Fig. 2: Measured and numerically simulated linear and nonlinear response of LLTO at the lithium K-edge.

a, The measured imaginary part of the refractive index of LLTO (brown shaded area, left axis) agrees well with the numerically retrieved linear response (dashed brown line, right axis) around the Li K-edge that appears around 61 eV in LLTO. The calculated curve is an equal-weight linear superposition of response from both LLTO and LCO. See Supplementary Fig. 13 for the effect of including the LCO contribution. The slight disagreement of the linear response at higher energies could have stemmed from broadening due to unrealistic sample structures in the simulation, where an ideal crystal geometry is assumed. b, Experimentally derived second-order nonlinear susceptibility χ⁽²⁾(2ω) response across the Li K-edge (blue open squares, blue solid line for visual clarity). The computationally simulated second-order nonlinear susceptibility χ⁽²⁾(2ω) for a LLTO at the surface is denoted by solid triangles, and is in good agreement with the measurements. Vertical error bars correspond to errors in the quadratic fit of the second-order response; horizontal error bars are a result of energy jitter of the FEL for which 72,000 shots were collected at each photon energy. a,b, Double-sided arrow highlights the difference in peak positions between linear absorption (a) and second-harmonic response (b). c,d, Representative wave function of the resulting lithium atom core-excited states in the XUV-SHG spectrum at ~61 eV (c) and ~64 eV (d). We adopt the convention that the positive and negative phases of wave function are coloured beige and teal, respectively.