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 Thu Nov 5 02:18:32 2015, gautam, Update, LSC, FSR and linewidth measurements with phase tracker Fri Nov 6 15:56:00 2015, gautam, Update, LSC, FSR and linewidth measurements with phase tracker Mon Nov 9 11:34:51 2015, yutaro, Update, LSC, FSR and linewidth measurements with phase tracker Mon Nov 9 14:24:36 2015, yutaro, Update, LSC, FSR and linewidth measurements with phase tracker Mon Nov 9 15:59:06 2015, ericq, Update, LSC, FSR and linewidth measurements with phase tracker Mon Nov 9 16:58:59 2015, gautam, Update, LSC, FSR and linewidth measurements with phase tracker Tue Nov 10 02:34:28 2015, gautam, Update, LSC, Updated interpretation of peaks Tue Nov 10 11:06:02 2015, yutaro, Update, LSC, Updated interpretation of peaks Tue Nov 10 11:40:03 2015, Koji, Update, LSC, Updated interpretation of peaks Tue Nov 10 16:34:00 2015, yutaro, Update, LSC, Updated interpretation of peaks Fri Nov 13 17:33:39 2015, gautam, Update, LSC, g-factor measurements Tue Nov 10 11:41:56 2015, Koji, Update, LSC, Updated interpretation of peaks Fri Nov 13 15:48:16 2015, gautam, Update, LSC, Phase tracker calibration using Rubidium standard 6x
Message ID: 11743     Entry time: Mon Nov 9 16:58:59 2015     In reply to: 11742     Reply to this: 11745
 Author: gautam Type: Update Category: LSC Subject: FSR and linewidth measurements with phase tracker
 Quote: There are two modulation frequencies that make it to the arm cavities, at ~11MHz and ~55MHz. Each of these will have their own modulation depth indepedent of each other. Bundling them together into one number doesn't tell us what's really going on.

Summary:

As an update to Yutaro's earlier post - I've done an independent study of this data, doing the fitting with MATLAB, and trying to estimate (i) the FSR, (ii) the mode matching efficienct, and (iii) the modulation depths at 11MHz and 55MHz.

The values I've obtained are as follows:

FSR = 3.9704 MHz +/- 17 kHz

Mode matching efficiency = 92.59 % (TEM00 = 1, TEM10 = 0.0325, TEM20 = 0.0475)

Modulation depth at 11MHz = 0.179

Modulation depth at 55MHz = 0.131

Details:

• To approximately locate the TEM10 and TEM20 resonances, I followed the methodology listed here (though confining myself to (m+n) = 1,2).
• To approximately locate the 11 MHz and 55 MHz sidebands, I used the mod command in MATLAB to locate approximately how far they should be from a carrier resonance.
• The results of these first two steps are demonstrated pictorially in Attachment #1. Red = carrier resonance, grey = 55MHz sideband resonance, cyan = 11MHz sideband resonance, green = TEM20 resonance, and yellow = TEM10 resonance
• The FSR was calculated by fitting the center frequencies of fits to the three carrier resonances with a lorentzian shape, vs their index. The quoted error is the 95% C.I.s generated by MATLAB
• The mode-matching efficiency was calculated by taking the fitted height of Lorentzian shapes to the TEM00, TEM10 and TEM20 shapes. The ratio of the peak heights was taken as a measure of the fraction of total power coupled into the TEM10 and TEM20 modes relative to TEM00. In calculating the final value, I took the average of the 3 available values for each peak to calculate the ratios.
• The modulation depth was calculated by approximating that the ratio   $\sqrt\frac{P_c}{P_s} = \frac{J_0(\beta)}{J_1(\beta)}$, and solving for $\beta$. Attachment #2 shows a plot of the RHS of this equation as a function of $\beta$ - the two datatips mark the location of the ratios on the LHS of the equation - both P_c and P_s were averaged over the 3 and 6 values available, respectively. The values I have obtained are different from those cited here - not sure why? The real red flag I guess is that I get the modulation depth at 11MHz to be larger than at 55MHz, whereas elog10211 reports the reverse... Do we expect a resonance for a 44MHz sideband as well? If so, it could be that the two peaks close to the carrier resonance is in fact the 55.30 MHz sideband resonance, and the peaks I've identified as 55MHz sideband resonances are in fact 44MHz sidebands.. If this were true, I would recover the modulation depth for 55.30 MHz sidebands to be approximately 0.22...

Misc Remarks and Conclusions:

• The y-scale in Attachment #1 is log(transmission) - the semilogy command in MATLAB messed up the rendering of the overlaid semi-transparent rectangles, hence the need for adopting this scale...
• I've attached the code used to split the entire scan into smaller datasets centered around each peak, and the actual fitting routine, in Attachment #3. I've not done the error analysis for the mode matching efficiency and the modulation depths, I will update this entry with those numbers as soon as I do.
• In my earlier elog11738, I had mislabelled some peaks as being sideband peaks - attachment #1 in this entry is (I think) a correct interpretation of the various peaks.
• There are two peaks on either side of every carrier resonance, spaced, on average, about 177kHz away from the resonance on either side. I am not sure what the interpretation of this peak should be - are they the 55.30 MHz resonances?
• These values should allow us to carry out alternative measurements of the round trip arm loss as estimating this from the cavity finesse seems to not be the best way to go about this.

 Attachment 1: Y_scan.pdf  191 kB
 Attachment 2: modDepth.pdf  40 kB
 Attachment 3: Matlab_code.zip  2 kB
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