With 3 or more optical heads it has been shown that a multi-link interferometer is sensitive to the pitch and yaw of an incoming wavefront. Although the multi-link interferometry concept uses a weighted average to cancel any pitch and yaw coupled error, it is possible that a multi-link architecture could be used in applications where differential wavefront sensing is required. One example of this is multi-degree of freedom readout of a test mass. This is important for intraspacecraft measurements in LISA [146], can be used as an auxiliary measurement in ground based detectors [150], and could even be used to build more sensitive accelerometers [151].
In [152] a number of different optical techniques, including DEHI and DEHoI, as well as deep phase modulation [153] and deep frequency modulation [154] were compared as potential techniques for multi-degree of freedom test mass position readout. It was noted that since quadrant photodiodes weren’t available at GHz frequencies DEHI was incom- patible with a differential wavefront scheme and that multiple longitudinal measurements would be needed to measure any tilt of the test mass.
The multi-link architecture is able to do exactly this. While it would require some modi- fications at the input, a multi-degree of freedom test mass readout could be realised using the waveguide chip described in Section 8.2.3 and some slight modifications to the signal processing described in Chapter 3.
Figure 8.4 shows an example multi-degree of freedom test mass readout system using the multi-link optical head waveguide. Since the system only uses one-side of the GRACE multi-link interferometer, the reflections off the optical head and test mass will have the same frequency. An acousto-optic modulator (AOM) has been added to provide a local oscillator for the reflections to beat with. In a test mass readout the sensors will need to be placed close to the test mass. Therefore any cross-coupling between optical heads will be minimal and GHz EOMs will be needed to obtain range gates of at most a few centimetres.
On the detector there will be 6 beatnotes: 3 reference reflections from the optical head and 3 reflections off the test mass. Each beatnote will have 2 PRN modulations from
8.3 Other applications laser z x y EOM EOM EOM AOM +fhet optical head waveguide chip reference reflection test mass reflection test mass x(t) c.m θpitch(t) θyaw(t) x0 cA cB cC re fe re n ce detector local oscillator acousto-optic modulator electro-optic modulator pseudo random code C
Figure 8.4: Using the multi-link architecture for multi-degree of freedom test mass readout.
Adding a local oscillator path it is possible to measure the displacementx(t) and rotationθpitch(t)
and θyaw(t) of a test mass. Additional sensors need to be added to measure the remaining test
mass degrees of freedom.
the double-pass of the EOM. The fibre fluctuations along each optical head path can be cancelled by subtracting the displacement measured from the reference beatnotes from the test mass measurements. Assuming the optical heads are arranged in an equilateral triangle with each optical head offset from the centre by dm the displacement along the three links, once fibre fluctuations have been cancelled, will be:
xA(t) ≈ x(t) +d·θpitch(t) (8.1) xB(t) ≈ x(t)−d/2·θpitch(t)−d √ 3/2·θyaw(t) (8.2) xC(t) ≈ x(t)−d/2·θpitch(t) +d √ 3/2·θyaw(t) (8.3) The test mass rotations are assumed to be small and therefore the optical heads are to first order only sensitive to the displacement, x(t), and the pitch and yaw of the test mass1. The rotation is canceled using a weighted average to find an estimate of the displace- ment, x(t). If this estimate,x(t), is subtracted from the optical head measurements thenˆ combinations can be formed to extract measurements of the pitch and yaw:
ˆ θpitch(t) = xA(t)−x(t)ˆ d (8.4) ˆ θyaw(t) = xC(t)√−xB(t) 3 (8.5)
Three degrees of freedom – translation, x(t), and rotations, θpitch(t) and θyaw(t) – have been measured using three optical heads. In Figure 8.4, one waveguide with 3 optical heads is shown. Two more will need to be positioned on the orthogonal faces of the test mass in order to sense all degrees of freedom. Since one 3-link interferometer is able to sense 1 translational degree of freedom and 2 rotational degrees of freedom, there will be some redundancy in the measurements.
1To simplify the equations it is assumed that the centre of the triangle is aligned with the centre of
Chapter 8 Conclusions, further work and other applications
In this analysis it was assumed that the rotations of the test mass are small and therefore second order effects have been ignored. More analysis is needed to determine if this is a reasonable assumption. If it is found that it is not reasonable then more optical heads may be needed to decouple these additional degrees of freedom. Regardless this analysis has shown that a multi-degree of freedom test mass readout could be realised using the multi-link architecture.