LETTER doi:10.1038/nature12449 Stimulated X-ray emission for materials science M.Beye,S.SchreckF.SorgenfreiC.TrabantN.PontiusC.Schufler-Langeheine,W.Wurth&A.Fohlisch low-energy ex tations nality in n ge,spex-rays ering has been the for tates and he crh s to which such techn approaches unity P=1=p dx,yielding: oving th desi able.Her ted radiative de of on length relative to the inter has to tion in the directed beam of stim lated to thefracti core-ex sion is require erg excitat ength ns ca teract with cs in the con d energ hence not suited for spectroscopi femte d system ent for other geon Along thelate ntly the m of the i e present her stimulated X-ray em rom a whereas the re excitation at a photon ene of ISeV. with adiation we produce igh 2p core of the RIX ergy of 85eV to nearly 100ev) itin ies are rum by the ed in d.This tion is ske typical resonar em ogethe caSrcoddh detected methods B carefully choosing the geometr which is the result of the balance be he mple isminimizd.By properly shaping the fre Vith a spect tal con ofr on ed a characte eof the total ns on the incoming photor in the y s up th on depth of the ex ss the relev anle he orption and stir probabilities P can be appre photon he explicit formulat in the Methods section We can henc foamdhteadiferentialcqg tion of the inter ngth P=odx and P= 1) his inhe ent Auge rcn G-ISKC,Helmh D0 MONTH 2013 I VOL 000 I NATURE I1 vedLETTER doi:10.1038/nature12449 Stimulated X-ray emission for materials science M. Beye1 , S. Schreck1,2, F. Sorgenfrei1,3, C. Trabant1,2,4, N. Pontius1 , C. Schu¨ßler-Langeheine1 , W. Wurth3 & A. Fo¨hlisch1,2 Resonant inelastic X-ray scattering and X-ray emission spectros￾copy can be used to probe the energy and dispersion of the elemen￾tary low-energy excitations that govern functionality in matter: vibronic, charge, spin and orbital excitations1–7. A key drawback of resonant inelastic X-ray scattering has been the need for high photon densities to compensate for fluorescence yields of less than a per cent for soft X-rays8 . Sample damage from the dominant non￾radiative decays thus limits the materials to which such techniques can be applied and the spectral resolution that can be obtained. A means of improving the yield is therefore highly desirable. Here we demonstrate stimulatedX-ray emission for crystalline silicon at pho￾ton densities that are easily achievable with free-electron lasers9 . The stimulated radiative decay of core excited species at the expense of non-radiative processes reduces sample damage and permits narrow￾bandwidth detection in the directed beam of stimulated radiation. We deduce how stimulated X-ray emission can be enhanced by seve￾ral orders ofmagnitude to provide, with high yield and reduced sam￾ple damage, a superior probe for low-energy excitations and their dispersion in matter. This is the first step to bringing nonlinear X￾ray physics in the condensed phase from theory10–16 to application. In the soft X-ray region, the use of nonlinear techniques to enhance the signal levels has been prevented by the small cross-sections and the short lifetimes of core-excited states in the regime of a few femtose￾conds. In the past few years, free-electron lasers have become available, producing ultrashort, intense soft X-ray pulses9,17–20. Recently, the stimu￾lation of emissionfrom a singlefluorescence line in a rare gas21 and hard X-ray/optical sum frequency generation22 have been demonstrated. We present here stimulated X-ray emission from a solid-state sam￾ple recorded at the free-electron laser in Hamburg (FLASH) for non￾resonant silicon L-edge excitation at a photon energy of 115 eV. With free-electron laser radiation we produce regions with high 2p core excitation densities. The spontaneously emitted radiation from recom￾bination of the 2p core holes (photon energy of 85 eV to nearly 100 eV) seeds the stimulated emission of soft X-ray photons. The emitted spec￾trum is determined by the spontaneous emission as observed in a typical resonant inelastic X-ray scattering (RIXS) or X-ray emission spectroscopy (XES) experiment and thus conserves all the information and specificity of these methods. By carefully choosing the geometry, we can significantly enhance the weakfluorescence signal at the expense of Auger decays. Fewer electrons are emitted and electronic damage to the sample is minimized. By properly shaping the free-electron laser beam footprint on the sample the detected signal can be enhanced by orders ofmagnitude because the usually isotropic emission can be direc￾ted towards the detector. This opens up the possibility of nonlinear spectroscopy in the X-ray region by combining concepts from non￾linear optics with the high information depth of X-ray spectroscopy, which is intrinsically able to resolve femtosecond dynamics10–16,23–25. The absorption and stimulated emission probabilitiesP can be appro￾ximated for infinitesimally thin samples where the photons interact on a short length dx as: P~sratomdx and P~srchdx ð1Þ where ratom is the number density of absorbing atoms and rch (for core hole) is the number density of excited atoms for stimulation. The X-ray absorption or stimulation cross-sections s are determined by the dipole transition matrix elements between the core level and unoccupied (absorption) or occupied (stimulated emission) valence states and are assumed at this point to be approximately the same. The cross￾section is connected with the absorption length l 5 (sratom) –1. The effect of stimulated emission is large, when the stimulation probability approaches unity P ~ ! 1~srch dx, yielding: rch ratom ~ l dx ð2Þ For effective stimulation, the absorption length relative to the inter￾action lengths of stimulating photons has to be similar to the fraction of core-excited atoms. For comparable interaction and absorption leng￾ths, every atom needs to be core-excited; a populationinversionis required. In the forward direction, stimulated X-ray emission has been demon￾strated in the gas phase21. In this geometry, the maximum interaction length at which stimulating photons can interact with core holes is intrinsically given by the absorption length. The sample has to absorb one X-ray photon per atom. If applied to solids, this energy usually destroys the bonding network and is hence not suited for spectroscopic studies of the undisturbed system. The situation is very different for other geometries. Along the lateral dimension of the X-ray focus, the interaction length can be signifi￾cantly larger. With soft X-rays on solids, the absorption length of the exciting radiation with a photon energy that is slightly above an absorp￾tion edge is typically tens or hundreds of nanometres, whereas the lateral dimensions of typical X-ray foci are a few to hundreds of micro￾metres. The interaction length is then limited by the absorption length of the RIXS and XES signals with a photon energy below the absorp￾tion edge, which are typically an order of magnitude larger than for the exciting radiation. Consequently, lower excitation densities are needed, when the observation direction is chosen to be along the lateral dimen￾sion of the projected focus instead. This situation is sketched in Fig. 1, together with measured data of the total emission signal detected at several angles to the surface. We observe an emission maximum at around 9u, which is the result of the balance between the absorption length of the exciting radiation and the interaction length of the emit￾ted radiation inside the excited volume. We also studied the emission with a spectrometer. Experimental con￾straints meant that it was placed at around 15u to the sample surface. We observed a characteristic dependence of the total number and the spectral distribution of the outgoing photons on the incoming photon flux (Figs 2 and 3). Before we analyse the experimental data, we discuss the relevant pro￾cesses that have to be considered to derive the number of stimulated photons. The explicit formulation is given in the Methods section. Stimulated emission can be treated along the same lines as absorption. We can henceformulate a differential equation as afunction of the inter￾action length in an ansatz similar to the derivation of the Lambert–Beer lawfor absorption. This approach is inherently time-independent.Auger 1 Institute for Methods and Instrumentation of Synchrotron Radiation Research G-ISRR, Helmholtz-Zentrum Berlin fu¨r Materialien und Energie GmbH, Albert-Einstein-Straße 15, 12489 Berlin, Germany. 2 Fakulta¨t fu¨r Physik und Astronomie, Universita¨t Potsdam, Karl-Liebknecht-Straße 24–25, 14476 Potsdam, Germany. 3 Institut fu¨r Experimentalphysik, Universita¨t Hamburg and Centre for Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany. 4 II. Physikalisches Institut, Universita¨t zu Ko¨ln, Zu¨lpicher Straße 77, 50937 Ko¨ln, Germany. 00 MONTH 2013 | VOL 000 | NATURE | 1 ©2013 Macmillan Publishers Limited. All rights reserved