12.2 secondary noise levels in tdi
We have so far described TDI in terms of a generic laser phaseΦor phase and frequency fluctuations pi and ˙pi, respectively.
As described in section7.3, all but one laser in LISA will be locked to one primary laser. The locked lasers frequency will be controlled to drive the phasemeter signal used for locking to a pre-programmed reference value. In that process, it will inherit any noise of the primary laser, optical pathlength fluctuations during propagation as well as readout noises in the phasemeter channel used for locking. This effect is compounded when multiple lasers are locked in a chain, which is required to reach all lasers in the constella-tion.
Therefore, each term pi in the phasemeter equations given in section7.2is actually a complicated combination of multiple noise terms with potentially several delays. At first sight, this seems to imply a complicated coupling of these noise terms into the final TDI observable.
However, it turns out that all of these additional noise terms are suppressed alongside the laser noise, such that the choice of the locking configuration does not significantly impact the secondary noise levels in the final variable.
12.2.1 Secondary noises in TDI with locked lasers
As described in section5.3, pij denotes the phase fluctuations of the laser associated to MOSAij. These can be either the inherent frequency fluctuations due to the cavity used for frequency stabilizations, or fluctuations imprinted on the laser due to laser locking.
If we neglect the onboard filter, the timestamping errors and any technical
imperfections in TDI for now, We use here
expressions in phase fluctuations, but the same reasoning can be extended to frequency fluctuations by using the Doppler-delays introduced in section12.4.
the intermediary variables (cf. section12.1) can be written as
ηij =Dijpj−pi+Nηij, (12.3)
where we dropped the second index of the respective pij still present after construction ofηij, andNηijaccounts for all secondary noise terms inηijwhich enter with different correlations than the laser noise terms.
If we were to replace the pi terms with the expressions derived from the locking rules given in section7.3, some of theηij would cancel at this point, and others would become a complicated combination of noise terms; however, it is useful to keep pi unevaluated for now.
We know from section 10.3 that we can factorize any TDI combination as given in eq. (10.32),
TDI=
∑
ij∈I2
Pijηij. (12.4)
148 tdi in practice
Furthermore, we know from chapter10that the TDI variable is constructed in such a way that to all6lasers. These additional terms are
only the laser phase fluctuations pi of a single laser appear in them, scaled by a difference term corresponding to photon paths with almost equal time delays. As we saw in section10.2, this difference of delays corresponds to a suppression of more than10 orders of magnitude across the whole LISA frequency band2. On the other hand, any noise sources enteringηij with different correlations than the laser noise terms will not be meaningfully suppressed, but simply have their spectrum modulated by the delay polynomialsPij.
Overall, we get
TDI= [DA1...An−DB1...Bn]pi+
∑
ij∈I2
PijNηij, (12.5)
where we used A1. . .An and B1. . .Bn as placeholder for the photon paths defining the combination. Which pi appears in our combination is deter-mined by the starting spacecraft of the TDI string
It is argued in noisepiappears in eq. (12.5).
used to define the combina-tion.
Following section7.3, eachpi can be written as a potentially delayed version of the primary lasers phase fluctuations, plus any secondary noises entering either in the propagation to or the readout at the interferometer used for locking.
We use a simplified notation to describe this relationship, by writing
pi =Dlockp0+Npi (12.6)
for a locked laser, with p0as the laser noise of the primary laser,Dlocka series of delays depending on the locking scheme andNpi a term summarizing all secondary noise terms entering due to the locking.
Inserting this expression into eq. (12.5), we get TDI= [DA1...An−DB1...Bn](Dlockp0+Npi) +
∑
ij∈I2
PijNηij. (12.7) We see that the secondary noise terms for eachηij (without considering the laser locking) appear scaled byPij, while all the additional secondary noise term imprinted on the laser beam are suppressed by the same delay term as the laser noise.
Since laser noise needs to be suppressed by roughly8order of magnitudes, we conclude that the additional secondary noise terms entering with the same suppression are completely negligible compared to the terms entering scaled by the delay polynomialsPij.
For example, if we assume each noise termNηijto be uncorrelated to the others, we can estimate the residual noise level in any TDI variable by computing
STDINη ≈
∑
2 The fundamental arm length mismatch of the photon paths making up a second generation TDI combination is∆τ≈10 ps, such that the residual laser noise is scaled by 2πf∆τ<10−10.
12.2 secondary noise levels in tdi 149
As argued in section10.3, it is here This is usually the case
as long as the
often sufficient to compute the Fourier transform of the delay polynomialsPeij under the assumption that all delays are equal.
12.2.2 Examples
As examples of transfer functions for secondary noises, we want to compute how the two limiting secondary noises included in our simulation, readout noise and test-mass acceleration noise, couple into the second generation Michelson variableX2.
Under the assumption of three equal arms, the Pij of X2 are simply given as Here, following [39], it is useful to define
CXX = (1−e−iω4d)(1−e−iω2d)2 =64 cos2(dω)sin4(dω). (12.13) For readout noise, we need to distinguish between readout noise in the ISC, reference and test-mass interferometers. We assume all of these to be uncorrelated, with noise levels given in appendix D.5.
Inserting the phasemeter equations given in chapter7into the intermediary variables given in section12.1, and ignoring all terms except for the readout noise, we get
All readout noise terms are assumed to be uncorrelated, but some of them appear in multiple ηij. For these, we cannot directly use eq. (12.8), but need to instead collect the correct factors and delays of all terms appearing in the TDI combinations before computing the squared magnitude.
Overall, this gives
150 tdi in practice
For the test-mass acceleration noise, we instead simply have
ηij =−ν0(N˙ijδ−DijN˙jiδ), ∀ij∈ I2. (12.17) We can assume all6test-masses to be uncorrelated and collect the respective terms, to get the PSD
SXtm2 =4ν02(3+cos(2ωd))CXXSN˙δ, (12.18) where SN˙δ is the jitter of a single test-mass expressed as a dimensionless fractional frequency shift.
These formulas are verified by the simulation results presented in fig. 9.1, where the secondary noise level inX,Y andZcould be perfectly explained by the sum of eqs. (12.16) and (12.18).