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Humboldt-Universität zu Berlin - Faculty of Mathematics and Natural Sciences - Optical Metrology

Optical subsystems for LISA

T. Schuldt, M. Gohlke, A. Peters

Click here for more information about LISA


In the LISA context an optical readout (ORO) of the test mass position is necessary. For more information about LISA and the strap-down architecture please read the LISA chapter (link above). However, in the current baseline the noise requirements are defined as the follow.

For translation measurements:

noise < 1 sqrt[1 + (2.8mHz/f)4 ] [pm/sqrt(Hz)]

For tilt measurements:

noise < 20 sqrt[1 + (0.1mHz/f)2 ] [nrad/sqrt(Hz)]

EADS Astrium (Friedrichshafen) in cooperation with the Humboldt University (Berlin) and the University of Applied Science Konstanz (HTWG-Konstanz) developed a highly symmetric heterodyne Michelson interferometer over the past several years as a possible ORO. This interferometer nearly fulfils the requirements shown above. In the following chapters we will give an overview about the interferometer concept, the experimental setup and the measured noise performance.
We also use this high precision optical metrology system for other questions of interest in the LISA-mission context. For example the linear coefficient of thermal expansion (CTE) of satellite structural materials as carbon fiber reinforced plastic (CFRP). Therefore we developed a dilatometer based on our interferometer. An overview about these application activities is given in the last section.


The way to our setup

In this section we describe the way from an ordinary Michelson Interferometer to our highly symmetric heterodyne Michelson Interferometer setup.


Simple Michelson Interferometer

A simple Michelson Interferometer is shown in part a in the picture below. The laser beam is divided at a non polarizing 50/50 beam splitter (BS). One beam is the measurement beam – reflected by the measurement mirror and in the case of LISA the proof mass – and the other is called reference beam – reflected by a fixed reference mirror. On there way back, both beams passed the BS the second time. Now one part of each beam runs back to the laser sources – not showed in the figure – and the other parts are superimposed on a photo diode (PD). The intensity on the photo diode is now directly connected (see formula in the figure above) to the displacement δ of the proof mass. The information about the displacement is a DC-signal. These DC-signals are hard to handle in the sub Hz frequency range because there are also many disturbances (e.g. long time gain drifts in amplifiers) in this DC-domain. For our experiment it is easier to have the displacement information in an AC-signal.

Grafik1



Heterodyne Interferometer

A way to generate the preferred AC-signal is a heterodyne setup, this means two laser frequencies f1 and f2 are necessary.
One possible heterodyne setup is shown in part b of the figure above. The beams with the different frequencies are superimposed before they run into the polarizing beam splitter (PBS). The laser at frequency f1 has a p-polarization consequently it first passed the PBS and gets reflected by the proof mass. One the way back to the PBS the measurement beam passed the quarter wave plate the second time, now it has an s-polarization and gets reflected by the PBS. The behavior of the reference beam is similar. First it is s-polarized and gets reflected by the PBS in the direction of the reference mirror. After passing twice the quarter wave plate, it runs through the PBS. After passing a polarizer (45°) both beams are now superimposed on the photo diode. Now the information about displacement of the proof mass is in the phase of the sinusoidal signal at the frequency fhet = |f1-f2| (cf. figure above, part B where Δω = 2 πfhet).
A problem in this setup is the clear separation of the different beams. In a lab experiment you neither have an ideal PBS nor a perfect prepared pair of beams of different frequencies. So in the end parts of the laser at the frequency f1 are in the reference arm and parts of the laser at the frequency f2 are in the measurement arm. These mixings leads to incorrect phase values and can become the limiting factor for setup like these.

An alternative heterodyne setup is depicted in part c of the figure above. In order to avoid the mixing problem the different laser frequencies are separated before the run through the beam splitter. This setup is more or less a theoretical concept. There is no reference mirror anymore consequently all changes of the optical pathlength of the laser at the frequencies f2 are leading directly to a measurement error.

A mix of both concepts leads to our interferometer concept.


Our Setup

Our experimental setup is based on a highly symmetric design and represents a heterodyne interferometer with spatially separated beams. 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 the basic idea of our interferometer setup is shown in the figure above. The laser at the frequency f1 (represented by the blue line) is split up into two parallel output beams at the so-called energy separator cube (ESC). The upper one is the reference beam and is reflected back at the reference mirror. The lower one represents the measurement beam and is reflected back at the proof mass. After passing twice the quarter wave plate (λ/4) both beams are reflected at the polarizing beam splitter (PBS) and become superimposed at the non-polarizing beam splitter (BS) with the beams at the frequency f2 (represented by the orange lines). Two photodiodes detect the heterodyne signals at the heterodyne frequency fhet = |f1-f2|.

Grafik2



The phase difference between the measurement and reference signal is proportional to the relative translation of the measurement mirror.

The signal at the measurement photodiode (Σ1) is given by

Imeas ~ AB cos (Δωt – φ(t));

and the signal at the reference photodiode (Σ2) by

Iref ~ AB cos (Δωt).

Here, A and B denote the amplitudes of the laser frequencies f1 and f2, respectively, and Δω = 2π |f1- f2|. The phase φ(t) between the signals on measurement and reference photodiode is proportional to the displacement δ of the measurement mirror:

φ(t) = 4πn/λ × δ(t)

where λ is the vacuum wavelength of the light and n the refractive index of the medium the light is traveling in.

We are also using the technique of the differential wavefront sensing (DWS) to measure the tilt of the mirrors. The single element photodiode are replaced by quadrant photodiodes. The phase difference between opposing halves of each quadrant diode is measured and the tilt calculated - for both reference and measurement mirror.


Experimental setup

The metrology setup can be divided into three parts. The frequency generation and laser control systems outside the vacuum chamber as depicted in the Figure below. The second part is the interferometer setup itself inside the vacuum chamber, and the third part is the data acquisition and signal post processing. Details of each part are described in the following.

Grafik3



Outside the vacuum chamber

An NPRO (non-planar ring oscillator) Nd:YAG laser at 1064 nm (Innolight , Mephisto, 1W) is used as light source in this setup. The laser is split and each beam runs through an acousto-optic modulator (AOM). The AOM shifts the frequency of the incoming light. It is also used for an intensity stabilization loop. The RF signals - 79.99 MHz and 80.00 MHz - driving the AOMs are generated by two phase locked direct digital synthesizers (DDS). The resulting heterodyne frequency is 10 kHz. Both beams are fiber coupled to polarizing maintaining single-mode fibers and sent via feed through into the vacuum chamber to the interferometer.


Inside the vacuum chamber

The interferometer board is placed in a vacuum chamber to avoid pathlength changes caused by air fluctuations and to provide better temperature stability. The pressure in the chamber is below 10-3 mbar. The breadboard (300 mm x 440 mm x 40 mm, M3 thread grid 10 mm x 10 mm) is made of cast aluminum which offers low internal stresses and therefore reduced long-term drifts of the material. After fiber out coupling a part of each beam is reflected to a single element photo diode (PD1 and PD2). Each one is used as the control input for the intensity stabilization. On a third photo diode (PD3) a part of the beams intensity are overlapped. The resulting heterodyne signal is the input for a phase control loop in order to ensure a stable phase relation after fiber out coupling. The core of the interferometer setup is similar to sketch shown in the section our setup. The distance between the reference and the measurement beam is 4 mm. The beams have diameter (1/e2) of 1.4 mm each. The parallelism - horizontal and vertical - of the beam pair is smaller then 25 µrad. As depicted in figure below measurement and reference mirror are represented by the same mirror in order to characterize the noise of the setup.

Grafik4



Data acquisition and post processing

The phasemeter is realized on a field programmable gate array (FPGA) PCI-board from National Instruments (NI-8333 PCI) and programmed with LabVIEW. After passing a 20 kHz anti-aliasing filter the pre-amplified signal of each quadrant gets sampled at 160 kHz in an onboard 16-bit analog digital converter (ADC). We realized two different ways to obtain the phase information.

One method to compute the phase information and in the end the information about the translation of the proof mass is shown in part A in the figure below. At the input we have the two analog sum signal Σ1 and Σ2 given by:

Σ1 = AB cos (Δωt – φ1(t)) and Σ2 = AB cos (Δωt – φ2(t))

After digitalization both input signal are multiplied with each other and after a 3 Hz low pass filter only the DC part of the product is left over. The resulting signal

S1 = 1/2 AB cos(Δφ(t)) with Δφ(t) = φ1(t) - φ2(t)

is written to a FIFO (first in first out) memory. For an in-quadrature measurement the input signal Σ2 gets shifted in phase by 90° and also multiplied with Σ1 and low pass filtered. Afterwards the signal

S2 = 1/2 AB sin(Δφ(t))

is also written to the FIFO. The LabVIEW Host-program reads the memory information and computes the phase difference between the input signals S1 and S2 via arctan.

Δφ(t) = arctan(S1/S2)

The amplitude of the signal can also be calculated.

AB = 1/sqrt(2) sqrt(S12 + S22)

The signal chain for the tilt measurement is similar.

Grafik5



An alternative phasemeter concept is depicted in part B in the figure above. The incoming signals are the same as before. Both of them are mixed with a local sine and cosine signal at the fixed heterodyne frequency – in our setup fhet = 10kHz. Now four signals are written to the FIFO memory.

S1 = 1/2 AB cos(φ1(t))
S2 = 1/2 AB sin(φ1(t))
S3 = 1/2 AB cos(φ2(t))
S4 = 1/2 AB sin(φ2(t))

The Host-program calculates only the phases φ1(t) and φ2(t). The translation or the tilt can now be calculated via post processing e.g. mathematic software MatLab.


Testing, calibration and noise performance

To ensure the functionality of the entire experimental setup and the data post processing, we began with some quick tests. First we tested the translation signal by placing a thin glass plate into the measurement arm. Both beams were reflected by the same mirror, the relative translation should be constant. Now we started to tilt the glass plate and consequently the optical pathlength in the measurement arm changes. The relation between the tilt α of the glass plate and the pathlength changes is given by

Δx0(α) ≈ d(n-1)α2/2n

where d is thickness of the glass plate and n the refractive index. The measured translation signal matches with the predicted values.
The tilt signal was tested by tilting the measurement mirror. The measured values agree with the adjusted within the error budgets. Caused by the basic principle of the DWS the tilt measurement is limited to absolute angular values smaller then 250 µrad, this effect was also observable.

After these pre-tests we performed long-time noise measurements where the measurement and the reference mirror are represented by the same fixed mirror. In the figure below the power spectrum densities (PSD) of the measurement series are depicted. In translation measurements, noise level below 200 pm/sqrt(Hz) for frequencies above 0.1 mHz and below 5 pm/sqrt(Hz) for frequencies above 10 mHz were obtained. For the tilt measurements the noise level is below 300 nrad/sqrt(Hz) for frequencies above 0.1 mHz and below 1 nrad/sqrt(Hz) for frequencies above 100 mHz. These noise level in tilt measurement fulfill the LISA DFACS requirements for frequencies above 1 mHz.

Grafik6



Applications

The interferometer setup as described before is a high sensitive measurement system which now can be used as an optical head for other applications (cf. figure below). Two examples will be discussed in the following. Other planned applications are the measurement of:
  • mirror surfaces in the pm-range
  • the point ahead mechanism
  • the in-field pointing mechanism
  • low noise piezo-actuators


Grafik7



Dilatometer

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 depicted in part C in the figure above. We change the temperature of the device under test via radiative heating/cooling and measure the linear expansion with our interferometer with sub-nm resolution. The devices under test are tubes made of various materials as carbon fiber reinforced plastic (CFRP) or Zerodur. However, the CTE is given as:

CTE = ΔL/L · 1/ΔT [K-1]

where L is the length of the tube, ΔL is the expansion and ΔT the temperature variation of the probe. Measurement/reference mirror is placed at the top/bottom inside the tube. We also developed a 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 device under test. They are made of Invar36, an iron nickel alloy with a very low CTE (1.8 10-6 K-1). The mirror is clamped inside the mirror mount, which on its part is clamped inside the 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 tube. For measurement, a sine thermal cycling with a period up to several hours is applied to the tubes.

Due to limitations of the vacuum chamber and the interferometer setup, we are restricted to tubes with an inner diameter of 16 mm, an outer diameter of ~27 mm and a length of 120 mm. 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 0.65 10-6 K-1. Our result with the interferometer test facility provided a CTE of 0.61 10-6 K-1. Currently we investigate various tubes made of Zerodur or CFRP. A typical measurement series is shown in the Figure below.

Grafik8



Profilometer

Another planned application is measurement setup for surface property measurement. Therefore a scan of the measurement beam must be implemented. A schematic of a possible profilometer implementation is shown in part B of the figure at the begin of this section. As the surface information is integrated over the beam diameter, a lens in the measurement beam will be needed placing the focus on the surface. A lateral resolution in the µm range should be possible. In order to overcome the restriction in dynamic range, the optical sensor can be combined with a three axes coordinate measuring machine offering a dynamic range up to several meters. The combination will transfer the high sub-nanometer resolution of the optical sensor to the dynamic range of the coordinate measuring machine.


Acknowledgements

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 Gerhard Heinzel and Karsten Danzmann from the Albert-Einstein-Institute Hannover.