STUDY OF UNIFORMLY BENT PERIODICALLY DIAPHRAGMED WAVEGUIDES FOR A CYCLIC ELECTRON ACCELERATOR

B. I. Morozov, V. M. Marenkov, B. G. Tsikin, L. G. Shishenina

The model of a waveguide cyclic electron accelerator \[1\] is intended to be built on the basis of the magnet of one of the betatrons for 15 MeV, i.e., with an external controlling magnetic field and preliminary betatron acceleration of the electrons to an energy of 2–3 MeV. The chamber of such an accelerator is a uniformly bent waveguide that provides a slowing of the phase velocity of the electromagnetic wave to the speed of light at the equilibrium radius \(R_0\), with an azimuthal component of the electric field \(E_\varphi\) sufficient for capture and acceleration of the electrons. For the model of a waveguide cyclic accelerator one may use a waveguide of rectangular cross section, periodically loaded with diaphragms along the bent walls (Fig. 1).

Fig. 1. General view of the diaphragmed waveguide.

The principal geometrical dimensions of the chamber are: the equilibrium radius \(R_0\), determined by the magnetic system of the betatron (for betatrons with energy 15 MeV, \(R_0\) varies within the limits 12–14 cm); the size \(2a\) of the interaction space, determined by the betatron oscillations of the electrons (of the order of 4–6 cm); the height of the waveguide \(b\), the maximum value of which is limited by the interpole space of the magnet (not more than 7 cm, allowing for the wall thickness).

With the chosen ten-centimeter wavelength range of the high-frequency generator, the slowing system of the waveguide accelerator will be a waveguide with strong bending (azimuthal field periodicity \(n=7\text{--}9\)). In this case [2] the dispersion properties of the waveguide may differ by 20–25% from the properties of an unbent slowing system of the “comb” type [3, 4], i.e., a rigorous treatment of the problem of wave propagation in a bent waveguide is necessary.

Most linear waveguide electron accelerators, for a number of reasons [5], operate with the \(\pi/2\)-type of high-frequency oscillations. With the same phase shift between adjacent cells, operation of a waveguide cyclic accelerator in the standing-wave regime is assumed. Therefore the results of solving the problem in the impedance approximation, i.e., under the assumption that an infinite number of thin diaphragms falls within one wavelength, may give discrepancies of the order of 10–15% in comparison with the real system [3, 4]. In application to electron devices of the LBE type, the indicated inaccuracy may prove insignificant, since there is always the possibility of achieving synchronism of the motion of the electrons and the wave by a corresponding change in the voltages on the control electrodes, which is not the case in accelerators.

Derivation of the dispersion equation

A rigorous solution of the problem of wave propagation in bent diaphragm-loaded waveguides [6] leads to an equation in the form of an infinite determinant, which makes it difficult to calculate the dispersion properties. It is known that the calculation of periodic structures is facilitated if, in a certain way, the field distribution is specified at the boundary separating the interaction space and the resonator space. The simplest approximation, widely used in magnetron theory [7, 8, 9], is \(E_\varphi=\mathrm{const}\). In this case all harmonics in the interaction space are taken into account.

Equating the input admittances

\[ Y=-\frac{\displaystyle \int\limits_{0}^{\varphi} E_\varphi^{*}H_z\,r\,d\varphi} {\displaystyle \left[\int\limits_{0}^{\varphi} E_\varphi\,r\,d\varphi\right]^{*}} \tag{1} \]

at the boundary gives the dispersion equation.

Let us determine the input admittance of the interaction space, where the field components have the form [6]:

\[ E_r=-\omega\mu\,\frac{1}{r}\sum\left[A_s J_{n_s}^{s}(\chi r)+B_s N_{n_s}^{s}(\chi r)\right] n_s \sin \gamma z\,e^{j n_s\varphi}; \]

\[ E_\varphi=-j\omega\mu\sum\left[A_s J'_{n_s}(\chi r)+B_s N'_{n_s}(\chi r)\right]\sin\gamma z\,e^{j n_s\varphi}; \]

\[ E_z=0; \]

\[ H_r=\gamma\chi\sum\left[A_s J'_{n_s}(\chi r)+B_s N'_{n_s}(\chi r)\right]\cos\gamma z\,e^{j n_s\varphi}; \tag{2} \]

\[ H_\varphi=j\,\frac{1}{r}\gamma\sum\left[A_s J_{n_s}(\chi r)+B_s N_{n_s}(\chi r)\right] n_s \cos\gamma z\,e^{j n_s\varphi}; \]

\[ H_z=\chi^2 \sum \left[A_s J_{n_s}(\chi r)+B_s N_{n_s}(\chi r)\right]\sin \gamma z\, e^{j n_s \varphi}. \]

Here, as in what follows, the factor \(e^{j\omega t}\) is omitted; summation is carried out over \(s\) from \(-\infty\) to \(+\infty\); \(A_s\) and \(B_s\) are amplitude coefficients; \(J_{n_s}\), \(N_{n_s}\), \(J'_{n_s}\), \(N'_{n_s}\) are, respectively, the Bessel and Neumann functions of order \(n\) and their derivatives with respect to the argument;

\[ n_s=n+\frac{2\pi s}{\Theta}; \qquad n=\frac{c}{V_\varphi}-kR; \qquad s=0,\ \pm 1,\ \pm 2,\ldots \]

\[ \gamma=\frac{\pi}{b}; \qquad \chi=k\sqrt{1-\left(\frac{\lambda}{2b}\right)^2}; \qquad k=\frac{2\pi}{\lambda}. \]

Assuming that the oscillations in neighboring resonators differ only in phase, the following boundary conditions for \(E_z\) may be written:

\[ E_z(r_c,z)=E_1 e^{j\frac{2\pi q}{N}}, \]

\[ \text{for } \frac{2\pi q}{N}-\Phi<\varphi<\frac{2\pi q}{N}+\Phi \]

\[ E_\varphi(r_a,z)=E_2 e^{j\frac{2\pi q}{N}} \tag{3} \]

\[ E_z(r_c,z)=E_\varphi(r_a,z)=0 \quad \text{for the remaining values of } \varphi . \]

Here \(r_a=R+a\); \(r_c=R-a\); \(R=R_\varphi\).

From (2) and (3) it is easy to find \(A_s\) and \(B_s\), and, consequently, the tangential components of the field:

\[ E_z=\frac{N\Phi}{\pi}\sum M(n_s;\Phi)\left[ E_1 \frac{J'_{n_s}(\chi r)N'_{n_s}(\chi r_a)-J'_{n_s}(\chi r_a)N'_{n_s}(\chi r)} {J'_{n_s}(\chi r_c)N'_{n_s}(\chi r_a)-J'_{n_s}(\chi r_a)N'_{n_s}(\chi r_c)} - \right. \]

\[ \left. -\,E_2 \frac{J'_{n_s}(\chi r)N'_{n_s}(\chi r_c)-J'_{n_s}(\chi r_c)N'_{n_s}(\chi r)} {J'_{n_s}(\chi r_c)N'_{n_s}(\chi r_a)-J'_{n_s}(\chi r_a)N'_{n_s}(\chi r_c)} \right]e^{j n_s\varphi}; \tag{4} \]

\[ H_z=\frac{jN\Phi\chi}{\omega\mu\pi}\sum M(n_s;\Phi)\left[ E_1 \frac{J_{n_s}(\chi r)N'_{n_s}(\chi r_a)-J'_{n_s}(\chi r_a)N_{n_s}(\chi r)} {J'_{n_s}(\chi r_c)N'_{n_s}(\chi r_a)-J'_{n_s}(\chi r_a)N'_{n_s}(\chi r_c)} - \right. \]

\[ \left. -\,E_2 \frac{J_{n_s}(\chi r)N'_{n_s}(\chi r_c)-J'_{n_s}(\chi r_c)N_{n_s}(\chi r)} {J'_{n_s}(\chi r_c)N'_{n_s}(\chi r_a)-J'_{n_s}(\chi r_a)N'_{n_s}(\chi r_c)} \right]e^{j n_s\varphi}, \]

where

\[ M(n_s;\Phi)=\frac{\sin n_s\Phi}{n_s\Phi}. \]

Taking (1) and (4) into account, we find expressions for the conductance of the interaction space toward the resonators located along the large and small radii and, eliminating \(E_1\) and \(E_2\), obtain:

\[ \frac{Y_{r=r_c}\, r_c}{B} - \sum M^2(n_s,\Phi)\, \frac{ J_{n_s}(\chi r_c)N'_{n_s}(\chi r_a)-J'_{n_s}(\chi r_a)N_{n_s}(\chi r_c) }{ J'_{n_s}(\chi r_c)N'_{n_s}(\chi r_a)-J'_{n_s}(\chi r_a)N'_{n_s}(\chi r_c) } \over \sum M^2(n_s,\Phi)\, \frac{ J_{n_s}(\chi r_c)N'_{n_s}(\chi r_a)-J'_{n_s}(\chi r_c)N_{n_s}(\chi r_a) }{ J'_{n_s}(\chi r_c)N'_{n_s}(\chi r_a)-J'_{n_s}(\chi r_a)N'_{n_s}(\chi r_c) } } + \frac{ \sum M^2(n_s,\Phi)\, \frac{ J_{n_s}(\chi r_a)N'_{n_s}(\chi r_c)-J'_{n_s}(\chi r_a)N_{n_s}(\chi r_a) }{ J'_{n_s}(\chi r_c)N'_{n_s}(\chi r_a)-J'_{n_s}(\chi r_a)N'_{n_s}(\chi r_c) } }{ \frac{Y_{r=r_a}\, r_a}{B} + \sum M^2(n_s,\Phi)\, \frac{ J_{n_s}(\chi r_a)N'_{n_s}(\chi r_c)-J'_{n_s}(\chi r_c)N_{n_s}(\chi r_a) }{ J'_{n_s}(\chi r_c)N'_{n_s}(\chi r_a)-J'_{n_s}(\chi r_a)N'_{n_s}(\chi r_c) } } , \tag{5} \]

where

\[ B=j\,\frac{Nz}{2\pi \omega \psi}. \]

If in (2) one sets \(n_s=0\) and takes into account the boundary conditions at \(r=R_1=r_c-h_1\) and \(r=R_2=r_a+h_2\), then the input conductances for the resonators are found analogously:

\[ Y_1=Y_{r=r_c}=-\frac{B\pi}{r_c N\Phi}\, \frac{ J_0(\chi r_c)N_1(\chi R_1)-J_1(\chi R_1)N_0(\chi r_c) }{ J_1(\chi r_c)N_1(\chi R_1)-J_1(\chi R_1)N_0(\chi r_c) }; \]

\[ Y_2=Y_{r=r_a}=-\frac{B\pi}{r_a N\Phi}\, \frac{ J_0(\chi r_a)N_1(\chi R_2)-J_1(\chi R_2)N_0(\chi r_a) }{ J_1(\chi r_a)N_1(\chi R_2)-J_1(\chi R_2)N_1(\chi r_a) }. \tag{6} \]

Relation (5), taking (6) into account, is the required dispersion equation of a rectangular waveguide diaphragmed along the bent walls.

Special cases and examples of calculation

As a chamber model for a waveguide cyclic accelerator, three varieties of bent diaphragmed waveguide may be used: a symmetrically loaded waveguide \((h_1=h_2\) in Fig. 1), or waveguides diaphragmed along one of the bent walls.

In the first case, acceleration can be carried out either on a wave of the in-phase type (the \(H_{11}\) wave), or of the anti-phase type (the \(H_{10}\) wave); in the second case, on a wave of type \(H_{10}\).

If the waveguide is loaded with diaphragms along the wall with the larger radius \((h_2=h,\ h_1=0)\), then from (5) and (6) we obtain the dispersion equation

\[ \sum M^2(n_s,\Phi)\, \frac{ J_{n_s}(\chi r_a)N'_{n_s}(\chi r_c)-J'_{n_s}(\chi r_c)N_{n_s}(\chi r_a) }{ J'_{n_s}(\chi r_a)N'_{n_s}(\chi r_c)-J'_{n_s}(\chi r_c)N'_{n_s}(\chi r_a) } = \]

\[ =-\frac{\pi}{N\Phi}\, \frac{ J_0(\chi r_a)N_1(\chi R_2)-J_1(\chi R_2)N_0(\chi r_a) }{ J_1(\chi r_a)N_1(\chi R_2)-J_1(\chi R_2)N_1(\chi r_a) }. \tag{7} \]

Analogously, for a system loaded with diaphragms along the wall with the smaller radius, (5) becomes

\[ \sum M_s^2(n_s;\Phi)\, \frac{ J'_{n_s}(\chi r_c)N''_{n_s}(\chi r_a)-J''_{n_s}(\chi r_a)N'_{n_s}(\chi r_c) }{ J'_{n_s}(\chi r_c)N'_{n_s}(\chi r_a)-J'_{n_s}(\chi r_a)N'_{n_s}(\chi r_c) } \]

\[ = \frac{\pi}{N\Phi}\, \frac{ J_1(\chi r_c)N'_1(\chi R_1)-J'_1(\chi R_1)N_1(\chi r_c) }{ J_1(\chi r_c)N_1(\chi R_1)-J_1(\chi R_1)N_1(\chi r_c) }. \tag{8} \]

For large radii of curvature (or for resonators of the rectangular-slot type), the right-hand side in equations (7) and (8) may be replaced, respectively, by

\[ \frac{\pi}{N\Phi}\operatorname{ctg}\chi h . \]

Calculations by (5, 7, 8) show [10] that:

a) owing to the bending of a symmetrically loaded waveguide, the separation of the resonant frequencies of in-phase and anti-phase waves increases;
b) the smaller the waveguide height \(b\) for \(a=\mathrm{const}\), the greater must be the diaphragm height \(h\) in order to slow the wave to the velocity of light at the mean radius \(R\);
c) with a constant waveguide dimension in the radial direction, the interaction space can be increased without changing the phase velocity of the wave by increasing the waveguide height;
d) the phase velocity of all types of waves is more critical to a change in the waveguide height \(b\) than to the dimensions in the radial direction.

Any of the systems indicated can provide the necessary slowing of the wave; however, the choice of one or another variety of diaphragmed waveguide for a model of a cyclic accelerator can be made after comparing their efficiency. The efficiency of waveguides can be judged from the magnitude of the coupling resistance \(R_{cb}\). Strong bending leads to a substantial redistribution of the field in the transverse cross section of the waveguide and, consequently, to a change in \(R_{cb}\) in comparison with an unbent waveguide.

From Fig. 2 it is seen that, for \(kR<15\), the \(H_{10}\)-type wave of a waveguide loaded along the wall with the larger radius has the greatest coupling resistance.

The calculation of the dispersion curve for such a waveguide with dimensions

\[ R_1=8.05\ \mathrm{cm};\quad R_2=18.85\ \mathrm{cm};\quad h=5.15\ \mathrm{cm};\quad b=5.85\ \mathrm{cm};\quad N=36;\quad \Phi=0.08 \]

was carried out from the transcendental equation (7), graphically, for the wavelength range \(\lambda=9\text{--}11.7\ \mathrm{cm}\). In this case the arguments of the cylindrical functions vary within the limits from 0 to 6.11. This makes it possible to simplify equation (7), replacing the cylindrical functions on the left-hand side (for \(s\ne 0\)) by the asymptotic representations of Fock–Debye [11]:

\[ J_n(x)= \frac{e^{\,n(q-\operatorname{th}q)}}{\sqrt{2\pi n\,\operatorname{tg}q}} = \frac{e^{\,n\operatorname{arcch}\frac{x}{n}-p}}{\sqrt{2\pi p}}; \qquad n\,\operatorname{th}q=p=\sqrt{n^2-x^2}; \]

\[ J'_n(x)=J_n(x)\left[ \operatorname{sh}q+\frac{\operatorname{ch}q}{2n\,\operatorname{sh}^2 q} \right] =J_n(x)\left[ \frac{p}{x}+\frac{x}{p^2} \right]; \]

\[ q=\operatorname{arcch}\frac{n}{x}. \tag{9} \]

\[ N_n(x)=-\frac{e^{nq}\operatorname{th}q}{\sqrt{1+2n\,\operatorname{th}q}} -\frac{e^{\operatorname{arsh}\frac{n}{x}-\frac{n}{p}}}{\sqrt{1+2p}}; \]

\[ N'_n(x)=N_n(x)\left[\operatorname{sh}q-\frac{\operatorname{ch}q}{2n\,\operatorname{sh}^2 q}\right] -N_n(x)\left[\frac{p}{x}-\frac{x}{2p^2}\right]. \]

It turns out [10] that, beginning with \(s=-1\), each term of the sum in (7), taking (9) into account, can be written as

\[ -\frac{J_{ns}(x)}{J'_{ns}(x)} =-\left(\frac{p}{x}-\frac{x}{2p^2}\right)^{-1}. \tag{10} \]

Thus, in calculating the dispersion curve in the present case, it is possible, with a sufficient degree of accuracy, to restrict oneself to taking into account only the fundamental and the first negative harmonic.

Fig. 2. Influence of bending on the coupling resistance for \(c/v_\varphi=1\) for \(r=R\), \(kd=2\), \(1-(l/2b)^2=0.5\): 1 and 2, respectively, for waves of type \(H_{11}\), \(H_{10}\) of a symmetrically loaded waveguide; 3 and 4—for waves of type \(H_{10}\) of waveguides loaded along the walls with small and large radius, respectively; 5 for a wave of type \(H_{10}\) of a straight waveguide loaded on one side.

It is seen from Fig. 3 that, in the oscillatory regime, the discrepancies of the first and second approximations amount to \(1\%\), in the regime \(\pi/2\) (\(n=9\)) they are less than \(0.02\%\) of the generator wavelength; however, discrepancies in the wavelength in the waveguide at a radius \(r=12.5\ \text{cm}\) amount to \(9\%\) at \(\lambda=10.75\ \text{cm}\) and reach \(15\%\) at \(\lambda=10.6\ \text{cm}\).

Fig. 3. Dispersion curve of the wave of type \(H_{11}\) of a waveguide loaded along the wall with the large radius.

The simplifications of equation (7) presented above can be applied to any of the varieties of bent waveguide for the geometrical dimensions indicated in the introduction. However, in each particular case one must take into account the larger number of spatial harmonics, the greater the thickness of the diaphragms (as is seen from a comparison of the graphs in Figs. 3 and 7).

Experimental Data

The experimental investigation of the dispersion properties of waveguides was carried out in the standing-wave regime by two different methods: the active-probe method \[12\] with a measuring line, and the method of small perturbing bodies \[13\].

Both methods are described in sufficient detail in the literature \[5, 12, 13, 14\], and we shall not dwell here on their particular features. We note only that the latter method, although in principle more accurate than the former, gives reliable results only for measurements on resonant models with a sufficiently high \(Q\). Otherwise, monitoring small frequency changes is difficult.

The dispersion properties of a symmetrically loaded waveguide were investigated on two resonant models. One of the models (studied by the method of small perturbing bodies) consisted of one cell and two half-cells. Measurements on the second model, which was a half-wave resonator in the \(\pi/4\) mode, were carried out by the active-probe method. The resonators had dimensions \(R_1 = 5.2\ \mathrm{cm}\), \(R_2 = 20.8\ \mathrm{cm}\), \(b = 5.9\ \mathrm{cm}\), \(h = 6\ \mathrm{cm}\). Figure 4 gives the dispersion curves calculated in the first

Fig. 4. Dispersion curves for a symmetrically loaded waveguide: — theory; \(x\)—experiment by the active-probe method; \(o\)—experiment by the method of small perturbing bodies.

Fig. 5. Dispersion curve of the \(H_{10}\) waveguide wave, loaded along a small radius: 1. Calculation by formula (8); 2. Calculation for resonators of the rectangular-slot type; \(x\)—experiment by the active-probe method.

approximation according to (5), and the experimental points are indicated. The maximum discrepancy of the experimental data from the calculated ones does not exceed \(1\%\) of the generator wavelength, but may reach \(10\text{–}15\%\) of the wavelength in the waveguide. It should be noted that if, in calculations of the waveguide chamber of a cyclic accelerator, an error of \(1\%\) is allowed in determining the resonant generator wavelength \(\lambda\), then the radius \(R_0\) on which the wave will propagate with the speed of light will also be shifted by \(1\%\) relative to the calculated value, which is quite acceptable for a waveguide model of a cyclic accelerator. It is also seen from Fig. 4 that in-phase waves have a steeper dispersion characteristic than anti-phase waves, which advantageously distinguishes the former. The retardation of the \(H_{10}\)-type wave to the speed of light at radius

$r = 12.7$ is reached at wavelength $\lambda = 11.2\ \text{cm}$; waves of type $H_{11}$—at radius $r = 11.3\ \text{cm}$ for $\lambda = 10.81\ \text{cm}$.

Figure 5 gives the dispersion curves calculated from (8) ($s = 0$) for a waveguide loaded along the wall with the small radius. The waveguide was a closed ring with dimensions: $R_1 = 8.5\ \text{cm}$;

Fig. 6. Experimental model of a bent waveguide, diaphragmed along the walls with the small radius (without the upper end ring).

$R_2 = 16.5\ \text{cm}$; $h = 3\ \text{cm}$; $b = 5.16\ \text{cm}$; $N = 36$ (flat diaphragms 3 mm thick), and is shown in Fig. 6. The difference between the theoretical data and the experimental data also does not exceed 1% of the generator wavelength. A retardation equal to unity is reached at $\lambda = 8.75\ \text{cm}$ at radius $r = 12.5\ \text{cm}$.

Fig. 7. Dispersion characteristic of waves of type $H_{10}$ in a waveguide loaded along the larger radius. — calculation in the ninth approximation; - - - calculation in the first approximation; x—experiment by the active-probe method.

The experimental points and calculated curves in the first \((s=0)\) and ninth \((s=0;\ 1,\ 2,\ 3,\ 4)\) approximations for the wave of type \(H_{10}\) of a waveguide loaded with diaphragms along the wall with the larger radius are shown in Fig. 7. The waveguide model with dimensions \(R_1 = 10.0\ \text{cm}\), \(R_2 = 20.25\ \text{cm}\), \(h = 4.13\ \text{cm}\), \(b = 6.0\ \text{cm}\), \(N = 32\), \(\Phi = 0.0186\) (with resonators of the rectangular-chain type) consisted of three cells and two half-cells. The agreement between the calculated and experimental data is quite satisfactory. Deceleration of the wave to the speed of light at a radius \(r = 13.1\) is achieved in the oscillation mode \(\pi/2\) at the generator wavelength \(\lambda = 10.3\ \text{cm}\).

The last waveguide, under strong bending, is the most effective of the considered varieties of slow-wave systems of the “comb” type; in addition, it possesses a sufficiently flat dispersion characteristic, and the wave of type \(H_{10}\) is the fundamental wave among waves of type \(H\).

Conclusion

As a result of the foregoing, the following conclusions may be drawn.

1. Formulas have been obtained that make it possible to calculate, with accuracy sufficient for practical purposes, the dispersion properties of rectangular-cross-section waveguides periodically diaphragmed along curved walls. The calculation with allowance for higher harmonics is simplified by replacing cylindrical functions with asymptotic expressions.

2. It has been shown that, for a model of a waveguide cyclic accelerator with an iron magnetic system, the most suitable of the “comb”-type slow-wave systems is a waveguide diaphragmed along the wall with the larger radius.

3. The discrepancy between the theoretical and experimental data does not exceed 1% of the generator wavelength, which is entirely permissible for a model waveguide cyclic accelerator. On the samples studied, wave deceleration to the speed of light was obtained with waveguide geometrical dimensions acceptable for a \(10\ \text{MeV}\) accelerator model.

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