Physics

G. V. Voskresenskii, B. M. Bolotovskii

RADIATION OF A POINT CHARGED PARTICLE FLYING ALONG THE AXIS OF A SEMI-INFINITE CIRCULAR WAVEGUIDE

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

1. Consider a circular waveguide of radius $a$ with ideally conducting, infinitely thin walls. The waveguide is open at one end. For consideration of the problem it is convenient to use a cylindrical coordinate system $r,\varphi,z$, aligning the $z$ axis with the axis of the waveguide. We shall assume that the position of the waveguide walls is determined by the equations $r=a$, $z>0$.

Let a point particle of charge $q$ move along the axis of the waveguide with velocity $u$. The problem is to determine the radiation arising when the particle enters the waveguide ($u>0$) or when it exits from the waveguide ($u<0$).

2. We shall describe the field excited by the charge by the Hertz vector $\vec{\Pi}$. Let us represent $\vec{\Pi}$ in the form of a sum

\[ \vec{\Pi}=\vec{\Pi}^{\,0}+\vec{\Pi}^{\,1}, \tag{1} \]

where $\vec{\Pi}^{\,0}$ describes the field of a point charge moving in empty space, and $\vec{\Pi}^{\,1}$ describes the free field, which is to be determined with the aid of boundary conditions.* The function $\vec{\Pi}^{\,0}$ is determined as the solution of the inhomogeneous d’Alembert equation and is written in the form

\[ \Pi^{0}=\frac{iq}{\pi}\int_{-\infty}^{\infty} e^{i\frac{\omega}{u}(z-ut)}K_{0}(k\gamma r)\,\frac{d\omega}{\omega}, \tag{2} \]

where the following notation has been introduced:

\[ \gamma=\frac{\sqrt{1-\beta^{2}}}{|\beta|},\qquad \beta=\frac{u}{c},\qquad k=\frac{|\omega|}{c}. \tag{3} \]

The free field, determined by the function $\Pi^{1}$, is sought in the form

\[ \Pi^{1}=\int_{-\infty}^{\infty}\Pi_{\omega}^{1}e^{-i\omega t}\,d\omega. \tag{4} \]

In what follows we shall omit the index $\omega$ on the function $\Pi_{\omega}^{1}(r,z)$. It is convenient to represent the function $\Pi^{1}$ in the form of an expansion in a Fourier integral with respect to the variable $z$, writing this expansion in the following way:

\[ \Pi^{1}(r,z)=-\frac{2\pi^{2}a}{\omega}\int_{-\infty}^{\infty}F(w) \left\{ \begin{array}{l} J_{0}(va)\,H_{0}^{(1)}(vr)\\ J_{0}(vr)\,H_{0}^{(1)}(va) \end{array} \right\} e^{iwz}\,dw, \tag{5} \]

where $v=\sqrt{k^{2}-w^{2}}$ ($\operatorname{Im} v\geq 0$). For $r>a$ in the expression for $\Pi^{1}$ one should take the upper line in the braces under the integral sign, and for $r<a$ the lower line. The function $F(w)$, which remains to be determined, has the meaning of the Fourier component of the current induced by the moving charged par-

\[ \text{* The symmetry of the problem makes it possible to choose both vectors directed along the } z\text{ axis.} \]

… flowing on the walls of the semi-infinite waveguide:

\[ j(z)=\int_{-\infty}^{\infty} F(w)e^{iwz}\,dw . \tag{6} \]

As the boundary conditions of the problem we shall take the vanishing of the current density \(j(z)\) on the extension of the waveguide walls \((r=a,\ z<0)\) and the vanishing of the tangential component of the total electric field on the waveguide walls \((r=a,\ z>0)\). These two conditions lead to the following system of paired integral equations for the function \(F(w)\):

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

\[ \int_{-\infty}^{\infty} F(w)L(w)e^{iwz}\,dw = -\frac{iqk^{2}\gamma^{2}}{2\pi^{2}}K_{0}(k\gamma a)e^{i\frac{\omega}{u}z} \qquad \text{for } z>0, \tag{7} \]

where

\[ L(w)=\pi a v^{2}J_{0}(va)H_{0}^{(1)}(va). \tag{8} \]

1. We shall solve the system of equations (7) with kernel (8) by the Wiener–Hopf method, analogously to how this was done by L. A. Vainshtein \((^{1})\) for the case of diffraction of electromagnetic waves at the open end of a waveguide. We factor the kernel \(L(w)\):

\[ L(w)=v\varphi_{1}(w)\varphi_{2}(w), \tag{9} \]

where the function \(\varphi_{1}(w)\) is analytic and has no zeros in the upper half-plane of the complex variable \(w\), while the function \(\varphi_{2}(w)\) has the same properties in the lower half-plane \(w\). The explicit form of the functions \(\varphi_{1}\) and \(\varphi_{2}\) is given in \((^{1,2})\). They are connected by the relation \(\varphi_{1}(w)=\varphi_{2}(-w)\). The solution of the system of equations (7) has the form

\[ F(w)=-\frac{q}{8\pi^{3}aiI_{0}(k\gamma a)} \frac{\sqrt{k-\omega/u}\,\varphi_{2}(\omega/u)}{\sqrt{k-w}\,\varphi_{2}(w)} \frac{1}{w-\omega/u}. \tag{10} \]

Here the pole \(w=\omega/u\) should be regarded as lying above the real axis.

1. Let us now consider the radiation field inside the waveguide. Substitution of (10) into (5) gives

\[ \Pi^{1}= \frac{q\sqrt{k-\omega/u}\,\varphi_{2}(\omega/u)} {4\pi i\omega I_{0}(k\gamma a)} \int_{-\infty}^{\infty} \frac{J_{0}(vr)H_{0}^{(1)}(va)e^{iwz}} {\sqrt{k-w}(w-\omega/u)\varphi_{2}(w)}\,dw . \tag{11} \]

To compute the field inside the waveguide \((z>0,\ r<a)\), it is important to know the behavior of the integrand in the upper half-plane of the complex variable \(w\). The only singularities of the integrand in the upper half-plane are the poles located at the zeros of the function \(\varphi_{2}(w)\) and at the point \(w=\omega/u\). The integral (11) is readily evaluated by means of the residue theorem, which gives

\[ \Pi^{1}(r,z)= \frac{q}{\pi i\omega} \frac{K_{0}(k\gamma a)}{I_{0}(k\gamma a)} I_{0}(k\gamma r)e^{i\frac{\omega}{u}z} + \]

\[ + \frac{q\sqrt{k-\omega/u}\,\varphi_{2}(\omega/u)} {2\omega I_{0}(k\gamma a)} \sum_{m} \frac{J_{0}(v_{m}r)H_{0}^{(1)}(v_{m}a)e^{iw_{m}z}} {\sqrt{k-w_{m}}\,\varphi_{2}'(w_{m})(w_{m}-\omega/u)} . \tag{12} \]

The first term in (12), together with \(\Pi^{0}\), gives the field of a charge moving along the axis of an infinite circular waveguide. The sum over \(m\) gives the radiation into the waveguide caused by the entry or exit of the charge. The spectral density

of radiation into the waveguide at frequency \(\omega\) is

\[ W_\omega = \frac{q^2}{\pi \omega}\, \frac{|k-\omega/u|\,|\varphi_2(\omega/u)|^2}{I_0^2(k\gamma a)} \sum_m \frac{\mathfrak{w}_m}{(k-\mathfrak{w}_m)\,|\varphi'_2(\mathfrak{w}_m)|^2(\mathfrak{w}_m-\omega/u)^2}. \tag{13} \]

The summation is carried out over those waves for which the longitudinal wave number \(\mathfrak{w}_m\) is real.

1. Let us consider the radiation field outside the waveguide. We pass to spherical coordinates \(z=R\cos\theta\), \(r=R\sin\theta\). Evaluating expression (11) by the saddle-point method for large values of \(R\), we obtain

\[ \Pi^1 = -\frac{q}{2\pi\omega I_0(k\gamma a)} \frac{e^{ikR}}{R} \frac{\varphi_2(\omega/u)}{\varphi_2(k\cos\theta)} \frac{\sqrt{k-\omega/u}}{\sqrt{k-k\cos\theta}}\, \frac{J_0(ka\sin\theta)}{k\cos\theta-\omega/u}. \tag{14} \]

The intensity of radiation at frequency \(\omega\) into the solid angle \(d\Omega\) is

\[ W_\omega(\theta)\,d\Omega = \frac{q^2 |u|(1-\beta)\,|\varphi_2(\omega/u)|^2}{4\pi^2 c^2 I_0^2(k\gamma a)} \frac{J_0^2(ka\sin\theta)\sin^2\theta\,d\Omega} {(1-\beta\cos\theta)^2(1-\cos\theta)|\varphi_2(k\cos\theta)|^2}. \tag{15} \]

For small charge velocities the radiation spectrum lies in the region of low frequencies satisfying the inequality

\[ k\gamma a=\frac{\omega}{u}\,a<1. \tag{16} \]

For higher frequencies the radiation intensity decreases exponentially.

For large charge velocities \((\beta\simeq 1,\ \gamma\to 0)\), the radiation spectrum lies in the region of frequencies satisfying the inequality

\[ \omega<\frac{u}{\sqrt{1-\beta^2}\,a}=\frac{c}{\gamma a}. \tag{17} \]

For high frequencies one may use the circumstance that the function \(\varphi_2\), for large values of its argument, tends to unity. For definiteness let us consider the case of the charge emerging from the waveguide \((u<0)\) and take into account that the radiation of a fast charge is concentrated in the range of angles \(\pi-\theta\simeq \gamma\). In this case

\[ W_\omega(\theta)= \frac{q^2}{4\pi^2 c}\, \frac{\sin^2\theta}{(1-\beta\cos\theta)^2}. \tag{18} \]

The expression obtained coincides with the expression for the intensity of transition radiation when a charge emerges from a plane metal boundary \({}^{3}\). The expression for the radiation losses when a charge passes along the axis of a circular aperture in an ideally conducting plane screen \({}^{4,5}\) is also brought to an analogous form.

An estimate of the total losses to radiation into free space when a relativistic charge emerges from the waveguide gives

\[ W=\frac{2q^2}{\pi\gamma a}. \tag{19} \]

The geometrical-optics approximation, which is applicable as \(\gamma\to 0\), leads to the same qualitative results.

Received
29 X 1963

References Cited

\({}^{1}\) L. A. Vainshtein, Diffraction of Electromagnetic and Sound Waves at the Open End of a Waveguide, Moscow, 1953.
\({}^{2}\) B. Noble, Wiener–Hopf Method, IL, 1962.
\({}^{3}\) V. L. Ginzburg, I. M. Frank, ZhETF, 16, 15 (1946).
\({}^{4}\) V. I. Bobrinev, V. V. Braginskii, DAN, 123, No. 4, 634 (1958).
\({}^{5}\) Yu. N. Dnestrovsky, D. P. Kostomarov, DAN, 124, No. 4, 792 (1959); 124, No. 5, 1026 (1959).