Experimental confirmation of transverse focusing and adiabatic damping
in a standing wave linear accelerator
S. Reiche,* J. B. Rosenzweig, S. Anderson, P. Frigola, M. Hogan, A. Murokh, C. Pellegrini, L. Serafini,
²
G. Travish, and
A. Tremaine
Department of Physics and Astronomy, University of California, Los Angeles, Los Angeles, California 90095-1547
Received 28 February 1997
The measurement of the transverse phase-space map, or transport matrix, of a relativistic electron in a
high-gradient, radio-frequency linear accelerator rf linac at the UCLA photoinjector is reported. This matrix,
which indicates the effects of acceleration adiabatic damping, first-order transient focusing, and ponderomo-
tive second-order focusing, is measured as a function of both rf field amplitude and phase in the linac. The
elements of the matrix, determined by observation of centroid motion at a set of downstream diagnostics due
to deflections induced by a set of upstream steering magnets, compare well with previously developed ana-
lytical theory J. Rosenzweig and L. Serafini, Phys. Rev. E 49, 1599 1994. The determinant of the matrix is
obtained, yielding a direct confirmation of trace space adiabatic damping. Implications of these results on beam
optics at moderate energy in high-gradient linear accelerators such as rf photoinjectors are discussed.
S1063-651X9709508-1
PACS numbers: 41.75.Ht, 41.85.-p, 29.17.+w, 29.27.Bd
With the rise in use of high-gradient radio-frequency lin-
ear accelerators rf linacs in devices such as rf photoinjec-
tors 1 and linear collider test facilities 2, there has been
increased attention placed on the strong transverse focusing
effects present in these devices. These effects, which are due
both to first-order transient effects at the entrance and exit of
a linac and to second-order ponderomotive alternating gra-
dient effects in the body of the periodic linac structure, are
of primary importance in understanding the beam transport
in moderate energy sections (5 =E / m
e
c
2
100) of elec-
tron accelerators. While theoretical analyses of the focusing
properties of linacs date back to the 1960s 3,4, recent work
has produced a more detailed understanding of the pondero-
motive force 5 and analytical solutions of these equations
for arbitrary acceleration phase and spatial harmonic content
of the rf fields have been found 6, which led to a matrix
description of the trace space transport.
This matrix treatment of beam dynamics in high-gradient
rf linacs, as well as the underlying analytical model for the
averaged over a rf period transverse forces, has formed the
underpinning of much recent work, from the optics of linear
collider test facilities 2 to the full theory of space-charge-
dominated beam dynamics in rf photoinjectors 7. While
these implementations of the theory have been compared
positively with computer simulation, there has been, how-
ever, no effort to date, to the authors’ knowledge, to verify
the theoretical advances with experiment. This paper pre-
sents such a verification.
The trace space transport matrix corresponding to a rf
linac, which upon multiplication of a transverse trace space
vector, e.g. ( x , x ' ), gives the mapping of this vector through
the linac, has been derived recently for arbitrary rf phase,
amplitude, and spatial harmonic content in the linac. Includ-
ing all terms to second order in the average accelerating gra-
dient eE
0
cos()'m
e
c
2
, where E
0
is the amplitude of the
synchronous ( v
c ) spatial harmonic wave component of
the rf field and = t -kz is the phase defined with respect
to the maximum acceleration in this wave, the action of a
ponderomotive force can be obtained to second order by av-
eraging over the fast alternating gradient first-order forces
and the induced lowest-order oscillatory motion, as 5,6
F
¯
r
=
qE
0
2
8 m
0
c
2
r
n =1
b
n
2
+b
-n
2
+2 b
n
b
-n
cos 2
qE
0
2
8 m
0
c
2
r . 1
The coefficients b
n
are the Floquet amplitudes of the spatial
harmonics, defined by the expression, valid for an ultrarela-
tivistic ( v
b
c ) electron,
E
z
E
0
Re
n =-
b
n
e
i 2 k
0
nz +
, 2
where k
0
= / d = / c and is the rf phase shift per period of
the linac, with = in the structure considered here. The
coefficients b
n
have been determined for this structure by
mapping the on-axis longitudinal profile of the field using a
bead frequency perturbation technique. The fundamental ( n
=0) component of the field, with the coefficient normalized
to unity, provides the only significant secular averaged over
a period acceleration, with nearly negligible transverse ef-
fects. The other field components, the nonsynchronous spa-
tial harmonics, contribute almost no net secular acceleration,
but give rise to second-order focusing through an alternating
gradient or ponderomotive effect 5,6. Typically, forward
and backward wave components of the nonsynchronous har-
monics have degenerate frequencies in the frame of the rela-
*Permanent address: DESY, Hamburg, Germany.
²
Permanent address: INFN-Milano, Milan, Italy.
PHYSICAL REVIEW E SEPTEMBER 1997 VOLUME 56, NUMBER 3
56 1063-651X/97/563/35726/$10.00 3572 © 1997 The American Physical Society