Vector Lissajous laser beams

We consider a new type of vector beam, the vector Lissajous beams (VLB), which is of double order (p,q) and a generalization of cylindrical vector beams characterized by single-order p. The transverse components of VLBs have an angular relationship corresponding to Lissajous curves. A theoretical and numerical analysis of VLBs was performed, showing that the ratio and parity of orders (p,q) affect the properties of different components of the electromagnetic field (EF) (whether they be real, imaginary, or complex). In addition, this allows one to engineer the imaginary part of the longitudinal component of the electromagnetic field and control the local spin angular momentum density, which is useful for optical tweezers and future spintronics applications.