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The reflection coefficient phase is studied for four different artificial magnetic conductors (AMCs) having canonical frequency selective surface (FSS)-type two-dimensional periodic structures to be used as back reflectors for an aperture antenna. The bidirectional circularly polarised (CP) radiation of the octagonal-shaped aperture (OSA) antenna is made unidirectional using these AMC surfaces as ground planes. The antenna height measured from the upper surface of AMC reflector to the OSA radiator is chosen to be small to realise low-profile antenna: 0.0825λo at the lowest analysis frequency of 4.5 GHz. Different antenna parameters like voltage standing wave ratio (VSWR) of 2, 3-dB axial ratio (AR) bandwidth, gain, and front-to-back ratio are studied and compared for these four AMCs and the conventionally used perfect electric conductor (PEC) ground plane as back reflectors. Four different aperture shapes with fixed aperture perimeter are designed to integrate with the square-loop AMC as back reflector to realise a low-profile unidirectional wideband CP aperture antenna. Hexagonal-shaped aperture antenna over the square-loop AMC shows the largest measured 3-dB AR bandwidth of 23.33% (5.65-7.05 GHz), VSWR of 2 bandwidth of 36.67% (5.16-7.36 GHz), and the gain of around 7 dBic over the band for overall antenna volume of 0.72λo × 0.60λo × 0.19λo at 6.0 GHz.