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Entry  Mon Apr 29 09:17:20 2013, Evan, Notes, PMC, Choice of modulation frequency for PMC hom_vary_fpdh.pnghom_vary_L.png
    Reply  Mon Apr 29 23:13:36 2013, Evan, Notes, PMC, Choice of modulation frequency for PMC 
    Reply  Mon May 13 01:34:27 2013, Evan, Notes, PMC, Choice of modulation frequency for PMC minDetPlot3mirror.pdffsrSweep3a.pdffsrSweep3b.pdf
Message ID: 1174     Entry time: Mon May 13 01:34:27 2013     In reply to: 1161
Author: Evan 
Type: Notes 
Category: PMC 
Subject: Choice of modulation frequency for PMC 

The plots aren't right because I took the two-mirror mode spacing formula from Kogelnik and Li without applying the necessary modifications for a 3-ring cavity. The correct formula for mode (m,n) is $f_{mn} / f_\text{FSR} = (q+1) + (m + n + 1) \arccos(1-2L/R) / 2 \pi + \eta /2$, where $q$ is the axial mode number, $L$ is the half of the round-trip length, and $\eta$ is 1 if $m$ is odd and 0 if $m$ is even. (Note: for a 4-mirror cavity, $\eta$ is 0 always.) For reference, the K&L formula for a two-mirror cavity is $f_{mn} / f_\text{FSR} = (q+1) + (m + n + 1) \arccos(1 - L/R) / \pi$, where $L$ is half of the round trip length.

Instead of making more scatter plots, for each value of $g$ I computed the distance (in MHz) from the fundamental resonance to the nearest HOM resonance (up to order 20); the result is shown in the first attachment. I then picked the most promising $g$ factors and simulated a frequency sweep across a full FSR for a 3-mirror cavity with $L$ = 20 cm and $F \simeq 300$; the results are in the second and third attachments. Each mode is labeled with its order number, as well as 'e' or 'o' depending on whether $m$ is even or odd. I picked a arbitrary uniform amplitude for the HOMs, so these plots are only meant to indicate the locations and widths of the resonances. I've spot checked these plots against a Finesse model, so I'm reasonably confident that I've got the formula right this time.

I think the moral here is that the nearest HOM resonance is going to be about 16 MHz away from the fundamental, assuming $L$ = 20 cm. If we make $L$ = 10 cm, we can get to 32 MHz, but (depending on how bad the intensity noise at 30 to 50 MHz is) this potentially requires increasing the finesse to something like 600 to get the required intensity filtering.

If we go with a 4-mirror cavity, the modes don't have this $\eta$ degeneracy breaking, and there are $g$ factors for which the nearest HOM resonance is more like 30 MHz for $L$ = 20 cm. I have plots for this, but I want to check them against a Finesse model.

Attachment 1: minDetPlot3mirror.pdf  153 kB  | Hide | Hide all
minDetPlot3mirror.pdf
Attachment 2: fsrSweep3a.pdf  259 kB  | Hide | Hide all
fsrSweep3a.pdf
Attachment 3: fsrSweep3b.pdf  260 kB  | Hide | Hide all
fsrSweep3b.pdf
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