Doubly Periodic Staircase Systems

P. I. GOSKOV

(Presented by the scientific seminar of the Research Institute of Nuclear Physics)

Staircase slowing systems with a protrusion or a recess were investigated theoretically and experimentally in works [1, 2]. In the present paper, staircase systems with several protrusions distributed periodically along the pins are considered (Fig. 1).

Fig. 1. Doubly periodic staircase systems.

This system is periodic both along the \(z\)-axis (along the pins) and along the \(Y\)-axis (along the protrusions), i.e., it is a doubly periodic system. Its dispersion properties can be investigated by considering the staircase as a multiconductor line. We shall restrict ourselves to considering only the long-wavelength part of the spectrum with the fundamental TEM wave. In doing so, we assume that the phase angle \(\Theta\) between any two neighboring protrusions—

is strictly constant, i.e. \(\Theta=\dfrac{n\pi}{N}\), where \(N\) is the number of periods along \(z\), and \(n=0,1,2,3\ldots N\). In this case the boundary conditions for the potentials along the pins in the region of the ridges \((0 \leqslant z \leqslant d)\) and in the region between the ridges \((d \leqslant z \leqslant D)\) may be written as follows:

\[ \begin{gathered} U_1(0)=U_2(D)e^{j\Theta},\\ I_1(0)=I_2(D)e^{j\Theta},\\ U_1(d)=U_2(d),\\ I_1(d)=I_2(d), \end{gathered} \tag{1} \]

where the expressions for the potentials and currents in both regions have the form

\[ \begin{gathered} U_1=A\cos \kappa z+B\sin \kappa z,\\ U_2=C\cos \kappa z+F\sin \kappa z,\\ I_1=jM_1(\psi)\,[B\cos \kappa z-A\sin \kappa z],\\ I_2=jM_2(\psi)\,[F\cos \kappa z-C\sin \kappa z], \end{gathered} \tag{2} \]

where \(\kappa\) is the propagation constant in the \(z\)-direction, \(M_1(\psi)\) and \(M_2(\psi)\) are the wave admittances in the region of the ridges and in the region between the ridges, and \(\psi\) is the phase angle between neighboring pins in the \(Y\)-direction.

Substituting from (2) the expressions for the potentials and currents into the boundary conditions (1), we obtain

\[ \begin{gathered} A=(C\cos \kappa D+F\sin \kappa D)e^{j\Theta},\\ A\cos \kappa d+B\sin \kappa d=C\cos \kappa d+F\sin \kappa d,\\ M_1B=M_2(F\cos \kappa D-C\sin \kappa D)e^{j\Theta},\\ M_1(B\cos \kappa d-A\sin \kappa d)=M_2(F\cos \kappa d-C\sin \kappa d), \end{gathered} \tag{3} \]

Solving the system of equations (3) with respect to the coefficients \(A\), \(B\), \(C\), and \(F\), we arrive at the following dispersion equation of doubly periodic staircase systems

\[ \cos \kappa d \cos \kappa(D-d)-\frac{M_1^2+M_2^2}{2M_1M_2}\sin \kappa d \sin \kappa(D-d)=\cos \Theta . \tag{4} \]

Using this equation, dispersion curves were calculated for a uniform doubly periodic staircase system (Fig. 1-a), bounded along \(z\) by six periods and along \(Y\) by two periods.

Figure 2 gives dispersion curves for the modified TEM wave, which has a longitudinal component of the electric field along the \(z\)-axis (along the pins). Considering the properties of the doubly periodic staircase in this direction, we may compare it with a diaphragmed waveguide of the same dimensions, into which it turns if the thickness of the pins is made to tend to zero, i.e. \(p-q=0\).

The introduction of pins into the interaction space of the diaphragmed waveguide leads to the appearance of new passbands, whose number is determined by the number of periods along \(Y\), i.e. by the possible phase angles \(\psi\) between neighboring pins. In our case \(\psi=0,\dfrac{\pi}{2},\pi\).

In the diaphragmed waveguide, the fundamental wave with a longitudinal component along \(z\) is the \(LE_{11}\) wave. The introduction of pins causes a perturbation of the \(LE_{11}\) wave and shifts it into the short-wavelength region, i.e. in the doubly periodic staircase, for the same dimensions, the fundamental wav-

with a longitudinal component along \(z\) is a modified TEM wave.

Its characteristic feature is that, at a slowing

\[ \frac{c}{v_{\phi}}=1 \]

it becomes purely transverse, i.e., the longitudinal component (along the pins) of the modified TEM wave exists only for slowing

\[ \frac{c}{v_{\phi}}>1. \]

This means that this system, with the use of the longitudinal component along \(z\), cannot be applied in devices for which a slowing

\[ \frac{c}{v_{\phi}}=1 \]

is essential, for example, in waveguide synchrotrons. On the other hand, in microwave electronic devices using prolonged interaction of electrons with the field of a slowed electromagnetic wave, such a system may in a number of cases be of greater interest than the ordinary iris-loaded waveguide.

Fig. 2. Dispersion curves of waves with a longitudinal component along \(z\) (\(D=25\) mm, \(d=5\) mm, \(w_2=30\) mm, \(w=1\) mm, \(b=18\) mm, \(p=35\) mm, \(q=34\) mm). Points on the curves are experimental values.

In Fig. 2, for comparison, the dispersion curve \(I\) of the fundamental \(LE_{11}\) wave of an iris-loaded waveguide is shown, with the same dimensions as those of the doubly periodic staircase, but with the pins absent, i.e., \(p-q=0\). In the presence of pins the \(LE_{11}\) wave will evidently be excited somewhere in the three-centimeter range, since in carrying out the experiment oscillations of this type were not detected in the range \(\lambda>7.8\) cm, and the available experimental setup made it possible to carry out measurements only in the range \(7.8 \div 31\) cm. As follows from Fig. 2, the dispersion curves of the doubly periodic staircase have a considerably wider passband and, for phase angle \(\psi\) close to or equal to 0, also a greater slowing, irrespective of the operating mode of the system \(\Theta\).

In addition to measuring the dispersion, in two cases a measurement was made of the coupling resistance, which for \(\psi=\pi\) and \(\Theta=\frac{\pi}{3}\) proved to be

equal to 102 ohms, while for $\psi=0$ and $\Theta=\dfrac{5\pi}{6}$ it was 216 ohms, i.e., one may expect that for oscillations with $\psi=0$ the longitudinal component (along $z$) will have a greater magnitude than for the corresponding oscillations with $\psi=\pi$. The coupling resistance was measured on the axis of the system $(x=0, Y=0)$ in an electric-field antinode by the small-perturbation method.

The smallest discrepancy between the results of measuring the dispersion and the theoretical values is observed for $\Theta \to \pi$. This is explained by the fact that for $\Theta \to 0$, in addition to the fundamental TEM wave of the multiwire line, when calculating the dispersion it is evidently necessary to take into account higher wave types.

Fig. 3. Dispersion curves of waves with a longitudinal component along $Y$ (the dimensions of the system are the same).

If the properties of a doubly periodic staircase are considered in the direction along the row of pins (along $Y$), then in this case it can be compared with staircase slow-wave systems with one ridge or groove. In Fig. 3, in addition to the fundamental $(n=1)$, higher passbands $(n=2 \div 6)$ are also shown.

Curve 1 corresponds to a staircase system with one ridge of the same thickness $d=5$ mm, located symmetrically with respect to the pins, whose height (along $z$) was $N \cdot D = 150$ mm. Increasing the number of ridges to 6 at unchanged pin height leads to a widening of the fundamental passband $(n=1)$ and to its shift into a longer-wavelength range. Curve 1 was calculated from the dispersion equation of a single-row staircase with a ridge for the symmetric mode [1].

As the numerical calculations carried out show, the dependence (Fig. 4) of the dispersion properties of a staircase system with a large number of ridges (a doubly periodic staircase) on the height of the ridges ($w_1 \sim$, $w_2 = 30$ mm = const) and on their thickness ($d \sim$, $D = 25$ mm = const) has the same character as in staircases with one ridge. As the distance $w_1$ increases, the passband decreases and, in the limiting case $w_1=w_2$, the system resonates only at one wavelength, equal to twice the pin height (30 cm). As the ridge thickness $d$ is increased to a certain optimal value (in the present case $d_{\mathrm{opt}}$ is of the order of 15 mm), the dispersion curve, while broadening, shifts toward longer waves, and then toward shorter waves, with a simultaneous narrowing of the passband.

In calculating the dispersion curves, the expression for the wave conductance from [3] was used.

The dispersion equation (4) can also be used in calculating two-row doubly periodic staircases (Fig. 1, b).

Thus, such doubly periodic staircase systems, in comparison with a diaphragm-loaded waveguide of the same dimensions, owing to the introduction into the interaction space of the latter of plane pins—even comparatively thin ones $(p-q=1$ mm)—which lead to the appearance of new passbands and to a change in the type of the fundamental wave, make it possible to substantially broaden the passband of the fundamental wave and to increase the slowing. As applied to microwave devices, the pins in this case can play the role of focusing devices.

In comparison with staircases having a single ridge, a doubly periodic staircase also makes it possible to broaden the stop band of the fundamental wave and to shift it into a longer-wavelength range. In this case, when

Fig. 4. Dependence of the dispersion of a wave with a longitudinal component along \(Y\) on: a) the size of the ridge along \(x\)
(\(b=1\) mm; \(1\)—\(w_1=1\) mm, \(2\)—\(w_1=2.5\) mm, \(3\)—\(w_1=5\) mm, \(4\)—\(w_1=15\) mm, \(5\)—\(w_1=w_2=30\) mm; the remaining dimensions are the same as in Fig. 2); b) the sizes of the ridges along \(z\)
(\(b=1\) mm; \(1\)—\(d=1\) mm, \(2\)—\(d=5\) mm, \(3\)—\(d=15\) mm, \(4\)—\(d=24\) mm; the remaining dimensions are the same as in Fig. 2).

using a wave with a longitudinal component along \(Y\), the given doubly periodic system may find application both in waveguide synchrotrons and in microwave electronic devices.

LITERATURE

1. Yu. G. Altshuler, A. S. Tatarenko. Low-Power Backward-Wave Tubes, Sovetskoe radio, 1963.

2. P. I. Gos’kov, Izv. vuzov, Radiofizika, 5, 1964.

3. E. S. and V. S. Kovalenko. Radiotekhnika i elektronika, 8, 1963.