If an image, focused by a lens or a telescope, contains light from a bright source concentrated in a small region, the bright source always contaminates the surrounding pixels, beyond its geometric image, with diffracted light, thus affecting the detection of images from adjacent fainter sources. Part of this light is diffracted by the abrupt edges of the imaging lens or mirror and appears in the form of the concentric rings in the classical Airy pattern. Obscurations such as a spider structure supporting a secondary mirror, etc. disturb the ring structure. Periodic secondary peaks also appear if the primary mirror is a mosaic of segments. An addition to the contaminating diffracted light, appearing in the form of random speckles, is generated by phase disturbances such as the residual bumpiness of the main mirror or lens.

Apodization, literally "removal of the feet" of the point spread function, typically uses a continuous attenuation pattern with graded edges across the entrance aperture or a conjugate pupil, in order to reduce the contaminating light in the pixels surrounding the central peak of the diffraction-limited image. Such soft-edged apertures can be fabricated by degrading the reflectivity of the aluminum, t The parsec is defined as the distance at which the parallax of a star, with reference to a star at infinity, is one arcsec between points separated by the Earth's orbital radius. 1 parsec = 3.3 light-years.

or coating it with partially absorbing material. Various such patterns of radially graded intensity can be applied to the wavefront. A general effect is to attenuate the Airy diffraction rings, at the expense of somewhat enlarging the central peak. Apodization degrades somewhat the angular resolution and the light throughput, but improves the detectability of faint features near a star. For some applications, the resulting removal of the Airy rings in the spread function is so efficient that no additional coronagraphic mask might be needed, although an absorbing mask in the bright central peak is often useful to protect the camera from saturation or destruction. An important advantage of apodization, compared to the techniques of nulling and coronagraphy, is that it applies equally well to point sources anywhere in the field of view, and therefore very accurate pointing of the telescope toward the star is not a prime requirement.

In order to attenuate the "feet" of the point spread function isotropically, an apodization mask rotationally symmetric around the optic axis is required. But the use of an aperture mask which is not rotationally symmetric, and apodizes excellently along certain directions at the expense of others, is also possible and often easier in practice. This proved its worth in the discovery of the star Sirius B, and remains a good illustration of the sensitivity gain achievable with apodization. In the 1840s, the German astronomer F. W. Bessel observed a periodic motion of the bright star Sirius, and deduced from it the existence of a stellar companion Sirius B. This turned out to be a white dwarf companion about 10,000 (10 magnitudes) times fainter and was observed in 1972 by using a hexagonal mask on a telescope in order to attenuate the Airy rings around the image of Sirius A. The Fraunhofer diffraction pattern formed in the focal plane from a hexagonal aperture has six radial spikes on a background which is darker than the Airy pattern of rings from a circular aperture. When the companion star was in one of the dark spaces, it was bright enough to be visible. However, this example represents a weak form of apodization. Better results are obtainable with graded attenuation at the edge of the aperture. A solution to the general problem of finding the optimum linear and radial attenuation functions for a finite aperture, to give an arbitrarily dark surrounding field around a point image, was found by Slepian (1965) using prolate functions, and an example is shown in figure 10.2. This solution can in principle achieve the required background of 10-10 of the peak intensity, with an acceptable reduction of the resolution limit by a factor of about 4, but the manufacture of an accurately controlled gradation which is also resistant to degradation with time remains a problem. A combination of a square aperture and graded absorption was suggested by Nisenson and Papaliolios (2001), who also checked the sensitivity of their solution to inevitable wavefront errors (figure 10.3)

Fig. 10.2. An example of Slepian's prolate function apodization mask (intensity attenuation factor as function of radius) and the cross-section of the point spread function, shown on a logarithmic scale. The abscissa angle 0 is in units of X/D, so that the first zero of the Airy function for the full aperture would be at 1.22 (Kasdinetal. 2003).

Fig. 10.2. An example of Slepian's prolate function apodization mask (intensity attenuation factor as function of radius) and the cross-section of the point spread function, shown on a logarithmic scale. The abscissa angle 0 is in units of X/D, so that the first zero of the Airy function for the full aperture would be at 1.22 (Kasdinetal. 2003).

Fig. 10.3. Nisenson and Papaliolios (2001) considered apodization of a square aperture with the sonine function [(1 - x2)(1 - y2)]3. The figure shows diagonal cuts through the PSF in polychromatic light for a circular aperture, without apodization (1) and with sonine apodization (2), and a square aperture with sonine apodization (3) and with the addition of a planet of relative intensity 10-9 of the star (4). Absicissa angle 0 as in figure 10.2.

Fig. 10.3. Nisenson and Papaliolios (2001) considered apodization of a square aperture with the sonine function [(1 - x2)(1 - y2)]3. The figure shows diagonal cuts through the PSF in polychromatic light for a circular aperture, without apodization (1) and with sonine apodization (2), and a square aperture with sonine apodization (3) and with the addition of a planet of relative intensity 10-9 of the star (4). Absicissa angle 0 as in figure 10.2.

In order to solve the apodization problem without using graded attenuation, it is necessary to limit the search for suitable masks to functions which have either zero or unit value, and therefore can be manufactured by cutting holes in an opaque sheet. Ideally, the opaque part of the mask should be contiguous so as to avoid the need for a supporting structure, although a mask could be printed onto the primary mirror to solve the problem of mechanical support. For example, if a slit is cut out of such a sheet with variable width y(x) proportional to the value of Slepian's function for the graded attenuation mask in one linear dimension, the point spread function intensity has along its u axis the value of | / y(x)exp[-iux] dx |2 which is similarly dark outside the central peak. This mask would have to be rotated in its plane to allow scanning of the complete surroundings of a star. The scanning problem can be considerably alleviated by using an array of narrow slits of this type of construction which broadens the angular region within which the point spread function is darker than the required value. This type of mask has been suggested by Kasdin et al. (2003) and is illustrated in figure 12.6, where the overall shape of the mask is restricted by the demands of NASA's TPF-C project (section 12.2.3).

Some other possible solutions have been found for this problem with rotational and quasirotational symmetry (Vanderbei et al. 2003). For example, Slepian's solution can be "digitized" as a set of annular apertures. In this case, the fact that the set of rings is finite in number results in a finite outer limit to the dark part of the field of view, but this is hardly a problem if the limit is far enough out (figure 10.4). In another solution, it is digitized as a set of radial spoke-like slits with angular width suitably varying as a function of radius, which creates a point image surrounded by a completely dark field in the limit of an infinite number of spokes. Compromises result from using a finite number of spokes; apparently 120 are necessary to achieve the required background darkness.

A problem with aperture masks is that a significant amount of light is absorbed by the opaque part, more than 50% in the examples in section 10.3.1. An alternative is to make a rotationally symmetric phase distortion of the wavefront. Zernike's phase contrast, the classical technique used with microscopes to see transparent biological cells, transforms phase patterns into intensity patterns by means of a n/2 phase change at the zero order. This means that a phase mask corresponding to one of the designs in section 10.3.1 can be used together with a n/2 stop in the image plane. This type of approach has been developed by Martinache et al. (2004). A major problem is to carry out such an operation achromatically, taking into account the fact that every wavelength must be apodized differently. An alternative method of phase apodization, called "phase-induced amplitude apodization," was proposed by Guyon (2003) who calculated the figuring distortions of the primary and secondary mirrors of a Cassegrain telescope needed to create an intensity modulation in the pupil plane with properties similar to those of Slepian's function.

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