3 Experimental setup: Strategy and practical realization

We have built an experimental setup to validate these simulations and to test the efficiency of optical filtering with different devices, pinholes and dielectric waveguides. The measurement of the rejection rate allows us to quantify the filtering efficiency. The strategy adopted for the setup is simple:

• we control low spatial frequency defects that are not well corrected by pinhole filtering (mainly, OPD and tip-tilt for phase and global intensity of each beam for amplitude);
  
• we optically filter high frequency amplitude and phase defects.
  

The demonstration experiment uses an amplitude division Mach-Zehnder type interferometer in which several elements have been modified (Fig. 1).

Figure 1: Demonstration experiment scheme. On a classical Mach-Zehnder structure, several devices have been put: a beam intensity equalizing device, delay lines, a tip-tilt correction device and an optical filtering device located after the recombination and before the detection

We took into account neither the chromaticity of the rejection rate, nor its feasibility over a large range of wavelengths (typically 6-18 $\mu$m for the DARWIN space mission). The wavelength of 10.6 $\mu$m has been chosen because it was located roughly in the middle of DARWIN's spectral range. In addition, single mode lasers exist at this wavelength and can provide a higher signal than the thermal background. A red visible laser beam ( $\lambda \sim$ 630 nm) is superimposed on the infrared laser beam. We thus have a star model with visible and infrared monochromatic radiation. The ratio of 17 between visible (red) and infrared wavelengths allows a huge increase in the accuracy of the settings. For instance, a path difference of $\lambda_{\rm IR}$/3000 corresponds approximatively to a path difference of $\lambda_{\rm Vis}$/200.

All the optical elements used in this experiment should allow the propagation of visible and infrared wavelengths. With that goal, the refractive optical elements (beam-splitters and windows) are in zinc selenide (ZnSe) while the mirrors, generally in Zerodur, are coated with a single layer of gold deposited on a chrome coated blank to increase the quality of the gold layer.

Let us now consider, the different parts of the setup.

3.1 Light sources

The infrared source is a CO₂ laser, lasing on the P20 line, pumped by radio-frequency, linearly polarized (single polarization), transverse and longitudinal single-mode, with a maximum power of 250 mW at 10.59 $\mu$m. The frequency stability of the laser is better than 10 kHz (for $T<10~\mu$s). The source is tunable in a range of $\pm$100 MHz. It is placed at one edge of the optical table, and its beam is expanded by a 3 lens optical device. The final pupil is circular, with a radius of 12 mm and cuts the Gaussian laser beam at 1/e³. Taking into account the transmission of the injection device and the elimination of the ghost beams, about 150 mW are reaching the beam-splitter.

The visible source is a tunable dye laser (rhodamin 6G), pumped by a continuous argon laser. This dye laser is used in the red part of the spectrum where its power is a few tens of milliwatts. The beam is brought to the experiment bench by a single mode optical fibre that allows an additional cleaning of the beam (the dye laser is in a separate room). At the fibre output, the beam is collimated by a microscope objective. The mixture IR/Visible is performed on a dichroic plate located in front of the beam-splitter.

3.2 Beam separation and recombination

The beam-splitter and the beam-combiner have been designed to avoid multiple wave interferences (Perot-Fabry resonator) and the superposition of multiple reflections on the transmitted wavefront. The classical plane-parallel beam-splitter has been replaced by a device made of 2 prismatic windows, one is placed upside down with respect to the other, and separated by a gap that allows multiple reflections to escape without changing the direction of the principal beam. The 27 mm diameter, 7.5 mm thick windows, separated by 60 mm have an angle of 5 degrees that allows us to control the first extra reflections on a 12 mm beam (Fig. 2). These ZnSe plates are not coated, and are used near the normal incidence (5 degrees). They have been polished to an accuracy of $\lambda$/20 peak to valley ($\lambda$ = 632.8 nm). The global efficiency of such a device at 10.6 $\mu$m is about 47% for the transmitted beam and 17% for the reflected one. At 10.6 $\mu$m, the absorption of ZnSe is low. The size of the beam avoids localized concentrations of energy and a thermal gradient to develop in the material. The remaining 36% of the source energy is spread over all the ghost beams (reflections on the plates' interfaces) that propagate out of the main beam directions. The flux differences introduced by the beam-splitter are compensated by the beam-combiner. The difficulty of this solution is the handling of extra reflections that can either be used as control signals for the servo-control of the experiment or be trapped to avoid interferences with the main beam. The coherence length of the lasers is much higher than the experiment dimensions.

Figure 2: Principle of the beam-splitter and the beam-combiner. We represent the main beam with plain lines and the first ghost reflections with dashed lines. 47% of the input energy is in the transmitted beam, 17% in the reflected one, and 36% is spread over all the ghost beams

After alignment, the only degrees of freedom of the interferometer are the two degrees of rotation of the beam-combiner that allow an angular superposition of the wavefronts and the two degrees of rotation through a tip tilt correction device that allows lateral superposition of the wavefronts. Both angular and lateral position control devices are powered by piezo electric stacks that allow very fine settings and/or servo control. All the other optical elements are screwed onto aluminium plates after the settings have been done.

3.3 Wavefront quality control

Various wavefront control and cleaning elements are added in the optical path:

• an infrared flux balance device;
  
• delay lines;
  
• a wavefront angular and lateral superposition device;
  
• an optical filtering device.
  

$\bullet$ The infrared flux balance device must equalize the intensity between the two arms of the interferometer before recombination.

This is a pure absorbing device, composed of two barium fluoride (BaF₂) prisms, one placed upside down with respect to the other. The first prism is fixed while the other is free to move beside it. The total thickness of absorbing material is thus constant over the whole beam section, and can be set within a range of $\pm$3% of the total transmission ( $\tau _{i} \sim 0.72$).

The flux balance device is set by successive measurements of each of the beam's intensity. The measurement of infrared intensity of each beam is performed by pyroelectric detectors with AR coated germanium windows (P5236 detectors from GEC-Marconi). These measurements are performed after recombination and optical filtering but before detection.

$\bullet$ The delay lines allow the localization and the control of the zero path difference with a precision of less than 10 nm.

The optical part of a delay line is made by a cat eye, built with a parabolic primary mirror ($\phi$: 50 mm, focal length: 150 mm) and a secondary plane mirror ($\phi$ 4 mm). The secondary mirror is glued to a piezo-electric ceramic stack. It allows fast displacements over a total range of about 15 micrometers.

The cat eye is mounted on a frictionless table, which is necessary to control the displacements to a nanometer accuracy. As a consequence, this table involves only pure bending of metallic blades. Performances are:

total course	4 cm
angular error	3 arcsec (peak to valley)
lateral displacement	5 micrometers (peak to valley)

Linear motors provide low frequency translations (thermal drifts, slow fluctuations of path difference) while the piezo-electric ceramics provide high frequency (typically a few hundreds of Hz) small translations (vibrations).

The optical path difference control is based on a polarization rotation sensitive system (Connes & Michel 1975). A reference interferometer is included in the delay lines breadboard. It uses a helium-neon laser as a source and provides a signal proportional to the path difference with a resolution higher than 10 nm in a wide frequency band (50 kHz).

The path difference servo control is made up of two loops: a "local'' loop working with the He-Ne laser signal and a "vernier'' loop working with the dye laser signal. The error signal is obtained through a lock-in detection of the visible dark interference pattern on the beam-combiner.

Finally, the servo controlled system (linear motor and piezo-electric ceramics) has a frequency band of 1.3 kHz, that reduces residual vibration effects at a level of $\sigma_{\rm OPD} \ \sim 5$ nm (rms).

A PC type computer manages all the settings under LABVIEW and allows a flexible control of the system.

$\bullet$ The wavefront lateral superposition device is made up of a "tip-tilt'' mirror located well ahead of the beam-combiner. It mainly changes the lateral position of one beam with respect to the other, allowing their superposition. This mirror however leads to a small angular deviation of one wavefront from the other. This deviation is compensated, as are the other sources, of deviation by a rotation of the beam-combiner itself, equipped with the same type of mount. The rotation of the beam-combiner modifies the orientation of only one of the two beams (the reflected one), and allows the angular settings of the system to be done. The other beam is taken as a reference. These devices are servo-controlled by error signals coming from beam lateral and angular position sensors (two CCD cameras monitoring the position at two different positions).

$\bullet$ Several optical filtering devices have been considered or are under evaluation.

• Pinholes: it is made up of a hole with a diameter comparable to the pupil Airy disk diameter (first intensity minimum of the diffraction pattern) obtained in focusing the beam from the interferometer after recombination. As mentioned before, the main drawback of this type of filter is that it is not efficient for low spatial frequency defects such as geometric distortions (the efficiency of a pinhole optical filtering increases with the defect's spatial frequency);
  
• Another solution is to use single mode waveguides. Such waveguides are beginning to be available in that spectral range, particularly dielectric waveguides, and their huge absorption may still be acceptable given the small lengths and the powerful sources we use. The advantage of waveguides over pinholes is the possibility, at least in theory, to correct all phase defects (including low order defects such as tip-tilt, but not the optical path differences). Phase defects are transformed into coupling defects and path differences that can be monitored (Coudé du Foresto & Ridgway 1991). Path difference fluctuations are corrected in both cases by delay lines. Coupling efficiency differences can be compensated for by the infrared flux balance device.

Single-mode waveguides for astronomy are already available (Kern & Malbet 1996) in the visible and in the near infrared as single-mode fibres (Monerie et al. 1985) or integrated optics devices (Malbet et al. 1999). They should also soon be available in the thermal infrared range as fibres (Perrin et al. 2000), hollow waveguides, or integrated optics waveguides. These guides are manufactured by ion exchange in IR glasses (chalcogenides) and are under development (Laurent et al. 2000). One of the main difficulties is to reduce the loss in the guide, while still limiting its length to a few centimetres. However, considering a guide with a small index difference between core and cladding, it appears that a 1.5 m single-mode waveguide is necessary at 10 $\mu$m to attenuate all the non-propagating modes by a factor of 10⁴ (Leproux et al. 2001).

To be efficient, the filtering must be symmetric on both optical paths of the interferometer. In the demonstration experiment we chose to put the filtering device after the beam-combiner. The two beams cross a unique and common device. Formally, the results are similar to separated identical devices because of the linearity of Maxwell's equations describing the recombination.

The filtering device is composed of focusing optics (this optical element is common to pinhole and waveguide devices), and relay optics to the detector.

3.4 Detection

The maximum power of the laser is 250 mW. The total transmission of the interferometer is about 1.5%. The intensity after recombination should thus vary from a few milliwatts (constructive interferences) to a few nanowatts (destructive interferences) with a rejection rate of about 10⁶. Such a dynamic range cannot be reached by common infrared detectors.

In our case, we want an accurate measurement of the destructive interference intensity. For low level detections, we choose to use 77 K HgCdTe detectors. These detectors were built by the SAT company and were used on the SOIRDETE interferometer (Mekarnia & Gay 1990). They have a typical detectivity $D^{\star}=5~10^{10}$ cm Hz^1/2 W^-1 with a typical surface area of $50\times50~\mu$m². These detectors were included in an Infrared Lab cryostat by Jean Gay's team. The cryostat can function for about 24 hours without having to be refilled and this allows stable measurements to be made over a day. A custom made pre-amplifier allows accurate measurements.

3.5 Mechanical stability of the experiment

The experiment is implemented on a 4.2 m $\times$ 1.0 m granite optical bench. The position of the 2.4 ton bench is servo-controlled with 3 fixed and 3 pneumatic devices. The beams are located 190 mm above the surface of the table. Every optical element is mounted on a barrel screwed onto a 100 mm long aluminium column (with a diameter of 36 mm), which is screwed to an aluminium base and placed unscrewed on the marble bench on a kinematic device that allows smooth control of the position of the elements and avoids violent thermal relaxations. The temperature of the room varies over a range of 1 to 2 degrees during the course of a year. The stability of the whole device is good, although it has still not been quantified. However, fringes can be obtained over weeks without any supplementary modifications.

3.6 Laboratory environment and turbulence

The turbulence in the laboratory is harmful to the quality of measurements. It leads to local modifications of the air index on the optical paths and thus to permanent fluctuations of the global path difference (that could be corrected by delay lines) but also phase spatial fluctuations on the pupil. The path difference fluctuations should not be higher than $\lambda$/3000 at 10 $\mu$m, to get a 10⁶ rejection rate. Turbulence appears as soon as a horizontal or a vertical (directed downwards) temperature gradient appears. In our case, a temperature difference between the table and the top cover of the experiment may exist.

This laboratory turbulence may be approximated, in the case of a vertical temperature gradient by a Rayleigh-Bénard problem (plane turbulence between two horizontal boundaries) and characterized by a number (the Rayleigh number) that is in fact the ratio of the potential energy to the viscous dissipation energy. This number, written Ra, is given by the following relation:

$$Ra = \frac{g . \alpha . \Delta T . h^{3}}{\kappa . \nu}$$ (1)

with:
g: gravitational constant,
$\alpha$: fluid thermal expansion coefficient,
$\Delta T$: temperature difference,
h : fluid layer height,
$\kappa$: fluid heat diffusivity,
$\nu$: fluid kinematic viscosity.

The appearance of convective cells exhibits a threshold for a particular value of the Rayleigh number, called the critical Rayleigh number and written $Ra_{\rm c}$, of about 1700 in the case of a fluid layer between two horizontal solid boundaries (Drazin & Reid 1981). This threshold appears when the diffusion is not fast enough to thermally balance the gas. Hot bubbles, that are lighter, appear and a general circulation starts in the experimental enclosure.

In the case of a horizontal temperature gradient, a circulation phenomenon also occurs, but this time, its onset does not exhibit any threshold. Particular care must be taken in order to avoid strong horizontal temperature gradients along the optical path (e.g. control of instrumental thermal dissipation). In order to limit the effects of turbulence, the experiment is covered by a sealed helium, filled enclosure. This environment has two major advantages over air:

• The helium optical index fluctuations are 10 times lower than that in air when the temperature varies;
  
• The helium thermal conductivity is 6 times higher than that of air. This property allows a fast homogenization of temperature on all the optical paths, and delays the onset of turbulence (cf. Eq. (1)). As a comparison, the value of $Ra_{\rm c}=1700$ is obtained with h = 30 cm for $\Delta T= 6.9~10^{-4}$ K in air, while in helium $\Delta T$ must reach 4.5 10^-2 K before turbulence appears.

The airtightness constraints are more or less the same as those used for a setup under low-quality vacuum conditions, with the huge difference that the enclosure structure does not need to support the effects of atmospheric pressure and so can be built with polymer (and transparent) materials; the pressure in the interior part of the enclosure is a bit higher ($\sim$2 mbar) than the pressure in the air.

Another solution to avoid the appearance of turbulent cells (which can be coupled to the former one) is to add an upward oriented temperature gradient in the enclosure. This gradient can be obtained by slowly heating ( $\Delta T \sim 0.1{-}0.5~^{\circ}$C) the top of the enclosure. Horizontal temperature gradient problems are not solved but this effect can be strongly minimized by reducing the internal heat sources as much as possible. In addition, the ascending thermal gradient tends to stop gas from moving when circulation starts.

Figure 3 gives a general view of the optical bench seen from the injection part towards the beam combiner device.

Figure 3: General view of the testbed. From foreground to background: beam-splitter, flux balance device, delay lines, beam-combiner and optical filtering device