real-time.
We analyze the output in total frequency, to which we apply a third order polynomial least-square fit to remove large trends. Figure8.1shows the total frequency before the fit, while Figure 8.2 and Figure 8.3 show the post-fit residuals for a locking and a non-locking ISC beatnote, respectively.
As expected, the non-locking beatnotes carry large frequency fluctuations due to unsuppressed laser frequency noise. The locking beatnotes residuals, on the other hand, appear to be dominated by numerical noise and a residual trend after the polynomial fit.
8.2 noise levels
We estimate PSDs using the log-scale power spectral density (LPSD) method described in [89]2.
We overlay all plots with a 10 pm noise allocation curve, which is a typical target noise level for a single link in LISA [10]. It is given in units of frequency as
pSIFO(f) = 2πf
1064 nm ·10 pm√ Hz ·
s 1+
2 mHz f
4
(8.1)
8.2.1 Non-locking ISC interferometer
We show in fig.8.4the noise level in a non-locking ISC interferometer, isc12. We observe that it is dominated by laser noise.
We can derive a simple model for the residual noise level by considering only laser noise in eqs. (7.7b) and (7.28b), which yields
isce12 ≈D12D21p˙12−p˙12, (8.2)
where ˙p12is the laser noise of the primary laser, whose noise level is shown in grey in fig.8.4. The PSD can be estimated to be proportional to the squared magnitude of the fourier transform (cf. appendix C for more information PSD estimation), which gives
|F [isce12]|2(f)≈(ei2π(d12o+d21o)f −1)F[p˙12] (f)2
≈4 sin2(2πf d)|F[p˙12] (f)|2 ,
(8.3) wheredis the average arm length in seconds. We overlay this model with our simulated data in fig. 8.4, which shows perfect agreement.
1 This is with the highest level of C++compiler optimizations (-O3), on a Linux workstation equipped with an AMD Ryzen3700x. Compilation took an additional 2 min 5 s.
2 We use a python implementation of this method developed by C. Vorndamme at AEI, with the following parameters:olap="default", bmin=1, Lmin=0, Jdes=2000, Kdes=1000, order=-1, win=np.kaiser, psll=300.
94 simulation results
We show in fig.8.5the noise level in a non-locking reference interferometer, ref12.
In our locking scheme N1, one of the two reference interferometers on each spacecraft is always used for locking, such that all noise is cancelled in the adjacent ref13. Since both reference interferometers on the same spacecraft interfere the same lasers, any common noise between ref13and ref12will be cancelled in both interferometers. This applies for example to laser frequency noise, but not to noise sources unique to the two reference interferome-ters.
Inspecting fig.8.5, we observe a colored noise which can be explained by these non-common noise sources, which in our model are optical pathlength noise (cf. appendix D.6), backlink noise (cf. appendix D.4), and readout noise (cf. appendix D.5). We can add up the PSDs for these noise terms in both
8.2 noise levels 95
Figure 8.6:Residual noise in locking beat-note. We are limited by numerical effects.
Simulating just fluc-tuations allows for a significantly lower numerical noise floor.
10−4 10−3 10−2 10−1 100
10−15 10−11 10−7 10−3 101
Fourier frequency in Hz ASDinHz/√ Hz
Raw laser noise 10pm allocation Locking ISC beatnote Numerical noise (lower limit) Locking ISC fluctuations
interferometers to explain the observed noise floor. Note that this includes a factor √
2 to account for non-common noise terms inbothinterferometers, which due to the laser locking condition are both transferred to ref12.
This estimate is only valid for locking configurations N1, N3and N5, which use one reference interferometer per spacecraft for locking. In configura-tions N2, N4and N6, on the other hand, we expect a laser noise dominated residual for one of the reference beatnotes, similar to that described in sec-tion8.2.1.
8.2.3 Locking interferometer
Since we model a perfect frequency lock in section7.3, we would expect no noise in the locking beatnotes. In practice, the noise level will be limited by numerical effects, such as the limited dynamic range of our variables. We can give a rough estimate for a lower limit of the expected numerical quantization noise by estimating the least significant bit of a double precision variable as
LSB=1.1×10−16×Mag , (8.4)
where Mag is the magnitude of the variable. Mag is given in our case as the value of the respective beatnote frequency before detrending, so around 10 MHz. Following [50], numerical quantization noise causes a white noise at the level
pSLSB(f) = pLSB
6fs . (8.5)
We show in fig. 8.4 the noise level in a locking interferometer, isc31. We observe that our noise floor is not perfectly white, and is about one order of magnitude above the lower limit derived above.
Since the shift to THE, all onboard filtering as well as the polynomial detrend-ing was performed on this variable in total frequency, we would expect some
96 simulation results resid-ual noise level is due to numerical effects.
accumulation of numerical errors, which can explain this increased noise level.
Overall, the noise is more than a factor ten below the 10 pm allocation, such that we are confident the simulated variables can be used for data analysis studies.
To verify that our locking conditions are implemented correctly, we also overlay the PSD computed from the variable containing just the frequency fluctuations, as described in section4.3.2. We see that we get a white noise floor at 2×10−15Hz/√
Hz, many orders of magnitude below the required levels.