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The limitations on the performance of a reversible lens are analysed by mock ray tracing. To al the design of reversible lenses is a theme marred by false notes. Maxwell proved proved that "it is impossible, by means of any combination of reflections and refractions, to produce a perfect image of an object at two different distances, unless the instrument be a telescope," a telescope which must have unit angular magnification. T. Smith 2 as well as Herzberger3 found flaws in Maxwell's proof, but showed that his results were correct. Maxwell's proof does not depend on the symmetries of the system; lenses with particular symmetries were studied in detail by Buchdahl. He also comes to the conclusion that a lens such as the conference lens cannot be free from aberrations. From a practical standpoint this is not the end of the matter. How serious are the imperfections? The residual aberrations might be smaller than the unavoidable diffraction effects and therefore of no consequence. To the best of our knowledge the existing literature does not provide numerical estimates for the best image quality that can be realized. In this paper we begin to address this question; we show how a judicious mixture of analytic and computational techniques reduces the complexity of the problem to a manageable size. It turns out that for the conference lens the theoretical limitations are inconsequential as long as we allow the distortion to be large. If, however, the lens specifications are made more demanding, the fundamental restrictions become at times a serious matter, as we shall show by an example. One approach to the calculation of the residual aberrations is a frontal attack by algebraic methods in the style of Smith and Buchdahl. This method has the advantage that the relations between the aberrations and the design parameters can be expressed in analytical form. However, it usually remains unclear how far the series expansions should be extended, and the amount of algebra required is formidable because we must not only determine the unavoidable aberrations, but also balance them as best we can by choosing the optimum values for other aberrations that we do have under our control. We have chosen an alternative technique, mock ray tracing, 5 which consists of tracing rays through a lens specified by one of its eikonal functions rather than by its curvatures, thicknesses, and indices. As the physical symmetries of a lens can be easily translated into mathematical symmetries of the eikonal, this method is well suited to the problem at hand.
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