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One of the most popular methods for high precision lens assembly of an optical system is using an autocollimator and a rotation stage. Some companies provide software for calculating the state of the lens along with their lens assembly systems, but the calculation algorithms used by the software are unknown. In this paper, we suggest a calculation method for lens alignment errors using ray transfer matrices. Alignment errors resulting from tilting and decentering of a lens element can be calculated from the tilts of the front and back surfaces of the lens. The tilt of each surface can be obtained from the position of the reticle image on the CCD camera of the autocollimator. Rays from a reticle of the autocollimator are reflected from the target surface of the lens, which rotates with the rotation stage, and are imaged on the CCD camera. To obtain a clear image, the distance between the autocollimator and the first lens surface should be adjusted according to the focusing lens of the autocollimator and the lens surfaces from the first to the target surface. Ray propagations for the autocollimator and the tilted lens surfaces can be expressed effectively by using ray transfer matrices and lens alignment errors can be derived from them. This method was compared with Zemax simulation for various lenses with spherical or flat surfaces and the error was less than a few percent.
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