Results and Discussion

For an arbitrary electromagnetic wave propagating in a homogeneous medium, the curl of the energy flow density can be presented as (SI Appendix, section I)

The spin–momentum locking in an evanescent plane wave (Fig. 1A) as demonstrated in previous work (9) is a special, one-dimensional (1D) case of spin–momentum locking with the SAM vector aligned along the interface. Assuming the guided mode propagating along the y direction and evanescently decaying along the z direction, one can deduce the Poynting vector P = $$ \widehat{y}$$ωε/(2β)exp(−2k_zz) and the SAM S = $$ \widehat{x}$$εk_z/(2ωβ)exp(−2k_zz), where ε denotes the permittivity of the medium and β and ik_z stand for the in-plane and out-of-plane wave vector components, respectively. The energy flow density and SAM of the evanescent plane wave are connected through the generalized spin–momentum relation: S = ∇ × P/(2ω²) = ‒$$ \widehat{x}$$(∂P_y/∂z)/(2ω²). For structured evanescent modes with spatially varying intensity distribution, the inhomogeneity of energy flow density can induce several SAM components in different directions. The variation of the energy flow density in the z direction induces an in-plane component of the SAM, while its in-plane variations induce a z component. Both are perpendicular to the local energy propagation direction. The relationship between the two components leads to a chiral spin texture with spin vectors swirling around the energy flow (Fig. 1B). More importantly, its tendency of directional variation (i.e., the chirality) is locked to the momentum direction. This is a manifestation of the generalized optical spin–momentum locking associated with an electromagnetic evanescent wave.

Fig. 1.

Generalization of spin–momentum locking for structured guided modes. (A) In unstructured, plane guided wave, optical spin–momentum locking results in the transverse spin (S) uniformly distributed and parallel to the interface. The spin vector direction is perpendicular to the wave vector k and flips if the propagation direction flipped from +k to −k. (B) In an arbitrary structured guided wave, the optical spin is related to the vorticity of the energy flow density P. The transverse spin vector varies from the “up” state to the “down” state around the energy flow density, remaining perpendicular to the local wave vector. This forms a chiral swirl of the 2D transverse spin which is locked to the energy propagating direction and fulfills a right-handed rule. The direction of the local transverse spin vector flips if the energy flow density flipped from forward (+P) to backward (−P).

It should be noted that the transverse spin discussed here is different from the “spins” in conventional topological photonics, typically called a “pseudospin.” For a pseudospin, the spin–momentum locking is achieved by engineering the spin–orbit interaction in artificial photonic structures in order to break the time-reversal symmetry (8). In the case of the optical transverse spin of an evanescent wave, the generalized spin–momentum locking is an “intrinsic” feature of the spin–orbit interaction governed solely by the Maxwell’s theory. The nonzero spin Chern number for the structured waves (SI Appendix, section IV) implies the existence of nontrivial helical states of electromagnetic waves which are strictly locked to the energy propagation direction. However, since the topological $$ \mathrm{\mathbb{Z}}$$₂ invariant of these states vanishes owing to the time-reversal symmetry of the Maxwell’s equations, there is no protection against (back)scattering. Although the transformation of the two helical states of evanescent waves is not topologically protected against scattering, the spin–momentum locking and the induced unidirectional excitation and propagation are the intrinsic feature of the Maxwell’s theory and are topologically nontrivial, possessing the $$ \mathrm{\mathbb{Z}}$$₄ topological invariant.

To demonstrate the spin–momentum locking features described by Eq. 2, four types of the electromagnetic modes exhibiting evanescent field with inhomogeneous spatial energy distribution were investigated, including the solutions of a wave equation in Cartesian coordinates (Cosine beam) (34), in cylindrical coordinates (Bessel beam) (35), in parabolic coordinates (Weber beam) (36), and in Cartesian coordinates but with a parabolic path (Airy beam) (37) (SI Appendix, section V). The spatial distributions of their energy flow densities are shown in Fig. 2 A–D, while the beams’ propagation directions can either be forward (+P) or backward (–P). The corresponding cross-sections along the dashed lines are shown in Fig. 2 E–H and I–L for the beams with opposite propagation directions, respectively, together with the SAM distributions and the spin vector variations. For all four different types of the beams, the spin orientation varies progressively from the “up” state to the “down” state when the energy propagates along the forward direction (Fig. 2 E–H). The intrinsic spin–momentum locking present in evanescent waves ensures the topological protection in terms of spin vector swirl being completely determined by the energy flow density. Thus, to observe the reversal of the spin swirling from the “down” state to the “up” state, the propagation direction must be reversed (Fig. 2 I–L). This spin–momentum locking is preserved even for surface modes suffering from the ohmic losses (12), which influence only the intensity of the wave but not the orientation of photonic spin vector. Note that the spin vector has orientation along the interface at the maxima of the energy flow density and is normal to it at the nodes. Therefore, a period of spin variation can be defined between the two adjacent nodes of energy flow density which exhibits a similar feature to a topological soliton (38394041.–42).

Fig. 2.

Spin–momentum locking in various surface structured waves. (A–D) The spatial distributions of the energy flow density for different structured surface waves: (A) surface Cosine beam, (B) surface Bessel beam with topological charge l = ±1, (C) surface Weber beam, and (D) surface Airy beam. These beams can either propagate in the forward (labeled +P) or backward (labeled –P) directions. (E–H) Transverse SAM components Sz and Sx and the cross-sections of the energy flow density distributions along the green dashed lines in A–D for the beams propagating in the direction indicated with the arrow labeled +P. (I–L) The same as E–H for the beams propagating in the opposite direction indicated with the arrow labeled –P in A–D. The inserts at the top of E–L show the local transverse spin vector orientations (cf. Fig. 1). The spin vectors are swirling around the energy flow density and their local orientations vary from the “up” to the “down” states, fulfilling the right-handed rule. These orientations are seen inverted for the waves with the opposite direction of the energy propagation. Note that for the beams with curved trajectory, the spin variation is considered in the plane perpendicular to the local tangential direction of the energy flow density. The distance unit is the wavelength of light in vacuum.

In order to experimentally observe the spin–momentum locking features associated with the structured surface waves and out-of-plane transverse SAM, the experiments were performed on the example of SPPs (SI Appendix, section VII). SPPs were excited under the condition of a total internal reflection using a microscope objective with high numerical aperture (NA) = 1.49. Spatial light modulator and amplitude masks were employed to modulate the phase and wavevector of the excited SPPs to generate the desired plasmonic modes. A scanning near-field optical microscope, which employs a dielectric nanosphere to scatter the SPPs to the far field, and a combination of a quarter waveplate and a polarizer to extract the two circular polarization components (I_RCP and I_LCP) of the far-field signal, were used to measure the out-of-plane SAM component S_z = εβ²/[4ωk_z²(I_RCP ‒ I_LCP)]. The corresponding in-plane spin components were also reconstructed from the measurements (SI Appendix, section VIII). The measured distributions of the SAM components are shown in Fig. 3 and SI Appendix, Figs. S16–S19 for the four types of structured SPP waves propagating in the forward and backward directions. All the predicted SAM and spin–momentum locking features are observed experimentally: 1) the variation of SAM from the positive/negative state to the negative/positive across the beam profile and 2) the reversal of spin variation when inversing the beam propagation direction.

Fig. 3.

Experimental validation of the spin–momentum locking. The measured out-of-plane SAM components (S_z) for (A, E, and I) surface Cosine beam, (B, F, and J) surface Bessel beam, (C, G, and K) surface Weber beam and (D, H, and L) surface Airy beam: the spatial distributions of S_z spin components for the beams with (A–D) forward (+P) and (E–H) opposite (−P) energy propagating direction, (I–L) the cross-sections of A–H. The direction of the out-of-plane transverse SAM is seen inverted for the waves propagating in opposite directions. The distance unit is the wavelength of light in vacuum.

Since the kinetic momentum density can be divided into the spin (p_s) and orbital (p_o) parts, p = p_s + p_o, where p_s = ∇ × S/2, and obeys the spin–momentum relationship (Eq. 2), we can formulate a set of the Maxwell-like equations linking the transverse spin and the momentum density (and the Poynting vector) of evanescent electromagnetic fields (Table 1). This formulation provides comprehensive and intuitive understanding of the boundary conditions and the dynamical properties of the spin, momentum, and energy flow in guided waves (SI Appendix, section III). For example, the flip of the out-of-plane spin and the in-plane Poynting vector of a SPP wave across the metal/dielectric interface immediately follows from the boundary conditions due to the opposite sign of the permittivities on the different sides of the guiding interface. The same as variations of E field induces H field in the Maxwell’s equations, equation ∇ × p = 2k²S indicates that the spatial variations of the momentum/energy flow density induce the transverse SAM. In the same manner, equation ∇ × S = 2p_s = 2(p − p_o) tells us that the spin variation in turn contributes to the momentum/energy flow density, with the remainder provided from the orbital part (p_o). Consolidating spin–momentum equations results in an analog of a Helmholtz equation ∇²S + 4k²S = 2∇ × p_o, which describes spin–orbit interaction in evanescent waves, linking transverse spin and orbital part momentum density. In both the Helmholtz equation and the last Maxwell’s equation in Table 1, current J is an external source of magnetic field; similarly, in the corresponding spin–momentum equations, p_o, which determines the orbital angular momentum, influences the spin. Since an electromagnetic wave in a source-free and homogenous medium can be described with Hertz potential (Ψ) satisfying the Helmholtz equation, and the Poynting vector can be calculated from the Hertz potential as P ∝ (Ψ*∇Ψ−Ψ∇Ψ*) (43), one can obtain the spin and orbital properties of the electromagnetic guided waves directly from the spin–momentum equations without any knowledge on the electric and magnetic fields (SI Appendix, section VI).

Table 1.

Spin–momentum equations and the analogy to the Maxwell’s equations

Maxwell's equations	Spin–momentum equations
$$ \nabla \cdot \mathbf{E}=0$$	$$ \nabla \cdot \mathbf{p}=0$$
$$ \nabla \cdot \mathbf{H}=0$$	$$ \nabla \cdot \mathbf{S}=0$$
$$ \nabla \times \mathbf{E}=i\omega \mu \mathbf{H}$$	$$ \nabla \times \mathbf{p}=2k^2\mathbf{S}$$
$$ \nabla \times \mathbf{H}=\mathbf{J}-i\omega \epsilon \mathbf{E}$$	$$ \nabla \times \mathbf{S}=2\left(\mathbf{p}-{\mathbf{p}}_{\text{o}}\right)$$
Helmholtz equation
$$ {\nabla }^2\mathbf{H}+k^2\mathbf{H}=-\nabla \times \mathbf{J}$$	$$ {\nabla }^2\mathbf{S}+4k^2\mathbf{S}=2\nabla \times {\mathbf{p}}_{\text{o}}$$

Conclusion

We have demonstrated an intrinsic spin–momentum law which governs the transverse spin dynamics of guided electromagnetic waves. It was shown that the 1D uniform spin of surface plane wave evolves in a 2D chiral spin swirl for structured guided modes, providing a manifestation of the generalized photonic spin–momentum locking. Four types of structured surface waves, including the Cosine beam, Bessel beam, Weber beam, and Airy beam, have been investigated both theoretically and experimentally to demonstrate the concept of the generalized spin–momentum locking. Applying this spin-momentum locking, we obtained a set of spin–momentum equations that are analogous to the Maxwell’s equations and the boundary conditions. This optical spin framework can be used to evaluate the spin–orbit coupling in the electromagnetic guided waves and for designing specific transverse spin structures, without a priori information on the electric and magnetic fields. The generalized intrinsic spin–momentum features could also appear in other types of waves with evanescent field, such as fluid, surface elastic, acoustic, and gravitational waves. The effect could be of importance to the development of spin optics for quantum technologies and topological photonics.

Materials and Methods