4.5 Main challenges in interferometry

In this section, I address the main difficulties encountered in optical interferometry. The main one is the effect of turbulence due to the atmosphere. I will also tackle the limitation in terms of performance due to the various types of noise.

4.5.1 Atmosphere turbulence

Figure 4.12: Effect of the atmosphere turbulence on the incoming wavefronts.

The main effect of the presence of the atmosphere is the corrugation of the incoming wavefronts. Due to difference of temperature between the ground and the upper layers of the atmosphere, convection occurs and creates turbulent eddies. These eddies are characterized by different temperature and therefore different refractive indices. They move up and down with different spatial scales. When looking to objects through the atmosphere, the light rays are deviated randomly, i.e. the plane incoming wavefront is corrugated by different phase delays depending on the total optical thickness of the atmosphere along the propagation path (see Fig. 4.12). In optical interferometry, this phenomenon called seeing yield two main consequences:

• at each telescope pupil entrance, the local wavefront is corrugated. One does not get diffraction-limited images of the object, but seeing-limited ones. In short exposure images and for multi-mode interferometers, the image is formed of various speckles (see left panel of Fig. 4.13) and in long-exposure images its typical size is given by the characteristic size of turbulent cells where the light is coherent, i.e. several tenths of meters. In addition, these features move with time at timescale of several milliseconds. In the case of single-mode interferometers, the speckles are spatially filtered at the entrance of the interferometer and are translated in fluctuations of the light coupling. Therefore one has to correct from the photometric fluctuations induced by the turbulence (see right part of Fig. 4.13 and Sect. 4.5.3).

  Figure 4.13: Effect of the turbulence of each aperture. Left: in multi-mode beam combination, the image is formed of speckles with fringes at different phases (GI2T, adapted from [Labeyrie 1978]). Right: in single-mode beam combination, the wavefront corrugation is translated into photometric fluctuations (IONIC, [Berger et al. 2001]). Upper curve: raw interferometric signal; center curves: photometric signals; lower curve: photometry-corrected interferometric signal (see also Sect. 4.5.3)

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• the atmosphere induces fluctuations of the optical path differences for each baseline. The result is that the fringes are not stable at a given optical delay but randomly move around the sidereal position. It is impossible to calibrate the phase of the fringes and their amplitude is also decreased due to smearing during the integration time.

  Figure 4.14: Optical path fluctuations due to the turbulence. Left: simulations of the fluctuations for two Unit Telescopes at the VLTI with typical seeing parameters (in microns). Right: the power spectrum of the fluctuations follows a Kolmogorov law with a saturation at low frequency due to the outer scale.

     

The atmosphere decorrelates the phase between different points of the incoming wavefronts both in space and in time. That is why we usually present the turbulence as yielding coherence volumes inside which the wavefront can be considered as a non-disturbed plane wave. The geometric parameters of the coherence cell are called the Fried's parameters:

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the size of the cell, $r_0$, typically $10$ cm at $\lambda=0.5\ensuremath{\mbox{ $\mu$m}}$ and $1$ m at $\lambda=2.5\ensuremath{\mbox{ $\mu$m}}$ depending of the turbulence strength.

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the coherence time, $t_0$, typically $10$ ms at $\lambda=0.5\ensuremath{\mbox{ $\mu$m}}$ and $100$ ms at $\lambda=2.5\ensuremath{\mbox{ $\mu$m}}$.

The turbulence occurs at different spatial scales and a popular model is the Kolmogorov model that represents the power spectrum of the turbulence as a power law [Roddier 1981]. However the turbulence at small spatial frequencies saturates, i.e. the size of the largest eddies is limited. This limit, $L$ is called the outer scale of the turbulence and is important in interferometry since we estimate its size to be of the order of $10-100$ m, typical lengths of most interferometer baselines. If the baseline is larger than the outer scale then the effect of turbulence is less than predicted by the Kolmogorov model.

4.5.2 Other atmosphere systematics

Figure 4.15: Effect of the atmosphere refraction. The object image at each telescope entrance is in fact a small spectrum. However there is no effects on the optical delay since the optical path for the two beams in the atmosphere is the same at each wavelength $\lambda _1$ and $\lambda _2$, and, the resulting OPD does not depend on wavelength. Even if $L_1 \neq L_2$, we have $L_1=L_1'$ and $L_2=L_2'$, and $\delta _1=\delta _2$.

Optical interferometers must also take into account the atmospheric refraction and the longitudinal spectral dispersion.

The wavelength dependence of the air refractive index implies wavelength-dependent refraction angles when the light enters the atmosphere. The atmosphere acts like a prism and the images at each aperture are spectrally dispersed. Single mode interferometers spatially filter the incoming wavefront, that means they select a part of the image. Therefore this refraction effect decreases the coupling factor in the interferometer. The larger the telescope size, the smaller the diffraction-limited images: this effect begins to be important for large telescopes. Atmospheric dispersion correctors are classical devices made of two prisms that can rotate and compensate the dispersion due to the atmosphere refraction.

The refraction induces chromatic arrival angle, but does it result in chromatic OPD. Fig. 4.15 shows that even if the optical paths are different for two different wavelengths, they are the same for each aperture and the OPD remains zero for all wavelengths. The refraction does not yield a chromatic OPD.

However, the optical delay due to the zenith angle of the observed object has to be compensated by delay lines. If the delay lines are located in vacuum the compensation matches exactly the geometrical delay above the atmosphere, but if the optical delay in performed in air, then the compensation is performed only for one wavelength because of the chromatism of the air refractive index $n(\lambda)$. The resulting OPD given by a delay $L$ of the delay line that matches the geometrical delay $\delta$ at $\lambda=\lambda_0$ is then:

$$OPD(\lambda) = \left( \frac{n(\lambda)}{n(\lambda_0)} - 1 \right) \delta$$ (4.5)

Therefore the location of the zero-OPD changes with wavelength. At high spectral resolution, the main effect is to twist the fringes, whereas at low spectral dispersion the contrast of the fringes can be severely decreased. To overcome this effect and besides using vacuum delay lines, two translating prisms produce a variable glass thickness that compensates exactly the chromatic OPD. This device is called a longitudinal dispersion compensator.

4.5.3 Fighting the atmosphere: complexity and accuracy

The previous sections show that the propagation in the air implies several problems. The chromatic effect of the air refractive index can be compensated by an atmospheric dispersion compensator and a longitudinal dispersion corrector. These phenomena are completely predictable and therefore can easily be controlled by computer in function of the zenith angle.

However the effect of the turbulence is much more difficult to control since the time scale is of the order of several milliseconds and the spatial scales are small. Adaptive optics (or tip-tilt compensation) and fringe trackers are therefore required to increase the sensitivity of optical interferometers (see Sect. 4.3). However the correction is never perfect and some residuals can still affect the signal.

One solution is to use speckle techniques to calibrate those residuals. GI2T has proven that one can use several speckles to calibrate the visibility of an object. Another method is to filter out the incoming wavefront. The principle is well-known by the opticians: they clean up images by placing so-called spatial filters in the Fourier plane associated to the images. With a pinhole at the focus of a telescope, the wavefront in the exit pupil is then cleaned up and flat. The wavefront corrugation is transfered in intensity fluctuations since the speckle image on the pinhole is not stable. Optical waveguides, like optical fibers, are optical devices that behave like infinitively small pinholes but with a high coupling efficiency (typically 70%). The signal that exits from a waveguide is an electric field whose shape is given by the geometry of the waveguide (and therefore is fixed) and for which only two parameters can vary: the global amplitude and the phase. By measuring the variations for the photometry for each beam, we can compute a visibility corrected from the atmospheric perturbations. The visibility estimator,

$$V_{\rm corr} = \frac{2\sqrt{I_A I_B}}{I_A+I_B} \; V_{\rm mes},$$ (4.6)

little depends on the turbulence with $V_{\rm mes}$ the raw visibility, $I_A$ and $I_B$ the intensities measured for each beam.

Figure 4.16: The FLUOR fiber beam combiner (right). Example of incoming signals: left top panel shows the raw signal with two photometry channels that monitors the coupling in the fibers; left bottom panels shows the corrected interferogram from [Coudé Du Foresto, Ridgway, & Mariotti 1997].

$\parbox{0.4\hsize}{\centering \includegraphics[width=\hsize]{fluor-uncor}\ *[1ex] \includegraphics[width=\hsize]{fluor-cor}}$ $\parbox{0.55\hsize}{\includegraphics[width=\hsize]{fluor}}$

This method has proven to be very accurate in measuring visibilities. FLUOR reaches 0.3% for some targets [Coudé Du Foresto, Ridgway, & Mariotti 1997]. However, one should keep in mind that the measured visibility is not exactly the object visibility except if the objet is not resolved by the individual apertures. When the object is larger than the projected size of the spatial filter on the sky, one has to apply a visibility correction using the information given by the image obtained with the resolution of the individual apertures.

4.5.4 Noise sources - Sensitivity

The signal measured with an optical interferometer is affected by several sources of noises. In the case of the instrument AMBER on the VLTI these sources of noises are:

Figure: Signal-to-noise ratio computed for the AMBER instrument at the VLTI in the K band (2.2  $\ensuremath{\mbox{ $\mu$m}}$) for an object of visibility of 1, a seeing of 0.5$''$, a low dispersion, and a long exposure using fringe tracking.

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the photon noise is the fundamental noise associated to the detection of the photons. It follows a Poisson-type statistics.

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the detector read-out noise is the noise of the electronics that reads the signal. It is an additive Gaussian noise with a characteristic level called the read-out noise (RON).

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the thermal noise is the noise that comes from the detection of background photons. The background level is measured and subtracted. The background estimation gives an error due to the photon statistics.

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the instrument OPD stability is not a noise that affects the detected photons, but the measured visibility. The residual motions of the optics in the instrument induce a blurring of the fringes at a small level. In AMBER, we expect this level to be lower than $10^{-4}$ of the unit visibility.

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the atmosphere fluctuations, even corrected by simultaneous measurements of the photometry induce degradation of the signal-to-noise ratio. The obvious situation is the case where no photons are coupled into the interferometer.

Computing the error propagation in the final signal allows to calculate a signal-to-noise ratio (SNR) for different type of situations. To illustrate the consequence of the source brightness on the performance of an interferometer, I show in Fig. 4.17 the SNR curve for different star magnitudes in the case of the AMBER instrument on the unit telescope of the VLTI with different typical values of the site.

For bright objects the dominant noise source is the instrument stability. When observing faintest objects, the limitations become first the photon noise, then depending on the integration time, either the thermal background noise or the read-out noise.