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
).
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
.
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
]:
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.
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