With synthesis of antenna array you can not get so high direction in smallest dimension. I'm get wanted characteristic only with H-type of horn antenna (1x6 cm open end), but you need smaller than 6cm.
Try to do that with one gunnplexer on 24Ghz, this is resonator with gunn, pin and varactor diode (receiver, transmiter and mixer in resonator hole). On easy way you can comparing the phase. Standard WR-42(UG-595/U) flange is output open end.
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Sensitivity of gunnplexer is about -90dBm and 24Ghz is standard freq.