EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
CERN - SL DIVISION
SL-Note 96-03 (RF)
Longitudinal echo in a continuous beam.
(Extension to more general case).
E. Shaposhnikova
January 1996
First measurements of echo signals associated with two RF excitations of the continuous beam have been made in the SPS, [1]. While this new data has confirmed understanding of the phenomenon in general, [2]-[4], some unexpected results still await an explanation. This short note is an extension of [4] and presents calculations of the echo effect in a more general way, without using perturbation theory. Only the assumption about the length of the RF kicks (they must be short) is still necessary. Below we use results and definitions from [4].
The longitudinal
echo can be observed in the continuous beam
at some well defined time
and frequency
after two consecutive
RF kicks,
with time delay 
between them,
have occurred
at different harmonics (h1 and h2)
of the revolution frequency.
At the moment t (the first RF kick is at t=0) the beam current in the machine is
which can be expanded in the Fourier series
where
and 
is the longitudinal distribution function
in the beam at the moment t.
According to Liouville's theorem phase space density doesn't change along the particle trajectories
where 
is the initial distribution function.
Below
we assume that
the initial distribution function of the continuous beam is
a function of p0 only.
Then if later coordinates of the particles 
can be expressed as functions of the
initial coordinates 
and time,
,
we can write for
the n-th harmonic
of the beam current at the moment t the following expression
The solution for
the particle motion
found in [4]
can be presented
for 
in the form
where we introduced the definitions
and used the notations
Here Vn and Tn are the amplitude and the length of the n-th RF kick. This solution is valid only under the assumption that there is negligible change in the position of the particles during the kick, which leads to the requirements discussed in [2]-[4]:
where pmax is the maximum value of p in the beam.
Now using solution (6) we can rewrite formula (5) for the n-th harmonic of the beam current at the moment t as
To proceed further let us consider separately the following expressions:
After some transformation these expressions can be presented as
where Jk(x) is the Bessel function of order k.
For simplicity let us consider first the most interesting case when
Then in (15) we can keep
only the terms
with 
in the series
and rewrite it in the form
Multiplication of expressions A and B will give
Summation over l can be eliminated by taking into account the fact that we are interested only in harmonics with l-k=1. Then we obtain
where 
.
Using Neumann's theorem for the summation of Bessel functions:
we can sum over k in (18) and present (12) as
Here we dropped the index "0" for initial coordinates.
After integration over 
in (19)
we get
Finally after substituting expressions for z1, z2 and y in (20) we can write for the amplitude of the echo signal at harmonic (h2-h1)
where we used the notations
The echo signal can be observed at times
close to
, where
However as can be seen from the expression
(21) at exactly 
the
amplitude of the signal
is always zero.
The function 
can be defined from the
Fourier transform of the initial
distribution
For
an initial distribution which
is symmetric in energy,
F0(p)=F0(-p),
function 
) becomes
For small values of 
we can use the Bessel function expansion
with
. In this case
formula (21)
gives the same answer as found in [4].
In the more general (but nondegenerate) case when
the amplitude of the n-th beam current harmonic becomes
with
