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It is assumed that the powers and movements of the components in the zoom system are known. One position of the zoom is designated the reference position, and the primary (third order) aberrations, including the spherical aberration and longitudinal chromatic aberrations of the pupil, of each component are used as independent variables. For the reference position and for a series of other zoom positions, paraxial marginal and pupil (principal) rays are traced. Initially these rays are traced assuming each component to consist of thin lenses in contact: subsequently they are traced through designed components. Using the data from these rays, the primary aberrations of each component in any of the other zoom positions are expressed as linear combinations of their values in the reference position. Special cases can arise, such as when the pupil in some other zoom position coincides with the object position in the reference position. These special cases lead to the need for four separate sets of coefficients, and a simple initial test indicates which set is to be used. In cases where a given component acts nearly as a field lens, the primary aberrations of the pupil for the reference position are used as independent variables in place of those of the image. The total primary aberrations of the whole system in the selected zoom positions are set equal to zero, or are given target values, and the resulting set of equations is solved by the method of damped least squares, using weightings for the overall aberration residuals and damping factors for the independent variables. For the initial, thin-lens, design there are only three independent variables (the spherical aberration, coma, and longitudinal chromatic aberration). The values of these needed for stable correction of the primary aberrations are determined, and the different components are then designed. The resulting thick-lens components will usually show greater variations of the aberrations on zooming than those predicted by the thin-lens solution. The formulae employed, which are perfectly general, are then used to obtain changes in the values of the primary aberrations of each component in the reference position (possibly with target-values for the overall primary aberrations in each zoom position if ray-tracing has revealed the higher order aberrations present) prior to re-design of the components. The technique is also useful in studying ways of improving an existing design in an economical way; for example, by imposing the condition that only one component may be subjected to re-design.
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