A complete set of normal modes in three-layered electron waveguides

Mathematical expressions are given for guided and radiation modes including nonpropagating parts in three-layered electron waveguides, in addition to gaining a clear understanding of the difference between electron and optical waveguides in terms of normal modes. According to the dispersion diagrams, it is shown that a complete set of normal modes for electron waveguides depends on the electron´s total energy and that in the quantum-well-type waveguide the discrete modes and the continuous modes radiating to one side of the film are reduced to nonpropagating modes in a certain energy range, whereas a complete set of normal modes for optical waveguides always consists of three types of modes: guided modes, substrate radiation modes, and substrate-cover radiation modes. In addition, the discrete modes and the continuous modes radiating to one side of the film are not reduced to nonpropagating modes in the entire range of frequencies. Also, it is shown that the guided modes always exist in electron waveguides composed of arbitrarily different Al-concentration ratios in the film, substrate, and cover regions, and a new waveguiding quantum-step-type structure, which utilizes the total reflection at both the potential rise and drop is proposed, whereas in optical waveguides, the guided modes exist only in the structure in which the refractive index of the film is higher than those of the substrate and of the cover