Photonic spin–orbit coupling

We start by embedding the perylene crystal within a metallic microcavity (Fabry–Perot resonator) as sketched in Fig. 2a. The quantization of the z-component of the wave vector leads to the formation of 2D bands parametrized by the in-plane 2D wave vector $$ \mathbf{k}={\left(k_x,k_y\right)}^T$$. The modes we consider are above the light cone. They have small in-plane wavevector, parabolic dispersion, and are radiatively coupled to the outside of the cavity⁴⁴. The experimental setup (see Methods for details), allowing to make a full optical state tomography out of which the band geometry (Berry curvature and quantum metric) can be reconstructed²³, is shown in Fig. 2b. All experiments are performed at room temperature. The reflection of the sample is measured versus energy and in-plane wave vector for the six different light polarizations (left and right circular, horizontal-vertical, and diagonal-anti-diagonal), which allows a full determination of the three Stokes vector components (light polarization pseudo-spin) of the modes. The axes of the polarizer are aligned with the crystal axes and with the axes of the reciprocal space defined by the CCD camera. The orientation of the Stokes vector S(k) on the Poincaré sphere is given by the polar angle θ(k) and the azimuthal angle ϕ(k). The measurement of these quantities allows to extract the Berry curvature⁴⁵ B_z and the quantum metric⁴⁶ g_ij of a given mode as⁴⁷:

Fig. 2

Combining the spin-orbit couplings.

a The sample consists of a perylene crystal embedded in a microcavity. b Experimental setup allowing to obtain polarization-resolved complete state tomography. BS: beam splitter; L1–L4: lenses; M1: mirror. The red beam traces the optical path of the reflected light from the sample at a given angle. c Embedding the perylene crystal in a planar cavity combines TE-TM splitting βk², linear birefringence β₀, and emergent OA ζk_x. d Reflection of an empty cavity versus energy and wave vector, which evidences the k-dependent TE-TM splitting βk². e Reflection of a cavity filled with perylene, which is inducing linear birefringence β₀ on top of the cavity-induced TE-TM splitting, leading to emergent OA ζk_x (black arrows). Dashed lines show a fit of the two coupled bands with the Hamiltonian (4). The axes of the polarizer and the excitation direction k_x are aligned with the fast axis of the crystal. The figures are measured for a fixed angle range; white regions appear due to angle-to-wave vector conversion.

The Stokes vector can be found as an eigenstate of an effective 2 × 2 Hamiltonian, accounting for the polarization effects arising in the structure. It can describe two modes which are close to each other, whether they are of the same parity (as in inorganic microcavities) or not (as is the case here). To facilitate the understanding, we decompose the combination of polarization effects into a set of individual contributions. The first contribution is the TE-TM splitting, ubiquitous in 2D photonic systems²⁴. In general, the TE-TM splitting appears in any inhomogeneous system, in presence of any gradient allowing to define the transverse directions for the field³². In particular, in planar cavities it appears because of the polarization-dependent reflection coefficients⁴⁸. In an ideal case, the above-mentioned quantized modes of an empty cavity are doubly polarization degenerate at zero in-plane wavevector. This polarization degeneracy is lifted by different contributions sketched in Fig. 2c. The TE-TM splitting of the bare cavity^21,48 characterized by β is zero at k = 0 and then grows quadratically with k:

The two contributions introduced above cancel each other at two points $$ \pm k_0=\sqrt{{\beta }_0/\beta }$$ along the k_x axis. Together, they determine the linear polarization of the modes (S₁, S₂ Stokes components). Finally, the third contribution is the emergent optical activity (chirality) of the structure, described by the parameter ζ. As shown in Rechcińska et al.³⁵, for almost degenerate modes of opposite parity, the induced chirality appears as a splitting between the circular-polarized modes, linear in k_x:

Figure 2d shows the total reflection coefficient of an empty cavity with bare TE and TM modes with different effective masses (because of β), but degenerate at k = 0. The cavity filled with the active material (Fig. 2e) shows a radically different mode dispersion. The first visible consequence is that the cavity is optically thicker due to the perylene crystal refractive index (n ≈ 2). More importantly, the splitting between linearly polarized modes at k = 0 becomes comparable with that of the modes of different orders, because of the significant linear birefringence of perylene (refractive indices 1.7 and 2.5⁵²). These modes anticross at a finite wave vector (instead of simply crossing) because of the emergent OA ζ which can be associated with an effective optical activity coefficient: α ≈ ζk_xn²/2ℏc (see Methods for more details and Supplementary Fig. 6 for the polarization-resolved image of the dispersion). The key difference with respect to a cavity with TRS broken by the Faraday effect²⁴ is that here, the effective Zeeman field changes sign with k_x. This Hamiltonian shows two gapped tilted Dirac cones at the two reciprocal space points where the in-plane components of the field cancel. The sign of the mass term, opposite for the two cones, is given by the sign of the effective Zeeman field. We note that Fig. 2e corresponds to the direction with the smallest gap, k_x. We have mapped the whole reciprocal space and we can therefore exclude the possibility that the observed anticrossing is simply due to a tilt of an optical axis which could shift the crossing point away.

The Stokes vector and the quantum geometric tensor

The validity of the effective Hamiltonian is confirmed by the measured 2D wave vector maps of the Stokes vector components of the lower branch, shown in Fig. 3a–c compared with theoretical predictions shown in panels (d–f). We note that the experimentally measured Stokes components are zero outside an elliptic region where the detection is efficient. Inside this region (marked with a white dashed line), the experiment and the theory exhibit a good agreement. As expected, the linear birefringence is compensated by the k-dependent TE-TM field at the anticrossing points⁵³. The two components S₁ and S₂ of the Stokes vector cancel and change sign around these points, forming 2D monopoles. The third Stokes vector S₃ component is maximal at these two points but is changing sign at k_x = 0 because of the TRS. The cross-sections of the dispersion close to the anticrossing points together with the pseudospin orientation for the two branches are shown in Supplementary Fig. 1. The opposite circular polarization of the branches is also directly visible in Supplementary Fig. 6, showing the circular polarization degree of the reflection at different energies without any extraction.

Fig. 3

Experimentally measured and theoretically calculated Stokes parameters.

Measured Stokes parameters of the mode E₀ from Fig. 2e: a S₁, b S₂, c S₃. The boundary of the meaningful (non-zero) signal is shown with a white line; Theoretically calculated Stokes parameters for the lowest eigenstate of (5): d S₁, e S₂, f S₃.

The measurements of the Stokes vector allow to extract the Berry curvature and the quantum metric of the modes, as shown in Fig. 4a, b. As expected from the TRS (encoded in the Zeeman SOC and the S₃ texture), the Berry curvature shows two maxima of opposite signs, which means that the integrated Berry curvature over the band is zero. However, by separating the reciprocal space in two regions, like in the quantum valley Hall effect²⁰, it is possible to associate non-zero pseudo-spin Chern numbers to each of these two regions. These regions can viewed as being analogs of valleys, which emerge in this optically active system without the need of using a lattice. The trace of the quantum metric (Fig. 4b) is maximal in the regions of the anticrossing, where the Stokes vector rotation is the fastest. The experimentally extracted geometry (a,b) corresponds very well to the theoretical predictions (c, d) based on the 2 × 2 effective Hamiltonian (5) with parameters β₀ = 0.18 eV, β = 9 × 10⁻⁴ eVμm², ζ = 2.5 × 10⁻³ eVμm, extracted from the experimental dispersion (Fig. 2e). The distortion of the maxima of the Berry curvature in experiment could be explained by the contribution of the exciton resonance (see Methods section and Supplementary Fig. 3) and by the difference of the experimental resolution in the two directions. The OA coefficient obtained from the splitting at the anticrossing point demonstrates a remarkably high value of α = 1.4 × 10⁴ degrees/mm. This is a crucial ingredient which has allowed room temperature measurements. Indeed, for a typical OA material such as the tartaric acid, the mode splitting at the anticrossing point would be of the order of 10⁻⁴ meV, much smaller than the broadening k_BT_RT ≈ 26 meV. The room temperature operation at optical wavelengths is highly favorable in the prospect of using such non-trivial band geometry for implementing practical topological photonic devices.

Fig. 4

Experimentally extracted and theoretically calculated Berry curvature and quantum metric.

Quantum geometry extracted from the measured Stokes vector: a Berry curvature B_z, b trace of the quantum metric g_xx + g_yy; Calculated quantum geometry for the lowest eigenstate of (5): c Berry curvature B_z, d trace of the quantum metric g_xx + g_yy.

Material structure and fabrication

The perylene (99%+) and TBAB (Tetrabutylammonium bromide) used in the experiment were purchased from Acros and Innochem respectively without further purification. The 2D square sheet of crystalline perylene was prepared by space-confined strategy⁷⁰: 50 μL of 0.5 mg/ml perylene/chlorobenzene was first added onto 1mg/ml TBAB/water solution. After the complete evaporation of chlorobenzene, 2D square sheets of crystalline perylene were formed on the solution surface, exhibiting a thickness of 200–1000 nm. The molecule arrangement of the perylene film is illustrated in Supplementary Fig. 2.

Cavity structure

For the empty microcavities characterized by Fig. 2d, 80 nm silver film was first evaporated on a glass substrate, followed by a spin-coating of 300 nm polystyrene (PS) film and a final vacuum evaporation of 50 nm silver film. The reflectance of the 80-nm and 50-nm silver films were 99.4% and 87% respectively, enabling easier light extraction from the top mirror of the cavity. For the active microcavity embedding perylene, 25 nm PS film was first spin-coated on 80 nm silver film, then the prepared 2D perylene sheets (thickness ~ 750 nm for the studied cavity) were transferred onto the PS film by bringing them in contact at the surface of the TBAB/water solution. A final evaporation of 20 nm SiO₂ and 50 nm silver film was made to form the microcavity sketched in Fig. 2a.

Spectroscopy

The angle-resolved spectroscopy was performed at room temperature by the Fourier imaging using a ×100 objective lens of a NA 0.95, corresponding to a range of collection angle of ±70^∘. As sketched in Fig. 2b, an incident white light from a Halogen lamp was focused on the area of the microcavity containing perylene, and the k-space or angular distribution of the reflected light was located at the back focal plane of the objective lens. Lenses L1-L4 formed a confocal imaging system together with the objective lens, by which the k-space light distribution was first imaged at the right focal plane of L2 through the lens group of L1 and L2, and then further imaged, through the lens group of L3 and L4, at the right focal plane of L4 on the entrance slit of a spectrometer equipped with a liquid-nitrogen-cooled CCD. The use of four lenses here provided flexibility for adjusting the magnification of the final image and efficient light collection. Tomography by scanning the image (laterally shifting L4) across the slit enabled obtaining spectrally resolved 2D k-space images. In order to investigate the polarization properties, we placed a linear polarizer, a half-wave plate and a quarter-wave plate in front of the spectrometer to obtain the polarization state of each pixel of the k-space images in the horizontal-vertical (0^∘ and 90^∘), diagonal (±45^∘), and circular (σ⁺ and σ⁻) basis^71,72.

Extraction of the Stokes parameters and the quantum geometry

The experimental tomography images represent a set of intensities in six polarization components measured in reflection as a function of in-plane wave vector (k_x, k_y) and wavelength λ. To obtain the maps of the Stokes vector components for a given mode, shown in Fig. 3a–c, we proceed as follows. We consider a reflectivity spectrum measured under white light excitation for total intensity, such as shown in Supplementary Fig. 7 (black circles) for a single point of the reciprocal space. We first determine the wavelength λ₀ and the energy E₀ corresponding to the particular mode, by fitting the total reflection spectrum with Lorentzian-broadened resonances over an approximately linear background (red solid line). We then fit the individual intensity components to determine the relative weight of resonance (taking into account the magnitude and the width of the peak) in each of the six polarizations (blue and violet triangles in Supplementary Fig. 7 for experimental circular polarization and red dashed and dash-dotted lines for theory), which allows finally to determine the three components of the Stokes vector. In our example, we show only two polarization projections of the six (to avoid overloading the figure). We note that the positions of the reflectivity minima detected in two polarizations under a non-polarized excitation do not necessarily correspond to the positions of the modes: their position depends on the linewidth and the polarization degree, and the maximal deviation can be of the order of the linewidth. As an example, the reflectance in the two circular polarizations is given by:

The quantum geometric tensor components g_ij and B_z are extracted from the Stokes vector according to Eq. (1) of the main text. Lowpass Fourier-transform smoothing is applied to the maps of the angles θ and ϕ before calculating the partial derivatives numerically.

Coupling with excitons

The high exciton oscillator strength in perylene is known⁵² to cause the strong coupling effect: in vicinity of the exciton resonance, the eigenstates are not purely photonic or excitonic, but become a mixture of the two. The strong coupling is characterized by the Rabi splitting, which is the splitting of the mixed exciton-polariton modes. In ref. ⁵², it has been estimated as V_R = 140 meV, more than enough for the observation of the effect at room temperature.

There are numerous consequences of the strong coupling: the non-parabolicity of the polariton dispersion and the dependence of the polarization splittings on the wave vector. While a detailed study of the strong coupling in a cavity with perylene is beyond the scope of the present work, we note that these effects do show up in our experimental measurements. The dispersions shown for the cavity with an active region deviate from parabolicity for high wave vectors, and the observed distribution of the Berry curvature and the quantum metric deviates from the predictions of a simple model based on a 2 × 2 effective Hamiltonian. The strong coupling could be one of the possible reasons of these deviations, which we demonstrate in the Supplemental Fig. 3, calculated with a 4 × 4 Hamiltonian taking into account the strong coupling of the excitonic and photonic states. Indeed, this figure exhibits a better agreement with the experiment (Fig. 4a) thanks to the elongated shape of the modes. We note, however, that all the essential physics of the system is already captured by the 2 × 2 effective Hamiltonian, which is why we have used it throughout in our work.

Optical activity

Different formalisms have been developed for the description of the optical properties of crystals. We start with the equation of the optical indicatrix or the index ellipsoid^32,33, obtained from the relations between D and E in the medium, which for a crystal with indices n₁ and n₂ in the absence of the OA reads

The gyration tensor value corresponding to the experimentally measured dispersion is η_xz ≈ 0.14, much larger than any of the tensor components of quartz η_Q ~ 10⁻⁵, which guarantees that our observations are not due to the optical activity of quartz, a layer of which is present inside the microcavity. At the anticrossing wavelength λ ≈ 500 nm, the splitting between the energies is E₊ − E₋ ≈ 50 meV, which gives an equivalent value of α = 500 rad/mm or α = 2.8 × 10⁴ degrees/mm. This is comparable with the values observed in metamaterials, such as chiral photonic crystals⁷⁴. It starts to approach the optical activity of chiral stacks of 2D materials, where the rotation of tens of degrees can be observed for only 10 monolayers of material, and the corresponding α_chir ~ 5 × 10⁶ degrees/mm³⁴.

Origin of the optical activity

The optical activity is an effect which stems from non-locality³². As such, it can also stem from inhomogeneities in non-chiral structures, for example, from a transverse thickness gradient⁷⁵. We have made all possible efforts to exclude the possibility that the optical activity observed in our experiments stems from this origin. For this, on the one hand, we have measured the thickness gradient of the structure experimentally, obtaining an average value of ≈1.3 nm/μm (see Supplementary Fig. 4). On the other hand, we have estimated the energy splitting which could stem from such gradient analytically and numerically, using full 3D electrodynamic simulations in COMSOL(R), and obtained no measureable splitting. To play any role in the effect, the gradient has to be at least 10 times higher.

As discussed in the main text, the OA arises in our sample at the scale of the whole structure, which becomes chiral because of the degeneracy of the modes of opposite parity. To confirm this statement, we perform a full numerical simulation with COMSOL(R), taking the parameters of the structure (perylene and mirror thickness) shown in Fig. 2. We did not include any microscopic optical activity in the model. In particular, we did not add the quartz layer, which could provide this activity (albeit small). We take the ordinary and extraordinary refractive indices of 1.7 and 2.5, respectively⁵². The excitonic effects are neglected. The results of the simulations are shown in Supplementary Fig. 5, showing the transmittivity as a function of in-plane wave vector k_x and energy (similar to Fig. 2e). A clear anticrossing of the modes of opposite parity is observed at approximately the same energy and wave vector as in experiment. This confirms our interpretation of the effect in question.