LETTER RESEARCH out a facto This is h 。 div 6 a high more effective for emis n tastest,and th ning photonso a.The e emitedclectronsandee ctrons he maximun s for different spec om Auger decays e emi ck.71-s4eV).The spectrum act Never Typica ion shown in Fig.2b). the tha g tor of five at grazing angle ) hi and is indeed the result of a noni m leads to the sam tei ionofcstiitaie at th e centre of the be isas large as possible ved pho ton numbers directions the signal will be s in In Fig.2 light. the fo along the sample s hmen mized while -ra DO MONTH 2013 I VOL 000 I NATURE I 3becomes important at about 1011 incoming photons per pulse. Typical soft X-ray free-electron lasers produce up to 1013 photons per shot, so that stimulated emission has to be considered as a dominating effect. We now turn to the experimental observations. In Fig. 1b, we show the angular dependence of the emission signal. The incidence angle is chosen to be 45u and we find enhanced intensity for the specular geo￾metry with the detector at 45u. Additionally, we find a strong enhance￾ment of the signal by about a factor of five at grazing angles around 9u, far from the specular reflection. It appears in a narrow angular window and is indeed the result of a nonlinear effect. With the used excitation conditions, we are deep in the regime where the stimulated emission saturates and the observed effects are variations in the angular distri￾bution of the stimulated emission. The maximum signal is expected in a directionforwhich the interaction lengthinside the core excited volume is as large as possible and for which the reabsorption probability is minimized. Because the absorption length for the incoming radiation is about an order of magnitude shorter than the reabsorption length, the optimal angle is expected around arcsin(0.1) < 6u which is in reasonably good agreement with the experiment. In Fig. 2a, we show the total number of observed photons in our spec￾trometer placed at 15u to the surface plane, along with a fit to the expected dependence. We also include the linear limits derived from thefit.We observe that already in the lowerfluence range studied, there is a deviation from linearity and the stimulated emission quickly satu￾rates to the upper limit. In the chosen experimental geometry, the enhan￾cement of the emission between purely spontaneous and saturated stimulation is only about a factor of two. This is because the majority of the stimulated photons are radiated more grazing to the surface and do not reach our detector, as shown in Fig. 1b. Still, the nonlinear enhan￾cement and saturation of the signal through stimulated emission are clearly observed. It is instructive to display the total number of observed photons divi￾ded by the number of incoming photons (see Fig. 2b). This yields the conversion efficiency from incoming photons to emitted photons, as seen by the detector—the detected conversion. The limiting linear dependences turn into limiting constant conversion efficiencies and are determined by the experimental parameters. The nonlinearity of the curve becomes obvious as a monotonous change from the lower to the upper limit. The photon number needed for the nonlinear part between the limiting lines is directly related also to the emission-energy-dependent stimulation cross-section. For larger stimulation cross-sections, the nonlinearity and thus the saturation of stimulated emission will occur at smaller incoming photon numbers. In Fig. 3a, we show the spectral evolution of the silicon L-edge emission depending on the number of incoming photons and how we separate the spectrum into parts of different emission intensity. The stimulation cross-section contains the same matrix elements as the spontaneous emission probability, which is in turn proportional to the measured spectral emission intens￾ity, so the stimulation of emission becomes more effective for emis￾sion-energy regions that already show a high intensity at low fluences. Figure 3b displays the dependence of the normalized detected conver￾sion on the number of incoming photons in specific emission-energy regions. The peak region around an emission energy of 90 eV, as the most intense feature, approaches saturated stimulation fastest, and the other regions follow in the order of spectral intensity. The emission energy region for usually unoccupied states above the bandgap shows a strikingly different behaviour. Here, a secondary effect sets in. Photo-emitted electrons and electrons from Auger decays scat￾ter inelastically in the sample, very quickly creating a multitude of delo￾calized electron–hole pair excitations around the bandgap. This effect becomes stronger with an increasing number of incoming photons and the connected radiative decays annihilate core holes that contrib￾uted to the stimulated emission of other parts of the spectrum at lower photon numbers. Therefore, the detected conversion in other parts of the spectrum actually shrinks rather than becoming constant. Never￾theless, because all emission processes saturate when all the core holes are stimulated down, the total detected conversion still becomes con￾stant (compare the integral detected conversion shown in Fig. 2b). In the direction of 15u from the surface, the detected conversion increases by a bit more than a factor of two between the limiting cases of spontaneous emission and saturated stimulated emission. By using a shallower angle, we can increase the gain further by another factor of five (see the difference in the saturated emission signal in Fig. 1b). This value is limited by the round shape of the free-electron laser beam foot￾print on the sample. The shape of the beam leads to the same maximi￾zed interaction lengthin every azimuthal direction in the sample surface plane and thus to a stimulated emission profile which is rotationally symmetric around the sample normal at the centre of the beam. The observed gain can clearly be further increased when the foot￾print of the irradiated photons is specifically shaped. An elongation of thefootprint in the observation directionwillfurther enhance the obser￾ved photon numbers, whereas in other directions the signal will be reduced. To ensure temporal overlap of the core excitations in the sam￾ple and the majority of the stimulated photons travelling with the speed of light, the footprint of the exciting beam needs to be elongated by placing the sample grazing to the incident beam. In this way, the exciting wavefront will travel along the sample surface in time with the stimu￾lated photons and the enhancement of the emission signal will be maximized while X-ray-induced sample damage is further minimized. The same applies to soft X-ray experiments from other systems, because the determining parameters are very similarfor most materials. Incoming photons (1011) 1 0.8 0.6 0.4 Detected spectral conversion (normalized) 4 812 b a 80 90 100 8 4 12 Emission energy (eV) Incoming photons (1011) 80 90 10 8 4 12 g photons (10 Detected conversion (arbitrary units) Figure 3 | Spectrally resolved stimulated emission. a, The detected conversion is shown, highlighting the integration regions for data plotted in b. b, The maximum detected conversions for different spectral regions are marked with arrows. The most intense features saturate first (green, 87–95 eV), followed by the emission shoulder (blue, 95–99.5 eV) and the weak multiple scattering background (black, 71–84 eV). The onset of saturation can be connected with the stimulation cross-section. Emission above the band gap (red, 99.5–108 eV) is not observed at low intensities. The signal is connected with X-ray-induced electronic excitations. LETTER RESEARCH 00 MONTH 2013 | VOL 000 | NATURE | 3 ©2013 Macmillan Publishers Limited. All rights reserved