# Plasma-based Accelerator with Magnetic Compression Page: 4 of 8

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Consider homogeneous, uniformly magnetized plasma,

i.e., B= B(t)i, where B(t) changes with time. For ex-

ample, this could be realized within a plasma column

confined inside a solenoid carrying a time-varying cur-

rent. Magnetization implies that the plasma density, n,

varies with B1. For slow variation of plasma parame-

ters, a plasma wave with wavevector k B obeys the

eikonal equation [20]: 2 P + 3k2)vT, where og is

the electron thermal velocity parallel to the magnetic

field. In the case of an ultrarelativistic plasma wave, i.e.,

tph =w/k ~ c, the cold plasma eikonal equation suffices,

rw =oP. Since wP a n a B, we have Lp =oP(t), while k

remains constant (neglecting nonlinear effects [3]), since

the compression is perpendicular to the wavevector. In

the neighborhood of w/k c, changes in n result in large

changes in w/k, so that minimal compression is needed

to retain proper phasing of relativistic particles.

The axial dynamics of a relativistic electron interact-

ing with a sinusoidal potential exhibiting a dynamically

changing vph(t) are captured by the equations [21]:d-y

dtdP

(1

ck

where ke - T(t), ,ymec2 is the electron energy in

the laboratory frame, e is the elementary charge, and

S(t) f0 w(k; t') dt'. Because minimal compression is

anticipated to correct the wave phase velocity, the electric

field amplitude Eo const.

The compression profile required to overcome phase

slippage in PBWA can be calculated from Eqs. (1) and

(2). Suppose T(t) is configured such that a stable fixed

point arises in the phase space associated with the rest

frame of the accelerating plasma wave. Then, combin-

ing Eqs. (1) and (2) by eliminating the square root gives

4 4o( + 41) cos (, with 4Po _eEo/kmec2. If

corresponding to the fixed point, there exists an energy-

like integral of motion: d/dt(y - 4oI cos o) 0. Equa-

tion (2) yields the necessary plasma compression profile.

Noting that 4' ,p(t), and that the distance traversed

by a phase-locked particle D(t) (T - To)/k, the re-

quired normalized density profile, = wg/wjo, is:

n[D(t)] ='o ('o + 4okDcos o) - ] (3)

(-y- 1) (yo + okD cos o)2'

where yo =Ypho (1 -- Ph /c2)-1/2 implies exact

initial wave-particle resonance. This expression for h

is monotonic in D, asymptotically approaching hmax

yo/(yo - 1) as D -> o (and, hence, -y -> o). For in-

stance, to accelerate a 2 MeV electron bunch ('o ~ 4)

to arbitrarily high energies requires a peak density shift

hmax ~ 1.07, demonstrating that only small changes indensity are needed to accelerate relativistic electrons to

arbitrarily high energies while maintaining the proper

wave-particle phase relationship. Physically, the quan-

tity hmax represents the total change in density required

for vph -> c. To express Eq. (3) as an explicit function

of time, one integrates D(t) f4 v dt' for a relativistic

particle at fixed phase in the accelerating potential:

D(t) = - act + - 1 + 1 -y , (4)

with a eEo cos o/mec2. The fixed point assumes zero

transverse momentum, which equates to a compression

profile optimized to trap relativistic particles with a nar-

row transverse energy spread.

Peak acceleration occurs when o 0, for which an

electron starting at = 0 obeys y - 40'= const. In

fact, for a wave of specified Eo and k, this is the maximum

achievable acceleration, in which the electron experiences

the peak accelerating field as the wave and electron ac-

celerate together. More generally, choosing o : (0, 7r/2)

in Eq. (3) leads to compression profiles enabling electron

trapping over a broader range of initial electron energies,

at the cost of slower acceleration. This is because the

fixed point in the wave rest frame, o, lies ahead of the

peak accelerating field, at 0, allowing some particles

that slip behind o to catch up to o once again.

Wakefield acceleration with compression. Mitigating

phase slippage through magnetic compression in (linear)

wakefield acceleration, including LWFA and PWFA, is a

somewhat different process. Here, a time-varying density

profile during wake excitation results in an axial gradient

in the plasma wake parameters, which was not the case

with PBWA. Electron dephasing is often the dominant

effect limiting energy gain in wakefield acceleration when

the driver amplitude is no more than weakly relativistic,

i.e., eA/mec2 < 1 for LWFA, where A is the peak vector

potential of the laser driver [13], or nb/n < 1 for PWFA,

where nb is the peak driver beam density [3].

In wakefield acceleration, a subluminal wakefield is

excited by an ultrarelativistic driver, i.e., Yd (1 -

02)-1/2 1, with d vd/c, and vd is the driver

pulse velocity. For PWFA, the longitudinal velocity of

the electron beam driver, vb, is unaffected by the per-

pendicular magnetic compression. Because only modest

density changes will be needed, the laser pulse group ve-

locity, tgr, is mostly unaffected as well, since a change

in plasma frequency, AOP, leads to a change in wave

phase velocity Avph/ph A p/op, which is large com-

pared to the change in laser group velocity, Avgr/vgr ~

(Lp/wd)2AwP/wP, where Wd is the characteristic laser fre-

quency, and wp/wd < 1 in underdense plasma. Thus,

both PWFA and LWFA can be treated similarly, where

the characteristic driver velocity vd const.

We follow the technique of Ref. [9] to derive the re-

quired density profile (in the 1D limit) to maintain a1/ eI cos (

/ 1 \ 1/2 c

C1 - -W,

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Schmit, Paul F. & Fisch, Nathaniel J. Plasma-based Accelerator with Magnetic Compression, report, June 28, 2012; Princeton, New Jersey. (digital.library.unt.edu/ark:/67531/metadc827807/m1/4/: accessed December 12, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.