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Entry  Sun Oct 9 17:13:46 2011, kiwamu, Update, LSC, First attempt to estimate mode matching efficiency using interferometer mm_reflection.png
    Reply  Mon Oct 10 00:01:26 2011, Koji, Update, LSC, First attempt to estimate mode matching efficiency using interferometer 
       Reply  Mon Oct 10 13:52:04 2011, kiwamu, Update, LSC, RE: First attempt to estimate mode matching efficiency using interferometer 
Message ID: 5639     Entry time: Sun Oct 9 17:13:46 2011     Reply to this: 5640
Author: kiwamu 
Type: Update 
Category: LSC 
Subject: First attempt to estimate mode matching efficiency using interferometer 

The efficiency of the mode matching (MM) to PRC (Power-Recycling Cavity) has been estimated by using the interferometer.

The estimated MM efficiency is about 74 % when losses in the cavity are assumed to be zero.



 There had been an issue that the recycling gain didn't go to the designed high value of about 42  (#5541).
One of the possibilities is a low efficiency in the MM to PRC (also see #5541).
Although the MM efficiency had been measured using a beam scan ( see a summary on the wiki) a long time ago, it haven't been verified.
Therefore the MM has to be reviewed by using the real interferometer.


 The concept of this measurement is observe the amount of the reflected light from a power-recycled cavity and estimate the MM efficiency based on the measured reflectivities.
 Since using the real PRC (consisting of BS, ITMs and PRM) could be a too complicated system for this measurement,
simpler cavities, namely Power-Recycled ITMX and ITMY (PRX and PRY), were used to examine the MM efficiency.
 The measurement goes in the following order :
    (1) Measurement of the amount of the single-bounce reflection from PRM with BS and ITMs misaligned.
    (2) Lock PRX or PRY to carrier resonance.
    (3) Alignment of PRX/Y to maximize the intracavity power. This time ASDC was used as a monitor of the intracavity power.
    (4) Measurement of the amount of the reflected light when the cavity is in resonance. The value in REFLDC was averaged in 100 sec.
     => done by tdsavg 100 C1:LSC-REFLDC_OUT
The same measurement was performed for both PRX and PRY.
 - locking parameters -
  Sensor = REFL11_I
  Whitening gain = 10 (30 dB)
  UGF ~ 200 Hz


In order to estimate the relation between the MM efficiency vs. the reflected light, two models are considered:
   (1) simple model => no loss and no sidebands
   (2) sideband-included model => no loss but sidebands are taken into the account of the reflection.
(1) In the simple model the reflectivity Prefl / Pin is expressed by
         [Reflectivity]  = Prefl / Pin = Z * Rcav +  (1- Z) * Rprm
where Z is MM efficiency and Rprm is the reflectivity of PRM
and Rcav is the reflectivity of PRX/Y when it's resonance and it is defined by
         Rcav = | rprm - ritm t2BS|2 / |1 -rprm ritm t2BS |2
Tprm = 5.75% and Titm = 1.4 % are assumed in all the calculations.
In the first equation the first term represents the mode matched light and hence it couples with the cavity through Rcav.
The second term is the non-mode-matched light and because they are not interacting with the cavity they will be simply reflected by PRM through Rprm.
(2) In reality two phase-modulated light (11 MHz and 55 MHz) will behave differently from the carrier.
For example when the carrier is in resonance the sidebands will be anti-resonance against the cavity.
So that the amount of REFLDC will be slightly bigger because of the reflection of the sidebands.
      Prefl = Z * Rcav * Pc + Z * Ranti * Ps +  (1- Z) * Rprm * (Pc + Ps)
where Pc and Ps are the power in the carrier light and the sidebands respectively.
And Ranti is the reflectivity of the anti-resonance PRX/Y, which can be obtained by replacing the minus sign by the plus sign in the equation of Rcav shown above.
It is assumed that the sum of the carrier power and sidebands power is the incident power Pin = Pc + Ps.
The power in the carrier and the sidebands were estimated based on the OSA measurement (#5519), so that
          Pc / Pin = |J0(0.14)|2 * |J0(0.17)|2 =  0.976
          Ps / Pin = 2 * |J1(0.14)|2 + 2 * |J1(0.17)|2 =  0.024


Here are the measured values in REFLDC

 -- Measurement 1 : PRX
    Single bounce from PRM = 4802.27 counts
       ==> the incident power = 5095.25 counts
    Reflected light from PRX = 4433.88 counts
      ==> Reflectivity = 0.8702
-- Measurement 2 : PRY
    Single bounce from PRM = 4833.05 counts
       ==> the incident power = 5127.05 counts
    Reflected light from PRX = 4444.48 counts
      ==> Reflectivity = 0.86672
On average the reflectivity of power-recycled ITM cavity was 0.868 with a standard deviation of  0.001744.
Actually the standard deviation estimated here is not fair because the measurement was done by only twice,
but my intention was that I wanted to see how the error can affect the estimation of the MM efficiency.
Here is a plot comparing the model curves and the measured values with 5 sigma error box (5 times of measured standard deviation).


It is shown that the mode matching efficiency is 73.7 % when the sideband-included model is considered.
With the 5 sigma deviation it can go from 65% to 82% but it is still low and lower than the beam scan measurement ( see a summary on the wiki). 

Anyways the estimated MM efficiency with the sidebands effect included and without loss effect is

        MM efficiency = 73.7 +/- 1.7 % (1 sigma error)  or +/- 8.7 % (5 sigma error)

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