Search for the optimal shape of the envelope of a microwave pulse of a powerful generator in the maximum permissible radiation mode

In the article, the problem of choosing the shape of the envelope of the microwave pulse of a powerful generator, which provides the maximum radiated surface antenna energy in the pre-breakdown mode, is formulated in the form of an isoperimetric problem of the calculus of variations, and its solution is obtained. Optimization was carried out at a fixed effective duration of the envelope. It has been established that the optimal shape depends on the effective pulse duration and is a truncated oval. As the pulse duration increases, the optimal shape tends to be rectangular. Analytical expressions are obtained that relate the effective duration and maximum allowable energy to the breakdown field for pulses with optimal, rectangular, Gaussian, and triangular envelopes. The dependences of the breakdown field and the maximum allowable energy of pulses with the indicated envelope shapes on the effective pulse duration are calculated in the range of realizable durations of powerful relativistic microwave generators. A comparative analysis of the obtained results is given.