PHYSICS

G. V. VOSKRESENSKII, B. M. BOLOTOVSKII

THE FIELD OF A CHARGED FILAMENT MOVING UNIFORMLY NEAR A SYSTEM OF PERFECTLY CONDUCTING HALF-PLANES

(Presented by Academician M. A. Leontovich, 25 I 1964)

Radiation of charged particles in linear periodic media has previously been considered by approximate methods by a number of authors \((^{1-5})\). It is of interest to consider one of the problems of this type that admits an exact solution.

Let a uniformly charged filament, parallel to the \(x\)-axis, move with constant velocity \(u_z=u\) along a system of parallel perfectly conducting half-planes described by the equation \(z=na\) \((n=0,\pm1,\pm2,\ldots)\), \(y>0\). The position of the filament in space is specified by the relations \(y=-b,\ z=ut\). Denote the charge per unit length of the filament by \(q\). In the problem under consideration the polarization is such that the magnetic field has only one component \(H_x\), while the electric vector lies in the plane normal to the charged filament \((E_y,E_z)\). It is convenient to describe this field by the Hertz vector \(\vec{\Pi}\), through which the fields are expressed by the well-known formulas
\[ \mathbf{E}=\left(\operatorname{grad}\operatorname{div}-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)\vec{\Pi},\qquad \mathbf{H}=\frac{1}{c}\frac{\partial}{\partial t}\operatorname{rot}\vec{\Pi}. \]
We shall expand all quantities in a Fourier integral with respect to time; the subsequent consideration refers to the Fourier components corresponding to the frequency \(\omega\). Represent the Hertz vector \(\vec{\Pi}_\omega\) in the form of the sum
\[ \vec{\Pi}_\omega=\vec{\Pi}^{\,0}_\omega+\vec{\Pi}^{\,1}_\omega, \tag{1} \]
where
\[ \Pi^0_\omega=\Pi^0_{\omega z} = -\frac{q}{i\omega k\gamma} e^{-k\gamma |y+b|+i\frac{\omega}{u}z} \qquad \left( k=\frac{|\omega|}{c},\ \gamma=\frac{1}{\beta}\sqrt{1-\beta^2},\ \beta=\frac{u}{c} \right) \tag{2} \]
is the Hertz vector describing the field of the charged filament moving in free space, and \(\vec{\Pi}^{\,1}_\omega\) describes the free field that must be added to \(\vec{\Pi}^{\,0}_\omega\) in order to satisfy the boundary conditions on the metallic plates. The vector \(\vec{\Pi}^{\,1}_\omega\) can be expressed in terms of the currents flowing on the plates,
\[ \Pi^1_{\omega y}(y,z) = \frac{i}{\omega} \sum_{m=-\infty}^{\infty} \int_0^\infty \int_{-\infty}^{\infty} \frac{e^{ikR_m}}{R_m} j_m(\eta)\,d\eta\,d\xi \tag{3} \]
\[ \left( R_m=\sqrt{(x-\xi)^2+(y-\eta)^2+(z-am)^2} \right). \]

Using the relation
\[ j_{\omega m}=e^{i\frac{\omega}{u}am}j_{\omega0}, \]
which connects the Fourier components of the current \(j_m(y)\) on plate number \(m\) and the current \(j_0(y)\) for the plate \(z=0,\ y>0\), and also representing the current \(j_0(y)\) in the form of the Fourier integral
\[ j_0(y)=\int_{-\infty}^{\infty} F(w)e^{iwy}\,dw, \tag{4} \]

we obtain

\[ \Pi_{\omega y}^{1}=-\frac{2\pi}{\omega}\int_{-\infty}^{\infty}F(w)\frac{e^{iwy}}{v}\sum_{m=-\infty}^{\infty}e^{iv|z-am|+i\frac{\omega}{u}am}\,dw . \tag{5} \]

The sum over \(m\) is evaluated if we restrict ourselves to considering the range of values \(z\) satisfying the inequalities

\[ na \leq z \leq (n+1)a . \tag{6} \]

Summation gives

\[ \Pi_{\omega}^{1}(y,z)= \tag{7} \]

\[ =-\frac{2\pi i}{\omega}\int_{-\infty}^{\infty}F(w) \frac{\sin v[z-a(n+1)]-e^{i\frac{\omega}{u}a}\sin v(z-an)} {\cos av-\cos a\frac{\omega}{u}}\, e^{i\frac{\omega}{u}an+iwy}\frac{dw}{v}, \]

\[ v=\sqrt{k^{2}-w^{2}},\qquad \operatorname{Im}v \geq 0 . \]

Requiring that the tangential component of the total electric field vanish on the plates (for \(z=an,\ y>0\)) and that there be no current on the continuations of the plates (\(y<0\)), we arrive at a system of paired integral equations for the Fourier amplitude of the current \(F(w)\):

\[ \int_{-\infty}^{\infty}F(w)e^{iwy}\,dw=0 \qquad \text{for } y<0, \]

\[ \int_{-\infty}^{\infty}F(w)L(w)e^{iwy}dw =\frac{q\omega}{2\pi iu}e^{-k\gamma(y+b)} \qquad \text{for } y>0, \tag{8} \]

where

\[ L(w)=\frac{v\sin va}{\cos av-\cos a\frac{\omega}{u}} =\frac{2}{a}v^{2}\frac{\sin va}{va} \frac{\frac{a}{2}\left(v-\frac{\omega}{u}\right)\frac{a}{2}\left(v+\frac{\omega}{u}\right)} {\sin\frac{a}{2}\left(v-\frac{\omega}{u}\right)\sin\frac{a}{2}\left(v+\frac{\omega}{u}\right)} \frac{1}{w^{2}+k^{2}\gamma^{2}} . \tag{9} \]

The solution of system (8) can be obtained by the Wiener–Hopf method, analogously to how this was done in papers \((^{6-9})\). Let us represent \(L(w)\) in (9) in the form

\[ L(w)=\frac{2}{a}v^{2}\frac{L_{1}(w)L_{2}(w)}{w^{2}+k^{2}\gamma^{2}}, \tag{10} \]

where \(L_{1}(w)\) is holomorphic in the upper half-plane of the complex variable \(w\) and has no zeros there, while \(L_{2}\) is a function possessing the same properties in the lower half-plane. Then the solution of the system of equations (8) has the form

\[ F(w)=\frac{aq\omega\gamma}{4\pi^{2}iu(1+i\gamma)L_{1}(ik\gamma)} e^{-k\gamma b}\, \frac{1}{(k-w)L_{2}(w)} . \tag{11} \]

The functions \(L_{1}(w)\) and \(L_{2}(w)\), entering into the factorization (10), can be represented in the form of infinite products as follows:

\[ \begin{aligned} L_{1}(w)=& \prod_{1}^{\infty} \left(\frac{wa}{n\pi}+i\sqrt{1-\left(\frac{ka}{n\pi}\right)^{2}}\right) e^{i\frac{aw}{n\pi}} \times\\ &\times \left\{ \prod_{-\infty}^{-1} \left[ \frac{wa}{2n\pi} -i\sqrt{\left(1-\frac{\omega a}{u2n\pi}\right)^{2} -\left(\frac{ka}{2n\pi}\right)^{2}} \right] e^{\frac{a}{2n\pi}\left(\frac{\omega}{u}-iw\right)} \times \right.\\ &\left. \times \prod_{1}^{\infty} \left[ \frac{wa}{2n\pi} +i\sqrt{\left(1-\frac{\omega a}{u2n\pi}\right)^{2} -\left(\frac{ka}{2n\pi}\right)^{2}} \right] e^{\frac{a}{2n\pi}\left(\frac{\omega}{u}+iw\right)} \right\}^{-1}, \tag{12} \end{aligned} \]

\[ L_{2}(w)=-L_{1}(-w). \]

The relations (9)—(12) obtained completely determine the sought function \(F(w)\), and hence also the radiation field of the charged filament. It may be noted that the solution of the problem of exciting the system of plates under consideration by a plane charge distribution modulated according to a harmonic law has an analogous form. In that case the quantity \(q\) is proportional to the modulation coefficient.

The only nonzero component of the radiated magnetic field, \(H_x^1\), is determined (for the strip \(an<z<a(n+1)\)) by the formula

\[ H_x^1 = -\frac{qk\gamma e^{-k\gamma b+i\frac{\omega}{u}an}} {\pi i u(1+i\gamma)L_1(ik\gamma)} \int_{-\infty}^{\infty} \frac{\cos v[z-a(n+1)]-e^{i\frac{\omega}{u}a}\cos v(z-an)} {k-w} \times \frac{v}{\sin va}\, \frac{L_1(w)e^{iwy}}{w^2+k^2\gamma^2}\,dw . \tag{13} \]

The remaining components of the radiation field are readily determined from (13) by Maxwell’s equations.

The integral representation of the radiation field (13) can be reduced to a representation as a series of residues at the singularities of the integrand. In the problem under consideration the field has a different character in the region between the metal plates (\(y>0\)) and in the free half-space (\(y<0\)). For \(y>0\) the quantity \(H_x^1\) is determined by the simple poles of the integrand in (13) in the upper half-plane of the complex variable \(w\):

\[ w_0=ik\gamma,\qquad w_{01}=k,\qquad w_m=\sqrt{k^2-\left(\frac{m\pi}{a}\right)^2} \quad (m=1,2,\ldots). \tag{14} \]

The residue at the pole \(w_0=ik\gamma\) cancels the contribution of \(H_x^0\) in the expression for the total field, which is thus represented as a superposition of the fundamental wave of TEM type (corresponding to the pole \(w_{01}\)) and symmetric electric waves (corresponding to the zeros of the function \(\sin va\)),

\[ H_x=R_0e^{iky}+\sum_m R_m\cos\frac{m\pi}{a}z\,e^{iw_my}. \tag{15} \]

A TEM-type wave is excited at arbitrarily small frequencies and propagates between the plates with the velocity of light. The excitation coefficients of the indicated waves are determined by the formulas

\[ R_0 = -\frac{2q\gamma u e^{-k\gamma b}}{(1+i\gamma)kac^2} \frac{L_1(k)}{L_1(ik\gamma)} e^{i\frac{\omega}{u}an} \left(1-e^{i\frac{\omega}{u}a}\right), \]

\[ R_m = \frac{2qk\gamma e^{-k\gamma b}}{(1+i\gamma)uaL_1(ik\gamma)} e^{i\frac{\omega}{u}an} \frac{\left(\frac{m\pi}{a}\right)^2 L_1(w_m)(-1)^{m(n-1)} \left[(-1)^m-e^{i\frac{\omega}{u}a}\right]} {(k-w_m)w_m(w_m^2+k^2\gamma^2)} . \tag{16} \]

The excitation coefficient \(R_0\) of the TEM-type wave contains the factor \(\left(1-e^{i\frac{\omega}{u}a}\right)\), which vanishes when \(\frac{\omega}{u}a=2n\pi\); factors of the same type are also present in the expressions for the other excitation coefficients. The vanishing of the excitation coefficient of the fundamental wave at the frequencies

\[ \omega=\frac{u}{a}\,2n\pi \]

has a simple physical meaning: if the time taken by the source to pass through one period of the structure is equal to, or is a multiple of, the period of the wave, then such a wave cannot be excited, since the work done by it on the source over a period of the structure is zero. From formulas (15), (16) it is easy to see that the fields in neighboring plane “waveguides,” as well as the currents on neighboring plates, differ—

are reduced to the phase factor \(e^{i\frac{\omega}{u}a}\). The total energy flux radiated into the “waveguide” can be obtained by calculating the integral

\[ W_{\omega}=c\int_{na}^{(n+1)a} E_{\omega z}H_{-\omega x}\,dz =\frac{ac}{2k}\sum_m w_m |R_m|^2 , \tag{17} \]

where the summation extends only over those \(m\) for which the longitudinal wave number \(w_m\) is real at the given frequency \(\omega\).

The radiation field of the filament into the free half-space \((y<0)\) is determined by the poles of the integrand in (13) that lie in the lower half-plane of the complex variable \(w\):

\[ \hat w_0=-ik\gamma,\qquad \hat w_m=\sqrt{k^2-\left(\frac{2\pi m}{a}-\frac{\omega}{u}\right)^2} \quad (m=\pm1,\pm2,\ldots). \tag{18} \]

Evaluating the integral by means of residues, we obtain for the radiation field*

\[ H_x^{1}= -\frac{qce^{-k\gamma b}L_1(-ik\gamma)} {u^2(1+i\gamma)^2L_1(ik\gamma)} \,e^{k\gamma y+i\frac{\omega}{u}z} - \]

\[ -\frac{2iqk\gamma e^{-k\gamma b}}{u(1+i\gamma)L_1(ik\gamma)} \sum_m \frac{\operatorname{Res}L_1(\hat w_m)} {\hat w_m^2+k^2\gamma^2} \frac{\frac{\omega}{u}-\frac{2\pi m}{a}} {k-\hat w_m} e^{i\hat w_m y+i\left(\frac{\omega}{u}-\frac{2\pi m}{a}\right)z}. \tag{19} \]

The first term (corresponding to the pole \(w=\hat w_0\)) determines a surface wave, whose electromagnetic field propagates with the velocity of the source and decays exponentially with distance from the edges of the plates; the terms in the sum over \(m\) correspond to the poles \(\hat w_m\). For real values of \(\hat w_m\) they describe plane electromagnetic waves radiated by the source as it moves along the structure. For each such wave, the projection of the wave vector onto the axis is determined by the evident equality

\[ k_{zm}=k\cos\theta_m=\frac{\omega}{u}-\frac{2\pi m}{a}, \tag{20} \]

where \(\theta_m\) is the angle between the direction of propagation of the wave and the \(z\)-axis. From the last equality we obtain the following relation for the radiation frequency:

\[ \nu=\frac{\omega}{2\pi}=\frac{mu}{a} \left(1-\frac{u}{c}\cos\theta_m\right)^{-1}. \tag{21} \]

Here \(\nu_0=u/a\) gives the number of plates passed by the source per unit time (the “transit frequency”), while the factor in parentheses gives the characteristic Doppler frequency shift, dependent on the observation angle.

The solution of the problem of the field excited by a current filament uniformly flying along the diffraction grating considered differs, generally speaking, only in the type of polarization of the electromagnetic radiation field and is carried out in a completely analogous manner.

Received
29 X 1963

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* The expression (19) does not contain the plate number \(n\), and thus represents the field in the lower half-space for arbitrary values of \(z\).