Synthesis and Analysis of Polyfocal Constrained Lenses

Abstract

Three-dimensional problems of synthesis and analysis of polyfocal constrained lenses are considered. It is shown that such lenses can have up to five focal points on each side. The problem of synthesis of a pentafocal lens in the general case is reduced to three transcendental equations and, of a quadrifocal lens, to two transcendental equations. In the case of a pentafocal lens with three focuses in the symmetry plane of the lens, forming plane wave fronts at the output, the synthesis problem is reduced to one transcendental equation; in all other cases of plane front formation, an analytical solution of the problem is obtained. An analytical solution is also obtained for the problem of synthesis of a quadrifocal lens with three symmetry planes. An analytical solution is also found for the problem of synthesis of a quadrifocal lens with two symmetry planes forming in the plane of symmetry four plane fronts with simultaneous fulfillment of the aplanatic conditions (Abbe sines) in the orthogonal plane. Examples of solving the synthesis problems with the optimization of parameters in order to minimize the mean square aberration of the eikonal are given. In the two-dimensional case, the dependence of the mean square aberration of the eikonal on focal distances is studied and it is shown that there is an optimal ratio of focal distances providing its minimum.