It is significantly challenging to protect the power source against the reflected power when standing-wave rf structures operate, in particular under overcoupled conditions. Conventionally, nonreciprocal circulators, usually operating under pressurized $S{F}_6$ atmosphere, are employed to isolate the power source from the reflected power. However, breakdowns, rf losses, and thermal stress make these devices unusable in rf structures operating at high frequencies (6–12 GHz) and can limit the maximum power or the repetition rate, all of which are critical aspects for next-generation accelerators or rf photoinjectors, as an example. Furthermore, the use of $S{F}_6$ gas is becoming increasingly problematic due to its environmental impact. In this paper, two novel rf networks are proposed to compensate reflections from standing-wave structures, thus avoiding the use of circulators. They are based on specifically designed hybrids that, combined with rf structures, allow to strongly reduce the reflected power. The design of hybrids with arbitrary power ratios between different ports constitutes a multiobjective optimization problem. To address this challenge, a new optimization methodology is proposed to simplify the optimization process into a single-objective problem. These novel rf networks are then applied to a new C-band photoinjector recently developed at the Italian National Institute of Nuclear Physics INFN (Frascati, Rome, Italy) and to a new proposal of X-band rf photoinjector for National Synchrotron Radiation Laboratory (NSRL).

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• Radio frequency calculations
• Room temperature RF
• Electron sources
• Methods in electromagnetism