Stimulated X-ray Raman scattering in Free Electron Lasers with incoherent spectrum
SStimulated X-ray Raman scattering in Free Electron Lasers with incoherent spectrum
Gennady Stupakov
SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
Max Zolotorev
Lawrence Berkeley National Laboratory, One Cyclotron Road, Berkeley, California,94720, USA (Dated: November 7, 2018)The single-pulse spectrum of self-amplified spontaneous emission (SASE) free electron lasers (FELs) is char-acterized by random fluctuations in frequency. The typical spectrum bandwidth for a hard x-ray FEL is in therange of 10-20 eV and is comparable with the distance between energy levels of valence electrons in atoms anmolecules. We calculate the rate of transitions in a quantum three-level system with the energy di ff erence ofseveral eV caused by such radiation and show that for x-ray intensities in the range of 10 W / cm the probabil-ity of the transition over the duration of the x-ray pulse is large. We argue that this e ff ect can be used to modifythe spectrum of a SASE FEL potentially making it more narrow. X-ray free electron lasers (FELs) [1–3] can generate highlyintense beams of radiation in which nonlinear x-ray–matter in-teraction plays a dominant role [4, 5]. Radiation pulses fromFELs of short, sub-femtosecond duration t have the coher-ent bandwidth (cid:126) / t of several eV which is commensurate withthe energy associated with electronic structure of atoms andmolecules. Attosecond pulses open a new path for creation ofcoherent localized valence electronic wave packets for studyof the energy transport in pump-probe experiments in molec-ular systems [6, 7].Generation of subfemtosecond pulses with a large coherentbandwidth requires operation of an FEL in a special mode [8–10], while nominally x-ray pulse duration is in the range oftens, or hundreds, of femtoseconds. Spectra of the typicalSASE pulses exhibit fine structure with narrow spikes fluctu-ating in positions and heights within the relative incoherentbandwidth on the order of 10 − . This structure is due to thefact that the radiation in a SASE FEL is initiated by the in-trinsic shot noise of the electron beam. For the x-ray energy E xr ≈
10 keV, the incoherent bandwidth is in the range of 10-20 eV, thus exceeding the distance between the energy levelsof valence electrons in atoms in molecules. In this paper, weshow that in combination with a high intensity of a focusedx-ray beam, through the mechanism of the stimulated Ramanscattering, FEL x-rays can excite quantum levels with the en-ergy distance between them on the order of a few eV. We alsoargue that the same mechanism leads to ionization of the va-lence electrons with the cross section that can be many ordersof magnitude larger than the direct photoionization cross sec-tion for hard x-rays. This e ff ect can play an important role fortightly focused x-ray beams required for single particle imag-ing [11].We consider a quantum system that has three energy levels E , E and E , as shown in Fig. 1, and assume that there areelectric dipole transitions from level 1 to 2 and from 2 to 3, butno direct transitions from 1 to 3 (e.g., 1 s , 2 p and 3 s subshellsin an atom). In the initial state, the system is at the lowest level E . Stimulated by the incident photons, the system makes avirtual transition to level 2 (which does not require the photonenergy to be equal to E − E ) and then another virtual transi- � � � � � � �� ���� FIG. 1. Three energy levels in a quantum system. Direct dipoletransitions are allowed between levels 1 and 2, and 2 and 3, but notbetween 1 and 3. tion from 2 to 3. If the spectral width of x-rays is greater that E − E , the transition probability 1 → → E ( t ) cos( ω t ), where ω = E xr / (cid:126) is the central frequency and E ( t ) is a slow varyingamplitude of the incident FEL radiation. Variation of E ( t ) intime determines the width of the x-ray spectrum; because ofthe stochastic nature of the SASE radiation we treat E ( t ) as astationary random process with a given statistical properties.We start from the equations for the time evolution of theprobability amplitudes a , a and a for the corresponding en-ergy levels [12]:˙ a = (cid:15) V ( t ) a cos ( ω t ) , (1a)˙ a = − (cid:15) V ∗ ( t ) a cos ( ω t ) + (cid:15) U ( t ) a cos ( ω t ) , (1b)˙ a = − (cid:15) U ∗ ( t ) a cos ( ω t ) , (1c)where the matrix elements U ( t ) and V ( t ) are given by V ( t ) = − i (cid:126) (cid:104) | H int | (cid:105) e − i ( E − E ) t / (cid:126) = − i (cid:126) d E ( t ) e − i ω t , U ( t ) = − i (cid:126) (cid:104) | H int | (cid:105) e − i ( E − E ) t / (cid:126) = − i (cid:126) d E ( t ) e − i ω t , (2) a r X i v : . [ phy s i c s . acc - ph ] D ec with H int the interaction Hamiltonian in the dipole approxima-tion, and d and d the matrix elements of the dipole opera-tor between the corresponding levels. We assume that the timeevolution of the amplitudes a i is slow in comparison with ω − ;this is indicated by the formal small parameter (cid:15) in Eqs. (1): (cid:15) ∼ E d (cid:126) ω (cid:28) , (3)where d is the characteristic dipole matrix element involvedinto the transitions. Eq. (3) states that the Rabi frequency E d / (cid:126) is much smaller than the x-ray frequency ω . Note thatEqs. (1) conserve the total probability | a | + | a | + | a | .Using the smallness of (cid:15) we now average Eqs. (1) overthe rapid oscillations with frequency ω , and obtain simpli-fied equations for a slow variation of amplitudes a i on a timeinterval much larger than 1 /ω . The starting point for this ap-proximation is the following representation of the amplitudes: a i ≈ α i ( (cid:15) t ) + (cid:15)β i ( (cid:15) t ) sin ( ω t ) + (cid:15) γ i ( (cid:15) t ) cos ( ω t ) + . . . (4)where α i , β i and γ i are slow varying functions of time which isindicated by their argument being (cid:15) t . Strictly speaking, on theright-hand side, there should also be terms with harmonics ofthe frequency ω , however, they do not contribute to the finalresult, and are neglected. Note that β i and γ i are small correc-tions to the zeroth order amplitudes α i , which are the subjectsof our interest. Substituting Eq. (4) into (1a) and collectingterms in front of cos( ω t ) and sin( ω t ) yields β = V α ω , γ = ˙ β ω . Averaging Eq. (1a) over the fast period 2 π/ω shows that therate of change of α is of the second order˙ α = (cid:15) V γ . Repeating the same analysis for the second and third equationsin (1) we arrive at the following set of di ff erential equationsfor the slow varying parts of the amplitudes:˙ α = (cid:15) ω V ddt ( − V ∗ α + U α ) , (5a)˙ α = − (cid:15) ω (cid:32) V ∗ ddt V α + U ddt U ∗ α (cid:33) , (5b)˙ α = (cid:15) ω U ∗ ddt ( V ∗ α − U α ) . (5c)Note that the second equation is decoupled from the first andthe third ones; it will be omitted from the subsequent analysis.The system of equations (5) conserves the probability | a | + | a | only approximately. It is easy to derive from Eqs. (5a)and (5b) that the following combination remains constant, | α | + | α | + (cid:15) ω (cid:104) | α | | V | + | α | | U | − (cid:0) U α α ∗ V (cid:1)(cid:105) , which di ff ers from the sum of probabilities | a | + | a | by thelast term. This term is small for (cid:15) (cid:28)
1. We will drop theformal smallness parameter (cid:15) in what follows.We now use the initial condition that at time t = E , α ( t = = α ( t = =
0. Considering time intervals small enough thatthe probability to find the system at level 3 remains small, wehave | α ( t ) | (cid:28)
1, and α ( t ) ≈
1. To the lowest order, we sub-stitute α = α = α = − (cid:126) ω d d E ( t ) e i ω t ddt E ( t ) e i ω t , where we have also used Eqs. (2) for the matrix elements.Integrating this equation over time and calculating the proba-bility to find the system at level 3 at time t gives w ≡ | α ( t ) | = (cid:126) ω d d (cid:90) t (cid:90) t dt (cid:48) dt (cid:48)(cid:48) E ( t (cid:48)(cid:48) ) e − i ω t (cid:48)(cid:48) × E ( t (cid:48) ) e i ω t (cid:48) ddt (cid:48)(cid:48) (cid:16) E ( t (cid:48)(cid:48) ) e − i ω t (cid:48)(cid:48) (cid:17) (cid:32) ddt (cid:48) E ( t (cid:48) ) e i ω t (cid:48) (cid:33) . (6)We now assume that the electric field E ( t ) is a stationaryrandom function which, after decomposition into the Fourierintegral, E ( t ) = (cid:82) ∞−∞ d ω ˆ E ( ω ) e − i ω t , can be characterized by thecorrelator (cid:104) ˆ E ( ω ) ˆ E ( ω (cid:48) ) (cid:105) = W ( ω ) δ ( ω + ω (cid:48) ) , (7)where the brackets denote an ensemble averaging and W ( ω )is the spectrum of the electric field measured relative to thecentral frequency ω . To carry out the statistical averaging ofthe probability w , one has to substitute the Fourier represen-tation for E ( t ) into (6) and calculate the fourth order correla-tors (cid:104) ˆ E ( ω ) ˆ E ( ω ) ˆ E ( ω ) ˆ E ( ω ) (cid:105) . With an additional assumptionthat E ( t ) is a Gaussian random process, these correlators areexpressed as a sum of the products of the second order correla-tors [13] for which we can use Eq. (7). After a straightforwardcalculation one finds w = π d d c (cid:126) ω t ( ω − ω ) (cid:90) ∞ d ω P ( ω − ω ) P ( ω ) , (8)where instead of the spectral function W ( ω ) we now usethe spectral power of the x-ray radiation P ( ω ), P ( ω ) = ( c / π ) W ( ω ). It is important to emphasize here that P ( ω ) isthe FEL spectrum averaged over many pulses; while a single-pulse SASE spectrum exhibits many spikes, the average oneis a smooth function of frequency.Note that the probability (8) vanishes if level 2 is in themiddle between levels 1 and 3, that is ω = ω . This isa well known e ff ect of vanishing Raman scattering in three-level system(see, e.g., [14], p. 185).To illustrate the feasibility of the stimulated Raman scat-tering for typical FEL parameters, we will now estimate theprobability of transitions in a hydrogen atom from level 1 s (level 1) to level 3 s (level 3) through level 2 p (level 2). Mea-suring the energy from the lowest level, we have E = E = Ry and E = Ry, and for the dipole moments, d = / − / ea B , d = / ea B , where a B is the Bohrradius [15]. We take the FEL parameters close to the ones inthe experiment [16], assuming the pulse energy of 1.0 mJ, thepulse duration of 50 fs with a flat temporal pulse profile, and (cid:126) ω =
10 keV. The beam is focused onto 150 nm ×
150 nm spotsize with the intensity P = . × W / cm . To simplifycalculations, we take for the averaged x-ray spectrum a Gaus-sian profile with the rms spread ∆ ω ≈ × − ω =
20 eV / (cid:126) , P ( ω ) = (2 π ) − / ∆ ω − P e − ω / ∆ ω (we remind the reader thatthe frequency ω in this equation is measured relative to thecentral frequency ω ). Carrying out the integration in Eq. (8)we find for the probability w : w = tP π / d d c (cid:126) ω ∆ ω ( ω − ω ) e − ω / ∆ ω , (9)which for our example gives for the transition probability w ≈ . t [fs] . (10)Since our calculations assume w (cid:28)
1, this formula is validfor t (cid:46)
20 fs. Note that the smallness parameter (3) estimatedwith
E ∼ √ π P / c ≈ × V / m and d ∼ √ d d = . ea B equals 0.37 which is not small compared to unity.To test the accuracy of our approximate analysis we nu-merically integrated Eqs. (1) for 200 realizations of the ran- � �� �� �� �� ����������������� � ( �� ) � � FIG. 2. Time evolution of the probability w averaged over 200 re-alizations of the random field E ( t ): 1)- P = . × W / cm , 2)- P = . × W / cm and 3)- P = × W / cm . The red dashedlines show the small-time approximation (9) for each case. dom electric field E ( t ) with a Gaussian spectrum describedabove, for three di ff erent intensities: P = . × W / cm , P = . × W / cm and P = × W / cm . The plotsof w ( t ) as a function of time for the three cases are shown inFig. 2. The red dashed lines near the origin show the small-time approximation calculated with Eq. (9) for each case. Re-markably, even though the parameter (3) is not really small for cases 1 and 2, Eq. (9) gives a relatively good approximationfor the initial slope of w ( t ). One can also see that, for thesetwo cases, after an initial, approximately linear, growth w ( t )saturates at the level w ( t ) ≈ . P ( ω ) in (8) by c (cid:126) ω n ph ( ω ) where n ph ( ω ) is thedensity of photons in the beam per unit frequency interval. Wethen re-write Eq. (8) as an expression for the probability perunit time w t = c (cid:90) ∞ d ω σ ( ω ) n ph ( ω ) , (11)where σ ( ω ) has a meaning of the di ff erential cross section forthe scattering, σ ( ω ) = π d d c (cid:126) ω ( ω − ω ) P ( ω − ω ) . (12)Note that this cross section is proportional to the incident in-tensity of x-rays at the frequency shifted by the distance be-tween the level 1 and 3. For our numerical example (10), themaximum cross section at ω = ω , is σ ≈ × − cm .This cross section is almost five orders of magnitude largerthan the ionization cross section of hydrogen by 10 keV pho-tons, σ ion ≈ × − cm , and three orders of magnitudelarger than the Thomson cross section for elastic scattering.Examination of Eq. (12) shows that for radiation with abandwidth smaller than ω the cross section vanishes because P ( ω − ω ) lies outside of the bandwidth if ω is inside it. Fora spectrum wider than ω the scattering is di ff erent at thelow-energy and high-energy parts of the spectrum. Indeed,assuming for illustration a flat spectrum occupying the inter-val [ ω , ω ] of width ∆ ω = ω − ω > ω (see Fig. 3) wesee that the scattering occurs only in the region [ ω + ω , ω ],while at the low-energy end of the spectrum [ ω , ω + ω ] thecross section (12) is zero. Taking into account that in an actof scattering a photon of frequency ω changes its frequency to ω − ω , we expect that, given enough scattering events, thestimulated Raman scattering would lead to a noticeable mod-ification, and possible shrinking, of the incident spectrum ofx-rays. For a numerical example, let us assume that x-rays Δωω �� ω - ω � ���������� FIG. 3. Illustration of the x-ray spectrum evolving due to the stimu-lated Raman scattering. The scattering processes occur only to theright of the red vertical line located at the distance ω from theleft edge of the spectrum. The scattering downshifts photons in fre-quency as indicated by the curved arrows. are passing through a frozen solid hydrogen with the density5 . × atom / cm . For the cross section σ ≈ × − cm estimated above, one needs the hydrogen target thickness of ≈ . ∼
100 nm can be neglected.While the above analysis indicates the feasibility to modifythe spectrum of the x-rays through the mechanism of stimu-lated Raman scattering, a more accurate, quantum treatmentof the problem is needed to be able to draw quantitative con-clusions about the e ff ect.To elucidate the underlying physical mechanism, in ouranalysis above, we have considered a model of a three-levelquantum system. In reality, in atoms and molecules, the stim-ulated scattering would cause multi-level transitions occur-ring at the same time with di ff erent frequencies and at variousrates. Our numerical results should then be considered as aguide only; a more accurate analysis is required of the quan-tum dynamics in a multi-level system interacting with stochas-tic incident field. We would also point out, that it seemshighly plausible that the same mechanism of the stimulatedRaman scattering will lead to electron transitions into con-tinuum part of the spectrum, e ff ectively ionizing atoms andmolecules with the cross section much larger that the directphotoionization by x-rays.In conclusion, we have shown that, in a three-level sys-tem, tightly focussed SASE FEL radiation can lead to exci-tation, and likely ionization, of valence electrons in atoms andmolecules through the mechanism of the stimulated Ramanscattering. The cross section for the scattering can be largeenough to be used for modification of the FEL spectrum bysending the x-ray beam through a medium with properly se-lected energy levels, thus opening up an opportunity to modifyand control the SASE spectrum before it is used in an experi-ment.The authors would like to thank P. Bucksbaum, D. Bud-ker and J. Hastings for stimulating discussions. G. S. ac-knowledges support from the DOE grant No. DE-AC02-76SF00515, and M. Z. acknowledges support from DOE grantNo. DE-AC02-05CH11231. [1] P. Emma, A. Akre, J. Arthur, R. Bionta, C. Bostedt, J. Bozek,A. Brachmann, P. Bucksbaum, R. Co ff ee, F.-J. Decker, Y. Ding, D. Dowell, S. Edstrom, A. Fisher, J. Frisch, et al. Nat. Photon-ics , 641 (2010).[2] W. Ackermann, G. Asova, V. Ayvazyan, A. Azima, N. Baboi,J. Bahr, V. Balandin, B. Beutner, A. Brandt, A. Bolzmann,R. Brinkmann, O. Brovko, M. Castellano, P. Castro, L. Catani, et al. Nature Photonics , 336 (2007).[3] T. Ishikawa, H. Aoyagi, T. Asaka, Y. Asano, N. Azumi,T. Bizen, H. Ego, K. Fukami, T. Fukui, Y. Furukawa, S. Goto,H. Hanaki, T. Hara, T. Hasegawa, T. Hatsui, et al. Nat Photon , 540 (2012).[4] L. Young, E. P. Kanter, B. Kr¨assig, Y. Li, A. M. March, S. T.Pratt, R. Santra, S. H. Southworth, N. Rohringer, L. F. Di-Mauro, G. Doumy, C. A. Roedig, N. Berrah, L. Fang, et al. Nature , 56 (2010).[5] G. Doumy, C. Roedig, S.-K. Son, C. I. Blaga, A. D. DiChiara,R. Santra, N. Berrah, C. Bostedt, J. D. Bozek, P. H. Bucksbaum,J. P. Cryan, L. Fang, S. Ghimire, J. M. Glownia, et al.
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