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Message ID: 1115     Entry time: Wed Nov 5 12:41:36 2008
Author: Alberto 
Type: Update 
Category: LSC 
Subject: Absolute Length and g-factor measurements conclusions 
Absolute Length and g-Factor Measurement for the 40m Arm Cavities, Summary of Results

MOTIVATION OF THE EXPERIMENT
Lately locking the interferometer in the so called spring configuration (SRC on +166 MHz sideband) has been difficult and a possible resonance of an higher order mode of the +166 MHz sideband in the arms was
hypothesized as the cause. We wanted to know the frequencies of the HOMs of the sidebands and see where they are, relatively to the carrier resonance.

THE EXPERIMENTAL TECHNIQUE IN BRIEF
A second laser beam from an NPRO is injected into the interferometer through the AS port. The beam is mode matched to the arm cavities so that it can resonate inside of these. The secondary beam interferes with
the PSL beam and the incident intensity on one end mirror, excluding by now any higher mode, is I(t)=I1+I2+(interference terms)*exp[-i*(f1-f2)*t]. The last term comes from the beat between the two fields at the
relative frequency of the two lasers. For beating frequencies multiple of the FSR of the cavity, the beat gets transmitted and appears at the trans PD.
Whereas the PSL has a constant frequency, the NPRO frequency fluctuates, so that the relative phase between the two is not constant. To prevent that, a PLL servo locks the phase of the NPRO to that of the PSL.
The result is a beat frequency at the steady and tunable value set by the local oscillator of the PLL.

Length Measurement
One arm at a time, the cavity is locked to the TEM00 mode of the main laser. The beat frequency is then scanned for a few cavity FSRs and the transmitted power is measured. A linear fit of the resonant frequencies gives
us the FSR of the cavity.

g-factor Measurement
For non-planar Fabry-Perot cavities, the HOMs of the laser are not degenerate and resonate in the cavity at frequencies different from the correspondent fundamental mode. The shift in frequency is measured by the
Transverse Mode Spacing (TMS) and it is a function of the g-factors of the cavity:

TMS=FSR*acos[sqrt(g1*g2)]/pi

with g1=1-L/R1, where L is the cavity absolute length and R1 the radius of curvature of the input mirror, and similarly for g2 for the end mirror.
We measured the TMS by means of the beat between an HOM of the main laser and the TEM00 of the secondary beam. To do that we locked the cavity to either TEM01/10 and looked at the transmitted power for frequencies
of the beat around the TMS expected from the design parameters of the cavity.
Since the phase of the intensity of the beat between TEM01/10 and TEM00 has only DC components if measured across a symmetric portion of the spot, it is necessary to brake the symmetry of the incident beam on the
PD by chopping it just before it hits the sensor.
We approximated g1=1 for the ITMs. The effect of an astigmatic ETM is to brake the degeneracy of the TEM10 and TEM01 modes and split their resonant frequencies. By measuring that shift, we can evaluate the radius
of curvature of the mirror for the axis of the two transverse modes.

EXPERIMENTAL RESULTS
X Arm
FSR     =  (3897627 +/- 5 )   Hz
L       = (38.45833  +/- 0.00005) m
g2x     =   0.31197  +/- 0.00004
g2y     =   0.32283  +/- 0.00004
R-ETM_x = (55.8957   +/- 0.0045) m
R-ETM_y = (56.7937   +/- 0.0038) m

Y Arm
FSR     = ( 3879252 +/- 30 )  Hz
L       = (38.6462   +/- 0.0003) m
g2x     =   0.31188  +/- 0.00004
g2y     =   0.32601  +/- 0.00004
R-ETM_x = (56.1620   +/- 0.0013) m
R-ETM_y = (57.3395   +/- 0.0011) m


CONCLUSIONS
The attached graphs,one for the X arm and the other for the Y arm, plot the distributions of the first HOMs of the sidebands near the carrier resonance in the arm cavities. As it appears, the resonances of
the +166 sideband are far enough for not resonating in the arm cavities if the arms are locked to the carrier.
We have to look for something else to explain the locking problem of the interferometer in the spring configuration.
Attachment 1: 2008-11-04_file_02-05.png  20 kB  | Hide | Hide all
2008-11-04_file_02-05.png
Attachment 2: HOM_resonances_Xarm.png  24 kB  | Hide | Hide all
HOM_resonances_Xarm.png
Attachment 3: HOM_resonances_Yarm.png  24 kB  | Hide | Hide all
HOM_resonances_Yarm.png
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