Digital Sky Survey Telescope Technical Note 19910912
The optical design for the DSS telescope is a fast, simple optical system with a nearly flat focal plane and low distortion. However, focal plane curvature is such that an attempt must be made to approximately place the ends of the optical fibers feeding the spectrographs on the best focal surface rather than simply a plane. Figure 1 is a graph of the best focal surface for spectroscopy (Try54d, from the August 1991 NSF proposal). To further complicate matters, the principle ray is only normal to the focal surface to ±30 mrad. Figure 2 shows the principle ray angle (with respect to the normal to a plane at the focus) as a function of linear distance from the center of the field. The situation is clarified in Figure 3. The best focal surface and the paraxial surface, which is normal to the principle ray, are shown together.
Figure 1. Best focal surface for the spectrographic mode. The displacement of this surface from a reference plane is plotted as a function of linear distance from the field center. The radius of the 3° field of view is 326 mm and the best focal surface deviates by about 2.9 mm peak to valley from a plane. The coefficients of an even polynomial fit to the raytrace results are given.
In the August 1991 NSF proposal, we describe one way to deal with this problem. We deform the plate during drilling so that all the holes can be drilled with their axes parallel. Each hole is drilled with a shoulder at a precalculated depth. The fiber plugs are inserted into the holes and registered against the shoulders. On the telescope, the plate is allowed to relax flat, and if the plate was deformed properly during drilling and the shoulders located correctly, the tips of the fibers should lie along the best focal surface and the fibers should be aligned with the principle ray.
Figure 2. Principle ray angle (with respect to the normal to a plane) as a function of linear distance from the center of the field. The coefficients of an odd polynomial fit to the raytrace results is given.
In this approach, about 3 mm of plate thickness is allocated to accommodate the range in depths of the locating shoulders (2.9 mm). An additional 3 mm of plate thickness is needed for plug designs that reference the fiber angle to the hole axis. (The exception is the magnet plug approach which references fiber angle from the flat shoulder surface. It requires that steel be used as a plate material and that the shoulder bore be oversized to get good magnetic retention.) The resulting plate thickness is 6 mm. However, 3 mm thick aluminum plate is adequate to provide adequate stiffness against gravity and forces exerted by the fibers. Currently, we estimate that 3 mm aluminum blanks will cost $120k and that the drilling of 3000 plates will cost $600k. The cost of blanks scales linearly with thickness. Drilling may scale faster than linearly if bit clogging in the deeper hole is a problem. These costs could easily become dominant if thicker plates are used.
Other considerations include the fact that less storage space is needed for 3 mm plates. Less massive bending drilling fixtures are required. The bending strain is linear with material thickness and is higher for thicker plates. Finally, the stress in the plate in the telescope tends to reduce the deflection of the plate due to out of plane forces.
In this note, I explore a different approach wherein the plate is deformed both for drilling and use in the telescope. The original plate surface is used for locating the plugs axially, i.e., no shoulder is produced during the drilling process.
Figure 3. The best focal surface, the paraxial surface (the surface normal to the principle ray) and the shape of an ideal mandrel are shown. The mandrel is the shape that the plate should have for drilling if all holes are drilled with their axes parallel and if the plate is deformed to match the best focal surface in the telescope. Thus, the mandrel curve is the difference between the other two curves.
An axis symmetric solid element model was used to investigate the bending of the plate. This approach, in effect, reduces a 3-dimensional axis symmetric problem with axis symmetric loading to a two dimensional problem. Solid elements are used. In two dimensions, these are rectangular, but represent three-dimensional annuli with rectangular cross-sections. The mesh is 89 elements wide by 5 high. The material is aluminum 2024, 3.18 mm thick and 419 mm in radius. To match the focal plane shape, displacement constraints were applied at 343 mm and 384 mm radius to force the plate to match the focal plate slope at the edge of the field. A further displacement constraint was applied at radii of 0, 50, or 100 mm to force the plate sag to match that of the best focal surface. Without the latter constraint, the plate sags about 0.5 mm too much. To match the shape of the ideal drilling mandril, similar constraints are used. However, only radii of 0 and 50 mm for the central constraint were examined.
Figure 4. Axis symmetric finite element model showing constraints (triangles) used to force plate to match the telescope best focal surface. The model consists of 5x89 mesh of axis symmetric solid elements. The plate radius is 419 mm (16.5") and the thickness is 3.18 mm (0.125"). The constraints are applied at radii of 384 mm (15.1") and 343 mm (13.5"). An additional constraint is applied at the center of the plate.
The best match to the best focal surface occurred with the central constraint applied at a radius of 100 mm (Figure 5). For this case, the best focal surface is matched to an accuracy of 50 microns peak to valley. With the constraint applied at the center, the best focal surface is matched to an accuracy of 250 microns peak to valley. By way of comparison, with no constraint at the center, the mismatch was 830 microns peak to valley. Table 1 gives the RMS fit error of the deformed plate to the best focal surface. Only model plg41-3 (central constraint applied at a radius of 100 mm) satisfies the error budget of 25 microns that we have established for this item.
Model Central constraint Mean surface Fit error Name radius (mm) (microns) (RMS) plg31 0 -65.0 83.0 plg41 50 -16.0 46.0 plg41-3 100 4.7 7.9
Table 1. Quality of fit to the best focal surface. The mean error and root-mean-square (RMS) error are weighted by focal surface area. The RMS is calculated with respect to the mean surface.
Figure 5. The fit of three models to the best focal surface is plotted. The central constraint was applied at three different locations; r = 0 mm, r = 50 mm, and r = 100 mm. The fit with the constraint applied at a 100 mm radius is excellent.
These results are fairly insensitive to variations in the constraints. Figure 6 shows results for three models. Plg32 is the baseline case. In plg31, the displacement of the outer constraint was increased with respect to the inner constraint 10% increasing the slope at the edge of the focal plane by the same amount. In plg33, the thickness of the plate was increased 10%. Assuming that the telescope is focused on a probe at 70% of the radius, the maximum focus error does not vary much. Changes in the plate thickness would have a larger effect if the bending were done with forces rather than displacements. However force errors would have less effect than displacement errors on plate shape.
Figure 6. Fit error sensitivity to two parameters is plotted. Plg32 is the baseline case. In plg31, the displacement of the outer edge constraint is increased 10% with respect to the inner edge constraint. In plg33, the plate thickness is increased 10%. If the telescope is focused using a probe at r = 230 mm, the maximum focus error for each case is not changed much.
It appears to be an advantage to drill a plate with a simple three axis machine, if this is possible. Such a machine produces holes with their axes parallel. So the question is, can the plate be deformed for drilling so that when it is deformed to match the best focal surface and mounted in the telescope, the hole axes are aligned with the principle rays everywhere? It turns out that this is possible as shown in Figure 3. The desired shape for drilling is the difference between the paraxial surface and the best focal surface and is labeled "ideal mandrel" in the figure. Actually, it doesn't matter much if the plate in the telescope does not conform exactly to the best focal surface. The model of plg31 matches the slope of the best focal surface to 2 mrads or better.
Figures 7 and 8 shows results for two models. The constraints were basically the same as shown in Figure 4. The pair of edge constraints were chosen to match the shape of the ideal mandrel at the field edge. The central constraint was set to produce the overall sag of the ideal mandrel. In one case, the central constraint was applied at the plate center. In the other case, it was applied at a radius of 50 mm. The principle ray angle error was 12 mrad RMS for the first case and 6.7 mrad RMS for the second. We have established an error budget of 10 mrad RMS for this item, so the first case exceeds the error budget somewhat.
One problem with plg34 is that the maximum tensile stress is 389 MPa (56 kpsi) at the plate center. Since the yield strength of aluminum alloy 2024 T3 is only 340 MPa (50 kpsi), this is a problem. For plg44, the maximum stress is 107 MPa (16 kpsi), well below the yield strength of the material. Thus it appears that some attempt to distribute the central force will be necessary.
Figure 7. Two attempts to produce the optimal shape for plate drilling are shown (indicated by the "ideal mandrel" curve). In plg34, the central force is applied at the center of the plate. In plg44, the force is applied at a radius of 50 mm.
Figure 8. The predicted errors in hole alignment with the principle ray are plotted for the models of Figure 7. The curves are labeled with the radius of the central constraint. For r = 0, the hole misalignment is 12 mrad RMS. For r = 50 mm, the error is 6.7 mrad RMS.
Table 2 gives the forces required for each model. The forces are given in newtons/radian. The tabular values should be multiplied by 2p to get the total force on each annulus. The forces are modest but will have to be considered in the design of the bending fixtures.
Model Central Inner edge Outer edge Name Constraint force force plg31 34.69 -976.0 941.3 plg32 26.99 -810.0 783.0 plg33 40.42 -1032.0 991.4 plg41-3 52.41 -1021.0 968.2 plg34 -293.70 1193.0 -898.9 plg44 -357.30 1370.0 -1013.0
Table 2. Forces applied to model in N/rad. To get total force on annulus multiply by 2p.
It is a pity that an adequate match to the best focus surface does not occur with the central force applied to the center of the plate. Applying a force on an annulus with a radius of 100 mm is probably not something we want to do, even by use of 4 to 6 tension rods attached to the plate at discrete locations. However, if it could be done, the plate could be drilled from the sky-facing side since no counterboring would be needed. In this configuration, the plate could be supported by a mandrel with the ideal shape and this would give excellent control of the hole axis angles.
The simple scheme with the support at the center does provide a fair match to the best focus surface. If this were used with the counterbore scheme of the August 1991 NSF proposal, only 250 microns of plate thickness would be needed to generate the locating shoulder and 3.2 mm thick plate could still be used.
The constraint at the center does comes close to meeting our error budget for slope. It is likely that it could be improved or that the error budget could be modified. I should point out that the error budget is very tentative at this point since little analysis has been completed.
If the plate were used flat in the telescope, no moment would be required at the plate edge and the reaction at the plate center would be smaller. The result is a better match to the paraxial surface. Increasing the thickness of the plate should help as well.
