LETTER RESEARCH METHODS This sion also a counts for the l 2 of the number of cor hole poanoesdhay,aeedanenihidaheeho ded th The i 1.200. -per-m nize the p the The of 10 Hz e ava- -per- late the h holes (and Ktimesm are obse of st emitted n ded nd 40.000 single sh the detector),the higher stimulatio for each bin.Wit q-Aon ions in nle also reflect the of th sdlghydierentnonlneai in the of the s is the and th ts as the actual di is alre dy around 170 fs.)Th only be e by using a different equation.In the differ ntilcgtionforhcgcoho Na( -N-回 ting ph time Th For aussian ofi on is th d by the dete we mus on that is the tot sion and the reabsorption on the interaction) m the ++ hatisncedc 2013 Mac millan PubMETHODS Photon parameters. The experiments have been conducted during two separate runs at the free-electron laser FLASH at Hamburg, Germany. The machine was ope￾rated at a central photon energy of about 115 eV. In one run, we recorded the spectral resolved data. The machine was operated at a repetition rate of 5 Hz with typically 30-fs pulses17,27. We used beamline PG 2 with the 1,200-lines-per-millimetre grat￾ing in zero order, thus reflecting the incident beam without dispersion28. To maximize the photon flux, a grazing angle of 3.5u was used on the grating. The spot on the sample was round with a diameter of about 45 mm, determined through measurements on a fluorescence screen and permanent imprints studied under a microscope. These parameters lead to fluences up to 1 J cm22 . The incoming photon numbers have been measured for each shot by a gas monitor detector29. In another experimental campaign, the angular dependence of the stimulated emission signal was studied. Our setup was placed at beamline BL 2 (ref. 29). After an upgrade of the FLASH accelerator, a burst repetition rate of 10 Hz became avai￾lable, each burst including 100 pulses at 250 kHz. The pulse length was, at about 50 fs, slightly longer and the spot size on the sample was adjusted to be about 40 mm for compensation. With a gas attenuator, we set the fluence at around 1 J cm22 , the upper limit of the spectrally resolved data. Experimental setup. Hydrogen-passivated silicon (100) surfaces were scanned through the beam. The spectra have been recorded with a commercial Scienta XES 355 spectrometer30using the 300-lines-per-millimetre grating. The counts have been detected by a multi-channel-plate, phosphor-screen combination that can count sin￾gle photons and is multi-hit capable. The detection window was centred at 92 eV and the resolutionwas set to 0.4 eV. The scattering planewas the horizontal plane of pola￾rization of the incoming photons. The spectrometer was mounted at around 85uto the beam (15uto the sample surface), while the pulses impinged at an angle of 80uto the surface. For the angular dependence, we used an avalanche photodiode with an alumi￾niumfilter to block optical light. The incidence angle on the samplewas 45u, whereas the detection angle was varied. Data analysis. With our spectrometer, we recorded around 40,000 single shots. The data was sorted and binned and the standard deviations of the averaged values were computedfor each bin.With the photodiode, we analysed 15,000 bursts. During free-electron laser irradiation, the emission from a plasma plume was optically visible at the sample and slight ablation was observed. The instantaneous core-hole density. As described in the main text, the instant￾aneous core-hole density contains the main approximations in our theoretical treatment. Without stimulation, the core-hole density is assumed to be constant inside a cylinder formed by the exciting radiation, and outside it is assumed to be zero. The dimensions of this cylinder in the surface plane of the sample are given by the distribution of the incoming photons: the measured focal width and height (w and h). The depth of the cylinder is taken as the tabulated absorption length of the incoming radiation l. By choosing 1/e dimensions for this volume, the integral number of core holes is the same as for the actual distributions and given through the number of incoming photons. We thus scale the core-hole density to yield the same integrals and the same second moments as the actual distributions. We treat the temporal distribution of the core holes as follows: Core-hole decays during the pulse length reduce the instantaneous core-hole density. The temporal distribution of the number of core holes in the sample at a given time t, that is, Nch(t), is the solution of the following differential equation: dNchð Þt dt ~Ninð Þt {Nchð Þt tlt with the temporal distribution of the exciting photons Nin(t) as the source term of this inhomogeneous differential equation and the core-hole lifetime tlt. For a tem￾poral Gaussian pulse of incoming photons, the solution is the product of an expo￾nential decayingfunction and an errorfunctionfor the creation of core holes. To cast the analytical solution into the same approximation as above, where the core-hole density (without stimulation) is constant during a specific time period, we analyse Nch(t) as follows. The integral of Nch(t) per incoming photon, that is, the total number of core holes per photon present at each moment in time is proportional to tlt. The longer the lifetime, the more core holes are present, because they have not yet decayed. The second central moment of Nch(t) for temporal Gaussian pulses is t2 ltzt2 pl with the pulse length tpl. The width of this function is approximated by the square root of the second moment. The constant function that yields the same integral over time (tlt), and is zero outside a window as large as the square root of the second moment of Nch(t), is thus given by: T~ tlt ffiffiffiffiffiffiffiffiffiffiffiffiffiffi t2 ltzt2 pl q ~ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1zt2 pl t2 lt s This expression also accounts for the lowering of the number of core holes through spontaneous decay, because the decays are included in the core-hole lifetime. The instantaneous core-hole density without stimulation is taken to be con￾stant; only stimulation is taken to annihilate core holes. The number of total stimulated photons is taken to be K times bigger than observed (accounting for angular dependences and the finite detector acceptance and efficiency). The instantaneous core-hole density rch for the differential equation is thus given by: rch~p Nð Þ in{KNstim,obs p~ 4T pwhl The number of stimulating photons. The number of stimulating photons in the observation direction has been termed Nobs. This value is given by the sum of the spontaneously emitted photons in the observation direction and the stimulated photons in the observation direction Nstim,obs. We introduce the acceptance and detection efficiency A of our spectrometer, weighted with the possible angular distributions of the spontaneous emission. The total number of photons that are spontaneously emitted in the observed direction increases across the interaction region, starting from zero at the far end of the observed volume to the number of incoming photons Nin times the fluor￾escence yield vfy and the acceptance (AvfyNin).We approximate this increase with a constant across the interaction region. We take half the maximal value to yield the correct average value for a linear increase. This approximation ensures integr￾ability and neglects the aspects around the temporal evolution, travel times of photons inside the interaction region and so on. Given that stimulated emission annihilates core holes (and K times more sti￾mulated photons are emitted than are observed because of the finite acceptance of the detector), the number of spontaneously emitted photons is reduced with higher stimulation. We thus introduce the source term for stimulated emission as: Nobs~q Nð Þ in{KNstim,obs zNstim,obs q~1 2 Avfy The full source term should in principle also reflect the spatial growth of the seeding spontaneous emission across the interaction region, which would lead to a slightly different nonlinearity in the stimulated emission signal, especially for low photon numbers. In reality, this effect is more than compensated by the bigger effect of the temporal mismatch of the seeding spontaneous emission (for near￾normal impinging irradiation, the seed appears at the same time across the whole interaction region in a time window of around ffiffiffiffiffiffiffiffiffiffiffiffiffiffi t2 ltzt2 pl q <35 fs (see above), whereas the travel time of the stimulated field across the 50-mm-long interaction region is already around 170 fs.) This can only be overcome by using a different geometry, as proposed in the main text. The full differential equation. In the differential equation for the growth of stimulated emission along the interaction region, we also treat the reabsorption of emitted photons explicitly, with reabsorption length L. With the above intro￾duced abbreviations, we thus solve: dNstim,obs dx ~ð Þ qNinzð Þ 1{qK Nstim,obs ð Þ pNin{pKNstim,obs sstim{Nstim,obs L with Nstim,obsð Þ x~0 ~ ! 0 To get the total number of photons measured by the detector Ntot,obs, we must add the spontaneous emission, including the reduction through stimulated emis￾sion and the reabsorption along the interaction length x, which is exp(–x/L): Ntot,obs~2qeð Þ {x=L ð Þ Nin{KNstim,obs zNstim,obs The solution and its limits. In the solution of the differential equation further combinations of parameters appear. X 5 x/L is the ratio between the actual inter￾action length and the reabsorption length. This number is given by the experi￾mental geometry and can thus be derived from tabulated values. W 5 pLsstim, a dimensionless number that relates the excited volume p–1 to a stimulation volume Lsstim. The inverse value ofW is connected to the number of photons that is needed to observe the nonlinear increase in signal due to stimulated emission. LETTER RESEARCH ©2013 Macmillan Publishers Limited. All rights reserved