Cylindrical cavities

Let us apply the methods of the previous section to the TM modes of a right circular cylinder of radius $a$. We can write

$$E_z = A f(r,\varphi) \cos(k_3 z)\, {\rm e}^{-{\rm i}\,\omega t},$$ (1084)

where $f(r,\varphi)$ satisfies the equation

$$\frac{1}{r} \frac{\partial}{\partial r}\!\left(r\,\frac{\par... ...r^2} \frac{\partial^2 f}{\partial \varphi^2} + k_s^{~2} f = 0,$$ (1085)

and $(r,\varphi,z)$ are cylindrical polar coordinates. Let

$$f(r,\varphi) = g(r) \,{\rm e}^{{\rm i}\,m\varphi}.$$ (1086)

It follows that

$$\frac{1}{r} \frac{d}{d r} \!\left(r\,\frac{d g} {d r}\right) + \left(k_s^{~2}-\frac{m^2}{r^2}\right) g = 0,$$ (1087)

or

$$z^2\,\frac{d^2 g}{dz^2} + z\,\frac{dg}{dz} + (z^2 - m^2) \,g = 0,$$ (1088)

where $z=k_s r$. The above equation is known as Bessel's equation. The two linearly independent solutions of Bessel's equation are denoted $J_m(z)$ and $Y_m(z)$. In the limit $\vert z\vert\ll 1$ these solutions behave as $z^m$ and $z^{-m}$, respectively, to lowest order . More exactly¹⁶

$$J_m(z) = \left(\frac{z}{2}\right)^m \sum_{k=0}^\infty \frac{(-z^2/4)^k}{k!(m+k)!} ,$$ (1089)

$$Y_m(z) = -\frac{(z/2)^{-m}}{\pi} \sum_{k=0}^{m-1} \frac{(m-k-1)!\, (z^2/4)^k}{k!} + \frac{2}{\pi}\,\ln(z/2) \,J_m(z)$$

$$-\frac{(z/2)^m}{\pi} \sum_{k=0}^\infty \left[\psi(k+1)+\psi(m+k+1)\right] \frac{(-z^2/4)^k}{k! (m+k)!}$$ (1090)

for $\vert z\vert\ll 1$, where

$$\psi(1) = -\gamma,$$ (1091)

$$\psi(n) = -\gamma + \sum_{k=1}^{n-1} k^{-1},$$ (1092)

and $\gamma=\sum_{k=1}^{\infty} k^{-1}=0.57722$ is Euler's constant. Clearly, the $J_m$ are well behaved in the limit $\vert z\vert\rightarrow 0$, whereas the $Y_m$ are badly behaved.

The asymptotic behaviour of both solutions at large $\vert z\vert$ is

$$J_m(z) = \sqrt{\frac{2}{\pi z}} \cos(z-m \,\pi/2 - \pi/4) + O(1/z),$$ (1093)

$$Y_m(z) = \sqrt{\frac{2}{\pi z}} \sin(z- m\,\pi/2 -\pi/4) + O(1/z).$$ (1094)

Thus, for $\vert z\vert\gg 1$ the solutions take the form of gradually decaying oscillations which are in phase quadrature. The behaviour of $J_0(z)$ and $Y_0(z)$ is shown in Fig. 21.

Figure 21: The Bessel functions $J_0(z)$ (solid line) and $Y_0(z)$ (dotted line)

Since the axis $r=0$ is included in the cavity the radial eigenfunction must be regular at the origin. This immediately rules out the $Y_m(k_s r)$ solutions. Thus, the most general solution for a TM mode is

$$E_z = A\, J_m(k_l r) \,{\rm e}^{{\rm i}\,m\varphi} \cos(k_3 z)\, {\rm e}^{-{\rm i}\,\omega t}.$$ (1095)

The $k_l$ are the eigenvalues of $k_s$, and are determined by the solutions of

$$J_m(k_l a) = 0.$$ (1096)

The above constraint ensures that the tangential electric field is zero on the conducting walls surrounding the cavity ($r=a$).

The most general solution for a TE mode is

$$H_z = A\, J_m(k_l r) \,{\rm e}^{{\rm i}\,m\varphi} \sin(k_3 z)\, {\rm e}^{-{\rm i}\,\omega t}.$$ (1097)

In this case, the $k_l$ are determined by the solution of

$$J_m'(k_l a) = 0,$$ (1098)

where $'$ denotes differentiation with respect to the argument. The above constraint ensures that the normal magnetic field is zero on the conducting walls surrounding the cavity. The oscillation frequency of both the TM and TE modes is given by

$$\frac{\omega^2}{c^2} = k^2 = k_l^{~2} + \frac{n^2\pi^2}{L^2}.$$ (1099)

If $l$ is the ordinal number of a zero of a particular Bessel function of order $m$ ($l$ increases with increasing values of the argument), then each mode is characterized by three integers, $l$, $m$, $n$, as in the rectangular case. The $l$th zero of $J_m$ is conventionally denoted $j_{m,l}$ [so, $J_m(j_{m,l})=0$]. Likewise, the $l$th zero of $J_m'$ is denoted $j_{m, l}'$. Table 2 shows the first few zeros of $J_0$, $J_0'$, $J_1$, and $J_1'$. It is clear that for fixed $n$ and $m$ the lowest frequency mode (i.e., the mode with the lowest value of $k_l$) is a TE mode. The mode with the next highest frequency is also a TE mode. The next highest frequency mode is a TM mode, and so on.

Table 2: The first few values of $j_{0,l},$ $j_{1,l}$, $j_{0,l}'$, and $j_{1,l}'$

$l$	$j_{0,l}$	$j_{1,l}$	$j_{0,l}'$	$j_{1,l}'$
1	2.4048	3.8317	0.0000	1.8412
2	5.5201	7.0156	3.8317	5.3314
3	8.6537	10.173	7.0156	8.5363
4	11.792	13.324	10.173	11.706