John Lugten
April 22, 1999

SMX Corp and Leica each manufacture a coordinate measuring machine (CMM) which should be capable of measuring the primary mirror surface of an ALMA telescope to an accuracy of better than 10 microns rms. These systems are portable, accurate, and operate within a volume extending only about 1 dish diameter in front of the dish surface.

Perhaps the most crucial performance test for the prototype antenna which is related to primary mirror accuracy is to confirm that the gravitational deformation with elevation change is as expected; these CMMs provide the best possibility for measuring the dish surface at high elevation prior to building a working mm interferometer (unless the LES-9 satellite is kept operating beyond Jan. 2000). Measuring the primary mirror at high elevation (say 60 and 120 degrees) requires a very stable measuring tower about 15 to 16 meters high; the low elevation measurement requires a stable tower of about 6 to 8 m height. The antenna performance tests of verifying good primary mirror stability with thermal and wind loading are very difficult to attack quantitatively, but the laser CMM method will probably be as effective as any.

A second goal of adjusting the surface of the prototype antenna to the accuracy required to achieve a good aperture efficiency at 300 GHz (about 40 to 50 microns rms is desired) can also be achieved with the laser CMMs.

These systems have a tracker head which tracks the movements of a retro-reflector and periodically records its location. The retro-reflector location is measured by measuring the angular location of the reflector as seen from the tracker head and by measuring the radial distance to the retro-reflector using a laser interferometer. The retro-reflector can be moved along any path over the surface which is being measured.

The quantitative parts of the following discussion are based on the performance specifications of SMX Corporation's Tracker 4000. The SMX Tracker 4000 quotes a radial accuracy (1 sigma) of 2 microns plus 0.8 microns per meter of optical path. Transverse accuracy is 12 microns plus 5 microns per meter of optical path. This transverse accuracy corresponds to about 1 arc second accuracy of the angle encoders in the tracker head.

With a desired measurement accuracy of 10 microns rms over the dish surface, the quoted transverse accuracy will not be adequate for the general case. However, for our special case of measuring a concave paraboloid, the transverse accuracy is of no consequence if we are just a bit careful. To minimize the significance of the tangential accuracy contribution, we simply place the tracker head near the average center of curvature of the paraboloid. As a particular example, for a 12 m diameter f/0.38 paraboloid, if we place the tracker head on the dish center line, a distance 10.46 m from the vertex, the maximum rate of change of radial distance with angle is only 1.5 microns per arc second. Thus a 1 arc second angular measurement error is of negligible contribution to the total measurement error. For our special case, the total accuracy of the measurement is limited entirely by the radial measurement accuracy of the CMM.

In further discussion with an SMX scientist, I learned that the actual radial accuracy is dominated by systematic uncertainties such as inaccurate air temperature compensation (the 0.8 microns/m radial accuracy corresponds to a 1 C error in air temperature compensation), and (at a lower level) imperfect axis alignment compensation in the tracker head and imperfect centering of the retro-reflector in the target sphere. Typical performance in the SMX laboratory where the instrument is calibrated against HP and Renishaw HeNe laser interferometers shows errors which are less than half of the quoted 1 sigma accuracy specifications. The SMX calibration laboratory is not a particularly nice environment, having typical temperature gradients of 1 C per meter.

To attain the accuracy which is possible with this CMM requires mechanical stability of the telescope (which we already expect for a well designed telescope which meets our performance specifications), mechanical stability of the tower supporting the tracker head, and careful measurement of air temperature and control of air temperature gradients during the surface measurement process. Long term drifts in measured distances can be mostly removed by repeated measurements of several fiducials located on the dish (a "nest" can be defined by temporarily locating two pins on the surface against which the target sphere can be precisely placed). Additional checks of tower stability can be made by measuring distances to reference markers on the ground. The VLA antenna barn can easily house the required test towers and the subject antenna and would provide protection from wind and solar loading. Thermal gradients may be more severe in the antenna barn than outdoor night time conditions however.

Using this system does not require modifying the quadripod, secondary mirror drive, or changing the telescope weight and balance. Cost to buy the Tracker 4000 is $161,000, but many consulting firms exist which specialize in making these measurements; it is probably wise to demonstrate the system performance in a suitable test before considering a purchase of the instrument.