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4.7 Physical interpretation of the TDI combinations

It is important to notice that the four interferometric combinations (a,b, g,z), 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 a, for instance, in the following form,
a = [j1'+ D2j3'+ D1D2'j2'] - [j1 + D3j2 + D1D3j2], (42)
and noticing that the first square bracket on the right-hand side of Equation (42View Equation) 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 a, b, and g, z 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 a, b, and g [1Jump To The Next Citation Point]:

z- D1D2D3z = D1a - D2D3a + D2a - D3D1b + D3g - D1D2g. (43)
As a consequence of the time-structure of this relationship, z 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 (X, Y,Z), the Beacons (P, Q,R), the Monitors (E,F, G), and the Relays (U, V,W ). Contrary to the Sagnac combinations, these only use four of the six data combinations ji, j'i. For this reason they have obvious utility in the event of selected subsystem failures [7Jump To The Next Citation Point].

These observables can be written in terms of the Sagnac observables (a,b, g,z) in the following way,

D1X = D2D3a - D2b - D3g + z, P = z - D1a, E = D1 - D1z, (44) U = D1g - b,
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 5View Image.
View Image

Figure 5: Schematic diagrams of the unequal-arm Michelson, Monitor, Beacon, and Relay combinations. These TDI combinations rely only on four of the six one-way Doppler measurements, as illustrated here.
In the case of the combination X, in particular, by writing it in the following form [1Jump To The Next Citation Point],
' ' ' ' X = [(j1 + D2'j3) + D2'D2(j1 + D3j 2)]- [(j1 + D3j2) + D3D3'(j 1 + D2'j3)], (45)
one can notice (as pointed out in [26] and [25Jump To The Next Citation Point]) 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 f1. The first square-bracket term in Equation (45View Equation) 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 X combinations should therefore be regarded as the response of a zero-area Sagnac interferometer.


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