Strong spin–orbit interaction of photonic skyrmions at the general optical interface

1 Introduction

Light carries spin angular momentum (SAM) and orbital angular momentum (OAM) [1], [2], [3], [4], [5], [6]. The SAM is associated with the ellipticity of polarization [1], and the OAM is determined by the vortex phase or the trajectory of optical beam [2], [3]. The interplay of these two angular momenta results in many fascinating phenomena [7], [8], [9], including spin–orbit coupling and conversion [10], [11], [12], the spin Hall effect of light [13], [14], [15], [16], [17], [18], the optical Magnus effect [19], [20], [21], the Coriolis effect [22], [23] and the Aharonov–Bohm effect [24]. In particular, the strong spin–orbit interactions between the evanescent waves and interfaces in the optical near-field lead to the discovery of the optical transvers spin [25], [26], [27], [28], [29], [30], [31], [32], [33] and the photonic analog to the quantum spin Hall effect [34], [35], [36], which have potential in the applications of directional scattering and transportation of photons, nanometrology [37], [38], [39], chiral quantum optics [40] and spin-based robust optical surface [41], [42], [43], [44]. Very recently, by considering the spin and complex vectorial properties of surface electromagnetic wave, a photonic skyrmion [45], which has chiral spin texture in line with the topological soliton named from the physicist Tony Skyrme, was discovered and attracts wide interests in the field of spin and topological optics. Most of the relevant research until now has focused on the topological number and the experimental measurement of photonic skyrmion [46], [47], [48], [49], [50]; and its spin and orbital features still need to be explained explicitly.

Here, we give an insight into the spin and orbital features of photonic skyrmions by analyzing the decomposition of energy flow density (EFD) and momentum density (MD) into the spin and orbital parts. Firstly, we investigate the energy flow densities and spin textures on both sides of an optical interface and find the EFD and skyrmion number reversal on the two sides. Secondly, we study the chirality feature of photonic skyrmion and find that their time-averaging chirality density vanishes; hence this spin texture is constituted by the optical transverse spin. In addition, we discuss and demonstrate the spin-dependent force, the topological origins and the resulted spin-momentum locking of photonic skyrmions. Finally and more importantly, we demonstrate that the physically meaningful equations of continuity for the EFD and MD lead to four ‘spin–orbit interaction’ terms, which are in line with electric-magnetic terms of the chirality interactions [51], [52], [53]. For photonic skyrmions constructed by the p-polarized surface waves, the electric polarizability related term governs the spin–orbit interaction of surface waves and results in the EFD reversal and an inverted out-of-plane orientation of spin texture through the optical interface. If a variation of permeability is introduced into the material in lower half-space, the magnetic susceptibility related term would lead to the MD reverse and the accompanied in-plane orientation of spin texture through the optical interface. This theoretical analysis, which is thorough and can be extended to the other classical waves, helps in understanding the spin–orbit features of photonic skyrmions and in constructing further optical manipulation, sensing, quantum and topological techniques.

2 Spin and orbital features of photonic skyrmions

Here, we firstly consider a p-polarized evanescent wave propagating at the z = 0 interface between an isotropic dielectric (in the z > 0 half-space with real permittivity ε₊ and permeability μ₊) and an isotropic medium (in the z < 0 half-space with real permittivity ε₋ and permeability μ₋). The corresponding results for the s-polarized evanescent wave at a general interface, including the dispersion relation, field distributions and spin–orbit features, can be found in Supplementary materials II, IV and VI, respectively. The time-averaged Poynting vector Π of this Bessel-type evanescent wave in cylindrical coordinates (r, φ, z) with unit vector (rˆ,φˆ,zˆ) is [54]

The EFD propagates in the azimuthal direction and decays exponentially with factor Ψ=1/e±2κ±z in the z-direction, which results in the vanishing of out-of-plane EFD. Here, E(r) and H(r) denote respectively the electric and magnetic field; A is the normalized complex amplitude; J_l is the l-order Bessel function of first kind; the superscript ∗ indicates the complex conjugate operator of a vector; the symbols ‘+’ and ‘−’ indicate physical quantities in the z > 0 and z < 0 half-spaces; ω is the angular frequency; β is the in-plane propagation constant of the evanescent wave; ±iκ_± denotes the out-of-pane wave vectors, which satisfy the dispersion relation ω2ε±μ±=β2−κ±2 (Supplementary material I). All the quantities, including β, ω, κ_±, ε_± and μ_±, are considered as real numbers throughout the paper. The time-dependent quantity e^−iωt is ignored because only the time-averaging properties are considered here.

Generally, a Bessel-type surface wave can be excitated by a focused circularly polarized light (carry longitudinal spin) with an additional vortex phase at a dielectric/metal interface. The imaginary characters of the field components arise from the transversality condition ∇·D = 0 [55], which results in the ∓π/2 phase difference between the field in propagating direction E_φ and two transverse direction E_r, E_z. Thus, the electric field vectors rotate in the (φ, z) and (r, φ)-planes simultaneously. In other words, a Bessel-type evanescent wave is elliptically polarized in the two propagation planes. The SAM in a complex medium can be expressed as [56], [57], [58]

where ε′±=∂[ωε±(r,ω)]/∂ω and μ′±=∂[ωμ±(r,ω)]/∂ω are respectively the group permittivity and permeability; the contributions of electric Φ^E and magnetic Φ^H parts to the SAM, which are essentially related to the polarization ellipticities, are

and

respectively. Here, the last term of Equation (2) is stemmed from the dispersion and η±=ε±′/μ±′, Z±=μ±/ε±. The SAM components in the upper and lower half-spaces can be seen in Figure 1(a–f). For a dielectric material in the upper half-space, the out-of-plane and radial SAM components are positive at r = 0 for the topological charge l > 0. Thus, the spin state is ‘up’ in the center, and varies from ‘up’ state to ‘down’ state gradually as shown in Figure 1(g) and (j). Whereas in the lower space with permittivity ε−′ < 0 and permeability μ−′ > 0, the out-of-plane SAM component reverses, which makes the spin texture reversal as shown in Figure 1(h) and (k). In addition, if there is a magnetic material with ε−′ > 0 and μ−′ < 0, the out-of-plane SAM component is unchanged and the radial SAM component reverses due to the evanescent property in the lower half-space. Thus, although the spin state also varies from ‘up’ state to ‘down’ state gradually, the chirality of spin texture is reversal owing to the inversion of radial SAM component as exhibited in Figure 1(i) and (l).

Figure 1:

The spin properties of p-polarized Bessel-type surface mode with topological charge l = +2. The radial spin angular momentum (SAM) (a) and out-of-plane SAM (d) components of surface mode in the upper half-space (at z = 10 nm) as the relative permittivity and permeability are 1; the corresponding 1D contour and normalized spin texture can be found in (g) and (j), respectively. The radial SAM (b) and out-of-plane SAM (e) components of surface mode in the lower half-space (at z = −10 nm) as the relative permittivity and permeability are respectively −11.8 and 1; the corresponding 1D contour and normalized spin texture can be found in (h) and (k), respectively. The radial SAM (c) and out-of-plane SAM (f) components of surface mode in the lower half-space (at z = −10 nm) as the relative permittivity and permeability are respectively 11.8 and −1.5; the corresponding 1D contour and normalized spin texture can be found in (i) and (l), respectively. The sign of out-of-plane spin orientation is determined by the permittivity, while the direction of in-plane spin orientation is related to the permeability. The wavelength is 633 nm and all quantities are normalized to the maximal absolute value of total SAM in the upper half-space.

Recently, it is widely accepted that the electric field vector in the upper half-space has a Néel-type skyrmion-like distribution for the topological charge l = 0 [48], whereas for the topological charge l ≠ 0, the spin texture of this Bessel-type evanescent field is analogous to that of a magnetic skyrmion [45]. Here, we only discuss the skyrmion-like spin texture (l ≠ 0). Since the spin vectors only have radial and out-of-plane components and depend on the radial coordinate r, the directional unit vector of SAM can be converted to sr±=Sr±/|S±|=cosΘr± and sz±=Sz±/|S±|=cosΘr± with |S±|=Sr±2+Sz±2. Simple derivation yields the skyrmion number of the normalized SAM:

Note here that the integral region C, which is defined by the centroid of beam r = r_i and the boundary r = r_i+1, is circular symmetrical and chosen precisely from the ‘up’ to ‘down’ states of spin vectors. These interesting results indicate the skyrmion-like spin textures are determined by the signs of vortex topological charge and permittivity. As the sign of permittivity reverses in the lower half-space, the spin vectors also reverse, which can be intuitively seen in Figure 1(j–l).

Through these Equations (2)–(4), we can recognize the spin properties of photonic skyrmions in three aspects. Firstly, this great electric-magnetic asymmetry of SAM is originated from the breaking of the inverse symmetry owing to the presence of the optical interface. Naturally, the spin Chern number of photonic skyrmions equals 4 (exactly speaking, +4 in dielectric material, −4 in left-handed material and imaginary in non-Hermitian system; see the derivations in Supplementary material IX), and hence the optical modes are double-degenerate and there would be two pairs of surface modes (s-mode and p-mode). However, the existence of an interface between media with different permittivity and permeability breaks the dual symmetry between the electric and magnetic features and only one polarized state of surface evanescent modes is allowed due to the breaking of the polarization degeneracy [59], [60], [61], [62], [63]. Thus, photonic skyrmions are governed by the topological origin of nontrivial spin Chern number and suffer from spin-momentum locking. Secondly, there are radial and out-of-plane components exist for this Bessel-type evanescent waves, which are corresponding to the rotation of electric field vectors in the (φ, z) and (r, φ)-planes, respectively. Moreover, the directions of SAM are perpendicular to the propagating direction of EFD, which indicates that the SAM of these photonic skyrmions can be regarded as transverse spin.

Thirdly, we now explore an important connection between the spin and chirality of the surface wave to explain this transverse feature of spin. Similar with the energy density and EFD Π, one can derivate that the chirality density and chirality flow Ʃ also satisfy a continuity equation, which can be utilized to characterize the chirality or spin properties of the electromagnetic field [53], [64], [65]. In a homogeneous medium, the time-averaged chirality density and chirality flow can be written as

Here, n±′=∂[ωn±(r,ω)]/∂ω is the group refractive index. Substituting here the Bessel-type field distributions (Supplementary material III), we immediately reach

Thus, the chirality density vanishes since Im{H*·E}=0 in the whole space independent of the optical interface. A nonzero chirality flow is determined by the ellipticities of the field polarization stemming from the transversality condition. To further understand helical properties of photonic skyrmions, one can compare these results to those of the propagating fields in free space. For the propagating fields, the integral of chirality density in a transverse plane is proportional to the averaged helicity of photons, which is determined by the longitudinal spin: ⟨Sˆ⋅pˆ⟩/|n|k0 [31]. Here, Sˆ and pˆ are the spin and canonical momentum operators; k₀ is wave vector of photons in vacuum; |n| is the absolute value of refractive index; the Dirac notation denotes the inner product. Obviously, the vanishing of the chirality density definitely indicates that the SAM is orthogonal to the momentum and can be regarded as transverse spin universally. On the other hand, the chirality flow is proportional to the SAM [26]:

where v±′=1/ε±′μ±′. Thus, although the chirality density vanishes, the appearance of ‘local’ chirality flow can induce helix-dependent torque, which has potential in the applications of optical spinning and manipulation of nanoparticles [66], [67], [68], [69], [70].

To demonstrate this helix-dependent mechanical effect, the optical torque experienced by the particle in the upper half-space can be written as [71], [72]:

Here, p = α^eE (m = α^hH) is the electric (magnetic) dipole induced by the electric (magnetic) field of the incident electromagnetic wave with α^e and α^h the electric and magnetic polarizabilities. The optical torque for photonic skyrmion is obviously helix-dependent. Note here that the SAM and the optical torque are perpendicular to the propagating direction of photonic skyrmion, which indicates the transverse property of SAM and optical torque. In addition, the total optical force experienced by the particle can be written as (Supplementary material VII):

where ε^ijk is the Levi-Civita tensor, and i, j, or k stands for either x, y or z. For simplicity, we only consider the nonmagnetic spherical Rayleigh particle and ignore the crossover term in Eq. (10). The azimuthal optical force (along the energy propagation direction) is depended by the longitudinal spin (helix) of the incident light. For example, owing to the conservation law of total angular momentum, the photonic skyrmions of ±1 order can be excitated by a circularly polarized light with helical degree σ = ±1. Thus, parts of the longitudinal optical forces are also helix-dependent.

3 Spin and orbital features of photonic skyrmions

Finally, we investigate the spin–orbit coupling in the optical interface. The Poynting vector determines the spin flow density (SFD: P_s) and orbital flow density (OFD: P_o) of electromagnetic wave in complex medium (Supplementary material V):

The SFD is determined by the vorticities of the polarization ellipticities ΦE and ΦH, whereas the OFD is essentially related to the canonical momentum of a single wavepacket Po±∝p±〈ψ′±|pˆ|ψ′±〉/ℏω with pˆ=−iℏ∇ the momentum operator and |ψ′±〉=[ε′E±,iμ′H±] the photon wave function and ℏ the reduced Planck constant (Supplementary material V). The distributions of EFD, OFD and SFD in the upper and lower half-spaces with vortex topological charge l = +2 can be found in Figure 2. Naturally, the EFD is determined by sgn(l/ε) and the OFD is related to sgn(l/μ′). Thus, the EFD and OFD are parallel in the dielectric material, whereas there are antiparallel in the noble metal or negative magnetic right-handed materials. Moreover, in the left-handed material, there are transformed into parallel. In addition, the SFD is always antiparallel to the OFD, which makes the group velocity of light subluminal.

Figure 2:

Energy flow density (EFD), orbital and spin flow densities of p-polarized Bessel-type surface mode with topological charge l = +2. The azimuthal EFD (a), orbital flow density (OFD) (b) and spin flow density (SFD (c) of surface mode in the upper half-space (at z = 10 nm) as the relative permittivity and permeability are 1; the azimuthal EFD (d), OFD (e) and SFD (f) of surface mode in the lower half-space (at z = −10 nm) as the relative permittivity and permeability are respectively −11.8 and 1; the azimuthal EFD (g), OFD (h) and SFD (i) of surface mode in the lower half-space (at z = −10 nm) as the relative permittivity and permeability are respectively 11.8 and −1.5. The EFD is determined by the sgn(l/ε), while the OFD is related to the sgn(l/μ).The wavelength is 633 nm. All quantities are normalized to the maximal absolute value of energy flow in the upper half-space.

It is worth noting that, in the complex medium, the relation Π = P_o + P_s does not satisfy because there will be an addition dispersive term existing:

and

where the superscript “D” indicates the dispersion. Thus, there must be

Moreover, the dispersive terms also play critical roles in the spin–orbit interaction. Thus, one can summarize the contributions of spin and orbital parts to the EFD as: Πs=Ps+PsD and Πo=Po+PoD. As in the plane wave cases given in a study by Bliokh et al. [26], the decomposition of EFD into spin and orbit parts works well in a homogeneous medium, but in the presence of inhomogeneity, the spin and orbit parts acquire nonzero divergences, ∇⋅Πs=−∇⋅Πo≠0, which does not make physical sense since ∇⋅Π=0. One can modify the separation of the spin and orbital parts of EFD to satisfy ∇⋅Πo′=∇⋅Πs′=0. In this manner, we obtain Πs′=Πs−Ξ and Πo′=Πo+Ξ with Π=Πo′+Πs′:

These quantities Ξ^HE and Ξ^HE arise from the optical potential and can describe the spin–orbit interaction, which vanishes in a homogeneous medium but would play a critical role in optical interfaces. Owing to the abovementioned polarization properties of the Bessel-type surface waves, the electric and magnetic field ellipticities do not vanish. In our case, the inhomogeneity introduced by the optical interface is along the z-direction, which means that ∇(1/ε) and ∇(1/μ) only have out-of-plane components. Thus, the term Ξ^EH ∝ ∇(1/ε) × Φ^H always vanishes for the p-polarized modes, whereas the other term Ξ^HE ∝ ∇(1/μ) × Φ^E is along the azimuthal direction, which results in the strong spin–orbit interaction and even the reversal of EFD. From Eq. (9) with fixed vortex topological charge l, it can be found that the direction of Poynting vector is determined by the permittivity ε. If the ε₋ has an opposite sign with the ε₊ (such as surface plasmon polaritons at the air/metal interface), the direction of EFD in the z < 0 half-space is reversed comparing to that in the z > 0 half-space, while the reversal of permeability would not affect the energy propagating direction. However, this term Ξ^HE is intensely determined by the gradient of permeability. It is nonlogical because there also would be spin–orbit coupling for the nonmagnetic surface mode, for example, the photonic skyrmion suffers from great twisting and the skyrmion number reverses through the optical interface with invariable permeability. Thus, we introduce another two physical quantities to characterize these permittivity or permeability depended spin–orbit interaction.

The Poynting vector determines the density of the MD of the field [73]:

In our case, for the Bessel-type surface wave, the direction of MD is definitely determined by the permeability μ_± from the Equation (17). The sign of MD need not reverse through the nonmagnetic optical interface and hence it is continuous ∇·π = 0. As acted in the homogeneous space, the MD can be decomposed into the contributions of spin and orbital parts. Note here that we do not follow the analysis above in the EFD because it is more concise to employ the spin and orbital parts of MD instead of defining spin and orbital momentum densities or dispersion related spin and orbital momentum densities to analyze the spin–orbit terms. Note here the spin parts and orbital parts are related to the decomposition of EFD into the spin and orbital parts by

These two quantities suffer from nonzero divergences: ∇⋅πs=−∇⋅πo≠0, which do not make physical sense since ∇⋅π = 0. Here, we modify the separation of the SMD and OMD, π=π′o+π′s, such that ∇⋅π′o=∇⋅π′s=0. In this manner, we obtain π′s=πs−Ξ′ and π′o=πo+Ξ′ with

It is worth noting that we utilize a factor 1/εμ to unify the dimensions of physical quantities Ξ and Ξ′. For the p-polarized surface waves, due to the discontinuity of ∇ε at the optical interface, strong counter-propagating boundary spin and orbital energy flows arise there. By examining the spin–orbit interaction terms given by Eqs. (16) and (20), we can indicate that Ξ^EE and Ξ^HH are the electric polarizability and the magnetic susceptibility related spin–orbit interaction terms, respectively; Ξ^HE and Ξ^EH are the mixed electric-magnetic terms. Meanwhile, the boundary EFD is

where δ(z) is the Dirac function. On the other hand, there is only Ξ^EE exist for the p-polarized surface wave at the interface of dielectric and nonmagnetic metal, and it can be expressed as

The scaling factor −κ+ε−/(2ω2ε+2μ+) is positive and hence we can conclude that the strong z gradient of permittivity for p-polarized field leads to the strong spin–orbit coupling and the reversal of EFD, which results in the reversal of out-of-plane spin orientation of photonic skyrmion as shown in Figure 1(b) and (e) at the same time. In addition, if the medium of lower half-space is magnetic with positive permittivity, the boundary MD and the spin–orbit interaction term caused by the z-gradient of permeability are:

and

Obviously, the scaling factor −κ+/(2ω2ε+2μ+μ−) is positive. Thus, this quantity is also proportional to the boundary MD and can lead to the reversal of MD and the accompanied in-plane spin orientation of photonic skyrmion as shown in Figure 1(c) and (f). In addition, it worth noting that, for the s-polarized surface modes, these spin–orbit interaction processes would be governed by the terms Ξ^EH and Ξ^HH(Supplementary material VI). We summarize the physical quantities discussed above, including the spin–orbit interaction terms, EFD, MD, OFD and skyrmion topological number of p-polarized and s-polarized surface waves in various electric, magnetic and left-handed interfaces in Figure 3.

Figure 3:

Summary of physical quantities for the p-polarized and s-polarized surface waves in various electric, magnetic and left-handed interfaces. Here, topological charge l > 0; if the l is inverse, the direction of all quantities are inverted. The corresponding available “spin–orbit interaction” terms are given in the end.

Taking into account the “spin–orbit” correction given in Eq. (20), the boundary MDs are

Thus, in the upper half-space, the evanescent waves possess a backward spin part. This backward spin part is subtracted from the forward orbital part to give the total MD and ensures proper local energy transport in evanescent electromagnetic waves. On the other hand, in the lower half-space, the intensity of orbital part (corresponding to the canonical momentum) is much weaker than that of spin part. This is logical because the electromagnetic field in the lower half-space is much more localized and cannot be regarded as propagating field.

4 Conclusion

In conclusion, we investigate the spin and orbital features of photonic skyrmions in both sides of optical interface. For the photonic skyrmion constructed by the p-polarized surface wave, the directions of EFD and spin vector would be reversal due to the inverse of permittivity through the interface, whereas the MD reverse owing to the opposite magnetic permeability in the two sides of interface. By considering the chirality feature of photonic skyrmion, we find that the time-averaging chirality density of photonic skyrmion vanishes, which means that this spin texture is constituted by the optical transverse spin. In addition, by investigating the physically meaningful equations of continuity for the SFD, OFD, spin MD and orbital MD, we obtain four ‘spin–orbit interaction’ terms given by Ξ^HE, Ξ^EH, Ξ^EE and Ξ^HH, respectively. For the p-polarized surface wave in the dielectric/metal interface, the term Ξ^EE, which is related to the electric polarizability, governs the spin–orbit interaction and results in the reversal of EFD and the accompanied out-of-plane orientation of skyrmion-like spin texture through the optical interface. If a variation of magnetic permeability is introduced into the material in lower half-space, the term Ξ^HE, which is determined by the z gradient of the magnetic permeability, would also affect the reverse of the MD and the accompanied in-plane orientation of skyrmion-like spin texture through the optical interface. The presented theoretical analysis have beneficial to understand the spin–orbit features for photonic skyrmions and can be extended to the other classical waves [74], [75], [76], [77], [78], [79], such as: acoustic and fluid waves (Supplementary material VIII). Our findings can construct further applications in the fields of optical manipulation, sensing, quantum and topological techniques.