In this chapter two space interferometers were discussed: LISA and GRACE Follow-On. While the goals of the two missions differ, the basic measurement principles are the same. The heterodyne interferometry and digitally implemented phasemeters used to infer the interspacecraft displacement from the phase of optical signals traveling between spacecraft were explained. The main sources of noise that limit the sensitivity of the displacements were then discussed as well as currently used, and potential future techniques, that can be used to mitigate their effect.
Chapter 3
Multi-link GRACE
The multi-link interferometry concept is simple: the line-of-sight displacement between two spacecraft centres of mass can be recovered in post-processing by forming linear com- binations of multiple interspacecraft link measurements. This chapter focuses on the multi-link concept, explaining the signal processing steps required to transform multiple one-way link measurements into a round-trip displacement, immune from pathlength noise, laser frequency displacement noise and rotation-to-pathlength coupled error.
The chapter starts with a discussion of the multi-link concept, explaining how the interfer- ometer takes advantage of geometry to suppress rotation. The concept is then compared with some existing techniques that operate on similar principles. The requirements of a multi-link interferometer are then presented before an example multi-link implementation is introduced. Taking the example implementation as reference, the signal processing re- quired to synthesise a line-of-sight displacement measurement is outlined. The chapter ends with a discussion of the main challenges of this implementation, indicating three experimental results that are needed before the feasibility of a multi-link GRACE can be determined.
This chapter is based on the following publication:
S.P. Francis, T.T-Y. Lam, D.E. McClelland, and D.A. Shaddock, “Multi-link laser in- terferometry architecture for interspacecraft displacement metrology,” Journal of Geodesy (2017).
3.1
Multi-link interferometry concept
Interspacecraft laser links, offset from the line-of-sight between the spacecraft centres of mass, will be sensitive to rotation and jitter of both spacecraft. To sense the wavefront tilt due to these rotations, multiple laser links can be formed between spacecraft. In a multi-link interferometer, each spacecraft will be equipped with a number of spatially offset optical heads. Optical links are formed between all of the optical heads on one spacecraft with all of the optical heads on the distant spacecraft. Each optical head is a fibre collimator used to both transmit light to, and receive light from, the distant spacecraft. The output surface of each optical head is used as a reference, with the light coupling in from the distant spacecraft interfering with small back reflections from each optical head. This allows the measurement to be immune to the noise internal to the interferometer. A similar concept was proposed in [76].
In Chapter 1, the multi-link concept was introduced with a simple example. Links po- sitioned symmetrically either side of the line-of-sight between the spacecraft centres of
3.1 Multi-link interferometry concept multi-link interferometer optical head dA ∆xtilt,B(t) θ(t) line-of-sight laser link B SC1 xR(t) z x y SC2 laser link A A B A B ∆xtilt,A(t) dB multi-lin k interf erome ter c.m plane SC1 rotated c.m plane SC2 unrotated c.m plane SC2 c.m c.m
Figure 3.1: A simple multi-link interferometer. The optical heads are positioned arbitrarily within the centre of mass plane on each spacecraft. With the optical heads positioned on the same side of the spacecraft centre of mass, the interferometer cannot use symmetry to cancel rotation to pathlength coupled error. It is possible however to find weights in post processing that will cancel the effect of spacecraft rotation.
mass experienced an asymmetric lengthening and shortening as the local spacecraft ro- tated. By averaging the two link measurements the rotation-to-pathlength coupling was shown to cancel. This required the links to be placed symmetrically which would make the spacecraft integration challenging. The real advantage of the multi-link architecture however is that the linear combinations are formed offline in post-processing and there- fore, it is possible to adjust the weights of individual link measurements. Consequently, the optical heads do not need to be placed symmetrically but can be positioned arbitrarily on the spacecraft. The linear combination that cancels the rotation-to-pathlength coupled displacement can be determined after the optical heads have been positioned.
In Figure 3.1 the optical links have not been placed symmetrically but are instead posi- tioned on the same side of the spacecraft centre of mass. Link A is offset a distance dA and Link B is offsetdB from the line-of-sight. The one-way displacement measured along each link will be:
xA(t) = xR(t) +dAsinθ(t) (3.1) xB(t) = xR(t) +dBsinθ(t) (3.2) wherexR(t) is the interspacecraft displacement signal and both measurements have rota- tion coupled error, albeit with different strength.
The rotation coupled error can be cancelled, recovering the interspacecraft displacement, by forming a linear combination of the two link measurements with weights wA and wB. The weights need to cancel the rotation while preserving the range measurement. In this example it is straightforward to show that if the weights are:
wA = dB dB−dA (3.3) wB = −dA dB−dA (3.4)
Chapter 3 Multi-link GRACE
then the rotation-to-pathlength coupled error will cancel in the linear combination. The linear combination is:
wAxA(t) +wBxB(t) =xR(t) (3.5) Therefore, despite being positioned arbitrarily, it is possible to find weights such that the rotation-to-pathlength coupled error will cancel. As this example shows, the weights can be calculated from the positions of the optical heads relative to the centre of mass. As it will not always be practical to measure the positions of the optical heads, there are fortunately other methods to determine the weights. These are discussed in Section 3.5.2. In this example, since the optical heads are placed on the same side of the centre of mass one of the weights will need to be negative. This affects the signal to noise ratio (SNR) of the combination, leading to a lower SNR compared with when positive weights are used. This is discussed further in Section 3.5.4.