Minimum-noise maximum-gain telescopes and relaxation method for shaped asymmetric surfaces

For future low-noise receivers, future radio telescopes must avoid the pickup of ground radiation, and for obtaining the maximum gain with a given diameter, they should be shaped two-mirror systems. A configuration with a double asymmetry is suggested, which would improve the signal-to-noise ratio by a factor of two to three. The asymmetric shaping problem is solved by a method of iterative relaxation. Starting with an original paraboloid-hyperboioid system, the primary surface is changed in each iteration for a gradual approach to the illumination demand, including the boundary condition that the circular aperture rim is represented by an exact circle of the feed pattern, and an exactly focussing secondary mirror is calculated in each iteration. The well-documented Fortran program can be obtained from the National Radio Astronomy Observatory (NRAO). All calculated examples, for various strongly asymmetric configurations, converged after 6-10 iterations (within a few minutes of computer time) to an aperture efficiency of \eta \geq 99.97 percent (for geometrical optics). The maximum change of the primary surface, from the original paraboloid to its final shape, was always less than 3 percent of the aperture diameter.