Experimental Aspects of TDI

7 Experimental Aspects of TDI

It is clear that the suppression of the laser phase fluctuations by more than nine orders of magnitude with the use of TDI is a very challenging experimental task. It requires developing and building subsystems capable of unprecedented accuracy and precision levels, and test their end-to-end performance in a laboratory environment that naturally precludes the availability of $5 × 106$ km delay lines! In what follows we will address some aspects related to the experimental implementation of TDI, and derive the performance specifications for the subsystems involved. We will not address, however, any of the experimental aspects related to the verification of TDI in a laboratory environment. For that, we refer the interested reader to de Vine et al. [8 *, 32 *], Spero et al. [48 *], and Mitryk et al. [33 *].

From simple physical grounds, it is easy to see that a successful implementation of TDI requires:

accurate knowledge of the time shifts, $′ L i(t),Li(t) i = 1,2,3$ , to be applied to the heterodyne measurements $si(t),s′i(t),τi(t),τ′i(t) i = 1,2,3$ ;
accurate synchronization among the three clocks onboard the three spacecraft as these are used for time-stamping the recorded heterodyne phase measurements;
sampling time stability (i.e., clock stability) for successfully suppressing the laser noise to the desired level;
an accurate reconstruction algorithm of the phase measurements corresponding to the required time delays as these in general will not be equal to integer multiples of the sampling time;
a phase meter capable of a very large dynamic range in order to suppress the laser noise to the required level while still preserving the phase fluctuations induced by a gravitational-wave signal in the TDI combinations.

In the following subsections, we will quantitatively address the issues listed above, and provide the reader with a related list of references where more details can be found.

7.1 Time-delays accuracies

The TDI combinations described in the previous sections (whether of the first- or second-generation) rely on the assumption of knowing the time-delays with infinite accuracy to exactly cancel the laser noise. Since the six delays will in fact be known only within the accuracies $δLi ,δL′ i = 1,2,3 i$ , the cancellation of the laser frequency fluctuations in, for instance, the combinations ( $α, β,γ,ζ$ ) will no longer be exact. In order to estimate the magnitude of the laser fluctuations remaining in these data sets, let us define $ˆLi , ˆL′i i = 1,2, 3$ to be the estimated time-delays. They are related to the true delays $Li ,L ′i = 1,2,3 i$ , and the accuracies $δLi ,δL′ i = 1,2,3 i$ through the following expressions

and similarly for the primed delays. In what follows we will limit our derivation of the time-delays accuracies to only the first-generation TDI combinations and treat the three common delays $Li , i = 1,2,3$ as constants equal to 16.7 light-seconds. We will also assume to know with infinite accuracies and precisions all the remaining physical quantities (listed at the beginning of Section 7) that are needed to successfully synthesize the TDI generators.

If we now substitute Eq. (98 *) into the expression for the TDI combination $α$ , for instance, (Eq. (43 *)) and expand it to first order in $δLi$ , it is easy to derive the following approximate expression for $ˆα(t)$ , which now will show a non-zero contribution from the laser noises

where the symbol $˙$ denotes time derivative. Time-delay interferometry can be considered effective if the magnitude of the remaining fluctuations from the lasers are much smaller than the fluctuations due to the remaining (proof mass and optical path) noises entering $α(t)$ . This requirement implies a limit in the accuracies of the measured delays.

Let us assume the laser phase fluctuations to be uncorrelated to each other, their one-sided power spectral densities to be equal, the three armlengths to differ by a few percent, and the three armlength accuracies also to be equal. By requiring the magnitude of the remaining laser noises to be smaller than the secondary noise sources, it is straightforward to derive, from Eq. (99 *) and the expressions for the noise spectrum of the $α$ TDI combination given in [15 *], the following constraint on the common armlength accuracy $|δL α|$

with similar inequalities also holding for $β$ and $γ$ . Here $S ϕ$ , $S proof mass$ , $Soptical path$ are the one-sided power spectral densities of the relative frequency fluctuations of a stabilized laser, a single proof mass, and a single-link optical path respectively. If we take them to be equal to the following functions of the Fourier frequency $f$ [53 , 15 ]

Sϕ(f) = 10 −28 f −2∕3 + 6.3 × 10 −37 f −3.4 Hz− 1 (101 ) proof mass −48 −2 −1 S (f) = 2.5 × 10 f Hz (102 ) Soptical path(f) = 1.8 × 10−37f2 Hz −1 (103 )

(where $f$ is in Hz), we find that the right-hand side of the inequality given by Eq. (100 *) reaches its minimum of about 30 meters at the Fourier frequency $fmin = 1.0 × 10−4 Hz$ , over the assumed ( $10− 4,1$ ) Hz LISA band. This implies that, if the armlength knowledge $|δL | α$ can be made much smaller than 30 meters, the magnitude of the residual laser noise affecting the $α$ combination can be regarded as negligible over the entire frequency band. This reflects the fact that the armlength accuracy is a decreasing function of the frequency. For instance, at $10−3 Hz$ the armlength accuracy goes up by almost an order of magnitude to about 155 meters.

A perturbation analysis similar to the one described above can be performed for $ζ$ , resulting into the following inequality for the required delay accuracy, $|δL ζ|$

Equation (104 *) implies a minimum of the function on the right-hand side equal to about 16 meters at the Fourier frequency $fmin = 1.0 × 10−4 Hz$ , while at $10−3 Hz$ the armlength accuracy goes up to 154 meters.

Armlength accuracies at the centimeters level have already been demonstrated in the laboratory [16 *, 50 *, 64 *, 26 *], making us confident that the required level of time-delays accuracy will be available.

In relation to the accuracies derived above, it is interesting to calculate the time scales during which the armlengths will change by an amount equal to the accuracies themselves. This identifies the minimum time required before updating the armlength values in the TDI combinations.

It has been calculated by Folkner et al. [21 *] that the relative longitudinal speeds between the three pairs of spacecraft, during approximately the first year of the LISA mission, can be written in the following approximate form

where we have denoted with $(1, 2),(1, 3),(2, 3)$ the three possible spacecraft pairs, $Vi(0,j)$ is a constant velocity, and $Ti,j$ is the period for the pair $(i,j)$ . In reference [21 *] it has also been shown that the LISA trajectory can be selected in such a way that two of the three arms’ rates of change are essentially equal during the first year of the mission. Following reference [21 ], we will assume $V (1,0)2 = V (1,0)3 ⁄= V (2,0)3$ , with $V1(0,)2 = 1 m ∕s$ , $V (2,03)= 13 m ∕s$ , $T1,2 = T1,3 ≈ 4$ months, and $T2,3 ≈ 1$ year. From Eq. (105 *) it is easy to derive the variation of each armlength, for example $ΔL (t) 3$ , as a function of the time $t$ and the time scale $δt$ during which it takes place

Equation (106 *) implies that a variation in armlength $ΔL3 ≈ 10 m$ can take place during different time scales, depending on when during the mission this change takes place. For instance, if $t ≪ T1,2$ we find that the armlength $L3$ changes by more than its accuracy ( $≈ 10 m$ ) after a time $δt = 2.3 × 103 s$ . If however $t ≃ T ∕4 1,2$ , the armlength will change by the same amount after only $δt ≃ 10 s$ instead. As this value is less than the one-way-light-time, one might argue that the measured time-delay will not represent well enough the delay that needs to be applied in the TDI combinations at that particular time.

One way to address this problem is to treat the delays in the TDI combinations as parameters to be determined by a non-linear least-squares procedure, in which the minimum of the minimizer is achieved at the correct delays since that the laser noise will exactly cancel there in the TDI combinations. Such a technique, which was named time-delay interferometric ranging (TDIR) [60 ], requires a starting point in the time-delays space in order to implement the minimization, and it will work quite effectively jointly with the ranging data available onboard.

7.2 Clocks synchronization

The effectiveness of the TDI data combinations requires the clocks onboard the three spacecraft to be synchronized. In what follows we will identify the minimum level of off-synchronization among the clocks that can be tolerated. In order to proceed with our analysis we will treat one of the three clocks (say the clock onboard spacecraft 1) as the master clock defining the time for LISA, while the other two to be synchronized to it.

The relativistic (Sagnac) time-delay effect due to the fact that the LISA trajectory is a combination of two rotations, each with a period of one year, will have to be accounted for in the synchronization procedure and, as has already been discussed earlier, will be accounted for within the second-generation formulation of TDI.

Here, for simplicity, we will analyze an idealized non-rotating constellation in order to get a sense of the required level of clocks synchronization. Let us denote by $δt2$ , $δt3$ , the time accuracies (time-offsets) for the clocks onboard spacecraft 2 and 3 respectively. If $t$ is the time onboard spacecraft 1, then what is believed to be time $t$ onboard spacecraft 2 and 3 is actually equal to the following times

If we now substitute Eqs. (107 * and 108 *) into the TDI combination $ζ$ , for instance, and expand it to first order in $δti , i = 2, 3$ , it is easy to derive the following approximate expression for $ˆζ(t)$ , which shows the following non-zero contribution from the laser noises

By requiring again the magnitude of the remaining fluctuations from the lasers to be smaller than the fluctuations due to the other (secondary) noise sources affecting $ζ (t)$ , it is possible to derive an upper limit for the accuracies of the synchronization of the clocks. If we assume again the three laser phase fluctuations to be uncorrelated to each other, their one-sided power spectral densities to be equal, the three armlengths to differ by a few percent, and the two time-offsets’ magnitudes to be equal, by requiring the magnitude of the remaining laser noises to be smaller than the secondary noise sources it is easy to derive the following constraint on the time synchronization accuracy $|δtζ|$

with $S ϕ$ , $S proof mass$ , $S optical path$ again as given in Eqs. (101 * – 103 *).

We find that the right-hand side of the inequality given by Eq. (110 *) reaches its minimum of about 47 nanoseconds at the Fourier frequency $fmin = 1.0 × 10− 4 Hz$ . This means that clocks synchronized at a level of accuracy significantly better than 47 nanoseconds will result into a residual laser noise that is much smaller than the secondary noise sources entering into the $ζ$ combination.

An analysis similar to the one described above can be performed for the remaining generators ( $α, β,γ$ ). For them we find that the corresponding inequality for the accuracy in the synchronization of the clocks is now equal to

with equal expressions holding also for $β$ and $γ$ . The function on the right-hand side of Eq. (111 *) has a minimum equal to 88 nanoseconds at the Fourier frequency $fmin = 1.0 × 10−4 Hz$ . As for the armlength accuracies, also the timing accuracy requirements become less stringent at higher frequencies. At $10− 3 Hz$ , for instance, the timing accuracy for $ζ$ and $α,β, γ$ go up to 446 and 500 ns respectively.

As a final note, a required clock synchronizations of about 40 ns derived in this section translates into a ranging accuracy of 12 meters, which has been experimentally shown to be easily achievable [16 , 50 , 64 , 26 ].

7.3 Clocks timing jitter

The sampling times of all the measurements needed for synthesizing the TDI combinations will not be constant, due to the intrinsic timing jitters of the onboard measuring system. Among all the subsystems involved in the data measuring process, the onboard clock is expected to be the dominant source of time jitter in the sampled data. Presently existing space qualified clocks can achieve an Allan standard deviation of about $10 −13$ for integration times from 1 to 10 000 seconds. This timing stability translates into a time jitter of about $10−13$ seconds over a period of 1 second. A perturbation analysis including the three sampling time jitters due to the three clocks shows that any laser phase fluctuations remaining in the four TDI generators will also be proportional to the sampling time jitters. Since the latter are approximately four orders of magnitude smaller than the armlength and clocks synchronization accuracies derived earlier, we conclude that the magnitude of laser noise residual into the TDI combinations due to the sampling time jitters is negligible.

7.4 Sampling reconstruction algorithm

The derivations of the time-delays and clocks synchronization accuracies highlighted earlier presumed the availability of the phase measurement samples at the required time-delays. Since this condition will not be true in general, as the time-delays used by the TDI combinations will not be equal to integer-multiples of the sampling time, with a sampling rate of, let us say, 10 Hz, the time delays could be off their correct values by a tenth of a second, way more than the 10 nanoseconds time-delays and clocks synchronization accuracies estimated above.

Earlier suggestions [27 *] for addressing this problem envisioned sampling the data at very-high rates (perhaps of the order of hundreds of MHz), so reducing the additional error to the estimated time-delays to a few nanoseconds. Although in principle such a solution would allow us to suppress the residual laser noise to the required level, it would create an insurmountable problem for transmitting the science data to the ground due to the limited space-to-ground data rates.

An alternate scheme for obtaining the phase measurement points needed by TDI [59 *] envisioned sampling the phase measurements at the required delayed times. This scheme naturally requires knowledge of the time-delays and synchronization of the clocks at the required accuracy levels during data acquisition. Although such a procedure could be feasible in principle, it would still leave open the possibility of irreversible corruption of the TDI combinations in the eventuality of performance degradation in the ranging and clock synchronization procedures.

Given that the data will need to be sampled at a rate of 10 Hz, an alternative options is to implement an interpolation scheme for reconstructing the required data points from the sampled measurements. An analysis [59 *] based on the implementation of the truncated Shannon [47 ] formula, however, showed that several months of data were required in order to reconstruct the phase samples at the estimated time-delays with a sufficiently high accuracy. This conclusion implied that several months (at the beginning and end) of the entire data records measured by LISA would be of no use, resulting into a significant mission science degradation.

Although the truncated Shannon formula was proved to be impracticable [59 *] for reconstructing phase samples at the required time-delays, it was then recognized that [46 *] a more efficient and accurate interpolation technique [31 ] could be adopted. In what follows, we provide a brief account of this data processing technique, which is known as “fractional-delay filtering” (FDF).

In order to understand how FDF works, let’s write the truncated Shannon formula for the delayed sample, $yN (n − D )$ , which we want to construct by filtering the sampled data $y(n)$

where, as usual, $sinc(x) ≡ sin(x)∕x$ , and $N$ is an integer at which the Shannon formula has been truncated to. As pointed out in [46 *], although the truncated Shannon formula is optimal in the least-squares sense, the sinc-function that appears in it is far from being ideal in reconstructing the transfer function $e2πifD ∕fs$ , where $fs$ is the sampling frequency. In fact, over the LISA observational band the sinc-function displays significant ringing, which can only be suppressed by taking $N$ very large (as the error, $𝜖$ , decays slowly as $1∕N$ ). It was estimated that, in order to achieve an $−8 𝜖 < 10$ , an $8 N ≥ 10$ is needed.

If, however, we give up on the requirement of minimizing the error in the least-squares sense and replace it with a mini-max criterion error applied to the absolute value of the difference between the ideal transfer function (i.e., $2πifD∕fs e$ ) and a modified sinc-function, we will be able to achieve a rapid convergence while suppressing the ringing effects associated with the sinc function.

One way to achieve this result is to modify the Shannon formula by multiplying the sinc-function by a window-function, $W (j)$ , in the following way

where $W (j)$ smoothly decays to zero at $j = ±N$ . In [46 *] several windows were tested, and the resulting values of $N$ needed to accurately reconstruct the desired delayed samples were estimated, both on theoretical and numerically grounds. It was found that, with windows belonging to the family of Lagrange polynomials [46 ] a delayed sample could be reconstructed by using $N ≃ 25$ samples while achieving a mini-max error $𝜖 < 10−12$ between the ideal transfer function $e2πifD∕fs$ , and the kernel of the modified truncated Shannon formula.

7.5 Data digitization and bit-accuracy requirement

It has been shown [59 ] that the maximum of the ratio between the amplitudes of the laser and the secondary phase fluctuations occurs at the lower end of the LISA bandwidth (i.e., 0.1 mHz) and it is equal to about $1010$ . This corresponds to the minimum dynamic range for the phasemeters to correctly measure the laser fluctuations and the weaker (gravitational-wave) signals simultaneously. An additional safety factor of $≈ 10$ should be sufficient to avoid saturation if the noises are well described by Gaussian statistics. In terms of requirements on the digital signal processing subsystem, this dynamic range implies that approximately 36 bits are needed when combining the signals in TDI, only to bridge the gap between laser frequency noise and the other noises and gravitational-wave signals. More bits might be necessary to provide enough information to efficiently filter the data when extracting weak gravitational-wave signals embedded into noise.

The phasemeters will be the onboard instrument that will perform the phase measurements containing the gravitational signals. They will also need to simultaneously measure the time-delays to be applied to the TDI combinations via ranging tones over-imposed on the laser beams exchanged by the spacecraft. And they will need to have the capability of simultaneously measure additional side-band tones that are required for the calibration of the onboard Ultra-Stable Oscillator used in the down-conversion of heterodyned carrier signal [57 , 27 ].

Work toward the realization of a phasemeter capable of meeting these very stringent performance and operational requirements has aggressively been performed both in the United States and in Europe [43 , 23 , 22 , 6 , 63 ], and we refer the reader interested in the technical details associated with the development studies of such device to the above references and those therein.

	Abstract
1	Introduction
2	Physical and Historical Motivations of TDI
3	Time-Delay Interferometry
4	Algebraic Approach for Canceling Laser and Optical Bench Noises
	4.1	Cancellation of laser phase noise
	4.2	Cancellation of laser phase noise in the unequal-arm interferometer
	4.3	The module of syzygies
	4.4	Gröbner basis
	4.5	Generating set for the module of syzygies
	4.6	Canceling optical bench motion noise
	4.7	Physical interpretation of the TDI combinations
5	Time-Delay Interferometry with Moving Spacecraft
	5.1	The unequal-arm Michelson
	5.2	The Sagnac combinations
	5.3	Algebraic approach to second-generation TDI
	5.4	Solutions with one arm nonfunctional
6	Optimal LISA Sensitivity
	6.1	General application
	6.2	Optimization of SNR for binaries with known direction but with unknown orientation of the orbital plane
7	Experimental Aspects of TDI
	7.1	Time-delays accuracies
	7.2	Clocks synchronization
	7.3	Clocks timing jitter
	7.4	Sampling reconstruction algorithm
	7.5	Data digitization and bit-accuracy requirement
8	Concluding Remarks
	Acknowledgement
A	Generators of the Module of Syzygies
B	Conversion between Generating Sets
	References
	Footnotes
	Updates
	Figures