Resonant modes in cilindrical cavities

We shall calculate modes of the most common resonant structure encountered in particle accelerator applications, the cylindrical cavity. If the cylinder has end surfaces, it is called a cavity; otherwise, a wave guide. In our discussion of this problem the boundary surfaces will be assumed to be perfect conductors, otherwise the losses occurring in practice can be accounted for adequately by evaluating the time average power loss per unit of area by equation :

$$\frac{{dP_{loss} }}{{da}} = \frac{1}{{2\sigma \delta }}\left\vert {K_{eff} } \right\vert^2$$ (1)

where $\sigma$ is the conductivity and $\delta$ is the socalled skin depth, the profondity of penetration of the wave in conductor medium and $K_{eff}$ is a effective surface current density. For simplicity, the cross-sectional size and shape are assumed constant along the cylinder axis.

In this problem we must find solutions of the wave equations for both $E$ and $B$ with boundary conditions given by the metallic cilinder closed on its extremity. Both $E$ and $B$ satisfy wave equation

$$\begin{array}{l} \left( {\nabla ^2 E + \mu \varepsilon \f... ...n \frac{{\omega ^2 }}{{c^2 }}B} \right) = 0 \\ \end{array}$$

In the physics of accelerated particle only longitudinal components of electric field is of interest, so that the following assumptions are adopted:

1. Modes of interest have azimuthal symmetry $\partial / \partial \theta=0$ .
2. The electric field has no longitudinal variation $\partial E /\partial z=0$ .
3. The only component of electric field is longitudinal, $E_z$ .
4. Fields vary in time as $e^{i\omega t}$ .

The solutions of wave equation gives only discrete modes of $E$ and $B$ . We have

$$\begin{array}{l} E_{zn} \left( {r,t} \right) = E_{0n} J_0... ...lon \mu } E_{0n} J_1 \left( {k_n r} \right) \\ \end{array}$$

because magnetic field is directed along the $\theta$ direction. $J_n$ is the $n$ -th order Bessel function, it can be evaluated on many handbooks. Allowed values of $k_n$ are determined by the zeroes of $J_0$ .

This type of oscillations are called TM$_{0n0}$ modes. The term TM (transverse magnetic) indicates that magnetic fields are normal to the longitudinal direction. The other class of oscillations, TE modes, have longitudinal components of $B$ and $E_z=0$ . The subscript are the azimuthal, radial and longitudinal mode number (see figure )

$TM_{0n0}$ is the fundamental mode in particle acceleration. $\omega_{010}$ can be evaluated by the formula

$$\omega_{mnp}=\frac{1}{\sqrt{\epsilon \mu}}\sqrt{\frac{x_{m,n}^2}{R_0^2}+\frac{\pi^2 p^2}{d^2}}$$ (2)

where $x_{m,n}$ is the $n$ -th root of the equation $J_n(x)=0$ , suitable on many handbook of mathematical functions. The mode of our interest is TM$_{010}$ . This lowest mode has an angular frequency

$$\omega_{010}=\frac{2.405}{\sqrt{\epsilon \mu }R_0}$$

The longitudinal electric field is uniform along the propagation direction of the beam and its magnitude is maximum on axis. The transverse magnetic field is zero on axis; this is important for electron acceleration where transverse magnetic fields could deflect the beam.

Figura: Dashed lines indicates displacement currents, solid line real currents. (a) TM$_{010}$ (b) TM$_{020}$