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5.1 The unequal-arm Michelson

The unequal-arm Michelson combination relies on the four measurements j 1, j ' 1, j ' 2, and j 3. Note that the two combinations j1 + j2',3, j1'+ j3,2' represent the two synthesized two-way data measured onboard spacecraft 1, and can be written in the following form,
j1 + j2',3 = (D3D3' - I)f1, (50) j1'+ j3,2'= (D2'D2 - I)f1, (51)
where I is the identity operator. Since in the stationary case any pairs of these operators commute, i.e. DiDj' - Dj'Di = 0, from Equations (50View Equation, 51View Equation) it is easy to derive the following expression for the unequal-arm interferometric combination X which eliminates f1:
X = [D2'D2 - I](j1 + j2',3)- [(D3D3' - I)](j1'+ j3,2'). (52)
If, on the other hand, the time-delays depend on time, the expression of the unequal-arm Michelson combination above no longer cancels f 1. In order to derive the new expression for the unequal-arm interferometer that accounts for “flexing”, let us first consider the following two combinations of the one-way measurements entering into the X observable given in Equation (52View Equation):
[(j1'+ j3;2') + (j1 + j2;3);22'] = [D2'D2D3D3' - I]f1, (53) [(j + j ' ) + (j '+ j ') '] = [D D 'D 'D - I]f . (54) 1 2 ;3 1 3;2 ;33 3 3 2 2 1
Using Equations (53View Equation, 54View Equation) we can use the delay technique again to finally derive the following expression for the new unequal-arm Michelson combination X1 that accounts for the flexing effect:
' ' ' ' X1 = [D2D2 D3 D3 - I][(j21 + j12;3) + (j31 + j13;2);33] - [D3'D3D2D2' - I][(j31 + j13;2) + (j21 + j12;3');2'2]. (55)
As usual, X2 and X3 are obtained by cyclic permutation of the spacecraft indices. This expression is readily shown to be laser-noise-free to first order of spacecraft separation velocities Li: it is “flex-free”.
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