from spacecraftin, received at spacecrafti1, and travel among the spacecraft as indicated by the list of spacecraft indices appearing in the nested delays, from right to left. The problem of laser noise suppression is now translated into the problem of finding the correct photon paths, which itself depends on the orbital dynamics governing the light travel times for each inter-satellite link.
There is of course a trivial solution to laser noise cancellation, which is i1. . .in =j1. . .jm, such that the two beams travel exactly the same path. This amounts to directly subtracting our measurement from itself, yielding exact cancellation of all noises, but also all signals. We are therefore only searching for non-trivial solutions, such thati1. . .in6=j1. . .jm
10.2 tdi generations
Traditionally, the solutions of the problem of finding laser noise reducing TDI combinations have been categorized into different simplified scenarios:
• 0th generation TDI, considering all delays to be constant and equal,
• 1st generation TDI, considering all delays to be constant, equal along two direction of the same link (e.g., D12 =D21), but unequal for different links (e.g.,D126= D13),
• 1.5th generation TDI, considering all delays to be constant and unequal, and
• 2nd generation TDI, considering all delays to be linearly evolving func-tions of time.
As we already saw in section3.2, second generation TDI is sufficient for the laser noise suppression requirements in LISA. We will still give an example of a third generation combination in section10.2.5.
Note that since higher generations impose increasingly strict conditions, any TDI solution at a given generation N is also a solution of lower generation M < N.
As an example, we give for each generation The three Michelson
combinations centered on spacecraft1,2and3 are traditionally called X, Y and Z.
the Michelson X combination.
None of them are perfect equal length interferometers, and their arm-length mismatch∆τis dominated by the next-to-leading order contributions to the arm length neglected in their derivation. We can model this up to2nd generation by considering a third order expansion for the light travel times,
DAx(t) =x(t−dA−d˙At− 12d¨At2), (10.14) withdA, ˙dAand ¨dAconstant. We will then give a model for the leading order arm length mismatch terms up to2nd generation. Following section2.3.1, we
112 introduction to time-delay interferometry (tdi)
Figure 10.4:Schematic overview of the0th and first generation Michelson combina-tions. Two beams are emitted from space-craft1. For the0th generation, they are recombined after a sin-gle round-trip, while for1st generation, they each travel to both spacecraft2and3 be-fore being recombined, cancelling constant armlength mismatches.
SC1
SC2 SC3
SC1
SC2 SC3
expect a laser noise residual in the TDI variable (expressed in frequency) of SXνi(ω) =∆τi2ω2Sν(ω), (10.15) whereω is the angular Fourier frequency,ian index denoting the generation, andSνas the raw laser frequency noise PSD.
We show a simplified numerical simulation in section10.2.6 to verify that these models are accurate.
10.2.1 0th generation TDI
If all delays are constant and equal, the condition for laser noise cancellation is simply that we need the same number of delays for each beam. This corresponds to n = min eq. (10.13). An example of a 0th generation TDI combination is the simple Michelson (sketched in fig.10.4(left)),
X0= η12+D12η21−η13−D13η31
= (D121−D131)Φ1. (10.16)
This is equivalent to the simple Michelson combination we considered in the beginning of this chapter, with the only difference being that we can construct such a signal
See section12.2for more information how the different locking schemes impact TDI.
regardless of the laser locking configuration, for each of the three spacecraft.
The leading order contributions to the armlength mismatch is simply
∆τ0≈2(d12−d13), (10.17)
which as we saw before leads to laser noise residuals far above the require-ments.
10.2.2 1st generation TDI
First generation combinations were first proposed in [82].
Since all delays are constant, their order does not matter, and we have
[DA,DB] =DADB−DBDA=0 (10.18)
10.2 tdi generations 113
for the commutator of As described in
appendix A, we sometimes use upper latin letters as placeholder for any index pairij.
any two delays. This means we can freely exchange the order in which the spacecraft indicies appear in the two nested delays in eq. (10.13). This yields the simple condition that the visited spacecraft have to be identical for both paths. Formally, we have
{ik|k=1 . . .n}={jk|k=1 . . .n} (10.19) for the sets of all indicesik andjk in eq. (10.13).
An example of a1st generation TDI combination is a variant of the Michelson, where each of the two beam visits both spacecraft, but in opposite order (sketched in fig.10.4(right)):
X1=η12+D12η21+D121η13+D1213η31
−[η13+D13η31+D131η12+D1312η21]
= (D12131−D13121)Φ1.
(10.20)
As argued below,X1 is actually also automatically TDI generation1.5. There-fore, the leading order contributions to the armlength mismatch are propor-tional to ˙dA, and we can compute them via a first order expansion in ˙d as
∆τ1 ≈4d(d˙31−d˙12), (10.21)
wheredis the average arm length in seconds, and we assumed ˙dij ≈d˙ji.
10.2.3 1.5th generation TDI
In 1.5th generation TDI (also called ’modified’ 1st generation TDI), all the conditions of1st generation apply, but we further distinguish the direction in which the delays are computed. This gives the more restrictive condition that the individual delays appearing in the expanded nested delays of both paths must be identical.
Expanding the nested delays inX1, we observe that it also satisfies this stricter condition, so it is automatically1.5th generation.
Generalized1.5th generation combinations were first proposed in [84].
An example of a1st generation TDI combination which is not1.5th generation is the Sagnac combination, where the two beams travel in a circle around the constellation, in opposite directions:
α1 =η12+D12η23+D123η31−[η13+D13η32+D132η21]
= (D12D23D31−D13D32D21)Φ1. (10.22) Indeed, the leading-order contribution to the arm-length mismatch forα1is simply
∆τα1 ≈(d12+d23+d31)−(d13+d32+d21), (10.23) which only vanishes under the conditiondij =dji.
114 introduction to time-delay interferometry (tdi)
In second generation TDI, the delays are no longer modelled as constants, but instead as linear functions of time,
DAx(t) =x(t−dA−d˙At), (10.24) withdA and ˙dA constant. Therefore, we can no longer freely commute two delays, since nested delays yield non-symmetric terms (we neglect any terms of the form ˙dAd˙B):
DADBx(t)≈ x((1−d˙A−d˙B)t−dA−dB+dAd˙B) (10.25) This is easily generalized to an arbitrary number of delays, where each delay couples to all the linear terms in the delays to it’s right:
DA1...Anx(t)≈x Note that only the last summand,
∑
n k=1dAk
∑
nm=k+1
d˙Am, (10.27)
depends on the order of the delays. Therefore, in order to achieve laser noise cancellation to first order in ˙d, the two beams must be chosen in such a way that
• Both beams contain the same overall delays, and
• they are ordered in such a way that eq. (10.27) yields the same result for both.
10.2 tdi generations 115 An example of a2nd generation TDI combination is yet another variant of the
Michelson, where each of the two beam visits both distant spacecraft twice (sketched in fig.10.5):
X2=η12+D12η21+D121η13+D1213η31+D12131η13
+D121313η31+D1213131η12+D12131312η21
−η13+D13η31+D131η12+D1312η21+D13121η12
+D131212η21+D1312121η13+D13121213η31
= (D121313121−D131212131)Φ1.
(10.28)
These combinations were first described in [85].
For its armlength mismatch, we consider terms to first order in ¨dij ≈d¨jiand second order in ˙dij ≈d˙ji and again use the average armlengthd, to get
∆τ2 =8d d˙212−d˙231
−2d d¨12−d¨31
. (10.29)
Here, the first term matches the previous result from the literature [24].
The second term is proportional to ¨d12−d¨31, which we will estimate to be dominant for the ESA provided orbits (cf. section10.2.5).
10.2.5 3rd and higher generation TDI
In principle, the above reasoning could be continued to include higher and higher polynomial orders. Third generation TDI would mean that the arms are equal to first order in all contributions given in eq. (10.14).
However, as we saw in fig.3.4,2nd generation TDI is sufficient to reach the required laser noise suppression in LISA.
Indeed, we can compute numerically from the orbits shown in fig.3.3that the orders of magnitude for the arm length derivatives can be approximated asdA ≈8.3 s, ˙dA≈10−9 and ¨dA≈10−15s−1. Since laser noise ’only’ needs to be suppressed by roughly8orders of magnitude, we see that the armlength mismatch ∆τ2 given in eq. (10.29) should be sufficient. In addition, we can confirm that the contribution of ¨dA should be the dominant contribution to
∆τ2.
In addition to not being required, the analytical problem of finding third generation TDI combinations also seems rather challenging. Due to the quadratic time dependence, the number of terms in a nested delay goes with the square of the number of delays, which means we would have to solve analytical formulas with a very large number of terms.
As a an example for a third generation combination, we can take an ed-ucated guess based on an argument from [92]. They conjecture that each TDI combinations of 2nd generation can be constructed from 1st genera-tion combinagenera-tions by ’splicing’. For example, each of the two beams in the 2nd generation Michelson combination contains exactly the same delays as
116 introduction to time-delay interferometry (tdi)
both beams combined in the first generation version, such that each arm is transversed twice, with a reversed direction for the second pass.
Extending that idea, we can guess the third generation Michelson combination to be the interferometer evaluating to
X3= (D12131312131212131−D13121213121313121)Φ1, (10.30) which is a splicing of the2nd generation combination, where each arm is now traversed four times by each beam.
To
This is by no means a conclusive study, but merely a quick test to check the general feasability of this combination.
test this hypothesis, we can numerically evaluate the difference in light travel time for the first and second beam, assuming a2nd order fit of realistic orbits given in fig. 3.3. The computations where performed using Mathe-matica, with the working precision of all numerical values set to 50digits precision.
For reference, using this method, theX2combinations time difference between the two paths evaluates to 9.7×10−12s, which exactly what we would expect from inserting the values of d, ˙d and ¨d determined by our orbital fit into eq. (10.29). Our ’third generation’ MichelsonX3, on the other hand, evaluates to just 8.6×10−20s, an improvement of8 orders of magnitude, and indeed below the level we would expect from terms proportional to either ˙d2A or d¨A.
10.2.6 Simulation comparison across generations
To verify the laser noise suppressing capabilities of the different TDI genera-tions, we present results using a simplified simulation usingLISA Instrument, where we disable all noises except laser noises. We simulate 105samples, and do not simulate laser locking. The same realistic ESA orbits and the same sampling rates are used as in chapter8. In addition, we use the same transi-tion band for the anti-aliasing filter, but increase the attenuatransi-tion to 320 dB in order to not be limited by aliasing1.
We then compute the0th,1st,2nd and3rd generation Michelson variables described above, using PyTDI, where we chose an
Cf. appendix B for
The results are shown in fig. 10.6. We observe that our model matches the achieved laser noise level well for0th,1st and2nd generation, although the 2nd generation variable is limited by a white numerical noise floor below 2×10−3Hz. This is even more pronounced for the3rd generation variant, which is completely limited by numerical noise, with an estimated theoretical
1 The attenuation of the filter given in chapter7was intentionally chosen to limit the level of aliased noise to just below the 1 pm allocation (with some margin). A stronger filter would be computationally more expensive, which might impose additional requirements on the instrument hardware. In this chapter, we are more interested in the fundamental limits of TDI than being true to the instrument design, such that we can use a higher attenuation in order to highlight the maximum achievable laser noise suppression.