标题： 非中心对称材料中的二阶微观非线性极化率：用于成像价电子运动 显示英文标题

标题： Second-order microscopic nonlinear susceptibility in a centrosymmetric material: application to imaging valence electron motion

摘要： 我们报告了使用亚共振0.95 eV激光脉冲和来自LCLS自由电子激光器的9.5 keV硬X射线脉冲，从单晶硅中相位匹配的非线性X射线和光学和频生成的测量结果。 和频信号表现为弹性布拉格峰的能量和动量侧带。 它与光学诱导的微观电荷/电流的相关时间和空间傅里叶分量的模平方成正比。 我们测量了220布拉格峰的第一和第二阶侧带，并发现当施加的场沿着倒格子矢量时效率最大。 对于光学强度为$\sim10^{12} \text{W}/\text{cm}^2$，我们分别测得第一和第二阶侧带的峰值效率为$3\times 10^{-7}$和$3\times 10^{-10}$（相对于弹性布拉格峰）。 第一阶侧带与施加电场方向的诱导微观电流一致（与各向同性的响应一致）。 第二阶侧带非平凡地依赖于光学场方向，并与沿具有C$_{3v}$点群对称性的键上诱导电荷产生的各向异性响应一致。 结果与第一性原理布洛赫-弗洛凯计算结果非常吻合。 显示英文摘要

摘要： We report measurements of phase-matched nonlinear x-ray and optical sum-frequency generation from single-crystal silicon using sub-resonant 0.95 eV laser pulses and 9.5 keV hard x-ray pulses from the LCLS free-electron laser. The sum-frequency signal appears as energy and momentum sidebands to the elastic Bragg peak. It is proportional to the magnitude squared of the relevant temporal and spatial Fourier components of the optically induced microscopic charges/currents. We measure the first- and second-order sideband to the 220 Bragg peak and find that the efficiency is maximized when the applied field is along the reciprocal lattice vector. For an optical intensity of $\sim10^{12} \text{W}/\text{cm}^2$, we measure peak efficiencies of $3\times 10^{-7}$ and $3\times 10^{-10}$ for the first and second-order sideband respectively (relative to the elastic Bragg peak). The first-order sideband is consistent with induced microscopic currents along the applied electric field (consistent with an isotropic response). The second-order sideband depends nontrivially on the optical field orientation and is consistent with an anisotropic response originating from induced charges along the bonds with C$_{3v}$ site symmetry. The results agree well with first-principles Bloch-Floquet calculations.