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In this paper, the input impedance and radiation properties of the L-probe fed thick-substrate patch antennas with finite ground plane are studied using the finite-difference time-domain method. For validation, the theoretical predictions are compared with measurement. Good agreement is obtained. Initially, the effects of ground plane size are investigated. For the case of foam substrate, the input impedance almost remains constant for different ground plane sizes as the foam substrate does not support any surface wave. The dependence of the gain on the ground plane size is found to be periodic as the radiation patterns strongly depend on the ground plane size. The ground plane is then modified to reduce the backward radiation, or to shape the radiation beam. The investigations are extended to cases of the antennas surrounded fully or partially by metallic sidewalls, including sidewalls on both of the E- and H-plane edges, sidewalls only on the E-plane edges, and sidewalls only on the H-plane edges, respectively. New results are obtained to improve the radiation properties of the antenna. Finally, the sidewalls are angled to shape the beamwidth to a larger extent.