Figuring and Testing The Mirror
Making a 4.25 Inch Dobsonian Reflector Telescope
Once your mirror is polished, you need a way to test it in order to adjust or figure the mirror. Figuring is done by polishing the mirror in a way that guides it toward an optically perfect parabaloid. At this stage, the size of the mirror and the F number will become important. If you have a small mirror and/or a large F number, your job may already be done. There are a few different ways to test your mirror.
The Star Test
To do a star test, you do not actually need a star, however you will need to complete the assembly of your telescope. This is because you will need to point it at something in order to evaluate the image you are seeing. Instead of a star, you can point your scope at a telephone pole insulator that is positioned to reflect the sun. You may also use a bright star. The technique is to defocus the telescope and look at the fuzzy disk. If the disk is uniform, then you have a fairly well figured mirror. If it is uneven, the location of the dark and light rings tell you what you need to do to correct it. I cannot tell the details. I think this is a reasonable method for a small mirror or one with a large F number.
The Ronchi Tester
A Ronchi tester is very similar to the Foucault tester that I will describe below. The difference is that instead of a knife edge, the Ronchi tester uses a grating. The image you get consists of curved dark and light lines on the surface of the mirror. The catch is that you need to know what those lines should look like for your specific mirror. I'm told that there is free software on the web to answer this question.
A Ronchi Eyepiece
Yes, you can buy an eyepiece with a grating. While probably not as sensitive as a Ronchi tester, there is one advantage to using an eyepiece. You are looking at a star at infinity so the pattern you are looking for is always the same, straight parallel lines. I think this method is probably as good as the star test, at the disadvantage of needing a special piece of equipment.
The Foucault or Knife Edge Tester
Building your own Foucault tester is not all that difficult. There are many designs you can find on the web. The basic tester needs two axes to move a small table. The forward/backward movement is crucial. Usually a long bolt with threads of known size is used. By turning the bold, you can move the table a very precise known distance. The other axis is used to move back and forth to find a null point. Other components you need are an LED for illumination and a straight edge razor for the knife edge.
I'm only going to give a brief overview here. I recommend you check out a good resource on this subject. One I found useful was "Understanding Foucault" by David A HarBour.
Consider what would happen if you had a pin point light source at the focal point of your spherical mirror. The light from the pinpoint would bounce back from the mirror and cross at the original source. Now if you were to move this pin point slightly off center, the light would again converge at a point, but one that was accessible. Now imagine you are trying to find this point by standing behind it, and placing a knife across the light path. If the knife edge is between you and focal point, you will see a dark shadow cross in the same direction as the knife edge is moving. If the knife edge is behind the focal point, you will again see a shadow, but it will move in the opposite direction. So by moving the knife edge in and out, and then running it across your sight line, you can find the focal point. The closer you are to this point, the faster the shadow moves. When you are very very close to the point, instead of seeing a shadow, the image of the mirror surface will go dark all of a sudden.
That's the basic idea of finding the null point of a spherical mirror. The catch is, you don't want a spherical mirror, you want one in the shape of a parabaloid. Remember that the sphere and the parabola are already very close to being one and the same. You can model a corrected parabaloid by thinking of the mirror as divided into a number of rings called zones, which each is a sphere, but at a slightly different focal length. You put a mask over the mirror called a Couder mask, seen in the photograph above, and then find the null point for each zone. There is free software available on the web to create Couder masks.
Having taking careful measurements, you now can compare them to a set of calculated measurements. The difference tells you which rings of the mirror need to be worn down more by polishing.