RF Cavity resonators are used in different applications in the accelerator physics. Often a mode naming convention is used in the literature in the form of TM or TE followed by three numbers, like TM010, etc. In this short note, we review the mode naming and numbering conventions used.
When Maxwell’s equations are solved in three dimensional space inside of a cavity, the resulting equations of the electric and magnetic field will have 3 component along the axes each, summing up to 6 components in total. Either the Cartesian, cylindrical, spherical, elliptical or even confocal coordinate system is used to suit the application at hand. In the most general case, each of these 6 components are non-zero and are a function of all three coordinates.
The naming convention in cavity resonators is based on that of RF wave guides, with an added boundary to make standing waves in the 3rd dimension. Usually a propagation direction is assigned. While this direction is somewhat arbitrary for rectangular resonators, usually the longer side is taken as the $z$ direction, since it mostly resembles an RF wave guide that has been terminated on both sides. For the cylindrical cavities, the choice of the $z$ axis is obvious.
The mode names are either TM for transverse magnetic or TE for transverse electric. This indicates whether the magnetic or electric field consists of only transversal components, so that their third component is zero valued. The three index numbers describe the mode geometry. The first two belong the transversal plane and the third one shows the field variations in the 3rd dimension.
The mode numbers describe the minimum number of needed variations (extrema) of the field within that boundary in order to build up a standing wave pattern. This applies to regions that have metal boundaries i.e. all 3 directions in a rectangular cavity, and the radial and axial directions in a cylindrical cavity. The starting point of the azimuthal field component is arbitrary and does not have a metal boundary. So the phase of the field component must repeat after $2\pi$ i.e. one full wavelength. Therefore the azimuthal index shows the number of full wave lengths. Note that a zero would not mean a zero field component but one with a constant value along that dimension. To determine whether a field component has vanishing value or not, one must take a look at the full field equations.
For example, a TE101 mode in a rectangular cavity with lengths $a$, $b$ and $d$ along the coordinates axes $x$, $y$ and $z$ has no electric field components in the $z$ direction. So the remaining components are $E_x$ or $E_y$. The first two indexes show that in the transversal plane, the remaining components of the electric field have one extremum in the $x$ and a constant value in the $y$ directions. The 3rd index indicates that they have one extremum in the axial direction $z$. The components of the magnetic field follow the electric field components respectively.
While we know that some fields may have a constant value, at this point we can not say whether that constant is zero or not. This needs to be checked with the field equations. In the special case of TE101, not only $E_z$ is zero valued by definition, but also $E_x$ and $H_y$. So only three components out of 6 are actually non zero i.e. $E_y$, $H_x$ and $H_z$.
The case for circular cylindrical cavities is similar. The TM110 mode in a cylinder with radius $\varrho$ and depth $d$ along the cylindrical coordinate system $\varphi$, $\varrho$ and $z$ shows one complete wavelength in the azimuthal direction, one extremum in the radial direction and a constant field variation of the non-zero transversal components in the axial direction. After checking the field equations, it turns out that in this special case, not only the $H_z$ is zero valued by definition, but also $E\varrho$ and $E_\varphi$, so the mode contains only three non-zero field components $H_\varrho$, $H_\varphi$ and $E_z$.
The above discussion can be extended for other coordinate systems as well.