Consider any two optical systems which are axially symmetrical and composed of spherical surfaces. A general contribution formula is derived relating any tangential ray in one system with any tangential ray in the other system. As special cases of the general formula, an exact spherical contribution formula, an exact coma contribution, an exact contribution formula for the final image height, and others are derived. Exact spherical and coma contributions should be particularly useful for designing systems of very high aperture and small field, where zonal aberrations demand special consideration. The computing formulas are especially adapted to a well-known ray-tracing scheme.
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