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3 Performance: Optical quality and sensitivity

In this section, the computation of the performance of the optical configuration is presented in terms of optical quality and sensitivity to misalignments. The calculations for optical quality were brought together in a dedicated software called SIVA. Results are given in terms of contrast estimation and displacement of the centroid of the point spread function in the field of view. In addition, sensitivity to misalignments (tilt and decenterings) is performed with CodeV.

3.1 Computation of the optical quality

We developed an algorithm, called SIVA (Software for Interferometer Visibility Analysis), dedicated to assess the performance of optical configurations for a 2-aperture interferometer. The two identified criteria to evaluate the optical quality are:

$\cdot$ fringe contrast;

$\cdot$ displacement of the Airy disk.

SIVA estimates the fringe contrast from the wavefront decomposition in Zernike polynomials. The coefficients of the Zernike polynomials are given by CodeV which calculates them for different points in the field of view.

To compute the equations, the main hypotheses are a monochromatic observation and an unresolved star. An aberrated wavefront is expressed in the 2-aperture interferometer plane by the following equation:

\Psi(u,v)=\left[ \Pi\left( \frac{f}{f_{\rm c}}\right) \;\oti...
u,v+\frac{f_{\rm ci}}{2}\right) \right] \;.\;h_{1}(u,v)+\end{displaymath}

\left[ \Pi\left( \frac{f}{f_{\rm c}}\right)
\;\otimes\;\delta\left( u,v-\frac{f_{\rm ci}}{2}\right) \right] \;.\;h_{2}(u,v)\end{displaymath} (10)

where the symbols are:

f is equal to $\sqrt{u^{2}+v^{2}}$,u and v being the spatial frequencies;

$f_{\rm c}$ is equal to $D/\lambda F$, D being the pupil diameter and F the instrument's focal length;

$f_{\rm ci}$ is equal to $B/\lambda F$, B being the baseline;

$\Pi$ is the pupil function defined as $\Pi$ = 0 for $f/f_{\rm c}$ > 1 and = 1 elsewhere;

$\otimes$ is the convolution operator;

$\delta$ is the Dirac's function;

hi is equal to $\exp\left( j~.~\Delta_{ i}(u,v)\right) $, where $\Delta_{ i}(u,v)$ is the aberrated phase contained in the ith aperture expressed in terms of Zernike polynomial coefficients and $j=\sqrt{-1}$.

The Point Spread Function (I) is the square modulus of the diffracted wave complex amplitude, the latter being given by the Fourier Transform of the incident wave amplitude. The PSF is so expressed by:  
I(x,y)=\widetilde{P}(x,y)~.~\widetilde{P}^{\ast}(x,y)\end{displaymath} (11)

where $r=\sqrt{x^{2}+y^{2}}$ , $\sim$ denotes the Fourier Transform and * the complex conjugate. The Fourier Transform of the incident aberrated wave amplitude is equal to:

\widetilde{\Psi}=\frac{\pi.f_{\rm c}^{2}}{2}\left[\left(\fra...
 ...^{-j\pi f_{\rm ci}}\right)\otimes\widetilde{h}_{1}(x,y)+\right.\end{displaymath}

\hspace*{1.5cm}\left.\left(\frac{J_{1}(\pi.f_{\rm c}.r)}{\pi...
 ...e}^{j\pi f_{\rm ci}}\right)\otimes\widetilde{h}_{2}(x,y)\right]\end{displaymath} (12)

where J1 is the first order Bessel function of the first kind. The normalized Optical Transfer Function is given by:

\cdot\end{displaymath} (13)

As in the ideal case (i.e. no aberrations in the pupil), the OTF consists of three peaks from which the fringe contrast can be evaluated: two high frequency peaks, $O_{\rm HF}$, containing the coherent energy, and one low frequency peak, $O_{\rm LF}$, containing the incoherent energy.

The fringe contrast C is defined as the ratio of the coherent to the incoherent energies:

C=2\;.\frac{\;\left\vert \int O_{\rm HF}(u,v).{\rm d}u.{\rm ...
 ...left\vert \int
O_{\rm LF}(u,v).{\rm d}u.{\rm d}v\;\right\vert }\end{displaymath} (14)

C, equal to 1 in the ideal case, is in practice degraded by optical aberrations and is always smaller than 1.

SIVA uses the wavefront output file of CodeV (Zernike polynomial coefficients) to estimate fringe contrast and Airy disk displacement as a function of star position in the field of view. The Airy disk displacement is computed considering the centroid of the point spread function.

SIVA was validated by comparison with CodeV results in terms of calculation of the Modulation Transfer Function, and real and imaginary parts of the Optical Transfer Function.

3.2 Sensitivity analysis

We performed analyses of the Korsch configuration with CodeV to assess its performance in terms of sensitivity to misalignments. This sensitivity is expressed by the centroid displacement of the Airy disk, more severe than visibility criterion, induced by a tilt or decentering of one mirror. This can be considered as relevant enough for these kinds of misalignments, although it is the fringe position that would matter if one wanted to express the tolerances in optical aberrations as piston.

An instantaneous astrometric accuracy of 100 $\mu$as is required in order to achieve the final 10 $\mu$as accuracy on the measurements. Considering the 30-m focal length, this accuracy can be achieved if the centroid displacement induced by spurious effects is less than 15 nm. The misalignments that we analyze here are the followings:

$\Delta\alpha$: tilt around the x-axis (normal to baseline);
$\Delta y$: decentering along the y-axis (parallel to baseline);

$\Delta z$: decentering along the z-axis.

These effects are studied for each aspherical mirror (M1, M2 and M3). The mirrors M2 and M3 are monolithic, whereas there are two independent parts of M1 for each arm of the interferometer. The contributions of each mirror misalignment to the astrometric accuracy degradation are assumed to be equivalent. The final centroid displacement is assumed to vary as the sum in quadrature of the displacements induced by each contribution. Therefore, each of these contributions must be smaller than 5 nm.

3.3 Performance results

3.3.1 Optical quality

\resizebox {\hsize}{!}{
}\end{figure} Figure 5: Visualisation of fringe patterns, contrast and Airy disk centroid displacement computation

Figure 5 shows an example of the contrast estimate and Airy disk displacement for two different positions: centre of the field of view and field edge. The hypotheses are:

$\cdot$ pupil diameter: 60 cm;

$\cdot$ baseline: 2.55 m;

$\cdot$ wavelength: 550 nm;

$\cdot$ field of view: 0.8 $\times$ 0.8 deg2.

Table 3 shows the estimated monochromatic contrast and Airy disk displacement for different positions in the field of view for the Korsch configuration. The focal length varies in the field due to the optical distortion. The contrast remains high across the whole field (> 95%). There is a displacement of the centroid, 4.48 $\mu$m and 11.26 $\mu$m respectively at the ($0.0\hbox{$^\circ$}$, $0.2\hbox{$^\circ$}$) and ($0.4\hbox{$^\circ$}$, $0.4\hbox{$^\circ$}$) points, which has to be taken into account for data calibration.

Table 3: Contrast and centroid displacement estimated for different points in the field of view
Field point & Contrast & \multicolumn{2}{c}{Centroi...
 ... 0.4)&0.991&20.2 &3.02 \\  
(0.4, 0.4)&0.975&74.8 &11.26 \\  \hline\end{tabular}

3.3.2 Sensitivity to misalignments

Table 4 gives the estimated maximum tilts and decenterings, for which centroid displacements are maintained smaller than 5 nm. The results show that nanometric metrology is required to control the positions of the mirrors if the astrometric precision must be preserved. The control of the primary mirror (M1) seems to be the most stringent, as it requires a less than 20 $\mu$as tilt and 1 nm decentering. The control of the other mirrors is less constraining.

Table 4: Tolerances of the Korsch configuration for a maximum centroid shift of 5 nm (34 µas)

 &$<$5 nm centroid shift \\ \hline
Tilt $\Delta\alpha...
 ...ace &2 nm \\ $M_{2}$\space &3 nm \\ $M_{3}$\space &40 nm \\  \hline\end{tabular}

3.3.3 Distortion

At the edge of the field ($0.4\hbox{$^\circ$}$), the change $\Delta F$ in focal length is 0.06 m, giving a ratio $\Delta F/F$ equal to 2.0 10-3. Although this value is 4 times over the requirement defined by Eq. (9), it is not considered as critical as it is possible to overcome it by technical solutions.

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