An interference wiggler for precise diagnostics of electron beam energy Page: 3 of 4
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Kwang-Je Kim
Lawrence Berkeley Laboratory
University of California
Berkeley, California 94720Summary
Relativistic electrons passing through two dentical magnetic sec-
tions generate synchrotron radiation whose spectrum is strongly modu-
lated as the photon energy vanes The modulation is caused by the
interference of radiation from each section, and has been observed I in
the spectrum of spontaneous radiation from transverse optical klstron
which utilizes two undulators. In this paper, we analyze and apply
another device based on two simple wigglers. The device, which will be
called the interference wiggler, can be used for precise diagnostics of elec-
tron beam energy, by analyzing the modulated spectrum with a mono-
chromator. the electron energy can be determined up to an accuracy of
10t- or 10- In this paper we develop general design cntena for
interference wigglers. We also give several example designs to measure
the electron energy to an accuracy 10 for the planned electron beam
lacilil at CEBAF 2. and to an accuracy 10-i for the 1-2 GeV Light
Source at Berkeley 3
Spectrum of Interference Wiggler
The electron trajectors in an interference wiggler is shown in Fig I
The trajecton will be assumed to le on the horizontal plane In wiggler
approximation .. the radiation in the direction 1a.41. where e and S are
respecetily the horizontal and 'erical angles, comes matoly from small
segments of electron trajectors about the points where the siope is parallel
to a For the trajectors in Fig I there arc in general four such points
labelled I. 2. 3 and 4. Among these we 'ill consider only I and 2.
assuming either that 3 and 4 are suiucntly separated transversely from I
and 2 so that they can be considered separate, or that the radiation
mtensil' from 3 and 4 is much weaker due to weaker magnetic field
( omputing the electric field from I and and squarng it cne obtains the
angular density of flux The result when the effect of the electron beam
angular divergence is taken ino axeuni is as follows
da 3 d___ - ff ,cose . I)
dob4 -hdr,
wheref - esp -ykUI - g) Vi'
f - a p ktana 2I + i lb
Ir(an '1; + tan 'biy .(2)
(3)LBL--23007
DE87 011095In Eq. (1), d2 /dedd is the angular density of flux from point I or
2 alone, which is a smooth function of photon energy represented by the
dotted curse in Fig 2 The term proportional to cosn is due to interfer-
ence and causes the modulation of the spectrum represented schemati-
cally as the solid curse in Fig 2 An equa:- similar to Eq (1) was first
derived by Ellaume I in the analysis of the spontaneous radiation from
transverse opical klystron which is a two undulator system
For a complete characterization of source. it is necessary to calcu-
late the flux density in phase space known as the brghtness by using the
method discussed in Ref 5 The results are in accord with the expecia-
tion that the sources at I and 3. for example. appear to be separated
transversely In forward direction the source separation is given by the
maximum excursion amplitude a of electron trajectory (see Fig 1)
Method of Dete_rrnmmg t, and n,
The modulated spectrum has peaks when a - 2rn. n being an
integer In this paper, we consider only the forward direction 6 - o - 0
Using Eq (5). and neglecting for the moment the last two terms, the loca-
tion of nth peak k, is found to bek~ = t UI - g)) n
From this it follows for any pairs of integers (n.m) that
n = mk (k,,. - krl(b)
The location of peaks k, and k can be determined by analynng the
spectrum with a monochromator The integer m can be determined by
counting the number of peaks between k and k We can thus deter-
mine the integer n associated with k, The electron energy ', is then
determined from Eaq. (6).
To discuss the measurement accuracy. let ] indicate the error in the
measurement We obtain from Eqs ') and (6) that
An _ k, Ak_, - k~)
n k, k_ - k~ (8)(9)
yo 2 ku I g1- g)
For an unambiguous determination of n it follows from Eq (8) thai the
monochromator bandwidth Ak./k needs to he smaller than I 'n and that
the spectrum needs to be observed oser a wide range of k so that k -
(4) k0 is of order k From Ea (9). it follows that both the monochromator
bandwidth and the errors in the magnet parameters should be about the15)
and k - 2w/A. A - radiation wavelength. , - electrons' relative energy
spread (rms). L - the distance between two wigglers (see Fig 1), Y. - the
as erage electron energy in unit of rest energy g is defined so that ULI +
g l - is the arc length of electron tralectory between the two cess m
Fig I a and ~ are respecrir ihe horizontal and vertical angular
spread of electron beam mst
Fig I A schematic representation of the electron trajectory in the
interference w'ggler Radiation in the direction o is generated
from small segments of the trajectory around the tangent points.
marked as 1. 2. 3 and 4u-
ikL w cr.
Fig 2 A schematic representation of the specter im from the inieri-r
ence wiggler The solid line shows the modulated speirum
while the dotted toe represents the smooth background due I
radiation from idividual wigglers For the enampes ;m.-
sidered in the paper, hundreds of peaks would appear in th
operating spectrum range
DISTRIBUTION OF HIS DOICUMENT IS UNUMITED
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Kim, Kwang-Je. An interference wiggler for precise diagnostics of electron beam energy, article, March 1, 1987; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc1210059/m1/3/: accessed April 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.