A formalism is developed for determining the possibility of operation of a gyrotron at both the fundamental, w = wc, in a TE mode and at the second harmonic, w' = 2wc, in a TE , mode. This formalism is then applied to two physical problems. First, the conditions are derived for conversion of a gyrotron from operation at w = w to operation at w' 2w . Such a conversion c c is shown to be most pra...