Second-Order Microscopic Nonlinear Optical Susceptibility in a Centrosymmetric Material: Application to Imaging Valence Electron Motion

We report measurements of phase-matched nonlinear x-ray and optical mixing from single-crystal silicon using subresonant 0.95 eV laser pulses and 9.5 keV hard x-ray pulses from the Linear Coherent Light Source free-electron laser. The mixing 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 and 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 approximately <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:msup> <a:mrow> <a:mn>10</a:mn> </a:mrow> <a:mrow> <a:mn>12</a:mn> </a:mrow> </a:msup> <a:mtext> </a:mtext> <a:mtext> </a:mtext> <a:mi mathvariant="normal">W</a:mi> <a:mo>/</a:mo> <a:msup> <a:mrow> <a:mi>cm</a:mi> </a:mrow> <a:mrow> <a:mn>2</a:mn> </a:mrow> </a:msup> </a:mrow> </a:math> , we measure peak efficiencies of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mn>3</d:mn> <d:mo>×</d:mo> <d:msup> <d:mn>10</d:mn> <d:mrow> <d:mo>−</d:mo> <d:mn>7</d:mn> </d:mrow> </d:msup> </d:math> and <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mn>3</f:mn> <f:mo>×</f:mo> <f:msup> <f:mn>10</f:mn> <f:mrow> <f:mo>−</f:mo> <f:mn>10</f:mn> </f:mrow> </f:msup> </f:math> 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 and 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 <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mrow> <h:msub> <h:mrow> <h:mi mathvariant="normal">C</h:mi> </h:mrow> <h:mrow> <h:mn>3</h:mn> <h:mi>v</h:mi> </h:mrow> </h:msub> </h:mrow> </h:math> site symmetry. The results agree well with first-principles Floquet-Bloch calculations.