A method for generating a single broadband radiation pulse from a strongly chirped electron pulse is described. The evolution of the chirped electron pulse in an undulator may generate a pulse of coherent spontaneous radiation of shorter duration than the FEL cooperation length. An analytic expression for the emitted radiation pulse is derived and compared with numerical simulation

Campbell, L. T.

McNeil, B W. J.

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryCampbell, L. T. and McNeil, B W. J. (2013) A simple model for the generation of ultra-short radiation pulses. In: FEL 2012. JACoW, pp. 626-629. ISBN 9783954501236 , This version is available at http://strathprints.strath.ac.uk/57081/Strathprints is  designed  to  allow  users  to  access  the  research  output  of  the  University  of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may  not  engage  in  further  distribution  of  the  material  for  any  profitmaking  activities  or  any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge. Any  correspondence  concerning  this  service  should  be  sent  to  Strathprints  administrator: strathprints@strath.ac.ukA SIMPLE MODEL FOR THE GENERATION OF ULTRA-SHORTRADIATION PULSESL.T. Campbell and B.W.J. McNeilSUPA, Department of Physics, University of Strathclyde, Glasgow, UKAbstractA method for generating a single broadband radiationpulse from a strongly chirped electron pulse is described.The evolution of the chirped electron pulse in an undulatormay generate a pulse of coherent spontaneous radiation ofshorter duration than the FEL cooperation length. An an-alytic expression for the emitted radiation pulse is derivedand compared with numerical simulation.INTRODUCTIONFree Electron Lasers based on the SASEmechanism pro-duce intense, uncorrelated radiation pulses of width ap-proximately equal to the FEL cooperation length, lc [1].For sufficiently short electron pulses only one single pulsemay be emitted. However, such a mode of operation re-sults in output pulses that contain significant temporal jitter.Other methods are being considered to improve the tempo-ral coherence, and to further reduce the temporal width ofthe radiation pulses [2].The method discussed here predicts emission of a singleradiation pulse of width< lc from a short, strongly-chirpedelectron beam evolving in an undulator. The method is de-scribed from a general perspective without reference to anyspecific electron beam source.A sufficiently strong negative chirp in an electron pulsewill cause the pulse to shorten in longitudinal phase spacewhen propogating in an undulator as shown in Fig. 1. Af-ter a sufficient propagation distance, the pulse length willcome to a minimum, the energy chirp will then becomepositive and the pulse will start to lengthen.If the electron pulse length evolves to the order of a res-onant FEL wavelength λr, it will generate a short pulseof coherent radiation. This coherent spontaneous emissionmay occur for a number of undulator periods dependentupon the magnitude of the chirp and the length of the elec-tron pulse. For example, for a stronger chirp will causethee electron pulse to radiate coherently for fewer undula-tor periods, resulting in a shorter radiation pulse of broaderbandwidth.This is not an FEL effect, rather, it is a coherent ef-fect arising purely from the linear evolution of the electronbeam due to the chirp. Nevertheless, an unaveraged systemof FEL equations is used to describe the effect, as it is ca-pable of describing both the evolution of the longitudinalshape of the electron pulse, and the consequent emission ofCSE from the short electron pulse of length < λr, neitherof which can be described using an averaged set of undula-Figure 1: Schematic of the evolution of short negativelychirped electron pulse propagating through an undulatorshowing the current I and scaled energy γ.tor/FEL equations.Here, a 1D analytic solution of the expected coherent ra-diation output from such a chirped electron pulse is derivedand compared with both a 1D and 3D numerical simulationusing the unaveraged FEL code Puffin [3].ANALYTIC SOLUTIONAn analytic solution is derived from the un-averaged sys-tem of FEL equations presented of [4]. These equations areidentical to those of the code Puffin [3] in the 1D Comptonlimit. The electrons are assumed uncoupled from the radi-ation so that only the wave equation need be solved:(∂∂z¯+∂∂z¯1)A(z¯, z¯1) =1n¯p‖N∑j=1e−iz¯12ρ δ(z¯1− z¯1j), (1)where the scaling of [4] has been used. As in [4], the sourceterm can be put into a more convenient functional form bydiscretising z¯1 into intervals of width ∆z¯1 << 4πρ, i.e.much less than a radiation wavelength. In this way, thesource term of (1) becomes a sum over discrete intervals:∞∑n=−∞χn (z¯) e−iz¯12ρ δ(z¯1 − z¯1n)∆z¯1, (2)where: χn (z¯) ≡ In (z¯) /Ipk is the current in the n-th in-terval scaled with respect to the peak current of the electronpulse at the beginning of the undulator, z¯ = 0. The solu-tion proceeds by then taking the unitary Fourier transform,THPD41 Proceedings of FEL2012, Nara, JapanISBN 978-3-95450-123-6626Copyrightc ○2012bytherespectiveauthorsFEL Technology III: Undulators, Monitors, Beam diagnosticsFigure 2: These 8 plots show the intensity and electron beam phase space at 4 different propagation distances. The initialchirp in the plot at z¯ = 0 in the top left causes the electron pulse to compress and then decompress. When it is bunchedto a length in z¯2 of . 4πρ (that is to say, less than a resonant radiation wavelength) it generates a single coherent spike inintensity.Fω¯{X (z¯, z¯1)} = X˜(z¯, ω¯), of (1), redefining the radiationfield envelope so that A′(z¯, ω¯) = A˜(z¯, ω¯)eiω¯z¯ and takingthe limit ∆z¯1 → 0 in the source term to obtain:∂A′(z¯, ω¯)∂z¯=√2πeiω¯z¯χ˜(z¯, ω¯′), (3)where ω¯′ ≡ ω¯+1/2ρ. Under the assumptions of [4], this isa general equation for the wave envelope for a given scaledcurrent χ(z¯, z¯1).It is now assumed that the electron current has a Gaus-sian distribution with a linear, negative energy chirp and inscaled form is given by:χ(z¯, z¯1) =σ0σ(z¯)exp( −z¯212σ2(z¯))(4)where σ(z¯) =√σ20(1 + g0z¯)2+ σ2r is the z¯ dependentwidth due to a linear scaled energy chirp. The chirp is de-fined in electron phase space at the start of the undulator by:p0(z¯1) ≡ (γ(z¯1) − γr)/ργr = g0z¯1 where g0 is the initialchirp gradient in z¯1 . Hence, the the range of electron ener-gies in the pulse is determined by the product g0σ0. Here itis assumed g0 < 0. The constant σr describes a minimumresidual width of the pulse when g0z¯ = −1.This evolution of the electron pulse as a function of z¯ al-lows the field equation (3) to be integrated over −∞ <z¯ < ∞ to obtain an analytic solution for the field. Toenable a solution, a transform to a new independent vari-able zˆ = 1 + g0z¯ is made so that g0 = 0 is not allowed.Note also, that the evolution as a function of z¯ is not given,only the result of the electron emission in an undulator ofsufficient length z¯ ≫ |g0|−1. The solution is obtained inFourier space, and the final expression for the envelope asa function of f = ω/ωr may be written as:A˜(f) =4πρfexp(−if − 12ρg0− f2σ2i8ρ2− (f − 1)22g20σ20f2)(5)It is clear from the final term in the exponent of the expo-Proceedings of FEL2012, Nara, Japan THPD41FEL Technology III: Undulators, Monitors, Beam diagnosticsISBN 978-3-95450-123-6627 Copyrightc ○2012bytherespectiveauthorsnential term that the greater the total energy range withinthe electron pulse, as defined by g0σ0, the broader the ra-diation spectrum and the shorter the radiation pulse gen-erated. In what follows, both 1D and 3D simulation arepresented and compared with the above analysis.SIMULATIONSThe non-averaged FEL simulation code Puffin [3] wasused to simulate typical output from an helical undulatorusing the following parameters: ρ = 8 × 10−3, a¯u = 1.5,γ = 707 and λw = 1.5cm. An electron beam energy chirpof g0 = −0.192 was applied to a Gaussian current distri-bution with scaled rms width of σ0 = 0.333. An homoge-neous energy spread of σγ/γ = 0.17ρ was also used.In what follows, the scaled units of [3] are used. Themain difference from the results above derived from (1) isa change from the independent variable z¯1 to z¯2 = z¯−z¯1. Aresonant electron would propagate along z¯1 =constant i.e.z¯1 measures distance in units of lc in the resonant electronrest-frame away from the tail of the electron pulse. How-ever, a radiation wavefront propagates along z¯2=constantand z¯2 measures distance in units of lc in the light rest-frame away from the head of the light pulse. Note that p2jis the rate of change of the j-th electron’s position in z¯2 , sothat a larger values of p2j means the electron will propagatefaster away from the head of the light pulse - i.e. larger p2jmeans a lower electron energy.The results of a 1D simulation using the above parame-ters and shown in Fig. 2 which plots the scaled radiation in-tensity and the evolution in (z¯2, p2) phase space of the elec-tron bunch as it propagates through the undulator. (Notethat in the 1D limit, |A|2 is the scaled intensity, not power,generated by an electron beam of uniform transverse den-sity.) Electron bunch compression and decompression dueto the energy chirp is seen as the interaction progresses.As the length of the electron pulse approaches that of theresonant radiation wavelength (4πρ ≈ 0.1), it starts to ra-diate strongly and coherently. This is seen in the phasespace plot at z¯ = 5.22 which shows the compressed elec-tron bunch emitting a coherent radiation spike in the cor-responding intensity plot. Further propagation begins tode-bunch the electrons and the coherent emission reduces,so forming the short radiation pulse. Note that the final ra-diation pulse width, from the intensity plot at z¯ = 7.25 ofFig. 2, is slightly less than a cooperation length. The radia-tion pulse width may be significantly less than one cooper-ation length, which is smaller than pulse widths producedby the FEL process in SASE.The radiation intensity spectrum was calculated using anumerical Fourier transform and is plotted in Fig. 3. Thisspectrum is in very good agreement with the analytical so-lution of equation (5).A 3D simulation using the same parameters as above wascarried out. A scaled emittance of ǫ¯ = 4πǫ/λr = 1.6 wasused and the beam radius matched to the undulator usingnatural focussing. This gives an effective energy spread,Figure 3: The intensity spectrum |A˜|2 plotted as a functionof scaled frequency f = ω/ωr at z¯ = 7.6676 as shown inFig. 2.from [5], of σǫ = 0.00064, which is very small comparedto the homogeneous energy spread as used in the 1D sim-ulation above. The betatron wavelength in units of z¯ isλ¯β = 4√2πγρ/a¯u = 67.01, and the Rayleigh length of theresonant wavelength for the matched beam radius, again inunits of z¯, is z¯R = ǫ¯λ¯β/4π = 8.532 [3], both of whichare relatively large with respect to the FEL gain length.With these parameters it may be expected, and as is shownbelow, that diffraction and emittance effects are relativelysmall so that the results will not deviate significantly fromthe 1D simulation of above.From the 3D simulations, the transverse plot of thescaled intensity at z¯ = 7.6676 and z¯2 = 7.0456, closeto the coherently radiated peak is plotted in Fig. 4. A sharppeak is seen on-axis at (x¯, y¯) ≈ (0, 0) where the electronpulse is propagating. The scaled power |A|2 is calculatedby integrating the scaled intensity over the transverse re-gion of the simulation and is plotted in Fig. 5 as a functionof z¯2. As with the 1D case, the sharp peak in the powerin the region 6 ≤ z¯2 ≤ 7 indicates where the electronbeam became strongly bunched and emitted a strong co-herent pulse.The scaled spectral intensity was calculated from thedata along the center of the transverse plane at (x¯, y¯) ≈(0, 0) and is plotted in Fig. 6. It can be seen that the spec-trum is in good agreement with that of the analytical solu-tion and the 1D simulation of Fig. 3. As expected from thecalculated betatron wavelength and Rayleigh range above,the 3D effects have not significantly reduced the spectralintensity, and the emission of the radiation from the chirpedpulse has not been significantly degraded.CONCLUSIONSUsing a relatively simple 1D model, a method wasdescribed which appears capable of generating a single,intense, broad-bandwidth radiation pulse from a shortTHPD41 Proceedings of FEL2012, Nara, JapanISBN 978-3-95450-123-6628Copyrightc ○2012bytherespectiveauthorsFEL Technology III: Undulators, Monitors, Beam diagnosticsFigure 4: The transverse intensity profile near the peak atz¯ = 7.6676 and z¯2 = 7.0456 as shown in Fig. 5.Figure 5: The scaled power |A|2 is calculated by integrat-ing the scaled intensity of Fig. 4 over the transverse areaand is plotted as a function of z¯2 for z¯ = 7.6676.Figure 6: The spectral intensity calculated on-axis alongthe z¯2 axis at z¯ = 7.6676.chirped electron pulse in an undulator. Analysis and simu-lations in both 1D and 3D using the unaveraged modellingof [3] show good agreement. It is noted that the gaussianpulse current used in the model is not best suited to gen-erating strong coherent emission and other current profilescan be expected to generate more intense output.REFERENCES[1] R. Bonifacio, L. de Salvo, P. Pierini, N. Piovella and C. Pel-legrini, Phys. Rev. Lett. 73, 70 (1994)[2] Brian W. J. McNeil and Neil R. Thompson, Nature Photon.4, 814 (2010)[3] L.T. Campbell and B.W.J. McNeil, ‘Puffin: A three dimen-sional, unaveraged Free Electron Laser simulation code’,MOPD12, 34th International FEL Conference, Nara, Japan(2012)[4] B.W.J. McNeil, M.W. Poole and G.R.M. Robb, Phys. Rev.ST-AB 6, 070701 (2003)[5] R. Bonifacio, L. De Salvo Souza and B.W.J. McNeil, OpticsComm, Vol 93, 179-185 (1992).Proceedings of FEL2012, Nara, Japan THPD41FEL Technology III: Undulators, Monitors, Beam diagnosticsISBN 978-3-95450-123-6629 Copyrightc ○2012bytherespectiveauthors

A simple model for the generation of ultra-short radiation pulses

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A simple model for the generation of ultra-short radiation pulses

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