As discussed in section10.2, TDI has a fundamental limit to the achievable level of noise suppression determined by the armlength mismatch of the corresponding interferometer.
However, there are additional technical noise couplings which can limit the achievable overall noise suppression.
Contrary to the secondary noises described in section 12.2.1, the residual laser noise will depend on the locking scheme. We will derive all couplings below for the special case of locking scheme N1-L12, but remark that the same principle is applicable to all other locking schemes. In addition, it should be noted that the locking schemes break the usual triangle symmetry, such that the cyclic permutations of the same variable (e.g., the three Michelson variablesX,Y andZ) will have different laser noise residuals.
We consider only the primary lasers phase fluctuations p0. Following sec-tion7.3, this laser noise term is distributed throughout the whole constellation, such that we get in our locking scheme
p12 = p13= p0, (12.19a)
p21 = p23= D21p0, (12.19b)
p31 = p32= D31p0, (12.19c)
for the different laser fluctuations.
12.3.1 Fundamental armlength mismatch
In the absence of technical imperfections, we know from chapters10and11 that the TDI combinations are designed to give a laser noise residual of the form
TDI= [DA1...An−DB1...Bn]pi (12.20)
= [DA1...An−DB1...Bn]Dlockp0 (12.21)
12.3 laser noise residuals 151
Figure 12.1: Funda-mental laser locking limit for TDIX,Yand Zwith laser locking enabled. The same analytical model we derived without laser locking (in dotted grey) still applies.
10−4 10−3 10−2 10−1 100
10−14 10−12 10−10 10−8 10−6 10−4
Fourier frequency in Hz ASDinHz/√ Hz
X simulation Y simulation Z simulation 1pm Allocation
wherepi is the laser noise for the starting spacecraft of any combination, and all other laser noise terms cancel. Here, [DA1...An−DB1...Bn]is a placeholder for the actual delay commutator defined by the photon path of the particular combination under study.
Inserting eqs. (12.19a) to (12.19c) into this equation shows that we get the same fundamental laser noise suppression limit we had before, with the only difference that the primary lasers noise appears with up to one additional delay Dlock, which does not affect the noise level.
Thus, the Fourier transform of this term can be approximated as
F [[DA1...An−DB1...Bn]Dlockp0]≈ ω∆τe−iωQdpe0(ω), (12.22) withQ=n if the photon path starts at spacecraft1andQ=n+1 otherwise.
∆τis the armlength mismatch of the TDI combination.
To verify this, we run a simulation of 105s using LISANode, with only laser noise enabled and no bias in the ranging. The light-travel time between the spacecraft are computed based on the ESA orbits, as shown in fig.3.3. We use the locking scheme N1-LA12, and a stronger anti-aliasing filter with 320 dB attenuation in order to not be limited by aliasing. We then compute the second generation TDI X,YandZusing a high interpolation order of 65, such that interpolation errors appear off-band. The result is plotted in fig. 12.1, and agrees well with the analytical model given in section10.2, overlayed in grey.
At low frequencies, we are limited by numerical errors.
12.3.2 Technical imperfections
The algorithm outlined in chapters10and11works under the assumption that the delay operators applied in post-processing are identical to those modelling the actual propagation between the spacecraft. In addition, they
152 tdi in practice
did not account for the filter operator present in the phasemeter equations given in section7.2.
We can model these effects in multiple steps. First, we can insert eqs. (12.19a) to (12.19c) into the phasemeter equations given in chapter7and those for the intermediary variables given in section12.1to get3
η12 =F(D121−1)p0, (12.23a)
η13 =F(D131−1)p0, (12.23b)
η23 =F(D231−D21)p0, (12.23c)
η32 =F(D321−D31)p0, (12.23d)
η21 =0 , (12.23e)
η31 =0 . (12.23f)
Here, we neglected the impact of the timestamping operator, which we will discuss in detail in chapter13.
As described above, we can write any TDI combination as TDI=
∑
ij∈I2
Pijηij, (12.24)
which now simplifies to
TDI=P12F(D121−1) +P13F(D131−1)
+P23F(D231−D21) +P32F(D321−D31)p0. (12.25) The delay polynomials Pij contain the delays we apply in post-processing, for which we use the different symbol Dij. They will be affected by any ranging errors in our MPR (cf. section12.3.2.1) as well as errors due to the interpolation (cf. section12.3.2.2).
Instead of modelling these errors in each delay appearing in eachPij, we will go the other way around, and replace all propagation delaysDij appearing in eq. (12.25) with offline delaysDij. Each of these replacement will introduce an additive error term, as described below. In addition, we can commute the filter operatorFwith all delays to its right, such that it appears right next to the laser noise p0.
Overall, this will yield a laser noise residual as in eq. (12.21), where the delay commutator now consists of only offline delays and is applied to the filtered laser noise, in addition to a number of additional noise terms as described below.
For each of these effects, we will run a simulation to highlight it by choosing appropriate parameters. Not that these parameters are usually not realistic, and lead to higher residual laser noise levels than we will have in the actual mission. In all cases, we simulate 105s using LISANode, with laser noise enabled using the locking scheme N1-LA12and a strong anti-aliasing filter
3 Note that with this locking scheme, and considering only laser noise, we have simplyηij≡iscij, as described in section12.1.
12.3 laser noise residuals 153 with an attenuation of 320 dB. We will overlay in each case the analytical
model given in section12.3.2.4, but considering only the single effect under study.
12.3.2.1 Ranging errors
We call ranging errors any mismatch between the delays we apply in
post-processing and the real Depending on in
which reference frame the data is given, the real delays we should apply can be either light travel times computed in the TCB or they could be MPRs including the clock desynchronizations, cf.
chapter13.
propagation delays. This is not to be confused with the fundamental armlength mismatch presented in section12.3.1, which only depends on the orbital mechanics.
In our case, we will consider as a first step offline delays Dijµ which carry a constant bias4 µwith respect to the true delays, but are still considered free of interpolation errors, such that we have
Dijf(t) = f(t−dij) and Dµijf(t) = f(t−dij−µij). (12.26) Assuming that µij is small, we can introduce the operator ∆µij, defined by
∆µijf(t) = (Dij− Dijµ)f(t)≈µijf t˙ −dij
. (12.27)
In the Fourier domain, we simply get
F h∆µiji(ω)≈µijωe−iωdij. (12.28) To test this in the simulation, we use a large ranging bias of 10 m in each arm and again use a high interpolation order of65. The result is shown in fig.12.2, where the overlayed analytical model in dotted grey agrees perfectly with the simulated result, except at very low frequencies where we are again limited by numerical effects. Interestingly, TDIXappears to be significantly less sensitive to ranging noise in this locking scheme, presumably due to the special role spacecraft 1 has in both the locking scheme and the TDI combination.
12.3.2.2 Interpolation errors
As described in detail in appendix B, the interpolation with Lagrange polyno-mials causes an error depending on the fractional delay. We will track this error term by introducing the operator∆Iij, given as
∆Iij = (Dµij− Dij). (12.29)
It is derived in appendix B.2, with its Fourier transform given in eq. (B.26). It can be written as
F h∆ijI
i(ω) =e−iωdijeeij(ω), (12.30)
4 As described in section13.5, the ranging data can be filtered to remove most in-band fluctu-ations. A detailed analysis of the coupling of in-band ranging noise is under development inside the LDPG with contributions from the author of this thesis, but was not finalized at the time of writing. It should be available in the future in an upcoming publication.
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Figure 12.2: Laser noise residual due to a constant ranging bias for TDIX,YandZ with laser locking en-abled. To highlight the effect, we use a large bias of 10 m in each arm. The coupling intoXis strongly sup-pressed compared to
whereeeij is a term depending on the Lagrange interpolation coefficientscek,
eeij(ω) =
See appendix B.2for more details.
Together with eq. (12.27), we then get
Dij =Dij+∆ijI +∆µij. (12.32)
For a nested delay, we instead get
Dijk= (Dij+∆ijI +∆µij)(Djk+∆jkI +∆µjk) (12.33a)
To test this in the simulation, we use the same input data as in section12.3.1, but this time we compute the TDI variables with a very low interpolation order of just5. The result is shown in fig.12.3, again overlaying the analytical model with the simulation results. We see that our model in dotted grey reproduces the general trend and noise level, but is unable to capture the finer structure.
12.3.2.3 Flexing-filtering coupling
We need to move all filter operators next to the laser noise term p0. It was argued in [24] that the filter operator does not commute with a time-varying
12.3 laser noise residuals 155 we use a very small interpolation order of just5. The model overlayed in dotted grey captures the general trend and noise level, but is not able to explain the fine structure of the noise.
delay, giving rise to the so-called flexing-filtering coupling. We can keep track of each commutation by introducing a filter-delay commutator:
F,Dij
=FDij− DijF. (12.34)
This definition is straightforwardly applied to nested delay operators by commutingFwith each individual delay. For example,
FDijDjk= DijFDjk+F,Dij
The Fourier transform of F,Dij
is given in [24,20], and can be approximated as
F F,Dij
≈ −ωe−iωdijd˙ijKF(ω), (12.37) whereKFis the frequency domain derivative of the filter transfer function HF (cf. eq. (7.3)),
Following [24], we assumed that the group delay of the filter HF has been compensated in a first processing step, by shifting all filtered time series by N−1 samples, such that the sum does not start atk= 0. This significantly reduces the impact of this effect.
To check the validity of this model for the flexing-filtering coupling, we run a simulation with the filter design parameters set to a transition band spanning from 1 mHz to 1 Hz, while keeping the attenuation at 320 dB. We observe that the analytical model can explain the extra noise present at high frequencies between 0.1 Hz and 1 Hz, while we are limited by noise due to the
156 tdi in practice the effect, we used a very wide transition band spanning from 1 mHz to 1 Hz together with a strong attenu-ation of 320 dB. The model overlayed in dotted grey captures the general trend and noise level, but is not able to fully explain
fundamental armlength mismatch at lower frequencies. However, similar to the interpolation errors, the model fails to fully reproduce the fine structure of the observed noise.
12.3.2.4 Summary of technical imperfections
We observe that each propagation delay in eq. (12.25) yields a delay-filter commutator in addition to the terms given in eq. (12.32). It is therefore useful to define We can now insert these expressions into eq. (12.25) to get
TDI=hP12(D121−1) +P13(D131−1) +P23(D231− D21)
+P32(D321− D31)iFp0 (12.43a)
+hP12R121+P13R131+P23[R231− R21]
+P32[R321− R31]ip0. (12.43b)
12.3 laser noise residuals 157
Figure 12.5: Aliasing during downsampling from 16 Hz to 4 Hz.
Signals in the red, yel-low and green areas
We want to analyze the residual noise in the frequency domain, for which we will assume all delays appearing in thePij to be constant. The term labelled eq. (12.43a) has the Fourier transform as given in eq. (12.22), such that we get
F [TDI] (ω)≈hω∆τe−iωQdHF(ω) +Pe12Re121+eP13Re131 Since we replaced all delays appearing in the TDI combination by those applied in post-processing, ∆τ is now the armlength mismatch computed using these offline delays5.
In terms of PSD, since we only have a single laser noise term, we can compute
STDIp0 ≈ω∆τe−iωQdHF(ω) +eP12Re121+Pe13Re131
As described in chapter7, the phasemeter signals will be filtered and down-sampled before being transmitted down to earth. The anti-aliasing filter used in this step has a strong, but finite attenuation at high frequencies, such that there will be a small residual noise level left after filtering, which gets aliased into our measurement band when downsampling. In the real mission, this will happen in multiple steps going all the way from the high phasemeter sampling rate of 80 MHz down to the same rate used for transmission. We
5 This includes the potentially large ranging bias. However, since TDI combinations are by construction insensitive to constant armlength mismatches, we expect no big change in the resulting values.
158 tdi in practice
work under the assumption that the last filtering step in the real phasemeter will be the dominant effect, and that it is comparable to that used in the simulation6.
We can model how this effect enters our simulation in the following steps:
• Compute the Fourier transform of the laser noise in eachηij up to the physical sampling rate. This depends on the locking scheme.
• Apply the filter transfer function as given in eq. (7.3).
• Compute the aliased noise in eachηij by reflecting the spectrum across the Nyquist frequency. Since we downsample by a factor 4, we get a total of three aliased noise terms, NijA1, NijA2 and NijA3, for each ηij. Here,NijAk is the noise in the band[kπfsmeas,(k+1)πfsmeas], expressed in angular frequency.
• Apply the usual delay polynomialsPij to each term, treating the NijAk as uncorrelated for differentkbut fully correlated for the samek.
We get for the Fourier transform of eachηij
ηe12(ω) =HF(ω)(e−iω(d12+d21)−1)p˜0, (12.46a) ηe13(ω) =HF(ω)(e−iω(d13+d31)−1)p˜0, (12.46b) ηe23(ω) =HF(ω)(e−iω(d23+d31)−e−iωd21)p˜0, (12.46c) ηe32(ω) =HF(ω)(e−iω(d32+d21)−e−iωd31)p˜0, (12.46d)
ηe21(ω) =0 , (12.46e)
ηe31(ω) =0 , (12.46f)
from which we can compute the aliased terms as
NeijA1(ω) =ηeij(2πfsmeas−ω), (12.47a) NeijA2(ω) =ηeij(2πfsmeas+ω), (12.47b) NeijA3(ω) =ηeij(4πfsmeas−ω), (12.47c) which is to be evaluated for values ofω
The Nyquist frequency is here given in angular frequency, diving it by 2πgives the more familiar fsmeas2 .
up to the Nyquist rate,πfsmeas. Note that noise in the bands[πfsmeas, 2πfsmeas]and[3πfsmeas, 4πfsmeas]appears reflected in our measurement band. The noise in the band[2πfsmeas, 3πfsmeas] is reflected twice, such that the two effects cancel and it is simply shifted without reflection, see fig.12.5.
For eachk, the aliased terms represent signals which are physically at different frequencies. This is why we assume that they are fully un-correlated to each other and to the in-band laser noise described above.
6 The phasemeter design is not finalized yet, but the sampling rates used in our simulation (16 Hz and 4 Hz) are under consideration for the final two downsampling steps of the phasemeter [47]
12.3 laser noise residuals 159
Figure 12.6: Laser noise residual due to aliasing in TDIX, YandZwith laser
We then compute for each set of aliased noise terms how they couple into TDI,
which we evaluate numerically. The overall PSD of the aliased noise is then computed as
STDIA =
∑
3 k=1STDIAk. (12.49)
To test this model, we run a simulation with the same parameters as those in section12.3.1, but with the filter attenuation set to just 120 dB. The result is shown in fig.12.6, where we see that our model can mostly explain the ob-served aliased noise well. However, as we also obob-served for the interpolation errors and the flexing filtering coupling, the model is not perfect, and some parts of the spectrum are not fully reproduced7.
12.3.4 Overall laser noise level
To estimate the final residual laser noise level, we run a simulation with the default parameters as described in part ii, but disabling all noises except laser noise, plus an additional bias of 1 m in each ranging measurement. We simulate again 105s, using the locking scheme N1-LA12. For TDI, we use
7 The error is within an order of magnitude, such that all models presented here should still be useful to estimate the residual noise levels. It would be interesting to further explore the source of the discrepancy between the simulation and our models as a follow-up work. Such studies are ongoing inside the LISA consortium, with contributions from the author of this thesis, and might be published in the future.
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Figure 12.7: Overall laser noise residual in TDIXwith laser locking enabled, using noise floor at low fre-quencies.
interpolation order418, and compute just TDI Xfor clarity. We overlay the different contributions of the analytical models.
The residual laser noise is well explained by our models, down to around 1 mHz where we hit the usual noise floor of our simulations. If not for this numerical deviation, the fundamental armlength mismatch would limit at frequencies below 1 mHz, while aliasing is limiting in the band between 1 mHz and 4 mHz. Above 4 mHz, the most significant residual is caused by the 1 m ranging bias, until the interpolation errors take over close to the Nyquist frequency. The strong increase around 1 Hz is a combination of interpolation errors and aliasing. Flexing-filtering coupling is sub-dominant throughout the whole measurement band, and the overall laser noise residual is comfortably below the 1 pm allocation curve for the whole measurement band up to 1 Hz.