Quantum-noise limited sensitivity (of Advanced LIGO)

(An optics simulation with Finesse)

The following shows how to compute the quantum-noise limited sensitivity of an interferometric gravitational wave detector with Finesse.

The input file

%------------------------------------------------------------------------
% Finesse input file to model a Michelson interferometer
% with power and signal recycling. The setup is based on
% the aLIGO reference design, DCC number M060056
% Daniel Brown 17.05.2014
%------------------------------------------------------------------------ 

l l1 $Pin 0 nin
s s1 0 nin nprc1
# Power recycling mirror
m1 prm $prmT 37.5u 90 nprc1 nprc2
s  prc $lprc nprc2 nbsin
# Central beamsplitter
bs bs1 .5 .5 0 45 nbsin n0y n0x nbsout

# X-arm
s ichx $lmichx n0x n1x
m1 itmx $itmT 37.5u 90 n1x n2x
s armx $Larm n2x n3x
m1 etmx 5u 37.5u 89.999875 n3x n4x
attr itmx mass $Mtm zmech sus1
attr etmx mass $Mtm zmech sus1

# Y-arm
s  ichy $lmichy n0y n1y
m1 itmy $itmT 37.5u $michy_phi n1y n2y
s  army $Larm n2y n3y
m1 etmy 5u 37.5u 0.000125 n3y n4y
attr itmy mass $Mtm zmech sus1
attr etmy mass $Mtm zmech sus1

# Signal recycling mirror
s  src $lsrc nbsout nsrc1
m1 srm $srmT 37.5u $srm_phi nsrc1 nsrc2

# Force-to-position transfer function for longitudinal 
# motions of test masses
tf sus1 1 0 p $mech_fres $mech_Q
const mech_fres 1  # 9 sus-thermal spike
const mech_Q    1M # Guess for suspension Q factor
# DC readout: 100mW = michy_phi 0.07 _or_ darm_phi .00025
const michy_phi 0  
const darm_phi  .00025

const Larm 3995
const itmT 0.014
const srmT 0.2
const prmT 0.03
const Pin  125 
const Mtm  40
const srm_phi -90 
const lmichx 4.5
const lmichy 4.45
const lprc   53
const lsrc   50.525 

# A squeezed source could be injected into the dark port
sq sq1 0 0 90 nsrc2

# Differentially modulate the arm lengths
fsig darm  armx 1 0
fsig darm2 army 1 180

# Output the full quantum noise limited sensitivity
qnoisedS NSR_with_RP    1 $fs nsrc2
# Output just the shot noise limited sensitivity
qshotS   NSR_without_RP 1 $fs nsrc2

# We could also display the quantum noise and the signal 
# separately by uncommenting these two lines.
# qnoised noise 1 $fs nsrc2
# pd1     signal  $fs nsrc2

xaxis darm f log 5 5k 1000
yaxis log abs
 

The file will firstly setup all the various optical cavities (using a plane waves model). It then proceeds to suspend the arm cavity mirrors whilst setting the mechanical suspension transfer functions to a simple pendulum with a resonance at 1 Hz. Next, a gravitational wave signal is injected as a modulation to both arm `spaces', out of phase by 180 degrees. Lastly we use the qnoisedS and qshotS detectors to output the noise-to-signal ratio, or the sensitivity.

The optical layout

The optical layout is a very much simplified version of the Advanced LIGO interferometer, a Michelson interferometer with Fabry-Perot cavities in the arms, power recycling and signal recycling. Squeezed light is injected into the so-called dark port, which is also the main detection port, in which we measure the sensitivity.

Output graph

Shot-noise and quantum-noise limited sensitivity of a (simplified) Advanced LIGO setup.

The model is loosely based on the Advanced LIGO design file and thus we expect to see the peak sensitivity around 100 Hz at a sensitivity of about 10^(-23)/sqrt(Hz). We can see the both the qnoised and qshot agree at high frequencies, because they both model shot noise correctly. At low frequencies we see that they differ only qnoised takes into account the radiation pressure effects.