Helicity of EM Pulses with Gaussian Envelope

The intrinsic rotational content of electromagnetic fields can be described in terms of the helicity and its associated flow. The helicity density continuity equation is obtained here with the complementary fields procedure. This formalism allows for very general EM fields that are neither limited to monochromatic nor transverse fields. An important asset of this approach is that no temporal nor spatial averages are performed. This feature is shown using real fields in order to avoid the possible averaging due to complex conjugate products. Nonetheless, the helicity of a plane wave derived from the complementary fields is a time independent quantity. The plane wave solution can be modulated with a Gaussian function in time as well as its direction of propagation. In this way, it is possible to localize the wave in the propagation direction. Waves with Gaussian temporal envelope can be endowed with linear, circular or elliptical polarization. The time and propagation evolution of the rotational content of the wave will be monitored and described via the helicity density and its corresponding flow. The Gaussian pulse parameters are related to h, the elementary quantum helicity of the photon.