Instantaneous field singularities in electromagnetic waves

Singularities, i.e. places of discontinuity of physical parameters are extremely general objects appearing in both stationary and wave processes. They commonly occur in purely coherent, highly directional electromagnetic waves, such as laser beams, determining additional spatial or 'topological' properties of the beams independently of their propagational dynamics. For instance, a helicoidal structure of the wavefront curved around a line of undefined phase, so-called phase singularity, adds an orbital degree of freedom to electromagnetic waves. The phase singularities are typical to all types of scalar and one-component vectorial waves where the wave field can be defined only by intensity and phase distributions in space and time. The situation becomes more complex when the electromagnetic wave, as a vectorial object, depends in a different way on coordinates in its field components. This leads to a nonuniform field pattern containing singular points of an undefined instantaneous orientation of the electromagnetic field. In contrast to the phase singularities in a scalar wave, the points of instantaneous field (IF) singularities can be completely isolated by surrounding fields in 3D space at any fixed moment of time. Here we present a systematic description of the IF singularities of the electromagnetic waves in a paraxial approximation. Based on the IF distributions around the singularities, we provide a general qualitative classification of the wave beams. We also detail some of the common types of singularities where the transverse components of electric and magnetic fields have a form of the field sources and show the compensation mechanism of such vectorial field distributions by the longitudinal field components.