The Twyman-Green interferometer is a very important instrument in modern optical systems for observing the surfaces of optics. It is essentially a Michelson interferometer with lenses to produce plane waves instead of spherical waves in the interferometer. Two example setups are shown in the following figure (from RP Photonics, which is technically my rival (can you have a rival with zero viewers?) but also you should check out the website!)
Now, let’s walk through each of these one by one. In the upper setup, the surface we are trying to inspect is the “inspected mirror” (who’d have guessed?), so the other mirror serves as a reference flat surface. We can see via tracing the path of the waves that if both mirrors are completely flat, we will have either a uniformly bright or dark screen at the image sensor depending on if the conditions for constructive or destructive interference are met (if the optical path length is exactly the same, we would have a bright screen). If neither condition is met, then we would have fringes because of the path length difference. We would not study the latter case, though, because the beauty of the Twyman-Green is that you can easily see if you have a disturbance on the surface of an object. For example, If we are trying to inspect a flat mirror, we are likely trying to determine if there is any surface roughness we need to take care of. In this case, if we have an interference pattern that shows some fringes (called Frizeau fringes) like in the following image, then there is likely some residual roughness on the mirror.
Now, if we are trying to study a curved surface, like in the second setup from the Figure above, we will need to put in a compensator lens with a curved surface such that the waves reflected back towards the detector are planar. If the two radii of the curved surfaces are exactly matched, then you could determine if there is anything left on the surface of the inspected in much the same way as in the first scenario. If, however, there is a difference between them, then you would observe rings like in the figure below.