### 4.7 Physical interpretation of the TDI combinations

It is important to notice that the four interferometric combinations , which
can be used as a basis for generating the entire TDI space, are actually synthesized Sagnac
interferometers. This can be seen by rewriting the expression for , for instance, in the following form,
and noticing that the first square bracket on the right-hand side of Equation (42) contains a combination of
one-way measurements describing a light beam propagating clockwise around the array, while the other
terms in the second square-bracket give the equivalent of another beam propagating counter-clockwise
around the constellation.
Contrary to , , and , can not be visualized as the difference (or interference) of two
synthesized beams. However, it should still be regarded as a Sagnac combination since there exists a
time-delay relationship between it and , , and [1]:

As a consequence of the time-structure of this relationship, has been called the Symmetrized Sagnac
combination.
By using the four generators, it is possible to construct several other interferometric combinations, such
as the unequal-arm Michelson , the Beacons , the Monitors , and the
Relays . Contrary to the Sagnac combinations, these only use four of the six data
combinations , . For this reason they have obvious utility in the event of selected subsystem
failures [7].

These observables can be written in terms of the Sagnac observables in the following way,

as it is easy to verify by substituting the expressions for the Sagnac combinations into the above equations.
Their physical interpretations are schematically shown in Figure 5.
In the case of the combination , in particular, by writing it in the following form [1],
one can notice (as pointed out in [26] and [25]) that this combination can be visualized as the difference of
two sums of phase measurements, each corresponding to a specific light path from a laser onboard
spacecraft 1 having phase noise . The first square-bracket term in Equation (45) represents a
synthesized light-beam transmitted from spacecraft 1 and made to bounce once at spacecraft 2 and 3,
respectively. The second square-bracket term instead corresponds to another beam also originating
from the same laser, experiencing the same overall delay as the first beam, but bouncing off
spacecraft 3 first and then spacecraft 2. When they are recombined they will cancel the laser phase
fluctuations exactly, having both experienced the same total delay (assuming stationary spacecraft).
The combinations should therefore be regarded as the response of a zero-area Sagnac
interferometer.