Technology Nugget—Mirror Segment Manufacturing
  Jerry Nelson
  University of California, Santa Cruz

Polishing the primary mirror segments to the desired shape is the most expensive single part of the construction of the TMT. This is because the polishing process is difficult and we need about 700 square meters of mirror surface, spread out over 738 segments. Why is this so hard?

The peculiar surface shape of the segments is probably the key issue here, but the mere act of polishing glass to any shape is pretty amazing. To make our mirrors reflect light to the needed accuracy, the mirrors need to be incredibly smooth. For good reflectance, we need the optical surface to be smooth to better than ~ 100 atoms in thickness, else the light will be scattered objectionably. We don’t know how to polish most materials to this level of smoothness, but glass is one of the few materials that can be polished to such a precise level, if we are clever enough.

A basic idea helps us appreciate how to make smooth surfaces. We want to polish the surface with some kind of polishing tool that abrades away glass. The resulting surface will be smooth only if the tool fits the glass extremely well. Polishers then rub the tool against the glass in a random pattern, so no grooves get polished into the glass. The only way this can happen is if both surfaces are the same spherical surface. So, at least locally, we need the glass and the polishing tool to be spherical. As a result of this, spherical surfaces are by far the easiest and quihttp://www.tmt.org/newsletter/img/tn-0609-fig1-th.jpgckest shape to polish. Because of this, some telescopes are made with spherical mirrors.

Unfortunately, spherical mirrors do not bring starlight to a well-defined focus. The images experience “spherical aberration,” which can degrade the image quality quite severely. In order to bring starlight to a good focus, a mirror must be parabolic in shape, not spherical. For many large telescopes, the basic focus occurs after light is reflected from the primary mirror and a secondary mirror. It turns out that in this circumstance, it’s advantageous to make the primary either elliptical or hyperbolic, but very close to a parabola. For this discussion I will assume our primary is parabolic.

A paraboloid is the shape made by rotating a parabola about its optical axis. However, if one examines a piece of this surface that is away from the optical axis, the local surface has a slight saddle-shaped error relative to a spherical shape, and this error increases as the piece is examined further off axis. This makes it hard to polish! Over the years, opticians have become skilled at polishing figures of revolution on a polishing table, rotating the table while artfully moving the polishing tool, and thus making an optic whose shape is a figure of revolution where the rotation axis of the mirror coincides with the rotation axis of the polishing table. But our piece (or segment) has its axis of revolution as far as 15 meters away from its center, quite inconvenient for a small polishing machine.

Quantitatively, the surfaces we want to polish differ from a sphere (the easy shape to polish) by quadratic terms (combinations of x2, y2 and xy in Cartesian coordinates, where z is the local axis perpendicular to the segment surface). Opticians make these surfaces with great difficulty, and their smoothness is limited.

So, how do we plan to polish our segments? For many reasons, we want our mirror segments to be quite thin (cost of materials, thermal inertia, mass to be supported). It turns out that glass is a wonderfully elastic material. This means that if one pushes on it somehow, the resulting deformations are linearly proportional to how hard one pushes, and when one stops pushing, the glass immediately springs back to its original shape. Push too hard and it breaks, but until then it is ideally elastic.

Since our mirrors segments are very thin (1.2 meters in diameter and only 0.040 meters thick) they can be considered as thin plates (the curvature is slight; the curve of the segment is only 3 millimeters, compared to its 40-millimeter thickness). It turns out that in the 19th century, the theory of elastic deformations of thin plates was worked out by physicists, so we can readily calculate how they will deform under load.

Given this background, here is the clever idea for polishing our segments. If we can apply forces to elastically deform our mirror segments so their normal shape is deformed into a sphere, then we can polish the loaded segments as spheres (remember this is the easiest shape to polish by far). This makes a very smooth surface since the tool fit is so good. Once the sphere is polished into the glass, we then remove the forces and the surface elastically relaxes to the off-axis shape we desire.

From the equations describing the deformations of elastic plates it turns out that to make the desired quadratic shape change requires that we apply ONLY shear forces (forces perpendicular to the surface) around the perimeter of the segment, and bending moments or torques around the edge.

This approach was used quite successfully to polish the mirror segments for the Keck telescopes (see figure 1, above).

This isn’t the entire story. We polish circular mirrors this way, but we want hexagonal segments. So there is another interesting tale to understand manufacturing hexagonal segments. That story will have to wait for next time.