If the aberrations of any centered optical system are known both for an object which intersects all rays transmitted by the system and also for the centre of the effective stop, the position in the image space of the emergent portion of a given incident ray is known, and the aberrations in the image of any other object for any stop position can be expressed in terms of those for the first object. The investigation aims to express the relations in the second case in terms of those present in the first when the objects are planes normal to the axis of symmetry whatever the order of the aberration may be. The investigation is based upon a convenient potential function, and the Eikonal, with the principal foci as reference points, is chosen as the standard function. First a distinction is drawn between aberrational and non-aberrational terms, and the discussion leads to the general form taken by the sine-condition in the outer parts of the field of view; it also becomes evident that there is only one stop aberration of each order to be considered, and that this is of spherical aberration form. The transformation is achieved by a change of variables, and to secure simple forms the aberrations of any order are arranged in series, each series forming a separate group as regards transformations. The rays in a plane containing the axis belong entirely to one series, which is called the zero series; in this series the coefficients in general must have finite values depending on the position of the object and of the stop if the aberrations are to be removed. In all the other series the coefficients should be zero for freedom from the aberrations, the conditions thus being independent of the positions of the stop and object. As the number of the series increases the expressions for the calculation of the coefficients become more simple. These facts are illustrated by the conditions for first aberrations, all of which are members of series 0 except the Petzval sum which belongs to series 1. The formulae obtained enable the aberrations to be expressed in terms of the coefficients of the standard Eikonal or vice versa , or alternatively the aberrations for any object and stop positions to be given in terms of those for standard positions. For use in the latter case certain advantages are secured by choosing for the stop the position where the magnification is + 1 and for the object the surface for which the magnification is - 1. As a very simple application the principles underlying the use of the formulae to obtain with the minimum of labour the effects of small variations in the constructional data of the system are illustrated.
No takes yet. Share an insight, caveat, or question.
T Smith (1922) studied this question.