Maxwell's equations, formulated 150 years ago, ultimately describe properties of light, from classical electromagnetism to quantum and relativistic aspects. The latter ones result in remarkable geometric and topological phenomena related to the spin-1 massless nature of photons. By analyzing fundamental spin properties of Maxwell waves, we show that free-space light exhibits an intrinsic quantum spin Hall effect, i.e., surface modes with strong spin-momentum locking. These modes are evanescent waves that form, e.g., surface plasmonpolaritons at vacuum-metal interfaces. Our findings illuminate the unusual transverse spin in evanescent waves and explain recent experiments demonstrating the transverse spin-direction locking in the excitation of surface optical modes. This deepens our understanding of Maxwell's theory, reveals analogies with topological insulators for electrons, and offers applications for robust spin-directional optical interfaces.Solid-state physics exhibits a family of Hall effects with remarkable physical properties. The usual Hall effect (HE) and quantum Hall effect (QHE) appear in the presence of an external magnetic field, which breaks the time-reversal ( T ) symmetry of the system. The HE induces charge current orthogonal to both the magnetic field and an applied electric field, whereas the QHE [1] involves distinct topological electron states, with unidirectional edge modes (chargemomentum locking), characterized by the topological Chern number [2].The intrinsic spin Hall effect (SHE) can occur in T -symmetric electron systems with spinorbit interactions. It produces a spin-dependent transport of electrons orthogonal to the external driving force [3,4]. There is also the quantum spin Hall effect (QSHE) [5,6], which is characterized by unidirectional edge spin transport, i.e., edge states with opposite spins propagating in opposite directions. Such topological states with spin-momentum locking gave rise to a new class of materials: topological insulators [7,8].Alongside the extensive condensed-matter studies of electron Hall effects, their photonic counterparts have been found in various optical systems. In particular, both the HE [9] the QHE with unidirectional edge propagation [10,11] have been reported in magneto-optical systems with broken T -symmetry. Furthermore, because photons are relativistic spin-1 particles, they naturally exhibit intrinsic spin-orbit interaction effects, including Berry phase [12] and the SHE [13][14][15] stemming from fundamental spin properties of Maxwell equations [16].The only missing part in the above optical Hall effects is the QSHE for photons. Recently, it was suggested that photonic topological insulators can be created in complex metamaterials structures [17][18][19]. Here we show that pure free-space light already possesses intrinsic QSHE, and simple natural materials (such as metals supporting surface plasmon-polariton modes) exhibit some features resembling topological insulators. We show that the recently discovered transverse spin in evanescent wav...

Nonlinear nanophotonics is a rapidly developing field with many useful applications for a design of nonlinear nanoantennas, light sources, nanolasers, sensors, and ultrafast miniature metadevices. A tight confinement of the local electromagnetic fields in resonant photonic nanostructures can boost nonlinear optical effects, thus offering versatile opportunities for subwavelength control of light. To achieve the desired functionalities, it is essential to gain flexible control over the near-and far-field properties of nanostructures. Thus, both modal and multipolar analyses are widely exploited for engineering nonlinear scattering from resonant nanoscale elements, in particular for enhancing the near-field interaction, tailoring the far-field multipolar interference, and optimization of the radiation directionality. Here, we review the recent advances in this recently emerged research field ranging from metallic structures exhibiting localized plasmonic resonances to hybrid metal-dielectric and alldielectric nanostructures driven by Mie-type multipolar resonances and optically-induced magnetic response.

Rapidly growing demands for fast information processing have launched a race for creating compact and highly efficient optical devices that can reliably transmit signals without losses. Recently discovered topological phases of light provide novel opportunities for photonic devices robust against scattering losses and disorder. Combining these topological photonic structures with nonlinear effects will unlock advanced functionalities such as magnet-free nonreciprocity and active tunability. Here, we introduce the emerging field of nonlinear topological photonics and highlight the recent developments in bridging the physics of topological phases with nonlinear optics. This includes the design of novel photonic platforms which combine topological phases of light with appreciable nonlinear response, self-interaction effects leading to edge solitons in topological photonic lattices, frequency conversion, active photonic structures exhibiting lasing from topologically protected modes, and many-body quantum topological phases of light. We also chart future research directions discussing device applications such as mode stabilization in lasers, parametric amplifiers protected against feedback, and ultrafast optical switches employing topological waveguides.

The discovery of two-dimensional topological photonic systems has transformed our views on electromagnetic propagation and scattering of classical waves, and a quest for similar states in three dimensions, known to exist in condensed matter systems, has been put forward. Here we demonstrate that symmetry protected three-dimensional topological states can be engineered in an all-dielectric platform with the electromagnetic duality between electric and magnetic fields ensured by the structure design. Magneto-electric coupling playing the role of a synthetic gauge field leads to a topological transition to an "insulating" regime with a complete three-dimensional photonic bandgap. An emergence of surface states with conical Dirac dispersion and spin-locking is unimpeded. Robust propagation of surface states along two-dimensional domain walls defined by the reversal of magneto-electric coupling is confirmed numerically by first principle studies. It is shown that the proposed system represents a table-top platform for emulating relativistic physics of massive Dirac fermions and the surface states revealed can be interpreted as Jackiw-Rebbi states confined to the interface between two domains with opposite particle masses.Following the footsteps of condensed matter topological systems, a significant progress has been recently made in understanding and realizing topological states for bosons [ 1,2,3,4,5] and in classical systems [ 6, 7, 8, 9, 10, 11, 12, 13, 14 15, 16, 17, 18, 19, 20, 21, 22, 23]. Unlike fermionic systems, achieving topological order for bosons meets several limitations. In particular, while fermions can support robust topological phases protected by time-reversal (TR) symmetry alone, in bosonic and classical systems TR symmetry is not sufficient to protect any nontrivial topological phases [ 24]. Consequently, the traditional approach to engineering topological order in classical systems relies on removal of TR symmetry. This approach requires either using magnetic materials, e.g. in magnetic-photonic crystals emulating Quantum Hall effect (QHE) which were successfully realized in 2D [ 9,25] and has also been recently extended theoretically to three dimensions [ 26], or temporal modulation emulating the effect of external magnetic field [ 27,28,29,17,30,13].

The quest for nanoscale light sources with designer radiation patterns and polarization has motivated the development of nanoantennas that interact strongly with the incoming light and are able to transform its frequency, radiation and polarization patterns. Here, we demonstrate dielectric AlGaAs nanoantennas for efficient second harmonic generation, enabling the control of both directionality and polarization of 1