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A compact bidirectional polarization splitting antenna (BPSA) composed of a patterned metallic structure coated by a thin dielectric film is proposed and theoretically investigated. Using a polarization-selective array of grooves, the backside illuminated light transmitted through the central nanoslit is split and coupled into TM- and TE-polarized modes supported by the metal-dielectric-air (MDA) configuration. The operation principle of the structure is clarified and theoretically illustrated by utilizing the fully vectorial aperiodic Fourier modal method (a-FMM). Numerical simulations show that insertion losses (ILs) less than 4 dB, polarization extinction ratios (PERs) better than 18 dB, and crosstalk (CR) less than -18 dB are achieved for both polarizations in the wavelength range 1510-1570 nm. The structure will find potential applications in highly integrated polarization diversity systems and photonic integrated circuit.