Self-Amplified Spontaneous Emission in
Smith-Purcell Free-Electron Lasers
Kwang-Je Kim and Su-Bin Song 1
Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439
Abstract
We present an analysis of a Smith-Purcell system in which a thin current sheet
of electrons moves above a grating surface in the direction perpendicular to the
gratinggrooves. We develop a complete theory for evolution of the electromagnetic
field and electron distribution in the exponential growth regime starting from the
initial electron noise and the incoming amplitude. The dispersionrelation for the
complex growth rate for this system is a quadratic equation.
Keg words: Smith-Purcell,FEL, SASE, grating
PACS: 41.60.Cr, 41.60.-m, 42.79,Dj, 52.75.Ms
1 Introduction
Beginning with the
The submitted manuscript has been create{
by the University of Chicago as Operator o
Argonne National Laboratory (“Argonne”
under Contract No, W-31 -?09-ENG-38 witt
the U.S. Department of Energy. The U.S
Government retains for itself, and others ac{
ing on its behalf, a paid-up, nonexclusive
irrevocable worldwide license in said artick
to reproduce, prepare derivative works, dis
tribute copies to the public, and perform pub-
licly and display publicly, by or on behalf O(
the Government.
work by Smith and Purcell [1], the radiation generated by
an electron beam passing over a grating surface has been studied for decades
[2-4]. Recently there has been a renewed interest in the Smith-Purcell sys-
tem with the observation of possible exponential gain in an experiment using
electron microscope beams [5,6].
In this paper, we study the Smith-Purcell system in the exponential gain
regime including self-amplified spontaneous emission (SASE). We study a sim-
plified system where the electrons are line charges oriented paraIlel to the di-
rection grooves and move perpendicular to the grating grooves. The electron
beam is furthermore assumed to be confined to a thin sheet. The configuration
is translationally invariant in the direction of the grooves, and is schematically
illustrated in Fig. 1.
1 E-mail:sbsong@aps.anl.gov
Preprintsubmit
ted to EXsevier Preprint
1 September 2000
The radiation field is expanded in terms of the plane wave modes discussed
by van den Berg [2]. The fundamental evanescent mode is synchronous with
the electron beam and induces electron bunching through the longitudinal
electric field. To take into account the interaction, the mode amplitudes are
assumed to be slowly varying in the direction “perpendicular” to the wave
direction. Using the boundary conditions across the electron beam and on
the grating surface, the outgoing mode amplitudes are completely determined
from the incoming mode amplitudes and the electron current. The evolution
of the electromagnetic field and the electron distribution can be described by
the coupled Maxwell-Klimontovich equations, as in the analysis of the usual
free-electron laser (FEL) system [7]. The growth rate for this system satisfies
a dispersion relation that turns out to be quadratic, rather than cubic as in
the usual FEL process.
We obtain a formula for the growth rate that differs from that derived pre-
viously {3]. The discrepancy appears to be due to an unnecessary additional
assumption employed in reference [3]. Our formula gives rise to a gain length
comparable to the length of the grating for the experiment reported in refer-
ence [5].
2 Smith-Purcell FEL Equation
2.1 Current
Figure 1 shows the Smith-Purcell system studied in this paper. The surface of
the metallic grating consists of a perfect conductor whose grooves are parallel
and uniform in the y direction. Electrons move parallel to the z axis in a thin
sheet along the grating surface. The current density is therefore
where vi is the velocity of the ith electron, q is electron charge, LAyis the
length in the y direction, ti(z) = ti(0) + f(l/vi)dz is the time when the zth
electron passes through Z, and z is the unit vector in the z direction. The
Fourier transform of the current density is given by
The z-dependent part of the phase in Eq. (2) has the average value
(2)
(3)
DISCLAIMER
This repofl was prepared as an account of work sponsored
by an agency of the United States Government. Neither
the United States Government nor any agency thereof, nor
any of their employees, make any warranty, express or
implied, or assumes any legal liability or responsibility for
the accuracy, completeness, or usefulness of any
information, apparatus, product, or process disclosed, or
represents that its use wouid not infringe privately owned
rights. Reference herein to any specific commercial
product, process, or service by trade name, trademark,
manufacturer, or otherwise does not necessarily constitute
or imply its endorsement, recommendation, or favoring by
the United States Government or any agency thereof. The
views and opinions of authors expressed herein do not
necessarily state or reflect those of the United States
Government or any agency thereof.
DISCLAIMER
Portions of this document may be illegible
in electronic image
produced from the
document.
products. Images are
best available original
where a. = k/,3’o, Do= vo/c is the normalized average velocity, k = w/c is the
wave number, and c is the speed of light. We then write Eq. (2) as
Jz(~,~;~) = 6(x)K(.Z,w)e2a0’,
(4)
where
K(,z, w) =
L Xeiw(t’(z)-z/vo)
Ay
(5)
Since the overall phase factor ezao’
is taken out, we expect that K(.z, w) is a
slowly varying function of .2.
R’EC!:: }! ~~:
2.2
Maxwell Equations
O(H’062800
Q.s1-j
The electric and magnetic fields in the frequency domain are defined as
From the Maxwell equation it follows that the magnetic field H is in the g di-
rection for the present system [2]and that
Hv satisfies the following Helmholtz
equation:
(32 82
(— —
8X2
+ ~z2
+
k2)Hv(x, .z;u) = :Jz(x, z?;W).
The electric field is then determined by the Faraday equation:
la
E.(z, Z;LJ) =-
—Hv(fc, z; L4J),
2LSOW 6’.2
E,(X, .2; ~) =*
[
1
JZ(Z, z; u) – &(z, Z;W) ,
(6)
(7)
(8)
where co is the vacuum dielectric constant.
2.3
Modes
Van den Berg has studied modes for the magnetic field Hy in the absence of
the interaction between the current and the field [2]. The plane wave solution
of the Helmholtz equation is of the form [2]
ei(chz+?w)
>
(9)
where
p. .
J=
n“
(lo)
3