Directional difference of the spin angular-momentum density and spin vectors in tightly focused bichromatic optical Lissajous beams

The spin angular-momentum density vector and the so-called spin vector reflecting the polarization properties of a light field are parallel for a monochromatic light field whose polarization ellipse at a point necessarily lies in a fixed plane. In contrast, the electric field of polychromatic light may, in general, evolve in three directions such that the average intensity in all of them is nonzero for any orientation of the reference frame. Consequently, the two vectors are no longer necessarily parallel. In this work we consider tightly focused bichromatic Lissajous beams and show that the spin angular-momentum density vector and the spin vector in such three-dimensional light fields may point in markedly different directions, almost orthogonal in some cases, and thus generally provide different information on the spin of light.