq 1,\qquad J_1(x)\simeq \frac{1}{2}x, \]

\[ N_0(x)\simeq -\frac{2}{\pi}\ln\frac{2}{\gamma x},\qquad N_1(x)\simeq -\frac{2}{\pi x}, \]

(\(\gamma\) is Euler’s constant), we obtain from (1.8) the following expression for the phase velocity, valid for \(a\ll b\),

\[ v_\varphi=c\left\{\varepsilon-\left(\frac{\omega_0}{\omega}\right)^2\left[1+3.7\left(1-\frac{1}{\varepsilon}\right)\frac{a^2}{b^2}\right]\right\}^{-1/2}. \tag{1.9} \]

This quantity, as was to be expected, exceeds the phase velocity for complete filling of the waveguide with dielectric; however, in the frequency range

\[ \omega>\frac{\omega_0}{\sqrt{\varepsilon-1}} \left[1+1.85\left(1-\frac{1}{\varepsilon}\right)\frac{a^2}{b^2}\right] \qquad v_\varphi<c. \]

Putting \(v_\varphi=\infty\), we determine the cutoff frequency of the waveguide \(\omega_g\). If \(a\ll b\), then the cutoff frequency is

\[ \omega_g=\frac{\omega_0}{\sqrt{\varepsilon}} \left[1+1.85\left(1-\frac{1}{\varepsilon}\right)\frac{a^2}{b^2}\right]. \tag{1.10} \]

Waves with frequencies lower than \(\omega_g\) cannot propagate in the waveguide. The cutoff frequency of a waveguide partially filled with dielectric always lies between the cutoff frequency of a waveguide completely filled with dielectric and the cutoff frequency of a waveguide containing no dielectric at all:

\[ \frac{\omega_0}{\sqrt{\varepsilon}}<\omega_g<\omega_0. \]

In the case where the radius of the channel in the dielectric is not small compared with the radius of the waveguide, the investigation of equation (1.8) is a complicated problem\(^2\). A rather convenient

it turns out to be a graphical solution of equation (1.8) (for this purpose one must draw the curve determining the dependence of \(k_2\) on \(k_1\), and find the points of its intersection with the hyperbola

\[ k_2^2-k_1^2=(\varepsilon-1)\frac{4\pi^2}{\lambda_0^2}, \]

where \(\lambda_0\) is the wavelength in vacuum).

In Fig. 2 is presented\(^2\) the dependence of the phase velocity \(v_\varphi\) and of the wavelength in the waveguide \(\lambda=\dfrac{2\pi}{k_3}\) on the wavelength in vacuum for \(\dfrac{a}{b}=\dfrac{2}{3}\) and \(\varepsilon=57\).

Thus, in waveguides partially filled with a dielectric, the phase velocity of wave propagation may be

Fig. 2.

reduced to values smaller than the velocity of light in vacuum. Therefore the fields of such waveguides can interact effectively with charged particles.

Let us dwell in somewhat greater detail on the question of the topography of the fields. Two cases should be distinguished, depending on whether the quantity

\[ k_1=\sqrt{k^2-k_3^2} \]

is real or imaginary.

Since \(k_3=\dfrac{\omega}{v_\varphi}\), then

\[ k_1=\frac{\omega}{c}\sqrt{1-\frac{c^2}{v_\varphi^2}} \]

will be real,

if \(v_\varphi>c\). In this case, in the region free of dielectric, \((r\leq a)\), the component of the electric field along the waveguide axis, proportional to \(J_0(k_1 r)\), attains a maximum on the waveguide axis.

If \(v_\varphi<c\), then \(k_1\) will be imaginary and \(E_z\) for \(r\leq a\) will be proportional to \(I_0(|k_1|r)\), where \(I_0\) is the modified Bessel function. In this case \(E_z\) is a monotonically increasing function of \(r\), taking its minimum value on the waveguide axis.

Fig. 3.

waveguide. Note in this connection that the interaction of particles with the field will be most effective not when they move along the waveguide axis, but when they move near the boundary with the dielectric.

Finally, let us consider the case when \(v_\varphi=c\). Then \(k_1=0\), and \(E_z\) for \(r\leq a\) does not depend on \(r\) at all. In Fig. 3 the dependence\(^1\) of \(E_z\) on \(r\) is shown for \(v_\varphi=5c\), \(v_\varphi=c\), and \(v_\varphi=0.2c\) (in the first case \(\varepsilon=31.82\), in the second \(\varepsilon=32.8\), and in the third \(\varepsilon=57.25\); the ratio of the channel radius to the wavelength in vacuum is \(5\cdot10^{-2}\), \(b/a=2\)).

Let us note that the constants \(A_2\) and \(B_2\), entering formulas (1,6), can, for \(v_\varphi<c\), be expressed in the following way through \(E_0\):

\[ \left. \begin{aligned} \frac{A_2}{E_0} &= \frac{\pi}{2}\, k_2 a \left\{ \frac{k_2}{|k_1|\varepsilon}\, I_1(|k_1|a)\,N_0(k_2 a) - I_0(|k_1|a)\,N_1(k_2 a) \right\}, \\[4pt] \frac{B_2}{E_0} &= \frac{\pi}{2}\, k_2 a \left\{ I_0(|k_1|a)\,J_1(k_2 a) - \frac{k_2}{|k_1|\varepsilon}\,J_0(k_2 a)\,I_1(|k_1|a) \right\}. \end{aligned} \right\} \tag{1,11} \]

1. Let us now determine the energy flux in a waveguide whose walls are covered with a layer of dielectric. Using the expressions obtained above for the field components, it is easy to calculate the density of the energy flux along the axis of the waveguide, \(p\). We obtain the following result:

\[ \begin{aligned} p&=\frac{E_0^2}{8\pi}\,c\,\frac{kk_3}{k_1^2}\,J_1^2(k_1r), \qquad r\leqslant a,\\[6pt] p&=\frac{E_0^2}{8\pi}\,c\,\frac{\varepsilon kk_3}{k_2^2} \left[\frac{A_2}{E_0}J_1(k_2r)+\frac{B_2}{E_0}N_1(k_2r)\right]^2, \qquad a\leqslant r\leqslant b, \end{aligned} \tag{1,12} \]

where \(\dfrac{A_2}{E_0}\) and \(\dfrac{B_2}{E_0}\) are determined by formulas (1,11). In Fig. 4 is shown\(^1\) the dependence of

\[ \ln \frac{8\pi p}{cE_0^2} \]

on \(r\) for \(v_\varphi=5c\), \(v_\varphi=c\), and \(v_\varphi=0.2c\) (in the first case \(\varepsilon=31.82\), in the second \(\varepsilon=32.8\), and in the third \(\varepsilon=57.25\); the ratio of the channel radius to the wavelength in vacuum is \(5\cdot10^{-2}\), \(b/a=2\)). The curves presented show that practically all the electromagnetic energy flows in the region occupied by the dielectric.

Fig. 4.

The total energy flux through the cross section of the waveguide is expressed by a rather complicated formula (see\(^2\)). We shall therefore give the expression for the total flux only in the limiting case \(a\ll b\). In this case

\[ P=2\pi\int_0^b pr\,dr = \]

\[ =5.83\cdot10^{-3}E_0^2\,ckk_3\varepsilon b^4 \left[1-7.4\left(1-\frac{1}{\varepsilon}\right)\frac{a^3}{b^2}\right] \frac{\text{erg}}{\text{sec}} \tag{1,13} \]

(the field \(E_0\) is expressed here in CGSE units).

This relation shows that, as the dielectric constant increases, the energy flux increases greatly. A decrease in the phase velocity also leads to an increase of \(P\).

Summarizing\(^ {1,2}\), we may say that partial filling of a waveguide with dielectric makes it possible, by varying the thickness

of the dielectric layer and the value of the dielectric constant, to reduce the phase velocity of wave propagation to arbitrarily small values. An advantage of this method of obtaining slow waves is also the lowering of the limiting frequency of the waveguide (in comparison with a waveguide of the same radius that contains no dielectric). The disadvantages of the method are the large power flux in the dielectric, energy losses in the dielectric, and, moreover, the possibility of charging the surface of the dielectric, especially when charged particles move near the surface of the dielectric. The latter circumstance leads to distortion of the electric field and to clogging of the waveguide.

The disadvantage associated with the possibility of charging the dielectric surface can apparently be eliminated by covering the dielectric with a metal layer whose thickness is less than the skin-depth.

1. The effectiveness of the method described for obtaining slow waves can be considerably increased by using anisotropic dielectrics with small losses\(^4\).

If the waveguide is completely filled with an anisotropic dielectric with dielectric constant \(\varepsilon_z\) along the \(z\)-axis and \(\varepsilon_r\) in the radial direction, then Maxwell’s equations have the form:

\[ \left. \begin{aligned} \frac{1}{r}\frac{\partial}{\partial r}(rH_\varphi) &= ik\varepsilon_z E_z,\\ k_3H_\varphi &= k\varepsilon_r E_r,\\ ik_3E_r+\frac{\partial E_z}{\partial r} &= ikH_\varphi. \end{aligned} \right\} \tag{1.14} \]

Eliminating \(H_\varphi\) and \(E_r\) from these, we obtain the wave equation

\[ \frac{\partial^2 E_z}{\partial r^2}+\frac{1}{r}\frac{\partial E_z}{\partial r} +\left(k^2\varepsilon_z-k_3^2\frac{\varepsilon_z}{\varepsilon_r}\right)E_z=0. \tag{1.15} \]

In the case of an \(E_0\)-wave, the field components are expressed by the following formulas:

\[ \left. \begin{aligned} E_z &= E_0J_0(k_1r),\\ E_r &= i\frac{k_3}{k_1}\frac{\varepsilon_z}{\varepsilon_r}E_0J_1(k_1r),\\ H_\varphi &= i\frac{k}{k_1}\varepsilon_z E_0J_1(k_1r) \end{aligned} \right\} \tag{1.16} \]

(the factor \(e^{i(\omega t-k_3z)}\) is not written here).

The total power flux along the axis of a waveguide filled with an anisotropic dielectric is equal to

\[ P_{\mathrm{aniz}}=\frac{cE_0^2}{8}\frac{|kk_3|}{k_1^2}\frac{\varepsilon_z^2}{\varepsilon_r}J_1^2(k_1b)b^2. \tag{1.17} \]

Let us compare this expression with the expression for the power flux in a waveguide filled with an isotropic dielectric, assuming that \(\omega\), \(v_\varphi\), and \(E_0\) are the same in both cases. In the case of an isotropic dielectric, the power flux is

\[ P_{\mathrm{isotr}}=\frac{cE_0^2}{8}\,\frac{kk_3}{k_1^2}\,\varepsilon J_1^2(k_1 b)\,b^2 . \]

From the condition of equality of the phase velocities it follows that

\[ \varepsilon=\varepsilon_r+\frac{\omega_0^2}{\omega^2}\left(1-\frac{\varepsilon_r}{\varepsilon_z}\right). \]

If \(\omega \approx \dfrac{\omega_0}{\sqrt{\varepsilon_z}}\), then \(\varepsilon \approx \varepsilon_z\). The ratio of the energy flux in the anisotropic dielectric to the flux in the isotropic dielectric is equal to

\[ \frac{P_{\mathrm{anis}}}{P_{\mathrm{isotr}}}\approx \frac{\varepsilon_z}{\varepsilon_r}. \]

Thus, if \(\varepsilon_r \gg \varepsilon_z\), then this ratio is considerably less than unity. It can be shown that the losses in the walls of a waveguide when it is filled with an anisotropic dielectric decrease in the same ratio.

II.—CHAIN OF ENDOVIBRATORS

To obtain slow waves, it is not necessary to fill the waveguide with a dielectric. Slow waves may also be obtained in waveguides that contain no dielectrics, if metallic partitions with apertures are placed in the path of wave propagation. In this section we shall consider the production of slow waves in linear periodic structures consisting of collections of identical elementary cells—endovibrators—connected with one another through apertures and arranged along a common axis. Such a structure may also be regarded as a waveguide loaded with disks having apertures for coupling*).

Let the system of endovibrators under consideration be formed by rotating about the axis \(z\) a periodic curve with period \(l\). The cavity of the resulting body of revolution must be simply connected, but otherwise the form of the generating curve in the interval \(0\leq z\leq l\) is arbitrary. We shall consider an axially symmetric wave (\(E_0\)-wave) with components \(E_z\), \(E_r\), \(H_\varphi\) different from zero, these quantities being independent of the angle \(\varphi\). Owing to the periodic—

*) The method of obtaining slow electromagnetic waves in periodic structures was proposed by V. V. Vladimirsky in 1946.\(^5\) The content of work\(^5\) is set out below.

ness of the structure, all components of the field must satisfy the following condition:

\[ F(r,z+l)=e^{i\psi}F(r,z), \tag{2.1} \]

where \(\psi\) is the phase difference between the oscillations in two neighboring cavity cells. This quantity depends on the frequency \(\omega\) and, for waves propagating in the direction of positive \(z\), lies in the interval \((0,\pi)\).

The phase velocity of the wave \(v_\varphi\) is related to \(\psi\) by

\[ v_\varphi=\frac{\omega l}{\psi}. \tag{2.2} \]

To make sure that this quantity can be made smaller than \(c\), it is sufficient to show that sufficiently low frequencies can pass through the chain of cavity cells. In this connection we shall consider the question of the frequency pass band of a chain of cavity cells. To estimate the width of this band there is no need to solve Maxwell’s equations completely with the corresponding boundary conditions. As was shown by V. V. Vladimirskii\(^5\), it is sufficient to make use of certain extremal properties of the eigenfrequencies.\(^6\) For this purpose let us introduce the function \(U=rH_\varphi\). It satisfies the equation

\[ r\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial U}{\partial r}\right)+\frac{\partial^2 U}{\partial z^2}+k^2U=0, \]

where \(k=\frac{\omega}{c}\). Assuming the walls of the cavity cells to be ideally conducting, we have the boundary condition \(E_t=0\), which, as is easy to see, reduces to the vanishing of the normal derivative of the function \(U\):

\[ \frac{\partial U}{\partial n}=0. \]

Let us divide the system of coupled cavity cells under consideration into separate cells by infinitely thin planes perpendicular to the \(z\)-axis and situated at distances \(l\) from one another. Then at the boundary dividing the cavity cells the condition \(\frac{\partial U}{\partial n}=0\) will be satisfied. Denote by \(\omega_s^e\) \((s=1,2,\ldots)\) the sequence of eigenfrequencies of an individual cavity cell. For the entire chain of cavity cells each of these values will be \(N\)-fold degenerate (\(N\) is the number of cells in the chain). When coupling is introduced through apertures, the degeneracy is removed, and from each frequency \(\omega_s^e\) there arises an entire frequency band \(\omega_s(\psi)\), where \(\psi\) is the phase difference introduced above between the oscillations in neighboring cavity cells (for waves propagating in the positive direction of the \(z\)-axis, \(\psi\) takes the values \(\pi\frac{p}{N}\), where \(p\) is an integer ...).

number varying from zero to \(N\)). As is known, when coupling is introduced the eigenvalues increase (see \({}^{6}\), theorem 4). Hence it follows that, for the propagation of an \(E_0\)-wave along a chain of endovibrators, it is necessary that the eigenfrequency of the endovibrators obtained by dividing the chain by ideally conducting planes be lower than the frequency of the propagating waves.

Let us now estimate the upper boundary of the frequency band obtained when coupling is introduced. For this purpose let us imagine that the chain is divided into separate cells by ideally magnetic partitions. This means that at the boundary between them the condition \(H_t=0\), i.e. \(U=0\), is satisfied. On the remaining surface of the endovibrators the condition \(\dfrac{\partial U}{\partial n}=0\) is preserved. The eigenfrequencies of these cells \(\omega_s^m\) will be higher than the eigenfrequencies \(\omega_s^e\) (see \({}^{6}\), theorem 5).

Each of the values \(\omega_s^m\) will be expressed \(N\)-fold. If coupling is introduced, the expression is removed, and moreover the eigenfrequency decreases (see \({}^{6}\), theorem 2). Hence it follows that, for the propagation of an \(E_0\)-wave along the chain, it is necessary that the frequency of the individual endovibrators obtained by dividing the chain by ideally magnetic partitions be higher than the frequency of the propagating waves.

Thus, the eigenfrequencies of the chain \(\omega_s(\psi)\) (\(s\) is the number of the frequency “band”) satisfy the inequality

\[ \omega_s^e \leq \omega_s(\psi) \leq \omega_s^m . \tag{2,3} \]

It can be shown that the bounds of this inequality are attained in the case of endovibrators possessing a plane of symmetry perpendicular to the \(z\)-axis\({}^{5}\).

Let us determine within what limits the phase velocity of the wave varies inside the \(s\)-th frequency band. For this note that the boundaries of the frequency passband correspond to values of \(\psi\) equal to \(\psi=0\) and \(\psi=\pi\). Therefore at the lower boundary the phase velocity is equal to

\[ v_\varphi=\frac{\omega_s^e l}{0}=\infty . \]

At the upper boundary of the \(s\)-th passband the phase velocity is equal to

\[ v_{\varphi \min}=\frac{\omega_s^m l}{\pi}. \tag{2,4} \]

This is the minimum value of the phase velocity of propagation of the wave in the \(s\)-band. For sufficiently small \(l\), \(v_{\varphi \min}\) can be much less than \(c\).

We have found the limits within which the phase velocity of waves belonging to a definite frequency band and capable of propagating in a chain of coupled endovibrators varies.

Often, however, this is insufficient. It is very important to know the exact dispersion dependence, i.e., the dependence of the phase velocity on frequency.

This problem can be solved if the dependence of the frequency on \(\psi\) is known within a certain frequency band.

Determining the dispersion dependence is an extremely complicated mathematical problem. We shall present here approximate methods for solving this problem, beginning with the simplest case of weak coupling between endovibrators*).

Consider two endovibrators with perfectly conducting walls. The endovibrators communicate with one another through a small circular opening of radius \(a\) in an infinitely thin flat wall; the dimensions of the opening are assumed small in comparison with the dimensions of the wall and with the wavelength; the linear dimensions of the endovibrators are of the same order as the wavelength. If the opening is filled by a perfectly conducting partition, then we obtain two uncoupled endovibrators, whose natural oscillations are assumed known. Each of the endovibrators has a series of natural frequencies, but under weak coupling appreciable resonance phenomena occur only for nearby frequencies. Therefore we shall consider one natural oscillation in each endovibrator (for simplicity we assume that degeneracy is absent).

Coupling through the openings causes a strong perturbation of the field near the openings. However, this perturbation is confined to a small region near the opening and does not extend over the whole volume of the endovibrators. Therefore it causes only a small change in frequency, which is an integral effect. We can, consequently, determine the change in frequency by using perturbation theory and assuming that the boundary conditions have undergone a small perturbation.

Let \(\mathbf{E}_1, \mathbf{H}_1\) and \(\mathbf{E}_2, \mathbf{H}_2\) be the fields in the uncoupled endovibrators. They satisfy Maxwell’s equations

\[ \begin{aligned} \operatorname{rot}\mathbf{E}_\alpha &= ik_\alpha \mathbf{H}_\alpha, \qquad &\operatorname{div}\mathbf{H}_\alpha &= 0,\\ \operatorname{rot}\mathbf{H}_\alpha &= -ik_\alpha \mathbf{E}_\alpha, \qquad &\operatorname{div}\mathbf{E}_\alpha &= 0, \end{aligned} \tag{2,5} \]

where \(\alpha=1,2\), \(k_\alpha=\dfrac{\omega_\alpha}{c}\), and \(\omega_\alpha\) are the natural frequencies of the two endovibrators.

Integrating by parts the expression \(\operatorname{rot}\mathbf{E}\operatorname{rot}\mathbf{E}_1^{*}\) over the volume of the first endovibrator \(V_1\) (\(\mathbf{E}\) is the field arising when the—

*) This case was studied by V. V. Vladimirskii\({}^{7}\) (see also \({}^{8,9}\)). Below we present the content of work \({}^{7}\).

...of the coupling between the endovibrators), we find:

\[ \int_{V_1}\left(\mathbf{E}\,\operatorname{rot}\operatorname{rot}\mathbf{E}_1^{*} -\mathbf{E}_1^{*}\operatorname{rot}\operatorname{rot}\mathbf{E}\right)\,dv+ \]

\[ +\int_{S_1}\left([\mathbf{E}\operatorname{rot}\mathbf{E}_1^{*}]_n- [\mathbf{E}_1^{*}\operatorname{rot}\mathbf{E}]_n\right)\,ds=0 \tag{2,6} \]

(\(S_1\) is the surface bounding the first endovibrator). Using (2,5) and noting that the fields \(\mathbf{E}_1\) and \(\mathbf{E}\) satisfy the boundary conditions

\[ E_{1t}=0 \quad \text{on the surface } S_1, \]

\[ E_t=0 \quad \text{on the surface } S_1-S' \]

(\(S'\) is the plane surface closing the aperture), we obtain:

\[ (k^2-k_1^2)\int_{V_1}\mathbf{E}\mathbf{E}_1^{*}\,dV +ik_1\int_{S'}[\mathbf{E}\mathbf{H}_1^{*}]_{n_1}\,ds=0. \tag{2,7} \]

In analogous fashion we obtain the second equation:

\[ (k^2-k_2^2)\int_{V_2}\mathbf{E}\mathbf{E}_2^{*}\,dV +ik_2\int_{S'}[\mathbf{E}\mathbf{H}_2^{*}]_{n_2}\,ds=0 \tag{2,8} \]

(\(n_1\) and \(n_2\) are the outward normals to \(S_1\) and \(S_2\)). Here \(k=\dfrac{\omega}{c}\), and \(\omega\) is the frequency of the coupled endovibrators. These equations as yet contain no approximations. For an approximate determination of the frequency we substitute here the approximate value of the field \(\mathbf{E}\). As the initial approximation for \(\mathbf{E}\) and \(\mathbf{H}\) we take \(q_1\mathbf{E}_1\) and \(q_1\dfrac{k}{k_1}\mathbf{H}_1\) in the volume of the first endovibrator, and \(q_2\mathbf{E}_2\) and \(q_2\dfrac{k}{k_2}\mathbf{H}_2\) in the volume of the second endovibrator, where \(q_1\) and \(q_2\) are certain constants. Since the perturbation of the fields is confined to a small region adjacent to the aperture, this initial approximation may be used for computing the volume integrals in (2,7) and (2,8). Assuming that the fields \(\mathbf{E}_1\) and \(\mathbf{E}_2\) are normalized according to the condition

\[ \int_{V_1}|\mathbf{E}_1|^2\,dv = \int_{V_2}|\mathbf{E}_2|^2\,dV =1, \tag{2,9} \]

we obtain:

\[ \int \mathbf{E}\mathbf{E}_1^{*}\,dV=q_1,\qquad \int \mathbf{E}\mathbf{E}_2^{*}\,dV=q_2. \tag{2,10} \]

We now turn to the calculation of the surface integrals in (2,7) and (2,8). The initial approximation evidently makes the surface integrals vanish, since only the tangential components of the fields enter them, while \(E_{1t}\) and \(E_{2t}\) vanish on the surface \(S'\). Noting that the tangential component...

... that \(H_\alpha\) is regular both on the surface \(S'\) and in a neighborhood of the order of the wavelength \(\lambda\), we may expand \(H_1, H_2\) under the integral sign in a series in powers of the coordinates (as for the component \(E_t\), it must be regular on the surface \(S'\), except at the edges of the aperture, where it has an integrable singularity). Thus we obtain an expansion of the surface integrals in a series in powers of \(\dfrac{a}{\lambda}\), where \(a\) is the radius of the aperture*).

Introducing a Cartesian coordinate system with origin at the center of the aperture and with the \(z\)-axis directed along the normal \(n_1\), and restricting ourselves in the expansion of \(H_1\) to the first two terms, we obtain:

\[ \left. \begin{aligned} ik_1 \int_{S'} [\mathbf{E}\mathbf{H}_1^{*}]_{n_1}\,ds &= ik_1 \int_{S'} [\mathbf{E}\mathbf{H}_1^{*}(0)]_{n_1}\,ds \\ &\quad - ik_1 \int_{S'} \left( E_x x\,\frac{\partial H_{1x}^{*}}{\partial x} + E_x y\,\frac{\partial H_{1y}^{*}}{\partial y} \right. \\ &\qquad\qquad \left. - E_y x\,\frac{\partial H_{1x}^{*}}{\partial x} - E_y y\,\frac{\partial H_{1y}^{*}}{\partial y} \right)\,dx\,dy, \end{aligned} \right\} \tag{2,11} \]

where all derivatives of \(\mathbf{H}_1^{*}\) are taken at the origin.

Since the dimensions of the aperture are small in comparison with the wavelength, it may be assumed that in the neighborhood of the aperture the field \(\mathbf{E}\), in the first approximation, has an electrostatic character and is determined from the following electrostatic problem. Far from the aperture \(E_x \to 0\), \(E_y \to 0\), while the component \(E_z\) tends to \(q_1 E_{1z}(0)\) in the upper half-space and to \(q_2 E_{2z}(0)\) in the lower half-space. In the plane \(z=0\) the condition \(E_t=0\) is satisfied everywhere except the aperture. It is easy to see that the first integral entering into (2,11) vanishes in the electrostatic approximation. It cannot, however, be discarded, since it contains the small parameter \(\dfrac{a}{\lambda}\) to a lower power than the second integral. Into the first integral, therefore, one must substitute a more exact value of the field \(\mathbf{E}\). This can be done by first transforming the first integral. Noting that the zeroth approximation \(q_1\mathbf{E}_1\),

*) If the distance \(l\), at which the bends of the separating wall nearest to the aperture or other singularities are located, is appreciably less than \(\lambda\), then we obtain an expansion not in powers of \(\dfrac{a}{\lambda}\), but in powers of \(\dfrac{a}{l}\); this quantity, as well as \(\dfrac{a}{\lambda}\), we shall regard as small in comparison with unity.

\(q_1\dfrac{k}{k_1}\mathbf H_1\) exactly satisfies the first Maxwell equation and the boundary condition \(E_t=0\) on the surface \(S'\); the first integral can be transformed from a surface integral into a volume integral:

\[ ik_1\int_{S'}[\mathbf E\mathbf H_1^{*}(0)]_{n_1}\,ds = ik_1\int_{S'}[(\mathbf E-q_1\mathbf E_1),\mathbf H_1^{*}(0)]_{n_1}\,ds = -kk_1\int_V(\mathbf H-\mathbf H_0)\mathbf H_1^{*}(0)\,dv, \tag{2,12} \]

where \(\mathbf H_0=q_1\dfrac{k}{k_1}\mathbf H_1(0)\).

As the distance from the aperture increases, the integrand in (2,12) rapidly decreases. Therefore the principal role is played by the region near the aperture. In this region the magnetic field \(\mathbf H\) evidently has a magnetostatic character, since \(a\ll\lambda\). It can be approximately determined from the following magnetostatic problem. It is required to find the magnetic field \(\mathbf H=\operatorname{grad}\chi\), where \(\chi\) is the magnetic potential satisfying the equation \(\Delta\chi=0\) and the boundary condition \(\dfrac{\partial\chi}{\partial n}=0\) on the surface of the conductor. Far from the aperture the condition \(\mathbf H\to q_1\dfrac{k}{k_1}\mathbf H_1(0)\) must be satisfied in the upper half-space, and the condition \(\mathbf H\to q_2\dfrac{k}{k_2}\mathbf H_2(0)\) in the lower half-space.

Solving these static problems and calculating the integrals (2,11), (2,12), we obtain from (2,7) and (2,8) the following system of equations for determining \(q_1\), \(q_2\), and \(k^{7,9}\):

\[ \left. \begin{aligned} &(k^2-k_1^2)q_1-\frac{2}{3}k_1^2a^3\bigl(|\mathbf E_1|^2q_1-\mathbf E_1\mathbf E_2^{*}q_2\bigr)+\\ &\qquad\qquad+\frac{4}{9}k_1^2a^3\bigl(|\mathbf H_1|^2q_1-\mathbf H_1\mathbf H_2^{*}q_2\bigr)=0,\\[4pt] &(k^2-k_2^2)q_2-\frac{2}{3}k_2^2a^3\bigl(|\mathbf E_2|^2q_2-\mathbf E_1^{*}\mathbf E_2q_1\bigr)+\\ &\qquad\qquad+\frac{4}{9}k_2^2a^3\bigl(|\mathbf H_2|^2q_2-\mathbf H_2\mathbf H_1^{*}q_1\bigr)=0. \end{aligned} \right\} \tag{2,13} \]

The method presented can easily be generalized to the case when there are not two, but any number of endovibrators.\(^{7}\) Following V. V. Vladimirskii,\(^{7}\) let us consider a chain of identical cylindrical endovibrators of radius \(b\) and length \(l\), which touch one another with the bases of the cylinders. The coupling between neighboring endovibrators is effected through circular apertures of radius \(a\), whose centers lie on the axis of the system. We shall confine ourselves to considering the lowest frequency of the electric waves of the cylinder.

In this case the nonzero components of the field have the form

\[ \left. \begin{aligned} E_z &= \frac{J_0(k_0 r)}{\sqrt{\pi b^2 l J_1^2(\rho_1)}},\\ H_\varphi &= -i\,\frac{J_1(k_0 r)}{\sqrt{\pi b^2 l J_1^2(\rho_1)}} , \end{aligned} \right\} \tag{2,14} \]

where \(J_0\) and \(J_1\) are Bessel functions, \(\rho_1\) is the first root of the function \(J_0\), and

\[ k_0=\frac{\rho_1}{b}. \]

Equations (2,13) lead to the infinite system of equations

\[ (k^2-k_0^2)q_n-\alpha k_0^2(2q_n-q_{n-1}-q_{n+1})=0, \tag{2,15} \]

where

\[ \alpha=\frac{2}{3\pi J_1^2(\rho_1)}\,\frac{a^3}{b^2 l}. \]

This quantity characterizes the coupling between endovibrators and is proportional to the third power of the radius of the aperture. The system of equations (2,15) admits a solution in the form of a traveling wave

\[ q_n=q_0 e^{in\psi}, \tag{2,16} \]

where \(\psi\) is the phase difference between the oscillations of neighboring endovibrators. Substituting (2,16) into (2,15), we obtain:

\[ \frac{\omega-\omega_0}{\omega_0}=\alpha(1-\cos\psi). \tag{2,17} \]

This is the dispersion equation of the problem under consideration.

According to (2,2), the phase velocity in the propagation of waves in the chain is equal to

\[ v_\varphi=\frac{\omega_0 l}{\psi}\,[1+\alpha(1-\cos\psi)]. \tag{2,18} \]

The phase velocity varies from \(\infty\) at \(\psi=0\) to \(\dfrac{\omega_0 l}{\pi}(1+2\alpha)\) at \(\psi=\pi\).

The group velocity of the wave is equal to

\[ v_g=l\frac{d\omega}{d\psi}=l\omega_0\alpha\sin\psi =\frac{2}{3\pi}\,\frac{\rho_1}{J_1^2(\rho_1)}\,\frac{a^3}{b^3}\,c\sin\psi. \tag{2,19} \]

It is proportional to the cube of the ratio \(\dfrac{a}{b}\). At the boundaries of the passband \(v_g\) vanishes.

Expression (2,18) for the phase velocity shows that, for small distances between the walls separating the endovibrators, the phase velocity can take values smaller than the velocity of light in vacuum. However, as \(l\) decreases, or, in other words, as the number of separating walls increases, the losses increase.

energy in the walls, and consequently the attenuation of waves propagating in the chain increases. The attenuation of the wave can be reduced by increasing the radius of the aperture, since in this case the group velocity increases, and together with it the attenuation length (the attenuation length is the distance over which the amplitude of the traveling wave decreases by a factor of \(e\); it is equal to \(2\dfrac{Qv_g}{\omega}\), where \(Q\) is the quality factor of the system\({}^{10}\)). Therefore it is of considerable interest to find the dispersion relation for a chain with not a small coupling between the individual endovibrators, otherwise speaking, the case when \(a \gtrsim l\). In this case one usually speaks not of a chain of endovibrators, but of a waveguide loaded with disks with apertures\({}^{1,2}\). We now turn to the study of such a waveguide\(*\).

III. WAVEGUIDES LOADED WITH DISKS

1. Let us consider first of all the case, according to\({}^{11,12}\), when the waveguide is loaded with disks situated sufficiently close to one another, so that the condition is satisfied

\[ l \ll \lambda_b, \tag{3,1} \]

where \(\lambda_b=\dfrac{1}{k_z}\) is the wavelength in the waveguide\({}^{11}\). The disks, as in the preceding section, will be assumed infinitely thin. Since \(l \ll \lambda_b\), it may be assumed that the fields in the region \(a \le r \le b\) (\(b\) is the radius of the waveguide) between adjacent disks practically do not depend on the coordinate \(z\) (Fig. 5). In this case, however, the field changes slowly in passing from one cell to another. In other words, we assume that the dependence of the fields on the coordinate \(z\) in the region \(a \le r \le b\) has the form \(e^{-ik_z \bar z}\), where \(\bar z\) denotes the \(z\)-coordinates of the centers of the cells. If the \(z\)-coordinates of the disks are equal to \((2n+1)\dfrac{l}{2}\) (\(n\) an integer), then \(\bar z = nl\).

Fig. 5.

In the region \(r \le a\) the fields have the form

\[ \begin{aligned} E_z &= E_0 J_0(k_1 r)e^{i(\omega t-k_z z)},\\ E_r &= iE_0 \frac{k_z}{k_1} J_1(k_1 r)e^{i(\omega t-k_z z)},\\ H_\varphi &= iE_0 \frac{k}{k_1} J_1(k_1 r)e^{i(\omega t-k_z z)}. \end{aligned} \tag{3,2} \]

\(*\) Methods for solving boundary-value problems in complex waveguide systems and endovibrators were developed by G. V. Kisun’ko, P. E. Krasnushkin, Ya. N. Fel’d, and others.

We shall assume that in the region \(a \leq r \leq b\) the component \(E_z\) has the form

\[ E_z=qE_0Z_0(kr)e^{i(\omega t-k_3z)}, \tag{3.3} \]

where the function

\[ Z_0(kr)=N_0(kb)J_0(kr)-J_0(kb)N_0(kr) \]

is chosen so as to satisfy the boundary condition \(E_z=0\) on the surface of the waveguide; \(q\) is a certain constant. Let us note the circumstance that the argument of the function \(Z_0\) is \(kr\), and not \(k_1r\). This is connected with the fact that \(E_z\), within one cell, does not depend on the coordinate \(z\).

We shall now find \(E_r\) and \(H_\varphi\). To determine \(H_\varphi\) we use Maxwell’s equation

\[ \frac{1}{r}\frac{\partial}{\partial r}(rH_\varphi)-\frac{1}{r}\frac{\partial H_r}{\partial \varphi}=ikE_z. \]

Since we are considering an axially symmetric field, then

\[ \frac{\partial H_r}{\partial \varphi}=0 \]

and

\[ H_\varphi=iqE_0Z_1(kr)e^{i(\omega t-k_3z)}, \tag{3.4} \]

where

\[ Z_1(kr)=N_0(kb)J_1(kr)-J_0(kb)N_1(kr). \]

It is easy to see that, in the approximation under consideration, when \(E_z\) within the cell does not depend on \(z\), \(E_r=0\). Indeed, from Maxwell’s equations it follows that

\[ \frac{1}{r}\frac{\partial H_z}{\partial \varphi}-\frac{\partial H_\varphi}{\partial z}=ikE_r. \]

Since within the cell \(H_\varphi\) does not depend on \(z\), then

\[ E_r=0. \tag{3.5} \]

Thus, in the region \(a \leq r \leq b\),

\[ \left. \begin{aligned} E_z&=qE_0Z_0(kr)e^{i(\omega t-k_3z)},\\ E_r&=0,\\ H_\varphi&=iqE_0Z_1(kr)e^{i(\omega t-k_3z)}. \end{aligned} \right\} \tag{3.6} \]

We must now “match” the solutions (3.2) and (3.6) at the boundary between the regions, i.e. at \(r=a\). Strictly speaking, this cannot be done, since the fields in the regions \(r<a\) and \(a<r<b\) depend on \(z\) in different ways. We shall “sew together” the field at the midpoints of the cells, i.e. at \(z=\bar z\). With such “matching,” the components \(E_z\) and \(H_\varphi\) will obviously be continuous only on average, since equality of the fields will hold only at the centers of the cells.

Having eliminated the constants \(q\) and \(E_0\) from the relations

\[ E_0 J_0(k_1 a)=qE_0 Z_0(ka), \qquad E_0 \frac{k}{k_1} J_1(k_1 a)=qE_0 Z_1(k_1 a), \]

we obtain the following equation, relating \(k_1=\sqrt{k^2-k_3^2}\) and \(k\):

\[ \Phi(k_1 a)=\frac{1}{k_1 a}\frac{J_1(k_1 a)}{J_0(k_1 a)}= \]

\[ =\frac{1}{ka}\, \frac{J_1(ka)N_0(kb)-N_1(ka)J_0(kb)} {J_0(ka)N_0(kb)-N_0(ka)J_0(kb)} \equiv \alpha(ka,kb). \tag{3,7} \]

Let us emphasize that this dispersion equation is valid, strictly speaking, only in the case of closely spaced disks \((k_3 l \ll 1)\). Indeed, we derived equation (3,7) from the condition of field continuity only at the midpoints of the cells. If the distance between the disks is small, then the fields at the middle and edge points of a cell in the region \(r<a\) will differ little from one another. In this case the condition of field continuity will be approximately satisfied at all points of the cell. Nevertheless, the dispersion equation (3,7) may be used for rough calculations up to \(k_3 l \sim \pi\).

We assumed that in the region \(r<a\) the field has the form of a traveling wave with one definite value of the wave vector \(k_3\). Such an assumption is approximately valid only in the case of large apertures\({}^{12}\), when \(a \gg l\). This is connected with the fact that in reality the field has the form of a superposition of a series of waves propagating with different phase velocities. Only by such a superposition of waves can the boundary conditions be satisfied exactly. It can be shown\({}^{12}\) that the waves superposed on the principal wave considered by us (with wave vector \(k_3\)) decay as \(r\) changes from \(r=a\) to \(r=0\) and penetrate over a distance of order \(l\). Therefore, if \(a \gg l\), the influence of these secondary waves is inessential and we may use, in the region \(r<a\), the representation of the field in the form (3,2).

1. Let us now proceed to the analysis of the dispersion equation (3,7). First consider the case when \(ka \ll 1\) and \(kb \ll 1\), i.e. the case of very long waves. Then

\[ \alpha(ka,kb)\simeq \frac{1}{(ka)^2 \ln \dfrac{b}{a}} \gg 1 . \]

From the dispersion equation it is easy to conclude that \(k_1\) must be real, and moreover \(k_1 a \gg 1\). This follows from the fact that the function \(\Phi(k_1 a)\), equal to \(\alpha\), for purely imaginary \(k_1\) lies within the limits \((0,\,1/2)\), while for real \(k_1\) it is determined by the following

by the asymptotic relations:

\[ \Phi(k_1a)\simeq \frac{1}{2}, \quad \text{if } k_1a \ll 1, \]

\[ \Phi(k_1a)\simeq \frac{1}{k_1a}\operatorname{ctg}\left(k_1a-\frac{\pi}{4}\right), \quad \text{if } k_1a \gg 1. \]

Since \(k_1a \gg 1\), we have \(a^2k_3^2=a^2k^2-a^2k_1^2\simeq -a^2k_1^2\), i.e. \(k_3\) is a purely imaginary quantity. This means that the waveguide, in the frequency region \(ka \ll 1\), behaves as an attenuator.

Fig. 6.

When, with increasing \(k\), the quantity \(kb\) reaches the value \(2.405\), which is the first root of the Bessel function \(J_0\), the dispersion equation takes the form

\[ \frac{J_1(k_1a)}{k_1aJ_0(k_1a)} = \frac{J_1(ka)}{kaJ_0(ka)}, \]

from which it follows that \(k_1=k\), i.e. \(k_3=0\). In other words, the lower boundary of the pass band of the waveguide loaded with

disks coincides with the lower boundary of the waveguide without disks of the same radius.

With further increase of \(ka\), the function \(\alpha(ka,kb)\) decreases and tends to zero (see Fig. 6\({}^{11}\), which shows a family of curves \(\alpha(ka,kb)=\mathrm{const}\)). As follows from the dispersion equation, for \(\alpha \to 0\) the quantity \(k_1\) becomes purely imaginary, and \(|k_1|a \gg 1\).

Since the value of \(ka\) which makes \(\alpha\) vanish is of order unity (see Fig. 6), then \(k_3 a \sim |k_1|a \gg 1\). Hence it follows that

\[ \frac{v_\varphi}{c} \simeq \frac{ka}{k_3 a} \ll 1. \]

Thus we see that, in a waveguide loaded with disks, waves may propagate whose phase

Fig. 7.

velocity is less than the velocity of light. Figure 7 presents the dependence of \(\dfrac{v_\varphi}{c}\) on \(kb\) for various values of \(\dfrac{b}{a}\).

1. As we have already indicated above, the dispersion equation (3.7) is valid in the case of a waveguide loaded with infinitely thin disks, the distance between which is sufficiently small. We shall now set forth a method that makes it possible, in principle, to determine the dependence of the phase velocity on frequency for arbitrary distances between the disks and for finite disk thickness\({}^{13}\).

Assuming the field to be axially symmetric, we shall start from the following expressions for the component \(E_z\):

\[ \left. \begin{aligned} E_z&=\sum_{m=-\infty}^{\infty} a_m J_0(\chi_m r)e^{i(\omega t-\beta_m z)}, \qquad r\leq a,\\ E_z&=\sum_{s=0}^{\infty} Z_0(\Gamma_s r)(b_s\cos\gamma_s z+d_s\sin\gamma_s z), \qquad a\leq r\leq b, \end{aligned} \right\} \tag{3,8} \]

where

\[ \chi_m^2=k^2-\beta_m^2,\qquad \Gamma_s^2=k^2-\gamma_s^2, \]

\[ Z_0(\Gamma_s r)=J_0(\Gamma_s r)N_0(\Gamma_s b)-N_0(\Gamma_s r)J_0(\Gamma_s b) \]

(with this choice of \(Z_0\) the boundary condition \(E_z=0\) at \(r=b\) is satisfied).

Using Maxwell’s equations, we obtain the following expressions for the components \(E_r\) and \(H_\varphi\):

\[ \left. \begin{aligned} E_r&= i\sum_{-\infty}^{\infty}\frac{\beta_m}{\chi_m}a_m J_1(\chi_m r)e^{i(\omega t-\beta_m z)}, \qquad r\leq a,\\ E_r&=\sum_{0}^{\infty}\frac{\gamma_s}{\Gamma_s}Z_1(\Gamma_s r)(b_s\sin\gamma_s z-d_s\cos\gamma_s z), \qquad a\leq r\leq b,\\ H_\varphi&=ik\sum_{-\infty}^{\infty}\frac{1}{\chi_m}J_1(\chi_m r)e^{i(\omega t-\beta_m z)}, \qquad r\leq a,\\ H_\varphi&=ik\sum_{0}^{\infty}\frac{1}{\Gamma_s}Z_1(\Gamma_s r)(b_s\cos\gamma_s z+d_s\sin\gamma_s z), \qquad a\leq r\leq b, \end{aligned} \right\} \tag{3,9} \]

where

\[ Z_1(\Gamma_s r)=J_1(\Gamma_s r)N_0(\Gamma_s b)-N_1(\Gamma_s r)J_0(\Gamma_s b). \]

On the surface of the disks, i.e. at \(z=\pm \dfrac{l}{2}\) (Fig. 8), \(E_r\) must

Fig. 8.

vanish. Hence it follows that the quantity \(\gamma_s\), entering

in the argument of \(\cos \gamma_s z\) in the expression for \(E_r\), may take the values \((2s-1)\dfrac{\pi}{l}\), while the quantity \(\gamma_s\), entering into the argument of \(\sin \gamma_s z\), the values \(\dfrac{2s\pi}{l}\). Thus, the field in the region \(a \leq r \leq b\) can be represented in the form

\[ \left. \begin{aligned} E_z&=\sum_{s=0}^{\infty} Z_0(\Gamma_s r)b_s\cos\frac{2\pi s}{l}\,z +\sum_{s=1}^{\infty} Z_0(\Gamma_s r)d_s\sin\frac{(2s-1)\pi z}{l},\\ H_\varphi&=ik\sum_{s=0}^{\infty}\frac{1}{\Gamma_s}Z_1(\Gamma_s r)b_s\cos\frac{2\pi s}{l}\,z +ik\sum_{s=1}^{\infty}\frac{1}{\Gamma_s}Z_1(\Gamma_s r)d_s\sin\frac{(2s-1)\pi z}{l}, \end{aligned} \right\} \tag{3.10} \]

where

\[ \Gamma_s^2=k^2-\left(\frac{2\pi s}{l}\right)^2, \]

for terms containing \(\cos\dfrac{2\pi s}{l}z\), and

\[ \Gamma_s^2=k^2-\frac{(2s-1)^2\pi^2}{l^2}, \]

for terms containing \(\sin\dfrac{(2s-1)\pi}{l}z\).

Let us now consider in more detail the field in the region \(r<a\). From the periodicity condition of the structure under study it follows that any component of the field of the propagating wave may be represented in the form

\[ f(z,r)=e^{-i\beta_0 z}f_0(z,r), \]

where \(f_0(z,r)\) is a periodic function of \(z\) with period \(L\) (see Fig. 8), and \(\beta_0\) is a real quantity lying within the limits \(\left(-\dfrac{\pi}{L},\,\dfrac{\pi}{L}\right)^{14}\). Hence it is easy to conclude that

\[ \beta_m=\beta_0+\frac{2\pi m}{L}. \tag{3.11} \]

The expressions we have obtained for the fields must be “matched” at the boundary of the two regions, i.e. at \(r=a\). Such “matching” means that the component \(E_z\) must be continuous at \(r=a\) and \(|z|\leq \dfrac{L}{2}\) (if \(\dfrac{l}{2}\leq |z|\leq \dfrac{L}{2}\), then \(E_z{}_{r=a}=0\)) and the component \(H_\varphi\) at \(r=a\) and \(|z|\leq \dfrac{l}{2}\) (on the inner surface of the disks,

i.e., for \(-\dfrac{l}{2} \leqslant |z| \leqslant \dfrac{l}{2}\), there is a jump in \(H_\varphi\) associated with the surface currents).

Thus, the following relations hold:

\[ E_z{}_{r=a}=\sum_{-\infty}^{\infty} A_m e^{-i\beta_m z}= \begin{cases} \displaystyle \sum_{s=0}^{\infty} B_s \cos \frac{2\pi s}{l} z + \sum_{s=1}^{\infty} D_s \sin \frac{(2s-1)\pi}{l} z, & \text{if } |z| \leqslant \dfrac{l}{2},\\[1.2em] 0, & \text{if } \dfrac{l}{2} \leqslant |z| \leqslant \dfrac{L}{2}. \end{cases} \tag{3,12} \]

\[ H_\varphi{}_{r=a} = \sum_{-\infty}^{+\infty} i\,\frac{k}{\gamma_m}\, A_m \frac{J_1(\gamma_m a)}{J_0(\gamma_m a)} e^{-i\beta_m z} = \]

\[ = \sum_{s=0}^{\infty} i\,\frac{k}{\Gamma_s}\, B_s \frac{Z_1(\Gamma_s a)}{Z_0(\Gamma_s a)} \cos \frac{2\pi s}{l} z + \sum_{s=1}^{\infty} i\,\frac{k}{\Gamma'_s}\, D_s \frac{Z_1(\Gamma'_s a)}{Z_0(\Gamma'_s a)} \sin \frac{(2s-1)\pi}{l} z, \tag{3,13} \]

where

\[ A_m=a_m J_0(\gamma_m a),\qquad B_s=b_s Z_0(\Gamma_s a),\qquad D_s=d_s Z_0(\Gamma'_s a). \]

Multiplying (3,12) successively by \(\cos \dfrac{2\pi s}{l}z\) and \(\sin \dfrac{(2s-1)\pi z}{l}\), and integrating with respect to \(z\) over the limits from \(z=-\dfrac{l}{2}\) to \(z=\dfrac{l}{2}\), we obtain:

\[ \begin{aligned} B_0 l &= \sum_{-\infty}^{+\infty} A_m C_{0m},\\[0.4em] \frac{1}{2} B_s l &= \sum_{-\infty}^{+\infty} A_m C_{sm},\\[0.4em] \frac{1}{2} D_s l &= \sum_{-\infty}^{+\infty} A_m S_{sm}, \end{aligned} \qquad \tag{3,14} \]

where

\[ \left. \begin{aligned} C_{sm} &= \int_{-\frac{L}{2}}^{\frac{L}{2}} \cos \frac{2\pi s}{l}\, z e^{-i\beta_m z}\, dz \\ &= L\left[ \frac{\sin\left(\beta_m+\frac{2\pi s}{l}\right)\frac{L}{2}} {\left(\beta_m+\frac{2\pi s}{l}\right)L} + \frac{\sin\left(\beta_m-\frac{2\pi s}{l}\right)\frac{L}{2}} {\left(\beta_m-\frac{2\pi s}{l}\right)L} \right], \\[1.2em] S_{sm} &= \int_{-\frac{L}{2}}^{\frac{L}{2}} \sin \frac{(2s-1)\pi}{l}\, z e^{-i\beta_m z}\, dz \\ &= iL\left[ \frac{\sin\left(\beta_m+\frac{(2s-1)\pi}{l}\right)\frac{L}{2}} {\left(\beta_m+\frac{(2s-1)\pi}{l}\right)L} - \frac{\sin\left(\beta_m-\frac{(2s-1)\pi}{l}\right)\frac{L}{2}} {\left(\beta_m-\frac{(2s-1)\pi}{l}\right)L} \right]. \end{aligned} \right\} \tag{3.15} \]

Multiplying, further, (3.13) by \(\cos \frac{2\pi s z}{l}\) and \(\sin \frac{(2s-1)\pi z}{l}\) and integrating with respect to \(z\) over the limits from \(z=-\frac{l}{2}\) to \(z=\frac{l}{2}\), we obtain, using (3.14), the following infinite system of equations for determining the coefficients \(A_m\):

\[ \sum_{m=-\infty}^{\infty} A_m\left[ \frac{k}{\Gamma_s}\frac{Z_1(\Gamma_s a)}{Z_0(\Gamma_s a)} C_{sm} - \frac{k}{\gamma_m}\frac{J_1(\gamma_m a)}{J_0(\gamma_m a)} C'_{sm} \right]=0, \tag{3.16} \]

\[ \sum_{m=-\infty}^{\infty} A_m\left[ \frac{k}{\Gamma_s}\frac{Z_1(\Gamma_s a)}{Z_0(\Gamma_s a)} S_{sm} - \frac{k}{\gamma_m}\frac{J_1(\gamma_m a)}{J_0(\gamma_m a)} S'_{sm} \right]=0, \tag{3.17} \]

where \(C'_{sm}\) and \(S'_{sm}\) are obtained from \(C_{sm}\) and \(S_{sm}\) by replacing \(L\) by \(l\). Equating to zero the determinant of the infinite system (3.16) and (3.17), we obtain the dispersion equation relating \(\beta_0\) to the frequency and the parameters of the system.

As calculations show\(^{18}\), in this determinant it is sufficient to restrict oneself to three rows and columns corresponding to \(m=0,-1,+1\) and \(s=0,1\) in equation (3.16) and \(s=1\) in equation (3.17), in order to obtain a sufficiently good approximation.

It is easy to see that the method just described leads, in the case of closely spaced thin disks, to the dispersion ...

to equation (3.7). Indeed, if \(l \ll \lambda\) and \(a \gtrsim l\), then of all equations (3.16) one may retain only one equation, corresponding to \(s=0\). The remaining equations contain the factor \(l/\lambda\) and therefore, in the approximation under consideration, are automatically satisfied. (If \(l \ll \lambda\) and \(a \gtrsim l\), then \(|\Gamma_s|\sim 2\pi s/l\), \(|\chi_m|\sim 2\pi m/l\), \(m,s\ne 0\), and all coefficients at \(A_m\) in the equations with \(s\ne 0\), proportional to

\[ \frac{k}{\Gamma_s}\frac{Z_1(\Gamma_s a)}{Z_0(\Gamma_s a)} - \frac{k}{\chi_m}\frac{J_1(\chi_m a)}{J_0(\chi_m a)}, \]

will be of order \(l/\lambda \ll 1\).)

Rewriting equation (3.16) with \(s=0\) in the form \((C'_{sm}\simeq C_{sm},\ C_{0m}\simeq 0\) for \(m\ne 0)\)

\[ W_0 A_0 C_{00}=0, \]

where

\[ W_0 \equiv \frac{Z_0(ka)}{Z_1(ka)} - \frac{k}{k_1}\frac{J_1(k_1a)}{J_0(k_1a)}, \]

we obtain the relation

\[ W_0=0, \]

coinciding with equation (3.7) \(\left(k_1=\sqrt{k^2-k_3^2},\ k_3=\beta_0\right)\).

Thus we see that the dispersion equation (3.7) is valid in the case of very closely spaced thin disks with not very small apertures \(a\).

Let us note that the representation of the fields for \(r<a\) in the form of a single traveling wave is valid if \(a\gg l\), since only in this case do the amplitudes of the additional traveling waves, proportional to \(J_0(|\chi_m|r)\), decrease from the boundary separating the regions \((r=a)\) to the axis as \(e^{-r/l}\).

In summary, one may say that a waveguide loaded with disks is an effective method of obtaining slow waves in the region of phase velocities \(v_\varphi\sim c\), since in this case, owing to the comparatively sparse arrangement of the disks, the attenuation of the waves is considerably reduced. Moreover, in this case the axial component of the field in the region \(r<a\) changes little (increases) with radius.

IV. SPIRAL WAVEGUIDES

1. Slow electromagnetic waves can also be obtained by means of a spiral waveguide, which is an open conducting spiral (Fig. 9). If a wave propagates along the spiral with a velocity close to \(c\), then the phase velocity of the wave along the axis of the spiral will be less than \(c\). Roughly speaking,

SLOW ELECTROMAGNETIC WAVES

it is proportional to the pitch of the helix and inversely proportional to the radius of the helix.

A large number of works has been devoted to the method of obtaining slow waves by means of helical waveguides\(^{15,16,17,18}\). In this paragraph we shall set forth the principal results of these works.

In this section let us consider the propagation of axially symmetric waves in a sufficiently dense helix\(^{15,16,17,18}\). The components of the fields \(E_z\) and \(H_z\) in this case, for \(v_\varphi \ll c\), are determined by the following expressions:

\[ E_z=\left[AI_0(k_1r)+BK_0(k_1r)\right]e^{i(\omega t-k_3z)}, \]

\[ H_z=\left[CI_0(k_1r)+DK_0(k_1r)\right]e^{i(\omega t-k_3z)}, \]

where \(I_0\) and \(K_0\) are the modified Bessel functions, and \(k_1=\sqrt{k_3^2-k^2}\). The field inside the helix \((r\le r_0)\) obviously cannot contain \(K_0\), since at \(r=0\) \(K_0(k_1r)\) becomes infinite. The field outside the helix must not contain \(I_0\), since \(I_0\) is a monotonically increasing function tending to infinity as \(r\to\infty\). Therefore the components \(E_z, H_z\) inside the helix (region I) have the form

\[ \begin{aligned} E_{zI}&=E_0 I_0(k_1r)e^{i(\omega t-k_3z)},\\ H_{zI}&=C_1 I_0(k_1r)e^{i(\omega t-k_3z)} \end{aligned} \tag{4,1} \]

(\(E_0\) is the amplitude of \(E_z\) on the axis of the helix). Outside the helix (region II) \(E_z\) and \(H_z\) are equal to:

\[ \begin{aligned} E_{zII}&=C_2K_0(k_1r)e^{i(\omega t-k_3z)},\\ H_{zII}&=C_3K_0(k_1r)e^{i(\omega t-k_3z)}. \end{aligned} \tag{4,2} \]

Fig. 9.

The remaining components of the field can be determined with the aid of Maxwell’s equations. As a result we obtain:

In region I:

\[ \begin{aligned} E_{zI}&=E_0 I_0(k_1r),\\ E_{rI}&=iE_0\frac{k_3}{k_1}I_1(k_1r),\\ H_{\varphi I}&=iE_0\frac{k}{k_1}I_1(k_1r),\\ H_{zI}&=C_1 I_0(k_1r),\\ E_{\varphi I}&=-iC_1\frac{k}{k_1}I_1(k_1r),\\ H_{rI}&=iC_1\frac{k_3}{k_1}I_1(k_1r). \end{aligned} \tag{4,3} \]

In region II

\[ \begin{aligned} E_{z\mathrm{II}}&=C_2K_0(k_1r),\\ E_{r\mathrm{II}}&=-iC_2\frac{k_3}{k_1}K_1(k_1r),\\ H_{\varphi\mathrm{II}}&=-iC_2\frac{k}{k_1}K_1(k_1r),\\ H_{z\mathrm{II}}&=C_3K_0(k_1r),\\ E_{\varphi\mathrm{II}}&=iC_3\frac{k}{k_1}K_1(k_1r),\\ H_{r\mathrm{II}}&=-iC_3\frac{k_3}{k_1}K_1(k_1r). \end{aligned} \tag{4,4} \]

(in these formulas we have omitted the factor \(e^{i(\omega t-k_3z)}\)).

Let us now proceed to the derivation of the dispersion equation connecting \(k_3\) and \(k\). Since the tangential component of the electric field along the turns of the helix is equal to zero, the relation*) holds

\[ E_{\varphi \mathrm{I}}\cos\psi+E_{z\mathrm{I}}\sin\psi=0,\qquad r=r_0, \tag{4,5} \]

where \(\psi\) is the winding angle (see Fig. 8). From the continuity of \(E_z\) and \(E_\varphi\) it follows that, for \(r=r_0\),

\[ \begin{aligned} E_{z\mathrm{I}}&=E_{z\mathrm{II}},\\ E_{\varphi\mathrm{I}}&=E_{\varphi\mathrm{II}}. \end{aligned} \tag{4,6} \]

Since there is no current in the direction perpendicular to the helix, the tangential component of the magnetic field along the turns of the helix is continuous, i.e.,

\[ H_{\varphi\mathrm{I}}\cos\psi+H_{z\mathrm{I}}\sin\psi = H_{\varphi\mathrm{II}}\cos\psi+H_{z\mathrm{II}}\sin\psi, \qquad r=r_0. \tag{4,7} \]

Substituting into these boundary conditions the expressions for the fields (4,3) and (4,4) and eliminating the constants, we obtain the dispersion equation

\[ (kr_0\operatorname{ctg}\psi)^2 = (k_1r_0)^2 \frac{I_0(k_1r_0)K_0(k_1r_0)} {I_1(k_1r_0)K_1(k_1r_0)}. \tag{4,8} \]

1. Let us examine in greater detail two limiting cases, when \(k_1r_0\ll1\) and when \(k_1r_0\gg1\). Using the asymptotic expressions for the Bessel functions at small values of the argument,

\[ I_0(x)\simeq1,\qquad I_1(x)\simeq\frac{1}{2}x,\qquad K_0(x)\simeq-\ln\frac{x}{2},\qquad K_1(x)\simeq\frac{1}{x}\quad (x\ll1), \]

we obtain, in the case \(k_1r_0\ll1\), the following expression for the phase velocity:

\[ v_\varphi = c\left(1-\frac{\operatorname{ctg}^2\psi}{2\ln\dfrac{k_1r_0}{2}}\right)^{-\frac12}, \qquad k_1r_0\ll1. \tag{4,9} \]

*) Here it is assumed that the helix may be replaced by an equivalent cylindrical surface conducting current only in the direction of the helix.

This expression shows that if \(k_1 r_0 \ll 1\), then dispersion occurs, and the course of the dispersion is normal. Therefore the group velocity is less than the phase velocity. If \(k_1 r_0 \to 0\), then \(v_\varphi \to c\).

In the case \(k_1 r_0 \gg 1\), the expressions

\[ I_0(k_1 r_0) \simeq I_1(k_1 r_0) \simeq (2\pi k_1 r_0)^{-\frac12} e^{k_1 r_0}, \]

\[ K_0(k_1 r_0) \simeq K_1(k_1 r_0) \simeq \left(\frac{2}{\pi} k_1 r_0\right)^{-\frac12} e^{-k_1 r_0}, \]

are valid; using them we obtain:

\[ \frac{k}{k_1}\operatorname{ctg}\psi = 1. \]

For a tightly wound helix \(\operatorname{ctg}\psi \gg 1\), and therefore \(k_1 = \sqrt{k_3^2 - k^2} \gg k\), whence \(k_1 \simeq k_3\). Thus, the phase velocity is

\[ v_\varphi = c\frac{k}{k_3} = c\cdot \operatorname{tg}\psi \ll c,\qquad k_1 r_0 \gg 1. \tag{4,10} \]

This formula has a simple meaning: the ratio of the phase velocity to the velocity of light in vacuum is equal to the ratio of the paths traversed by the wave along the helix and along its axis.

Let us note that in the case \(k_1 r_0 \ll 1\) the field slowly increases with increasing \(r\) from the axis of the helix to its surface and slowly decreases outside the helix. In the case \(k_1 r_0 \gg 1\) the field rapidly increases from the axis of the helix to its surface and rapidly decreases outside the helix.

A distinctive and very essential feature of the helical waveguide is the absence of a cutoff frequency.

We have considered the limiting cases \(k_1 r_0 \ll 1\) and \(k_1 r_0 \gg 1\). A graph showing the dispersion dependence in the general form is presented in Fig. 10.

1. Using the boundary conditions (4,5) and (4,6), let us express the coefficients \(C_1, C_2, C_3\) in terms of \(E_0\). We obtain the following expressions for the fields:

In region I \((r \leq r_0)\):

\[ \left. \begin{aligned} E_{z1} &= E_0 I_0(k_1 r),\\ E_{r1} &= i\,\frac{k_3}{k_1} E_0 I_1(k_1 r),\\ H_{\varphi 1} &= i\,\frac{k}{k_1} E_0 I_1(k_1 r),\\ H_{z1} &= -i\,\frac{k_1}{k}\operatorname{tg}\psi\, \frac{I_0(k_1 r_0)}{I_1(k_1 r_0)} E_0 I_0(k_1 r),\\ E_{\varphi 1} &= -\operatorname{tg}\psi\, \frac{I_0(k_1 r_0)}{I_1(k_1 r_0)} E_0 I_1(k_1 r),\\ H_{r1} &= \frac{k_3}{k}\operatorname{tg}\psi\, \frac{I_0(k_1 r_0)}{I_1(k_1 r_0)} E_0 I_1(k_1 r). \end{aligned} \right\} \tag{4,11} \]

In region II \((r \geqslant r_0)\):

\[ \left. \begin{aligned} E_{z\mathrm{II}}&=\frac{I_0(k_1 r_0)}{K_0(k_1 r_0)}\,E_0 K_0(k_1 r),\\ E_{r\mathrm{II}}&=-i\,\frac{k_3}{k_1}\,\frac{I_0(k_1 r_0)}{K_0(k_1 r_0)}\,E_0 K_1(k_1 r),\\ H_{\varphi\mathrm{II}}&=-i\,\frac{k}{k_1}\,\frac{I_0(k_1 r_0)}{K_0(k_1 r_0)}\,E_0 K_1(k_1 r),\\ H_{z\mathrm{II}}&=i\,\frac{k_1}{k}\,\operatorname{tg}\psi\, \frac{I_0(k_1 r_0)}{K_1(k_1 r_0)}\,E_0 K_0(k_1 r),\\ E_{\varphi\mathrm{II}}&=-\operatorname{tg}\psi\, \frac{I_0(k_1 r_0)}{K_1(k_1 r_0)}\,E_0 K_1(k_1 r),\\ H_{r\mathrm{II}}&=\frac{k_3}{k}\,\operatorname{tg}\psi\, \frac{I_0(k_1 r_0)}{K_1(k_1 r_0)}\,E_0 K_1(k_1 r) \end{aligned} \right\} \tag{4,12} \]

(we have omitted the factor \(e^{i(\omega t-k_3 z)}\)).

Fig. 10.

Let us now compute the total flux of electromagnetic energy in the direction of the waveguide axis:

\[ P=\frac{c}{8\pi}\operatorname{Re}\int [\mathbf{E}\mathbf{H}^{*}]_z\,ds =\frac{c}{4}\operatorname{Re}\left\{ \int_0^{r_0}(E_{r\mathrm{I}}H_{\varphi\mathrm{I}}^{*}-E_{\varphi\mathrm{I}}H_{r\mathrm{I}}^{*})\,r\,dr +\int_{r_0}^{\infty}(E_{r\mathrm{II}}H_{\varphi\mathrm{II}}^{*}-E_{\varphi\mathrm{II}}H_{r\mathrm{II}}^{*})\,r\,dr \right\}. \tag{4,13} \]

Using formulas (4.10) and (4.11), we obtain

\[ P=\frac{c}{8\pi}E_0^2 r_0^2\,\frac{kk_3}{k_1^2} \left\{ \left(1+\frac{I_0K_1}{I_1K_0}\right)(I_1^2-I_0I_2) + \left(\frac{I_0}{K_0}\right)^2 \left(1+\frac{I_1K_0}{I_0K_1}\right)(K_0K_2-K_1^2) \right\}, \tag{4.14} \]

where \(I_i, K_i\) are the values of the corresponding Bessel functions for the value of the argument equal to \(k_1r_0\). Here the first term in braces determines the energy flux inside the waveguide, and the second term outside the waveguide.

For small values of \(k_1r_0\), almost the entire energy flux is concentrated outside the helix. As \(k_1r_0\) increases, the energy flux inside the helix increases and decreases outside it. At the same time the total flux decreases and, for a given value of \(r_0\), reaches a minimum near \(k_1r_0\approx 1.2\). (The energy flux outside the helix also reaches a minimum near \(k_1r_0\approx 1.2\).) Such a dependence of the flux on \(k_1r_0\) is connected with the fact that for \(k_1r_0\ll 1\) the fields decrease slowly outside the helix, while for \(k_1r_0\gg 1\) they rapidly increase from the axis to its surface and rapidly fall off outside the helix.

1. We saw above that a significant part of the energy flux passes outside the helix. In order to redistribute the energy flux and reduce the fraction of the flux outside the helix, one can surround the helix by a metallic cylinder coaxial with the helix \(^{17}\). In this case the dispersion properties of the helix also change. In particular, it turns out that the phase velocity of the wave decreases in comparison with an open helix with the same parameters.

We shall give here the dispersion equation for a helix in a cylindrical waveguide, which is derived in the same way as the equation for an open helix \(^{17}\) (in deriving the equation it is necessary to take into account that on the surface of the cylinder the boundary conditions \(E_{z\parallel}=0,\ E_{\varphi\parallel}=0\) are satisfied):

\[ (kr_0\operatorname{ctg}\psi)^2 = (k_1r_0)^2 \frac{I_0(k_1r_0)K_0(k_1r_0)} {I_1(k_1r_0)K_1(k_1r_0)} \,\eta(k_1r_0,k_1R), \tag{4.15} \]

where

\[ \eta(k_1r_0,k_1R) = \frac{ 1-\dfrac{I_0(k_1r_0)K_0(k_1R)} {I_0(k_1R)K_0(k_1r_0)} }{ 1-\dfrac{I_1(k_1r_0)K_1(k_1R)} {I_1(k_1R)K_1(k_1r_0)} }. \]

(\(R\) is the radius of the cylinder).

If \(k_1r_0\ll 1\) and \(k_1R\ll 1\), then from (4.15) it follows that

\[ v_\varphi = c\left( 1+ \frac{\operatorname{ctg}^2\psi\left[1-\left(\dfrac{r_0}{R}\right)^2\right]} {2\ln\dfrac{R}{r_0}} \right)^{-\frac12}. \]

Thus, in this case \(v_\varphi\) does not depend on the frequency and is determined only by the parameters of the helix; the group velocity in this case coincides with the phase velocity.

If

\[ k_1 r_0 \gg 1, \]

then

\[ v_\varphi = c \operatorname{tg} \psi . \]

This relation coincides with formula (4,10), valid in the case of an open helix. Such a coincidence is explained by the fact that the fields for \(k_1 r_0 \gg 1\) rapidly decrease outside the helix, and therefore the outer cylinder has no influence on the topography of the field.

Fig. 11.

The phase velocity for \(k_1 r_0 \gg 1\), just as in the case \(k_1 r_0 \ll 1\), does not depend on the frequency.

Figure 11 presents the dependence of \(\dfrac{k}{k_1}\operatorname{ctg}\psi\) on \(k r_0 \operatorname{ctg}\psi\) for an open helix (curve \(II\)) and for a helix in a cylindrical waveguide (curve \(I\)), when \(\dfrac{R}{r_0}=2\). We see that the presence of the cylinder leads to a decrease in the phase velocity, and this decrease is especially significant in the region of long waves.

Calculation shows that the flow of energy in a helix surrounded by a cylinder is smaller than in an open helix for given frequency, helix radius, and field strength on the axis of the waveguide.

The method of obtaining slow waves with the aid of a helix is effective in the region of small phase velocities.

V. INTERACTION OF A BEAM OF CHARGED PARTICLES WITH SLOW WAVES

1. We now turn to the study of the interaction of charged particles with slow waves. Let us first note that a charged particle moving uniformly in a periodic structure can, under certain conditions, radiate electromagnetic waves.

It is known that, in the uniform motion of a charge in a dielectric, radiation occurs only if the velocity of the charge exceeds the phase velocity of waves in an unbounded dielectric (the Cherenkov effect) ^19, 20, 21. It can be shown ^22 that the presence of a dielectric is not a necessary condition for radiation. If the phase velocity of propagation of a wave in a complex waveguide containing no dielectric is less than the velocity of light in vacuum, then a charged particle moving uniformly in such a structure will also radiate electromagnetic waves, provided only that the velocity of the particle exceeds some critical value.

This phenomenon, which may be called the complex Cherenkov effect, plays an important role in the mechanism of generation and amplification of slow waves by means of beams of charged particles.

The radiation of a particle may be interpreted ^19a, 22 as a resonance between the natural oscillations of the periodic structure and the driving force associated with the moving charged particle. The natural frequencies of the system \(\omega_\lambda\) are determined by the set of parameters \(\lambda = \gamma, s\), one of which \((\gamma)\) is continuous and lies within the limits \(-\pi/l, \pi/l\), where \(l\) is the period of the structure; the remaining parameters \((s)\) are discrete and serve to denote the “band” of frequency transmission. The driving force is characterized by the frequency spectrum

\[ \Omega_{\gamma n} \equiv \left(\gamma + \frac{2\pi n}{l}\right)v, \]

where \(n = 0, \pm 1, \pm 2, \ldots\) and \(v\) is the particle velocity. The resonance condition, which is therefore the condition for radiation, has the form

\[ \omega_\lambda = \omega_{\gamma s} = \Omega_{\gamma n} = \left(\gamma + \frac{2\pi n}{l}\right)v. \]

In unbounded vacuum \(\omega = \gamma c\), \(\Omega_{\gamma n} = \gamma v \cos\theta\) (\(\theta\) is the angle between the direction of wave propagation and the particle velocity, \(n = 0\)); since \(v < c\), the resonance condition cannot be satisfied. In an unbounded dielectric,

\[ \omega_\lambda = \frac{c\gamma}{\sqrt{\varepsilon}} \]

(\(\varepsilon\) is the dielectric permittivity), and the resonance condition leads to the familiar condition for Cherenkov radiation

\[ \cos\theta = \frac{c}{v\sqrt{\varepsilon}} < 1. \]

In periodic structures the resonance condition can be fulfilled for a wide variety of combinations of the quantities \(\gamma, n, s\).

1. A considerable number of works have been devoted to the study of the interaction of beams of charged particles with slow waves \(^{24,25,26,27,28}\). In these works it is shown that if the unperturbed velocity of the modulated beam exceeds a certain critical value, then the charge-density fluctuations arising in the beam propagate in the form of waves with increasing amplitude. At the same time electromagnetic waves are excited, also with increasing amplitude \(^{24-28}\).

In order to clarify this phenomenon, let us first consider the simplest case. Let an infinitely wide plane-parallel beam of electrons move in an unbounded homogeneous dielectric in the direction of the \(z\)-axis, its unperturbed state being determined by the velocity \(v_0\) and the charge density \(\rho_0\); we shall regard these quantities as constant and independent of the coordinates. (In order that such a state may exist, we shall assume that, along with the electron beam, there is also an ion beam with charge density \(-\rho_0\) and velocity \(v_0\).)

The particles of the beam obey the equation of motion

\[ \frac{\partial \mathbf{w}}{\partial t}+(\mathbf{w}\nabla)\mathbf{w}=\frac{e}{m}\mathbf{E} \]

and the continuity equation

\[ \frac{\partial \rho}{\partial t}+\operatorname{div}\mathbf{w}(\rho+\rho_0)=0, \]

where \(\mathbf{w}\) is the perturbed velocity of the beam particles, \(\rho\) is the deviation of the charge density from the equilibrium value \(\rho_0\), and \(\mathbf{E}\) is the electric field.

We shall consider small deviations from the unperturbed state, i.e. assume that \(\rho \ll \rho_0\) and \(|\mathbf{w}-\mathbf{v}_0| \ll v_0\). Taking \(\mathbf{w}\) to be parallel to \(\mathbf{v}_0\), we obtain the following linearized equations of motion and continuity:

\[ \frac{\partial v}{\partial t}+v_0\frac{\partial v}{\partial x}=\frac{e}{m}E_x, \tag{5,1} \]

\[ \frac{\partial \rho}{\partial t}+\frac{\partial}{\partial x}(\rho_0 v+v_0\rho)=0, \tag{5,2} \]

where \(v=w-v_0\). To these equations we adjoin the linearized equations for the vector and scalar potentials

\[ \begin{aligned} \Delta \mathbf{A}-\frac{1}{u^2}\frac{\partial^2 \mathbf{A}}{\partial t^2} &=-\frac{4\pi}{c}(\rho_0\mathbf{v}+\mathbf{v}_0\rho),\\ \Delta \varphi-\frac{1}{u^2}\frac{\partial^2 \varphi}{\partial t^2} &=-\frac{4\pi}{\varepsilon}\rho,\\ \operatorname{div}\mathbf{A}+\frac{\varepsilon}{c}\frac{\partial\varphi}{\partial t} &=0, \end{aligned} \tag{5,3} \]

where \(\varepsilon\) is the dielectric constant and \(u=\dfrac{c}{\sqrt{\varepsilon}}\) is the velocity of propagation of waves in the dielectric.

Equations (5,1), (5,2), (5,3) constitute a complete system of equations describing the interaction of the beam with the dielectric. We seek the solution of these equations in the form of plane waves \(e^{i(\omega t-g{\bf r})}\).

We obtain, evidently, the following relations:

\[ \left. \begin{aligned} \left[-\left(\gamma^2+\gamma_0^2\right)+\frac{\omega^2}{u^2}\right]\varphi &=-\frac{4\pi}{\varepsilon}\rho,\\[4pt] A_x&=\frac{\varepsilon\omega}{c\gamma}\varphi,\\[4pt] (\omega-v_0\gamma)v &=\frac{1}{\gamma}\left(\gamma^2-\frac{\omega^2}{u^2}\right)\frac{e}{m}\varphi,\\[4pt] (\omega-v_0\gamma)\rho&=\rho_0\gamma v,\\[4pt] E_x&=-\frac{1}{c}\frac{\partial A_x}{\partial t}-\frac{\partial\varphi}{\partial x} =\frac{i}{\gamma}\left(\gamma^2-\frac{\omega^2}{u^2}\right)\varphi, \end{aligned} \right\} \tag{5,4} \]

where \(\gamma\) is the projection of \({\bf g}\) on the \(x\)-axis, \(\gamma_0^2=g^2-\gamma^2\). Eliminating \(\varphi\), \(\rho\), and \(v\) from these equations, we obtain the dispersion equation relating \(\omega\) and \(\gamma\):

\[ \frac{\omega_0^2}{\omega^2-u^2\gamma^2} +\frac{\Omega^2}{(\omega-v_0\gamma)^2}=1 \tag{5,5} \]

\[ \left(\text{here } \omega_0=\gamma_0u \text{ and } \Omega^2=\frac{4\pi e^2\rho_0}{\varepsilon m}\right). \]
Considering \(\gamma\) as given, we shall regard (5,5) as an equation determining \(\omega\).

Passing to the investigation of equation (5,5), we shall assume the beam density to be sufficiently small, i.e. we shall regard the quantity \(\Omega\) as sufficiently small. We shall seek a root of equation (5,5) close to \(\omega=\gamma v_0\). Setting

\[ \omega=v_0\gamma+\eta, \]

where \(|\eta|\ll v_0\gamma\), we obtain the following expression for \(\eta\):

\[ \eta=-\frac{i\Omega}{\sqrt{\dfrac{\omega_0^2}{\gamma^2\left(v_0^2-u^2\right)}-1}}. \tag{5,6} \]

This expression shows that if \(\Omega\ll\gamma v_0\), then the root close to \(\gamma v_0\) indeed exists. Further, we see that if \(u\) is a real quantity, which corresponds to the transparency region of the dielectric, then for \(v_0>u\) the quantity \(\eta\) can become purely imaginary (if \(v_0<u\), then \(\eta\) is always, i.e. for all values of \(\gamma\), real). Hence it may be concluded\(^{31}\) that if the unperturbed velocity of the beam \(v\) exceeds the phase velocity of propagation of waves in the infinite dielectric, then the state of the beam characterized by constan-

... $\rho_0$ and $\tau_0$, will be unstable, and the fluctuations of density and velocity arising in the beam will propagate in the form of waves with an amplitude growing exponentially with time. At the same time, together with the charge-density waves, electromagnetic waves will also propagate with increasing amplitude.

We see that the condition for instability of the beam and for the occurrence of charge-density waves and electromagnetic waves with increasing amplitude coincides with the condition for Cherenkov radiation of an individual charged particle.

1. Formula (5.6) becomes inapplicable if $\gamma$ is equal to

\[ \tilde{\gamma}=\frac{\omega_0}{\sqrt{v_0^2-u^2}}, \]

since in this case $\eta$ tends to infinity.

Let us examine in more detail the case when $\gamma=\tilde{\gamma}$, which corresponds to the frequency

\[ \tilde{\omega}=v_0\tilde{\gamma}=\frac{\omega_0}{\sqrt{1-\frac{u^2}{v_0^2}}}. \tag{5.7} \]

Putting in equation (5.5) $\gamma=\tilde{\gamma}$ and $\omega=\tilde{\omega}+\xi$, we obtain the following equation for determining $\xi$:

\[ \xi^4+\frac{2\omega_0}{\sqrt{1-\frac{u^2}{v_0^2}}}\xi^3-\Omega^2\xi^2-\frac{2\omega_0\Omega^2}{\sqrt{1-\frac{u^2}{v_0^2}}}\xi-\Omega^2\omega_0^2=0. \]

Since $\Omega\ll\omega_0$, in this equation one may neglect the first, third, and fourth terms. As a result we obtain:

\[ \xi=-\frac{1+i\sqrt{3}}{2^{4/3}}\left(1-\frac{u^2}{v_0^2}\right)^{1/6}\omega_0\left(\frac{\Omega}{\omega_0}\right)^{2/3}. \tag{5.8} \]

Thus, if the projection of the wave vector on the $x$ axis is equal to $\tilde{\gamma}$, then the imaginary part of the frequency is proportional not to $\dfrac{\Omega}{\omega_0}$, but to $\left(\dfrac{\Omega}{\omega_0}\right)^{2/3}$.

Let us note that if equation (5.5) is regarded as an equation for determining $\gamma$ at a given frequency $\omega$, then, putting $\omega=\tilde{\omega}$, we obtain the following expression for the imaginary part of $\gamma$:

\[ \chi=\frac{3^{1/2}}{2^{4/3}}\frac{\omega_0}{u}\left(\frac{u}{v_0}\right)^{1/3}\left[1-\left(\frac{u}{v_0}\right)^2\right]^{1/6}\left(\frac{\Omega}{\omega_0}\right)^{2/3}. \tag{5.9} \]

The maximum value of this quantity as a function of \(\vartheta_0\) is

\[ \chi_{\max}=\frac{3^{\frac12}}{4}\frac{\omega_0}{u}\left(\frac{\Omega}{\omega_0}\right)^{\frac23} \tag{5,10} \]

\[ \left(\text{it is attained for } \frac{\vartheta_0}{u}=\sqrt{2}\right). \]

It is easy to understand the physical meaning of the frequency \(\widetilde{\omega}\) corresponding to the maximum instability of the beam. If an individual particle moves in the dielectric with a velocity \(\vartheta_0\) exceeding the velocity of wave propagation \(u\), then Cherenkov radiation arises, which may be interpreted as a resonance between the natural oscillations of the field in the dielectric and the driving force due to the moving particle. The frequency of the natural oscillations is \(gu\), while the frequency of the driving force is \(\vartheta_0\gamma\). Equating these quantities and noting that \(g=\sqrt{\gamma_0^2+\gamma^2}\), we obtain the frequency \(\widetilde{\omega}\), to which the wave with the greatest increase of amplitude corresponds.

Let us note that in the frequency region where the dielectric constant is complex, i.e. in the region of opacity of the dielectric, equation (5,5) has complex roots for arbitrarily small values of the unperturbed velocity \(\vartheta_0\); in other words, the unperturbed state of the beam is unstable for all values of \(\vartheta_0\).

1. We have considered the case of an unbounded dielectric and an infinitely wide beam of particles; however, the conclusions obtained remain valid also for bounded beams. \(^{24-28}\)

Let us consider, \(^{27}\) for example, a cylindrical waveguide filled with a dielectric, through which, in the direction of the axis of the waveguide, a beam of charged particles passes. In order not to complicate the calculation, suppose that the beam fills the entire space inside the waveguide. In this case we must still proceed from equations (5,1), (5,2), (5,3), to which one need only add the boundary condition \(E_z=0\) at the boundary of the waveguide. The quantities \(\rho,\vartheta,\varphi,E_z\) will be sought in the form \(e^{i(\omega t-\gamma z)}f(r)\), where the function \(f\) satisfies the equation

\[ \Delta_r f+\gamma_0^2 f=0, \tag{5,11} \]

where \(\Delta_r\) is the two-dimensional Laplacian in the \(x,y\) plane, and \(\gamma_0\) is the boundary wave vector, equal to \(\dfrac{\omega_0}{u}\) (\(\omega_0\) is the cutoff frequency of the waveguide).

Taking (5,11) into account, from equations (5,1), (5,2), (5,3) we obtain the relations

\[ \left(-\omega^2+2i\gamma\vartheta_0\omega-\gamma^2\vartheta_0^2+\Omega^2\right)\rho =\frac{e}{m}\rho_0\gamma_0^2\varphi, \]

\[ \left(-\frac{\omega^2}{u^2}+\gamma^2+\gamma_0^2\right)\varphi =\frac{4\pi}{\varepsilon}\rho, \]

which lead to the dispersion equation (5.5). The quantity \(\omega_{0*}\) entering it now denotes the cutoff frequency of the waveguide.

Let us note that charge-density and velocity waves in waveguides without a dielectric were investigated in Ramo’s work\(^{23}\), in which the dispersion equation (5.5) was derived for the case \(u=c\).

The instability of the beam and the occurrence of charge-density waves and electromagnetic waves with increasing amplitude may take place not only when the beam moves in a dielectric. These phenomena also arise when a beam of particles moves in slow-wave waveguides not containing a dielectric, provided only that the unperturbed beam velocity exceeds a certain critical value (for a chain of endovibrators with very small apertures this velocity is proportional to \(\sqrt{a}\), where \(a\) is determined by formula (2.15)).

From this point of view the mechanism of excitation of oscillations in a multicavity magnetron can be explained. The circumstance that in a multicavity magnetron we deal not with a linear, but with a circular, closed chain of endovibrators does not change the essence of the phenomenon: an infinite linear chain is, so to speak, a straightened multicavity magnetron. The closedness of the chain leads to the fact that the propagation constant \(\psi\) can assume only discrete values, equal to \(\psi_n=\dfrac{2\pi}{N}n\), where \(N\) is the total number of cavities, and \(n\) is an integer not exceeding \(N\). As follows from formula (5.7), the maximum growth of amplitude will occur for the oscillation for which \(\psi=\gamma l\) is equal to

\[ \tilde{\psi}=\frac{\omega_0 l}{\sqrt{v_0^2-u^2}} \]

(\(u\) is the critical velocity).

VI. INTERACTION OF A BEAM OF CHARGED PARTICLES WITH AN ELECTRON PLASMA

1. It is known that longitudinal electric waves can propagate in an electron plasma, whose frequency is equal to\(^{29,30}\)

\[ \omega=\sqrt{\omega_0^2+u^2\gamma^2}, \quad \text{where } \omega_0=\left(\frac{4\pi e^2 n_0}{m}\right)^{\frac12} \]

(\(n_0\) is the equilibrium density of plasma electrons), \(\gamma\) is the wave vector, and \(u=\left(\dfrac{3T}{m}\right)^{\frac12}\) is the mean thermal velocity of the electrons (\(T\) is the temperature, expressed in ergs). Formally, a dispersion relation of the same type occurs in a waveguide filled with a dielectric, with phase velocity of wave propagation equal to \(u\) (in this case \(\omega_0\) is the cutoff frequency of the waveguide). Therefore the question arises whether the interaction of an electron beam with a plasma will obey the same laws as the interaction of a beam

charged particles with slow waves in waveguides filled with a dielectric, and in chains of cavity resonators. It turns out \(^{27}\) that in many respects the interaction of a beam of charged particles with plasma oscillations is indeed analogous to the interaction of a beam with slow waves in complex waveguides; in particular, there is beam instability and the appearance of charge-density waves and electric longitudinal waves with increasing amplitude. However, in contrast to dielectrics and complex waveguides, in passing through which the beam becomes unstable at velocities exceeding the phase velocity of wave propagation, beam instability in plasma occurs for all values of the unperturbed beam velocity. The instability of the beam and the appearance of waves with increasing amplitude become especially significant if the beam velocity exceeds the mean thermal velocity of the plasma electrons \(^{27}\).

1. Let us consider \(^{27,31}\) an unbounded homogeneous plasma in which, in the direction of the \(z\)-axis, an infinitely wide plane-parallel beam of electrons is moving. The unperturbed state of the beam is characterized by a constant, coordinate-independent velocity \(v_0\) and charge density \(\rho_0\). The state of the plasma is determined by the distribution function \(F(\mathbf r,\mathbf w,t)\), which at high oscillation frequencies, when the role of pair collisions may be neglected, satisfies the following kinetic equation \(^{29}\):

\[ \frac{\partial F}{\partial t}+\mathbf w\nabla_r F+\frac{e}{m}\mathbf E\nabla_w F=0. \]

Here \(\mathbf w\) is the velocity of the plasma electrons, \(\mathbf r\) is the radius vector determining their position, and \(\mathbf E\) is the self-consistent electric field acting in the plasma. We shall consider states of the plasma close to the equilibrium state, and states of the beam close to the unperturbed state. Denoting by \(f(z,w,t)\) the deviation of the distribution function from the equilibrium Maxwellian function \(f_0(w)\), where \(w\) is the projection of the velocity of the plasma electrons on the \(z\)-axis (in the case studied by us of an unbounded plasma and an infinitely wide beam it is sufficient, as is easy to see, to consider only this projection of the velocity), we obtain the following equation for determining \(f\):

\[ \frac{\partial f}{\partial t}+w\frac{\partial f}{\partial z}+\frac{e}{m}E\frac{\partial f_0}{\partial w}=0. \tag{6,1} \]

The electric field satisfies the equation

\[ \frac{\partial E}{\partial z}=4\pi e\int_{-\infty}^{\infty} f\,dw+4\pi\rho, \tag{6,2} \]

where

\[ e\int_{-\infty}^{\infty} f\,dw \]

is the charge density uncompensated by the ions

of plasma electrons, and \(\rho\) is the deviation of the beam density from the unperturbed value \(\rho_0\). (We assume here that the equilibrium electron charge \(e\int f_0\,dv\) and the unperturbed charge of the beam are compensated by the charge of the ions.)

The particles of the beam obey the equation of motion

\[ \frac{\partial \mathbf{w}_p}{\partial t}+(\mathbf{w}_p,\nabla)\mathbf{w}_p=\frac{e}{m}\mathbf{E} \]

and the equation of continuity

\[ \frac{\partial \rho}{\partial t}+\operatorname{div}\mathbf{w}_p(\rho+\rho_0)=0 \]

(\(\mathbf{w}_p\) is the perturbed velocity of the beam). Assuming that \(\mathbf{w}_p\) is parallel to \(\mathbf{v}_0\), and regarding \(\mathbf{w}_p\) as differing only slightly from \(\mathbf{v}_0\), we obtain the following linearized equations of motion and continuity:

\[ \frac{\partial v}{\partial t}+v_0\frac{\partial v}{\partial z}=\frac{e}{m}E, \tag{6,3} \]

\[ \frac{\partial \rho}{\partial t}+\frac{\partial}{\partial z}(\rho_0v+v_0\rho)=0, \tag{6,4} \]

where \(v=w_p-v_0\).

Equations (6,1), (6,2), (6,3), (6,4) constitute a complete system of equations determining small perturbations of the plasma and the beam.

The linearized equations (6,1)—(6,4) admit solutions containing \(z\) and \(t\) in the form \(e^{i(\omega t-\gamma z)}\), where \(\gamma\) and \(\omega\) are constants. We shall prove that among these solutions there are solutions which, for real \(\gamma\), are characterized by complex \(\omega\) with negative imaginary part. The presence of such growing solutions shows that the initial state of the beam is unstable and that the fluctuations of the field and of the charge densities arising in the plasma and beam will propagate in the form of waves with increasing amplitude.

Solutions in the form of plane monochromatic waves (with complex frequency) are determined, according to (6,1), (6,3), (6,4), by the following formulas:

\[ \left. \begin{aligned} f&=i\frac{e}{m}E\,\frac{\dfrac{df_0}{dv}}{\omega-\gamma v},\qquad E=\mathrm{const}\,e^{i(\omega t-\gamma z)},\\ v&=-i\frac{e}{m}\frac{E}{\omega-\gamma v_0},\\ \rho&=-i\gamma\frac{e\rho_0}{m}\frac{E}{(\omega-\gamma v_0)^2}. \end{aligned} \right\} \tag{6,5} \]

Substitution of (9,5) into (6,2) leads to a relation connecting \(\omega\) and \(\gamma\):

\[ -\frac{4\pi e^2}{m\gamma}\int_{-\infty}^{\infty}\frac{f_0'(w)\,dw}{\omega-\gamma w} +\frac{\Omega^2}{(\omega-\gamma v_0)^2}=1, \tag{6,6} \]

where \(\Omega=\left(\dfrac{4\pi e^2\rho_0}{m}\right)^{\frac12}\) is the Langmuir frequency for the beam, and

\[ f_0(w)=n_0\left(\frac{m}{2\pi T}\right)^{\frac12} e^{-\frac{mw^2}{2T}} \]

is the equilibrium distribution function of the plasma electrons (\(n_0\) is the electron density, \(T\) the temperature, expressed in ergs).

The study of the dispersion equation (6,6) constitutes our main problem.

1. Introducing, instead of \(w\), the new variable \(y=\left(\dfrac{m}{2T}\right)^{\frac12}w\), we rewrite (6,6) in the form

\[ \left. \begin{aligned} &(2\pi)^{-\frac12}\left(\frac{m}{T}\right)^{\frac32}\frac{\omega_0^2}{\gamma^3} \int_{-\infty}^{\infty}\frac{e^{-y^2}\,dy}{z-y} -\frac{\omega_0^2}{\gamma^2}\frac{m}{T} \\ &\qquad\qquad\qquad +\frac{\Omega^2}{(\omega-\gamma v_0)^2}=1, \end{aligned} \right\} \tag{6,7} \]

where \(\omega_0=\left(\dfrac{4\pi e^2 n_0}{m}\right)^{\frac12}\) is the Langmuir frequency of the plasma, and

\[ z=\frac{\omega}{\gamma}\left(\frac{m}{2T}\right)^{\frac12}. \]

The integral entering here,

\[ J(z)=\int_{-\infty}^{\infty}\frac{e^{-y^2}\,dy}{z-y}, \]

can be represented in the form

\[ J(z)=\sqrt{\pi}\,z e^{-z^2}\int_0^1 e^{\xi z^2}\xi^{-\frac12}\,d\xi -\pi i e^{-z^2}\operatorname{Sg}\operatorname{Im} z. \tag{6,8} \]

This relation is very useful for further investigations, since in it one can pass to the limit of real \(z\).

Let us note two limiting cases of relation (6,8).

If \(|z|\gg 1\) and \(\operatorname{Im}z\to -0\), then

\[ J(z)=\frac{\sqrt{\pi}}{z}\left(1+\frac{1}{2z^2}+\frac{3}{4z^4}+\frac{15}{8z^6}+\cdots\right)+\pi i e^{-z^2}. \tag{6,9} \]

If \(|z| \ll 1\), then

\[ J(z)=2\sqrt{\pi}\,z-\pi i\,\operatorname{Sg}\operatorname{Im}z. \tag{6.10} \]

Using (6.8), one may write the dispersion equation (6.7) in the following form:

\[ \frac{1}{a^{2}\gamma^{2}} \left\{ \frac{3}{2}\left(\frac{\omega}{u\gamma}\right)^{2} e^{-\frac{3}{2}\left(\frac{\omega}{u\gamma}\right)^{2}} \int_{0}^{1} e^{\frac{3}{2}\left(\frac{\omega}{u\gamma}\right)^{2}y} y^{-\frac{1}{2}}\,dy + i\left(\frac{3\pi}{2}\right)^{\frac{1}{2}} \frac{\omega}{u\gamma} e^{-\frac{3}{2}\left(\frac{\omega}{u\gamma}\right)^{2}} -1 \right\} + \frac{\Omega^{2}}{(\omega-\gamma v_{0})^{2}} =1, \tag{6.11} \]

where

\[ a=\left(\frac{T}{4\pi n_{0}e^{2}}\right)^{\frac{1}{2}} \]

and

\[ u=\left(\frac{3T}{m}\right)^{\frac{1}{2}} \]

(\(u\) is the mean thermal velocity of the plasma electrons).

1. Let us now proceed to the investigation of the dispersion equation. Assuming the beam density to be sufficiently small, i.e. taking \(\Omega\) to be sufficiently small, we shall seek a root of equation (6.11) close to \(\omega=\gamma v_{0}\). Setting in (6.11)

\[ \omega=\gamma v_{0}+\varepsilon, \]

where \(|\varepsilon|\ll \gamma v_{0}\), we obtain the following expression for \(\Omega^{2}/\varepsilon^{2}\):

\[ \frac{\Omega^{2}}{\varepsilon^{2}} \approx 1- \frac{1}{a^{2}\gamma^{2}} \left[ \frac{3}{2}\left(\frac{v_{0}}{u}\right)^{2} e^{-\frac{3}{2}\left(\frac{v_{0}}{u}\right)^{2}} \int_{0}^{1} e^{\frac{3}{2}\left(\frac{v_{0}}{u}\right)^{2}y} y^{-\frac{1}{2}}\,dy + i\left(\frac{3\pi}{2}\right)^{\frac{1}{2}} \frac{v_{0}}{u} e^{-\frac{3}{2}\left(\frac{v_{0}}{u}\right)^{2}} -1 \right]. \tag{6.12} \]

This expression shows that, if \(\Omega\ll \gamma v_{0}\), then a root close to \(\omega=\gamma v_{0}\) does indeed exist; moreover, since the square of \(\varepsilon\) enters equation (6.12), there always exists a root with \(\operatorname{Im}\omega<0\), corresponding to a wave with increasing amplitude.

Thus we have proved that fluctuations of the field and density propagate in the form of waves with increasing amplitude, and that the initial state of the beam is unstable for all velocities \(v_{0}\).

Expression (6.12) can be considerably simplified in two limiting cases:

\[ v_{0}\ll u \quad\text{and}\quad v_{0}\gg u. \]

In the case when \(v_{0}\ll u\), we obtain, on the basis of (6.10),

\[ \frac{\Omega^{2}}{\varepsilon^{2}} \approx 1+\frac{1}{a^{2}\gamma^{2}} -i\sqrt{\frac{3\pi}{2}}\, \frac{1}{a^{2}\gamma^{2}}\, \frac{v_{0}}{u}, \qquad v_{0}\ll u, \tag{6.13} \]

whence

\[ \varepsilon \approx -\frac{\Omega}{\sqrt{1+(a\gamma)^{-2}}} \left[ 1+\frac{i}{2}\sqrt{\frac{3\pi}{2}}\frac{v_0}{u}\, \frac{(a\gamma)^{-2}}{1+(a\gamma)^{-2}} \right], \qquad \operatorname{Im}\omega<0 . \]

We see that for \(v_0 \ll u\) the frequency shift with respect to \(\gamma v_0\) is mainly real and is of the order of magnitude \(\Omega\). The imaginary part of the frequency, equal to

\[ \Gamma=\frac{1}{2}\sqrt{\frac{3\pi}{2}}\frac{v_0}{u}\,\Omega\, \frac{(a\gamma)^{-2}}{\left[1+(a\gamma)^{-2}\right]^{3/2}}, \]

reaches a maximum at \(a\gamma=1/\sqrt{2}\), which corresponds to the frequency

\[ \omega=\sqrt{\frac{3}{2}}\frac{v_0}{u}\,\omega_0, \]

and the maximum value of \(\Gamma\) is

\[ \Gamma_{\max}=\frac{\sqrt{2\pi}}{6}\frac{v_0}{u}\,\Omega, \qquad v_0\ll u . \tag{6.14} \]

It is considerably smaller than \(\Omega\).

Let us now consider the limiting case \(v_0 \gg u\). Using formula (6.9), we write (6.12) in the form

\[ \frac{\Omega^2}{\varepsilon^2}\simeq 1-\frac{\omega_0^2}{\gamma^2 v_0^2} \left[ 1+\frac{u^2}{v_0^2}+O\left(\frac{u^4}{v_0^4}\right) \right] - i\sqrt{\frac{3\pi}{2}}\frac{v_0}{u}\, e^{-\frac{3}{2}\left(\frac{v_0}{u}\right)^2}. \tag{6.15} \]

Neglecting terms \(O\left(u^4/v_0^4\right)\), we obtain:

\[ \varepsilon=-i\frac{\Omega} {\sqrt{\dfrac{\omega_0^2}{\gamma^2\left(v_0^2-u^2\right)}-1+i\eta}}, \tag{6.16} \]

where

\[ \eta=\sqrt{\frac{3\pi}{2}}\, \frac{v_0}{u(a\gamma)^2}\, e^{-\frac{3}{2}\left(\frac{v_0}{u}\right)^3}. \]

Since \(\eta\ll1\), we may replace (6.16) by the approximate expression

\[ \varepsilon\approx -i\frac{\Omega} {\sqrt{\dfrac{\omega_0^2}{\gamma^2\left(v_0^2-u^2\right)}-1}} . \tag{6.17} \]

Thus, the deviation from the frequency \(\omega=\gamma v_0\) for \(v_0\gg u\) is almost purely imaginary.

Let us note that for \(v_0 \gg u\) the dispersion equation can be approximately represented in the form

\[ \frac{\omega_0^2}{\omega^2-u^2\gamma^2}+\frac{\Omega^2}{(\omega-\gamma v_0)^2}=1, \tag{6,18} \]

where we have neglected, on the left-hand side of this equality, the term

\[ i\sqrt{\frac{3\pi}{2}}\,\frac{\omega}{u\gamma}\,e^{-\frac{3}{2}\left(\frac{\omega}{u\gamma}\right)^2}, \]

which contains an exponential factor, since

\[ \left(\frac{\omega}{u\gamma}\right)^2 \simeq \left(\frac{v_0}{u}\right)^2 \gg 1. \]

Equation (6,18) has the same form as the dispersion equation (5,5), which characterizes the interaction of a beam of charged particles with slow waves in complex waveguides and dielectrics, where the role of the phase velocity is played by the mean thermal velocity of the plasma electrons and the role of the waveguide cutoff frequency by the Langmuir frequency of the plasma \(\omega_0\). It is interesting to note that equation (6,18) (without the term containing the exponential factor) can be obtained if the plasma is described purely hydrodynamically and the pressure is taken to be equal to \(m u^2 n\), where \(n\) is the electron density.

1. Whereas at small beam velocities (\(v_0 \ll u\)) the imaginary part \(\omega\) is very small and is of the order of

\[ \frac{v_0}{u}\,\Omega, \]

at velocities exceeding \(u\), the imaginary part of \(\omega\) can take much larger values. According to (5,7) and (5,8), if the wave vector and the frequency become equal to

\[ \tilde{\gamma}=\frac{\omega_0}{\sqrt{v_0^2-u^2}},\qquad \tilde{\omega}=\frac{\omega_0}{\sqrt{1-\frac{u^2}{v_0^2}}}, \]

the imaginary part of the frequency is equal to

\[ \Gamma=\frac{3^{\frac{1}{2}}}{2^{\frac{4}{3}}} \left(1-\frac{u^2}{v_0^2}\right)^{\frac{1}{6}} \omega_0\left(\frac{\Omega}{\omega_0}\right)^{\frac{2}{3}}. \]

Thus, in the range of velocities exceeding the mean thermal velocity of the plasma electrons, for \(\gamma=\tilde{\gamma}\) and \(\omega=\tilde{\omega}\) the imaginary part of the frequency is proportional not to

\[ \frac{v_0}{u}\,\frac{\Omega}{\omega_0}, \]

as is the case in the range of small velocities, but to

\[ \left(\frac{\Omega}{\omega_0}\right)^{\frac{2}{3}}. \]

It follows from this that ...

the stability of the beam and the occurrence of waves with increasing amplitude become especially significant in the region of velocities exceeding \(u\). These results are valid if, in equation (6,18), the term with the exponential factor is not taken into account and, moreover, pair collisions in the plasma are not considered*).

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*) Phenomenologically, collisions in a plasma can be taken into account if one adds to (6,1) the term \(\dfrac{1}{\tau} f\), where \(\tau\) is the mean time between two collisions. Equation (6,18) then takes the form

\[ \frac{\omega_0^2}{\left(\omega-\frac{i}{\tau}\right)^2-u^2\gamma^2} + \frac{\Omega^2}{(\omega-\gamma v_0)^3} =1. \]

The frequency shift is equal to

\[ \varepsilon = -i\, \frac{\Omega}{ \sqrt{ \dfrac{\omega_0^2}{\left(\gamma v_0-\dfrac{1}{\tau}\right)^2-u^2\gamma^2} -1 } }; \]

for all \(v_0\), it always has a negative imaginary part.

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