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考虑了当射频调制的超短脉冲电子束径向长度远小于辐射波长时，将其理想化为δ时间函数，提出了用波导本征模展开的方法来计算矩形波导中超短脉冲电子束的相干同步辐射及其频率特性。结果发现：①当谐振频率等于调制电子束微脉冲间隔的射频的整数倍时，辐射模式表现为“纯”的波导本征模。②波导效应使得超短脉冲电子束在两个频率处发生相干同步辐射，而且低频支的辐射功率高于高频立的。最后指出了波导自由电子激光器单横模运行的条件。
Considering that the radial length of the ultrashort-pulsed electron beam modulated by radio frequency is idealized as a function of δ when its radial length is much less than the wavelength of radiation, a method of waveguide eigenmode expansion is proposed to calculate the coherence of the ultrashort pulsed electron beam in a rectangular waveguide Synchrotron radiation and its frequency characteristics. The results show that: (1) when the resonant frequency is equal to an integral multiple of the radio frequency of the modulated electron-microbeam interval, the radiation mode shows a “pure” waveguide eigenmode. ② waveguide effect makes the ultrashort pulse electron beam occurs at two frequencies coherent synchrotron radiation, and low-frequency branch of the radiation power is higher than the high-frequency legislation. Finally, the conditions for the free-electron laser waveguide to run in a single transverse mode are pointed out.