The alternative to the temporal aperture modulation is the spatial aperture modulation. It requires two-dimensional position sensitive detectors. The spatial modulation of the aperture is achieved by a pattern of holes in an otherwise X-ray absorbing plate, providing a unique spatial code. The combination of such a mask with a detector constitutes a coded-aperture- or coded-mask telescope. The use of coded apertures for X-ray astronomy was proposed in 1968 (independently of one another) by Ables [1] and by Dicke [5]. The most popular masks are those with a transmission of one-half, that is, the total open and blocked areas are equal. For an overview of coded-mask techniques see [4] and Imaging in High Energy Astronomy, edited by Bassani and di Cocco in 1995 [2].

The basic principle of coded-mask imaging is that the mask pattern (in the form of the shadow produced by the parallel beam of an X-ray source) is recognized by a two-dimensional position sensitive detector (Fig. 5.2). Any shift of the pattern on the detector surface is directly related to a shift in source position. There are different types of mask patterns and mask/detector configurations, which will be discussed in the next paragraph. The "fully-coded field of view" (FCFOV) is defined as follows: photons from any source within this area of the sky cannot reach the detector without passing through the mask (that is the complete detector surface is "coded").

The boundaries of the "partially-coded field of view" (PCFOV) are reached when the fraction of the detector area which is coded reaches zero. In real systems, the effective FOV may be limited by an additional conventional collimator (generally to a FOV smaller than the total PCFOV). The instrumental angular resolution of such a telescope is given by d/D, with d being the characteristic length scale of the mask pattern (the width of the holes) and D being the distance between the mask and the detector (here it is assumed that the spatial resolution of the detector is equal or better than d). Note, however, that the accuracy with which the position of a point source is determined can be substantially better than d/D (depending on photon statistics).

The ultimate goal is to produce an image of the sky, represented by a two-dimensional array of pixels filled with intensity values. To construct this image, the observed intensity distribution on the detector surface must be interpreted ("unfolded") using the coding function provided by the mask pattern. This is a classical "inversion problem" in which an unknown cause (the sky distribution) is to be reconstructed from a measurement (the detector distribution). The basic coding equation is given by with D(x) being the observed detector distribution as the result of the "folding" of the sky distribution S(x) by the aperture modulation function M(x) (x is the vector describing the two-dimensional coordinate in the respective plane). For the inversion of this equation, several techniques are available. It is, however, important to understand that - even with "optimum patterns" (see below) - the resulting sky image is not unique, but rather subject to uncertainties that can be quite large. The main

reason for this is the uncertainty from counting (Poisson) statistics, which is generally substantial because of the presence of background: the sky distribution S(x) is actually the superposition of the uniform diffuse X-ray background radiation and all existing sources (point-like or extended) in the total FOV (5sky(x) + sum(S;(x))), both coded by M(x)). In addition there is the so called "detector background," counts in the detector, indistinguishable from photon events, which are actually caused by charged particles and locally produced secondary photons. Also, for each individual source, all the other sources in the FOV constitute background. In general, the counting rate in the detector due to any single source is small compared with the combined background. For sources in the partially-coded field of view (PCFOV), the incomplete coding adds a "coding noise."

The reconstruction of the image (the best estimate of the sky distribution S(x)) can be achieved by several different methods. Only for special codes, it is possible to invert the coding matrix M(x) directly. The most widely used techniques are based on correlation procedures, that is, by cross-correlating the aperture code with the (suitably binned) observed intensity distribution. Alternative techniques are "mismatched filtering" (employing the inverse of the Fourier transform of the point spread function) or "Backprojection." In backprojection, one starts with the position of each detected photon and projects the mask pattern back onto the sky, marking all areas from which this photon could have originated. The superposition of all backprojections leads to the repeated marking of positions from which the photons could have come and produces an image of the source distribution in the sky. It can be shown that for optimum masks all methods are equivalent [21]. Figure 5.4 shows an image of a point source in the center of the FOV plus uniform background, reconstructed from simulated data by the cross-correlation technique. For the simulation, a four times replicated 11 x 13 URA mask (see below) was used. The optimum-coded FCFOV in the center is a flat top of a pyramid, which falls off to the edges of the PCFOV. By subtracting a "flat field" the image can be "balanced" over the complete FOV. In the PCFOV, so called "ghost images" (of the central point source) are visible, which are due to the incomplete coding.

5.1.2.2 Telescope Configurations and Types of Masks

The geometry of coded-mask telescopes can be quite different: for a given detector, the mask can be (a) smaller, (b) equal, or (c) larger than the detector. Configurations (b) and (c) are shown in Figs. 5.2 and 5.3. For configuration (b), the so called "box camera," full coding is only for on-axis sources (FCFOV equals zero). Here, it is necessary to have a closed (X-ray opaque) telescope tube (an effective collimator, defining the zero response field of view - ZRFOV), such that no photons can reach the detector without passing through the mask. A wide FOV is achieved by configuration (c). Often the mask is chosen to be twice as wide as the detector. Also here an additional collimator limiting the FOV is generally employed. While configuration (a) has some advantage over (b) (e.g., nonzero FCFOV), it suffers from a smaller effective area and therefore reduced sensitivity.

A coded aperture is defined by four parameters: the dimension of the mask elements (pixels), the number of pixels, the fraction of open pixels, and the coding pattern. The pixel size is dictated by the desired angular resolution, the spatial resolution element of the detector (which should be smaller than the mask pixel), and the desired or allowed geometry (the size of the mask and its distance D to the detector). The number of pixels is constrained by the overall dimension of the mask and the pixel size as well as by the design of the coding pattern (e.g., if certain prime numbers are to be used). In the case of background dominated measurements, the optimum for the open fraction is 50% (this is equivalent to the above discussed ON/OFF observation where the optimum on-source time is 50%). For observations with negligible background, e.g., in observing gamma-ray bursts, the optimum open fraction is close to 33%, as was chosen for the BeppoSAX WFC [12]. For the coding pattern of the mask, numerous configurations have been proposed (for a compendium of coded-mask designs see [4] and [22] (and references therein)). Although the original proposal by [1] and [5] was to use either regular grids or truly random hole positions, it has become clear that "optimum masks" (those which allow to construct images with a flat background and well-defined source peaks) can be found following certain mathematical construction procedures, which ensure that the reconstructed images carry the desired characteristics. In one-dimension sequences of n (with n being a prime number) randomly selected binary values (0 or 1 for "open" and "closed" mask pixels) in cyclic repetition are the basis for many coding patterns. For two-dimensional aperture codes often twin-primes (certain combinations of number of elements in x and y) are chosen. When such basic patterns are repeated (except for one element), so called "uniform redundant arrays" (URA) are formed [6]. A more general procedure, allowing to use other combinations of prime numbers (and to construct quadratic masks) was introduced by [8] under the name of "modified uniform redundant arrays"(MURA). Summaries of the mathematical foundation of mask generation techniques can be found in [4,10,17] and in references given in Imaging in High Energy Astronomy [2]. Figure 5.5 shows the mask of the imager on INTEGRAL: it is a four times replicated 53 x 53 MURA (with 11 rows and 11 columns cut off).

The performance of an actual coded-mask telescope can be much worse than theoretically expected, e.g., if there is a variable background: a nonuniform background across the detector surface (caused by a nonregular distribution of matter in the spacecraft next to the detector) and/or a time variable background, which is usually present (certainly in satellites with nonequatorial orbits). These effects lead to false features in the reconstructed images and can severely limit the sensitivity of the instrument. The artifacts can in part be removed by special analysis software (as an example see [9]). An alternative is to perform the observation with a "variable aperture," such that a large fraction of the total detector area is alternately illuminated and not illuminated. One way to do this is to not stare at the sky in a fixed orientation, but rather step the optical axis of the telescope through a pattern of positions. This dithering technique is quite usefully applied even with truly imaging telescopes. An elegant (but not easy to realize) technique is to alternately observe through a mask and an "antimask" (where the open and closed pixels are inverted with respect to the mask). This assures that those areas of the detector, which are illuminated observing through the mask, are not illuminated when observing through the antimask. This technique is equivalent to the ON/OFF observations and allows direct background subtraction [3].

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