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Nuclear Instruments and Methods in Physics Research A 393 (1997) 376-379 NUCLEAR

EI.SE&IER

INSTRUMENTSa METHODSIN PHYSICSRESEARCHSecton A

Space-charge oscillations in a self-modulated electron beam inmulti-undulator free-electron lasersJ. Rosenzweig*, C. Pellegrini, L. Serafini, C. Ternieden, G. Travish

UCLA Departm ent of Physics and Astronomy , 405 Hilgard A ve., LOS Angeles, CA 90095-1547, UsA

AbstractWe examine here the oscillation of electron-beam density perturbations (longitudinal plasma oscillations) produced at

the exit of a high-gain free-electron laser (FEL) by the action of the FEL instability. These oscillations, which areanalyzed in the case of both a free-space drift and a dispersive section, can degrade the bunching of the beam in the driftbetween undulator sections in multi-stage FELs. The impact of these oscillations on the gain of an FEL in an undulatorfollowing such a drift, as well as the case of an optical klystron is studied.

1. Longitudinal space-chargeIn space-charge dominated beams derived from RF

photoinjectors, such as typically moderate energy driven(few tens of MeV) free-electron lasers (FELs), the collec-tive transverse motion is characterized by single-com-ponent plasma behavior. The free expansion due to therepulsive self-forces is controlled by the externally im-posed focusing lattice, producing beam envelope, or sur-face plasma oscillations. In contrast, for many beams, thelongitudinal space charge forces do not produce signifi-cant debunching, and small efforts (running slightly off ofRF crest) are necessary to compensate the energy slewintroduced by the space-charge forces. This suppressionof longitudinal forces is due to the fact that, in the beamrest frame, the bunch length is much larger than theradius y60,>>~,., and the longitudinal surface plasmaoscillation frequency 01,,, s much smaller than the trans-verse frequency O,~.For short-wavelength longitudinal density perturba-tions (microbunching) such as are introduced by the FELprocess, however, the field is no longer transverse, as thelongitudinal density gradients may have a rest framescale length shorter than the beam width. In the ultra-

* Corresponding author. Permanent address: INFN-Milano, Via Celoria 16, 20133Milan, Italy

short wavelength limit, the electric field associated witha pure harmonic density perturbation n, = n,, - nbD =6nb cos(k,[) of wavelength 1, = 2 n / k , , ith nbO he unper-turbed beam density, and < = z - vbot (distance meas-ured in the Galillean frame moving with the beam) is

E,,r - 47ce6nb~ sm(k,i).k,For a finite, constant density bunch distribution witha hard edge at radius a, the longitudinal electric fieldinside of the beam is given by

In the limit that the beam radius is large compared to theoscillation wavelength in the beam rest frame (k,a>>y,),the correction factor to Eq. (1) found in Eq. (2) ap-proaches unity. In the case of the FEL the radiationwavenumber has a strong dependence on the normalizedenergy, k, cc Yg, and so this situation, where the longitu-dinal space-charge force attains its one-dimensional limi-ting strength, is to be expected for any beam of moderateenergy or above (i.e. for the UCLA IR FEL k , c r , y o ,while for the TTF-FEL k,o,>>y,). We shall thereforebe concerned for the remainder of this work with one-dimensional plasma oscillations in FEL-microbunchedbeams.

0168-9002/97/$17.00 Copyright 0 1997 Published by Elsevier Science B.V. All rights reservedPII SO168-9002(97)00516-O

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J. Rosenzweig et al. /Mud. Instr. and Meth. in Phy. Res. A 393 (1997) 376-374 3772. Longitudinal plasma oscillations

The derivation of the longitudinal plasma oscillationequations in the linear, small amplitude (In, I

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378 J. Rosenzweig et al. /Nucl. Instr. and Meth. in Phys. Rex A 393 (1997) 376-379one-dimensional laminar flow holds), at maximizes againafter an additional one-quarter plasma period. We planto investigate these phenomena at the UCLA IR FELusing the coherent transition radiation-based micro-bunching diagnostic we have developed, with thetransition radiator placed at a number of positionsdownstream of the undulator [4]. The experimentalsetup for this measurement is discussed in Ref. [S].

For the TTF-FEL in its initial stage [6], we take forthe sake of example the energy as 500 MeV, the rmsbunch length beam cZ = 50 urn, and the RMS beam sizerr, = 50 urn. In this case, we have lipi = 300 cm, which isa relatively short length considering the energy of thebeam. The additional bunching factor for these TTF-FEL parameters is k,/k, pFEL/yi = 4.5 (twice that of theUCLA case, as pFEL is four times smaller), and the bunch-ing factor increases in a inter-undulator module drift byapproximately this factor if the drift length is chosen tobe approximately z = 3n/7k, = 400 cm. One is likely tohave shorter drifts in practice, which produce smallerdensity enhancements. One possible way to circumventthis is to use an optical klystron configuration, in whicha short dispersive section allows enhanced pulse com-pression in a shorter propagation distance.

4. Space-charge oscillations in dispersive sections: theoptical klystron

The use of a dispersive section at the end of an FELundulator serves to enhance the phase bunching of theelectron distribution by introducing a change in the rela-tionship (see Eq. (7)) between the longitudinal velocityand the momentum,

(12)For high energy electrons, the second, dispersive path-length term on the right side of Eq. (12) is dominant incases of interest. A chicane, which is basically one periodof a long wavelength undulator, is typically used to givean average horizontal dispersion which is negative, andso the bunching proceeds in the same direction as purevelocity bunching.

The introduction of this additional effect changes theplasma oscillation equation (and expressions derivedfrom it), by a redefinition of the plasma frequency,

This frequency has an explicit dependence on z, whichmakes the oscillation equation not easily integrable. If

we average the dispersion over the chicane section,we can assign an average value to it in Eq. (13)( - q,/R) 2 af,,/2yi z 812, where ach>>l is the equi-valent undulator parameter, and Q is the maximum bendangle in the chicane section. Taking this average value ofthe dispersion, we can again integrate the oscillationequations to obtain the equivalent of Eq. (lo),

%(z, ) = hub0 Wd) cos(kp,,bz)I k, db/kX.

k - sln(kp,chz)p.ch 24 1 (14)where kp,ch = cop,c&+,O. The maximum enhancement ofthe bunching is now approximately larger by a factor ofa,,/& from the case of a pure drift, the same factor bywhich the plasma frequency is raised.Because the plasma oscillation wavelength 2n/k,,,, isshortened, in the case of the UCLA IR FEL it may bedifficult to attempt chicane bunching after the initial60 cm undulator section. For the TTF-FEL example,however, there may be much to be gained by employinga chicane section in the so-called optical klystron mode[7]. Assuming a chicane section 66 cm long, which witha, = 10 corresponds to one-quarter of a plasma oscilla-tion length, the bunching enhancement is then raised toapproximately 31 at the exit of the chicane.

5. Future workIt is interesting to note that the results obtained above

are independent of microbunching amplitude as long asi%i

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It should also be pointed out that the study of thegeneral subject of space-charge oscillations in relativisticbeams in both drifts and chicanes occurs outside thecontext of FELs. An example of current interest is theso-called plasma klystron [9], which has been proposedas an injector for ultra-short wavelength accelerationschemes such as the plasma beatwave accelerator.References

[l] J. Rosenzweig et al., Nucl. Instr. and Meth. A, submitted. [8] J.B. Rosenzweig, Phys. Rev. A 38 (1989) 3634.[2] R. Bonifacio. C. Peliegrini, L. Narducci. opt. Commun. 50 [9] T. Katsouleas et al., IEEE Trans. Plasma Sci. 24

(1984) 313. (1996).

[3] G. Travish, M. Hogan, C. Pellegrini. J. Rosenzweig, Nucl.Instr. and Meth. A 358 (1995) 75.

[4] J. Rosenzweig, G. Travish, A. Tremaine. Nucl. Instr. andMeth. A 365 (1995) 255.

[S] J. Rosenzweig. A. Tremaine, G. Travish. Presented at 18thFEL Conf., Rome, Italy, 1996.163 J. Rossbach et al., Conceptual design report of the TESLA-FEL, in preparation.

[7] J.C. Gallardo. C. Pellegrini, Optical klystron configurationfor a high gain X-ray free-electron laser, Opt. Commun. 77(1990) 45.

V. PRE-BUNCHINGSUPERRADIANCE