# 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?

• "why three indices instead of two" It's a three-dimensional cavity, right? Jan 22 '13 at 19:20
• Yes, but looking at the wikipedia page, it only uses two. Now i'm noticing that it's for laser beams. Perhaps the two things are interconnected, i.e. when I have a stationary field in a cavity, then no direction of propagation => three indices? Jan 22 '13 at 19:26
• I think that the laser modes in the wikipedia article are not bounded in the beam direction, so there is no quantization in that direction--which cooresponds to the $z$ direction in my (rather incomplete) answer. Jan 22 '13 at 19:33
• Yes, exactly what I was thinking about. It wasn't difficult after all. Thanks. Jan 22 '13 at 19:35

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.

• It says it's gaussian. It also describes the cavity, but it means little to me. However, don't worry too much. Jan 22 '13 at 19:48

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.