Question
The “Degrees of freedom transfer functions” task in notebook 14 involves applying signals to the cavity mirrors or the power recycling or signal recycling mirror to modulate the differential arm, common arm, power recycling and signal recycling cavity lengths, respectively. We look at a photodetector at the output port of the interferometer, which is demodulated at the signal frequency for each step on the x-axis.
Here is the differential arm response:
Differential arm length to output port response
l LI 125.0 0.0 0.0 nin
s s1 0.0 nin nprc1
m1 PRM 0.03 3.75e-05 90.0 nprc1 nprc2
s prc 53.0 nprc2 nbsin
bs bs1 0.5 0.5 0.0 45.0 nbsin n0y n0x nbsout
s sx 4.5 n0x n1x
m1 ITMX 0.014 3.75e-05 90.0 n1x n2x
attr ITMX mass 40.0
s Lx 3995.0 n2x n3x
m1 ETMX 0.0 3.75e-05 90.0 n3x n4x
attr ETMX mass 40.0
s sy 4.45 n0y n1y
m1 ITMY 0.014 3.75e-05 0.0 n1y n2y
attr ITMY mass 40.0
s Ly 3995.0 n2y n3y
m1 ETMY 0.0 3.75e-05 0.0 n3y n4y
attr ETMY mass 40.0
s src 50.525 nbsout nsrc1
m1 SRM 0.2 3.75e-05 90.0 nsrc1 nout
l LO 1.0 0.0 90.0 nLO2
s sLO1 0.0 nout nLO3
bs bsLO 0.5 0.5 0.0 0.0 nLO3 nHD1 nHD2 nLO2
hd TF 180.0 nHD1 nHD2
xaxis sig f log 10 10000 1000
fsig sig ETMX z 1.0 0.0 1.0
fsig sig ETMY z 1.0 180.0 1.0
yaxis log abs- Plot
- Script
The response is 1 × 10^11 W/m at low frequencies, i.e. very strong. This is expected since the Michelson is configured to sit at the dark fringe condition and therefore couples differential arm length changes most strongly to the output port. So this looks good.
The problem is that when we modulate the other degrees of freedom, we still see a non-zero signal at the same port. Here is the common arm length change:
Common arm length to output port response
l LI 125.0 0.0 0.0 nin
s s1 0.0 nin nprc1
m1 PRM 0.03 3.75e-05 90.0 nprc1 nprc2
s prc 53.0 nprc2 nbsin
bs bs1 0.5 0.5 0.0 45.0 nbsin n0y n0x nbsout
s sx 4.5 n0x n1x
m1 ITMX 0.014 3.75e-05 90.0 n1x n2x
attr ITMX mass 40.0
s Lx 3995.0 n2x n3x
m1 ETMX 0.0 3.75e-05 90.0 n3x n4x
attr ETMX mass 40.0
s sy 4.45 n0y n1y
m1 ITMY 0.014 3.75e-05 0.0 n1y n2y
attr ITMY mass 40.0
s Ly 3995.0 n2y n3y
m1 ETMY 0.0 3.75e-05 0.0 n3y n4y
attr ETMY mass 40.0
s src 50.525 nbsout nsrc1
m1 SRM 0.2 3.75e-05 90.0 nsrc1 nout
l LO 1.0 0.0 90.0 nLO2
s sLO1 0.0 nout nLO3
bs bsLO 0.5 0.5 0.0 0.0 nLO3 nHD1 nHD2 nLO2
hd TF 180.0 nHD1 nHD2
xaxis sig f log 10 10000 1000
fsig sig ETMX z 1.0 0.0 1.0
fsig sig ETMY z 1.0 0.0 1.0
yaxis log abs- Plot
- Script
The signal is definitely not zero, rather it’s just over 2500 W/m at low frequencies.
The PRC and SRC length responses below are almost zero at low frequencies, but not quite zero.
Power recycling cavity length to output port response
l LI 125.0 0.0 0.0 nin
s s1 0.0 nin nprc1
m1 PRM 0.03 3.75e-05 90.0 nprc1 nprc2
s prc 53.0 nprc2 nbsin
bs bs1 0.5 0.5 0.0 45.0 nbsin n0y n0x nbsout
s sx 4.5 n0x n1x
m1 ITMX 0.014 3.75e-05 90.0 n1x n2x
attr ITMX mass 40.0
s Lx 3995.0 n2x n3x
m1 ETMX 0.0 3.75e-05 90.0 n3x n4x
attr ETMX mass 40.0
s sy 4.45 n0y n1y
m1 ITMY 0.014 3.75e-05 0.0 n1y n2y
attr ITMY mass 40.0
s Ly 3995.0 n2y n3y
m1 ETMY 0.0 3.75e-05 0.0 n3y n4y
attr ETMY mass 40.0
s src 50.525 nbsout nsrc1
m1 SRM 0.2 3.75e-05 90.0 nsrc1 nout
l LO 1.0 0.0 90.0 nLO2
s sLO1 0.0 nout nLO3
bs bsLO 0.5 0.5 0.0 0.0 nLO3 nHD1 nHD2 nLO2
hd TF 180.0 nHD1 nHD2
xaxis sig f log 10 10000 1000
fsig sig PRM z 1.0 0.0 1.0
yaxis log abs- Plot
- Script
Signal recycling cavity length to output port response
l LI 125.0 0.0 0.0 nin
s s1 0.0 nin nprc1
m1 PRM 0.03 3.75e-05 90.0 nprc1 nprc2
s prc 53.0 nprc2 nbsin
bs bs1 0.5 0.5 0.0 45.0 nbsin n0y n0x nbsout
s sx 4.5 n0x n1x
m1 ITMX 0.014 3.75e-05 90.0 n1x n2x
attr ITMX mass 40.0
s Lx 3995.0 n2x n3x
m1 ETMX 0.0 3.75e-05 90.0 n3x n4x
attr ETMX mass 40.0
s sy 4.45 n0y n1y
m1 ITMY 0.014 3.75e-05 0.0 n1y n2y
attr ITMY mass 40.0
s Ly 3995.0 n2y n3y
m1 ETMY 0.0 3.75e-05 0.0 n3y n4y
attr ETMY mass 40.0
s src 50.525 nbsout nsrc1
m1 SRM 0.2 3.75e-05 90.0 nsrc1 nout
l LO 1.0 0.0 90.0 nLO2
s sLO1 0.0 nout nLO3
bs bsLO 0.5 0.5 0.0 0.0 nLO3 nHD1 nHD2 nLO2
hd TF 180.0 nHD1 nHD2
xaxis sig f log 10 10000 1000
fsig sig SRM z 1.0 0.0 1.0
yaxis log abs- Plot
- Script
Our first hypothesis was that the asymmetries between the arms led to coupling between the bright and dark ports of the main Michelson beam splitter. Removing the difference in length between the beam splitter and ITMs in both arms gets rid of most of the signal in the common arm length signal at the output port, but we still see non-zero signal (similarly, PRC and SRC signals are also still non-zero):
Common arm length to output port response
l LI 125.0 0.0 0.0 nin
s s1 0.0 nin nprc1
m1 PRM 0.03 3.75e-05 90.0 nprc1 nprc2
s prc 53.0 nprc2 nbsin
bs bs1 0.5 0.5 0.0 45.0 nbsin n0y n0x nbsout
s sx 4.5 n0x n1x
m1 ITMX 0.014 3.75e-05 90.0 n1x n2x
attr ITMX mass 40.0
s Lx 3995.0 n2x n3x
m1 ETMX 0.0 3.75e-05 90.0 n3x n4x
attr ETMX mass 40.0
s sy 4.5 n0y n1y
m1 ITMY 0.014 3.75e-05 0.0 n1y n2y
attr ITMY mass 40.0
s Ly 3995.0 n2y n3y
m1 ETMY 0.0 3.75e-05 0.0 n3y n4y
attr ETMY mass 40.0
s src 50.525 nbsout nsrc1
m1 SRM 0.2 3.75e-05 90.0 nsrc1 nout
l LO 1.0 0.0 90.0 nLO2
s sLO1 0.0 nout nLO3
bs bsLO 0.5 0.5 0.0 0.0 nLO3 nHD1 nHD2 nLO2
hd TF 180.0 nHD1 nHD2
xaxis sig f log 10 10000 1000
fsig sig ETMX z 1.0 0.0 1.0
fsig sig ETMY z 1.0 0.0 1.0
yaxis log abs- Plot
- Script
Ok, so we get some of what looks like numerical junk at low frequencies which might be the suspensions. Let’s remove the mirror masses so suspension effects don’t apply. What does common arm look like now?
Common arm length to output port response
l LI 125.0 0.0 0.0 nin
s s1 0.0 nin nprc1
m1 PRM 0.03 3.75e-05 90.0 nprc1 nprc2
s prc 53.0 nprc2 nbsin
bs bs1 0.5 0.5 0.0 45.0 nbsin n0y n0x nbsout
s sx 4.5 n0x n1x
m1 ITMX 0.014 3.75e-05 90.0 n1x n2x
s Lx 3995.0 n2x n3x
m1 ETMX 0.0 3.75e-05 90.0 n3x n4x
s sy 4.5 n0y n1y
m1 ITMY 0.014 3.75e-05 0.0 n1y n2y
s Ly 3995.0 n2y n3y
m1 ETMY 0.0 3.75e-05 0.0 n3y n4y
s src 50.525 nbsout nsrc1
m1 SRM 0.2 3.75e-05 90.0 nsrc1 nout
l LO 1.0 0.0 90.0 nLO2
s sLO1 0.0 nout nLO3
bs bsLO 0.5 0.5 0.0 0.0 nLO3 nHD1 nHD2 nLO2
hd TF 180.0 nHD1 nHD2
xaxis sig f log 10 10000 1000
fsig sig ETMX z 1.0 0.0 1.0
fsig sig ETMY z 1.0 0.0 1.0
yaxis log abs- Plot
- Script
It looks like we removed most of the numerical junk now. But it’s still non-zero.
It looks to us like the path that the light takes after it splits at the beam splitter is identical. The arm cavity mirror parameters are identical, the lengths are identical. The x-arm mirrors are tuned to 90° and the y-arm 0°, but that’s just to set the Michelson to destructively interfere common motion at the beam splitter’s output port. What else could be causing this non-zero response from common (and PRM and SRM) length changes to the output port?
Answer
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