Sharp electromagnetic resonances play an essential role in physics in general and optics in particular. The last decades have witnessed the successful developments of high-quality (Q) resonances in microcavities operating below the light line, which however is fundamentally challenging to access from free space. Alternatively, metasurface-based bound states in the continuum (BICs) offer a complementary solution of creating high-Q resonances in devices operating above the light line, yet the experimentally demonstrated Q factors under normal excitations are still limited. Here, we present the realizations of quasi-BIC under normal excitation with a record Q factor up to 18 511 by engineering the symmetry properties and the number of the unit cells in all-dielectric metasurface platforms. The high-Q quasi-BICs exhibit exceptionally high conversion efficiency for the third harmonic generation and even enable the second harmonic generation in Si metasurfaces. Such ultrasharp resonances achieved in this work may immediately boost the performances of BICs in a plethora of fundamental research and device applications, e.g., cavity QED, biosensing, nanolasing, and quantum light generations.

We give new characterizations of the algebra L n (F q n ) formed by all linearized polynomials over the finite field F q n after briefly surveying some known ones. One isomorphism we construct is between L n (F q n ) and the composition algebra F ∨ q n ⊗ Fq F q n . The other isomorphism we construct is between L n (F q n ) and the so-called Dickson matrix algebra D n (F q n ). We also further study the relations between a linearized polynomial and its associated Dickson matrix, generalizing a well-known criterion of Dickson on linearized permutation polynomials. Adjugate polynomial of a linearized polynomial is then introduced, and connections between them are discussed. Both of the new characterizations can bring us more simple approaches to establish a special form of representations of linearized polynomials proposed recently by several authors. Structure of the subalgebra L n (F q m ) which are formed by all linearized polynomials over a subfield F q m of F q n where m|n are also described.

The Berry curvature in the band structure of transition metal dichalcogenides (TMDs) introduces a valley-dependent effective magnetic field, which induces the valley Hall effect (VHE). Similar to the ordinary Hall effect, the VHE spatially separates carriers or excitons, depending on their valley index, and accumulates them at opposite sample edges. The VHE can play a key role in valleytronic devices, but previous observations of the VHE have been limited to cryogenic temperatures. Here, we report a demonstration of the VHE of interlayer excitons in a MoS2/WSe2 heterostructure at room temperature. We monitored the in-plane propagation of interlayer excitons through photoluminescence mapping and observed their spatial separation into two opposite transverse directions that depended on the valley index of the excitons. Our theoretical simulations reproduced the salient features of these observations. Our demonstration of the robust interlayer exciton VHE at room temperature, enabled by their intrinsically long lifetimes, will open up realistic possibilities for the development of opto-valleytronic devices based on TMD heterostructures.

A class of bilinear permutation polynomials over a finite field of characteristic 2 was constructed in a recursive manner recently which involved some other constructions as special cases. We determine the compositional inverses of them based on a direct sum decomposition of the finite field. The result generalizes that in [R.S. Coulter, M. Henderson, The compositional inverse of a class of permutation polynomials over a finite field, Bull. Austral. Math. Soc. 65 (2002) 521-526].