If you are building a telescope or if you want to see if a telescope primary mirror has a good 'figure' before you buy it, you need to perform an optical test of the mirror.
A properly figured primary Newtonian mirror is a near-perfect parabola. How close to perfect does it need to be? A good telescope mirror has less than 1/4 wavelength of light error on its surface! This means that the surface imperfections should be smaller than 100 nm. This is a very small margin for error but amateurs telescope makers can achieve this and better if they are persistent and careful.
In order to test the figure of the primary mirror we must build the following:
Then I'll describe how to use these tools to determine the figure.
Step 1: Theory of Operation
The 100 nm accuracy of the surface requirement is a very tight tolerance and can not be measured directly using such things as rulers or depth gauges. These tight tolerances can indeed be achieved using normal hand tools by magnifying the surface angles. We can find the focus point of the mirror by shining a narrow slit of light onto the mirror and observing the shadows as a knife edge is passed in front of the returning light.
Let us assume that we have a perfectly shaped spherical mirror and our knife edge is near the focus point of the mirror.
See Figure A.
Looking down from the top, if we illuminate the mirror at the focus point the light will hit the mirror and return exactly to the focus point.
See Figure B.
If we move a knife edge inside the focus and on the right, the knife edge blocks the light returned from the right side of the mirror and does not affect the light from the left side of the mirror.
See Figure C.
If we move the knife edge outside the focus and on the right, the knife edge blocks the light returned from the LEFT side of the mirror.
So, when the knife edge is inside the focus, moving the knife edge left and right makes a shadow move across the mirror in the same direction as the movement of the knife edge.
When knife edge is outside the focus, moving the knife edge left and right makes a shadow move across the mirror going in the opposite direction of the knife edge.
When the knife edge is exactly at the focus point and the knife edge is moved from left and right, the mirror dims uniformly, i.e. the shadow doesn't appear to move left or right but the entire mirror dims.
A good telescope mirror does not have a spherical shape but a parabolic shape. For a parabola, the outer areas of the mirror focuses farther away from the inner areas of the mirror. We can therefor use this knowledge to determine the precise shape of the mirror by finding the focus points of each area of the mirror.