Cassegrain radio telescope geometry Aperture diameter, D 100.000 m Focal length of primary mirror, f 33.000 m Focal ratio of primary mirror, f/D 0.330 Telescope F-number (or F/#): F_eff/D 2.881 Dish depth 18.939 m Dish subtended angle 148.587 deg Subreflector subtended angle 19.837 deg Effective focal length, F_eff = M*f 288.109 m Cassegrain telescope magnification, M = f2/f1 8.731 Eccentricity of the hyperboloid, e = c/a 1.259 Subreflector diameter, d 8.000 9.000 10.000 11.000 m Height of secondary focus above paraboloid vertex, h 9.000 6.000 3.000 0.000 m Prime focus to hyperboloid vertex, f1 = c - a 2.466 2.775 3.083 3.391 m Secondary focus to hyperboloid vertex, f2 = c + a 21.534 24.225 26.917 29.609 m Secondary interfocal length, f1 + f2 = 2c 24.000 27.000 30.000 33.000 m Ray path difference to the foci, (f - h)a/c 19.067 21.450 23.834 26.217 m Dish edge angle as seen directly from secondary focus 78.757 75.491 72.318 69.254 deg Distance: prime focus – subreflector edge 4.155 4.675 5.194 5.713 m Subreflector depth at centre 1.342 1.509 1.677 1.845 m Diameter of blind spot on subreflector 0.598 0.757 0.935 1.131 m Ray tracing data Observing frequency [GHz] and wavelength 22.207 and 1.350 cm HPBW (beamwidth) 0.539 ' Assumed z-coordinate for aperture plane -23.530 m Number of rays traced 1300 Number of pattern bins in u.v dimensions 25.025 Span of u coordinate [spacial cycles] 5.000 Span of v coordinate 5.000 Free space illumination taper or asymmetry 0.064 dB Feed illumination taper at the edge (Gaussian function) -12.000 dB

Some characteristics based on analytical solutions (classical optics approximation for Gaussian illumination, d = 10 m) Half power beam width (Baars 2003) .............. 0.0090 [deg] First side lobe power level due to taper (Baars) 0.5351 [%] Gain loss due to tapered illumination ........... 13.3611 [%] Gain loss due to diffraction at subrefl. (Lamb) . 1.0840 [%] Petzval surface radius (inverse field curvature) 2.8196 [m] Beam dev. due to primary translation (Baars) .... -1.3160*Xoff/m [deg] Beam dev. due to primary rotation ............... 1.7580*tilt/deg [deg] Beam dev. due to secondary translation .......... -1.1193*Xoff/m [deg] Beam dev. due to secondary rotation about vertex 0.1632*tilt/deg [deg] Beam dev. due to secondary rotation about focus . 0.1030*tilt/deg [deg] Beam dev. due to secondary rotation about "z_piv" 0.1030*tilt/deg [deg] Beam dev. due to feed translation in sec. focus . -0.1967*Xoff/m [deg] Feed offset per beam width, prime focus (Baars) . -0.0068*NoOfBeams [m] Feed offset per beam width, secondary focus ..... -0.0458*NoOfBeams [m]

Fig. 1: Minimum gain losses of symmetrical Cassegrain antenna as a function of feed lateral offset from the secondary focus. The minimum was searched for by varying the feed z coordinate (along the telescope symmetry axis; see the next figure for the amount of the z-shift required) and adjusting the feed tilt. The four curves marked with 'd = ... m' correspond to four different subreflector sizes (diameters, d) with the height of secondary focus above the dish vertex (h equal to 0, 3, 6 or 9 m for d equal to 11, 10, 9 or 8 m, respectively) chosen so as to maintain the effective focal length constant (at 288.109 m). The blue curve represents similar losses assuming the feed direction is kept parallel to the telescope axis of symmetry and for subreflector diameter of 10 m. Avalaible is also a postscript version of this figure.

Fig. 2: The height of feed above the secodary focal plane (towards the subreflector) for which the least aberration and spillover losses (shown in Fig. 1) are found while simultaneously adjusting the direction of feed power pattern. Avalaible is also a postscript version of this figure.

The next table presents some data used for plotting Figure 1. and 2. In it, the feed coordinates and its pattern direction (relative to direction towards the subreflector vertex) are given in the first three columns (x, z [in metres] and dtx [in degrees], respectively). The further five columns contain losses [in percent of the antenna efficiency] due to aberration (Aberr), spillover (Spill), aberration and spillover together (A&S), illumination decentering (IllDec) and the three losses together (Totl). In columns ComaLow and ComaHi are given comatic distortions of the power pattern (these are the minimum and maximum values of the distortion pattern, i.e. the difference between patterns of the antenna in question and one without the offsets) in percent of the pattern maximum. The last column gives the level of the first side-lobe [in percent of the maximum].

Data used for plotting Fig. 1 and Fig. 2

x z dtx Aberr Spill A&S IllDec Totl ComaLow ComaHi 1stLob d = 8 m, h = 9 m 0.0 0.000 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.3417 0.4 0.031 0.116 0.0779 0.2853 0.3630 0.0000 0.3630 -0.1662 0.6659 0.5175 0.8 0.127 0.245 0.3942 0.6610 1.0525 -0.0134 1.0392 -0.3324 1.7048 0.7246 1.2 0.287 0.361 1.2628 1.0802 2.3293 -0.0522 2.2784 -0.5486 3.1284 1.0246 1.6 0.514 0.510 3.2298 1.3655 4.5511 -0.0565 4.4972 -0.6265 4.5356 1.5739 2.0 0.810 0.707 7.0319 1.7598 8.6680 -0.1025 8.5743 -0.6536 7.1470 2.4879 2.4 1.175 0.981 13.7486 2.0553 15.5213 -0.0106 15.5124 -0.6894 11.7671 4.2588 2.8 1.615 1.167 23.7469 2.4599 25.6226 -0.1414 25.5175 -0.4129 17.9269 8.0506 d = 9 m, h = 6 m 0.0 0.000 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.3417 0.4 0.027 0.109 0.0767 0.2548 0.3313 0.0034 0.3347 -0.2004 0.5929 0.5336 0.8 0.112 0.204 0.3716 0.6053 0.9747 -0.0095 0.9653 -0.4001 1.5400 0.7586 1.2 0.252 0.319 1.1059 1.0248 2.1193 -0.0473 2.0731 -0.5519 2.9224 1.0364 1.6 0.451 0.454 2.7599 1.2499 3.9753 -0.0278 3.9486 -0.7268 4.1313 1.4477 2.0 0.708 0.620 5.6898 1.6293 7.2264 -0.0810 7.1512 -0.6666 6.1401 2.2383 2.4 1.028 0.821 10.8656 2.0028 12.6508 -0.1092 12.5554 -0.7353 9.7548 3.5975 2.8 1.401 1.179 19.3196 2.1450 21.0502 0.3300 21.3107 -0.6220 15.4302 6.1827 d = 10.0 m, h = 3 m 0.0 0.000 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.4093 0.4 0.025 0.095 0.0771 0.1954 0.2723 0.0086 0.2809 -0.3096 0.4404 0.5608 0.8 0.100 0.174 0.3518 0.6092 0.9589 -0.0128 0.9463 -0.3841 1.5148 0.8596 1.2 0.225 0.279 1.0344 0.8815 1.9068 -0.0096 1.8974 -0.7596 2.5336 1.1320 1.6 0.401 0.409 2.4208 1.2340 3.6250 -0.0379 3.5884 -0.7427 3.8756 1.5318 2.0 0.630 0.537 4.9695 1.5195 6.4135 -0.0367 6.3791 -0.7755 5.4899 2.2102 2.4 0.911 0.717 9.1075 1.8538 10.7925 -0.0546 10.7437 -0.8067 8.3873 3.2825 2.8 1.236 1.021 15.7083 2.1534 17.5234 0.1038 17.6091 -0.7179 13.0932 5.2617 d = 11 m, h = 0 m 0.0 0.000 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.5615 0.4 0.022 0.093 0.0765 0.1949 0.2713 0.0094 0.2806 -0.3048 0.4326 0.6676 0.8 0.089 0.162 0.3439 0.5336 0.8756 -0.0021 0.8735 -0.4482 1.4153 0.8905 1.2 0.202 0.257 0.9519 0.8507 1.7945 -0.0118 1.7829 -0.7427 2.4706 1.1957 1.6 0.360 0.351 2.1511 1.2426 3.3670 -0.0608 3.3083 -0.7181 3.7217 1.5565 2.0 0.565 0.483 4.4234 1.4113 5.7723 -0.0147 5.7585 -0.8793 5.0427 2.2095 2.4 0.817 0.630 7.8702 1.7819 9.5118 -0.0387 9.4768 -0.9021 7.4399 3.2484 2.8 1.115 0.848 13.5184 2.0114 15.2578 0.0875 15.3319 -0.8425 10.8538 4.8181

An error of one wavelength (1.35 cm) in placement of the feed at the secondary focus along the z-axis results in additional gain loss of about 0.47 %, either side. For offset positions the tolerance of placement is higher. For example, at x = 2.8 m and z = 1.24 m, an upward (towards the subreflector) displacement of one wavelength causes a loss of 0.16 % and downward one only about 0.10 %.

The tolerance is gradually increasing towards lower frequencies (i.e. the longer waves the smaller total gain lost and the smaller losses per unit feed displacement). At 5 GHz (wavelength of 6 cm) at the secondary focus the z-error of 6 cm results in only 0.065 % gain loss. For the feed position at x = 2.8 m and z = 1.25 m (about optimum 'z' there, with a minimum of gain losses of about 3.2 %) the rate of gain degradation is slightly smaller than at the focus and is about 0.01 % per 1 centimetre error in the z-coordinate (upward or downward).