4. Circular Waveguides

4.1 Why circular?

A circular pipe is mechanically attractive in some applications. It is rotationally symmetric, which is what you want when the next stage rotates relative to this one (a mechanically scanning radar antenna, a gyrotron). It can be machined with high precision on a lathe. And for very high power, the absence of corners reduces local field enhancement, raising the breakdown power.

4.2 Bessel-function modes

The cross-section is a disk of radius $a$. Solving Helmholtz in cylindrical coordinates leads to Bessel-function solutions:

$E_z \text{ or } H_z \propto J_n(k_c \rho)\,e^{jn\phi}$

where $J_n$ is the order-$n$ Bessel function of the first kind. The boundary condition (tangential $E = 0$ at $\rho = a$) gives the cutoff equation: for TM modes, $J_n(k_c a) = 0$ (zeros of $J_n$); for TE modes, $J'_n(k_c a) = 0$ (zeros of the derivative).

The lowest zeros are tabulated. Lowest of all is the first zero of $J'_1$, at $k_c a = 1.841$, giving the dominant TE$_{11}$ mode with cutoff

$f_c^{(TE11)} = \frac{1.841 c}{2\pi a}$

For a 1-cm-radius circular guide, $f_c = 8.79$ GHz. Above this, TE$_{11}$ propagates. Higher modes follow: TM$_{01}$ at $f_c = 2.405 c/(2\pi a)$, TE$_{21}$ at $3.054 c/(2\pi a)$, etc.

4.3 Practical applications

Circular waveguide shows up in three places:

• Rotary joints for mechanically scanning radars. The transmitting side is fixed; the receiving side spins. A rotary joint is a circular waveguide with carefully tuned chokes that lets the joint rotate without disturbing the dominant TE$_{11}$ mode propagation.
• High-power transmission in some particle accelerators and fusion-research microwave sources, where the smooth circular cross-section minimizes local-field enhancement.
• Circular polarization handling. Adding a 90-degree corrugation or a polarizing iris in a circular guide can convert linear polarization into circular, useful for dual-polarization radar and satellite uplinks.

The tradeoff with circular guides is that the dominant TE$_{11}$ has only modest separation from the next propagating mode (TM$_{01}$ at 1.31 times the cutoff), so single-mode operation is over a narrower band than for rectangular guides.