Far-field properties in two off-axis superimposed Laguerre-Gaussian beams beyond the paraxial approximation

The analytical far-field expressions for the TE and TM terms and energy flux distributions in two off-axis superimposed nonparaxial Laguerre-Gaussian beams with azimuthal and radial indices l(1) = -l(2) = +1, p(1) = p(2) = 0 are derived and used to study the far-field properties including phase singularities and energy flux distributions of the resulting beam, where our main attention focuses on the dependence of phase singularities on the controlling parameters such as the off-axis distance, relative phase, amplitude ratio, and waist widths of superimposed beams, and the symmetry property of edge dislocations and energy flux distributions. The results are interpreted and compared with previous work.