Single-shot spectrum measurements of the radiation emitted by an electron bunch provide a novel way to characterize the bunch shape. Shot noise fluctuations in the longitudinal beam density result in radiation with a spectrum that consists of spikes with width inversely proportional to the bunch length. The variance of the Fourier transform of the spectrum is proportional to the convolution function of the beam current averaged over many bunches. After the convolution function is found, the phase retrieval technique can be applied to recover the bunch shape. This technique has been used to analyze the shape of the 4-ps-long bunches at the Low-Energy Undulator Test Line at the Advanced Photon Source

Sajaev, V.

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.- 0 f+ti~ /k5A/c% J5L99-3SDETERNHN.+TION OF LONGITUDINAL BUNCH PROFILE USINGSPECTRAL FLUCTUATIONS OF INCOHERENT RADIATION* g; .g~~,v~lV. Sajaev, Argonne National Laboratory, Argonne, IL 60439 U.S.A. AM 04 zu@NwT/AbstractSingle-shot spectrum measurements of the radiationemitted by an electron bunch provide a novel way tocharacterize the bunch shape. Shot noise fluctuations inthe longitudinal beam density result in radiation with aspectrum that consists of spikes with width inverselyproportional to the bunch length. The variance of theFourier transform of the spectrum is proportional to theconvolution function of the beam current averaged overmany bunches. After the convolution function is found,the phase retrieval technique can be applied to recover thebunch shape. This technique has been used to analyse theshape of the 4-ps-long bunches at the Low-EnergyUndulator Test Line at the Advanced Photon Source.1 INTRODUCTIONMeasurement of the longitudinal profile of a beam is animportant diagnostic tool for accelerators. For bunchlengths in the range of picosecond, such measurementscan be done by a streak camera. Shorter bunches usuallyrequire some kind of special techniques. It has beenrecently proposed that longitudinal properties of anelectron bunch can be obtained through measurement ofthe fluctuations of undulator radiation from the bunch [l].First measurements of the single-shot spectra of theundulator spontaneous emission at the Accelerator TestFacility at Brookhaven National Laboratory (ATF/BNL)with resolution required to demonstrate 100% fluctuationsof intensity were recently reported [2].This report presents results of single-shot spectrummeasurements obtained at the Low-Energy Undulator TestLine at the APS. The measurements repeat those made atthe ATF/BNL; however, data processing in theexperiment described here is expanded. The spectrumintensity fluctuations are used not only for the bunchlength extraction but for reconstructing the detailedlongitudinal bunch profile.This paper begins with a brief theoretical introduction.Then it describes the bunch length extraction followingthe ATF experiment. The rest of the paper is devoted tothe experiment~l recovering of the bunch profile.Let us consider the ml~ri~scopic picture of the electronbeam current at the en[r.irwe into the undulator. Theelectron bea,m current L(~II\Is[s ot’ electrons arriving at theentl-~incc of [he undul:[h~r .~[ w~me p:r[icular time f<:k=lwhere N is the number of electrons in a bunch. TheFourier transform of the current and the electric fieldemitted in the undulator can be written asi(tO)= Jeim l(t)dt=-e.’jeimk ,—m k=lE(u) = e(m). ~eiuk , (1)kwhere e(o) is the Fourier transform of an individualparticle traveling through the undulator and isproportional to sin( uq,)T/( u q)T. The summation oflarge numbers of exponential with different argumentsresults in an appearance of sharp spikes in the E(O)dependence. This leads to bunch-to-bunch fluctuations inintensity of the incoherent radiation spectrum.To extract the bunch length information from thespectrum, one can calculate the second order correlationof the Fourier harmonics ~(a):~“/ n,m. p.qThe second order is necessary because measurementsprovide the spectrum of intensity. The beam currentaveraged over many bunches can be written in the form(1(1)) = -eNF(t),where F(t) is the electron bunch form factor. For the caseN~F(0)l~ <<1(which is true for our situation) the expressionfor the correlation above can be simplified:So the spectrum correlation is related to the square of theFourier transform of the bunch form factor. This formulacan be used to calculate the bunch length using thecorrelation of the radiation spectrum.2 SPECTRUM MEASUREMENTS ANDBUNCH LENGTH EXTRACTIONR:ldiation spectra were measured using a high-resolution spectrometer [3] \vith a cooled CCD im~ger.which provides a resolution ot’ 0.4 .~ per pixel.Nle;isurement c)F the single-shot spectrum is shcwn inFigure I. The spectrn are comp{wed of spikes Of r:tndonlDISCLAIMERThis report was,.prepared as an account of work sponsoredby an agency of the United States Government. Neitherthe United States Government nor any agency thereof, norany of their employees, make any warranty, express orimplied, or assumes any legal Iiabiiity or responsibility forthe accuracy, completeness, or usefulness of anyinformation, apparatus, product, or process disciosed, orrepresents that its use wouid not infringe privateiy ownedrights. Reference herein to any specific commercialproduct, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constituteor impiy its endorsement, recommendation, or favoring bythe United States Government or any agency thereof. Theviews and opinions of authors expressed herein do notnecessarily state or refiect those of the United StatesGovernment or any agency thereof.DISCLAIMERPortions of this document may be illegibiein electronic image products. Images areproduced from the best available originaldocument...amplitude and frequency that have a characteristic widthAOJ- 11~ (where q defines the bunch length) and intensityfluctuation of almost 100%. The shape of an individualspectrum changes randomly from shot to shot, but theaverage of many shots approaches the familiar wigglerspectrum.Single shot spectrum515 520 525 530 535 540Wave ‘angth (rim)Figure 1. Typical single-shot spectrum.To extract the bunch length from the spectral data, Eq.(2) for the second-order spectrum correlation is used. Thenormalized autocorrelation of the spectral intensityaveraged over many shots is calculated from the spectrummeasurements:cn=(?p(mi)p(@i+tL))’(?p(47where P(q) is the signal in the i-th CCD channel and n isthe shift in channels. The correlation averaged over 100shots is plotted in Figure 2.For the ideal case of a zero emittance bezm anddiagnostics with sufficient spectral resolution, 100%fluctuation of the spectral intensity \vill occur. However.when the beam size is large or the detector spectralresolution is poor, the spectrum \vill be similar to aspectrum emitted by several independent sources. Thefluctuation level will be reduced, and a pedestal willappear in the spectrum.Spactrum autocorrelation109\I080.7.The spike width at half maximum according to Figure 2is about two pixels. The frequency step corresponding toone pixel is 60} = 2,4.10” rad/s. Therefore, assuming thebeam to be Gaussian, the sigma of the Gaussiandistribution is1-—=2ps.TbnibThis gives the FWHM length of the bunch to be equal to4.5 ps.Accuracy in determining the spike width is not so greatdue to the size only being a few pixels. This means theresolution of the spectrometer is not enough for our bunchlength. However, shorter bunches result in wider spikesand improved accuracy.3 PROFILE MEASUREMENTSMeasurement of spectral intensity fluctuations can beused not only to determine the bunch length but also torecover a longitudinal bunch profile. It has been shown[1] that the vmiance of the Fourier transform of thespectrum is proportional to the convolution function of thebeam current. After the convolution function is found, aphase retrieval technique can be used to recover the shapeof the pulse in many practical cases.These two steps are described below. For each stepboth simulation and experimental results are presented.3.1 Calculation of the Convolution Function ofthe Bunch ShapeLet us denote the detector signal in channel m formeasurement n as P,,,,,,. The Fourier transform of thespectrum is~=1where NC,, is the number of channels in the detector(CCD). After ~ccumulation of N, number of pulses largeenough for statistical analysis, the following quantity iscomputed:It can be described as the average deviation of the signalin the k-th channel from its average. The quantity dk givesthe convolution function of the particle density in thebunch averaged oler NObunches (the theoretical proof ofthis can be found in Ref. [l]).Figure ~ represents the convollltion function of thelon~itudinal buiwh distribution extracted from spectralme:lsurements described :Ibove. The ntimber of spectraused t’or :lverilging is 100 \\ith J slit size ot’ 50 m. Thispl,>[ c:~nalSO ty LINCI t{) de[errnine [k bunch length. If w’ewuine th:~t [ht bunch h.ts a Gaussian shape. then itS‘.(~lli ~>illtl{)n I, .II\, ) ~1 ~:lllssi:)n with R V2 ;,. A fittedG:lumi:ln t’un~’[:,’ni> :tlstl ~h~~i}n in Figure 3 m a solid line.. .Convolution of particle density,,!,$., -e !. .,,1 l..,, ,,! :, ,0, ... ,.. . . . .,,b: ‘+-;””:”-:”:””,!,0, . .. – .. . . . . . . . . .!,:.OSo!mls Zum. ?calsoFigure 3. Convolution of longitudinal particle densitycalculated using measured spectra. Solid line is a fittedGaussian function.Knowledge of the convolution function does not allowa unique restoration of I(t). However. as shown in Ref.[4], the use of a phase retrieval technique allows one torecover the beam profile in many practical cases.3.2 Reconstruction of the Bunch Shape from theConvolution FunctionIt is possible to extract both the amplitude and thephase information of the radiation source by applying aKramers-Kronig relation to the convolution function. Thistechnique of phase extraction was well developed in theoptics of solids for the problem of reflectivity. We willbriefly describe the technique following Ref. [4].We denote the longitudinal particIe density as S(Z) andits Fourier transform as S( o,). and write it in the followingformS(0) = p(~)eiw(’”) , (3)Here d@ corresponds to the Fourier transform of ourconvolution function. Phase v can be extracted using theexpression{)cu-mj+~aI. — ~(1) - w;where Vmis the minimal phase and the @j’s are the zerosof S( o) in the upper half of the complex frequency plane.If S(O) has no zeros, the contribution from YB,(,,.,,IC(0)equals zero, and the expression above gives the minimalphase. This minimal phase is a good approximation to theactual phase in cmes where the bunch density has nonearby zeros in the upper half of the complex frequencyplane. The final expression for mlculatin~ thephase is(‘“ Js(:) =qiu.p(fl)). cos !u ((0)-=~oOne can see that the plot of the convolution function isvery noisy despite averaging over 100 single shots. Thisnoise produces high frequency content in the spectrum;therefore, before taking the Fourier transform of theconvolution function. it has been smoothed by filtering thehigh frequencies. The result of calculating the bunchshape using the smoothed convolution is presented inFigure 4. A FWHM length of the bunch according to theplot is 4 ps, which corresponds to the estimates madeabove.Bunch shape>2- ,.. ,,!$11 ,11,,1 ,,, ,, ,1,11!!1!!M --- ––F–-,.---–* -__;-––roll0to 121413 !820 22242e28~Time (ps)Figure 4. Longitudinal bunch shape recovered fromspectrum measurements.4 CONCLUSIONSA new technique for recovering a longitudinal bunchshape from single-shot spectrum measurements has beenimplemented, and a bunch profile has been measured. Animportant feature of this method is that it can be used forbunches with lengths less than a millimeter – the shorterthe bunch, the less requirements for the spectrometer.To reconstruct the bunch shape from the convolutionfunction the technique suggested in Ref. [4] has beenutilized, but a different approach has been used to buildthe convolution. By using spectrum fluctuations toconstruct the convolution one can avoid measuring thespectrum of coherent far infrared radiation of the bunchand making any assumptions about the asymptoticbehavior of this spectrum.The author would like to thank Max Zolotorev for hisconsiderable help and many useful discussions during thiswork, and Efim Gluskin et al. for allowing the author touse the spectrometer system.REFERENCES[1] N1. Zolotorev. G. Stupakov. SLAC-PUB-7132,Nlarch 1996. See also in Prw. ot’ P.AC97.(2] P. Catr:l\m et al.. P//ys. /?c\. Letf. 82. N26. pp.5261-5:64.{.4] E. Gluskin et al.. IVIIW A 4?9. 19!)9. pp. 358-364./4] R. [ Ii A. Sle\~rs. 1 [Icr[) Bunches Workshop, A[PL un~ Pj[k. 367.’ 199J.’-. ‘.

Determination of longitudinal bunch profile using spectral fluctuations of incoherent radiation

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