Summary (Entry 1633 revised)

Analysis of the frequency modulation for the FSS loop has been done 

as we want to increase the frequency modulation from the current 14MHz to a

higher frequency in the range of 25 to 45 MHz in order to increase the FSS  unity

gain frequency. Below 25MHz we do not have too much benefit as currently we

are modulating at 14MHz. Above 45MHz we have photodiodes noise and we want to 

deal with that.

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 detuning frequency vs frequency modulations. We care that HOMs (Carrier and 

SBs) do not overlap with the carrier TEM00. The closest mode is the number 23 which

interfere with the TEM00 around 31MHz. I do not consider this to be a problem. The morde order

is ment to be as the sum of n and m. The legend shows color of the modes in a dashed style. However

each mode order is showed with carrier (solid line) a positive SB (dash-dot) and negative SB (dashed).

From this plot we can see that in the range 25MHz to 45MHz we are pretty safe from HOMs overlaps.

If we care of the mode 23 than we should consider modulation frequencies above ~35MHz at least.

I consider a cavity pole of 135kHz (see elog 1480 https://nodus.ligo.caltech.edu:8081/CTN/1480)

2.

Plot 2 shows a zoom-in version of plot 1.

3.

In order to better visualize the difference between the detuning frequency from TEM00 and the closest HOMs (either carrier or sidebands) 

I provide the following plot. Please keep in mind that in general this way to plot the differences does not take in account of

which mode order is the closest. However in this case it is only the mode order 23.

We see that at 31.3MHz we have a minimum as noticed previously and it is because of the mode 23.

Choice of the Modulation Frequency

Currently in the TCN lab we have two oscillators; one is at 36MHz and the other one is at 37MHz; I was tempted to use them

but Rana suggested to have 2MHz separation between the two modulation frequencies in order to avoid to see noise in the beat note 

after the demodulation process.

So if we want to use one of them we can pick 37MHz and buy a 40MHz one.

If we have to buy two of them anyway we can go for the same 37MHz and 40MHz or shift a bit the frequencies to 39MHz and 41MHz.

I would pick 37MHz and 40MHz leaving the option to buy or not buy the second 37MHz.

Before buying all the parts and do all the work which is consequent at this choice, it is good to have some comments/opinion!