Non-Equilibrium Modeling of the Fe XVII 3C/3D ratio for an Intense X-ray Free Electron Laser
aa r X i v : . [ phy s i c s . a t o m - ph ] J un Non-Equilibrium Modeling of the Fe XVII 3C/3D ratio for anIntense X-ray Free Electron Laser
Y. Li, ∗ M. Fogle, and S. D. Loch † Department of Physics, Auburn University, Auburn AL 36849, USA
C. P. Ballance
Queen’s University, Belfast, Belfast, BT7 1NN, UK
C. J. Fontes
Los Alamos National Laboratory, Los Alamos, NM 87545, USA (Dated: June 5, 2017)
Abstract
We present a review of two methods used to model recent LCLS experimental results for the3C/3D line intensity ratio of Fe XVII [1], the time-dependent collisional-radiative method and thedensity-matrix approach. These are described and applied to a two-level atomic system excited byan X-ray free electron laser. A range of pulse parameters is explored and the effects on the predictedFe XVII 3C and 3D line intensity ratio are calculated. In order to investigate the behavior of thepredicted line intensity ratio, a particular pair of A-values for the 3C and 3D transitions was chosen(2.22 × s − and 6.02 × s − for the 3C and 3D, respectively), but our conclusions areindependent of the precise values. We also reaffirm the conclusions from Oreshkina et al. [2, 3]: thenon-linear effects in the density matrix are important and the reduction in the Fe XVII 3C/3D lineintensity ratio is sensitive to the laser pulse parameters, namely pulse duration, pulse intensity, andlaser bandwidth. It is also shown that for both models the lowering of the 3C/3D line intensity ratiobelow the expected time-independent oscillator strength ratio has a significant contribution due tothe emission from the plasma after the laser pulse has left the plasma volume. Laser intensitiesabove ∼ × W/cm are required for a reduction in the 3C/3D line intensity ratio below theexpected time independent oscillator strength ratio. PACS numbers: 32.70.Cs ∗ [email protected] † [email protected] . INTRODUCTION Spectral emission from Fe XVII can be used as a valuable plasma diagnostic for bothlaboratory and astrophysical plasmas [4, 5]. The ratio of the 3C line intensity (transition2p
3d ( P ) → ( S )) to the 3D line intensity (transition 2p
3d ( D ) → ( S )) issensitive to the plasma electron temperature and has been the focus of much attention inthe literature. During the history of disagreement between theory and observation for thisline ratio, a number of underlying effects were found to be important, including blendingwith an inner shell satellite line of Fe XVI [6] and radiative cascades [7, 8]. In addition,Gu [9] explored the possibility that insufficient configuration-interaction was included in theatomic structure calculations leading to unconverged oscillator strengths. He then used anapproximate method to account for this lack of convergence to modify the atomic collisiondata used in Fe XVII spectral modeling. A full discussion of the comparison of theory andexperiment for this line ratio is outside of the scope of this article. Brown [10] presents areview of measurement results and Brown and Beiersdorfer [11] show a useful summary ofthe discrepancies and the effects that have been investigated. The focus of this article is onthe analysis of a recent experiment using an X-ray Free Electron Laser (XFEL) that soughtto identify the source of the aforementioned discrepancies [1].Bernitt et al. [1] used an intense XFEL at the Linac Coherent Light Source (LCLS),employing the laser to excite Fe ions in an Electron Beam Ion Trap (EBIT). The laserhas a narrow bandwidth and was tuned to only populate the upper level of either the 3Cor the 3D transition. In this two-level setup, the observed 3C/3D line intensity ratio wasexpected to be the same as the 3C/3D oscillator strength ratio, and any differences couldbe interpreted as an indicator of deficiencies in the current atomic structure calculations forFe . The experiment resulted in a much lower 3C/3D line intensity ratio (2.61 ± ∼ . II. THEORYA. C-R Method
The C-R method is used widely in laboratory and astrophysical plasma modeling. Thisapproach takes into account all of the atomic process in a rate matrix, from which thesteady-state and time-dependent populations can be evaluated. The laser bandwidth inthe LCLS experiment was sufficiently narrow to ensure that only one transition in Fe could be excited at a time, thus this could be treated as a two-level system. For both the3C and 3D lines, the only populating mechanism for the excited state is photo-absorptionfrom the ground level and the only associated depopulating mechanisms are stimulatedemission (sometimes referred to as the interacting process) and spontaneous emission (thenon-interacting process). The time-dependent population density for the excited state N e and ground state N g can be evaluated (see, e.g., Bethe and Jackiw [14] page 204–205): dN e dt = N g ( t ) ρ ( ω , t ) B g → e − N e ( t )( A e → g + ρ ( ω , t ) B e → g ) (1) dN g dt = − N g ( t ) ρ ( ω , t ) B g → e + N e ( t )( A e → g + ρ ( ω , t ) B e → g ) (2)3here B g → e , B e → g , A e → g are the Einstein photo-absorption, stimulated emission, and spon-taneous emission coefficients, respectively. ω is the angular frequency for the transitionbetween the two levels. These can be evaluated from atomic structure calculations. ρ is theradiation field density (J/m /Hz) and can be determined from the laser parameters. In theD-M approach the laser intensity I (W/cm ) is used, so it is beneficial to be able to convertbetween the two representations via ρ = I/ ( c · δν ). Here c is the speed of light and δν isthe bandwidth of the laser (e.g. I = 10 W/cm → ρ = 1 . × − J/m /Hz). In order tosolve the time-dependent Eqns. (1) and (2), the matrix form is used: dN g /dtdN e /dt = − ρ ( ω , t ) B g → e A e → g + ρ ( ω , t ) B e → g ρ ( ω , t ) B g → e − ( A e → g + ρ ( ω , t ) B e → g ) N g N e . (3)Initially, one hundred percent of the population is fixed to be in the ground state. Thus,the initial normalized population vector is h i T , where the superscript indicates the trans-pose. The excited state population N e ( t ) is evaulated for a given ρ ( t ) using Eqn. (3). Thiscan then be used to determine the photon emission for the time during which the laser pulseis in the plasma volume. Note that while stimulated emission is included in the modelingof the excited population density (see Eqn. (3)), these photons are not counted in the pre-dicted line intensity (see Eqn. (4)) since the stimulated emission photons are emitted in thedirection of the laser beam and not towards the detector. After the laser pulse has left theplasma volume, there will be a number of electrons left in the excited state. All of thesewill decay via spontaneous emission before the next laser pulse. Thus, there is a secondcontribution to the line emission with each of these excited state electrons producing onephoton. That is, the total photon energy detected in the spectral line will be proportionalto: I photone → g = ¯ hω A e → g Z T N e ( t ) dt + ¯ hω N e ( T ) . (4)The first term on the right hand side represents the emission during the time, indicated by T , that the laser pulse is interacting with the EBIT plasma and the second term representsthe contribution to the emission from the plasma after the laser pulse has passed. Clearlythe laser pulse temporal profile is a key factor in evaluating the time-dependent excitedpopulations. Various envelopes for ρ ( t ) have been considered and will be shown later in thisarticle. 4 . Density-Matrix Method The D-M approach is a different formalism compared to the C-R approach. For a two-level system in a stationary state, the ground and excited levels have eigenvalues ¯ hω g and¯ hω e , and wave functions Ψ g ( ~r ) and Ψ e ( ~r ) in the Heisenberg picture. The total wave functionof the system can be expressed as:Ψ( ~r, t ) = C g ( t )Ψ g ( ~r ) + C e ( t )Ψ e ( ~r ) . (5)The density operator is defined as ρ = | Ψ i h Ψ | , which has the form ρ = h g | ρ | g i h g | ρ | e ih e | ρ | g i h e | ρ | e i = N g N g + N e C g C ∗ e C e C ∗ g N e N g + N e (6)where N g N g + N e and N e N g + N e are referred to as the populations and the products C g C ∗ e and C e C ∗ g are referred to as the coherence terms. For systems interacting with a laser, the Hamiltonianof the system can be written as: H = H S + H I , (7)where the first term represents the stationary Hamiltonian given by H S = ¯ hω g | Ψ g i h Ψ g | + ¯ hω e | Ψ e i h Ψ e | (8)and the second term represents the interaction Hamiltonian H I = − ~D · ~E, (9)where ~D is the dipole moment and ~E is the radiation field. For a linearly polarized electricfield along the z -axis, it can be written as ~E = E ( t ) cos( ω L t + ψ ( t )) ~z , where E ( t ) is theelectric field amplitude. E ( t ) can be determined from the radiation field intensity I via I = cǫ nE , where c is the speed of the light, ǫ is the electric permittivity of free space,and n is the refractive index of the medium. ω L is the angular frequency of the laser and ψ ( t ) is the time-dependent phase of the laser field. Using the rotating wave approximation(RWA), the interaction Hamiltonian can be further expanded as H I = − ¯ h Ω ∗ | e i h g | e − iω L t − ¯ h Ω2 | g i h e | e iω L t , (10)5here Ω is the Rabi-frequency given by Ω = E ( t ) D eg e iψ ( t ) / ¯ h , with D eg = e h e | ˆ z | g i beingthe eletric dipole matrix element. The density operator ρ is governed by the equation: dρdt = 1 i ¯ h [ H, ρ ] + Λ ρ (11)where Λ is the decay term due to spontaneous emission. From Eqn. (11), one can show that: dρ gg dt = Γ ρ ee − i Ω ∗ e − iω L t ρ ge + i Ω2 e iω L t ρ eg (12a) dρ ee dt = − Γ ρ ee + i Ω ∗ e − iω L t ρ ge − i Ω2 e iω L t ρ eg (12b) dρ eg dt = i Ω ∗ e − iω L t ρ gg − i Ω ∗ e − iω L t ρ ee − ( iω + Γ2 ) ρ eg . (12c)By using ρ ge = ρ ∗ ge , it is straightforward to get the expression for dρ ge dt . By defining a newvariable ˜ ρ = e iω L t ρ and a detuning parameter ∆ = ω L − ω , Eqns. (12a) to (12c) can berewritten as follows: dρ gg dt = Γ ρ ee − i Ω ∗ ρ ge + i Ω2 ˜ ρ eg (13a) dρ ee dt = − Γ ρ ee + i Ω ∗ ρ ge − i Ω2 ˜ ρ eg (13b) d ˜ ρ eg dt = i Ω ∗ ρ gg − i Ω ∗ ρ ee + ( i ∆ − Γ2 ) ˜ ρ eg . (13c)From Eqns. (13a) to (13c) one can produce the Optical-Bloch equation dρ gg /dtdρ ee /dtd ˜ ρ ge /dtd ˜ ρ eg /dt = − i Ω ∗ i Ω2 − Γ i Ω ∗ − i Ω2 − i Ω ∗ i Ω ∗ − i ∆ − Γ2 i Ω ∗ − i Ω ∗ i ∆ − Γ2 ρ gg ρ ee ˜ ρ ge ˜ ρ eg . (14)The electric field amplitude E ( t ) should be a profile consistent with the laser pulse of theexperiment. Oreshkina et al. [2, 3] use a Gaussian envelope with a constant phase, anda Gaussian envelope with a random phase (evaluated with the partial coherent method(PCM) [15, 16]). These two cases are considered here, in addition to the case of the homo-geneous envelope.To solve Eqn. (14), it is assumed that initially one hundred percent of the population isin the ground state (i.e., one starts with h i T for the density vector). The energy6etected from the line emission can be expressed as a function of the detuning parameter E (∆) ∝ Γ ω Z + ∞−∞ ρ ee ( t ) dt, (15)with ρ ee ( t ) being evaluated from Eqn. (14). The line intensity is then evaluated from anintegral over the detuning parameter: L = Z E (∆) d ∆ . (16)Note that the laser pulse parameters are included in the D-M approach via the the electricfield ( ~E ), with the pulse envelope imposed on E ( t ) and the time dependence of the phaseof the electric field included in ψ ( t ). The C-R approach includes the intensity profile of thelaser via the radiation field density ( ρ ( t )) but does not include the phase of the electric field.The Einstein A and B coefficients are related via the detailed balance relationships and thusthe C-R method can be thought of as the limiting case for a perfectly incoherent field.As part of this work, codes were developed for both the C-R and D-M methods. TheC-R results have been presented in the literature [13]. Here we show the D-M results forthe same conditions as those of Oreshkina et al. [2, 3], to test their conclusions. Also, in thefollowing section C-R results will be shown which use identical Einstein A -coefficients as theD-M calculations and the radiation field densities will also be converted to the equivalentlaser intensities. Note that the two methods should not be expected to produce equivalentresults, even for low radiation field densities, as they treat the coherence effects differently. Itis nevertheless interesting to show the results from both approaches, and these are presentedin the next section. III. RESULTSA. LCLS parameter estimation
The LCLS XFEL parameters for the experiment are described by Bernitt et al. [1] andprevious publications [17]. The modeling results require the radiation field density param-eters (for the C-R results) and the laser intensity parameters (for the D-M results). FromBernitt et al. [1], the laser pulses vary in duration from 200 to 2000 fs, but mostly within therange of 200–500 fs (G.V. Brown, private communication). The total energy per laser pulse7n the experiment has an upper limit of 3 mJ. However the filtering and optical losses afterthe soft X-ray (SXR) monochromator are expected to reduce the total energy per shot to0.0013–0.39 mJ [13]. The LCLS XFEL focal diameter has a range of 3–10 µ m [18]. A valueof 10 µ m was chosen for the modeling to make the beam weakly focused. Note that the possi-bility that the beam had a much larger diameter will be considered later in this paper. Theseparameters result in a radiation field density ( ρ ) of 4.62 × − – 3.46 × − J/m /Hz, andusing a laser bandwidth of 1.0 eV the corresponding laser intensity would be in the range4.18 × – 3.14 × W/cm . Oreshkina et al. [2, 3] estimated the laser intensity tobe in the range 10 – 10 W/cm . They used a larger focal diameter than the one givenabove and also a larger energy per pulse (3 mJ).The other important characteristic about the LCLS XFEL pulses is their stochastic na-ture. Each pulse consists of many short spikes a few fs in duration, with gaps between thespikes also being a few fs long. The phase during each of the spikes is in general not coherentwith the previous spikes. Thus, both the intensity and the phase are stochastic in naturefor each pulse. In the case-studies presented below we first consider the line intensity ratiofor individual homogeneous pulses to illustrate the mechanism for the lowering of the lineintensity ratio. We then introduce stochastic pulses and evaluate the line ratio for a largenumber of stochastic pulses to simulate the experimental conditions as closely as possible. B. C-R model
The C-R results for these LCLS laser parameters using a number of pulse profiles for ρ ( t )are considered first. Einstein A -coefficients of 2.22 × s − and 6.02 × s − for the 3Cand 3D A -values were used, taken from the largest calculation shown in [2, 3]. The purposehere is to demonstrate the mechanism for the reduction in the 3C/3D line intensity ratio,with the conclusions being independent of the precise values chosen for the A -values.
1. Smooth homogeneous pulse
Considering first a pulse with a radiation field density that is homogeneous in time, thetime-dependent populations can be solved using Eqn. (3) and the 3C/3D line intensity ratiodetermined using Eqn. (4). Fig. 1 shows the excited states population for the upper levels8
Excited state population
Time (fs)
FIG. 1. Excited state fractional population ( N e / ( N e + N g )) as a function of time for a homogenousradiation field density using the C-R method. The solid lines shows the upper level populationsfor the 3C transition and the dashed lines show the upper level populations for the 3D transition.Results are shown for laser intensities of 10 W/cm (purple), 10 W/cm (green), 10 W/cm (red), 10 W/cm (yellow), and 10 W/cm (blue). of the 3C and 3D transitions for a range of pulse intensities. Both excited state populationsincrease towards a constant (steady-state) value during the homogeneous pulse. However,due to the different Einstein A coefficients for the 3C and 3D transitions, the two excitedstates converge onto this value at different rates. The excited state population for theupper level of the 3C line reaches steady-state in a shorter time than the corresponding3D population. For low radiation field densities the steady-state population value dependslinearly on the radiation field density and results in an excited state population fractionthat is less than 0.5. As the radiation field density increases, the excited states reach theirsteady-state value in a much shorter time and the steady-state value is no longer directlyproportional to the radiation field density. It can also be seen that the maximum value forthe steady-state excited population fraction is 0.5, the high radiation field density limit forthe excited population in the C-R method. In this case, the populating and depopulating ofthe excited states happen simultaneously, in other words the process is always incoherent,which leads to steady and non-oscillating excited state populations.The 3C/3D line intensity ratio for a homogenous radiation field density is shown inFig. 2. For laser intensities above approximately 1 × W/cm there is a reductionin the line intensity ratio below the oscillator strength ratio value. The reduction was9 Pulse duration (fs)
FIG. 2. The 3C/3D line intensity ratio as a function of pulse duration for a homogenous radiationfield density using the C-R method. Results are shown for laser intensities of 10 W/cm (solidpurple line), 10 W/cm (solid green line), 10 W/cm (solid red line), 10 W/cm (solid yellowline), 10 W/cm (solid blue line). shown previously [13] to be primarily due to contributions to the emission during the XFELinteraction with the plasma being different from the contribution after the pulse has left theplasma volume. For the intense pulses, the 3D intensity always has a larger fraction of itsemission coming from this ’after the pulse’ component than the 3C intensity. This resultsin a reduction in the line intensity ratio below the oscillator strength ratio value.
2. Stochastic pulse
Consider next the C-R results for a stochastic profile of the pulse. We generate a randomset of Gaussian profiles, each with 0.2 fs standard deviation and remove a random numberof Gaussians to produce a pulse profile similar to that shown on the LCLS web page, seeFig. 4 of Loch et al. [13]. We normalize the stochastic pulse profile so that the integratedintensity is equivalent to a homogeneous radiation field density. We then use this value tolabel the stochastic pulse, which allows us to compare the two sets of results.Fig. 3 shows the comparison of line ratio using the C-R method with both the homo-geneous and stochastic pulses. The stochastic features of the pulse profiles do not changethe overall trend of the line ratio using the C-R model. This is because the stochastic laserintensity spikes have only small (i.e., a few fs) gaps between them. Thus, for intense pulsesthe excited populations are still driven close to their steady-state values and do not have10
Pulse duration (fs)
FIG. 3. C-R values for the 3C/3D line intensity ratio as a function of pulse duration. The stochasticresults take an average of 80 stochastic pulses for each data point. The homogeneous results arethe same as those shown in Fig. 2. The solid lines show the stochastic results and the dashed linesshow the homogeneous data. Results are shown for intensities of 10 W/cm (purple), 10 W/cm (green), 10 W/cm (red), 10 W/cm (yellow), 10 W/cm (blue). time to decay significantly during the gap between the spikes. In the stochastic simulationswe use different pulses for the 3C and 3D transitions, and have many pulses for each set ofpulse parameters. Each point in Fig. 3 was generated using 80 stochastic pulse profiles forthe 3C and 80 pulses for the 3D. Note the stochastic pulse simulations produce a similarreduction in the line ratio to that obtained from the homogeneous pulse calculations, i.e.the 3C/3D line ratios are lower for shorter and intense pulses. Note that the experimentwould have involved a large number of pulses of different intensities and pulse durations. Ifthe distribution of pulse conditions was known, then it would be possible to compare with asimulated line ratio for the same set of pulse distributions. Such a simulation could be usedto explore the sensitivity to the A -values employed in the model, resulting in a recommendedrange of values on the A -value ratio. While the experimental distribution of pulse conditionsis not currently known well enough to perform such a comparison, it should be pointed outthat the C-R model implies that pulse intensities above 10 W/cm are required to producea reduction in the line ratio. 11 . D-M model We next consider the D-M approach for different pulse envelopes. The same laser band-width (1.0 eV) and A -values are used as those chosen by Oreshkina et al. [2, 3], to allow adirect comparison to be made with their results. As in the discussion of the C-R results, theconclusions that are drawn here will be general and not dependent upon the specific valueschosen for the A -values for Fe .
1. Smooth homogeneous pulse
In the D-M approach, the level populating and depopulating mechanisms are slightlydifferent from the C-R model, as the process involves an intermediate step which containstwo polarization states, ρ ge and ρ eg . This characteristic enables the Rabi-oscillation of thepopulations and is required for intense radiation fields and coherent systems.We consider first a homogeneous pulse, that is E ( t ) is a constant in time, with thevalue determined from the laser intensity. Eqn. (14) is used to evaluate the time-dependentpopulations and Eqn. (16) is used to evaluate the Fe XVII 3C/3D line intensity ratio. Fig. 4shows the excited state populations as a function of time using the D-M approach for a rangeof homogeneous pulse intensities. For low intensities the populations increase smoothly to asteady-state value, with a similar shape to the C-R results. There is, however, a noticeabledifference: the steady-state value can be different for the two transitions. It is still the casethat the 3C excited population reaches steady-state in a shorter time than the 3D excitedpopulation. At higher intensities ( ∼ W/cm and above), Rabi-flopping starts to becomeapparent in both the 3C and 3D populations. Thus, the duration of the pulse can make alarge difference in the relative emission for the two lines. One pulse could result in a 3Cexcited population that is greater than the 3D excited population, while a slightly longerpulse could lead to the opposite. It can also be seen that for the D-M method for coherentpulses, the 3C/3D line ratio could be higher than or smaller than the oscillator strengthratio, depending upon the relative populations of the two excited states. This will be shownin more detail in the next section. 12 Excited state population
Time (fs) 0 0.02 0.04 0.06 0.08 0.1 0.12 0 500 1000 1500 2000
Excited state population
Time (fs) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 500 1000 1500 2000
Excited state population
Time (fs) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 500 1000 1500 2000
Excited state population
Time (fs) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 500 1000 1500 2000
Excited state population
Time (fs) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 500 1000 1500 2000
Excited state population
Time (fs)
FIG. 4. Excited state fractional populations as a function of time under a continuous flat pulseusing the D-M approach. The solid purple lines show the excited 3C populations and the dashedgreen lines show the excited 3D populations. Results are shown for 10 W/cm (row 1, column 1),10 W/cm (row 1, column 2), 10 W/cm (row 2, column 1), 10 W/cm (row 2, column 2),10 W/cm (row 3, column 1), 10 W/cm (row 3, column 2).
2. Coherent Gaussian pulse
Oreshkina et al. [2, 3] modeled the Fe XVII experiment using a D-M approach with aGaussian profile as the pulse envelope. We consider the same case here, to allow us tocompare our D-M results with theirs. We start with pulses which have a coherent phasefor the duration of the pulse ( ψ ( t ) = 0). Fig. 5 shows the time evolution of the excitedpopulation fractions for a pulse with intensity of 1 × W/cm and two different pulsedurations (100 fs and 200 fs), showing characteristic Rabi-flopping. The Rabi-frequency ofthe 3C populations is more rapid than the 3D, due to the larger A -value for the 3C transition.13his difference in Rabi-frequency can result in quite different excited populations at the endof the laser pulse interaction with the plasma. Considering these two pulse durations asan illustrative example: for the 100 fs case, the 3D transition has a much larger excitedpopulation at the end of the pulse than the 3C excited population, while for the 200 fscase the two have almost the same population fraction. This behavior drives the 3C/3Dline intensity ratio for the 100 fs case to be much smaller than the oscillator strength value.For these coherent and intense laser conditions, the line intensity ratio produced from thesepopulations would not necessarily be equivalent to the oscillator strength ratio. Furthermore,the contribution to the emission from the time after the laser pulse has left the plasma volumeis quite sensitive to the population in the excited state at the end of the laser pulse. Againone has the scenario where the emission from the ‘after-the-pulse’ component will be quitedifferent in the two cases, producing quite different line ratio values for these two pulses.Fig. 6 shows the 3C/3D line ratio as a function of pulse duration for coherent Gaussianpulses. We obtain very similar line ratio results to those of Oreshkina et al. [2, 3]. It isuseful to consider the two pulse durations shown in Fig. 5. The 3C/3D line ratios for thetwo scenarios shown in Fig. 5 are shown by the purple and green squares in Fig. 6. For the100-fs pulse (where the 3D population fraction is greater than the 3C value at the end of thepulse), the line ratio is 1.55 which is much smaller than the 3C/3D oscillator strength ratio,as one might expect from the populations. For the 200 fs pulse (where the 3D populationfraction is about the same as the 3C at the end of the pulse), the ratio is 5.38. Fig. 6 alsoshows that for coherent pulses a change in the line ratio from the oscillator strength ratiorequires pulse intensities above about 1 × W/cm .
3. Stochastic Gaussian pulse
To model the LCLS pulse parameters more accurately, the stochastic features of thepulse need to be included. We use the PCM [15, 16] to model the stochastic nature of thepulse intensity and phase. Fig. 7 shows a stochastic pulse intensity generated using thePCM. Note that it still has a Gaussian envelope, but there are now many stochastic spikesof intensity throughout the pulse. Note also that the electric field strength and the phaseare both stochastic and complex. These stochastic pulses can now be modeled using theD-M formalism to produce a 3C/3D line intensity ratio. Fig. 8 shows the comparison of the14
Excited state population
Time (fs) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 120 140 160 180 200 220
Excited state population
Time (fs)
FIG. 5. Excited state fractional populations as a function of time for a Gaussian pulse with intensity10 W/cm using the D-M model. The left panel displays the 100-fs results: the solid (purple)line indicates the 3C population and the dashed (purple) line indicates the 3D population. Theright panel displays the 200-fs results: the solid (green) line indicates the 3C population and thedashed (green) line indicates the 3D population. Intensity (W/cm ) FIG. 6. 3C/3D line intensity ratio as a function of radiation field intensity under a Gaussian pulseusing the D-M model compared with Oreshkina el al. [2, 3]. In all cases the symbols show theresults from the work of this paper and the lines show the results of Oreshkina et al. [3]. Resultsare shown for 100 fs (purple), 200 fs (green), 400 fs (blue), 600 fs (yellow), 1200 fs (dark blue), and2000 fs (red). calculated 3C/3D line intensity ratio with the results of Oreshkina et al. [2, 3]. The lineratio results are calculated from an average of 80 pulses using a bandwidth of 1 eV, and theresults are in good agreement with Oreshkina et al. [2, 3]. We were, however, not able toachieve convergence within 10 or 20 pulses as stated in their paper; in general it took moreruns to achieve convergence on the average line ratio value. The calculated line ratios areall below the oscillator strength ratio for intensities above ∼ W/cm . The bandwidth15
0 20 40 60 80 100 120 140 160 180 200 220
Intensity (W/cm ) Time (fs)
FIG. 7. A sample stochastic pulse with Gaussian envelope for a 200 fs pulse duration. Intensity (W/cm ) FIG. 8. The 3C/3D line intensity ratio as a function of radiation field intensity for a stochasticGaussian pulse using the D-M model. The symbols show the current results and the lines showthe results of Oreshkina et al. [2, 3]. Results are shown for 100 fs (purple), 200 fs (green), 400 fs(blue), and 600 fs (yellow). of the pulse also affects the coherence of the pulse and the duration of the spikes in theintensity, thus it strongly affects the line ratio. If the bandwidth is very small, then thepulse profile becomes much more coherent and the spikes in intensity are wide. In this limitthe stochastic pulses produce line ratio values very close to the coherent Gaussian pulsesfrom Fig. 6.It should also be noted that the emission from the plasma after the pulse has left theplasma volume is still a strong factor in lowering the line intensity ratio below the oscillatorstrength value. In the D-M approach using Gaussian envelopes for the pulses, it is difficultto define a before- and after-the-pulse component to the emission as the Gaussian envelope16
Counting duration (fs) 2 4 6 8 10 12 14 0 200 400 600 800 1000 1200 1400 1600 1800 2000
Counting duration (fs)
Counting duration (fs) 2 4 6 8 10 12 14 0 200 400 600 800 1000 1200 1400 1600 1800 2000
Counting duration (fs) a) b)c) d)
FIG. 9. The measured line intensity ratio as a function of counting duration. D-M results areshown for Gaussian pulse envelopes with the following conditions: a) with 200 fs and an intensityof 1 × W/cm , b) 400 fs and an intensity of 1 × W/cm , c) 200 fs and an intensity of1 × W/cm , d) and 400 fs and an intensity of 1 × W/cm . The hollow circle shows whatthe value would be if one stopped counting photons after the time specified by the Gaussian widthand the solid circles show the results if one kept integrating until the final time. will continue far beyond the defined width of the pulse. However, with the laser intensitydropping off, one would expect the emission characteristics at later times to be quite dif-ferent from the emission when the pulse is at its peak intensity. Figure 9 shows what themeasured line intensity ratio would be if one stopped counting photons at different times,for 4 different pulse profiles. This was generated using the D-M code, with 80 stochasticpulses per datapoint, a Gaussian envelope of either 200 or 400 fs width, and intensities of1 × W/cm and 1 × W/cm . It can be seen that the contribution from the emissionafter the pulse has finished its strongest interaction with the plasma is an important factorin producing a 3C/3D line intensity ratio that is lower than the oscillator strength value. Infact, without this contribution in the D-M approach the results would often be above theoscillator strength ratio. Thus, for both the C-R and D-M approaches it is important tokeep counting the emission beyond the main interaction phase of the laser with the plasma.17s a final illustration of the results using the D-M approach, a simulation was carried outfor a distribution of pulse intensities and pulse durations. Using a laser bandwidth of 1.0 eV,a distribution of pulse intensities, with 10 evenly spaced points per decade from 10 to10 W/cm , and a distribution of linearly spaced pulse durations ranging from 200 to 500 fs,a total line intensity for the 3C and 3D lines was produced. The two total line intensities werethen used to produce a 3C/3D line intensity ratio, giving a value of 2.71. It should be notedthat the pulse parameters and distributions are not well known from the experiment, so thistype of investigation should not be considered to be a true simulation of the experiment, butan illustration that pulse parameters in this range of intensities and durations can producea line intensity ratio close to the value that was measured. For the A -values chosen for thissimulation, some pulse intensities at (or above) 10 W/cm are required to produce lineratios in the range measured by the experiment. It would clearly be very useful to be able touse the observed line intensity ratio, and knowledge of the pulse parameters, to determinewhat the 3C/3D A-value ratio would need to be to produce agreement with the experiment(i.e., to make no assumption about the A-values for either line, but to determine the ratiofrom the experiment). However, without more accurate knowledge of the pulse parameters,this does not currently appear to be possible. The next section explores this concept in moredetail.
4. Photon counts
If the laser intensity is significantly below 10 W/cm , one would expect the line intensityratio to be close to the oscillator strength ratio. In recent discussion with the experimental-ists, it was pointed out to us that the defocusing of the laser would produce a beam muchmore weakly focused than we assumed in our model. While we had assumed a beam radiusof 5 µ m, it was likely to be closer to 0.5 mm (FWHM), i.e. a factor of 100 times wider. Thischange would produce intensities a factor of 10 weaker, so the range of pulse intensitieswould be 4.18 × – 3.14 × W/cm . In this range, the measured line intensity ratiowould be expected to be the same as the oscillator strength ratio.It is instructive to consider the photon counts produced from each pulse, rememberingthat the LCLS experiment consisted of a large number of individual pulses, with the final lineintensity being the result from all of the pulses combined. Fig. 10 shows the photon emission18 Number of photon emissons
Intensity (Watts/cm ) FIG. 10. Averaged photon counts for the 3C line as a function of radiation field intensity forstochastic Gaussian pulses using the D-M model. Results are shown for 200 fs (solid purple line),300 fs (dashed green line), 400 fs (dotted blue line), and 500 fs (dot-dashed yellow line). as a function of pulse intensity. The weak pulses produce only a few photons, and the numberof photons produced increases linearly with pulse intensity until about 10 W/cm . Thus,the more intense pulses produce more photons from the plasma. For the line intensity ratioto be dominated by the pulse intensities in the 4.18 × – 3.14 × W/cm range, itwould be very important that no pulses had intensities above this range. It would only takea few pulses above 10 W/cm for those pulses to dominate the line intensities, and hencethe line ratio. This topic will be explored in future work. It would also be of great benefitif an experiment could be performed where no pulses with intensities above ∼ W/cm were allowed to interact with the plasma. In such an experiment, the observed line intensityratio is expected to be a good indication of the 3C/3D oscillator strength ratio. IV. CONCLUSIONS
A review has been presented of two time-dependent methods that have been used tomodel the Fe XVII 3C/3D line intensity ratio for an intense laser field, the C-R and D-Mapproaches. Both methods show a reduction in the line intensity ratio below the oscillatorstrength ratio for pulses with intensities above ∼ W/cm . A significant factor inlowering the line intensity ratio for both methods is the contribution to the emission fromthe plasma after the laser pulse has left the plasma volume. We confirm the importance ofthe effects previously reported by Oreshkina et al. [2, 3]: the non-linear effects in the D-M19ethod and the stochastic nature of the laser pulses. As stated earlier, it is likely that themajority of the FEL X-ray pulse intensities in the experiments presented by Bernitt et al.are below 1 × W/cm . Since the presence of even a small number of pulses above thisthreshold could lower the observed 3C/3D line intensity ratio below the oscillator strengthratio, an experiment which could ensure there were no pulses above this threshold, and withwell constrained pulse parameters, would allow a conclusive statement about the 3C/3Doscillator strength ratio to be made. ACKNOWLEDGMENTS
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