Fig. 3: Spin decomposition of an elliptically polarized plane wave.

a k-space representation of the plane waves, in which the wavenumbers confined within the circle \({k}_{x}^{2}+{k}_{y}^{2}={k}^{2}\) correspond to waves propagating in free space, whereas those outside the circle correspond to a single evanescent wave confined to an interface. The consistency of the elementary feature of L-spin for the same kind of wave-packets (Points “A” and “B”) leads to the classification of the T-spin into the normal helicity-independent component aligned perpendicular to the wavevector (the y-component) and the helicity-dependent component aligned anti-parallel to the wavevector. b, c Illustrations of the generic spin properties of right-handed and left-handed elliptically polarized evanescent waves propagating along the +x-direction. If the kinetic momentum is reversed, the T-spins (green arrows), including the helicity-independent T-spin (red arrows) and the helicity-dependent T-spin (yellow arrows), become opposite. Therefore, four spin-momentum locking states exist in a general EM system, consistent with the ℤ₄ topological invariance of the optical wave packet¹⁷.