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

$$E_z(r,\varphi,z,t) = A \,f(r,\varphi)\, \cos(k_3\, z)\, {\rm e}^{-{\rm i}\,\omega\, t},$$ (1344)

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

$$\frac{1}{r} \frac{\partial}{\partial r}\!\left(r\,\frac{\partial ... ...{1}{r^{\,2}} \frac{\partial^{\,2} f}{\partial \varphi^{\,2}} + k_s^{\,2} f = 0,$$ (1345)

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

$$f(r,\varphi) = g(r) \,{\rm e}^{\,{\rm i}\,m\,\varphi}.$$ (1346)

It follows that

$$\frac{1}{r} \frac{d}{d r} \!\left(r\,\frac{d g} {d r}\right) + \left(k_s^{\,2}-\frac{m^{\,2}}{r^{\,2}}\right) g = 0,$$ (1347)

or

$$z^{\,2}\,\frac{d^{\,2} g}{dz^{\,2}} + z\,\frac{dg}{dz} + (z^{\,2} - m^{\,2}) \,g = 0,$$ (1348)

where $z=k_s\, r$ . The above equation can be recognized as Bessel's equation. The independent solutions of this equation are denoted $J_m(z)$ and $Y_m(z)$ . The $J_m(z)$ are regular at $z=0$ , whereas the $Y_m(z)$ are singular. Moreover, both solutions are regular at large $\vert z\vert$ .

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

$$E_z(r,\varphi,z,t) = A\, J_m(k_l\, r) \,{\rm e}^{\,{\rm i}\,m\,\varphi} \cos(k_3 \,z)\, {\rm e}^{-{\rm i}\,\omega\, t}.$$ (1349)

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

$$J_m(k_l \,a) = 0.$$ (1350)

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

$$H_z(r,\varphi,z,t) = A\, J_m(k_l\, r) \,{\rm e}^{\,{\rm i}\,m\,\varphi} \sin(k_3 \,z)\, {\rm e}^{-{\rm i}\,\omega \,t}.$$ (1351)

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

$$J_m'(k_l \,a) = 0,$$ (1352)

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

$$\frac{\omega^{\,2}}{c^{\,2}} = k^{\,2} = k_l^{\,2} + \frac{n^{\,2}\,\pi^2}{L^2}.$$ (1353)

If $l$ is the ordinal number of a zero of a particular Bessel function of order $m$ (defined such that $l$ 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 $J_m(z)$ is conventionally denoted $j_{ml}$ [so, $J_m(j_{ml})=0$ ]. Likewise, the $l$ th zero of $J_m'(z)$ is denoted $j_{m l}'$ . Table 2 shows the first few zeros of $J_0(z)$ , $J_0'(z)$ , $J_1(z)$ , and $J_1'(z)$ . 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 a TM mode. The next highest frequency mode is a TE mode, and so on.

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

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