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Analytical vectorial structure of non-paraxial four-petal Gaussian beams in the far field

The analytical vectorial structure of non-paraxial four-petal Gaussian beams(FPGBs) in the far field has been studied based on vector angular spectrum method and stationary phase method. In terms of analytical electromagnetic representations of the TE and TM terms, the energy flux distributions of the TE term, the TM term, and the whole beam are derived in the far field, respectively. According to our investigation, the FPGBs can evolve into a number of small petals in the far field. The number of the petals is determined by the order of input beam. The physical pictures of the FPGBs are well illustrated from the vectorial structure, which is beneficial to strengthen the understanding of vectorial properties of the FPGBs.