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1. The 𝑚=2 amplituhedron and the hypersimplex: Signs, clusters, tilings, Eulerian numbers

   https://doi.org/10.1090/cams/23  

   Parisi, Matteo; Sherman-Bennett, Melissa; Williams, Lauren (July 2023, Communications of the American Mathematical Society)

   The hypersimplex ${\mathrm{\Delta <\#comment/>}}_{k+1,n}$is the image of the positive Grassmannian $G{r}_{k+1,n}^{\ge <\#comment/>0}$under the moment map. It is a polytope of dimension $n-<\#comment/>1$in ${\mathbb{R}}^n$. Meanwhile, the amplituhedron ${\mathcal{A}}_{n,k,2}\left(Z\right)$is the projection of the positive Grassmannian $G{r}_{k,n}^{\ge <\#comment/>0}$into the Grassmannian $G{r}_{k,k+2}$under a map $\stackrel{~<\#comment/>}{Z}$induced by a positive matrix $Z\in <\#comment/>Ma{t}_{n,k+2}^{>0}$. Introduced in the context ofscattering amplitudes, it is not a polytope, and has full dimension $2k$inside $G{r}_{k,k+2}$. Nevertheless, there seem to be remarkable connections between these two objects viaT-duality, as conjectured by Łukowski, Parisi, and Williams [Int. Math. Res. Not. (2023)]. In this paper we use ideas from oriented matroid theory, total positivity, and the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting outpositroid polytopes—images of positroid cells of $G{r}_{k+1,n}^{\ge <\#comment/>0}$under the moment map—translate into sign conditions characterizing the T-dualGrasstopes—images of positroid cells of $G{r}_{k,n}^{\ge <\#comment/>0}$under $\stackrel{~<\#comment/>}{Z}$. Moreover, we subdivide the amplituhedron intochambers, just as the hypersimplex can be subdivided into simplices, with both chambers and simplices enumerated by the Eulerian numbers. We use these properties to prove the main conjecture of Łukowski, Parisi, and Williams [Int. Math. Res. Not. (2023)]: a collection of positroid polytopes is a tiling of the hypersimplex if and only if the collection of T-dual Grasstopes is a tiling of the amplituhedron ${\mathcal{A}}_{n,k,2}\left(Z\right)$for all $Z$. Moreover, we prove Arkani-Hamed–Thomas–Trnka’s conjectural sign-flip characterization of ${\mathcal{A}}_{n,k,2}$, and Łukowski–Parisi–Spradlin–Volovich’s conjectures on $m=2$cluster adjacencyand onpositroid tilesfor ${\mathcal{A}}_{n,k,2}$(images of $2k$-dimensional positroid cells which map injectively into ${\mathcal{A}}_{n,k,2}$). Finally, we introduce new cluster structures in the amplituhedron. 

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2. Efficient and highly tunable second-harmonic generation in Z-cut periodically poled lithium niobate nanowaveguides

   https://doi.org/10.1364/OL.393445  

   Chen, Jia-Yang; Tang, Chao; Ma, Zhao-Hui; Li, Zhan; Meng_Sua, Yong; Huang, Yu-Ping (July 2020, Optics Letters)

   Thin-film lithium-niobate-on-insulator (LNOI) has emerged as a superior integrated-photonics platform for linear, nonlinear, and electro-optics. Here we combine quasi-phase-matching, dispersion engineering, and tight mode confinement to realize nonlinear parametric processes with both high efficiency and wide wavelength tunability. On a millimeter-long, Z-cut LNOI waveguide, we demonstrate efficient ( $1900\pm <\#comment/>500\mathrm{\%<\#comment/>}{\mathrm{W}}^{-<\#comment/>1}{\mathrm{c}\mathrm{m}}^{-<\#comment/>2}$) and highly tunable ( $-<\#comment/>1.71\mathrm{n}\mathrm{m}/\mathrm{K}$) second-harmonic generation from 1530 to 1583 nm by type-0 quasi-phase-matching. Our technique is applicable to optical harmonic generation, quantum light sources, frequency conversion, and many other photonic information processes across visible to mid-IR spectral bands. 

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3. High-performance electron-blocking-layer-free deep ultraviolet light-emitting diodes implementing a strip-in-a-barrier structure

   https://doi.org/10.1364/OL.400917  

   Velpula, Ravi_Teja; Jain, Barsha; Velpula, Swetha; Nguyen, Hoang-Duy; Nguyen, Hieu_Pham_Trung (September 2020, Optics Letters)

   In this Letter, the electron-blocking-layer (EBL)-free AlGaN ultraviolet (UV) light-emitting diodes (LEDs) using a strip-in-a-barrier structure have been proposed. The quantum barrier (QB) structures are systematically engineered by integrating a 1 nm intrinsic ${\mathrm{A}\mathrm{l}}_x{\mathrm{G}\mathrm{a}}_{(1-<\#comment/>x)}\mathrm{N}$strip into the middle of QBs. The resulted structures exhibit significantly reduced electron leakage and improved hole injection into the active region, thus generating higher carrier radiative recombination. Our study shows that the proposed structure improves radiative recombination by $\sim <\#comment/>220\mathrm{\%<\#comment/>}$, reduces electron leakage by $\sim <\#comment/>11$times, and enhances optical power by $\sim <\#comment/>225\mathrm{\%<\#comment/>}$at 60 mA current injection compared to a conventional AlGaN EBL LED structure. Moreover, the EBL-free strip-in-a-barrier UV LED records the maximum internal quantum efficiency (IQE) of $\sim <\#comment/>61.5\mathrm{\%<\#comment/>}$which is $\sim <\#comment/>72\mathrm{\%<\#comment/>}$higher, and IQE droop is $\sim <\#comment/>12.4\mathrm{\%<\#comment/>}$, which is $\sim <\#comment/>333\mathrm{\%<\#comment/>}$less compared to the conventional AlGaN EBL LED structure at $\sim <\#comment/>284.5\mathrm{n}\mathrm{m}$wavelength. Hence, the proposed EBL-free AlGaN LED is the potential solution to enhance the optical power and produce highly efficient UV emitters. 

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4. Determination of size and complex index of refraction of single particles with elastic light scattering

   https://doi.org/10.1364/AO.413675  

   Waez, Mir_Seliman; Eckels, Steven_J; Sorensen, Christopher_M (January 2021, Applied Optics)

   We show that for spherical particles greater than ca. 5 µm, the differential scattering cross section is only weakly dependent on the real and imaginary parts of the refractive index ( $m=n+i\mathrm{\kappa <\#comment/>}$) when integrated over angle ranges near $37\pm <\#comment/>5^{\circ <\#comment/>}$and $115\pm <\#comment/>5^{\circ <\#comment/>}$, respectively. With this knowledge, we set up an arrangement that collects scattered light in the ranges $37\pm <\#comment/>5^{\circ <\#comment/>}$, $115\pm <\#comment/>5^{\circ <\#comment/>}$, and $80\pm <\#comment/>5^{\circ <\#comment/>}$. The weak functionality on refractive index for the first two angle ranges simplifies the inversion of scattering to the particle properties of diameter and the real and imaginary refractive indices. Our setup also uses a diamond-shaped incident beam profile that allows us to determine when a particle went through the exact center of the beam. Application of our setup to droplets of an absorbing liquid successfully determined the diameter and complex refractive index to accuracies ranging from a few to ten percent. Comparisons to simulated data derived from the Mie equations yielded similar results. 

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5. Single-shot ultrafast visualization and measurement of laser–matter interactions in flexible glass using frequency domain holography

   https://doi.org/10.1364/OL.382645  

   Dempsey, Dennis; Nagar, Garima_C; Renskers, Christopher_K; Grynko, Rostislav_I; Sutherland, James_S; Shim, Bonggu (February 2020, Optics Letters)

   We perform single-shot frequency domain holography to measure the ultrafast spatio-temporal phase change induced by the optical Kerr effect and plasma in flexible Corning Willow Glass during femtosecond laser–matter interactions. We measure the nonlinear index of refraction ( $n_2$) to be $(3.6\pm <\#comment/>0.1)\times <\#comment/>{10}^{-<\#comment/>16}{\mathrm{c}\mathrm{m}}^2/\mathrm{W}$and visualize the plasma formation and recombination on femtosecond time scales in a single shot. To compare with the experiment, we carry out numerical simulations by solving the nonlinear envelope equation. 

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