The realization of lasers as small as possible has been one of the long-standing goals of the laser physics and quantum optics communities. Among multitudes of recent small cavities, the one-dimensional nanobeam cavity has been actively investigated as one of the most attractive candidates for effective photon confinement thanks to its simple geometry. However, the current injection into the ultra-small nano-resonator without critically degrading the quality factor remains still unanswered. Here we report an electrically driven, one-dimensional, photonic-well, single-mode, room-temperature nanobeam laser whose footprint approaches the smallest possible value. The small physical volume of ~4.6 × 0.61 × 0.28 μm3 (~8.2(λ n−1)3) was realized through the introduction of a Gaussian-like photonic well made of only 11 air holes. In addition, a low threshold current of ~5 μA was observed from a three-cell nanobeam cavity at room temperature. The simple one-dimensional waveguide nature of the nanobeam enables straightforward integration with other photonic applications such as photonic integrated circuits and quantum information devices.

If D is a tame central division algebra over a Henselian valued field F, then the valuation on D yields an associated graded ring GD which is a graded division ring and is also central and graded simple over GF. After proving some properties of graded central simple algebras over a graded field (including a cohomological characterization of its graded Brauer group), it is proved that the map D → GD g yields an index-preserving isomorphism from the tame part of the Brauer group of F to the graded Brauer group of GF. This isomorphism is shown to be functorial with respect to field extensions and corestrictions, and using this it is shown that there is a correspondence between F-subalgebras of D (with center tame over F) and graded GF-subalgebras of GD. Associated to the valuation on D there is a filtration of D by the principal fractional ideals of the valuation ring, which allows one to build an associated graded ring GD = γ∈ D GD γ , where GD 0 = D and the grade group of GD is precisely the value group D of D. Furthermore, GD is a graded division ring, i.e., its homogeneous elements are all units. In addition, as shown in [B2], the total ordering on D allows one to define a valuation on GD which extends to the ring of quotients QGD of GD, which is a division algebra. The valued division algebra QGD is usually not isomorphic to D, not even after Henselization, but we will see that their structures are closely related. The very presence of a valuation on QGD suggests that not so much is lost in the passage from D to its graded ring GD, even though GD appears to have a much simpler structure than D. We will show, in fact, that if D is tame then it is completely determined by GD, and its subalgebra structure is faithfully mirrored in that of GD.Specifically, let TBr F denote the tame part of the Brauer group of the Henselian field F and let GBr GF denote the graded Brauer group of the graded field GF determined by the valuation on F. We will show in Theorem 5.3 that the map D → GD g gives a Schur-index-preserving group isomorphism TBr F → GBr GF , which (see Corollary 5.7 and Theorem 6.1) is functorial with respect to scalar extensions and corestrictions. The index-preserving and functorial properties allow us to deduce (see Theorem 5.9) that if K is a tame valued field extension of F, and D and A are tame division algebras with center F, then K (resp. A) embeds in D iff GK (resp. GA) embeds in GD.These results show that much of what is known about tame valued division algebras can be carried over readily to graded division algebras finitedimensional over their centers, when the grade group is torsion-free (as has been done in several cases in [B2] and [B3]). Beyond that, it lays the foundation for proving theorems about valued division algebras by first proving corresponding results in the relatively easier setting of graded division algebras. This approach has previously been applied successfully for wildly ramified valued division algebras by Tignol in [T]. This paper is organized as follows: Before considering ...

demonstrated by integrating them into 1D silicon nitride cavities or open cavities. [17,18] On the other hand, hBN can be used in an entirely monolithic approach where the devices are fabricated from the parent material that hosts SPEs. [19,20] The monolithic approach is advantageous in that emitters can be located within the high-energy field of the cavities, and can therefore be positioned in the maxima of optical modes, thus enabling the realisation of optimal coupling efficiencies as previously done for diamond, GaAs, and SiC. [21][22][23][24] This in combination with recent reports on the tunability of the SPE emission wavelength will allow to directly tune emitters to the resonant modes of devices, [25] in order to reversibly investigate effects such as Purcell enhancement from hBN SPEs.In this work, we report detailed nanofabrication protocols used to realize photonic resonators from hBN. In particular, we demonstrate microring cavities and 1D photonic crystals that exhibit high-quality factors (Q) of ≈1500. We also discuss in detail the shortcomings and benefits of a hybrid reactive ion etching-electron beam induced etching (RIE-EBIE) process that is suitable for the fabrication of large-area, suspended, and supported device structures that are made from hBN, and contain nanoscale features and highly vertical sidewalls. The nanofabrication approach presented here is suitable for other layered materials, and therefore broadly relevant to the field of photonics, as well as polaritonic, nanoelectromechanical, optoelectronic, and optomechanic systems, [26] all of which require engineering of structures with nanoscale precision. Results and DiscussionThe fabrication process is outlined schematically in Figure 1. First, hBN flakes are transferred onto a silicon or a silicon dioxide substrate via mechanical exfoliation using sticky tape (Figure 1a). hBN has the advantage that it can be exfoliated from a larger parent crystal, to flakes that are chemically and mechanically stable/robust, with thicknesses as small as a single monolayer. Therefore, it does not require cumbersome preparation steps that are often necessary for classical bulk counterparts such as diamond and SiC. The substrate was prepatterned with trenches that are ≈5 µm deep to allow for a sufficient air gap below the suspended hBN structures. After hBN transfer (Figure 1b), residues and contaminants from the sticky tape were removed by calcination in air on a hot plate at 500 °C and subsequent annealing in Argon at 850 °C, which also increases adhesion of hBN flakes to substrates. An optical image of a suspended flake before the subsequent processing steps is shown in Figure 1c.Growing interest in devices based on layered van der Waals (vdW) materials is motivating the development of new nanofabrication methods. Hexagonal boron nitride (hBN) is one of the most promising materials for studies of quantum photonics and phonon polaritonics. A promising nanofabrication process used to fabricate several hBN photonic devices using a hybrid reactive ion et...

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Coupled metal nanostructures supporting localized surface plasmon resonances are represented as a nanoscale optical circuit that takes light fields as inputs and forms linear combinations of them with complex coefficients. The subwavelength arrays of these circuits form a metasurface that performs mathematical operations in two dimension on an incident light field. We demonstrate this concept with subwavelength scale plasmonic circuits that perform difference operations. The metasurface is fabricated from the arrays of coupled gold nanorods where each group of three rods forms the difference circuit. The operation of the metasurface is demonstrated experimentally.