Optical Quality

Not only the angular resolution but also the shape of the detector beams is an important contributing factor to the quality of the final mission results. The wish to crowd as many detectors as possible into the focal plane makes the Planck telescope a system with a very large field of view (FOV, of order 10 degrees on the sky), and therefore optical aberrations become important especially for detectors near the edges of the FOV. Figure 1 shows schematically the optical layout in the focal plane, with the HFI detectors occupying the center of the FOV, in turn surrounded by a ring of LFI detectors.

**Figure 1:** A sketch of the layout of detectors in the focal plane (HFI detectors in color, LFI detectors in black and white), reflecting also the location (but not the relative separation) of the arrays of beams on the sky. Each circle corresponds to one feed aperture. Note that each LFI horn feeds two detectors. The dimensions are in millimetres.

The specific telescope design determines the type of aberrations present in the beams. Typical distortions are sketched in Figure 2. The main effects observed are ellipticity of the 3 dB contours and significant coma at lower levels. An effort is currently underway to optimize the telescope configuration to reduce these effects. The optimization procedure involves the use of optical ray-tracing software to maximize a given criterion (i.e. the sum of the peak directivities of a subset of detectors), followed by verfication of the optimized design using physical optics software (i.e. GRASP 8). The latter involves very time consuming computations, therefore it is needed to use the ray-tracing software to carry out the actual optimization. In the optimization process, it is also necessary to take into account the spillover of radiation past the telescope reflecting surfaces, which must be kept at a level compatible with the stringent straylight rejection requirements (see Straylight Rejection) .

**Figure 2:** A sketch of the typical distortions produced by the Gregorian/Dragone-Mitsugushi design. Each square contains a Point Spread Function for a subset of detectors in the "standard design" (the one described in The Telescope and Baffling System), as calculated by Code V (a ray-tracing software package). Physical optics software (GRASP8) has confirmed the validity of these results. The contours correspond to 3 and 20 dB below peak.

The optical quality of the system is affected not only by the telescope and the off-axis location of the detectors in the focal plane (See Figure 2), but also by other sources of degradation such as alignment errors, cool-down distortions, etc. The impact of these sources on the WFE along the optical axis of the system, and on the depointing of the nominal line-of-sight has been evaluated numerically for the standard configuration during the Phase A study using the Nastran software, which incorporates the thermoelastic properties of the reflectors and payload elements. The results of this evaluation are summarized in Table 1, which shows only the displacements remaining after rigid body motions and telescope pure expansion effects have been removed (since these do not affect the image quality). This conservative calculation indicates that the on-axis system WFE is better than the goal of 60 $\mu$ m; since the assumed mirror manufacturing errors (which dominate the overall budget) are larger than what can in practice be achieved, it is expected that the final WFE will be well within the goal. It is also worth noting that the total expected depointing is low, and permits that on-ground alignment activities between the reflectors and detector boxes be carried out via mechanical means only. Finally, the WFE errors due to cool-down of the telescope to cryogenic temperatures do not dominate the overall budget, so that the optical quality of the telescope need only be verified at ambient temperature, thus simplifying considerably the testing activities.

**Table 1:** On-axis Wavefront Error and Pointing Budget
	WFE ( $\mu$ m rms)	LOS depointing (mrad)
		Scan angle	Spin Phase angle
Reflector Manufacturing Errors
Main reflector surface accuracy (10 $\mu$ m rms)	20	0	0
Subreflector surface accuracy (10 $\mu$ m rms)	20	0	0
Alignment Errors
Adjustment accuracy of 0.2 mm ^*	30	0.4	0.4
Adjustment accuracy of 1 arcminute ^*	25	0.6	0.6
Launch and In-orbit Distortions
Launch effects	5	0.1	0.1
Gravity release $^\dagger$	10	0.15	0.01
Moisture release $^\ddagger$	1.5	0.035	0
Cooling down (300 K to 100 K)	21	0.03	0.02
Temperature gradients ^#	0.5	0.015	0.003
Thermal fluctuations in orbit	0	0	0
Long term aging	5	0.1	0.1
Total	54	0.8	0.8
^* Includes reflectors and instruments
^# Assumes on-ground compensation during tests
$^\ddagger$ Assumes CME=10^-4, 30% desorption
$^\star$ Assumes 5 K/m on mirrors