LISA – the Laser Interferometer Space Antenna – is a ESA/NASA collaborative mission, planned to be launched around 2018. Its ambition is the detection of gravitational waves in the frequency range 30 μHz to 1Hz. With a strain sensitivity of 3.2 · 10–19 at a frequency of 10–3 Hz it will detect gravitational waves caused e. g. by neutron star binaries, white dwarf binaries, super-massive black hole binaries and super-massive black hole formations.
The LISA mission consists of three identical spacecraft which form an equilateral in a heliocentric Earth-trailing orbit, approx. 20° behind the Earth (cf. Fig. 1). Each edge of this triangle - i. e. one interferometer arm - has an arm lengths of about 5 million kilometers. Gravitational waves passing the LISA formation will be measured as changes in the length of the interferometer arms by use of laser interferometry with ~ 10pm/ sensitivity.
External disturbances like solar radiation pressure or solar wind will change the position of the spacecraft and thereby might affect the interferometer signals caused by gravitational waves. Therefore, a free-flying proof mass – as part of the gravitational reference sensor (GRS) - inside the satellites is taken as reference for a purely gravitational orbit. As the external disturbances only act on the surface of the satellite, the distance between proof mass and its housing (which is rigidly connected to the satellite) is changing. In case of a drag-free controlled satellite, any change of the proof mass position is measured and the satellite is controlled in such a way that it is centered around the proof mass at any time, canceling all non-gravitational forces acting on the spacecraft. The proof mass also acts as the end mirror of the interferometer arms.
In the so-called strap-down-architecture (cf. Fig. 2 and ) each interferometric measurement of the distance between two satellites is split into three independent measurement: (i) the distance between the proof mass and the optical bench on the first satellite, (ii) the distance between the optical benches aboard the two satellites, and (iii) the distance between the optical bench on the second satellite and its proof mass. The proof mass position sensor is part of the so called science interferometer and must fulfill requirements of 1pm/ (for frequencies above 2.8 mHz with an f−2 relaxation down to 30 μHz) for the translation measurement and at least 20 nrad/ for the tilt measurement (for frequencies above 0.1 mHz with an f-1 relaxation down to 30 μHz).
These requirements necessitate an optical readout (ORO) of the proof mass position utilizing laser in-terferometry. Our interferometer concept is a possible ORO for the LISA sensitive axes.
Our experimental setup is based on a highly symmetric design and represents a heterodyne interferometer with spatially separated beams [2, 3, 4, 5]. Reference and measurement beams of the interferometer have the same frequency and polarization. Also, their optical pathlengths (especially inside optical components) are similar. A schematic of our interferometer setup is shown in Fig. 3, a photograph in Fig. 4.
The signal at the measurement photodiode is given by
and the signal at the reference photodiode by
Here, A and B denote the amplitudes of the laser frequencies f1 and f2, respectively, and Δw = 2π|f1 — f2|. The phase φ(t) between the signals on measurement and reference photodiode is proportional to the displacement Δl of the measurement mirror:
where λ is the vacuum wavelength of the light and n the refractive index of the medium the light is traveling in.
For in-quadrature measurement, the two signals
are generated, where the phase measurement is obtained by
In order to measure the tilt of the proof mass the method of Differential Wavefront Sensing (DWS) [6, 7] is implemented. Therefore, we use quadrant photodiodes and the phase difference between opposing halves is measured. The interferometer setup can be subdivided into two parts: (i) the optical setup for frequency generation, and (ii) the interferometer board. The optics for frequency generation is placed on an optical table using standard optical components. The interferometer board is placed inside a vacuum chamber operated at pressures below 10–3 mbar. For mechanically stability and compactness, purpose-built optical component mounts with a beam-height of 2 cm are used.
Heterodyne Frequency Generation
An NPRO-type (non-planar ring-oscillator) Nd:YAG laser at a wavelength of 1064 nm is used as light source. Part of its output power is split and shifted in frequency by use of two acousto-optic modulators (AOMs). The AOMs are working at a frequency of 79.99 MHz and 80.00 MHz, respectively, resulting in a heterodyne frequency of 10 kHz. The two laser beams are fiber-coupled (cf. 5) and sent to the interferometer inside the vacuum chamber via optical single mode fibers. The RF-signals driving the AOMs are generated by use of two phase-locked direct digital synthesizers (DDS).
The 300mm × 440mm × 40mm interferometer board is made by cast aluminum which offers low internal stresses and therefore reduced long-term drifts of the material. The two frequencies are fiber-coupled to the interferometer board. Polarizers at the fiber inputs and outputs ensure a proper and clean polarization. The optical power of each beam is ∼ 2.5 mW at the fiber output, this is equivalent to 100μW on the proof mass. Both beams are split at a 6 mm thick wedged glass plate. The front side reflections are used in the interferometer while the back side reflections are sent to two monitor photodiodes (PD1 and PD2 in Fig. 3). Their signals are taken for intensity stabilization of the two laser beams after fiber outcoupling. The amplitudes of the RF signals driving the two AOMs are used for actuation of the light intensities. The beams transmitted at the glass plates are superimposed on a third photodiode (PD3 in Fig. 3). The generated heterodyne signal is phase-locked to a 10 kHz signal which is obtained by mixing the two DDS (direct digital synthesizer) output signals. The signal at the photodiode corresponds to an interferometer signal, where the splitting in measurement and reference beam takes place at the beamplitter before the AOMs outside the vacuum chamber. Therefore, it contains all differential phase effects caused by the fibers and the AOMs. A piezoelectric actuation of one mirror in one arm of the interferometer outside the vacuum chamber is used for actuation of the feedback loop. The two beams used for the interferometer are both split into two parallel output beams. A symmetric beamsplitter is used providing the same optical pathlengths inside the component. Also, ghost reflections of the input beams are not reflected back in direction of the input beam.
The beams with frequency f1 are representing measurement and reference beams of the interferometer, the beams with frequency f2 are used for generating heterodyne signals at the photodiodes. The beams with frequency f1 are first reflected by a polarizing beamsplitter (PBS) towards the measurement and reference mirror, which in our experiment is realized by one fixed mirror. After passing twice a quarter waveplate (λ/2), both beams are transmitted at the PBS and superimposed with the beams with frequency f2 at a (non-polarizing) beamsplitter (BS). Two quadrant photodiodes (QPD1 and QPD2 in Fig. 3) measure the heterodyne signals. Their sum signals are taken for translation measurement, the signals of opposing halves for tilt measurement.
Digital Phase Measurement
The signals of the quadrant photodiode are first pre-amplified inside the vacuum chamber. The sum signals and the signals for differential wavefront sensing are generated and amplified outside the vacuum chamber by use of analog electronics. All signals are anti-aliasing filtered by use of a 6th order Bessel filter (corner frequency: 20 kHz). The signals are then simultaneously digitized by a National Instruments field programmable gate array (FPGA) computer board at a frequency of 160 kHz and with a 16 bit resolution. The FPGA board is programmed by LabVIEW. A digital phasemeter is implemented where the corresponding input signals to the FPGA board are multiplied and low-pass filtered. For inquadrature measurement, the signals phase-shifted by 90° are generated. The data is reduced by a factor of 8000 and transferred to a LabVIEW host program which carries out the calculation as given in (6). The program also monitors phasejumps by π and therefore offers a dynamic range not limited by λ/2 in mirror translation. For the tilt measurement, the summed halves of the quadrant photodiode are input signals to the phasemeter. The obtained phase is converted to a tilt.
MEASURED NOISE PERFORMANCE
We performed a noise measurement where the measurement and the reference mirror are represented by the same fixed mirror (cf. Fig. 3). The power spectrum density (PSD) of the translation measurement is shown in Fig. 6, the PSD of the corresponding tilt measurement in Fig. 7. Also included in these graph are the LISA requirements. In these measurements, the intensity stabilization and the phase-lock of the heterodyne frequencies are operating. The interferometer board was not actively stabilized in temperature as it was seen, that a passive stabilization showed better results. The peak near the 3 Hz was caused by a missing synchronization of the FPGA clock and the DDS. It is removed in later measurements.
In translation measurements (cf. Fig. 6), noise level below 1nm/ for frequencies above 0.1 mHz and below 5pm/ for frequencies above 10 mHz were obtained. In tilt measurements (cf. Fig7) the noise level is below 1 μrad/ for frequencies above 0.1 mHz and below 10nrad/ for frequencies above 10 mHz. These noise level in tilt measurement fulfil the LISA requirements for frequencies above 1 mHz.
The interferometer setup as described before is a high sensitive measurement system which now can be used to investigate other key aspects in the LISA context. For example:
• adetermine the coefficient of thermal expansion of ultra-stable structural materials, e.g. carbon fiber reinforced plastic (CFRP),
• testing the noise performance and linearity of various LISA mechanisms, e.g. the actuator of the in-field pointing mechanism as well the mechanism itself, and
• investigate the quality of mirror surfaces in the pm-range
Based on the above presenting setup we developed a high sensitive dilatometer for characterizing the dimensional stability of ultra-stable materials. The principle of the dilatometer is shown in Fig.8. We are changing the temperature of the test tube via radiative heating/cooling and measure the linear expansion with our interferometer with sub-nm resolution. The CTE is given as:
Measurement and reference mirror are placed at the top resp. at the bottom inside a test tube made by the material under investigation (cf. Fig. 8). We also developed a test tube support, special mirror mounts and a heating/cooling system. The mirror mounts are critical components of the measurement facility as their thermal expansion should not affect the CTE measurement of the test tube. They are made of Invar36, an iron nickel alloy with a very low CTE (1.8 · 10–6K–1). The mirror is clamped inside the mirror mount, which on its part is clamped inside the test tube. The reflecting surface of the mirror and the clamping points of the mirror are all in the same plane - a thermal expansion of the mirror mount therefore should not affect the position of the reflective surface of the mirror with respect to the test tube. For measurement, a sine thermal cycling with a period up to several hours is applied to the CFRP test tube.
Due to limitations of the vacuum chamber and the interferometer setup, we are restricted to tubes with an inner diameter of 20 mm, an outer diameter of ~ 27 mm and a length of 10 cm. The temperature of the test tubes can be varied between 20° C and 60° C.
We first performed a measurement with a CFRP tube with a known CTE of ~ −6.47 · 10–7K–1 (cf. Fig. 10). Our result with the interferometer test facility provided a CTE of ~ −6.10 · 10–7K–1. In the next step, we procured a new CFRP tube with a theoretical CTE below −2.5 · 10–8K–1. We measured a CTE of ~ −3.6 · 10–7K–1, which not does not agree with the theoretical value .
Presently, we are and the company analyzing the manufacturing process as the determination of the CTE during production includes uncertainties.
SUMMARY AND OUTLOOK
We developed a compact setup of a high sensitivity heterodyne interferometer. Noise levels below 5 pm/ in translation measurement and below 10 nrad/ in tilt measurement – both for frequencies above 10 mHz – were demonstrated in lab experiments. In a current project, the interferometer is adapted as dilatometer for measurement of thermal expansion of carbon fiber reinforced plastics. Measurements were carried out validating the capability of our setup. In general, the test facility offers the possibility to characterize ultra-stable materials with a CTE down to 10–8 K–1.
Current activities also include the development of new quasi-monolithic setup using thermally highly stable glass ceramics as breadboard materials in the interferometer.
This work is partially supported by the German Aerospace Center (Deutsches Zentrum für Luft-und Raumfahrt e. V.) within the program LISA Performance Engineering (DLR contract number: 50OQ0701).
The authors thank the Albert-Einstein-Institute Hannover, Evgeny Kovalchuk and Klaus Palis from the Humboldt-University Berlin and Hans-Reiner Schulte from EADS-Astrium Friedrichshafen for their support and fruitful discussions.
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