What is a $TEM_{900}$ cavity radiation mode? I am reading a paper by Serge Haroche stating the cavity they use sustains a Gaussian mode of the e.m. field called $TEM_{900}$. I understand what Gaussian means. I found this explaining what TEM means, but if I am working in a cavity, what is the "direction of propagation"? And above all, why three indices instead of two?
 A: The solutions for a electric field in a perfectly conductive cylindrical cavity separate into the form
$$ E(r,\phi,z) = R(r)\Phi(\phi)Z(z)\quad ,$$
and quantize in each of the coordinate directions (periodic boundary conditions on $\phi$).
Likewise of the magnetic field.
The "T" in "TEM" is "transverse, which means that the intensity of the two fields must have the same dependence on the coordinates.
Accordingly we can label the modes with a set of three integers.
The thing I can't recall off the top of my head is the numbering scheme. Is it $(\text{radial},\text{angular},\text{longitudinal})$ or $(\text{longitudinal},\text{radial},\text{angular})$ which matters in your case. There may be a hint in the text.
A: I know this is very old, but here's an explanation anyways:
There are three indexes and the first one indicates the number of anti-nodes along the cavity direction (k vector of the field). In the specific paper that you showed the fields with frequencies of the order of 50 Ghz, which is equivalent to wavelengths of 6 mm. The cavity have 27 mm between the mirrors, fitting 4.5 wavelengths inside, that is, 9 anti-nodes.
The two other indexes indicates the mode pattern as usual, in this case $TEM_{00}$ indicates a Gaussian form.
