Based on field-matching theory, the general transmission matrix of pill-box windows and a new concept of hybrid transmission modes pill-box window were presented to extend the bandwidth of output window. With guidance of the new concept, the primitive structure and parameters of the extremely broad bandwidth pill-box output window, working over 34~52GHz, had been obtained. Afterward, HFSS was employed to optimize the parameters of the window and to examine the optimized mode. Through amount of thermodynamic analysis and optimization design, the extremely broad bandwidth pill-box window, with average power capacity 10kW over 40% bandwidth whose insert loss is less than －18dB, was attained. The cold test verifies the results of the design.