Generation of off-axis aspherics

A method for precise generation of optical surfaces is described. The work is turned about one axis, while a grinding head or cutting tool is gradually moved across by an arm turning about a second axis. This geometry can be used to directly generate spherical surfaces or aspherical surfaces presently termed 'hulahoids'. The properties of these surfaces are analyzed to determine how well large paraboloids and their off-axis segments can be generated. Very close approximations are possible. For example, 1.5-m segments of a 10-m paraboloid differ by only 0.6 microns peak-to-peak from the best fit hulahoid when the focal ratio is f/2, and 60 microns when it is f/0.4. For accurate generation the cutting tool may need numerically controlled travel of 1 mm or less, with the position set as a function of the swing arm angle.