developed by Morrison et al, Ref [27] and Ref [26] which uses multi-segment diodes to determine the radio frequency (RF) beat-note amplitude difference between the two halves of the wavefront and develops a control signal to adjust optical components and remove this difference Ref [55] and see section 2.2.1. The technique is well established and very accurate, but is limited by the number of segments on the diodes used to map wavefront tilt and therefore assumes that tilt to be linear. As explained in Chapter 3, LIGO also uses Hartmann sensors which measure higher-order distortions in the wavefront caused by thermal distortions of the mirror surfaces Ref [32]. These use a fine array of precisely positioned holes to project a small sample of a test laser beam onto a charged coupled device (CCD) and infer the shape of the wavefront by measuring the displacements of the individual image spots from their expected positions Ref [15] and figure 2.6. This technique can achieve an RMS wavefront error of λ/700 in a single-shot, but relies on having a pre-recorded reference wavefront for comparison, which must be updated offline. It has also been used to identify flaws in the test masses which were not otherwise visible, Ref [90], Ref [91] and Ref [92] for the general approach and Ref [33] for specifics.
In space, the current approach is to sense minute changes in wavefront tilt using segmented photodiodes (see Ref [93], Ref [94], Ref [82] and Ref [95]) and realign the receiving satellite so that it can reflect the signal back to the originating satellite successfully. This approach is also accurate and effective but suffers because signals are recorded by separate channels and then compared. Differential phase shifts in these channels produce errors. These may be due to temperature fluctuations as the instruments move in and out of the Earth’s shadow (GRACE) or other, internally generated temperature variations (LISA Pathfinder and LISA) Ref [96]. For example, the analogue-to-digital converters (ADCs) exhibit 1/f noise as their capacitances changes with temperature, decreasing the accuracy of the realignment signal. The capacitance of the photodiode segments themselves may also change with temperature, giving rise to spurious phase shifts between the segments which are indistinguishable from wavefront tilt. There have been attempts to reduce these errors using common pilot signals, but these must be separated in frequency from the heterodyne signal Ref [14] and so react differently to the analogue characteristics of the equipment, reducing their effectiveness.
This experiment presents a new technique based on a single-element photodiode and the spatial modulation of the local oscillator light. It encodes spatial information onto the light and uses this to separate the phases of different parts of the beam in post processing as outlined in Chapter 3. This technique shifts the complexity from the detection hardware into deterministic digital signal processing. Notably, the use of a single analogue channel (single photodiode and ADC) avoids the low-frequency error sources identified above. The technique can also sense the wavefront phase at many points, limited only by the number of pixels on the spatial light modulator in contrast to the standard 4 points from a quadrant diode. For ground-based systems, the new technique could be used to identify and eliminate higher-order modes, while, for space-based systems, it provides a measure of wavefront tilt which is less susceptible to low frequency noise. The technique involves only minimal changes to the physical set up of the alignment system of a satellite interferometer, Figure 5.1.
The experiment uses 1064 nm laser light which is in the near infra-red and is invisible to the human eye.
5.2
Experiment Goals
Chapter 5 Wavefront Sensing and Correction
Figure 5.1: A comparison of the current (left) and proposed (right) arrangements for space based interferometers. In the proposed set up, the steering mirror and segmented photodiode have been replaced by an SLM and single photodiode respectively. The separation of the wavefront phases is done in post processing.
1. Demonstrate that digitally enhanced heterodyne interferometry (DEHI) can be used to develop a detailed map of a laser beam wavefront with a single element photodiode. 2. Compare the performance of DEHI with the performance of the tradition, segmented photodiode approach and, in particular, to compare the noise performances of the two approaches.
3. Demonstrate the capacity to not only to detect the shape of the wavefront but, using these measurements (or indeed measurements made with a Hartmann sensor), make corrections to it as desired.
Conceptually the approach was straightforward, see 5.2. A laser beam was split and sent through the two arms of a Mach-Zehnder interferometer. One beam, the local oscillator, was passed through an Acousto-Optic Modulator (AOM) and directed onto a segmented spatial light modulator (SLM) where it was phase modulated with spatially-dependent codes while the other, the signal, was passed through a different AOM and then directed onto a fast steering mirror (FSM) which could apply wavefront tilts of known amplitude and frequency. They were recombined at a beam splitter where the difference between the frequencies of the two AOMs produced a radio frequency beat note which was sent to two segmented photodiodes and a single photodiode. The SLM was driven by a Multi-driver controller which in turn was driven by a Matlab program running on the host PC. Data from the photodiodes and the sync pulse were collected in a LabVIEW ADC and passed to the FPGA which did some initial processing (channel splitting, demodulation and integration) before passing the results to the host PC for analysis and data presentation and capture.
It is worth noting that encoding the local oscillator is mathematically the same as encoding the signal path.
At the highest level, the reason for undertaking this experiment was to see if DEHI can provide an alternative approach to wavefront sensing that has comparable or better noise