Exact analytical theory of aberrations of centered optical systems

In the paper we present a new theory designed to overcome the deficiencies of the existing aberration theories. The main advantage is the theory’s capacity to perform analytical, direct, and accurate radius, and thickness calculations of aberration corrections for optical components and systems. Methodologically, we derive exact formulas for bilinear transformations from an object to an image space. Next, we introduce relative parameters in the matrix coefficients and by combining bilinear and matrix transformations we develop exact corrections on-axis aberrations and field aberrations such as distortion, astigmatism, and curvatures (including sagittal and tangential). The theory with an accuracy of 10^?8 mm (except for tangential and sagittal coma) allows precise radius and thickness calculation of components and systems for aberration corrections, sets exact coordinates of free-form surfaces, and enables the study of the surface and component capacities for aberration correction. The theory has supported the design of more than 10 different patented optical systems.