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Entry  Tue Dec 8 15:57:16 2015, yutaro, Summary, LSC, Power recycling gain estimation from arm loss measurement 
    Reply  Fri Dec 11 09:35:44 2015, yutaro, Update, LSC, Power recycling gain estimation from arm loss measurement LaserOptikAR.png
       Reply  Fri Dec 11 13:28:36 2015, Koji, Update, LSC, Power recycling gain estimation from arm loss measurement 
          Reply  Fri Dec 11 15:37:50 2015, yutaro, Update, LSC, Power recycling gain estimation from arm loss measurement PRG_plot.png
Message ID: 11864     Entry time: Tue Dec 8 15:57:16 2015     Reply to this: 11872
Author: yutaro 
Type: Summary 
Category: LSC 
Subject: Power recycling gain estimation from arm loss measurement 

I estimated power recycling gain with the results of arm loss measurement.

From elog 11818 and 11857, round trip losses including transmittivity of ETM of Y arm and X arm (let us call them T_\mathrm{loss,Y} and T_\mathrm{loss,X}) are 229+13.7=243 ppm and 483+13.7=495 ppm, respectively.

 

How I calculated:

I used the following formula.

Amplitude reflectivity of an arm cavity r_\mathrm{FP}

r_\mathrm{FP}=\sqrt{1-\frac{4T_\mathrm{ITM}T_\mathrm{loss}}{T^2_\mathrm{tot}}}   (see elog 11816)

Amplitude reflectivity of FPMI r_\mathrm{FPMI}

r_\mathrm{FPMI}=\frac{1}{2}(r_\mathrm{FP,X}+r_\mathrm{FP,Y})

With power transmittivity of PRM T_\mathrm{PRM} and amplitude reflectivity of PRM r_\mathrm{PRM}, power recycling gain is

\mathrm{PRG}=\frac{T_\mathrm{PRM}}{(1-r_\mathrm{PRM}r_\mathrm{FPMI})^2}.

 I assumed T_\mathrm{ITM}\simeq T_\mathrm{tot}=\frac{2\pi}{401}=0.01566T_\mathrm{PRM}=0.05637, and r_\mathrm{PRM}=\sqrt{1-T_\mathrm{PRM}}, and then I got

PRG = 9.8.

Since both round trip losses have relative error of ~ 4 % and PRG is proportional to inverse square of T_\mathrm{loss} up to the leading order of it, relative error of PRG can be estimated as ~ 8 %, so PRG = 9.8 +/- 0.8

 

Discussion

According to elog 11691, which says TRX and TRY level was ~125 when DRFPMI was locked, power recycling gain was \mathrm{PRG}=125\times T_\mathrm{PRM}=7.0 at the last DRFPMI lock.

Measured PRG is lower than PRG estimated here, but it is natural because various causes such as mode mismatch between PRC mode and arm cavity mode, imperfect contrast of FPMI, and so on could decrease PRG, which Eric suggested to me. 

 

Added on Dec 9

If T_\mathrm{loss,X} were as small as T_\mathrm{loss,Y}, PRG would be 16.0. PRC would be still under coupled.  

ELOG V3.1.3-