## Making Metamaterials

Controlling the propagation of electromagnetic waves is a key requirement in communication technologies. The components tend to be bulky, however, which can make it difficult to integrate with microelectronics circuits. Using arrays of metallic nanoantennae patterned on a substrate surface, **Shitrit et al.** (p. 724) fabricated a novel class of metamaterials: anisotropic materials without inversion symmetry. The materials may pave the way to polarization-dependent nanophotonics.

## Abstract

Spin optics provides a route to control light, whereby the photon helicity (spin angular momentum) degeneracy is removed due to a geometric gradient onto a metasurface. The alliance of spin optics and metamaterials offers the dispersion engineering of a structured matter in a polarization helicity–dependent manner. We show that polarization-controlled optical modes of metamaterials arise where the spatial inversion symmetry is violated. The emerged spin-split dispersion of spontaneous emission originates from the spin-orbit interaction of light, generating a selection rule based on symmetry restrictions in a spin-optical metamaterial. The inversion asymmetric metasurface is obtained via anisotropic optical antenna patterns. This type of metamaterial provides a route for spin-controlled nanophotonic applications based on the design of the metasurface symmetry properties.

Metamaterials are artificial matter structured on a size scale generally smaller than the wavelength of external stimuli that enables a custom-tailored electromagnetic response of the medium and functionalities such as negative refraction (*1*), imaging without an intrinsic limit to resolution (*2*), invisibility cloaking (*3*), and giant chirality (*4*, *5*). An additional twist in this field originates from dispersion-engineered metamaterials (*6*, *7*). A peculiar route to modify the dispersion relation of an anisotropic inhomogeneous metamaterial is the spin-orbit interaction (SOI) of light; that is, a coupling of the intrinsic angular momentum (photon spin) and the extrinsic momentum (*8*–*10*). Consequently, the optical spin provides an additional degree of freedom in nano-optics for spin degeneracy removal phenomena such as the spin Hall effect of light (*9*, *11*–*14*). The chiral behavior originates from a geometric gradient associated with a closed loop traverse upon the Poincaré sphere generating the geometric Pancharatnam-Berry phase (*15*, *16*), not from the intrinsic local chirality of a meta-atom (*4*, *5*, *17*). Specifically, spin optics enables the design of a metamaterial with spin-controlled modes, as in the Rashba effect in solids (*18*–*21*).

The Rashba effect is a manifestation of the SOI under broken inversion symmetry [i.e., the inversion transformation **r** → **–r** does not preserve the structure (**r** is a position vector)], where the electron spin-degenerate parabolic bands split into dispersions with oppositely spin-polarized states. This effect can be illustrated via a relativistic electron in an asymmetric quantum well experiencing an effective magnetic field in its rest frame, induced by a perpendicular potential gradient ∇*V*, as represented by the spin-polarized momentum offset Δ*k* ∝ ±∇*V* (*18*–*21*). In terms of symmetries, the spin degeneracy associated with the spatial inversion symmetry is lifted due to a symmetry-breaking electric field normal to the heterointerface. Similar to the role of a potential gradient in the electronic Rashba effect, the space-variant orientation angle ϕ(*x*,*y*) of optical nanoantennas induces a spin-split dispersion of Δ*k* = σ∇ϕ (*22*–*24*), where σ_{±} = ±1 is the photon spin corresponding to right and left circularly polarized light, respectively. We report on the design and fabrication of spin-optical metamaterial that gives rise to a spin-controlled dispersion due to the optical Rashba effect. The inversion asymmetry is obtained in artificial kagome structures with anisotropic achiral antenna configurations (Fig. 1, A and B) modeling the uniform (*q* = 0) and staggered (*25*–*27*). In the geometrically frustrated kagome lattice (KL), the reorder of the local magnetic moments transforms the lattice from an inversion symmetric (IS) to an inversion asymmetric (IaS) structure. Hence, we selected the KL as a platform for investigating the symmetry influence on spin-based manipulation of metamaterial dispersion.

It was previously shown that the localized mode resonance of an anisotropic void antenna is observed with a linear polarization excitation parallel to its minor axis (*14*, *23*). We used this anisotropy in artificial kagome structures, where the anisotropic antennas are geometrically arranged in such a way that their principal axes are aligned with the original spin direction in the magnetic KL phases. These metamaterials with the nearest-antenna distance of *L* = 6.5 μm were realized using standard photolithographic techniques (*24*) on a SiC substrate supporting resonant collective lattice vibrations [surface phonon polaritons (SPPs)] in the infrared region. We measured angle-resolved thermal emission spectra by a Fourier transform infrared spectrometer at varying polar and fixed azimuthal angles (θ,ϕ), respectively, while heating the samples to 773 K (see Fig. 1C for the experimental setup). The dispersion relation ω(*k*) at ϕ = 60° of the *q* = 0 structure (Fig. 1D) exhibits good agreement with the standard momentum-matching calculation (*28*) [see (*24*) for the isotropic KL analysis]. However, the measured dispersion of the *S*_{3} component of the Stokes vector representing the circular polarization portion within the emitted light (*24*, *29*), we observed the *k* = 2π/3*L* (Fig. 1, F and G).

The removal of the spin degeneracy requires a spin-dependent correction to fulfill the momentum-matching equation. The spin-controlled dispersion of an IaS metamaterial obeys the spin-orbit momentum-matching (SOMM) condition *q* = 0 unit cell, whereas **k _{SPP}** is the SPP wave vector; (

*m*,

*n*) are the indices of the radiative modes; and

*i*∈ {1,2} is the index of the specific spin-dependent geometric Rashba term. Note that in the vector equation the sign of the orientational term is determined by the spin. Moreover, the specific geometric Rashba correction is a result of an arbitrary orientational vector choice from

**K**; based on this choice, the secondary vector linearly depends on the primary

_{1,2}**K**and both the structural vectors

_{i}**G**. Hence, the SOMM condition arises from the combined contributions of the structural and orientational lattices resembling the structural and magnetic unit cells in the kagome antiferromagnet; yet, this concept is general and can be tailored to metamaterials and metasurfaces (

_{1,2}*30*). Considering low modes of (

*m*,

*n*) ∈ {0,±1}, we calculated the spin-dependent dispersions at ϕ = 0° and 60° (Fig. 2, A and B, respectively), confirming the

*S*

_{3}measured dispersions (Fig. 1, F and G, respectively), with the optical Rashba spin split of 2π/3

*L*. Such a spin-split dispersion is due to a giant optical Rashba effect, as the geometric Rashba correction is in the order of magnitude of the structural term; in particular, the anomalous geometric phase gradient arising due to the space-variant antenna orientation in the investigated IaS photonic system resembles the potential gradient in the electronic Rashba effect.

The above condition can be also derived from symmetry restrictions, where the representation theory is applied to generate selection rules. If a given structure is invariant under a translation followed by a rotation and both operators commute, then a spin-orbit coupling is expected. By applying the representation theory formalism considering these symmetry constraints, a momentum selection rule with a spin-dependent geometric Rashba correction can be obtained. When this procedure is implemented for the *L* to the left followed by a rotation of 120° counterclockwise, the SOMM condition is realized [see (*24*) for the detailed discussion].

In addition to the spin-projected dispersions, selection rules can also specify the direction of the surface wave excited at a given frequency (Fig. 2, C to F). Hence, this concept serves as a platform for spin-controlled surface waves possessing excellent potential for manipulation on the nanoscale based on a geometric gradient. Particularly, the SOMM provides the basis for a new type of IaS spin-optical metamaterials, supporting spin-dependent plasmonic launching for nanocircuits (*31*, *32*) and multiport in on-chip photonics.

The observed optical Rashba spin-split dispersions reveal two obvious relations of (i) ω(*k*,σ_{+}) ≠ ω(*k*,σ_{–}), which is a signature of inversion symmetry violation, and (ii) ω(*k*,σ_{+}) = ω(–*k*,σ_{–}), which is a manifestation of time reversal symmetry (Fig. 1, F and G, and Fig. 2, A and B). Moreover, the spin-projected dispersions show a clear discrimination between the different IaS directions, which is not observed in the degenerated intensity dispersions. By measuring the angle-resolved emission spectra at varying ϕ and fixed θ, we obtained the strength of the optical Rashba effect pointing on the IS and IaS directions in the KL (*24*). The *S*_{3} dispersions of the

The symmetry-based approach offers an extended condition because it also recognizes the IS directions resulting in spin-degenerated dispersions. As a reference, we measured the intensity dispersions of the *q* = 0 structure at ϕ = 30°, verifying the standard momentum-matching calculation without the geometric Rashba correction. Note that this direction is arbitrary because this structure is IS in all directions; however, it is a specific IS direction in the

The spin degeneracy removal was also shown in the near-field associated with the orbital angular momentum variation along the IaS directions. We revealed a chain of vortices with alternating helicities in the artificial *24*) for the detailed analysis]. The reported spin-based phenomena in the near- and far-fields inspire the development of a unified theory to establish a link between the spin-controlled radiative modes and the metasurface symmetry properties to encompass a broader class of metastructures from periodic to quasi-periodic or aperiodic. The design of metamaterial symmetries via geometric gradients provides a route for integrated nanoscale spintronic spin-optical devices based on spin-controlled manipulation of spontaneous emission, absorption, scattering, and surface-wave excitation.

## Supplementary Materials

www.sciencemag.org/cgi/content/full/340/6133/724/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S6