N a

a) entrance pupil b) primary-image focal plane c) intermediate-pupil plane c) intermediate-pupil plane

Fig. 2.18 Lyot coronagraph: (a) and (b) wavefront and image at the pupil planes and successive images. (c) in red, residual light in the plane of the secondary pupil, which is a ring of light eliminated by the Lyot stop. All photons from a source on the optical axis are therefore, in theory, completely eliminated have attained remarkable performance (Malbet, 1996), such as the detection of a companion lying 10 Airy radii from a highly luminous star, which is 105 times as bright (Beuzit et al., 1997).

The stellar version of the Lyot coronagraph, even though it may be effective down to 2 arcsec of the star that is being occulted, when used with a telescope in the 4-m class, cannot observe beyond that limit. In fact, the closer to the star that one attempts to observe, the smaller the occulting disc needs to be, and the greater the diffraction that it creates, requiring a stop whose aperture rapidly becomes very small. In practice, the Lyot stop cannot cover less than the star's Airy disk. Yet it is precisely within this region that we want to observe. Short of using a large-diameter telescope which 'super-resolves' the system and eliminates the residual contrast, Lyot's original method cannot be used, as such, to search for terrestrial planets.

2.3.2.2 The Phase-Mask Coronagraph

An alternative to the Lyot coronagraph was suggested by Francois and Claude Roddier in 1997. It replaces the circular amplitude mask in the focal plane of the Lyot coronagraph with a circular phase mask. (The mask modifies the phase of the wavefront passing through it.) This mask is smaller, covering half (of the area) of the Airy disk, and applies an achromatic phase change (n) or, at least, one that is very weakly chromatic, given the low dispersion in the chosen spectral band. Light from the star is extinguished by destructive interference by virtue of the source's symmetry relative to the mask (which must be precisely located on the image spot), and rejected outside the geometric pupil, where it is blocked by the Lyot stop.

Several variants of a phase mask have been suggested. Among the most promising are the four-quadrant mask suggested by Daniel Rouan (Fig. 2.19). The theoretical performance of such a mask suggests that extinctions of 1010 may be attained, even inside the diffraction spot. The great advantage of this mask geometry is that it is 'spatially' achromatic (unlike Roddier mask, the size of which depends on the wavelength). Here, the efficiency does not depend on the size of the image spot but, uniquely, on the ability for the phase-change n, introduced by two of the four segments of the mask, to be achromatic.

Coronagraphs, whether of the amplitude mask or phase mask type, have requisites in common: the necessity for the stellar source to be perfectly centred on the centre of the mask, and for the stellar image to be as symmetrical as possible. Any deviation from symmetry results in a stellar residue, which may mask the image of the companion that is being sought. It is therefore essential:

• to have the best possible image of the star

• to permanently monitor the telescope's alignment relative to the centre of the star.

This is why all current coronagraphic systems operating from the ground are used in conjunction with adaptive optics.

Fig. 2.19 The principle of the 4-quadrant coronagraph: (a) appearance of the phase mask (white: no phase-change, black: phase-change n); (b) image in the primary focal plane; (c) the complex amplitude after passing through the mask; (d) secondary pupil plane before the Lyot spot; (e) secondary image plane; (f) central portion of the secondary image focal plane (After Rouan et al., 2000)

Fig. 2.19 The principle of the 4-quadrant coronagraph: (a) appearance of the phase mask (white: no phase-change, black: phase-change n); (b) image in the primary focal plane; (c) the complex amplitude after passing through the mask; (d) secondary pupil plane before the Lyot spot; (e) secondary image plane; (f) central portion of the secondary image focal plane (After Rouan et al., 2000)

2.3.2.3 The Phase-Induced Amplitude Apodization Coronagraph (PIAAC)

The PIAAC instrument has been proposed by Guyon et al., and is in fact the combination of two optical devices (Guyon et al., 2005):

• a high performance coronagraph

• a pupil 'remapper' to adapt the pupil shape of the telescope to the coronagraph.

The PIAAC concept comes from the observation that the diffraction pattern produced by a classical telescope leads to the presence of light (diffraction rings) far from the on-axis direction. Such diffraction rings are classically removed or reduced by 'apodized' apertures such as Gaussian apertures that can smoothen the aperture edge effects. The main drawback of such a technique is a drastic reduction of the source flux through modification of the pupil shape and thus its transmission, because of partial occultation of the pupil. Guyon proposes apodizing the pupil by a modification of the intensity distribution within the pupil This is done by a global wavefront phase distortion, using aspheric mirrors (Guyon, 2003). This method allows apodizing the pupil without changing its overall transmission.

The PIAAC combines a first stage of apodization, which optimizes the shape of the pupil, and a coronagraph (which may be either a phase- or an amplitude-coronagraph). The principle of PIAAC is given in Fig. 2.20.

This concept has been demonstrated in the laboratory and is currently under test on the sky. It could also be adapted to a space-borne mission concept.

2.3.2.4 Adaptive Optics and Accurate Pointing: The Keys to Performance

Because of atmospheric turbulence, the image of a star in a large-diameter telescope is never the theoretical diffraction image as set by the limiting size of the telescope's

Fig. 2.20 Principle of a Phase-Induced Amplitude Apodization Coronagraph. The beam is apodized before reaching the first focal plane where the coronagraph mask is located. The beam is then apodized backwards to revert to its initial form and to retain the initial phase information contained in the entrance pupil. It is then imaged in the detection focal plane (After Guyon, 2003)

Fig. 2.20 Principle of a Phase-Induced Amplitude Apodization Coronagraph. The beam is apodized before reaching the first focal plane where the coronagraph mask is located. The beam is then apodized backwards to revert to its initial form and to retain the initial phase information contained in the entrance pupil. It is then imaged in the detection focal plane (After Guyon, 2003)

aperture. The star's image is, in fact, the sum of multiple 'speckles', the number and size of which depend on the level of turbulence (which is a characteristic of the site and the construction of the telescope), and of the telescope's size. During a long exposure the speckles give rise to an integrated image that is distinctly spread out, and the size of which is determined by the turbulence. In terms of resolution, the degradation of the image reduces the performance of these telescopes to those of smaller instruments, typically by a few centimetres to some tens of centimetres, depending on the quality of the site and the wavelength under observation. (A thorough introduction to atmospheric optics is given by Lena, 1996). The degradation of the image into speckles results from aberration of the wavefront emitted by the star during its passage through the atmosphere. This aberration affects the phase of the wave emitted by the source. Adaptive optics (Fig. 2.21) is a system that allows:

• the analysis of the aberration of the wavefront caused by its passage through the atmosphere: this is the role of the wavefront sensor. To do this the sensor monitors a bright star within the field of view. This star serves as a point-source reference.

• the partial compensation of the degradation by means of one or more deformable mirrors, which, acting locally, will add or subtract a phase element to the wave-front in such a fashion that it will restore the wavefront's plane form that it had before passing through the atmosphere. These mirrors are actuated by a control system which uses, in real time, the information from the wavefront-sensor to calculate the correction that needs to be applied by means of the deformable mirrors.

Current 'mono-conjugate' systems correct atmospheric turbulence by assuming that it is limited to a specific atmospheric layer (and have a sole adaptive mirror). This solution proves to be inadequate as soon as one wishes to image the field around the reference object. Studies are under way to realize 'multi-conjugate' systems, which take into account turbulence at several different altitudes in the atmosphere.

Fig. 2.21 Schematic diagram of an adaptive-optics system. The wavefront that has been deformed by the atmosphere is analyzed, and the effects of turbulence are corrected by use of a deformable mirror (by courtesy of ONERA)

Several parameters may be used to quantify the efficiency of an adaptive-optics system. Among these, we may mention the Strehl ratio, which is defined as the ratio between the peak intensity of the image of a star (i.e., the point-spread function) and the theoretical peak intensity for an image solely limited by diffraction. A Strehl ratio of 1 corresponds to an image that is fully coherent and perfect. An image that is not corrected by adaptive optics has a low Strehl ratio, which depends on the severity of atmospheric turbulence. Current, fine adaptive-optics systems are able to attain a Strehl ratio of 0.6. Future projects for the detection and investigation of extrasolar planets from the ground will require Strehl ratios of at least 0.9.

Use of a coronagraph requires a high-quality wavefront, and which it is therefore essential to correct by adaptive optics when observations are made from the ground. In the specific case of a phase-mask coronagraph, the guiding constraints (on positioning and on centring the mask) are extremely tight (the functioning of these systems is based on the symmetry of the stellar image relative to the phase mask). These constraints are met by a fast-response, tip-tilt mirror ahead of a high-quality adaptive system which stabilizes the position of the image at the image plane.

Most 8-metre-class telescopes are now fitted with adaptive-optics systems. Some also incorporate new-generation phase-mask coronagraphs. We may mention in particular the NAOS-CONICA instrument on the VLTI. This instrument allowed Chauvin and his collaborators to obtain the first image of an exosystem (2M1207). In a few years' time, this instrument should be followed by the SPHERE instrument, which consists of a highly efficient adaptive-optics system coupled with a 4-quadrant coronagraph (see Chap. 8). These innovations, mainly developed in Europe, and more particularly in France, should, in due course, also be incorporated into American instruments.

2.3.3 Interferometry

As we have already seen in the section on astrometry, it is possible to measure the position and the proper motion of a star relative to a fixed reference by using in-terferometry. In that method, the position of the group of interference fringes and the length of the baseline are measured to derive the astrometric information. In this section we will show how analysis of the structure of the interference fringes allows us to extract information about the object itself and its environment. To understand this technique, one needs to understand some additional principles of interferometry.

2.3.3.1 Some Principles of Optical Interferometry

The Temporal-Spatial Coherence of Two Waves

When two sources of illumination are mutually coherent, superimposing the two waves produces a phenomenon known as 'interference'. Here, the intensity is not everywhere the sum of the intensities of the two beams taken separately, but is the result of an interference function, which we shall explain later.

Mathematically, the concept of temporal-spatial coherence is expressed by a quantity known as the 'complex degree of coherence', denoted y12(t), and which is expressed as the equation:

In this equation, E (P, t) is the electric field (complex notation) at point P at time t; < E (P, t) > is the temporal mean of this field, and t represents the delay of wave 1 relative to wave 2. j12(t) is a complex number where the modulus lies between 0 (no coherence) and 1 (complete coherence). In this expression, if Pi = P2 and t = 0, we are dealing with the wave's spatial coherence, and if t = 0, and P1 = P2, we are investigating the wave's spatial coherence.

Interferometry and Visibility

In observational astrophysics, an interferometer (Fig. 2.22) is, above all, an instrument that measures the degree of temporal-spatial coherence of the electromagnetic field emitted by a source and sampled by two telescopes.

Positions Pi and P2 are those of the two telescopes, the delay t is obtained by modifying the observational configuration (the position of the object on the sky and the setting of the delay line).

If I1 and 72 are the intensities in the two arms of the interferometer and if d = c.T is the path difference, then the intensity at the exit from the beam combiner may be written as:

Measurement of I(d) allows us to measure the complex degree of coherence (or at least its real part) for each of the positions P1 and P2. In the case of two monochromatic plane waves of wavelength X, spatially and temporally coherent, the preceding equation becomes:

This equation is interferometry's characteristic equation.

Fig. 2.22 A schematic diagram illustrating the principle of a twin-telescope interferometer of the VLTI type. The rays of light from the two telescopes are led to the beam combiner by a series of mirrors forming the optical train. The path difference is adjusted by using a delay line. (After ESO Internet site: www.eso.org.)

Fig. 2.22 A schematic diagram illustrating the principle of a twin-telescope interferometer of the VLTI type. The rays of light from the two telescopes are led to the beam combiner by a series of mirrors forming the optical train. The path difference is adjusted by using a delay line. (After ESO Internet site: www.eso.org.)

In practice, one records all or part of an 'interferogram' (Fig. 2.23), obtained by fixing positions P\ and P2 and varying t (by means of a delay line).

From the graph of /(d) we may determine a quantity known as the 'visibility', denoted V, and which is defined by:

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