Pillbox (cavity)

A pillbox ( English for tablets box ) or cookie jar is a circular cylindrical cavity resonator with conductive walls. In high-frequency technology it is of great importance as a simple model cavity.

Analytical consideration

The idealized pillbox can be calculated analytically using Maxwell's equations . The wave equations for the electric and magnetic fields are as follows

${\displaystyle \square {\vec {E}}=\left({\frac {\partial ^{2}}{c^{2}\partial t^{2}}}-\Delta \right){\vec {E}}=0}$

${\displaystyle \square {\vec {B}}=\left({\frac {\partial ^{2}}{c^{2}\partial t^{2}}}-\Delta \right){\vec {B}}=0}$

In the case of circular symmetry, the Laplace operator is noted in cylindrical coordinates and radial Bessel functions are found as solutions. With the help of the boundary conditions of the pillbox (metallic wall) one can find the permissible modes in the pillbox. ${\displaystyle J_{n}(r)}$

Fashions

Definition of the cylindrical coordinates

As is usual in (high-frequency) resonators, a distinction is typically made between TE modes (transverse electrical modes) and TM modes (transverse magnetic modes) in pillbox cavities. In connection with three indices, this clearly characterizes the oscillation in the pillbox. The usual notation follows the scheme ; each integer index m , n or p indicates the number of node lines of the field in the corresponding coordinate direction. In the pillbox with radius and height , these coordinate directions are defined by the cylinder coordinates as azimuth direction m (with ), radial direction n (with ) and axial direction p (with ). ${\displaystyle \mathrm {TE} _{mnp}}$ ${\displaystyle R}$ ${\displaystyle L}$ ${\displaystyle 0\leq \Phi \leq \pi }$ ${\displaystyle 0\leq r\leq R}$ ${\displaystyle 0\leq z\leq L}$

application

Pure, completely closed pillboxes are mainly considered in teaching. A practical application is the absorption frequency meter ( wave meter ). Many types of resonators in accelerator physics (see linear accelerators ) can be roughly described as cylinder-symmetrical, pillbox-like resonators. Examples are elliptical resonators.