ELOG - Power recycling gain estimation from arm loss measurement

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Tue Dec 8 15:57:16 2015, yutaro, Summary, LSC, Power recycling gain estimation from arm loss measurement

Fri Dec 11 09:35:44 2015, yutaro, Update, LSC, Power recycling gain estimation from arm loss measurement

Fri Dec 11 13:28:36 2015, Koji, Update, LSC, Power recycling gain estimation from arm loss measurement

Fri Dec 11 15:37:50 2015, yutaro, Update, LSC, Power recycling gain estimation from arm loss measurement

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 $T_\mathrm{ITM}\simeq T_\mathrm{tot}=\frac{2\pi}{401}=0.01566$ , $T_\mathrm{PRM}=0.05637$ , 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.

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