## A quantum twist on classical optics

Interpreting recent experimental results of light interactions with matter shows that the classical Maxwell theory of light has intrinsic quantum spin Hall effect properties even in free space. Complex effects in condensed-matter systems can often find analogs in cleaner optical systems. Bliokh *et al.* argue that the optical systems exhibiting such complex phenomena should also be simpler (see the Perspective by Stone). Their theoretical study shows that free-space light has a nonzero topological spin Chern number and thus should have counterpropagating surface modes. Such modes are actually well known and can be described as evanescent modes of Maxwell equations.

## Abstract

Maxwell’s equations, formulated 150 years ago, ultimately describe properties of light, from classical electromagnetism to quantum and relativistic aspects. The latter ones result in remarkable geometric and topological phenomena related to the spin-1 massless nature of photons. By analyzing fundamental spin properties of Maxwell waves, we show that free-space light exhibits an intrinsic quantum spin Hall effect—surface modes with strong spin-momentum locking. These modes are evanescent waves that form, for example, surface plasmon-polaritons at vacuum-metal interfaces. Our findings illuminate the unusual transverse spin in evanescent waves and explain recent experiments that have demonstrated the transverse spin-direction locking in the excitation of surface optical modes. This deepens our understanding of Maxwell’s theory, reveals analogies with topological insulators for electrons, and offers applications for robust spin-directional optical interfaces.

Solid-state physics exhibits a family of Hall effects with remarkable physical properties. The usual Hall effect (HE) and quantum Hall effect (QHE) appear in the presence of an external magnetic field, which breaks the time-reversal () symmetry of the system. The HE induces charge current orthogonal to both the magnetic field and an applied electric field, whereas the QHE (*1*) involves distinct topological electron states, with unidirectional edge modes (charge-momentum locking), characterized by the topological Chern number (*2*).

The intrinsic spin Hall effect (SHE) can occur in -symmetric electron systems with spin-orbit interactions. It produces a spin-dependent transport of electrons orthogonal to the external driving force (*3*, *4*). There is also the quantum spin Hall effect (QSHE) (*5*, *6*), which is characterized by unidirectional edge spin transport—edge states with opposite spins propagating in opposite directions. Such topological states with spin-momentum locking gave rise to a new class of materials: topological insulators (*7*, *8*).

Alongside the extensive condensed-matter studies of electron Hall effects, their photonic counterparts have been found in various optical systems. In particular, both the HE (*9*) and the QHE with unidirectional edge propagation (*10*, *11*) have been reported in magneto-optical systems with broken -symmetry. Furthermore, because photons are relativistic spin-1 particles, they naturally exhibit intrinsic spin-orbit interaction effects, including Berry phase (*12*) and the SHE (*13*–*15*) stemming from fundamental spin properties of Maxwell equations (*16*).

The only missing part in the above optical Hall effects is the QSHE for photons. Recently, it was suggested that photonic topological insulators can be created in complex metamaterials structures (*17*–*19*). Here, we show that pure free-space light already possesses intrinsic QSHE, and simple natural materials (such as metals supporting surface plasmon-polariton modes) exhibit some features that resemble topological insulators. We show that the recently discovered transverse spin in evanescent waves (*20*, *21*) and spin-controlled unidirectional excitation of surface or waveguide modes (*22*–*27*) can be interpreted as manifestations of the QSHE of light.

Propagating (bulk) free-space modes of Maxwell equations are polarized plane waves. Introducing the complex amplitude **E**(**r**) of the harmonic electric field **E**(**r**,*t*) = Re[(**r**)*e*^{–}^{i}^{ω}* ^{t}*], the plane-wave solution with wave vector is (1)Here,

*k*= ω/

*c*,

**e**is the complex unit polarization vector (), whereas , , and denote the unit vectors of the corresponding axes. The Jones vector ξ = (α,β)

*is a three-dimensional (3D) spinor, which describes the SU(2) polarization state of light. The spin states of propagating light are circular polarizations , with helicities σ ≡ 2Im(α*β) = ±1. According to the massless nature of photons, the plane-wave spin is directed along the wave vector:*

^{T}**S**= σ

**k**/

*k*[we consider the spin density per photon in units (supplementary text)].

Generalizing Eq. 1 to an arbitrary direction of propagation, the polarization vector becomes momentum-dependent: **e**(**k**). Namely, it is tangent to the **k***-*space sphere because of the transversality condition . This spherical **k**-space geometry underlies the spin-orbit interaction of light (*12*–*16*). In particular, introducing the helicity basis of circular polarizations **e**^{σ}(**k**) (*16*), one can calculate the Berry connection **A**^{σσ′} = –*i***e**^{σ} · (∇** _{k}**)

**e**

^{σ′}and curvature

**F**

^{σσ′}= ∇

**×**

_{k}**A**

^{σσ′}for photons. In agreement with the helicity-degenerate light-cone spectrum of photons, the Berry curvature is diagonal,

**F**

^{σσ′}= δ

^{σσ′}

**F**

^{σ}, and it forms two monopoles at the Dirac-point origin of the momentum space (

*12*–

*16*): (2)This curvature is responsible for the spin-redirection Berry phase and the SHE in optics (

*12*–

*16*).

We define the topological Chern numbers for the two helicity states , where the integral is taken over the **k**-space sphere. The Chern numbers are meaningful in systems with Abelian Berry phases, such as 2D systems with the conserved spin component along the third dimension (*7*, *8*, *28*). This is also the case for photons having Abelian Berry phase, 2D polarization on the **k**-space sphere, and conserved radial **k**-component of the spin (helicity) (*29*). The monopole curvature (Eq. 2) yields . The total Chern number and the spin Chern number characterize the photonic QHE and QSHE properties (*7*, *8*, *28*): (3)The physical meaning of the Chern numbers is the number of edge modes with fixed direction of propagation. The vanishing total Chern number (Eq. 3) reflects the -symmetry of Maxwell equations and the absence of the QHE for free-space photons. At the same time, the nonzero spin Chern number (Eq. 3) implies that free-space light has two pairs of QSHE modes—edge counterpropagating modes with opposite spins. Furthermore, the value implies that the topological ℤ_{2} invariant, associated with the -symmetry, vanishes: . This means that surface modes of Maxwell equations are not helical fermions (*30*) as, for example, surface states of the Dirac equation (*31*, *32*).

Nonetheless, nontrivial QSHE states of light exist, and they are well known. The photonic edge states of a bounded segment of free space are evanescent waves. For instance, assuming the boundary, with free space at , the generic evanescent-wave solution of Maxwell equations can be written as (*21*) (4)Here, the spinor still characterizes the wave polarization states. The wave (Eq. 4) propagates along the axis with wave number and decays exponentially away from the boundary with decrement .

One can consider the evanescent wave (Eq. 4) as a plane wave with the complex wave vector . The transversality condition **E** · **k** = 0 generates the imaginary longitudinal -component in the polarization vector **e**_{evan}, in contrast to the purely transverse polarization **e** in propagating waves (Eq. 1). This component produces a (*x*, *z*)–plane rotation of the electric or magnetic fields and generates unusual transverse spin in evanescent waves (Fig. 1) (*20*, *21*). This transverse spin is independent of the polarization and can be written as

Equation 5 demonstrates spin-momentum locking, similar to that in the QSHE and 3D topological insulators for electrons (*5*–*8*). In particular, the -propagating evanescent waves with and will have opposite transverse spins and (Figs. 2 to 4). Thus, any interface between free space and a medium supporting surface or guided modes with evanescent tails (Eq. 4) exhibits counterpropagating opposite-spin edge modes—the QSHE of light. This is the first key point of our work.

In agreement with , there are two pairs of QSHE modes in free space because the evanescent waves (Eq. 4) are double-degenerate with respect to the helicities . However, the existence of surface modes in Maxwell equations requires a planar interface between the vacuum and a medium characterized by a permittivity and permeability . Such interface breaks the dual symmetry between the electric and magnetic properties: (*29*). This breaks the polarization degeneracy, and only a single polarization survives in the surface modes. For example, only transverse-magnetic surface waves exist at the interface with a medium with and . Calculating the spectrum, polarization, and spin of these surface modes of Maxwell equations, we obtain (supplementary text):(6)Here, and are the unit vectors of the propagation direction and the outer normal of the medium, respectively, and we calculated the mean (integral) spin per one surface-mode particle. The momentum-dependent spin originates from the transverse spin (Eq. 5) of evanescent waves.

Equations. 5 and 6 determine the momentum locking of the spin **S** but not of the polarization spinor (Fig. 2A). Polarization specifically corresponds to spin for nonrelativistic electrons, but for relativistic particles these are different notions. The surface modes of Maxwell equations have momentum-dependent spin **S**_{surf} but fixed spinor ξ_{surf} (Eq. 6). The latter corresponds to the trivial ℤ_{2} invariant ν = 0 and shows that surface Maxwell modes are bosons rather than helical fermions (*30*). Nonetheless, these modes provide the unidirectional edge spin transport (QSHE) because of the spin **S**_{surf}. Precisely the opposite situation takes place in one of the main models for 3D electron topological insulators: the Dirac equation with surface modes at the interface between positive-mass and negative-mass regions (Fig. 2B) (*31*, *32*). In this case, spinor-momentum locking occurs, which corresponds to the topological ℤ_{2} invariant ν = 1. However, surprisingly, the expectation value of the spin of the surface Dirac modes vanishes because of the mutual cancellation of the polarization-dependent and polarization-independent (similar to Eq. 5) contributions (supplementary text). Thus, one can say that surface Maxwell modes exhibit unidirectional spin transport (QSHE) but with trivial ℤ_{2} spinor properties, whereas the surface Dirac modes are topologically protected helical fermions that, however, do not transport spin. This is the second key point of our work.

Optical spin-momentum locking was recently observed in several experiments (*22*–*27*). An important example is provided by surface plasmon-polaritons (SPPs) at the vacuum-metal interface (*33*). Real metals are dispersive media with permittivity , where is the plasma frequency. Metals are optical insulators at , and at (), the vacuum-metal interface supports surface Maxwell modes—the SPPs (Fig. 3A). The metal becomes transparent at , with bulk plasmons at and electromagnetic modes at . As shown in Fig. 3A, the vacuum-metal interface resembles, by using condensed-matter analogies, the interface between a semimetal and an insulator. The SPP modes demonstrate spin-momentum locking (Eqs. 5 and 6) and nonremovable (because of the light-cone spectrum in vacuum) spectral degeneracy at , which are typical for electron QSHE states. Furthermore, plotting the SPP spectrum for a 2D surface of a 3D metal (Fig. 3B), one can see the conical spectrum and vortex spin texture analogous to those in 3D electron topological insulators (*7*, *8*), but without the helical-fermion spinor properties (Fig. 2).

A schematic of the experiments (*22*–*27*) is shown in Fig. 4, revealing spin-controlled unidirectional transport in electromagnetic surface or guided waves. A transversely propagating free-space light beam with the usual spin (helicity ) was coupled to the evanescent tails of the SPP or waveguide modes via some scatterer (such as a nanoparticle or an atom). In doing so, the opposite incident-spin states excited the surface or guided modes running in the opposite directions: . This spin-direction correlation reached almost 100% efficiency in various systems, independently of their details. This proves the universal spin-momentum locking in optical surface waves—the QSHE of light.

Thus, we have shown that light has intrinsic QSHE features, which arise from the spin-orbit interactions of photons. The corresponding spin-momentum locking originates solely from the basic properties of evanescent waves in Maxwell equations and can be observed at any interface with the vacuum, which supports surface or guided waves. In particular, surface plasmon-polaritons at a metal-vacuum interface exhibit features similar to those of surface states of topological insulators (vortex spin texture at the conical dispersion). Because of their trivial spinor structure, surface electromagnetic states are not helical fermions and are not protected from backscattering. Nonetheless, they do provide robust unidirectional spin transport. Our work shows that recent experiments, demonstrating highly efficient spin-controlled unidirectional excitation of surface or guided modes, can be interpreted as observations of the QSHE of light. The transverse spin locked to the direction of propagation seems to be a universal feature of surface vector waves of different nature. It appears in Maxwell and Dirac equations, as well as in Rayleigh surface waves in elastic media and surface-water waves. This offers robust angular-momentum-to-direction coupling in various surface waves as well as analogies and generalizations involving quantum and classical wave theories.

## Supplementary Materials

## References and Notes

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
**Acknowledgments:**We are grateful to A. Furusaki, Y. Bliokh, E. Ostrovskaya, Y. Kivshar, and A. Khanikaev for fruitful discussions. This work was partially supported by the RIKEN iTHES Project, Multidisciplinary University Research Initiative Center for Dynamic Magneto-Optics (award number FA9550-14-1-0040), the Australian Research Council, Japan Society for the Promotion of Science–Russian Foundation for Basic Research contract 12-02-92100 and a Grant-in-Aid for Scientific Research (A).