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A complete solution is given for the problem of constructing a plane stratified dielectric medium having that the property that at a fixed frequency and polarization a plane wave at any angle of incidence will be transmitted without reflection by the medium. This problem can also be interpreted quantum‐mechanically. The dielectric medium of the first interpretation becomes, in the second interpretation, a potential for the one‐dimensional Schrödinger equation such that a particle which is sent in from ‐ ∞ is transmitted with probability one of getting through to + ∞, no matter what the initial kinetic energy of the particle may be. In addition, the results can be applied to the problem of finding perfectly matched nonuniform transmission lines.