3 Analyzed telescope designs and optical calculations

We have studied the far sidelobe levels and main beam distortions for several different optical designs (see also Appendix A). On the other hand, in the following sections we will focus only on the same kind of PLANCK telescope design but for three different primary mirror aperture, for sake of simplicity and to give a better emphasis to the relevant issues.

The first (PHASE A) is the design presented in the Phase A study, a $1.3\times1.5$m off-axis Gregorian telescope (Bersanelli et al. 1996). The second (BASELINE) is an enlarged version of the Phase A design with a primary mirror of 1.550 m aperture, and presents an aperture very close to the present ESA baseline telescope. The third (ENLARGED) is an enlarged version of the Phase A design with a primary mirror of 1.750 m aperture. The overall focal ratio is the same for the three configurations: the incident wave is scattered by the primary parabolic mirror to the secondary ellipsoidal mirror and then to the focal region where the detectors are located. The secondary reflector axis is tilted at 14^ $^{\circ}\;$with respect to the parabola axis in order to minimize the beam distortions at the center of the focal region. Table 1 gives the details of the designs.

   

Table 1: Parameters of telescope designs analyzed

Design	$D_{\rm p}$ (mm)	$F_{\rm p}$ (mm)	2a (mm)	2c (mm)	$d_{{\rm m}-{\rm s}}$ (mm)	$f_{{\rm eff}}/D$
PHASE A	1292.4	720	1200	514.29	1065.69	1.29
BASELINE	1550.0	863.51	1200	514.29	1209.20	1.29
ENLARGED	1750.0	974.93	1200	514.29	1320.62	1.29

The properties of the radiation pattern have been calculated for the three optical configurations proposed for PLANCK telescope. For each of them, we calculated the position and the shape of the focal surface, i.e. the surface of maximum directivity (Valenziano et al. 1998). Geometrical optics approximation has been used: circular bundles of rays from the rim of the primary mirror are propagated through the telescope optics. The points where the bundles converge to the minimum size (in least-square sense) trace the focal surface and also give the correspondence between angles from the optical axis and the linear displacements in the focal surface, often referred as the plate scale. The longer effective focal lengths of the BASELINE and ENLARGED designs compared to the PHASE A design gives more magnified images (i.e., smaller plate scales in the sense that the number of arcminutes per millimeter is smaller), as shown in Figs. 2, 3 and 4.

These results are used as input to calculate the response of the telescopes (the beam pattern) in a regular elevation-azimuth grid on the sky that covers the field of view (Villa et al. 1998b).

The detailed shapes of the main beams formed at each of the positions sampled on the sky were calculated using software based on Sletten (1988). Specifically, given the geometrical parameters of the optics, the far field is computed from the amplitude and phase distribution of currents on the main reflector surface. The amplitude is calculated by propagating the field from the horn to the main reflector, using geometrical scattering on the subreflector surface and taking into account the free space attenuation. The horn is modeled by a cos $(\theta)^N$ amplitude beam pattern. The phase distribution is calculated by geometrical optics given the far field scan angle and the feed position. The repointing of the feed is systematically considered. The averaged distance between the subreflector and feeds is 75 cm. The maximum diameter, $D_{\rm f}$, of LFI feeds is about 6.5 $\lambda$ and the Fraunhofer region (Far Field) starts at about $2 \times D_{\rm f}^2/\lambda$ from the feed. This means that the Far Field distance is about 85 cm and 30 cm respectively at 30 GHz and 100 GHz so the Near Field effects are expected to be small and are neglected in this study.

The cross-check between our results and those by TICRA (1997), based on the full Physical Optics analysis with the GRASP8 code, shows good agreement both for the predicted beam distortions, as shown in Fig. 5, and for the shape of the focal surface. All this validates the optical simulation codes considered here and allows us to use our simulated beams in the rest of the work.

Figure 5: Comparison of our simulated beams (labelled by "LFI team'') with those computed by TICRA (1997) for the PHASE A configuration at 100 GHz