Abstract

The robust spin and momentum valley locking of electrons in two-dimensional semiconductors makes the valley degree of freedom of great utility for functional optoelectronic devices. Owing to the difference in optical selection rules for the different valleys, these valley electrons can be addressed optically. The electrons and excitons in these materials exhibit the valley Hall effect, where the carriers from specific valleys are directed to different directions under electrical or thermal bias. Here we report the optical analog of valley Hall effect, where the light emission from the valley-polarized excitons in a monolayer ${{\rm WS}_2}$ propagates in different directions owing to the preferential coupling of excitonic emission to the high momentum states of the hyperbolic metamaterial. The experimentally observed effects are corroborated with theoretical modeling of excitonic emission in the near field of hyperbolic media. The demonstration of the optical valley Hall effect using a bulk artificial photonic media without the need for nanostructuring opens the possibility of realizing valley-based excitonic circuits operating at room temperature.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

In the monolayer limit, transition metal dichalcogenides (TMDs) such as ${{\rm WS}_2}$, ${{\rm WSe}_2}$, ${{\rm MoSe}_2}$, and ${{\rm MoS}_2}$ are direct-bandgap semiconductor materials with broken inversion symmetry [13]. These materials exhibit very strong interaction with light absorbing greater than 10% light at the monolayer limit close to the excitonic resonances. The excitons in these materials have unusually large binding energies (0.1–0.5 eV), can be developed into functional optoelectronic devices such as light-emitting diodes and detectors [48], and even demonstrate strong coupling to cavity photons at room temperature [911]. The direct bandgap in these materials occurs at the $K$ point in the momentum space in the monolayer limit. An intriguing property that arises from the broken inversion symmetry and the strong spin–orbit interaction arising from the heavy transition atoms is valley polarization—quantum mechanically distinct valleys $K$ and $K^{\prime}$ which can be selectively addressed by right-handed and left-handed circularly polarized light, as shown schematically in Fig. 1(a). Valley polarization has been studied extensively in these systems and has been touted as a path to encode and manipulate information [1214] resulting in the emerging field of valleytronics [1518]. Limiting factors for practical applications of the valley degree of freedom arises from the intervalley scattering and disorder-induced depolarization. Enhancement of valley polarization via strong coupling to the cavity photons as well as their routing using nanophotonic structures has recently garnered much attention and promise for the realization of functional valleytronic devices [1925]. In all these demonstrations, the enhancement of the valley degree of freedom and routing was achieved via optical cavities or very specific resonant plasmonic and dielectric nanostructures. A valley Hall effect in TMD monolayer materials was also demonstrated by observing carriers drift in the transverse direction of the electric field [26,27]. An optical valley Hall effect can be thought of as an optical analog of the conventional Hall effect with directional propagation of valley carriers under optical electric fields of opposite handedness. More recently the optical valley Hall effect was demonstrated in the context of strongly coupled exciton polaritons [28,29] through the observation of transverse separation of the spin-polarized polaritons under optical bias with opposite handedness. The excitonic valley Hall effect using thermal gradient produced by an excitation laser was also recently demonstrated [30].

 

Fig. 1. Optical valley Hall effect demonstration in HMM. (a) Schematic of Valley polarization in ${{\rm WS}_2}$ monolayer, where $K$ and $K^{\prime}$ valleys can be populated independently with $\sigma^{+}$ and $\sigma^{-}$ polarizations, respectively. (b) Schematic of the seven-period HMM composed of ${\rm Ag}/{{\rm Al}_2}{{\rm O}_3}$ shows a monolayer of ${{\rm WS}_2}$ transferred to its surface layer. Electric field intensity distribution of the high $k$-modes for the dipole radiating in the near field of the HMM surface for (c) linear polarization, and (d) and (e) are right- and left-handed circular polarizations, respectively.

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Here we report the use of a multilayered metamaterial medium with isofrequency contour of hyperbolic dispersion [3135] to realize an optical analog of the valley Hall effect, which does not rely on resonant structures. Instead, the near-field coupling of the valley-polarized excitonic emission to the high-$k$ states of this hyperbolic medium facilities the transverse separation of photons resulting in the directional propagation of the emission. This effect in the hyperbolic medium arises due to the nature of the light propagation inside such a medium and has previously been used to demonstrate an optical spin Hall effect in the microwave regime [36]. The spin Hall effect in these systems and other plasmonic structures arises due to the spin-momentum locking [3743]. While we do find weak optical valley Hall effect even with a simple metallic film, the observed valley polarization contrast is much stronger and has higher contrast in the case of our multilayered hyperbolic metamaterial (HMM). We use momentum and helicity-resolved spectroscopic techniques to demonstrate the optical valley Hall effect where we observe valley selective routing of excitonic emission in a 2D ${{\rm WS}_2}$ monolayer integrated with a HMM. The observed phenomena are modeled using both numerical and analytical techniques and agree with the experimental observations. The HMM essentially acts as an extremely broadband valley beam splitter for the circularly polarized valley exciton emission and a route to develop valley filters or valley-based signal processing circuits.

2. METHODS

A. Simulation Methods

Finite element method simulations were performed by using the COMSOL Multiphysics RF module. The HMM is designed with effective medium parameters of ${{\varepsilon}_{x}} = {{\varepsilon}_{y}} = - 6.246 + {i}1.081$ and ${{\varepsilon}_{ z}} = 7.76 + {i}0.2624$ for ${\rm Ag}/{{\rm Al}_2}{{\rm O}_3}$ multilayers volume fraction of 50%. The multilayered HMM is also designed with ${\rm Ag}/{{\rm Al}_2}{{\rm O}_3}$ multilayers in the ${XY}$ plane with periods extending in the ${z}$-direction. Simulations model the quantum emitter as a circular electric dipole oriented in the ${XZ}$ plane perpendicular to the metal–dielectric layers of the HMM with an emission spectrum at 620 nm. The dielectric constant for the Ag is obtained from the previous report of Palik et al. [44]. The dielectric constant for the ${{\rm Al}_2}{{\rm O}_3}$ layer is considered as ${{\varepsilon}_{{{\rm Al}_2}{{\rm O}_3}}} = 1.67$. Transfer matrix method simulations were implemented to obtain the HMM and one-period HMM samples mode dispersions for both linear and circular dipole excitations.

B. Sample Fabrication

A seven-period HMM substrate and one-period HMM control sample with Ag and ${{\rm Al}_2}{{\rm O}_3}$ multilayers were grown by a Kurt Lesker PVD 75 electron beam evaporation system on top of precleaned 120 µm thick glass substrates with surface roughness ${\lt}{1}\;{\rm nm}$. The pressure inside the vacuum chamber was kept to $\sim {2} \times {{10}^{- 6}}$ torr throughout the process. 1–2 nm of Ge seed layer was grown prior to each Ag layer to ensure good uniform distribution of the Ag thin layer on the ${{\rm Al}_2}{{\rm O}_3}$ layer. The top surface of the HMM was terminated with a ${{\rm Al}_2}{{\rm O}_3}$ layer to avoid the top Ag layer oxidation as well as the ${{\rm WS}_2}$ monolayer exciton emission quenching.

A monolayer of ${{\rm WS}_2}$ TMD material (HQ Graphene) was exfoliated onto a thick polydimethylsiloxane (PDMS) stamp and transferred to the HMM surface using the dry transfer technique on a home build transfer stage.

C. $k$-Space Photoluminescence Measurements

The ${{\rm WS}_2}$ monolayer on the HMM surface was excited from the top with a 0.25 NA, ${10\times}$ microscope objective with an excitation wavelength of 620 nm from a Toptica Photonics laser source with repetition rate of 80 MHz and pulse width of 1 ps. The polarization of the laser was set to circular polarization by using the combination of a linear polarizer and a quarter-wave plate. The near-field photoluminescence below the HMM surface was collected by using a ${100\times}$ oil immersion objective (Olympus PLN) with 1.4 NA. The whole back focal plane of the microscope objective was imaged onto the CCD camera (Princeton Instruments) operating at ${-}{{70}^\circ}{\rm C}$. For the helicity-resolved measurements, the photoluminescence signal was resolved into helicity angles by using the combination of another quarter-wave plate and a linear polarizer. The schematic of the experimental setup is shown in Supplement 1.

3. RESULTS

The artificial optical media was designed to have Type II hyperbolic (single sheet hyperbola) isofrequency dispersion in the visible region of the electromagnetic spectrum [33,34]. A schematic illustration of the HMM is shown in Fig. 1(b). The structure consisted of alternating layers of Ag and ${{\rm Al}_2}{{\rm O}_3}$ layers and the thickness of each layer was optimized to be 10 nm. The effective dielectric constants of the HMM, simulated using effective medium theory, is shown in Supplement 1 [Fig. S1(a)]. Such HMMs have been previously used to enhance spontaneous emission from emitters and tailor the TMD monolayer exciton dynamics owing to the enhanced local photonic density of states [4548].

Finite element method simulations were performed to investigate the near-field coupling of linear and circular polarization emissions to the plasmonic modes of the HMM. The HMM is taken to have effective dielectric permittivity values of ${\epsilon _x} = {\epsilon _y} = - 6.246+i 1.081$ and ${\epsilon _z} = 7.76 + i 0.2624$ at a wavelength of 620 nm as obtained from the effective medium theory calculations. As shown in Fig. 1(c), a linear dipole with dipole moment ${{P}_{2D}} = p[{1},\;{0}]$ with unit polarization $p$ along the $x$ axis shows a characteristic high-$k$ mode propagation in the HMM medium with symmetric electric field distribution in the $xy$ plane with symmetry about the $x$ axis. For right- and left-handed circular polarizations, two linear dipoles were assumed about the origin one in the $xy$ plane and another in the $xz$ plane with a phase difference of ${\pm {90}^\circ}$. The sign of the phase difference defines the right- and left-handedness of the generated circular polarization. A right-handed circularly polarized dipole ($\sigma^{-}$) with ${{P}_{2D}} = p(x - iz)/\sqrt 2$ is initially assumed in the $xz$ plane and swept for all azimuthal rotations about the $z$ axis and the resultant electric field distribution from all azimuthal rotations is plotted in Fig. 1(d). The high-$k$ mode electric field distribution is found to be asymmetric and strictly unidirectional in the ${-}x$ direction, as shown in Fig. 1(d). Similarly, this high-$k$ mode dispersion is toward the ${+}x$ direction for left-handed circular polarization ($\sigma ^{ +}$) with ${{P}_{2D}} = p(x + iz)/\sqrt 2$, as shown in Fig. 1(e). The fundamental principle behind such unidirectional excitation of the surface plasmon mode and the high-$k$ modes is governed by the selective vectorial excitation of the electric field associated with these modes. The handedness-dependent unidirectional coupling of chiral fields to the HMM can be thought of as the photonic analog of the spin-Hall-like effect for photons which was previously demonstrated in pioneering experiments for the surface plasmons in metals [36,37,49,50] and RF hyperbolic metamaterials and more recently shown theoretically for chiral emitters [51] and quantum dots on Type I HMM (two sheet hyperbola) in the visible frequency region [52]. TMDs owing to their strong valley polarization property emit circular polarized photoluminescence when excited with the same handedness owing to the valley specific excitation ($K$ or $K^{\prime}$). Thus, an HMM-TMD integrated system naturally lends itself to demonstrate the optical analog of a valley-Hall-like effect for valley-polarized emission of the excitons in TMDs and provides a path toward integrated valleytronics.

We proceed to model the chiral emission through the HMM by assuming a two-dimensional electric dipole source located on the upper surface of a multilayer structure. The electric field density in the lower half-space can be calculated by the simplified expression ${E_{x}} \propto ({\alpha + \beta {k_x}/| {{k_z}} |})$. Here, ${k_x}$ and ${k_z}$ are the $x$ and $z$ components of the wave vectors and $\alpha ,\beta = 0, \pm 1$ are constants that define the electric dipole polarization. Note that the expression for the electric field includes the contribution of the propagating [$\sqrt {k_x^2 + k_y^2} \lt {k_0}$] and evanescent [$\sqrt {k_x^2 + k_y^2} \gt {k_0}$] modes as shown previously [51]. The latter term is responsible for near-field interference effects, which is central to the chiral selectivity observed in the HMMs as discussed below.

If $\alpha \; \gt \;{0}$ and $\beta \; \gt \;{0}$, the spectral amplitude with ${k_x}\; \lt \; - {k_0}$ adds up destructively, whereas for ${k_x}\; \gt \;{k_0}$ constructive interference occurs. Therefore, the near-field interference effect lies at the heart of the selective directional excitation of guided modes illuminated by an elliptically polarized dipole. When an elliptically polarized dipole illuminates a multilayered structure, the inversion symmetry is broken, and the high-$k$ modes can be excited. The high-$k$ evanescent modes carry transverse spin angular momentum [39,49,50]. Therefore, the best coupling is achieved when the evanescent mode with transverse spin matches the same spin (helicity) of the incident/emitted field. By tuning the polarization of the dipole, the excitation and guidance of electromagnetic modes in the HMM can be controlled.

Using this theoretical model and finite element method simulations we show how to control the excitation direction of the electric field by tuning the polarization of the dipole emitter. The unit vectors for the horizontally and vertically polarized dipoles are ${{ p}_x} = { x}$ and ${{p}_z} = {z}$, respectively. For circularly polarized dipoles, we have unit vectors ${P_{{\pm}}} = (\alpha \;x + i\beta z)/\;\sqrt 2$, with $\alpha = {1}$ and $\beta = \pm {1}$. HMMs support two kinds of modes, namely the surface plasmon polaritons (SPPs) at the interfaces with the claddings and the high-$k$ modes confined to the bulk of the layered material [39,47,53]. The SPP modes in the HMM behave fundamentally different than in a single thin metallic film due to the fact that these modes originate from mutual repulsion of SPP modes at each metal–dielectric interface. The large values of the model index for the SPPs (called long- and short-range plasmon polaritons) in the HMM medium result in higher field confinement in comparison with SPPs in single metallic film [54]. As a result, the SPP modes in the layered medium show much reduced linewidths than the surface modes of a thin film, as shown in Fig. 2(a). For a linear dipole located at the top surface of the HMM is well known to show SPPs and high-$k$ wave vector modes [39,47,53]. The transfer matrix method (TMM) simulations for our seven-period HMM show a symmetric dispersion of the mode spectrum (Supplement 1, Fig. S2). Whereas the circular polarizations show an asymmetric dispersion as shown in Fig. 2(a) for $\sigma ^{ +}$ (red) and $\sigma ^{ -}$ (blue). For a thin metallic film with a dielectric corresponding to the one-unit cell of the HMM, a broad SPP mode is observed [Fig. 2(b)]. The asymmetric dispersion for the SPP mode (at ${{k}_x} = \;\pm {1.057}$) in the HMM is found to have almost similar contrast as the one-unit HMM with unidirectional optical routing for specific chirality. Whereas the contrast supported by a low-quality-factor SPP mode in a thin metallic layer is experimentally found to be indistinguishable. The electric field patterns for linear and circular polarizations are shown in Fig. 2. A distinct asymmetric field pattern is observed for a circular dipole [Figs. 2(e) and 2(f)] in comparison to the horizontal [Fig. 2(c)] and vertical [Fig. 2(d)] dipoles. Since a circular dipole is a linear combination of horizontal and vertical dipoles, the electric field pattern for the circular dipole can be interpreted as a linear combination of the linear dipoles. The phase relations of the fields generated by the horizontal and vertical dipoles of these fields can be summed up, and the qualitative result can be inferred as the observation of unidirectional propagation of the modes of the HMM for circular polarization dipole excitation. Finite element simulations were also carried out for the one-unit cell of the HMM consisting of a thin silver film with a dielectric layer. Figure S3 shows $\sigma ^+$ and $\sigma ^ -$ circular dipole emissions for the one-unit cell of the HMM. It is clear from these simulations that for the same circular dipole emission, the HMM shows more well-defined unidirectional coupling in comparison to the one-period HMM.

 

Fig. 2. Vectorial field of high-$k$ modes in the HMM. Mode spectrum calculated from TMM simulations for circular dipoles with opposite helicities in the $xz$ plane of (a) 7 P HMM sample and (b) one-unit HMM. The magnetic field ${{ H}_y}$ component intensity distribution in the HMM for linear dipole with polarization orientation along (c) horizontal $P=\hat x$ and (d) vertical $P=\hat z$. (e) Left-handed circularly polarized dipole $P=(\hat x + i\hat z)/\sqrt 2$ and (f) right-handed circularly polarized dipole $P=(\hat x-i \hat z \sqrt 2$. These simulations were carried out at the emission wavelength of the exciton (620 nm).

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To demonstrate the optical valley Hall effect of the ${{\rm WS}_2}$ monolayer, two energy degenerate $K$ and $K^{\prime}$ valley emissions need to be directed in two opposite directions. Since the $K$ and $K^{\prime}$ valleys of the ${{\rm WS}_2}$ emit light with specific handedness, the near-field interaction of the $K$ and $K^{\prime}$ valley emissions with high-$k$ and SPP modes of the HMM can be selectively routed to the ${+}x$ and ${-}x$ directions, respectively. Thus, the optical valley Hall effect in monolayers can be mediated through the robust optical spin Hall effect supported by the modes of the HMM. Since the optical spin Hall effect in the HMM is broadband, the optical valley Hall effect in the HMM also can be broadband over its broad hyperbolic regime, unlike the earlier studies using resonant nanostructured media [2023].

For the experimental verification of the optical valley Hall effect, we fabricated a 140 nm thick seven-period HMM structure on a 120 µm thick glass slide through sequential depositions of Ag and ${{\rm Al}_2}{{\rm O}_3}$ thin films each of 10 nm thickness with a 1 nm thick Ge wetting layer for each Ag layer using electron beam evaporation. A TMD monolayer of ${{\rm WS}_2}$ was exfoliated to a PDMS stamp and transferred to the HMM surface following our earlier reported process [23,55]. Helicity-resolved $k$-space measurements were performed in transmission configuration using a home-built microscope setup (Fig. S4). The polarization of the incident laser was set to circular polarization using a combination of a linear polarizer and quarter-wave plate. The collection path polarization was resolved into $\sigma^{ +}$ and $\sigma^{ -}$ helicities using another set of a quarter-wave plate and linear polarizer. The monolayer was excited at the exciton resonance wavelength of 620 nm from the top of the HMM surface with $\sigma ^{ +}$ and $\sigma^{-}$ polarizations using a ${10\times}$ microscopic objective with 0.25 NA. The emission from the bottom surface of the HMM is collected by using an oil immersion objective of ${100\times}$ magnification with a 1.4 numerical aperture. The back focal plane of the ${100\times}$ objective is imaged by using a Fourier space lens onto the CCD camera, as shown in the experimental setup (Supplement 1). The exciton emission was resolved to $\sigma^{+}$ and $\sigma^{-}$ emissions prior to the Fourier space lens and finally imaged to the CCD camera. From Fig. 3, it can be noticed that under $\sigma^{-}$ excitation, the $\sigma^{ -}$ emission corresponding to $K^{\prime}$ valley exciton is found to be in the ${-}{k_x}$ direction. Whereas the $\sigma^{ +}$ emission from $K$ valley for ${\sigma ^{+}}$ excitation is in the ${+}{k_x}$ direction. A complete set of helicities-resolved measurements are shown in Supplement 1, Fig. S5. These asymmetric intensity distributions in the $k$-space images can also be noticed from the line cuts made along the ${k_x}$ direction in the $k$-space images for ${\sigma ^ +}$ and ${\sigma ^ -}$ emissions in Figs. 3(c) and 3(d), respectively. The experimentally observed directional coupling of $K$ and $K^{\prime}$ valley emissions to the SPP mode are identical to the mode spectrums shown in Fig. 2(a). The simulated Fourier space images for $\sigma^{ +}$ and $\sigma^{ -}$ excitations in Figs. 3(e) and 3(f) were obtained by performing the Fourier transform over the field distributions at the bottom plane of the HMM medium from TMM simulations for chiral dipoles. The experimental observations agree with simulated results and indicate the routing of emission from the ${{\rm WS}_2}$ excitons using HMMs. Additionally, the photoluminescence (PL) emission is observed only at the SPP mode owing to the experimentally permitted modes of the HMM by the available oil immersion objective for the visible region. However, this asymmetric dispersion is true for both SPPs and high-$k$ modes, as shown in the simulations in Fig. 2. The phenomenon of unidirectional routing for $\sigma^{+}$ and $\sigma^{-}$ is the same for both SPP and high-$k$ modes; however, only the experimentally accessible (${k_{{\rm MAX}}}\sim{1.4}$) SPP modes were measured in this study.

 

Fig. 3. $k$-space imaging of the optical valley Hall effect in the HMM. $k$-space images of valley emissions under the helicity-resolved condition show the asymmetric intensity distribution for both (a) $\sigma^{+}$ and (b) $\sigma^{ -}$. (c) and (d) are the line cut profiles along the ${k_x}$ showing the asymmetric intensity for both (a) $\sigma^{+}$ and (b) $\sigma^{-}$, respectively. (e) and (f) show simulated $k$-space images for both $\sigma^{+}$ and $\sigma^{-}$ excitations, respectively.

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A plasmonic thin film is also known for the directional coupling of a SPP wave for the incident chiral electromagnetic wave [3739]. To compare the directional routing properties of a HMM to that of a planar silver film, we investigate the effect of directional coupling of ${{\rm WS}_2}$ emission on the one-unit cell of the HMM (10 nm silver film covered with 10 nm ${{\rm Al}_2}{{\rm O}_3}$ film) and subsequently a monolayer of ${{\rm WS}_2}$ transferred to its surface. Figure 4 shows helicity-resolved $k$-space images for a thin silver film sample for both $\sigma^{+}$ and $\sigma^{-}$ helicities. Helicity-resolved valley polarization measurements on the thin silver film show (Fig. 4) much lower contrast in the weight of valley-polarized emissions in two transverse directions as compared to that in the seven-period HMM (Fig. 3). The complete set of helicity-resolved valley emission measurements for the one-unit cell HMM sample are shown in Fig. S6 of Supplement 1. The line cuts for both $\sigma^{+}$ and $\sigma^{-}$ excitations are shown in Figs. 4(c) and 4(d), respectively. The central spot at ${k_x}/{k_0} = {k_y}/{k_0} = 0$ is due to the incident laser intensity transmission through the thinner one-unit cell HMM sample. This is hardly observed in the case of the seven-period sample owing to the low direct transmission at $k = {0}$ and removed to show similarity to the simulation results. The valley specific emission directionality is less visible in the one-unit cell HMM sample. It is evident that the seven-period HMM shows (Fig. 3) promising valley contrast in comparison to the one-unit cell sample [Fig. 4(a)]. The lower contrast valley emission routing in the one-unit cell HMM can also be understood by analyzing the simulated mode spectrum by using the electric field expression for the chiral dipole located at the surface of the thin film sample. The mode spectrum of the one-unit cell shows broader SPP modes with low quality factor $Q$ as shown in Fig. 2(b) in comparison to those in the seven-period HMM in Fig. 2(a). The broader linewidth of the SPP mode in the one-unit cell HMM is due to its lower model index value in comparison to that in the seven-period HMM medium. The lower mode index value modes are known to be relatively more lossy and leakier whereas the higher mode index value SPP modes of the HMM are of high-quality factor, less lossy, and tightly confined in the HMM medium [54]. These factors could be the reason to support the fast-decaying SPP mode and hence lesser contrasting valley-resolved directionality in the one-unit HMM sample.

 

Fig. 4. $k$-space imaging of optical valley Hall effect in thin metallic film. (a) and (b) show valley-resolved $k$-space maps for the one-unit cell HMM sample for both $\sigma^{+}$ and $\sigma^{ -}$ helicities, respectively. (c) and (d) show line cuts along the ${k_x}$ directions corresponding to (c) $\sigma ^+$ and (d) $\sigma^ -$ helicities, respectively.

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These experimental observations demonstrate that the $K$ and $K^{\prime}$ valley excitonic emissions unidirectionally couple to the SPP mode of the HMM analogous to the photonic spin Hall effect [36,39]. The contrast between the intensity along ${-}{k_x}$ and ${+}{k_x}$ for both $\sigma^ +$ (and $\sigma^-$) is found to be more than 20%. This contrast for selective valley directional guiding is found to be significantly higher in comparison to the one layer of silver film.

In summary, we demonstrate efficient routing of the valley exciton emission using hyperbolic media by exploiting the high-$k$ states and the higher $Q$ factor SPP modes in them. This effect relies on the photonic spin Hall effect in HMMs. The experimental demonstration is found to be in very good agreement with the theoretical calculations. Through Fourier space imaging we are able to establish a valley-dependent routing of emission from a monolayer ${{\rm WS}_2}$ placed in the near field of the HMM. While nanostructured photonic systems can be used to realize similar effects, the use of a bulk substrate to realize such valley-polarized routing presents a simple and broadband approach for practical realization of valley-dependent optoelectronics.

Funding

Welch Foundation (A-1943); National Science Foundation (DMR-1709996); Army Research Office (W911NF-16-1-0256).

Acknowledgment

S. G., M. K., and V. M. M. conceived the experiments. S. G., M. K., and N. Y. fabricated the devices and performed the measurements. S. G. and M. K. performed data analysis. S. G., W. L., and G. S. A. carried out the theoretical analysis. All authors contributed to writing the paper and discussing the results.

Disclosures

The authors declare no conflicts of interest.

 

Please see Supplement 1 for supporting content.

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23. S. Guddala, R. Bushati, M. Li, A. B. Khanikaev, and V. M. Menon, “Valley selective optical control of excitons in 2D semiconductors using a chiral metasurface,” Opt. Mater. Express 9, 536–543 (2019). [CrossRef]  

24. S. Kim, Y. Lim, R. M. Kim, J. E. Fröch, T. N. Tran, K. T. Nam, and I. Aharonovich, “A single chiral nanoparticle induced valley polarization enhancement,” Small 16, 2003005 (2020). [CrossRef]  

25. Y. Kawaguchi, S. Guddala, K. Chen, A. Alù, V. Menon, and A. B. Khanikaev, “All-optical nonreciprocity due to valley polarization in transition metal dichalcogenides,” arXiv:2007.14934 (2020).

26. K. F. Mak, K. L. McGill, J. Park, and P. L. McEuen, “The valley Hall effect in MoS2 transistors,” Science 344, 1489–1492 (2014). [CrossRef]  

27. J. Lee, K. F. Mak, and J. Shan, “Electrical control of the valley Hall effect in bilayer MoS2 transistors,” Nat. Nanotechnol. 11, 421–425 (2016). [CrossRef]  

28. O. Bleu, D. D. Solnyshkov, and G. Malpuech, “Optical valley Hall effect based on transitional metal dichalcogenide cavity polaritons,” Phys. Rev. B 96, 165432 (2017). [CrossRef]  

29. N. Lundt, Ł. Dusanowski, E. Sedov, P. Stepanov, M. M. Glazov, S. Klembt, M. Klaas, J. Beierlein, Y. Qin, S. Tongay, M. Richard, A. V. Kavokin, S. Höfling, and C. Schneider, “Optical valley Hall effect for highly valley-coherent exciton-polaritons in an atomically thin semiconductor,” Nat. Nanotechnol. 14, 770–775 (2019). [CrossRef]  

30. M. Onga, Y. Zhang, T. Ideue, and Y. Iwasa, “Exciton Hall effect in monolayer MoS2,” Nat. Mater. 16, 1193–1197 (2017). [CrossRef]  

31. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006). [CrossRef]  

32. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007). [CrossRef]  

33. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012). [CrossRef]  

34. C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14, 063001 (2012). [CrossRef]  

35. A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. De Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522, 192–196 (2015). [CrossRef]  

36. P. V. Kapitanova, P. Ginzburg, F. J. Rodríguez-Fortuño, D. S. Filonov, P. M. Voroshilov, P. A. Belov, A. N. Poddubny, Y. S. Kivshar, G. A. Wurtz, and A. V. Zayats, “Photonic spin Hall effect in hyperbolic metamaterials for polarization-controlled routing of subwavelength modes,” Nat. Commun. 5, 3226 (2014). [CrossRef]  

37. K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348, 1448–1451 (2015). [CrossRef]  

38. T. Van Mechelen and Z. Jacob, “Universal spin-momentum locking of evanescent waves,” Optica 3, 118–126 (2016). [CrossRef]  

39. F. J. Rodríguez-fortuño, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Electromagnetic guided modes,” Science 340, 328–331 (2013). [CrossRef]  

40. C. Triolo, A. Cacciola, S. Patanè, R. Saija, S. Savasta, and F. Nori, “Spin-momentum locking in the near field of metal nanoparticles,” ACS Photonics 4, 2242–2249 (2017). [CrossRef]  

41. K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Optical momentum, spin, and angular momentum in dispersive media,” Phys. Rev. Lett. 119, 073901 (2017). [CrossRef]  

42. F. Alpeggiani, K. Y. Bliokh, F. Nori, and L. Kuipers, “Electromagnetic helicity in complex media,” Phys. Rev. Lett. 120, 243605 (2018). [CrossRef]  

43. M. F. Picardi, K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Alpeggiani, and F. Nori, “Angular momenta, helicity, and other properties of dielectric-fiber and metallic-wire modes,” Optica 5, 1016–1026 (2018). [CrossRef]  

44. E. D. Palik, Handbook of Optical Constants of Solids, Five-Volume Set: Handbook of Thermo-Optic Coefficients of Optical Materials with Applications (Elsevier Science, 1997).

45. T. Galfsky, H. N. S. Krishnamoorthy, W. Newman, E. E. Narimanov, Z. Jacob, and V. M. Menon, “Active hyperbolic metamaterials: enhanced spontaneous emission and light extraction,” Optica 2, 62–65 (2015). [CrossRef]  

46. M. A. Noginov, H. Li, Y. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35, 1863–1865 (2010). [CrossRef]  

47. T. Tumkur, G. Zhu, P. Black, Y. A. Barnakov, C. E. Bonner, and M. A. Noginov, “Control of spontaneous emission in a volume of functionalized hyperbolic metamaterial,” Appl. Phys. Lett. 99, 151115 (2011). [CrossRef]  

48. J. Kim, V. P. Drachev, Z. Jacob, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Improving the radiative decay rate for dye molecules with hyperbolic metamaterials,” Opt. Express 20, 8100–8116 (2012). [CrossRef]  

49. K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 3300 (2014). [CrossRef]  

50. K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015). [CrossRef]  

51. W. Liu, V. M. Menon, S. Gao, and G. S. Agarwal, “Chiral emission of electric dipoles coupled to optical hyperbolic materials,” Phys. Rev. B 100, 245428 (2019). [CrossRef]  

52. R. K. Yadav, W. Liu, S. R. K. Chaitanya Indukuri, A. B. Vasista, G. V. Pavan Kumar, G. S. Agarwal, and J. K. Basu, “Observation of photonic spin-momentum locking due to coupling of achiral metamaterials and quantum dots,” J. Phys. Condens. Matter 33, 015701 (2021). [CrossRef]  

53. P. Shekhar, J. Atkinson, and Z. Jacob, “Hyperbolic metamaterials: fundamentals and applications,” Nano Converg. 1, 1–17 (2014). [CrossRef]  

54. I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75, 241402 (2007). [CrossRef]  

55. T. Galfsky, Z. Sun, C. R. Considine, C. Chou, W. Ko, Y. Lee, E. E. Narimanov, and V. M. Menon, “Broadband enhancement of spontaneous emission in two-dimensional semiconductors using photonic hypercrystals,” Nano Lett. 16, 4940–4945 (2016). [CrossRef]  

References

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    [Crossref]
  22. L. Sun, C.-Y. Wang, A. Krasnok, J. Choi, J. Shi, J. S. Gomez-Diaz, A. Zepeda, S. Gwo, C.-K. Shih, A. Alù, and X. Li, “Separation of valley excitons in a MoS2 monolayer using a subwavelength asymmetric groove array,” Nat. Photonics 13, 180–184 (2019).
    [Crossref]
  23. S. Guddala, R. Bushati, M. Li, A. B. Khanikaev, and V. M. Menon, “Valley selective optical control of excitons in 2D semiconductors using a chiral metasurface,” Opt. Mater. Express 9, 536–543 (2019).
    [Crossref]
  24. S. Kim, Y. Lim, R. M. Kim, J. E. Fröch, T. N. Tran, K. T. Nam, and I. Aharonovich, “A single chiral nanoparticle induced valley polarization enhancement,” Small 16, 2003005 (2020).
    [Crossref]
  25. Y. Kawaguchi, S. Guddala, K. Chen, A. Alù, V. Menon, and A. B. Khanikaev, “All-optical nonreciprocity due to valley polarization in transition metal dichalcogenides,” arXiv:2007.14934 (2020).
  26. K. F. Mak, K. L. McGill, J. Park, and P. L. McEuen, “The valley Hall effect in MoS2 transistors,” Science 344, 1489–1492 (2014).
    [Crossref]
  27. J. Lee, K. F. Mak, and J. Shan, “Electrical control of the valley Hall effect in bilayer MoS2 transistors,” Nat. Nanotechnol. 11, 421–425 (2016).
    [Crossref]
  28. O. Bleu, D. D. Solnyshkov, and G. Malpuech, “Optical valley Hall effect based on transitional metal dichalcogenide cavity polaritons,” Phys. Rev. B 96, 165432 (2017).
    [Crossref]
  29. N. Lundt, Ł. Dusanowski, E. Sedov, P. Stepanov, M. M. Glazov, S. Klembt, M. Klaas, J. Beierlein, Y. Qin, S. Tongay, M. Richard, A. V. Kavokin, S. Höfling, and C. Schneider, “Optical valley Hall effect for highly valley-coherent exciton-polaritons in an atomically thin semiconductor,” Nat. Nanotechnol. 14, 770–775 (2019).
    [Crossref]
  30. M. Onga, Y. Zhang, T. Ideue, and Y. Iwasa, “Exciton Hall effect in monolayer MoS2,” Nat. Mater. 16, 1193–1197 (2017).
    [Crossref]
  31. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006).
    [Crossref]
  32. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
    [Crossref]
  33. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012).
    [Crossref]
  34. C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14, 063001 (2012).
    [Crossref]
  35. A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. De Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522, 192–196 (2015).
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    [Crossref]
  37. K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348, 1448–1451 (2015).
    [Crossref]
  38. T. Van Mechelen and Z. Jacob, “Universal spin-momentum locking of evanescent waves,” Optica 3, 118–126 (2016).
    [Crossref]
  39. F. J. Rodríguez-fortuño, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Electromagnetic guided modes,” Science 340, 328–331 (2013).
    [Crossref]
  40. C. Triolo, A. Cacciola, S. Patanè, R. Saija, S. Savasta, and F. Nori, “Spin-momentum locking in the near field of metal nanoparticles,” ACS Photonics 4, 2242–2249 (2017).
    [Crossref]
  41. K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Optical momentum, spin, and angular momentum in dispersive media,” Phys. Rev. Lett. 119, 073901 (2017).
    [Crossref]
  42. F. Alpeggiani, K. Y. Bliokh, F. Nori, and L. Kuipers, “Electromagnetic helicity in complex media,” Phys. Rev. Lett. 120, 243605 (2018).
    [Crossref]
  43. M. F. Picardi, K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Alpeggiani, and F. Nori, “Angular momenta, helicity, and other properties of dielectric-fiber and metallic-wire modes,” Optica 5, 1016–1026 (2018).
    [Crossref]
  44. E. D. Palik, Handbook of Optical Constants of Solids, Five-Volume Set: Handbook of Thermo-Optic Coefficients of Optical Materials with Applications (Elsevier Science, 1997).
  45. T. Galfsky, H. N. S. Krishnamoorthy, W. Newman, E. E. Narimanov, Z. Jacob, and V. M. Menon, “Active hyperbolic metamaterials: enhanced spontaneous emission and light extraction,” Optica 2, 62–65 (2015).
    [Crossref]
  46. M. A. Noginov, H. Li, Y. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35, 1863–1865 (2010).
    [Crossref]
  47. T. Tumkur, G. Zhu, P. Black, Y. A. Barnakov, C. E. Bonner, and M. A. Noginov, “Control of spontaneous emission in a volume of functionalized hyperbolic metamaterial,” Appl. Phys. Lett. 99, 151115 (2011).
    [Crossref]
  48. J. Kim, V. P. Drachev, Z. Jacob, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Improving the radiative decay rate for dye molecules with hyperbolic metamaterials,” Opt. Express 20, 8100–8116 (2012).
    [Crossref]
  49. K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 3300 (2014).
    [Crossref]
  50. K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
    [Crossref]
  51. W. Liu, V. M. Menon, S. Gao, and G. S. Agarwal, “Chiral emission of electric dipoles coupled to optical hyperbolic materials,” Phys. Rev. B 100, 245428 (2019).
    [Crossref]
  52. R. K. Yadav, W. Liu, S. R. K. Chaitanya Indukuri, A. B. Vasista, G. V. Pavan Kumar, G. S. Agarwal, and J. K. Basu, “Observation of photonic spin-momentum locking due to coupling of achiral metamaterials and quantum dots,” J. Phys. Condens. Matter 33, 015701 (2021).
    [Crossref]
  53. P. Shekhar, J. Atkinson, and Z. Jacob, “Hyperbolic metamaterials: fundamentals and applications,” Nano Converg. 1, 1–17 (2014).
    [Crossref]
  54. I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75, 241402 (2007).
    [Crossref]
  55. T. Galfsky, Z. Sun, C. R. Considine, C. Chou, W. Ko, Y. Lee, E. E. Narimanov, and V. M. Menon, “Broadband enhancement of spontaneous emission in two-dimensional semiconductors using photonic hypercrystals,” Nano Lett. 16, 4940–4945 (2016).
    [Crossref]

2021 (1)

R. K. Yadav, W. Liu, S. R. K. Chaitanya Indukuri, A. B. Vasista, G. V. Pavan Kumar, G. S. Agarwal, and J. K. Basu, “Observation of photonic spin-momentum locking due to coupling of achiral metamaterials and quantum dots,” J. Phys. Condens. Matter 33, 015701 (2021).
[Crossref]

2020 (2)

H. Zhang, B. Abhiraman, Q. Zhang, J. Miao, K. Jo, S. Roccasecca, M. W. Knight, A. R. Davoyan, and D. Jariwala, “Hybrid exciton-plasmon-polaritons in van der Waals semiconductor gratings,” Nat. Commun. 11, 3552 (2020).
[Crossref]

S. Kim, Y. Lim, R. M. Kim, J. E. Fröch, T. N. Tran, K. T. Nam, and I. Aharonovich, “A single chiral nanoparticle induced valley polarization enhancement,” Small 16, 2003005 (2020).
[Crossref]

2019 (4)

L. Sun, C.-Y. Wang, A. Krasnok, J. Choi, J. Shi, J. S. Gomez-Diaz, A. Zepeda, S. Gwo, C.-K. Shih, A. Alù, and X. Li, “Separation of valley excitons in a MoS2 monolayer using a subwavelength asymmetric groove array,” Nat. Photonics 13, 180–184 (2019).
[Crossref]

S. Guddala, R. Bushati, M. Li, A. B. Khanikaev, and V. M. Menon, “Valley selective optical control of excitons in 2D semiconductors using a chiral metasurface,” Opt. Mater. Express 9, 536–543 (2019).
[Crossref]

N. Lundt, Ł. Dusanowski, E. Sedov, P. Stepanov, M. M. Glazov, S. Klembt, M. Klaas, J. Beierlein, Y. Qin, S. Tongay, M. Richard, A. V. Kavokin, S. Höfling, and C. Schneider, “Optical valley Hall effect for highly valley-coherent exciton-polaritons in an atomically thin semiconductor,” Nat. Nanotechnol. 14, 770–775 (2019).
[Crossref]

W. Liu, V. M. Menon, S. Gao, and G. S. Agarwal, “Chiral emission of electric dipoles coupled to optical hyperbolic materials,” Phys. Rev. B 100, 245428 (2019).
[Crossref]

2018 (6)

F. Alpeggiani, K. Y. Bliokh, F. Nori, and L. Kuipers, “Electromagnetic helicity in complex media,” Phys. Rev. Lett. 120, 243605 (2018).
[Crossref]

M. F. Picardi, K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Alpeggiani, and F. Nori, “Angular momenta, helicity, and other properties of dielectric-fiber and metallic-wire modes,” Optica 5, 1016–1026 (2018).
[Crossref]

S. H. Gong, F. Alpeggiani, B. Sciacca, E. C. Garnett, and L. Kuipers, “Nanoscale chiral valley-photon interface through optical spin-orbit coupling,” Science 359, 443–447 (2018).
[Crossref]

T. Chervy, S. Azzini, E. Lorchat, S. Wang, Y. Gorodetski, J. A. Hutchison, S. Berciaud, T. W. Ebbesen, and C. Genet, “Room temperature chiral coupling of valley excitons with spin-momentum locked surface plasmons,” ACS Photonics 5, 1281–1287 (2018).
[Crossref]

A. L. Rakhmanov, A. O. Sboychakov, K. I. Kugel, A. V. Rozhkov, and F. Nori, “Spin-valley half-metal in systems with Fermi surface nesting,” Phys. Rev. B 98, 155141 (2018).
[Crossref]

W. Zheng, Y. Jiang, X. Hu, H. Li, Z. Zeng, X. Wang, and A. Pan, “Light emission properties of 2D transition metal dichalcogenides: fundamentals and applications,” Adv. Opt. Mater. 6, 1800420 (2018).
[Crossref]

2017 (7)

S. Dufferwiel, T. P. Lyons, D. D. Solnyshkov, A. A. P. Trichet, F. Withers, S. Schwarz, G. Malpuech, J. M. Smith, K. S. Novoselov, M. S. Skolnick, D. N. Krizhanovskii, and A. I. Tartakovskii, “Valley-addressable polaritons in atomically thin semiconductors,” Nat. Photonics 11, 497–501 (2017).
[Crossref]

Z. Sun, J. Gu, A. Ghazaryan, Z. Shotan, C. R. Considine, M. Dollar, B. Chakraborty, X. Liu, P. Ghaemi, S. Kéna-Cohen, and V. M. Menon, “Optical control of room-temperature valley polaritons,” Nat. Photonics 11, 491–496 (2017).
[Crossref]

A. V. Rozhkov, A. L. Rakhmanov, A. O. Sboychakov, K. I. Kugel, and F. Nori, “Spin-valley half-metal as a prospective material for spin valleytronics,” Phys. Rev. Lett. 119, 107601 (2017).
[Crossref]

O. Bleu, D. D. Solnyshkov, and G. Malpuech, “Optical valley Hall effect based on transitional metal dichalcogenide cavity polaritons,” Phys. Rev. B 96, 165432 (2017).
[Crossref]

M. Onga, Y. Zhang, T. Ideue, and Y. Iwasa, “Exciton Hall effect in monolayer MoS2,” Nat. Mater. 16, 1193–1197 (2017).
[Crossref]

C. Triolo, A. Cacciola, S. Patanè, R. Saija, S. Savasta, and F. Nori, “Spin-momentum locking in the near field of metal nanoparticles,” ACS Photonics 4, 2242–2249 (2017).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Optical momentum, spin, and angular momentum in dispersive media,” Phys. Rev. Lett. 119, 073901 (2017).
[Crossref]

2016 (4)

T. Van Mechelen and Z. Jacob, “Universal spin-momentum locking of evanescent waves,” Optica 3, 118–126 (2016).
[Crossref]

T. Galfsky, Z. Sun, C. R. Considine, C. Chou, W. Ko, Y. Lee, E. E. Narimanov, and V. M. Menon, “Broadband enhancement of spontaneous emission in two-dimensional semiconductors using photonic hypercrystals,” Nano Lett. 16, 4940–4945 (2016).
[Crossref]

J. Lee, K. F. Mak, and J. Shan, “Electrical control of the valley Hall effect in bilayer MoS2 transistors,” Nat. Nanotechnol. 11, 421–425 (2016).
[Crossref]

K. F. Mak and J. Shan, “Photonics and optoelectronics of 2D semiconductor transition metal dichalcogenides,” Nat. Photonics 10, 216–226 (2016).
[Crossref]

2015 (5)

X. Liu, T. Galfsky, Z. Sun, F. Xia, E. Lin, Y.-H. Lee, S. Kéna-Cohen, and V. M. Menon, “Strong light–matter coupling in two-dimensional atomic crystals,” Nat. Photonics 9, 30–34 (2015).
[Crossref]

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. De Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522, 192–196 (2015).
[Crossref]

K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348, 1448–1451 (2015).
[Crossref]

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

T. Galfsky, H. N. S. Krishnamoorthy, W. Newman, E. E. Narimanov, Z. Jacob, and V. M. Menon, “Active hyperbolic metamaterials: enhanced spontaneous emission and light extraction,” Optica 2, 62–65 (2015).
[Crossref]

2014 (4)

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 3300 (2014).
[Crossref]

P. Shekhar, J. Atkinson, and Z. Jacob, “Hyperbolic metamaterials: fundamentals and applications,” Nano Converg. 1, 1–17 (2014).
[Crossref]

P. V. Kapitanova, P. Ginzburg, F. J. Rodríguez-Fortuño, D. S. Filonov, P. M. Voroshilov, P. A. Belov, A. N. Poddubny, Y. S. Kivshar, G. A. Wurtz, and A. V. Zayats, “Photonic spin Hall effect in hyperbolic metamaterials for polarization-controlled routing of subwavelength modes,” Nat. Commun. 5, 3226 (2014).
[Crossref]

K. F. Mak, K. L. McGill, J. Park, and P. L. McEuen, “The valley Hall effect in MoS2 transistors,” Science 344, 1489–1492 (2014).
[Crossref]

2013 (3)

O. Lopez-Sanchez, D. Lembke, M. Kayci, A. Radenovic, and A. Kis, “Ultrasensitive photodetectors based on monolayer MoS2,” Nat. Nanotechnol. 8, 497–501 (2013).
[Crossref]

R. S. Sundaram, M. Engel, A. Lombardo, R. Krupke, A. C. Ferrari, P. Avouris, and M. Steiner, “Electroluminescence in single layer MoS2,” Nano Lett. 13, 1416–1421 (2013).
[Crossref]

F. J. Rodríguez-fortuño, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Electromagnetic guided modes,” Science 340, 328–331 (2013).
[Crossref]

2012 (8)

J. Kim, V. P. Drachev, Z. Jacob, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Improving the radiative decay rate for dye molecules with hyperbolic metamaterials,” Opt. Express 20, 8100–8116 (2012).
[Crossref]

Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman, and M. S. Strano, “Electronics and optoelectronics of two-dimensional transition metal dichalcogenides,” Nat. Nanotechnol. 7, 699–712 (2012).
[Crossref]

D. Xiao, G. Bin Liu, W. Feng, X. Xu, and W. Yao, “Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides,” Phys. Rev. Lett. 108, 196802 (2012).
[Crossref]

T. Cao, G. Wang, W. Han, H. Ye, C. Zhu, J. Shi, Q. Niu, P. Tan, E. Wang, B. Liu, and J. Feng, “Valley-selective circular dichroism of monolayer molybdenum disulphide,” Nat. Commun. 3, 885–887 (2012).
[Crossref]

K. F. Mak, K. He, J. Shan, and T. F. Heinz, “Control of valley polarization in monolayer MoS2 by optical helicity,” Nat. Nanotechnol. 7, 494–498 (2012).
[Crossref]

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotechnol. 7, 490–493 (2012).
[Crossref]

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012).
[Crossref]

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14, 063001 (2012).
[Crossref]

2011 (1)

T. Tumkur, G. Zhu, P. Black, Y. A. Barnakov, C. E. Bonner, and M. A. Noginov, “Control of spontaneous emission in a volume of functionalized hyperbolic metamaterial,” Appl. Phys. Lett. 99, 151115 (2011).
[Crossref]

2010 (3)

M. A. Noginov, H. Li, Y. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35, 1863–1865 (2010).
[Crossref]

K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, “Atomically thin MoS2: a new direct-gap semiconductor,” Phys. Rev. Lett. 105, 136805 (2010).
[Crossref]

A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C.-Y. Chim, G. Galli, and F. Wang, “Emerging photoluminescence in monolayer MoS2,” Nano Lett. 10, 1271–1275 (2010).
[Crossref]

2008 (1)

W. Yao, D. Xiao, and Q. Niu, “Valley-dependent optoelectronics from inversion symmetry breaking,” Phys. Rev. B 77, 235406 (2008).
[Crossref]

2007 (2)

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[Crossref]

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75, 241402 (2007).
[Crossref]

2006 (1)

2005 (1)

K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov, and A. K. Geim, “Two-dimensional atomic crystals,” Proc. Natl. Acad. Sci. USA 102, 10451–10453 (2005).
[Crossref]

Abhiraman, B.

H. Zhang, B. Abhiraman, Q. Zhang, J. Miao, K. Jo, S. Roccasecca, M. W. Knight, A. R. Davoyan, and D. Jariwala, “Hybrid exciton-plasmon-polaritons in van der Waals semiconductor gratings,” Nat. Commun. 11, 3552 (2020).
[Crossref]

Agarwal, G. S.

R. K. Yadav, W. Liu, S. R. K. Chaitanya Indukuri, A. B. Vasista, G. V. Pavan Kumar, G. S. Agarwal, and J. K. Basu, “Observation of photonic spin-momentum locking due to coupling of achiral metamaterials and quantum dots,” J. Phys. Condens. Matter 33, 015701 (2021).
[Crossref]

W. Liu, V. M. Menon, S. Gao, and G. S. Agarwal, “Chiral emission of electric dipoles coupled to optical hyperbolic materials,” Phys. Rev. B 100, 245428 (2019).
[Crossref]

Aharonovich, I.

S. Kim, Y. Lim, R. M. Kim, J. E. Fröch, T. N. Tran, K. T. Nam, and I. Aharonovich, “A single chiral nanoparticle induced valley polarization enhancement,” Small 16, 2003005 (2020).
[Crossref]

Alekseyev, L. V.

Alpeggiani, F.

F. Alpeggiani, K. Y. Bliokh, F. Nori, and L. Kuipers, “Electromagnetic helicity in complex media,” Phys. Rev. Lett. 120, 243605 (2018).
[Crossref]

M. F. Picardi, K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Alpeggiani, and F. Nori, “Angular momenta, helicity, and other properties of dielectric-fiber and metallic-wire modes,” Optica 5, 1016–1026 (2018).
[Crossref]

S. H. Gong, F. Alpeggiani, B. Sciacca, E. C. Garnett, and L. Kuipers, “Nanoscale chiral valley-photon interface through optical spin-orbit coupling,” Science 359, 443–447 (2018).
[Crossref]

Alù, A.

L. Sun, C.-Y. Wang, A. Krasnok, J. Choi, J. Shi, J. S. Gomez-Diaz, A. Zepeda, S. Gwo, C.-K. Shih, A. Alù, and X. Li, “Separation of valley excitons in a MoS2 monolayer using a subwavelength asymmetric groove array,” Nat. Photonics 13, 180–184 (2019).
[Crossref]

Y. Kawaguchi, S. Guddala, K. Chen, A. Alù, V. Menon, and A. B. Khanikaev, “All-optical nonreciprocity due to valley polarization in transition metal dichalcogenides,” arXiv:2007.14934 (2020).

Atkinson, J.

P. Shekhar, J. Atkinson, and Z. Jacob, “Hyperbolic metamaterials: fundamentals and applications,” Nano Converg. 1, 1–17 (2014).
[Crossref]

Avouris, P.

R. S. Sundaram, M. Engel, A. Lombardo, R. Krupke, A. C. Ferrari, P. Avouris, and M. Steiner, “Electroluminescence in single layer MoS2,” Nano Lett. 13, 1416–1421 (2013).
[Crossref]

Avrutsky, I.

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75, 241402 (2007).
[Crossref]

Azzini, S.

T. Chervy, S. Azzini, E. Lorchat, S. Wang, Y. Gorodetski, J. A. Hutchison, S. Berciaud, T. W. Ebbesen, and C. Genet, “Room temperature chiral coupling of valley excitons with spin-momentum locked surface plasmons,” ACS Photonics 5, 1281–1287 (2018).
[Crossref]

Barnakov, Y. A.

T. Tumkur, G. Zhu, P. Black, Y. A. Barnakov, C. E. Bonner, and M. A. Noginov, “Control of spontaneous emission in a volume of functionalized hyperbolic metamaterial,” Appl. Phys. Lett. 99, 151115 (2011).
[Crossref]

M. A. Noginov, H. Li, Y. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35, 1863–1865 (2010).
[Crossref]

Basu, J. K.

R. K. Yadav, W. Liu, S. R. K. Chaitanya Indukuri, A. B. Vasista, G. V. Pavan Kumar, G. S. Agarwal, and J. K. Basu, “Observation of photonic spin-momentum locking due to coupling of achiral metamaterials and quantum dots,” J. Phys. Condens. Matter 33, 015701 (2021).
[Crossref]

Beierlein, J.

N. Lundt, Ł. Dusanowski, E. Sedov, P. Stepanov, M. M. Glazov, S. Klembt, M. Klaas, J. Beierlein, Y. Qin, S. Tongay, M. Richard, A. V. Kavokin, S. Höfling, and C. Schneider, “Optical valley Hall effect for highly valley-coherent exciton-polaritons in an atomically thin semiconductor,” Nat. Nanotechnol. 14, 770–775 (2019).
[Crossref]

Bekshaev, A. Y.

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Optical momentum, spin, and angular momentum in dispersive media,” Phys. Rev. Lett. 119, 073901 (2017).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 3300 (2014).
[Crossref]

Belov, P. A.

P. V. Kapitanova, P. Ginzburg, F. J. Rodríguez-Fortuño, D. S. Filonov, P. M. Voroshilov, P. A. Belov, A. N. Poddubny, Y. S. Kivshar, G. A. Wurtz, and A. V. Zayats, “Photonic spin Hall effect in hyperbolic metamaterials for polarization-controlled routing of subwavelength modes,” Nat. Commun. 5, 3226 (2014).
[Crossref]

Berciaud, S.

T. Chervy, S. Azzini, E. Lorchat, S. Wang, Y. Gorodetski, J. A. Hutchison, S. Berciaud, T. W. Ebbesen, and C. Genet, “Room temperature chiral coupling of valley excitons with spin-momentum locked surface plasmons,” ACS Photonics 5, 1281–1287 (2018).
[Crossref]

Bin Liu, G.

D. Xiao, G. Bin Liu, W. Feng, X. Xu, and W. Yao, “Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides,” Phys. Rev. Lett. 108, 196802 (2012).
[Crossref]

Black, P.

T. Tumkur, G. Zhu, P. Black, Y. A. Barnakov, C. E. Bonner, and M. A. Noginov, “Control of spontaneous emission in a volume of functionalized hyperbolic metamaterial,” Appl. Phys. Lett. 99, 151115 (2011).
[Crossref]

Bleu, O.

O. Bleu, D. D. Solnyshkov, and G. Malpuech, “Optical valley Hall effect based on transitional metal dichalcogenide cavity polaritons,” Phys. Rev. B 96, 165432 (2017).
[Crossref]

Bliokh, K. Y.

F. Alpeggiani, K. Y. Bliokh, F. Nori, and L. Kuipers, “Electromagnetic helicity in complex media,” Phys. Rev. Lett. 120, 243605 (2018).
[Crossref]

M. F. Picardi, K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Alpeggiani, and F. Nori, “Angular momenta, helicity, and other properties of dielectric-fiber and metallic-wire modes,” Optica 5, 1016–1026 (2018).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Optical momentum, spin, and angular momentum in dispersive media,” Phys. Rev. Lett. 119, 073901 (2017).
[Crossref]

K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348, 1448–1451 (2015).
[Crossref]

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 3300 (2014).
[Crossref]

Boltasseva, A.

Bonner, C. E.

T. Tumkur, G. Zhu, P. Black, Y. A. Barnakov, C. E. Bonner, and M. A. Noginov, “Control of spontaneous emission in a volume of functionalized hyperbolic metamaterial,” Appl. Phys. Lett. 99, 151115 (2011).
[Crossref]

M. A. Noginov, H. Li, Y. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35, 1863–1865 (2010).
[Crossref]

Booth, T. J.

K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov, and A. K. Geim, “Two-dimensional atomic crystals,” Proc. Natl. Acad. Sci. USA 102, 10451–10453 (2005).
[Crossref]

Bushati, R.

Cacciola, A.

C. Triolo, A. Cacciola, S. Patanè, R. Saija, S. Savasta, and F. Nori, “Spin-momentum locking in the near field of metal nanoparticles,” ACS Photonics 4, 2242–2249 (2017).
[Crossref]

Cao, T.

T. Cao, G. Wang, W. Han, H. Ye, C. Zhu, J. Shi, Q. Niu, P. Tan, E. Wang, B. Liu, and J. Feng, “Valley-selective circular dichroism of monolayer molybdenum disulphide,” Nat. Commun. 3, 885–887 (2012).
[Crossref]

Chaitanya Indukuri, S. R. K.

R. K. Yadav, W. Liu, S. R. K. Chaitanya Indukuri, A. B. Vasista, G. V. Pavan Kumar, G. S. Agarwal, and J. K. Basu, “Observation of photonic spin-momentum locking due to coupling of achiral metamaterials and quantum dots,” J. Phys. Condens. Matter 33, 015701 (2021).
[Crossref]

Chakraborty, B.

Z. Sun, J. Gu, A. Ghazaryan, Z. Shotan, C. R. Considine, M. Dollar, B. Chakraborty, X. Liu, P. Ghaemi, S. Kéna-Cohen, and V. M. Menon, “Optical control of room-temperature valley polaritons,” Nat. Photonics 11, 491–496 (2017).
[Crossref]

Chen, K.

Y. Kawaguchi, S. Guddala, K. Chen, A. Alù, V. Menon, and A. B. Khanikaev, “All-optical nonreciprocity due to valley polarization in transition metal dichalcogenides,” arXiv:2007.14934 (2020).

Chervy, T.

T. Chervy, S. Azzini, E. Lorchat, S. Wang, Y. Gorodetski, J. A. Hutchison, S. Berciaud, T. W. Ebbesen, and C. Genet, “Room temperature chiral coupling of valley excitons with spin-momentum locked surface plasmons,” ACS Photonics 5, 1281–1287 (2018).
[Crossref]

Chim, C.-Y.

A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C.-Y. Chim, G. Galli, and F. Wang, “Emerging photoluminescence in monolayer MoS2,” Nano Lett. 10, 1271–1275 (2010).
[Crossref]

Choi, J.

L. Sun, C.-Y. Wang, A. Krasnok, J. Choi, J. Shi, J. S. Gomez-Diaz, A. Zepeda, S. Gwo, C.-K. Shih, A. Alù, and X. Li, “Separation of valley excitons in a MoS2 monolayer using a subwavelength asymmetric groove array,” Nat. Photonics 13, 180–184 (2019).
[Crossref]

Chou, C.

T. Galfsky, Z. Sun, C. R. Considine, C. Chou, W. Ko, Y. Lee, E. E. Narimanov, and V. M. Menon, “Broadband enhancement of spontaneous emission in two-dimensional semiconductors using photonic hypercrystals,” Nano Lett. 16, 4940–4945 (2016).
[Crossref]

Coleman, J. N.

Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman, and M. S. Strano, “Electronics and optoelectronics of two-dimensional transition metal dichalcogenides,” Nat. Nanotechnol. 7, 699–712 (2012).
[Crossref]

Considine, C. R.

Z. Sun, J. Gu, A. Ghazaryan, Z. Shotan, C. R. Considine, M. Dollar, B. Chakraborty, X. Liu, P. Ghaemi, S. Kéna-Cohen, and V. M. Menon, “Optical control of room-temperature valley polaritons,” Nat. Photonics 11, 491–496 (2017).
[Crossref]

T. Galfsky, Z. Sun, C. R. Considine, C. Chou, W. Ko, Y. Lee, E. E. Narimanov, and V. M. Menon, “Broadband enhancement of spontaneous emission in two-dimensional semiconductors using photonic hypercrystals,” Nano Lett. 16, 4940–4945 (2016).
[Crossref]

Cortes, C. L.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14, 063001 (2012).
[Crossref]

Cui, X.

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotechnol. 7, 490–493 (2012).
[Crossref]

Dai, J.

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotechnol. 7, 490–493 (2012).
[Crossref]

Davoyan, A. R.

H. Zhang, B. Abhiraman, Q. Zhang, J. Miao, K. Jo, S. Roccasecca, M. W. Knight, A. R. Davoyan, and D. Jariwala, “Hybrid exciton-plasmon-polaritons in van der Waals semiconductor gratings,” Nat. Commun. 11, 3552 (2020).
[Crossref]

De Leon, N. P.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. De Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522, 192–196 (2015).
[Crossref]

Devlin, R. C.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. De Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522, 192–196 (2015).
[Crossref]

Dibos, A.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. De Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522, 192–196 (2015).
[Crossref]

Dollar, M.

Z. Sun, J. Gu, A. Ghazaryan, Z. Shotan, C. R. Considine, M. Dollar, B. Chakraborty, X. Liu, P. Ghaemi, S. Kéna-Cohen, and V. M. Menon, “Optical control of room-temperature valley polaritons,” Nat. Photonics 11, 491–496 (2017).
[Crossref]

Drachev, V. P.

Dryden, D.

Dufferwiel, S.

S. Dufferwiel, T. P. Lyons, D. D. Solnyshkov, A. A. P. Trichet, F. Withers, S. Schwarz, G. Malpuech, J. M. Smith, K. S. Novoselov, M. S. Skolnick, D. N. Krizhanovskii, and A. I. Tartakovskii, “Valley-addressable polaritons in atomically thin semiconductors,” Nat. Photonics 11, 497–501 (2017).
[Crossref]

Dusanowski, L.

N. Lundt, Ł. Dusanowski, E. Sedov, P. Stepanov, M. M. Glazov, S. Klembt, M. Klaas, J. Beierlein, Y. Qin, S. Tongay, M. Richard, A. V. Kavokin, S. Höfling, and C. Schneider, “Optical valley Hall effect for highly valley-coherent exciton-polaritons in an atomically thin semiconductor,” Nat. Nanotechnol. 14, 770–775 (2019).
[Crossref]

Ebbesen, T. W.

T. Chervy, S. Azzini, E. Lorchat, S. Wang, Y. Gorodetski, J. A. Hutchison, S. Berciaud, T. W. Ebbesen, and C. Genet, “Room temperature chiral coupling of valley excitons with spin-momentum locked surface plasmons,” ACS Photonics 5, 1281–1287 (2018).
[Crossref]

Elser, J.

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75, 241402 (2007).
[Crossref]

Engel, M.

R. S. Sundaram, M. Engel, A. Lombardo, R. Krupke, A. C. Ferrari, P. Avouris, and M. Steiner, “Electroluminescence in single layer MoS2,” Nano Lett. 13, 1416–1421 (2013).
[Crossref]

Feng, J.

T. Cao, G. Wang, W. Han, H. Ye, C. Zhu, J. Shi, Q. Niu, P. Tan, E. Wang, B. Liu, and J. Feng, “Valley-selective circular dichroism of monolayer molybdenum disulphide,” Nat. Commun. 3, 885–887 (2012).
[Crossref]

Feng, W.

D. Xiao, G. Bin Liu, W. Feng, X. Xu, and W. Yao, “Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides,” Phys. Rev. Lett. 108, 196802 (2012).
[Crossref]

Ferrari, A. C.

R. S. Sundaram, M. Engel, A. Lombardo, R. Krupke, A. C. Ferrari, P. Avouris, and M. Steiner, “Electroluminescence in single layer MoS2,” Nano Lett. 13, 1416–1421 (2013).
[Crossref]

Filonov, D. S.

P. V. Kapitanova, P. Ginzburg, F. J. Rodríguez-Fortuño, D. S. Filonov, P. M. Voroshilov, P. A. Belov, A. N. Poddubny, Y. S. Kivshar, G. A. Wurtz, and A. V. Zayats, “Photonic spin Hall effect in hyperbolic metamaterials for polarization-controlled routing of subwavelength modes,” Nat. Commun. 5, 3226 (2014).
[Crossref]

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S. Kim, Y. Lim, R. M. Kim, J. E. Fröch, T. N. Tran, K. T. Nam, and I. Aharonovich, “A single chiral nanoparticle induced valley polarization enhancement,” Small 16, 2003005 (2020).
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E. D. Palik, Handbook of Optical Constants of Solids, Five-Volume Set: Handbook of Thermo-Optic Coefficients of Optical Materials with Applications (Elsevier Science, 1997).

Supplementary Material (1)

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» Supplement 1       Supplementary information on Optical Analog of Valley Hall Effect of 2D Excitons in Hyperbolic Metamaterial

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Figures (4)

Fig. 1.
Fig. 1. Optical valley Hall effect demonstration in HMM. (a) Schematic of Valley polarization in ${{\rm WS}_2}$ monolayer, where $K$ and $K^{\prime}$ valleys can be populated independently with $\sigma^{+}$ and $\sigma^{-}$ polarizations, respectively. (b) Schematic of the seven-period HMM composed of ${\rm Ag}/{{\rm Al}_2}{{\rm O}_3}$ shows a monolayer of ${{\rm WS}_2}$ transferred to its surface layer. Electric field intensity distribution of the high $k$-modes for the dipole radiating in the near field of the HMM surface for (c) linear polarization, and (d) and (e) are right- and left-handed circular polarizations, respectively.
Fig. 2.
Fig. 2. Vectorial field of high-$k$ modes in the HMM. Mode spectrum calculated from TMM simulations for circular dipoles with opposite helicities in the $xz$ plane of (a) 7 P HMM sample and (b) one-unit HMM. The magnetic field ${{ H}_y}$ component intensity distribution in the HMM for linear dipole with polarization orientation along (c) horizontal $P=\hat x$ and (d) vertical $P=\hat z$. (e) Left-handed circularly polarized dipole $P=(\hat x + i\hat z)/\sqrt 2$ and (f) right-handed circularly polarized dipole $P=(\hat x-i \hat z \sqrt 2$. These simulations were carried out at the emission wavelength of the exciton (620 nm).
Fig. 3.
Fig. 3. $k$-space imaging of the optical valley Hall effect in the HMM. $k$-space images of valley emissions under the helicity-resolved condition show the asymmetric intensity distribution for both (a) $\sigma^{+}$ and (b) $\sigma^{ -}$. (c) and (d) are the line cut profiles along the ${k_x}$ showing the asymmetric intensity for both (a) $\sigma^{+}$ and (b) $\sigma^{-}$, respectively. (e) and (f) show simulated $k$-space images for both $\sigma^{+}$ and $\sigma^{-}$ excitations, respectively.
Fig. 4.
Fig. 4. $k$-space imaging of optical valley Hall effect in thin metallic film. (a) and (b) show valley-resolved $k$-space maps for the one-unit cell HMM sample for both $\sigma^{+}$ and $\sigma^{ -}$ helicities, respectively. (c) and (d) show line cuts along the ${k_x}$ directions corresponding to (c) $\sigma ^+$ and (d) $\sigma^ -$ helicities, respectively.