## 3 Time-Delay Interferometry

The description of TDI for LISA is greatly simplified if we adopt the notation shown in Figure 2, 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 [2452534].

Several notations have been used in this context. The double index notation recently employed in [25], 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 , , , with being opposite spacecraft . We orient the vertices 1, 2, 3 clockwise in Figure 2. Unit vectors between spacecraft are , oriented as indicated in Figure 2. We index the phase difference data to be analyzed as follows: The beam arriving at spacecraft has subscript and is primed or unprimed depending on whether the beam is traveling clockwise or counter-clockwise (the sense defined here with reference to Figure 2) around the LISA triangle, respectively. Thus, as seen from the figure, is the phase difference time series measured at reception at spacecraft 1 with transmission from spacecraft 2 (along ).

Similarly, 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: . We also adopt a notation for delayed data streams, which will be convenient later for algebraic manipulations. We define the three time-delay operators , , where for any data stream
where , , are the light travel times along the three arms of the LISA triangle (the speed of light is assumed to be unity in this article). Thus, for example, , , etc. Note that the operators commute here. This is because the arm lengths have been assumed to be constant in time. If the are functions of time then the operators no longer commute [534], 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 , , . The proof-mass-plus-optical-bench assemblies for LISA spacecraft number 1 are shown schematically in Figure 3. The photo receivers that generate the data , , , and at spacecraft 1 are shown. The phase fluctuations from the six lasers, which need to be cancelled, can be represented by six random processes , , where , are the phases of the lasers in spacecraft on the left and right optical benches, respectively, as shown in the figure. Note that this notation is in the same spirit as in [3325] in which moving spacecraft arrays have been analyzed.

We extend the cyclic terminology so that at vertex , , the random displacement vectors of the two proof masses are respectively denoted by , , and the random displacements (perhaps several orders of magnitude greater) of their optical benches are correspondingly denoted by , where the primed and unprimed indices correspond to the right and left optical benches, respectively. As pointed out in [7], the analysis does not assume that pairs of optical benches are rigidly connected, i.e. , 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 the phase fluctuations upon transmission through the fibers of the laser beams with frequencies , and . The phase shifts within a given spacecraft might not be the same for large frequency differences . For the envisioned frequency differences (a few hundred MHz), however, the remaining fluctuations due to the optical fiber can be neglected [7]. 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 .

The laser phase noise in is therefore equal to . Similarly, since 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: . Figure 3 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 and in Figure 3.

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 and in Figure 3.

The expressions for the , and , phase measurements can now be developed from Figures 2 and 3, and they are for the particular LISA configuration in which all the lasers have the same nominal frequency , and the spacecraft are stationary with respect to each other. Consider the process (Equation (13) below). The photo receiver on the right bench of spacecraft 1, which (in the spacecraft frame) experiences a time-varying displacement , measures the phase difference by first mixing the beam from the distant optical bench 3 in direction , and laser phase noise and optical bench motion that have been delayed by propagation along , after one bounce off the proof mass (), with the local laser light (with phase noise ). Since for this simplified configuration no frequency offsets are present, there is of course no need for any heterodyne conversion [33].

In Equation (12) the measurement results from light originating at the right-bench laser (, ), bounced once off the right proof mass (), and directed through the fiber (incurring phase shift ), to the left bench, where it is mixed with laser light (). Similarly the right bench records the phase differences and . 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 [7]:

Eight other relations, for the readouts at vertices 2 and 3, are given by cyclic permutation of the indices in Equations (11, 12, 13, 14).

The gravitational wave phase signal components , , in Equations (11) and (13) are given by integrating with respect to time the Equations (1) and (2) of reference [1], which relate metric perturbations to optical frequency shifts. The optical path phase noise contributions , , which include shot noise from the low SNR in the links between the distant spacecraft, can be derived from the corresponding term given in [7]. The , measurements will be made with high SNR so that for them the shot noise is negligible.