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3 Time-Delay Interferometry

The description of TDI for LISA is greatly simplified if we adopt the notation shown in Figure 2View Image, where the overall geometry of the LISA detector is defined. There are three spacecraft, six optical benches, six lasers, six proof-masses, and twelve photodetectors. There are also six phase difference data going clock-wise and counter-clockwise around the LISA triangle. For the moment we will make the simplifying assumption that the array is stationary, i.e. the back and forth optical paths between pairs of spacecraft are simply equal to their relative distances [24Jump To The Next Citation Point5Jump To The Next Citation Point25Jump To The Next Citation Point34Jump To The Next Citation Point].

Several notations have been used in this context. The double index notation recently employed in [25Jump To The Next Citation Point], where six quantities are involved, is self-evident. However, when algebraic manipulations are involved the following notation seems more convenient to use. The spacecraft are labeled 1, 2, 3 and their separating distances are denoted L1, L2, L3, with Li being opposite spacecraft i. We orient the vertices 1, 2, 3 clockwise in Figure 2View Image. Unit vectors between spacecraft are ^ni, oriented as indicated in Figure 2View Image. We index the phase difference data to be analyzed as follows: The beam arriving at spacecraft i has subscript i and is primed or unprimed depending on whether the beam is traveling clockwise or counter-clockwise (the sense defined here with reference to Figure 2View Image) around the LISA triangle, respectively. Thus, as seen from the figure, s 1 is the phase difference time series measured at reception at spacecraft 1 with transmission from spacecraft 2 (along L3).

View Image

Figure 2: Schematic LISA configuration. The spacecraft are labeled 1, 2, and 3. The optical paths are denoted by Li, L'i where the index i corresponds to the opposite spacecraft. The unit vectors ^ni point between pairs of spacecraft, with the orientation indicated.
Similarly, ' s1 is the phase difference series derived from reception at spacecraft 1 with transmission from spacecraft 3. The other four one-way phase difference time series from signals exchanged between the spacecraft are obtained by cyclic permutation of the indices: 1 --> 2 --> 3-- > 1. We also adopt a notation for delayed data streams, which will be convenient later for algebraic manipulations. We define the three time-delay operators Di, i = 1,2,3, where for any data stream x(t)
Dix(t) = x(t- Li), (10)
where Li, i = 1,2,3, are the light travel times along the three arms of the LISA triangle (the speed of light c is assumed to be unity in this article). Thus, for example, D2s1(t) = s1(t - L2), D2D3s1(t) = s1(t- L2- L3) = D3D2s1(t), etc. Note that the operators commute here. This is because the arm lengths have been assumed to be constant in time. If the Li are functions of time then the operators no longer commute [5Jump To The Next Citation Point34Jump To The Next Citation Point], as will be described in Section 4. Six more phase difference series result from laser beams exchanged between adjacent optical benches within each spacecraft; these are similarly indexed as ti, t'i, i = 1,2, 3. The proof-mass-plus-optical-bench assemblies for LISA spacecraft number 1 are shown schematically in Figure 3View Image. The photo receivers that generate the data s1, s'1, t1, and t'1 at spacecraft 1 are shown. The phase fluctuations from the six lasers, which need to be cancelled, can be represented by six random processes pi, ' pi, where pi, ' p i are the phases of the lasers in spacecraft i on the left and right optical benches, respectively, as shown in the figure. Note that this notation is in the same spirit as in [33Jump To The Next Citation Point25Jump To The Next Citation Point] in which moving spacecraft arrays have been analyzed.

We extend the cyclic terminology so that at vertex i, i = 1,2,3, the random displacement vectors of the two proof masses are respectively denoted by di(t), ' di(t), and the random displacements (perhaps several orders of magnitude greater) of their optical benches are correspondingly denoted by Di(t), D'i(t) where the primed and unprimed indices correspond to the right and left optical benches, respectively. As pointed out in [7Jump To The Next Citation Point], the analysis does not assume that pairs of optical benches are rigidly connected, i.e. ' Di /= D i, in general. The present LISA design shows optical fibers transmitting signals both ways between adjacent benches. We ignore time-delay effects for these signals and will simply denote by mi(t) the phase fluctuations upon transmission through the fibers of the laser beams with frequencies ni, and n' i. The mi(t) phase shifts within a given spacecraft might not be the same for large frequency differences ' ni - ni. For the envisioned frequency differences (a few hundred MHz), however, the remaining fluctuations due to the optical fiber can be neglected [7Jump To The Next Citation Point]. It is also assumed that the phase noise added by the fibers is independent of the direction of light propagation through them. For ease of presentation, in what follows we will assume the center frequencies of the lasers to be the same, and denote this frequency by n0.

The laser phase noise in s' 3 is therefore equal to D1p2(t) - p'(t) 3. Similarly, since s2 is the phase shift measured on arrival at spacecraft 2 along arm 1 of a signal transmitted from spacecraft 3, the laser phase noises enter into it with the following time signature: ' D1p 3(t) - p2(t). Figure 3View Image endeavors to make the detailed light paths for these observations clear. An outgoing light beam transmitted to a distant spacecraft is routed from the laser on the local optical bench using mirrors and beam splitters; this beam does not interact with the local proof mass. Conversely, an incoming light beam from a distant spacecraft is bounced off the local proof mass before being reflected onto the photo receiver where it is mixed with light from the laser on that same optical bench. The inter-spacecraft phase data are denoted s1 and ' s 1 in Figure 3View Image.

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Figure 3: Schematic diagram of proof-masses-plus-optical-benches for a LISA spacecraft. The left-hand bench reads out the phase signals s 1 and t 1. The right-hand bench analogously reads out ' s1 and ' t1. The random displacements of the two proof masses and two optical benches are indicated (lower case di,d'i for the proof masses, upper case Di,D'i for the optical benches).
Beams between adjacent optical benches within a single spacecraft are bounced off proof masses in the opposite way. Light to be transmitted from the laser on an optical bench is first bounced off the proof mass it encloses and then directed to the other optical bench. Upon reception it does not interact with the proof mass there, but is directly mixed with local laser light, and again down converted. These data are denoted t 1 and t ' 1 in Figure 3View Image.

The expressions for the si, ' si and ti, ' ti phase measurements can now be developed from Figures 2View Image and 3View Image, and they are for the particular LISA configuration in which all the lasers have the same nominal frequency n0, and the spacecraft are stationary with respect to each other. Consider the s'1(t) process (Equation (13View Equation) below). The photo receiver on the right bench of spacecraft 1, which (in the spacecraft frame) experiences a time-varying displacement ' D 1, measures the phase difference ' s1 by first mixing the beam from the distant optical bench 3 in direction n^2, and laser phase noise p3 and optical bench motion D3 that have been delayed by propagation along L2, after one bounce off the proof mass (d'1), with the local laser light (with phase noise p'1). Since for this simplified configuration no frequency offsets are present, there is of course no need for any heterodyne conversion [33].

In Equation (12View Equation) the t1 measurement results from light originating at the right-bench laser (' p1, D'1), bounced once off the right proof mass (d'1), and directed through the fiber (incurring phase shift m1(t)), to the left bench, where it is mixed with laser light (p1). Similarly the right bench records the phase differences ' s1 and ' t1. The laser noises, the gravitational wave signals, the optical path noises, and proof-mass and bench noises, enter into the four data streams recorded at vertex 1 according to the following expressions [7Jump To The Next Citation Point]:

gw opticalpath ' [ '] s1 = s1 + s1 + D3p 2- p1 + n0 -2^n3 .d1 + ^n3 .D1 + n^3 .D3D 2 , (11) ' ( ' ') t1 = p1 - p1 - 2n0 ^n2 . d1 - D 1 + m1. [ ] (12) s' = s'gw+ s'opticalpath + D p - p'+ n 2^n .d' - ^n .D' - ^n .D D , (13) 1 1 1 ( 2 3 ) 1 0 2 1 2 1 2 2 3 t1'= p1 - p'1 + 2n0 ^n3 . d1 - D1 + m1. (14)
Eight other relations, for the readouts at vertices 2 and 3, are given by cyclic permutation of the indices in Equations (11View Equation, 12View Equation, 13View Equation, 14View Equation).

The gravitational wave phase signal components gw 'gw si ,si, i = 1,2,3, in Equations (11View Equation) and (13View Equation) are given by integrating with respect to time the Equations (1) and (2) of reference [1Jump To The Next Citation Point], which relate metric perturbations to optical frequency shifts. The optical path phase noise contributions opticalpath si, 'opticalpath si, which include shot noise from the low SNR in the links between the distant spacecraft, can be derived from the corresponding term given in [7Jump To The Next Citation Point]. The ti, t'i measurements will be made with high SNR so that for them the shot noise is negligible.

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