Growth of transverse coherence in SASE FELs

Nuclear Instruments and Methods in Physics Research A 445 (2000) 77}83

Growth of transverse coherence in SASE FELs

Vinit Kumar*, Srinivas Krishnagopal

Beam Physics and FEL Laboratory, Centre for Advanced Technology, Indore 452013, India

Abstract

We introduce the correlation function between the electric "eld at two di!erent points in the transverse plane asa parameter to quantify the degree of transverse coherence. We also propose a more realistic model for the initializationof the radiation in computer codes used to study SASE FELs. We make these modi"cations in the code TDA and use it tostudy the growth of transverse coherence as a function of electron beam size, beam current and transverse emittance. Ourresults show explicitly that the onset of full transverse coherence in SASE takes place much before the power saturates.With the more realistic model the onset of the exponential growth regime is delayed, and to get a given power from theFEL one needs a longer undulator than would be predicted by the original TDA code. ( 2000 Elsevier Science B.V.All rights reserved.

1. Introduction

The coherence properties of Self-Ampli"edSpontaneous Emission (SASE) in free-electronlasers (FELs) are of considerable interest since onewould always prefer the output radiation froma laser to be coherent, both temporally as well asspatially. The temporal coherence of SASE is gov-erned by the longitudinal dynamics of the electronbeam and can be studied analytically under theframework of a one-dimensional theory using theMaxwell}Vlasov equation [1,2]. One can alsostudy it numerically [3] using one-dimensionaltime-dependent FEL simulation codes. On theother hand, the spatial (or transverse) coherence ofSASE is governed by the longitudinal as well as the

*Correspondence address. Accelerator Development Labor-atory, Room No. G-18, Centre for Advanced Technology, In-dore 452013, India. Tel.: #91-731-488-020; fax: #91-731-488-000.

E-mail address: [email protected] (V. Kumar).

transverse dynamics of the electron beam. The situ-ation here is di!erent from the conventional FELoscillator con"guration. In the latter, the presenceof the resonator helps in the build-up of transversecoherence by providing greater di!raction loss tohigher-order modes, so that after several roundtrips only the lowest-order TEM

00mode survives,

and the radiation has full transverse coherence.Since SASE FELs do not use mirrors, no suchmechanism is expected to work in the case of SASEFELs. In this paper, we focus on the evolution oftransverse coherence in SASE FELs.

In Section 2, we discuss analytic as well as nu-merical approaches to the study of transverse co-herence, and point out their limitations. We alsointroduce the correlation coe$cient as a quantitat-ive measure of the degree of transverse coherence.In Section 3, we propose a new, and more realistic,model for the initialization of the optical radiation,that is more appropriate for the study of SASEFELs. We present numerical results using thismodel, and study the dependence of transverse

0168-9002/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 8 - 9 0 0 2 ( 0 0 ) 0 0 1 1 7 - 0 SECTION I.

coherence on electron-beam parameters such as thecurrent, emittance and beam size.

2. Basic concepts

There have been earlier studies on the growth oftransverse coherence in SASE FELs where the fullthree-dimensional Maxwell}Vlasov equation is sol-ved under the linear approximation as an eigen-value problem [4], and it is shown that for most ofthe cases the growth rate of the lowest-orderTEM

00mode is greater than the higher-order

modes. Hence, after travelling a su$cient distancealong the undulator, the lowest-order modedominates, which makes SASE fully coherenttransversely. This analysis gives a good qualitativeunderstanding of the transverse coherence charac-teristics of SASE radiation, but it has certain limita-tions. It is valid only in the linear regime (since oneuses the linearized Maxwell}Vlasov equation), andin the high gain limit. This analysis cannot be usedto predict the growth and onset of transverse coher-ence in SASE for realistic cases where nonlineare!ects could be important.

In conventional lasers, the problem of the devel-opment of coherence is often addressed quantitat-ively in terms of correlation functions [5] which tellus how the electric "elds at two di!erent points inspace at two di!erent times are statistically corre-lated. It has been established that the two electric"elds are coherent if their correlation coe$cient, asde"ned in statistics, reaches unity. However, trans-verse coherence in SASE FELs has not generallybeen addressed in terms of correlation functions.

Earlier work [4] has addressed this issue in termsof the growth rate for di!erent transverse modes.An attempt has also been made [6] to quantify thedegree of transverse coherence in terms of a para-meter called M which is argued to be the number oftransverse radiation modes. However, there areproblems with this approach, as we discuss later inthis paper. We feel that a more appropriate andphysically meaningful way to address the issues oftransverse coherence in SASE FEL is in terms ofcorrelation coe$cients.

We start with the basic de"nition of the complexdegree of coherence of light vibrations at two di!er-

ent points P1(r1) at time t

1and P

2(r2) at time t

2, as

given in Ref. [5]

c(r1, r

2, t

1, t

2)

"

SEH(r1, t

1)E(r

2, t

2)T

[SEH(r1, t

1)E(r

1, t

1)T]1@2[SEH(r

2, t

2)E(r

2, t

2)T]1@2

(1)

where E(r1, t

1) is the complex electric "eld at the

point P1(r1) and E(r

2, t

2) is the complex electric

"eld at the point P2(r2). The symbol S2T repres-

ents the average over an ensemble, i.e. di!erentrandom microscopic replicas of the system havingidentical macroscopic parameters. If the pointsP1(r1) and P

2(r2) are identical, (and t

1Ot

2) then it

is the case of temporal coherence. On the otherhand, if the two times t

1and t

2are identical but the

points P1(r1) and P

2(r2) are spatially separated,

then it is the case of spatial coherence. Hence, asa special case of Eq. (1), the complex degree oftransverse coherence is de"ned as

cT(r1, r

2, t)

"

SEH(r1, t)E(r

2, t)T

[SEH(r1, t)E(r

1, t)T]1@2[SEH(r

2, t)E(r

2, t)T]1@2

. (2)

It can be shown that the absolute value ofcT(r1, r

2, t) lies between 0 and 1 and is a measure of

the degree of transverse coherence: cT"1 corres-

ponds to full transverse coherence.We would therefore like to use the correlation

coe$cient of Eq. (2) to study the growth of trans-verse coherence in SASE FELs. However, to studySASE in the saturation regime, where nonlineare!ects are important, one needs to use numericalsimulations. Since the de"nition of c

T(r1, r

2, t) in-

volves the correlation between electric "elds at twodi!erent points at the same instant of time, one canuse time-independent FEL simulation codes suchas TDA [7] to study it. The important thing is that ithas to be a three-dimensional simulation code. Theobvious advantage of using time-independentcodes is that they are simpler and have much small-er running times as compared to time-dependentcodes such as GINGER [8,9]. The fact that the simu-lation code TDA is only a single-frequency code doesnot a!ect us much since we are using it only to

78 V. Kumar, S. Krishnagopal / Nuclear Instruments and Methods in Physics Research A 445 (2000) 77}83

study transverse coherence. The simulation codeTDA3D [10] has earlier been used by Li-Hua Yu [11]to study the SASE process, where he has shownexplicitly that as long as the undulator length ismuch shorter than 16 power gain lengths, TDA3D

serves as a good approximation for the calculationof the output power in SASE even though it onlyhandles a single spectral line. We have used the axi-symmetric TDA instead of TDA3D, since it is simplerand serves our purpose of bringing out the essentialfeatures of the growth of transverse coherence.

There is an important computational issue here.In SASE FELs, there is no seed radiation, and it isthe incoherent spontaneous radiation that is ampli-"ed. Therefore, the initial radiation that is given tothe code must also have that character. On theother hand, TDA initializes the radiation ina TEM

00mode. Clearly, this is inappropriate, at

least in principle, for the study of SASE. In thiscontext, it is relevant to point out that simulationsof SASE FELs are often done with the codes TDA

and TDA3D, with the unphysical initialization of theradiation in a TEM

00mode. In the next section, we

propose a more realistic model for initializing theradiation in the code.

3. Numerical simulations

In the original version of TDA, one starts witha TEM

00mode for the radiation and the electric

"eld is initialized accordingly. The amplitude isassumed to have a Gaussian pro"le in the radialdirection at the entrance to the undulator. Thevariation of the phase along the radial directiondepends on the location of the waist and theRayleigh range and is parabolic.

To simulate the SASE process, where the build-up of radiation starts from noise, it would be moreappropriate to have a random transverse distribu-tion of the radiation phase. One expects morephotons to be radiated near the center of the elec-tron beam, and therefore the radiation amplitudecould be expected to be Gaussian, with a size equalto the electron beam size. In the work describedhere, we have used this model for the initializationof the radiation in TDA, and we study its e!ect ongain and coherence.

After making these changes in TDA, we have usedit to calculate the complex electric "eld at di!erentradial grid points as the electron beam travelsdown the undulator. We have repeated this calcu-lation for di!erent initializations (typically 50) ofthe random phases and electron beam distribution.In this way, we have constructed an ensemble of thephenomena we are studying. The average on theright-hand side of Eq. (2) is performed over thisensemble to calculate the correlation coe$cient.Hence, we "nally end up with a quantitativemeasure of the degree of transverse coherence inSASE as a function of distance along the undulator.

The FEL parameters we have chosen correspondto the 4 nm LCLS project as described in Ref. [12].The design parameters of the LCLS projecthave changed since then. We continue to use theearlier parameters since the later versions of thedesign have a multiple section undulator whichcannot (presently) be simulated in TDA. The param-eters we have used in our simulation are givenin Table 1.

Fig. 1 shows the growth of power along thelength of the undulator, starting with random andordered phases respectively, at the entrance to theundulator. A comparison of the two curves revealsthat the e!ect of having random phases at thebeginning of the undulator is that the onset ofthe exponential growth regime, i.e. the duration ofthe lethargy regime, gets prolonged compared tothe case where one starts with uniform or well-correlated phases (as in the original version of TDA).For the parameters under study, the plot tells usthat a 60 m undulator would be considered longenough to reach saturation if one started the radi-ation in a TEM

00mode. However, with the more

realistic initialization described here, with a ran-dom phase distribution, the FEL does not reachsaturation after 60 m; we have con"rmed that, witha 75 m long undulator, the FEL does reach satura-tion. The maximum power obtained is essentiallyequal in the two cases; with random phases it justtakes longer (around 15 extra metres) to reach thismaximum.

The reasons for this can be understood by study-ing the distribution of phases along the radial direc-tion, as one goes down the undulator. Initially, thedistribution is random. As the radiation propagates

V. Kumar, S. Krishnagopal / Nuclear Instruments and Methods in Physics Research A 445 (2000) 77}83 79

SECTION I.

Table 1Parameters used in the numerical simulations

Energy (GeV) 7Peak current (A) 2500Normalised emittance, rms (mm-mrad) 3Energy spread, rms 2]10~4

Electron beam size (lm) 57Undulator period (cm) 8.3Undulator parameter (rms) 4.285Undulator length (m) 60Radiation wavelength (nm) 4

Fig. 1. Plot of power gain along the length z of the undulator.The dashed curve was obtained using the usual TEM

00initia-

lization in TDA. The solid curve was obtained using the modi"edversion of TDA where we start with a random distribution ofphases at the entrance to the undulator.

Fig. 2. Plot of the correlation coe$cient along the length z ofthe undulator. The red curve corresponds to the correlationcoe$cient calculated between the 5th grid point (r

2"8.7 lm)

on the transverse plane and the point on the axis (r1"0) in the

same transverse plane. Similarly, the blue, green and blackcurves show the correlation coe$cient for the 50th, 75th and150th radial grid point, respectively, on the same transverseplane.

down the undulator, the phases start getting or-dered, corresponding to the development of trans-verse coherence. The time taken for this orderingresults in a delay in the onset of exponentialgrowth. Once transverse coherence has set in, thedynamics proceeds exactly as in the original TDA

code. We also "nd that the di!erent random distri-butions of phases lead to the same distribution ofphases along the radial direction after travellingsome distance along the undulator. This occurs

much before the FEL power saturates and is inter-preted as the onset of transverse coherence. Wehave checked in our simulations that this e!ectdisappears if we switch o! the electron beam cur-rent: the appearance of coherence is only due to theinteraction with the electron beam.

This e!ect can be studied quantitatively using thecorrelation coe$cient de"ned in Eq. (2). The cor-relation coe$cient c

T(r1, r

2, t) is to be calculated

between any two transverse points at a given time,i.e. at the same distance down the undulator. Fig. 2shows di!erent plots of the correlation coe$cientcalculated between di!erent points (r

2) on the

transverse plane and the point (r1"0) on the axis

of the undulator in the same transverse plane. Thefarthest point we have considered is the 150thradial grid point where the amplitude of the electric"eld falls to 1/e of its maximum (on-axis) value. We"nd that the degree of transverse coherence be-comes &1 after travelling about 35 m along the


undulator, which is much before the FEL saturates.We also "nd that the onset of transverse coherencetakes place almost simultaneously at all the pointson the radiation beam. This is probably because theonset of transverse coherence takes place when thefundamental TEM

00mode starts dominating sig-

ni"cantly over the higher-order modes and then allthe points on the radial grid point achieve fulltransverse coherence. We have checked, however,that this is not the case if we start with an opticalbeam which is much broader than the electronbeam. In that case, for the farthest points from theaxis, the onset of transverse coherence takes placelater since those points see no electron beam cur-rent.

Next, we have studied how the onset of trans-verse coherence depends on the electron beamparameters. For this we de"ne a quantity N

6,#0)which is the number of undulator periods the radi-ation beam has to travel down the undulator toachieve 90% transverse coherence, i.e., Dc

TD"0.9.

Fig. 3a shows the dependence of N6,#0)

on theelectron beam current. It is seen that as the electronbeam current is increased, N

6,#0)goes down. This is

as expected since more electron beam currentmeans better coupling between the electron beamand optical beam and hence faster build-up of co-herence. The dependence of N

6,#0)on electron

beam emittance is shown in Fig. 3b. We "nd thatN

6,#0)increases as the beam emittance is increased.

This is because the increase in beam emittanceresults in a drop in the optical power gain along theundulator, which prolongs the onset of full trans-verse coherence. Similarly, from Fig. 3c, it can beseen that N

6,#0)increases with increasing electron

beam radius. This is because for the same electronbeam current, a larger electron beam radius wouldmean smaller beam current density which leads toa weaker coupling between the electron beam andthe radiation beam, resulting in a slowing down ofthe build-up of transverse coherence.

4. Discussions and conclusions

In the preceding sections, we have shown that itis possible to implement the simple de"nition of thedegree of transverse coherence, in terms of correla-

tion coe$cients which are commonly used in con-ventional optics, into the analysis of SASE FELs tostudy the growth of transverse coherence. Pre-viously, a parameter called M [3,6], de"ned by

M"

SPT2

SP2!SPT2T(3)

has been used to study the growth of transversecoherence. M is just the inverse of the normalizedspread of the distribution of the instantaneous radi-ation power, where the averaging is again over anensemble comprising di!erent sets of random initialphases. It has been argued in Ref. [6] that M is alsoequal to the number of transverse radiation modes,and therefore full transverse coherence can be saidto have developed when M"1. It was then shown,using numerical simulations, that for the 70 nmSASE FEL under construction at the TESLA TestFacility (TTF), the parameter M approaches unityafter the radiation has traveled a su$cient distance(&6 m) along the 13 m long undulator, and hencefull transverse coherence is achieved halfway downthe undulator.

It is however to be noted that the calculation ofM has been done only up to the middle of undula-tor in Ref. [6], where the FEL is expected to be farfrom saturation. As coherence builds up, however,the spread in the powers from di!erent randominitializations will decrease towards zero, and Eq.(3) shows that M will then blow-up. To demon-strate this we have calculated the M parameteralong the full length of the undulator for the param-eters we have studied. Fig. 4 shows that M startswith a large value at the beginning of the undulator,and approaches unity near the middle, exactly asshown in Ref. [6]. However, towards the end of theundulator it blows up, as we have argued above,even though full transverse coherence has beenachieved. We therefore feel that the interpretationof M as the number of transverse radiation modesis not valid beyond the linear regime, and parti-cularly, when full transverse coherence sets in. It istherefore not as useful a parameter for studyingtransverse coherence as the correlation coe$cientwe have introduced.

The e!ect of starting with the random distribu-tion of phases along the radial direction is that the


SECTION I.

Fig. 3. Plots of the number of undulator periods N6,#0)

required to reach 90% transverse coherence as a function of (a) beam current, (b)beam emittance and (c) beam radius of the electron beam.

onset of power saturation for the FEL gets pro-longed. This suggests that one has to take care ofthis e!ect while designing a SASE FEL. The undu-lator has to be long enough such that even with arandom distribution of phases, the radiation powershould saturate. Simulations that assume an initial

TEM00

wave will tend to underestimate the lengthof undulator required to achieve a given powerlevel.

In conclusion, we have shown that the correctway to understand the degree and growth of trans-verse coherence in SASE FELs is to study it in


Fig. 4. Plot of the M-parameter, as de"ned in Eq. (3), along thelength z of the undulator.

terms of the correlation coe$cient between theelectric "eld at two di!erent points on the trans-verse plane, as is done in conventional optics. Wehave incorporated this de"nition explicitly into theFEL simulation code TDA and studied the growth ofthis parameter along the length of the undulator.We "nd that one has to take into account therandomness of phases for the optical "eld at theentrance to the undulator, in order not to underestimate the length of the undulator required forthe power to saturate. Our simulation results ex-plicitly show that the onset of full transverse

coherence in SASE takes place much before thepower saturates.

Acknowledgements

It is a pleasure to acknowledge the useful conver-sations with Kanan Puntambekar, Arup Banerjeeand Manoranjan Singh.

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SECTION I.

Documents

Growth of transverse coherence in SASE FELs