Testing the Paraboloid part 5
Removing the mask, we once more cut in the knife-edge. The first shadow to be seen moves in on the left-hand edge of the mirror, immediately followed by a second one appearing in a spot midway between the center of the mirror and the right-hand edge, in other words at the 50-per-cent zone. As the knife-edge progresses across the pinhole image, the second shadow expands, spreading quickly to the right, and more slowly to the left; at the same time the first shadow is moving farther in on the mirror.
When the knife-edge has reached the axis of the reflected cone of light (the actual point of intersection of rays from the 70-per-cent zone), the second shadow should extend from the center of the mirror to the 70-per-cent zone on the right; the first shadow should have reached the 70-per-cent zone on the left. The mirror now should have the appearance shown in Fig. 39c. The crests of the convex curves marking the 70-per-cent zone should form the boundaries of the shadows. If the knife-edge movement is now continued, the first and second shadows will approach each other, meeting at the 50-per-cent zone on the left side of the mirror, while the shadow that was stopped at the 70-per-cent zone on the right side will not quite have reached the edge of the mirror there.
Note that in Fig. 39c, the apparently depressed portions of the perfectly corrected mirror, namely, the center and extreme edges, are at the same level. And again note that the 70-per-cent zone marks the crests of the convex curves in the apparent cross section there.
This is the typical appearance of a paraboloid at that knife-edge setting. It is obvious that if the surface of a mirror is thus found to coincide with the paraboloid in the three zones tested, there can be little residual error. Now let us consider the most common cases of mirrors agreeing with the r2/R formula for the edge zones, but failing to conform at the 70-per-cent zone.
When the knife-edge is at the halfway setting, it may and frequently does happen that instead of the apparent cross section shown in Fig. 39c, shapes like those illustrated in either Figs. 42c or 43c are seen. It is apparent from these shapes that neither of the mirrors represented is a paraboloid, and the centers of curvature of their 70-per-cent zones are displaced. While in either case it is possible to cause the shadow boundaries or apparent crests to land on the 70-per-cent zones by positioning the knife-edge at these displaced centers of curvature, the setting will not be exactly halfway between the centers of curvature of the edge and center zones, nor will the apparently depressed parts of the mirror be at the same apparent level.
Other possible defects are too great a width of the curves at the 70-per-cent zone and insufficient breadth of the central trough, or a condition that is the opposite of this. These errors may escape the observation of the unpracticed eye at the halfway setting of the knife-edge, but their detection should be made possible by the discrepancy in the breadth of the central plateau in Fig. 39b. Location of the center, 50-per-cent, and 70-per-cent zones should not be left to guesswork. The zones and the center of the mirror can be exactly marked, just prior to testing, with tiny drops of ink de- posited on the mirror's surface from the end of a toothpick. When the mirror is set up for testing, the diameter along which the ink- spots lie should be in a horizontal plane; the spots will be conspicuously revealed by diffraction.
In all this shadow testing, no zonal irregularities should be visible; the shadows must flow smoothly from one zone to another. The depth of shading is not great, and so delicate are the transitions from highlight to shadow that it is difficult to reproduce them faithfully in the drawings. A mirror having the same contrasts as depicted in Fig. 39 would be slightly overcorrected, possibly bordering on the maximum allowable tolerance. Every effort must be made to bring the figure within these tolerances, and, in addition, you should strive for the ultimate in performance by bringing your mirror as close to a paraboloid as patience and knife-edge measurement will permit.
next- testing at focus |
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