Based on the theory of paraxial approximation of beam propagation, the analytical expression of the intensity of the off axial radially polarized beam (OARPB) is derived and the effect of the off axial magnitude on the distribution of intensity of the OARPB is studied. Meanwhile, according to the definition of the first-order moment of centroid, the coordinate of centroid of the OARPB is derived and the variation of cenreoid of the OARPB is studied. Simulation result shows that the intensity distribution of the OARPB is different from that of the radially polarized beam. The intensity distribution of the OARPB is not uniform in the near-field. With increasing propagation distance, the beam spreads and the uniformity of intensity of the OARPB is improved gradually. However, the intensity distribution of the radially polarized beam keeps the form of symmetric doughnut spot during propagation all the time. When the off axial magnitude is small, the intensity distribution of the OARBP is obviously asymmetric in the near-field, and it becomes nearly symmetric while the beam propagates a certain distance. The smaller the off axial magnitude, the shorter the required propagation distance to become symmetric for the OARPB. When the off axial magnitude is larger, the hollow part of intensity distribution disappears, and the doughnut beam of the OARPB changes into a Gaussian beam spot gradually during propagation. On the other hand, the centroid of the OARPB does not change with increasing propagation distance. The value of the ordinate of centroid is equal to zero all the time. And the value of the abscissa of centroid is related to the beam size and the off axial magnitude. While the beam size increases, the abscissa of centroid increases linearly at the same time. When the off axial magnitude is small, the abscissa of the centroid of the OARPB increases with the increase of the off axial magnitude, nonlinearly and slightly; however, when the off axial magnitude is larger, the abscissa of centroid of the OARPB increases with the increase of the off axial magnitude, linearly and significantly.
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