EVOLUTION OF ELECTRON BEAM PHASE SPACE DISTRIBUTION INA HIGH-GAIN FEL

Stephen D. Webb∗ Vladimir N. Litvinenko†

Brookhaven National Laboratory (BNL) Collider-Accelerator DepartmentStony Brook University Physics and Astronomy

Abstract

FEL-based coherent electron cooling (CEC) offers a newavenue to achieve high luminosities in high energy collid-ers such as RHIC, LHC, and eRHIC. Traditional treatmentsconsider the FEL as an amplifier of optical waves with spe-cific initial conditions, focusing on the resulting field. CECrequires knowledge of the phase space distribution of theelectron beam in the FEL. We present 1D analytical resultsfor the phase space distribution of an electron beam withan arbitrary initial current profile.

INTRODUCTION

Coherent electron cooling (CEC) ([1] and referencestherein) is a new proposed method of cooling hadron beamsin high energy storage rings. In contrast with electron cool-ing which becomes weaker with increasing energy, andstochastic cooling which is restricted by the bandwidth ofthe system, the cooling decrement for coherent electroncooling does not directly dependent on the hadron beamenergy. This makes CEC ideal for implementation in nextgeneration high energy hadron colliders, offering boosts inluminosity in colliders such as the LHC and Tevatron, andfuture electron-hadron colliders such as eRHIC, ELIC, andLHeC.

Figure 1: Schematic of CEC implementation.

CEC is structurally similar to stochastic cooling, andis schematically depicted in Figure 1. The hadron beamis combined with an electron beam with identical centerof mass velocity, and leaves an imprint of the individualhadrons as charge perturbations in the electron beam. Thatbeam is then passed through an FEL which amplifies theinitial signal and modulates it into a wave packet. In thekicker section, the hadrons are recombined with the elec-tron beam after passing through a chicane to create energy-dependent positioning. In this section, hadrons with toomuch or too little energy receive an energy-dependent kick,reducing the overall energy spread.

∗ swebb@bnl.gov† vl@bnl.gov

Table 1: Proposed Parameters of a Proof-of-principle ERLand for eRHIC

PoP eRHICEnergy ℰ0 22 MeV 136.2 MeV

Gain Length Γ 0.486 m 2 mElectron density 𝑛0 1× 1018 m−3 2.08× 1018 m−3

Pierce parameter 𝜌 0.0097 .004Space Charge Λ2

𝑝 0.155 .0097

To implement this scheme, a detailed understanding ofthe phase space evolution of the initial phase space pertur-bation created by the hadron beam must be developed. Inthe case of the proof-of-principle system, with 𝛾 ∼ 45,space charge effects will be important. We present initialresults for the perturbation’s evolution through a high-gainFEL using a 1D theory that accounts for space charge, aswell as a correction to the self-consistent FEL equations ofmotion to first order in the Pierce parameter.

THEORETICAL RESULTS

A focus on the evolution of the phase space distributionof an electron bunch in a high-gain FEL to obtain the equa-tions of motion for the output laser field [2]. This treatmentdevelops the couples Maxwell-Vlasov equations, and com-bines them into a single current equation. Beginning withan approximate one-dimensional hamiltonian,

ℋ = 𝑐

√(𝑝𝑧 − 𝑒

𝑐𝐴𝑧)2 +

(𝑒𝑐

)2 (��𝑤 + ��⊥

)2

+𝑚2𝑐2

(1)where ��𝑤 is the vector potential on axis for a helical wig-gler, ��⊥ is the laser field, 𝐴𝑧 is the longitudinal spacecharge field, and an appropriate gauge is chosen to makethe scalar potential zero. Linearizing in the laser field anddefining the phase space distribution 𝑓 = 𝑓0 + 𝑓1 where𝑓0 is the thermal background and 𝑓1 is the growing per-turbation and ∣𝑓1∣ ≪ ∣𝑓0∣, the resulting linearized Vlasovequation is

∂𝑓1∂𝑧

+1

𝑐

{1 +

1

𝛾2𝑧

(1 + 2

ℰℰ0

)}∂𝑓1∂𝑡

+

+

{− 𝑐

ℰ0

(1+

ℰℰ0

)[(𝑒𝑐

)2

��𝑤 ⋅ ∂��⊥∂𝑡

]− 𝑒𝐸𝑧

}∂𝑓0∂ℰ =0

(2)

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208

where ℋ = ℰ0 − ℰ , 𝛾−2𝑧 = (1 + 𝐾2)𝛾−2

0 and 𝐾2 =(𝑒𝐴𝑤/𝑚𝑐2)2. Introducing the general Fourier transform

𝐺(𝑧, 𝑡) =1√2𝜋

∫𝑑𝜈 𝑒𝚤𝑘𝑤𝑧+𝚤𝜈𝜔𝑟(𝑧/𝑐−𝑡) × ��(𝑧, 𝜈) (3)

the slow-varying Maxwell equation and space charge issolved as

��𝑤 ⋅ ��⊥(𝑧, 𝜈) = ��𝑤 ⋅ ��⊥(0, 𝜈)

+𝜋𝐾𝚤

𝜈𝜔𝑟𝛾0𝑒𝚤𝑘𝑤𝑧

∫ 𝑧

0

��1(𝑧′, 𝜈)𝑑𝑧′ (4)

and

��𝑧 =4𝜋𝚤

𝜈𝜔𝑟��1 (5)

By direct substitution into the Vlasov equation, this givesthe Fourier transformed linearized Maxwell-Vlasov equa-tion. Integrating over energy to obtain an equation for thecurrent, we obtain

𝒥 ′ = −𝑒𝑐 𝜌

1 + 𝐶𝜌

∫𝑑𝑃𝑓1∣𝑧=0𝑒

−𝚤(��−𝑃 )𝑧 +

∫𝑑𝑃

𝑑𝐹

𝑑𝑃

∫ 𝑧

0

𝑑𝑧′𝑒𝚤(𝐶−𝑃 )(𝑧′−𝑧) ×{(

1 +𝜌

1 + 𝐶𝜌

)(𝒰 (0)⊥ − 𝒥

)− 𝚤Λ2

𝑝𝒥 ′}

(6)

where 𝒥 =∫ 𝑧

0��1𝑑𝑧

′.The solution of this equation is carried out by use of

Laplace transform techniques, keeping terms to first orderin 𝜌. The Laplace transform yields an equation with thenumerator having as source terms an initial energy modu-lation and an initial laser field. For applications to CEC,the initial laser field term 𝒰 (0)

⊥ is set to zero, and we studythe amplification of the initial charge density perturbations.Dropping the laser field term, the Fourier transformed so-lution is given by Equation 7

Assuming that any energy distribution in the initial mod-ulation leads to an exponential damping term, the evolutionis determined by the roots in terms of 𝑠 of the equation

𝑠+

∫𝑑𝑃

𝑑𝐹

𝑑𝑃

1 + 𝜌

1−𝐶𝜌𝑃

𝑠+ 𝚤(𝐶 − 𝑃 )+

𝑠𝚤Λ2𝑝

∫𝑑𝑃

𝑑𝐹

𝑑𝑃

1

𝑠+ 𝚤(𝐶 − 𝑃 )= 0 (8)

The small correction due to 𝜌 is included, and results in asmall correction to the growth exponents and phase evo-lution. For CEC, the phase information is an importantquantity, and inclusion of some small correction may proveuseful in future work.

As a particular case, we look at the situation of a coldbackground beam, i.e. 𝐹 (𝑃 ) = 𝛿(𝑃 ). In this case, thedenominator becomes a cubic equation given by

𝑠(𝑠+ 𝚤𝐶)2 −(𝚤+

1

2

𝜌

1− 𝜌𝐶(𝑠+ 𝚤𝐶)

)+ 𝑠Λ2

𝑝 = 0 (9)

�8 �6 �4 �2 2 C�

�0.5

0.5

Re s

Figure 2: The real component of the roots for a cold beam.For the strong space charge effects present in the proof ofprinciple, there is no growth for short wavelengths.

�8 �6 �4 �2 2 C�

�2

2

4

6

8Im s

Figure 3: The imaginary component of the roots for a coldbeam. The space charge effects have little effect on thephase compared to the zero space charge case.

The inclusion of the 𝜌 parameter to first order offers smallcorrections to existing results in [2]. For particular appli-cation, we consider the eRHIC parameters given in Table1.

The resulting roots are depicted in Figures 2 and 3. Com-pared to standard results, the space charge effects stronglyreduces amplification at shorter wavelengths. The spacecharge effects have a minimal effect upon the phase evo-lution, but suppresses the peak growth, and narrows thebandwidth of growth substantially. This narrow bandwidthposes some potential troubles for the proof-of-principle,but will not be present at the operating energies for eRHIC.

APPLICATIONS TO ERHICeRHIC is a proposed upgrade that would allow colli-

sions of spin-polarized electrons with hadrons, with, for ex-ample, 325 GeV protons and 20 GeV spin-polarized elec-trons [4]. Due to technical restrictions in spin-polarizedelectron current, the application of CEC to reduce hadronbeam emittance is critical to reach the desired luminosi-ties for the eRHIC upgrade. The appeals of CEC are theweak dependence of cooling time on energy, compared totraditional electron cooling, and its large bandwidth com-pared to stochastic cooling. To achieve this cooling us-

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New and Emerging Concepts

209

∫��1𝑑𝑧

′ =∫ 𝛾+𝚤∞

𝛾−𝚤∞𝑑𝑠 𝑒𝑠𝑧 ×

− 𝑒𝑐2

𝜌

1−𝐶𝜌

∫𝑑𝑃𝑓1∣𝑧=0

1𝑠+𝚤(𝐶−𝑃 )

𝑠+∫𝑑𝑃 𝑑𝐹

𝑑𝑃

1+ 12

𝜌

1−��𝜌𝑃

𝑠+𝚤(𝐶−𝑃 )+ 𝑠𝚤Λ2

𝑝

∫𝑑𝑃 𝑑𝐹

𝑑𝑃1

𝑠+𝚤(𝐶−𝑃 )

(7)

ing traditional electron cooling methods would take over30 hours at eRHIC, whereas using CEC will take approx-imately six minutes for longitudinal cooling, and approxi-mately twenty minutes for both longitudinal and transversecooling [3].

This work begins the steps outlined in [5], providing aGreen’s function for the current growth given an initial sig-nal. Accounting for energy spread, which for the full eR-HIC upgrade is estimated at around 3 × 10−3, and around5× 10−3 for the proof of principle ERL design [5], as wellas the space charge effects in great detail, are necessary forworking implementation of coherent electron cooling.

FUTURE WORK

Further investigations these results to CEC will requirecalculation for a thermal beam, the explicit solution for aparticular initial condition 𝑓1(𝑧 = 0, 𝜈, 𝑃 ), and the nu-merical generalization of results to include the full three-dimensional case. The final objective of this work is tocalculate the cooling decrement of the FEL output signalon the hadron beam in the kicker of the CEC.

The first two requirements are coupled, as the input sig-nal from the hadron beam is analytically related to the ther-mal distribution of the electron beam. Analytical results forthe density perturbation generated by a charged hadron inan electron beam with Lorentzian and 𝜅 = 2 distributions,i.e.

𝑓0(𝑃 ) ∝ 1(1 + (𝑃/𝑃0)2

)𝜅

have been obtained [6], and these results are currently be-ing generalized to obtain the phase space distribution. Thecurrent distribution is already known (see figure 4) for a va-riety of distributions, and generalization to obtain the phasespace distribution is under way [7]. These results are three-dimensional, and will require some suitable reduction tothe one-dimensional picture of the results obtained for theFEL model presented here. Having a Green’s function fora thermal distribution and resulting imprinted signal is thenext step in the 1D theory, and would allow direct analyti-cal solution in frequency space for the wave packet outputof the current. Three-dimensional numerical simulationsfor a given input is the last phase for simulations of the FELoutput, and can be benchmarked against the 1D results.

Direct calculation of the cooling decrement and effectsof the kicker on the energy spread of the hadron beam inputis the ultimate goal of these calculations. Current calcula-tions using simple models for the output signal indicate acooling decrement weakly dependent on energy, but moreprecise calculations are desirable.

Figure 4: Taken from [6], the real space charge perturbationfor a single hadron in a Lorentzian electron background.Lengths are scaled to the Debye length.

ACKNOWLEDGEMENTS

The authors would like to thank Evgeny Saldin for usefulcorrespondence, and Jorg Kewisch, Mike Blaskiewicz, andGang Wang for their helpful comments, suggestions, anddiscussions. Work supported by Brookhaven Science As-sociates, LLC under Contract No. DE-AC02-98CH10886with the U.S. Department of Energy.

REFERENCES

[1] V. Litvinenko and Ya. Derbenev, “Coherent Electron Cool-ing”, PRL 102, 114801 (2009) and references therein.

[2] E. Saldin, E. Schneidmiller and M. Yurkov, The Physics ofFree Electron Lasers. Springer-Verlag 2000.

[3] V.N. Litvinenko et al., “Progress with FEL-Based CoherentElectron Cooling”. Talk. FEL ’08.

[4] T. Roser, “The Electron Ion Collider at BNL: eRHIC and itsStaging Options”. From EICAC Meeting, Feb 09.

[5] V. Litvinenko et al., “High Gain FEL Amplification of ChargeModulation Caused by a Hadron”, FEL’08, Gyeongju, Korea,MOPPH026, p. 51, http://www.JACoW.org.

[6] G. Wang and M. Blaskiewicz, “Dynamics of ion shieldingin an anisotropic electron plasma”, Phys. Rev. E 78, 026413(2008).

[7] G. Wang, “Velocity Modulation in CeC Modulator”, C-ADLecture. June 2009.

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