Remotely measuring mirror curvature

The primary of the James Webb Space Telescope (JWST) is a six-meter segmented mirror with each of the 18 hexagonal 1.5m lightweight mirror segments an off-axis parabola (OAP). All segments must have the same parent radius to within a fairly tight tolerance in order to ensure coincident images of the same size. In the 2007–2008 timeframe, the segments will be tested at Marshall Space Flight Center's X-Ray Calibration Facility, which contains a large cryovac chamber originally built to test the Chandra X-Ray Observatory (modified now so that optical metrology work can be conducted in the visible regime). Among the monitoring measurements needed for each JWST mirror segment is a way to remotely determine any changes in parent radius as the segment is dropped from ambient to deep space temperatures (≈30K). Presented here is a synopsis of one proof-of-concept empirical method for making this remote radius measurement.

From calculus, local radii at a point on a three-dimensional surface can be determined in different directions from the osculating circle. In a parabolic surface, the tangential and sagittal radii are of interest and are given by: where Î is the conic constant (and equal to −1 for a parabola). Equation (2) is easily solved for the parent radius, R:

Figure 1. This figure represents the relevant geometry for the off-axis parabola (OAP) with parent optical axis (radii R and center of curvature C) and the normal to the center of the OAP. Here, the sag of the mirror surface (as determined from the center of the OAP) equals δ_N, or the separation between C and C_S, the sagittal center of curvature (with radius ρ_S also equal to the length of the normal, L_N). While C_S and C_T (tangential center of curvature with radius ρ_T) lie along the local surface normal, C does not, with the difference between the angles θ and θ_N describing δ_N.

In our experiment, we used a 3″ (inches, or approximately 7.62cm) diameter OAP from Space Optics Research Labs. The mirror has a parent radius of 12″ and is separated from the parent vertex by y=3.5″ The surface normal of interest is the one located at the center of the OAP. Consequently, the local radii are:

Figure 1 illustrates the pertinent parameters. Note that the sagittal center of curvature, C_S, lies on the optical axis, and is a known distance from the parent's center, C (0.51″ in this case).

A top view of the experimental setup is illustrated in Figure 2. An alignment laser beam is normalized off the center of the OAP. Then, a thin metal mask plate in the shape of a cross is attached directly to the OAP and oriented to isolate the tangential and sagittal ray fans. An optical rail (parallel to the normalized beam) supports a micrometer carriage and point source (exit face of a single-mode optical fiber). The point source lies on the normal to the OAP center. Not shown in Figure 2 is a low power microscope (orthogonal to the page) that rides on the same carriage. The microscope is focused on the front face of the fiber chuck via a very-thin beam splitter. A return image from the OAP is also formed on the chuck face and observed via the microscope.

Figure 2. This experimental setup shows the alignment laser normalized to the center of the OAP inputting light via coupling optics into a single-mode optical fiber. The output tip (B) of this fiber is positioned at the sagittal image focus. A Leica Dista Pro distance-measuring instrument is used to determine the distance between the Leica (A) and B and A to the center (C) of the OAP. Subtracting these two measurements yields the distance between B and C, or the sagittal radius of curvature ρ_S, from which the parent radius may be determined.

During the experiment, both arms of the cross are illuminated simultaneously. There is enough focus offset between the tangential and sagittal images so no confusion arises as to the location of either image, i.e., the tangential image is grossly out of focus when the sagittal image is observed, and vice versa. Once these two images have been identified, the separation from the center of the OAP mask to the front face of the fiber chuck is measured using a Leica Disto Pro distance measuring instrument.

Using the Leica, we measure Ú_S and then use equation (3) to find the parent radius. In this experiment, measured Ú_S=12.53″, giving a parent radius 12.03″ This is 0.25% from the predicted value, showing the merit of this technique, which, with refinement, could be adapted for mirror segment testing. Full details of the experiment and analysis can be found in the reference.