PHYSICS

I. A. Gilinskii

INTERACTION OF ELECTRONS WITH THE FIELD OF THE \(H_{01}\) WAVE IN A CIRCULAR WAVEGUIDE

(Presented by Academician M. A. Leontovich, 28 V 1960)

In connection with the problem of generating millimeter waves, it is of interest to investigate the interaction of an electron beam with the field of an unretarded wave in a waveguide. This question was considered by A. V. Gaponov \((^1)\), with the assumption that the deviations of the electron trajectories from the equilibrium ones are small and that the equations of motion can be linearized. In \((^1)\) it was noted that another mechanism of energy transfer by electrons may also occur, associated with nonlinear effects. Such a mechanism is discussed in the present communication for the example of the interaction of a helical beam with the field of the \(H_{01}\) wave in a circular waveguide.

Let the wave propagate along the \(z\)-axis, coinciding with the axis of the waveguide, and let the static magnetic field have components \((0,0,H_0)\). An electron enters the waveguide at the point \((r_0,\varphi_0,0)\) at the instant \(t=0\). The field is determined by the \(\varphi\)-component of the vector potential:

\[ A_\varphi=\frac{1}{2}H_0 r - BgJ_1(gr)\sin(hz-\omega t+\alpha); \tag{1} \]

\(Bg\) is the amplitude of the high-frequency potential; \(h\) is the propagation constant of the (real) wave; \(g\) is the transverse wave number; \(\omega\) is the frequency; \(\alpha\) is the initial phase; \(J\) is the Bessel function of the first order.

Introduce the dimensionless variables

\[ x=gr;\qquad y=gz;\qquad \tau=\Omega t; \]

\[ P_x=gP_r/m\Omega;\qquad P_y=gP_z/m\Omega;\qquad \beta=-P_\varphi g^2/2m\Omega, \]

where \(\Omega=eH_0/mc\) is the cyclotron frequency; \(P_r, P_\varphi, P_z\) are the corresponding generalized momenta; \(\beta=\mathrm{const}\), since the potential does not depend on \(\varphi\). The dimensionless Hamiltonian of the problem has the form

\[ H=\frac{P_y^2}{2}+\frac{P_x^2}{2}+\frac{x^2}{8}-\frac{2\beta^2}{x^2}-\beta+\varepsilon f(x)\sin\left(\frac{h}{g}y-\frac{\omega}{\Omega}\tau+\alpha\right)+\ldots; \tag{2} \]

\[ f(x)=(2\beta/x-x/2)J_1(x);\qquad \varepsilon=Bg^2/H_0 \]

is a small parameter of the problem (\(Bg^2\) is the amplitude of the \(z\)-component of the high-frequency field). The term with \(\varepsilon^2\) is omitted, since it gives no contribution in the first approximation of perturbation theory.

There is an exact integral of the motion

\[ H_1=H-\frac{\omega g}{\Omega h}P_y=\mathrm{const}, \tag{3} \]

which is easily obtained by passing to new variables \(u, P_u\) with the aid of the generating function

\[ F=xP'_x+\left(y-\frac{\omega g}{\Omega h}\tau+\frac{g}{h}\alpha\right)P_u. \]

The solution of the unperturbed system \((\varepsilon=0)\) has the form:

\[ x=2\{W+\beta+\sqrt{W(W+2\beta)}\sin\psi\}^{1/2}, \tag{4} \]

\[ \dot{x}=\frac{\partial x}{\partial\psi} =\frac{\sqrt{W(W+2\beta)}\cos\psi} {\{W+\beta+\sqrt{W(W+2\beta)}\sin\psi\}^{1/2}}; \]

\[ P_y=\mathrm{const};\qquad \psi=\tau+\mathrm{const}. \tag{4'} \]

Here \(W=P_x^2/2+x^2/8+2\beta^2/x^2-\beta=\mathrm{const}\) is the transverse kinetic energy, equal to \(\tfrac12 x_c^2\); \(x_c\) is the dimensionless cyclotron radius of the electron orbit;

\[ \beta=\frac{X_o^2-x_c^2}{4}; \tag{5} \]

\(X_o\) is the dimensionless distance from the waveguide axis to the center of the orbit.

For \(\varepsilon\ne0\), in accordance with (2), we pass to the new variables \(W,\psi\) by formulas (4). The equations of motion in the new variables are obtained by differentiating the Hamiltonian \(H_1\). Next, we expand \(f[x(W,\psi)]\) in a Fourier series in \(\psi\):

\[ f=\sum_{k=0}^{\infty} a_k(W)\sin k\psi+\sum_{k=0}^{\infty} b_k(W)\cos k\psi \tag{6} \]

and consider resonance at the first harmonic. Using the averaging method (3) and taking into account that \(b_1(W)=0\), we obtain the equations of the first approximation in the form:

\[ \dot{W}=-\frac{\varepsilon}{2}a_1(W)\sin\theta;\qquad \dot{P}_u=-\frac{\varepsilon}{2}\frac{h}{g}a_1(W)\sin\theta; \tag{7} \]

\[ \dot{\theta}=1+\frac{h}{g}P_u-\frac{\omega}{\Omega} -\frac{\varepsilon}{2}\frac{da_1}{dW}\cos\theta;\qquad \theta=\psi+\frac{h}{g}u, \]

where

\[ a_1(W)=\frac{1}{\pi}\int_0^{2\pi} f[x(W,\psi)]\sin\psi\,d\psi . \]

From the first and second equations (7) it follows that

\[ W-\frac{g}{h}P_u=W_0-\frac{g}{h}P_u^0 . \tag{8} \]

We obtain another integral of system (7) by averaging (3):

\[ \frac{P_u^2}{2}-\frac{\omega g}{\Omega h}P_u+W -\frac{\varepsilon}{2}a_1(W)\cos\theta = \frac{P_u^{0\,2}}{2} -\frac{\omega g}{\Omega h}P_u^0+W_0 -\frac{\varepsilon}{2}a_1(W_0)\cos\theta_0 . \tag{9} \]

The investigation shows that the greatest transfer of energy takes place in the case of a strongly off-axis beam. In this case \(X_o\gg x_c\), \(W/2\beta\ll1\), and, expanding in powers of \(\sqrt{W/2\beta}\), we obtain

\[ a_1(W)\simeq C\sqrt{2W};\qquad C=-\sqrt{\beta}\,[J_2(2\sqrt{\beta})+J_0(2\sqrt{\beta})]; \tag{10} \]

\[ \dot{W}= \left\{ \frac{\varepsilon^2 C^2}{2}(W-W_0) +\frac{\varepsilon^2 C^2}{2}W_0 - \left[ \frac{h^2}{2g^2}(W-W_0)^2 +\Delta_0(W-W_0) +\frac{\varepsilon C}{2}\sqrt{2W_0}\cos\theta_0 \right]^2 \right\}^{1/2}. \tag{11} \]

\[ \Delta_0 = 1 \pm \frac{h}{g} P_u^0 - \frac{\omega}{\Omega} \]
is the initial detuning. The sign before the radical is determined by the value of \(\dot W|_{\tau=0}\).

Solving (11), one can, with the aid of (8) and (9), find the loss of the electron’s total kinetic energy \(\Delta E\) during the transit time,
\[ \Delta E = \frac{\omega}{\Omega}\Delta W. \]
If the relative energy transfer \((W-W_0)/W_0\) is small, then for the energy transfer averaged over the initial phases we have

\[ \overline{W-W_0} = -\frac{g^2}{h^2}\Delta_0 + \frac{g^2}{h^2}\frac{\Delta_0}{2\pi} \int_0^{2\pi} \frac{ \operatorname{dn}\left(\dfrac{h^2 b\tau}{2g^2},\,k\right)\,d\theta_0 }{ 1-k^2\cos^2\dfrac{\theta_0}{2}\, \operatorname{sn}^2\left(\dfrac{h^2 b\tau}{2g^2},\,k\right) }, \tag{12} \]

where

\[ b^2 = \frac{g^4}{h^4}\Delta_0^2 + \left| \frac{2eg^2}{h^2} C\sqrt{2W_0} \right| \cos^2\frac{\theta_0}{2}; \qquad k = \frac{ \left| \dfrac{2eg^2}{h^2} C\sqrt{2W_0} \right| }{b^2}. \]

\(\operatorname{dn}\), \(\operatorname{sn}\) are Jacobi elliptic functions.

The energy loss depends substantially on the magnitude and sign of the initial detuning. For \(\Delta_0=0\), \(\overline{W-W_0}=0\). If \(\Delta_0>0\), then \(\overline{W-W_0}<0\), i.e., the unbunched electrons on average give up energy. If, however, \(\Delta_0<0\), then \(\overline{W-W_0}>0\).

The case
\[ \frac{g^4\Delta_0^2}{h^4} \gg \left| \frac{2eg^2}{h^2} C\sqrt{2W_0} \right| \]
corresponds to the linear regime considered in (1). The optimal detuning is
\[ \Delta_0 \sim \frac{h^2}{g^2} \left| \frac{2eg^2}{h^2} C\sqrt{2W_0} \right|^{1/2}. \]
In this case the energy lost can be estimated by numerical integration of expression (12). Taking, for example, the waveguide radius \(a\sim 3\) cm, \(\omega=6\cdot 10^{10}\ \mathrm{s}^{-1}\), \(g/h=1\), \(\Delta_0^2=5\cdot 10^{-8}\),
\[ \left| \frac{2eg^2}{h^2} C\sqrt{2W_0} \right| = 10^{-7}, \]
the transverse velocity \(v_0=2\cdot 10^9\) cm/s, the longitudinal velocity \(v_z^0=3.5\cdot 10^8\) cm/s, and the transit length \(L=120\) cm, we find that
\[ \overline{W-W_0}/W_0 \sim 16\%. \]

The author expresses deep gratitude to P. A. Borodovskii, who drew his attention to the questions considered in the present communication, and to Yu. B. Rumer and V. L. Pokrovskii for discussion.

Institute of Radiophysics and Electronics
Siberian Branch of the Academy of Sciences of the USSR

Received
25 V 1960

CITED LITERATURE

1. A. V. Gaponov, Izv. vysshikh uchebn. zav., Radiophysics, 2, 3, 450 (1959).
2. B. V. Chirikov, DAN, 125, No. 5, 1015 (1959).
3. N. N. Bogolyubov, Yu. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations, Moscow, 1958.