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 Tue Jan 4 18:26:59 2022, Anchal, Update, BHD, Tested 2" PR2 candidates transmission Wed Jan 5 17:46:04 2022, Anchal, Update, BHD, Tested 2" PR2 candidates transmission Mon Jan 10 18:20:45 2022, Anchal, Update, BHD, Tested 2" PR2 candidates transmission Tue Jan 11 15:21:16 2022, Anchal, Update, BHD, PR2 transmission calculation Thu Jan 13 17:10:55 2022, Anchal, Update, BHD, PR2 transmission calculation Fri Jan 14 03:07:04 2022, Koji, Update, BHD, PR2 transmission calculation Fri Jan 14 11:00:29 2022, Anchal, Update, BHD, PR2 transmission calculation Fri Jan 14 13:46:25 2022, Anchal, Update, BHD, PR2 transmission calculation updated Wed Jan 19 16:22:48 2022, Anchal, Update, BHD, PR2 transmission calculation updated Thu Jan 20 01:48:02 2022, Koji, Update, BHD, PR2 transmission calculation updated
Message ID: 16598     Entry time: Wed Jan 19 16:22:48 2022     In reply to: 16587     Reply to this: 16602
 Author: Anchal Type: Update Category: BHD Subject: PR2 transmission calculation updated

I have further updated my calculation. Please find the results in the attached pdf.

Following is the description of calculations done:

### Arm cavity reflection:

Reflection fro arm cavity is calculated as simple FP cavity reflection formula while absorbing all round trip cavity scattering losses (between 50 ppm to 200 ppm) into the ETM transmission loss.

So effective reflection of ETM is calculated as

$r_{\rm ETMeff} = \sqrt{1 - T_{\rm ETM} - L_{\rm RT}}$

$r_{\rm arm} = \frac{-r_{\rm ITM} + r_{\rm ETMeff}e^{2i \omega L/c}}{1 - r_{\rm ITM} r_{\rm ETMeff}e^{2 i \omega L/c}}$

The magnitude and phase of this reflection is plotted in page 1 with respect to different round trip loss and deviation of cavity length from resonance. Note that the arm round trip loss does not affect the sign of the reflection from cavity, at least in the range of values taken here.

### PRC Gain

The Michelson in PRFPMI is assumed to be perfectly aligned so that one end of PRC cavity is taken as the arm cavity reflection calculated above at resonance. The other end of the cavity is calculated as a single mirror of effective transmission that of PRM, 2 times PR2 and 2 times PR3. Then effective reflectivity of PRM is calculated as:

$r_{\rm PRMeff} = \sqrt{1 - T_{\rm PRM} - 2T_{\rm PR2} - 2T_{\rm PR3}}$

$t_{\rm PRM} = \sqrt{T_{\rm PRM}}$

Note, that field transmission of PRM is calculated with original PRM power transmission value, so that the PR2, PR3 transmission losses do not increase field transmission of PRM in our calculations. Then the field gain is calculated inside the PRC using the following:

$g = \frac{t_{\rm PRM}}{1 - r_{\rm PRMeff} r_{\rm arm}e^{2 i \omega L/c}}$

From this, the power recycling cavity gain is calculated as:
$G_{\rm PRC} = |g|^2$

The variation of PRC Gain is showed on page 2 wrt arm cavity round trip losses and PR2 transmission. Note that gain value of 40 is calculated for any PR2 transmission below 1000 ppm. The black verticle lines show the optics whose transmission was measured. If V6-704 is used, PRC Gain would vary between 15 and 10 depending on the arm cavity losses. With pre-2010 ITM, PRC Gain would vary between 30 and 15.

### LO Power

LO power when PRFPMI is locked is calculated by assuming 1 W of input power to IMC. IMC is assumed to let pass 10% of the power ($L_{\rm IMC}=0.1$). This power is then multiplied by PRC Gain and transmitted through the PR2 to calculate the LO power.

$P_{\rm LO, PRFPMI} = P_{\rm in} L_{\rm IMC}G_{\rm PRC}T_{\rm PR2}$

Page 3 shows the result of this calculation. Note for V6-704, LO power would be between 35mW and 15 mW, for pre-2010 ITM, it would be between 15 mW and 5 mW depending on the arm cavity losses.

The power available during alignment is simply given by:
P_{\rm LO, align, PRM} = P_{\rm in} L_{\rm IMC} T_{\rm PRM} T_{\rm PR2}

P_{\rm LO, align, no PRM} = P_{\rm in} L_{\rm IMC} T_{\rm PR2}

If we remove PRM from the input path, we would have sufficient light to work with for both relevant optics.

I have attached the notebook used to do these calculations. Please let me know if you find any mistake in this calculation.

 Attachment 1: PR2transmissionSelectionAnalysis.pdf  6.131 MB  Uploaded Wed Jan 19 17:34:58 2022
 Attachment 2: PR2_Trans_Calc.ipynb.zip  2 kB  Uploaded Wed Jan 19 17:35:40 2022
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