1 Introduction

Future space missions dedicated to the imaging of the cosmic microwave background (CMB), like PLANCK and MAP, represent a powerful opportunity for cosmology and astrophysics. The nominal angular resolution and sensitivity of the two instruments on-board PLANCK, the High Frequencies Instrument (HFI, Puget et al. 1998) and Low Frequencies Instrument (LFI, Mandolesi et al. 1998), allow to determine the angular power spectrum, $C_\ell$, of CMB primordial fluctuations up to multipoles, $\ell$, larger than 10³, i.e. until secondary anisotropies do not largely overwhelm primordial signatures. On the other hand, experiences from the previous CMB experiments as well as a realistic analysis of PLANCK observational performances indicate that a stringent control of all the systematic effects is crucial to reach the mission objectives (e.g. Burigana et al. 1998a,b; Delabrouille 1998; Maino et al. 1999; Maino 1999). Galactic and extragalactic foregrounds are important sources of astrophysical contaminations at a level that depend on the frequency $\nu$and angular scale $\theta$(e.g. Danese et al. 1996; Toffolatti et al. 1998), but, from the opposite point of view, their study represents an important co-product of the PLANCK mission (e.g. De Zotti et al. 1999 and references therein).

Two complementary approaches have been proposed by both LFI and HFI teams for reducing the impact of instrumental systematic effects on anisotropy measurements: the "hardware'' approach, i.e. design mission strategy and instruments in order to minimize all the potential systematic effects, and the "software'' approach, i.e. develop data analysis methods to further reduce residual effects in the data.

The detailed design of the PLANCK telescope is continuously changing with the aim of optimizing its performance. On the other hand, the optical framework and the basic concepts to address the relation between optical performances and scientific goals are quite general and possibly applicable also to other CMB anisotropy experiments. We present here a methodological study on the optical performance of the PLANCK-like telescopes, by comparing a suitable options of optical designs and analyzing their impact on the observations of CMB anisotropy measurements. After a brief discussion of the relative importance of the optical distortions near and far from the central direction of each beam in the sky field of view, we focus on the impact of main beam distortions on PLANCK data. Main beam distortions may introduce a degradation of the angular resolution and of the sensitivity per resolution element. These two last effects can be seen as orthogonal to each other in the space $\theta - \Delta T$ of angular scales and temperature anisotropy or, equivalently, in the space of $\ell - C_\ell$(Mandolesi et al. 1997; Burigana et al. 1998a). The present analysis is devoted to recognize the "hardware'' requirements of PLANCK-like telescopes in order to keep at acceptable levels the effects of main beam distortions.

In Sect. 2 we present a summary of the recent developments (Mandolesi et al. 1998; Puget et al. 1998) in the design of the PLANCK mission since the Phase A study (Bersanelli et al. 1996) focussing on the aspects relevant for the optical performance. In Sect. 3 we present the set of three of optical designs assumed as references for the present discussion, the basic framework of our optical simulations and our main results for the beam shapes; other optical configurations concerning telescopes with worst optical quality at the primary mirror edges are considered in Appendix A. We consider here the case of the "clean'' 100 GHz channels which are the most important for the primary cosmological goal, having small foreground contaminations. Moreover, in the new design of the Focal Plane Unit, the channels at highest frequencies of HFI are located very close to the telescope optical axis, where optical distortions are expected to decrease in order to compensate their increasing with the frequency. We present also a brief discussion of the optical performance at 30 GHz in Appendix B. In Sect. 4 we estimate the implications of beam distortions by means of three different and complementary methods of analysis for quantifying the relevance of optical aberrations and by evaluating the final impact on CMB science. We set there the constraints to have a telescope "good enough'' to reach the key goal of $\simeq 10'$ resolution at 100 GHz. Section 5 concerns the limits on the edge taper for which the emission from the Solar System objects and from the Galaxy entering the sidelobes are acceptable without compromising the angular resolution. This is a first order analysis and therefore gives only an indication of the relevance of edge taper, sidelobes pick-up versus angular resolution. More detailed studies on the impact of sidelobes contamination have been done recently (e.g. De Maagt et al. 1998; Burigana et al. 1999b; Puget & Delabrouille 1999; Wandelt & Górski 1999). Finally, we discuss the mission impact of a such telescope, by dealing both with building problems and with cost problems, and draw out our main conclusions in Sect. 6.

Figure 1: New symmetric configuration accepted as current baseline (Mandolesi et al. 1998; Puget et al. 1998). The latter demands greater off-axis performance of the telescope due to the larger mean distance from optical axis for the LFI beams