2 Configuration for direct fringe detection

In this section, we give the requirements for deriving the optical configuration: estimated field of view, focal plane characteristics, instrument accommodation in the Ariane V fairing, optical quality and the sensitivity to misalignments.

The optical concept derived from these considerations is presented, as well as a possible associated focal plane and its characteristics according to available CCD Philips Imaging Technology's expertise.

2.1 Requirements

2.1.1 Field of view requirements

The major requirements to specify an optical configuration suitable for an astrometric mission come from the astrometric accuracy and the limiting magnitude imposed by the scientific objectives to be fulfilled by the mission.

The analytical relationships between the various observational parameters are provided by equations in Lindegren & Perryman (1996).

Assuming other parameters to be known, it can be shown that the requirement to achieve 10 $\mu$as accuracy for a visual 15th magnitude star will give the minimum dimension of the field of view (FOV) of each interferometer.

This is not the only requirement for specifying the interferometric FOV. The required sky coverage, and, in particular, the partial overlap between the sky strips covered by the FOV on successive great circles, will be achieved only if the across-scan dimension of the FOV is larger than the angular separation between two great circles. The latter depends on the spin rate of the spacecraft, the solar aspect angle, and the precession velocity $V_{\rm p}$ of the spin axis. Typical values (Fig. 1) for the spin rate and solar aspect angle are respectively 120 as/s and $55\hbox{$^\circ$}$ (Lindegren & Perryman 1996). For the precession velocity, it is deduced from the preceeding parameters, i.e. 4.8 revolutions per year. Given these values, it is possible to derive the displacement of the spin axis $D_{\rm s}$, after one revolution, (i.e, the angle between the spin axis at time t₁ and at time $t_{1}+\Delta t$): 

$$D_{\rm s}=V_{\rm p}~.~\Delta t~.~\sin(\xi)$$ (1)

where $\Delta t$ is the rotational period for the satellite. In this formula the contribution from the annual motion is neglected. From the preceeding values, $D_{\rm s}=0.48\hbox{$^\circ$}$. In order to achieve a good overlap by subsequent scans, a field size of $0.8\hbox{$^\circ$}$ was chosen (cf. Table 1).

  

Figure 1: The sky scanning law. The satellite is spinning at a constant rate and the spin axis revolves about the direction of the Sun

2.1.2 Focal plane assembly

The requirements on the focal plane are also derived from the mission's scientific objectives. The limiting magnitude and the brightest stars define the electronic dynamic range requirements of the pixels. The spin rate of the spacecraft defines the velocity at which fringes cross the focal plane, on which the instantaneous integration time directly depends. In this section, we translate these requirements in terms of integration time and number of lines on the detector that will have to be summed to reach a sufficient signal to noise ratio during the fringes displacement (TDI mode).

The instantaneous integration time $T_{\rm i}$ is derived from the along-scan size $\Delta x$ of the pixel and the fringe motion v. The parameter v depends on the spin rate $\omega$ of the satellite given in rad/s and on the focal length F of the interferometer given in metres. $T_{\rm i}$ is then given by:  

$$T_{\rm i}=\frac{\Delta x}{v}=\frac{\Delta x}{F.\omega}\cdot$$ (2)

The number of TDI lines ($N_{\rm TDI}$) required to obtain an acceptable value of the Signal to Noise Ratio (R) can also be estimated. If r is the read out noise of the CCD, the required number of photo-electrons can be written as:  

$$N_{\rm p}\approx\frac{R^{2}+\sqrt{R^{4}+4r^{2}R^{2}}}{2}\cdot$$ (3)

Given $N_{\rm s}$, the number of photo-electrons falling from the star per integration time $T_{\rm i}$, and contributing to a given fringe pattern, we can assume that the central fringe contains $N_{\rm s}$/5 from B/D ratio. Hence, if the number of pixels per fringe is given by P, a pixel in the central fringe collects:  

$$N_{\rm c}\approx\frac{N_{\rm s}}{5P}$$ (4)

photo-electrons. The required number of TDI lines is then given by:  

$$N_{\rm TDI}\approx\frac{N_{\rm p}}{N_{\rm c}} \cdot$$ (5)

The availability of suitable detectors for fringe detection has to be considered, in terms of:

$\cdot$

pixel size: rectangular pixels with 2.4 $\mu$m size in the scan direction seem to be achievable according to CCD Philips Imaging Technology's expertise (Peek et al. 1993; Peek et al. 1996);

$\cdot$

sampling rate: a minimum of about P=3 pixels per fringe period provides nearly theoretical performance for the fringe location process (the Shannon criterion being a minimum theoretical limit for an ideal signal). Considering an achievable pixel size $\Delta x$ equal to 2.4 $\mu$m, a baseline B of 2.55 m, and a focal length of 30 m, we obtain 2.7 pixels per fringe at 0.55 $\mu$m, which is close to the theoretical value for the fringe location process.

$\cdot$

Achievable quantum efficiency;

$\cdot$

focal plane dimensions: we considered a few tens of centimetres in linear size, typically 40 cm, as a reasonable size.

Taking into account the scan velocity of the satellite, the available integration time for a 2.4 $\mu$m detector is 133 $\mu$s at the most. The number of TDI steps compatible with the required signal to noise ratio (R>3), for a 15th visual magnitude star, is then about 1 500 (with the hypotheses of a 5 e^- noise level, 30% quantum efficiency and 100 nm bandwidth at 550 nm).

 

Table 1: Characteristics of the interferometer and focal plane assembly. Detector characteristics are our assumed values as projected from current Philips technology

 

2.1.3 Telescope accommodation requirements

For an assumed launch with the European launcher Ariane V, the volume of each telescope (in terms of diameter and height of its envelope) is limited by the internal dimensions of Ariane V Speltra fairing ($\phi$ = 4.57 m, h = 4.85 m for the cylindrical section) and by the number of instruments to be accommodated on the Payload Module (PLM). Assuming for the service module a height of 1.7 m, the PLM height cannot exceed, say 3 m (cylindrical section only). This limits the height available for each instrument (three have to be stacked on top of each other). An initial value for each interferometer can be taken as 0.7 m (internal height of the telescope compartment). This leaves enough room for the accommodation of an instrument as the Auxiliary Radial-Velocity Instrument (ARVI), (Favata & Perryman 1997), for which, however, the non-cylindrical upper region of the Speltra could be exploited (Fig. 2). The Speltra also limits the diameter of each interferometer. Considering a reasonable thickness of the PLM wall and its thermal cover, the maximum allowable diameter for each interferometer will be about 4.3 m.

Inside the PLM, each telescope shall be mounted with its baseline vector orthogonal to the nominal spin axis of the spacecraft, and with the line of sight oriented according to the values defined for the basic line-of-sight angles.

  

Figure 2: Telescope accomodation in the Ariane V Speltra fairing

2.1.4 Calibration and stability requirements

Calibration of residual aberrations has to respect the required value of fringe contrast and displacement of the Airy disk centroid, over the whole field of view. An interferometer can be considered diffraction-limited when the fringe visibility is at least 0.8 (Cecconi et al. 1997a). Particular attention is paid to the distortion introduced by the optical design, as its level must be compatible with a single TDI clocking rate applicable throughout the FOV (or at least to a single CCD chip).

The distortion introduces a loss of contrast due to the difference between the fringe pattern velocity and the charge transfer.

The evaluation of this effect is done using the desynchronization Modulation Transfer Function evaluated by considering that the fringes are equivalent to the modulation signal obtained using a grid on a single pupil. This MTF is given by the following expression:

 

$$M(\mu)=\frac{\sin(N_{\rm TDI}\pi\mu(\Delta x-vT_{\rm i}))}{N_{\rm TDI}\sin(\pi\mu(\Delta x-vT_{\rm i}))}$$ (6)

where v is the component of the diffraction pattern velocity in the along-scan direction, $T_{\rm i}$ the instantaneous integration time, $\Delta x$ the pixel size, $N_{\rm TDI}$ the number of TDI lines, $\mu$ the spatial frequency.

For this mission, we take v equal to $\omega F$, where $\omega$ is the spin velocity and F is the focal length. For the distortion at any given point within the field of view of the instrument, F becomes $F~+\Delta F$.Therefore, there is a difference between the diffraction pattern velocity and the charge transfer one. The latter is given by $\Delta x/T_{\rm i}=\omega F$. The expression of the desynchronization MTF becomes:

 

$$M(\mu)=\frac{\sin(N_{\rm TDI}\pi\mu\Delta x\frac{\Delta F}{F})}{N_{\rm TDI}\sin(\pi\mu\Delta x\frac{\Delta F}{F})}\cdot$$ (7)

Each fringe is sampled over 3 pixels, giving the frequency $\mu$ equal to 1/3$\Delta x$. $N_{\rm TDI}$ is equal to 1 500 lines (Sect. 2.1.2).

The degradation budget according to the defined independent sources of error must fall below 20%. The following five contributors to the contrast loss were assumed (Cecconi et al. 1997b):

$\cdot$

fringe polychromaticity;

$\cdot$

deviations of the optical system from its nominal configuration;

$\cdot$

jitter of the line of sight;

$\cdot$

TDI desynchronization due to the imperfect knowledge of the scan velocity;

$\cdot$

TDI desynchronization due to distortions.

Assuming that the sources of error contribute equally to the total degradation, the tolerance on fringe contrast for a 9% degradation is given by:

 

$$\frac{\sin\left( 1500\frac{\pi}{3}\frac{\Delta F}{F}\right)}... ....\sin \left( \frac{\pi}{3}\frac{\Delta F}{F}\right) }\geq0.91 .$$ (8)

This leads to the requirement in terms of distortion:

 

$$\frac{\Delta F}{F}\leq4.7\;10^{-4} .$$ (9)

The requirements on opto-mechanical tolerances, in terms of tilt and decenterings, have to be given for each mirror, in order to reach the desired performance in terms of Airy disk location. An instantaneous astrometric accuracy of 100 $\mu$as is required in order to achieve the final 10 $\mu$as accuracy on the measurements (estimated from Hipparcos results). Considering the 30-m focal length, this accuracy can be achieved if the centroid displacement induced by spurious effects is less than 15 nm.

Alcatel Space Industries has developped an algorithm called SIVA. Dedicated to fringe contrast computation, this algorithm is based on Zernike polynomials derived from residual wavefront abberations calculated by CodeV (a registred product of Optical Reasearch Associates). Opto-mechanical tolerances are directly derived from CodeV.

2.2 Optical configuration description

  

Figure 3: Korsch configuration and associated relay system

Figure 3 shows the scheme of the proposed optical configuration, which is a Korsch configuration, composed of three aspherical mirrors (M₁, M₂ and M₃). It allows the formation of a real exit pupil, making possible the insertion of both active optics and baffling.

An optical relay system is implemented, allowing to take into account dimensions of the ArianeV envelope and space dedicated to structure and mechanics. This relay system is the combination of two flat mirrors (M' and M''), as shown in Fig. 3.

The fraction of the obstructed field of view due to the optical relay system is about 0.15%. Table 2 gives the optical parameters of the configuration, in terms of mirror curvature radius, conical constants and the distance of the mirrors to the following surface. The rms spot diagrams given by CodeV using two 60 cm apertures are given in Fig. 4.

  

Figure 4: Spot diagrams for the Korsch configuration with two apertures as for the interferometer

 

Table 2: Optical parameters of the Korsch configuration