Comment to: Why do we see non-zero common, PRC and SRC length signals at the Michelson’s output port?

In the log post https://logbooks.ifosim.org/iucaa2019/2019/12/21/why-do-we-see-non-zero-common-prc-and-src-length-signals-at-the-michelsons-output-port/ there was uncertainty about where some unexpected common-mode arm motion coupling into the anti-symmetric output was coming from. We’ll focus on the last simulation repeated here:

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

Although it looks like the the model is symmetric there is still an additional 90 degrees of detuning on both ITMX and ETMX. Why does this cause a common mode coupling?

When we specify a detuning in degrees at a mirror what we are really specifying is a displacement of the mirror – this phase is relative to the default wavelength in the model. By default in Finesse this is 1064nm. This means the x-arm is 90 degrees shorter than the y-arm, albeit a microscopic length difference. The detuning phase in terms of the optical field frequency f is:

\phi = k z = \frac{2\pi}{\lambda} z = \frac{2\pi f}{c} z

So if we have a detuning in one arm different frequency components of the optical field will pick up slightly different phases. As we are generating sidebands in the arms by exciting the end mirrors the x-arm sidebands pickup this phase and do not destructively interfere anymore, hence a non-zero common-mode coupling.

To remove this length difference we can tune the beamsplitter instead with, keeping it at normal incidence and 45 degrees of tuning.

bs bs1 0.5 0.5 45.0 0 nbsin n0y n0x nbsout

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 45.0 0 nbsin n0y n0x nbsout
s sx 4.5 n0x n1x
m1 ITMX 0.014 3.75e-05 0.0 n1x n2x
s Lx 3995.0 n2x n3x
m1 ETMX 0.0 3.75e-05 0.0 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

Running this we now see it’s noisy throughout the frequency range, which is due to numerical errors not cancelling out correctly, but this is effectively a zero coupling.

Cross-references

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