The Fundamental and Second Harmonic

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 practical when the mode at w = w is a higher C order mode (m or p large) and/or the beam position is at a higher radial maximum. Several specific examples are considered. Second, the conditions are derived for oscillation of a parasitic mode at W = W in a gyrotron designed for operation at W' = 2w . Again, c c this problem is more severe for higher order modes and beams at larger radii. For a whispering gallery mode TEm'l1 oscillating at W' = 2w , if m' is odd, there is a resonant parasitic whispering c gallery mode TEmll at w = WC, with m = 0.5 (m'1). This would be expected to spoil the 2w operation of all TEm 1 modes, with m' odd.