OPTICAL-WAVELENGTH BUNCHING FOR
HIGH-ENERGY ACCELERATORS.
I. V. Smetanin, A. N. Oraevsky
P. N. Lebedev Physics Institute, Moscow, Russia,andKazuhisa Nakajima,High Energy Accelerator Research Organization (KEK), Tsukuba, Japan

Abstract
We propose the new technique for opticalwavelength bunching of an electron beam, which isbased on the dissipative instability of the beam in a gas
with resonant dielectric susceptibility. The instability isabsent for transparent media, and the growth rate isdetermined by the imaginary part of the gas dielectricsusceptibility. The effects of finite guiding magnetic
field are discussed, the conditions of resonant
enhancement of the instability increment are found.
Estimates show that the technique proposed is aneffective bunching mechanism for high-energy electronbeams.

1. INTRODUCTION
The problem of short-period bunching of particle
beams is very important for high-energy accelerator
physics. Besides the accelerator design and technology
itself, bunching is crucial for various applications, forthe free-electron lasers, for example. Preliminary
bunched electron beams can sufficiently enhance theoutput power and improve the coherent properties ofradiation in the SASE FEL schemes. The conventional
bunching technology is based on the interaction of beam
particles with undulator or electromagnetic wave fields.
Recently, a new concept of the free-electron laser,
the resotron, has been proposed [1-3], which is basedon the effect of amplification of an electromagnetic
wave in the presence of resonantly absorbing medium.
The physical mechanism of the gain in the resotron isthe electron - beam dissipative instability. Oscillation
frequency is determined by the material absorption
spectrum and does not depend on the energy of beamelectrons. Due to the resonant character of absorption,
the lasing spectrum width for this free-electron laser
scheme becomes very small, of the order of material
linewidth.
Lasing in the resotron FEL is accompanied with themodulation of electron beam density. As the most of
atomic and molecular transitions are in the IR andvisible spectral range, the bunching occurs at optical
wavelengths.

In this paper, we propose a new technique forbunching of the high-energy electron beam at optical
wavelengths, which is based on the gas-loaded resotron
free-electron laser scheme [3]. The bunchingmechanism is the electron beam dissipative instability at
the frequency, which corresponds to a resonanttransition of gas molecules. The effects of finite guiding
magnetic field are investigated. It is found that at adefinite cyclotron frequency, which is determined by thebeam energy and beam plasma frequency, the resonant
enhancement of
the instability increment occurs.
Estimates show that the technique proposed is aneffective bunching mechanism for high-energy electron
beams.

2. DISPERSION RELATIONLet us consider the process of co-propagation of therelativistic electron beam and an electromagnetic wave
in a waveguide, circular for definiteness (see Fig. 1).
Waveguide is filled by the gas, for which the wave
frequency corresponds to the resonant transition, anddielectric susceptibility is ε = ε1 + iε 2 , ε > > 0 . We
will consider the bunching at IR wavelengths, at which
absorption in the most of gases is determined byvibrational and rotational transitions and | ε − 1| << 1 .
The finite guiding magnetic field is applied along thewaveguide axis, a is the waveguide radius.

e2a−,γH0Fig. 1. The gas-loaded resotron free-electron laser
configuration.

For definiteness, we will consider the case of zeroorder TM 0n waveguide electromagnetic mode, forwhich
thefieldcomponentsare( E z , E r , Hϕ ) exp[i (hz − ωt )] .
Therelativistic
electron beam of the density nb , velocity v 0 , andenergy γ penetrates along the waveguide (z-)axis.
Strictly speaking, at finite guiding magnetic field H 0 ,zero-order TM mode is coupled with the correspondent
TE mode, but the coupling coefficient is sufficiently
small, and in the linear approximation, which isdeveloped below, TM 0 n can be treated as an eigen
mode of the system.
Equations of motion and continuity equation for the
beam electrons result in the following relations for thecomponents of high-frequency current perturbations

nb e 2
hv 0 − ω( E r − β0 H ϕ )mγ( hv 0 − ω ) 2 − ω c2

δj r = − iδj z = inb eωv0∂jr3 Ez
2 +imγhv 0 − ω ∂ r( hv 0 − ω )(1)2

Here ω c = eH 0 / mγc is the cyclotron frequency.
After substitution of this current perturbations into theMaxwell equations, we get for longitudinal componentof the wave electric field amplitude the wave equation

1 ∂r ∂r( r ∂∂Er ) + κ E = 0z

2(2)

z

where α n is the n-th root of the Bessel function,

J 0 (α n ) = 0 .

3. DISSIPATIVE INSTABILITY
The dispersion equation (4) has two pairs ofsolutions. Let us first consider the case of infinitewaveguide radius, a → ∞ and infinite guiding field,see Ref. [4]. Then we have(h − ε c ) (ω222)Ω b21− 2= 0.
εγ ( hv 0 − ω ) 2

The first pair of solutions, which is generated by thefirst term in brackets, h = εω / c , corresponds tothe free electromagnetic wave in a gas. It is easy to seethat these modes are rapidly damping in a few
222wavelengths, Im h ~ (1 / 2)(ε 2 / ε1 )The second pair of solutions ish=ωv0±1/ 2

ω / c.

Ω b −1/ 2ε ,γv 0(5)which represents two waves, fast and slow with respect
to the electron beam, one of which (slow) is growing
with the increment

Im h ≈Ωbv 0γ(| ε |− ε1 ) / 2 ~Ωb(Im ε ) 1/ 2 .
2v 0 γ(6)Here the characteristic transverse wave number is

h −ε2

κ

2= −

ω2c21+(+Ω 2b

Here and below, we assume the optical signal frequency
range, and moderate beam densities and guiding fields,
for which the inequalities( hv 0 − ω ) 2c 2 ( hv 0 − ω ) 2 − ω c2

Ω b2 ( β 02 − 1 / ε )ω >> Ω b , ω >> ω c×( hv 0 − ω ) 2 − ω c2

× 1−

Ω b2 / γ 2ε (hv 0 − ω ) 2)

(3)

Thus, the transverse dependence of the longitudinalcomponent of the wave electric field is guided by theusual formula, E z ~ J 0 (κ r ) , where J 0 ( x ) is thezero-order Bessel function. Taking into account theboundary conditions on the waveguide walls,
E z |r = a = 0 , we get the dispersion equation

κ = αn / a(4)are hold.
Physically, the above solutions represent the spacecharge potential waves which are coupled to theelectromagnetic mode. The growing solution exists onlyin an absorbing medium and vanishes at frequencies, atwhich the gas is transparent.
At finite waveguide radii, the geometrical factor
begins to play, and we have just from Eqs. (3) and (4) inthe limit of infinite guiding magnetic field ω c >> Ω b
the following instability increments:a) at sufficiently low ( | ε 1 − 1 | , ε 2 < < γ − 2 )electron energies
Im δ h ~Ωb2 γv 0(1+12(α n v 0γωa))
2Im ε ;

(7a)b) in the opposite case of high-energy electronbeam ( | ε1 − 1|, ε2 >> γ −2 ), the gain is determined bythe gas dielectric properties
Im δ h ~( (Ωb1+2γ v 0αncωa)2)

1Im ε| ε |− 1(7b)−2−1

the pressure region of our interest, 10 − 10 Torrand higher, the Bremsstrahlung radiation losses areinsignificant, and the energy loss rate is [5,6](∂γπmc 2γ 3/ 2~ −4πN 0 Zr02 ln∂zZWr)(13)

where Z is the mass number of the gas particle and Wr

4. RESONANT MAGNETIC GUIDINGFIELD EFFECTS
As one can easily see from Eq. (3), a specificresonance effects under the condition

R e( hv 0 − ω ) 2 ≈ ω c2(8)becomes possible. In fact, resonant magnetic guiding
field effects become significant at cyclotron frequencies( αa c ) | R e ( ε − β )|2ω c2 ≈−1n20(9)and moderate electron beam densities (with respect totransverse waveguide wave number)Ω b >>

α nca(10)γUnder these conditions, in contrast with the solution (5)and (7), the dissipative mode increments becomes
independent on electron beam density

Im δ h ≈(αna β0ε22 | ε |2)(11)

DISCUSSION

where J is the rotational quantum number, B is therotation constant of the molecule and T is thetemperature. For ortho-hydrogen, the most intense−1−1transition is at 587 cm , and at 814 cm for parahydrogen.
The maximum value of the imaginary part ofdielectric susceptibility is given by the relation

ε 2 max ≈ Fω / γ g(4π 2hcBhcBµ N 0 ( J + 1) 2 exp −J ( J + 1)
3TT

where µ is the oscillator strength, andcollisional linewidth,)(15)

γ g is theγ g ≈ 2.7 × 10 7 Hz / Torr forhydrogen.
It is easy to estimate that in the pressure region ofinterest we have for hydrogen ε 2 max ~ 0.2 and thisvalue is independent on gas pressure. The maximum
value of the instability increment, according to Eq. (11),is ~ 1cm at waveguide diameter of ~5mm, and thecharacteristic propagation distance with respect toenergy
spreadis~1mfortheenergy
spread ∆γ / γ ~ Im δh / h ~ 10 .
−3REFERENCES

In conclusion, let us make few estimates. The above
consideration is valid for the cold electron beam. It iseasy to show that the restrictions on the energy spreadfor the scheme proposed are of the standart form

∆γ / γ < Im δh / h(14)−1

According to the inequality (9), this increment
significantly exceeds the non-resonant value (5), (7).
This regime looks like the most prospective with respect
to the problem of beam bunching.

5.

( 2 J + 1)( J + 1) = 3T / ( hcB )F≈Ωb>>

is the ionization potential, Wr ~ 13eV
Let us make some estimates. In the case of hydrogenthe most intense rotation transitions are determined by
the relation(12)

The main source of the emittance and energy spread
growth in the scheme under consideration is thecollisions of beam electrons with the gas particles. In[1] A. N. Oraevsky, I. V. Smetanin JETP Lett.,vol.62, 258(1995);[2] A. N. Oraevsky, I. V. Smetanin Laser Phys.,vol.7, 155(1997)[3] I. V. Smetanin , A. N. Oraevsky, NIM A,vol.429, 440(1999)[4] A. N. Oraevsky. Sov. J. Quantum Electronics, vol.7, 495(1980), in Russian.
[5] E. P. Lee, R.K. Cooper, Part. Accel., vol.7, 83 (1976)[6] W. Sharp, M. Lampe, Phys. Fluids, vol. 23, 2383 (1980).