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2 Physical and Historical Motivations of TDI

Equal-arm interferometer detectors of gravitational waves can observe gravitational radiation by cancelling the laser frequency fluctuations affecting the light injected into their arms. This is done by comparing phases of split beams propagated along the equal (but non-parallel) arms of the detector. The laser frequency fluctuations affecting the two beams experience the same delay within the two equal-length arms and cancel out at the photodetector where relative phases are measured. This way gravitational wave signals of dimensionless amplitude less than 10-20 can be observed when using lasers whose frequency stability can be as large as roughly a few parts in 10- 13.

If the arms of the interferometer have different lengths, however, the exact cancellation of the laser frequency fluctuations, say C(t), will no longer take place at the photodetector. In fact, the larger the difference between the two arms, the larger will be the magnitude of the laser frequency fluctuations affecting the detector response. If L1 and L2 are the lengths of the two arms, it is easy to see that the amount of laser relative frequency fluctuations remaining in the response is equal to (units in which the speed of light c = 1)

DC(t) = C(t - 2L1) - C(t - 2L2). (1)
In the case of a space-based interferometer such as LISA, whose lasers are expected to display relative frequency fluctuations equal to about -13 V~ --- 10 / Hz in the mHz band, and whose arms will differ by a few percent [3Jump To The Next Citation Point], Equation (1View Equation) implies the following expression for the amplitude of the Fourier components of the uncancelled laser frequency fluctuations (an over-imposed tilde denotes the operation of Fourier transform):
|DC(f )| -~ |C(f )|4pf |(L1 - L2)|. (2)
At f = 10-3 Hz, for instance, and assuming |L - L | -~ 0.5 s 1 2, the uncancelled fluctuations from the laser are equal to -16 V~ --- 6.3 × 10 / Hz. Since the LISA sensitivity goal is about - 20V ~ -- 10 / Hz in this part of the frequency band, it is clear that an alternative experimental approach for canceling the laser frequency fluctuations is needed.

A first attempt to solve this problem was presented by Faller et al. [9Jump To The Next Citation Point1110], and the scheme proposed there can be understood through Figure 1View Image. In this idealized model the two beams exiting the two arms are not made to interfere at a common photodetector. Rather, each is made to interfere with the incoming light from the laser at a photodetector, decoupling in this way the phase fluctuations experienced by the two beams in the two arms. Now two Doppler measurements are available in digital form, and the problem now becomes one of identifying an algorithm for digitally cancelling the laser frequency fluctuations from a resulting new data combination.

View Image

Figure 1: Light from a laser is split into two beams, each injected into an arm formed by pairs of free-falling mirrors. Since the length of the two arms, L1 and L2, are different, now the light beams from the two arms are not recombined at one photo detector. Instead each is separately made to interfere with the light that is injected into the arms. Two distinct photo detectors are now used, and phase (or frequency) fluctuations are then monitored and recorded there.
The algorithm they first proposed, and refined subsequently in [14], required processing the two Doppler measurements, say y1(t) and y2(t), in the Fourier domain. If we denote with h1(t), h2(t) the gravitational wave signals entering into the Doppler data y1, y2, respectively, and with n1, n2 any other remaining noise affecting y1 and y2, respectively, then the expressions for the Doppler observables y1, y2 can be written in the following form:
y1(t) = C(t- 2L1) - C(t) + h1(t) + n1(t), (3) y2(t) = C(t- 2L2) - C(t) + h2(t) + n2(t). (4)
From Equations (3View Equation, 4View Equation) it is important to note the characteristic time signature of the random process C(t) in the Doppler responses y1, y2. The time signature of the noise C(t) in y1(t), for instance, can be understood by observing that the frequency of the signal received at time t contains laser frequency fluctuations transmitted 2L1 s earlier. By subtracting from the frequency of the received signal the frequency of the signal transmitted at time t, we also subtract the frequency fluctuations C(t) with the net result shown in Equation (3View Equation).

The algorithm for cancelling the laser noise in the Fourier domain suggested in [9] works as follows. If we take an infinitely long Fourier transform of the data y1, the resulting expression of y1 in the Fourier domain becomes (see Equation (3View Equation))

[ 4pifL1 ] y1(f) = C(f ) e - 1 + h1(f) + n1(f ). (5)
If the arm length L1 is known exactly, we can use the y1 data to estimate the laser frequency fluctuations C(f ). This can be done by dividing y1 by the transfer function of the laser noise C into the observable y1 itself. By then further multiplying y1/[e4pifL1 - 1] by the transfer function of the laser noise into the other observable y 2, i.e. [e4pifL2 - 1], and then subtract the resulting expression from y2 one accomplishes the cancellation of the laser frequency fluctuations.

The problem with this procedure is the underlying assumption of being able to take an infinitely long Fourier transform of the data. Even if one neglects the variation in time of the LISA arms, by taking a finite length Fourier transform of, say, y1(t) over a time interval T, the resulting transfer function of the laser noise C into y1 no longer will be equal to [e4pifL1 - 1]. This can be seen by writing the expression of the finite length Fourier transform of y1 in the following way:

integral +T integral + oo yT1 =_ y1(t) e2piftdt = y1(t)H(t) e2piftdt, (6) -T - oo
where we have denoted with H(t) the function that is equal to 1 in the interval [- T, +T ], and zero everywhere else. Equation (6View Equation) implies that the finite-length Fourier transform yT 1 of y1(t) is equal to the convolution in the Fourier domain of the infinitely long Fourier transform of y (t) 1, y 1, with the Fourier transform of H(t) [15] (i.e. the “Sinc Function” of width 1/T). The key point here is that we can no longer use the transfer function [e4pifLi- 1], i = 1,2, for estimating the laser noise fluctuations from one of the measured Doppler data, without retaining residual laser noise into the combination of the two Doppler data y1, y2 valid in the case of infinite integration time. The amount of residual laser noise remaining in the Fourier-based combination described above, as a function of the integration time T and type of “window function” used, was derived in the appendix of [29Jump To The Next Citation Point]. There it was shown that, in order to suppress the residual laser noise below the LISA sensitivity level identified by secondary noises (such as proof-mass and optical path noises) with the use of the Fourier-based algorithm an integration time of about six months was needed.

A solution to this problem was suggested in [29Jump To The Next Citation Point], which works entirely in the time-domain. From Equations (3View Equation, 4View Equation) we may notice that, by taking the difference of the two Doppler data y1(t), y2(t), the frequency fluctuations of the laser now enter into this new data set in the following way:

y1(t)- y2(t) = C(t - 2L1) - C(t - 2L2) + h1(t)- h2(t) + n1(t)- n2(t). (7)
If we now compare how the laser frequency fluctuations enter into Equation (7View Equation) against how they appear in Equations (3View Equation, 4View Equation), we can further make the following observation. If we time-shift the data y (t) 1 by the round trip light time in arm 2, y1(t- 2L2), and subtract from it the data y2(t) after it has been time-shifted by the round trip light time in arm 1, y2(t - 2L1), we obtain the following data set:
y1(t- 2L2) - y2(t- 2L1) = C(t- 2L1) - C(t - 2L2) + h1(t - 2L2) - h2(t - 2L1) +n1(t - 2L2) - n2(t - 2L1). (8)
In other words, the laser frequency fluctuations enter into y1(t)- y2(t) and y1(t - 2L2) - y2(t- 2L1) with the same time structure. This implies that, by subtracting Equation (8View Equation) from Equation (7View Equation) we can generate a new data set that does not contain the laser frequency fluctuations C(t),
X =_ [y1(t)- y2(t)]- [y1(t- 2L2) - y2(t - 2L1)]. (9)
The expression above of the X combination shows that it is possible to cancel the laser frequency noise in the time domain by properly time-shifting and linearly combining Doppler measurements recorded by different Doppler readouts. This in essence is what TDI amounts to. In the following sections we will further elaborate and generalize TDI to the realistic LISA configuration.

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