Summary
Analysis of the frequency modulation for the FSS loop has been done (may be I have repeated it )
as we want to increase the frequency modulation from the current 21.5MHz to something
bigger than something like 30MHz.
Higher order mode analysis
I have considered a cavity of length L=3.45’’=3.68cm and radius of curvatures R1=R2=1m.
1.
Plot 1 shows the frequency distance vs frequency modulations. I care that HOM (Carrier and
SBs) do not overlap with carrier and SBs TEM00. The closest mode is the number 23 which
interfere with the positive SB around 31MHz. I do not consider this to be a problem, but I go head
anyway. At that frequency we also see the mode 46 crossing the the SB (TEM00).
2.
In order to better visualize the distances shown in the previous plot I use the minimum distances from the TEM00 (carrier or SBs).
The minimum distances are referred to the closest mode which may change along the SBs scan.
We see that on 31.7mHz we have a minimum as noticed previously and it is because of the mode 23.
From here if we would like to avoid the mode 23 I would choose a frequency modulation of ~39MHz.
3.
I fix the frequency modulation at 39MHz
I wanted to check if the mode 23 can be “avoided”. I consider the tolerance on L as L+- 0.0005 and Rc of +-1cm
There is a chance that the mode 23 overlap depending on what the variations from the nominal values are.
However the same is going to happen if we fix the Frequency modulation at 35MHz. So between 35MHz and 39MHz
I do not see substantial differences. (Please note that the color are normalized at 1MHz detuning from carrier).
4. While the scan of L and Roc has been done I have checked that no other HOMs come into play. The mode 23 and 46 are
the modes which determins the red lines at the point 3.
5. Mode distances at 35MHz and 39MHz.
 
From what I see here 35MHz is fine as mode 23 is the closest one to TEM00. As a reminder the pole of the cavity is ~180kHz. In principle 35MHz would be 4MHz
apart considering the nominal values L=3.68cm and RoC=1m.
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