The interaction between the electrons and the radiation in a free-electron laser leads to a shift and a spread of the electron velocity distribution. The electron dynamics of a two-element system are studied in the small signal region. It is found that the efficiency and gain can be increased through introduction of an adjustable drift distance between two identical wigglers

Shih, Chun-Ching

Yariv, Amnon

Caltech Authors - Main

76 OPTICS LETTERS / Vol. 5, No. 2 / February 1980
Two-element free-electron lasers
Chun-Ching Shih and Amnon Yariv
California Institute of Technology, Pasadena, California 91125
Received October 8, 1979
The interaction between the electrons and the radiation in a free-electron laser leads to a shift and a spread of the
electron velocity distribution. The electron dynamics of a two-element system are studied in the small signal re-
gion. It is found that the efficiency and gain can be increased through introduction of an adjustable drift distance
between two identical wigglers.
Theoretical studies of free-electron lasers indicate
that the efficiency of such devices is limited by the small
fraction of energy loss of the electrons that can take
place in a single pass. The first operation of the free-
electron laser as an oscillator achieved an efficiency of
less than 0.01%.1 It has been proposed to recycle the
electron beam in order to increase the overall efficiency.2
However, the analysis of the electron dynamics in a
free-electron laser, using coupled Maxwell-Boltzmann
equations,3 showed that the velocity spread of an elec-
tron beam after the interaction is much larger than its
velocity shift. This broadening leads to a considerable
reduction of the gain and thus the usefulness of the
reused beam.
In this Letter we point out that it is possible to im-
prove the device efficiency by using multiple interaction
regions that are properly spaced. For a quantitative
analysis we use the single-electron model, which has
been proved very useful in describing the transverse4
and longitudinal5 free-electron lasers. In this model,
the trace of each electron is uniquely determined by its
initial velocity v and entry phase 0. The output ve-
locity of v' of an electron is found to second order in the
field amplitude:
v' = v + Av
Cos i . Si l
sin40+ Cost/H
- cos t- -sin r
2
+ sin ¢~ vsin (20 -
(1)
(l)
change of a single electron passing through an arbitrary
interaction region can be written as
Au = A + VX a cos 0, (2)
where A is the average velocity shift, a- represents the
velocity spread, and 0 = 0 + 5(t) still serves as a label
of individual electrons. 6 is a function depending on
only.
If the initial electron beam is monoenergetic and if
the electron distribution in j space is uniform each time
the electron beam re-enters the interaction region, we
find that the velocity shift is proportional to the number
of circulations N, while the velocity spreads to VN . As
the velocity spread becomes larger after each circula-
tion, the maximum number of circulations is thus de-
termined by the maximum allowable velocity spread
0lmax. The total velocity shift then becomes
Amax = (A/c 2 ) amax2. (3)
Since the extractable energy from the electron beam
is proportional to Amax, the overall efficiency of the
device depends on the value of the factor R (_ Ao/2).
Consequently, an increase in the efficiency can be
achieved by either enhancing the single-pass shift A or
reducing the single-pass spread a.
The velocity spread is due to the different entry
phases of electrons. If we can "invert" the interaction
between wave and electrons in a second interaction re-
gion, then we may expect a reduction in the velocity
spread. Such inversion can be achieved if the entry
phase of each electron is shifted by 7r radians with re-
spect to that of the first region. Thus two-stage devices
have been proposed6' 7 that consist of two identical in-
teraction regions separated by a drift distance between
them. Because of the velocity difference between wave
and electrons, the 7r shift of the entry phase can be ob-
tained by adjusting the value of LD (Fig. 1).8
where L is the length of interaction region, ¢ = [(I/up)
- (1/U)]wL is the phase slippage in one transit of an
electron with velocity v relative to the wave whose phase
velocity is vp. X is the field expansion constant, and its
explicit form depends on the type of free-electron laser.
For example, ij = eEwL2/mc3 ,y3 in the longitudinal
free-electron laser,5 while in = 2e 2BEL 2/m 2 c 4 )y2 in the
Stanford free-electron laser.4 In general, the velocity
interaction region drift space interaction region
__ ____ _ ___-___ : rz ///////I -_________-- -- -___  X
e- beam
I ' L *|- L D |!a - L
01 , VI V i2 1 V2
Fig. 1. A two-element free-electron laser.
0146-9592/80/020076-03$0.50/0 © 1980, Optical Society of America
= V +
- v72 -
V2
February 1980 / Vol. 5, No. 2 / OPTICS LETTERS 77
As a separate, but related, issue we consider the
problem of maximizing the single-pass velocity shift and
thus the single-pass gain by using two interaction re-
gions. The basic reasoning for this approach is derived
from the operation of the klystron.9 The first interac-
tion region acts as a buncher, giving rise to a strong ve-
locity modulation. In the drift space between the two
interaction regions, the velocity modulation gives rises
to bunchbs (current modulation). These bunches are
then made to enter the second interaction region at the
optimum phase for deceleration and energy extrac-
tion.
Using the single-electron formalism5 in the small-gain
regime, we are able to formulate the interaction process
in both interaction regions. The final velocity that is
to be expressed as a function of initial entry phase is
obtained by relating physical quantities in the second
region to those in the first region. The expression of
final velocity is then used to obtain the velocity spread
and shift.
In terms of the general expression (2), the velocity
shift and spread of a two-element free-electron laser are
found to be
_c 2n(wL A[(1 - cos ¢)(1 + cos a)]-1/2
c2=- () 2 + LD) sin a +g()(1 + cos a)
where
a = wLD ( +1 ) X ,
C V
f(r) = (1 - cos P/2
g(W) = (2-2 cos v-sin ~)/P3.
Since the device efficiency depends on A/o-2 , we tr
maximize A and temporarily neglect the problerr
velocity spread. Two terms with different depend
cies on QT are involved in A. The maximum of g(O
0.135 at P = 2.6, while f( ) = 0.5 at P = 0. Near rE
nance (= 0), the first term dominates over the secc
Furthermore, the presence of Lb in the first term ma
it possible to increase the shift by using longer d
distances. Choosing P = 0 and a = r/2, we have
A 2 CL(+ -L) and a =L7
Table 1. Gain Enhancement of a Two-Element
Device"
Conditions
Single-Element Two-Element
¢= 2.6 A=0; a = 7r/2
\ _~~0.54 c _2c21
wL wL
CI 1.05C - C2
wL
a The comparison of the velocity shift and spread is given for a
single-element device of length 2L, which is the total interaction
length of the two-element device. The drift distance is arbitrarily
chosen to be 2L, which results in a four times larger velocity shift.
3LD
X-Y2
(6)
The optimum operation can be achieved within a
change of 2i- in a, which means an adjustment of the
drift distance within IXLD,
IALD = 2X-y2. (7)
For an estimate of a typical value of ALD we consider
the Stanford device and find that ALD is about 5 cm.
This value is very reasonable for a practical experi-
mental setup.
Nekt we consider the problem of velocity spread. It
(4) is found that the first-order contribution to the spread
o can be rendered zero by choosing P = 27r or (x = 7r.
Unfortunately, we also find that the second-order shift
becomes zero under either of these two conditions.
Qualitatively, the spread is now of second order and the
shift is fourth order in the field amplitude.
This result follows directly from Madey's theorem,10
y to which can be written in the relativistic approximation
as
1 of
en-
) is
,so-
ind.
kes
.rift
(8)(Av)o = 1 a [(Av2 ),j,2 av
where triangular brackets represent the ensemble av-
erage over 0. Using the expression (4) for Av, we
have
A- (AV) = 0 d-.
(5)
for the shift and spread of the two-element device.
We next compare Eqs. (5) with the shift and spread
of a single-element device of length.2L, which is the total
interaction length in Eqs. (5). The result is shown in
Table 1, where we have arbitrarily chosen LD to be
2L.
Equations (5) show that for a given total interaction
distance the velocity shift can be enhanced by a two-
element system, especially when the electron-drift
distance is much larger than the length of each inter-
action region. Furthermore, it- is also shown that the
increase in the velocity shift is not accompanied by an
increase ih the spread of electrons.
The parameter, a is determined from the drift dis-
tance. In the high relativistic limit (-y >> 1),
(9)
It is obvious that A is identically zero if a vanishes. We
conclude that, up to second order, it is impossible to
eliminate the first-order spread without sacrificing the
second-order gain.
We have considered the problem of increasing the
shift and eliminating the spread separately. However,
the final purpose is to improve the value of R that can
be obtained from Eqs. (5),
R - 12 + L tan 2 +-cot 2¢1* (10)
The optimum condition for an efficient operation of the
device is then found by optimizing Eq. (10) with
changing a and A. Although the shift depends on a and
P in a complicated manner, the expression for R contains
two terms that depend exclusively on cv and A. So the
78 OPTICS LETTERS / Vol. 5, No. 2 / February 1980
optimum value for a and t can be found independently.
Consider the first term. It becomes infinitely large
when a - 7r. The second term, which depends only on
A, approaches infinity as . - 27r. As we have shown
before, at these two values the shift is actually zero.
However, U approaches zero at a faster rate than A,
which results in an increasing value of R.1"
In conclusion, we have shown that the efficiency of
a free-electron laser in beam circulation and the gain of
a single-pass device can be highly improved by using a
two-element system. For the gain enhancement, the
system is operated at the resonance and a is equal to
r/2. The gain is found linearly related to the drift
distance. For the efficiency improvement we choose
has close as possible to 2r and a as close as possible to
-r. However, the choice of t and a must be such that
single-pass gain is higher than the threshold condi-
tion.
The research reported in this Letter was supported
by the U.S. Air Force Office of Scientific Research.
References
1. D. A. G. Deacon, L. R. Elias, J. M. J. Madey, G. J. Ramian,
H. A. Schwettman, and T. I. Smith, Phys. Rev. Lett. 38,
892 (1977).
2. L. R. Elias, W. M. Fairbank, J. M. J. Madey, H. A.
Schwettman, and T. I. Smith, Phys. Rev. Lett. 36, 717
(1976).
3. H. Al Abawi, F. A. Hopf, and P. Meystre, Phys. Rev. A 16,
666 (1977).
4. W. B. Colson, Ph.D. Thesis, Stanford University
(1977).
5. A. Yariv and C. Shih, Opt. Commun. 24, 233 (1978).
6. C. Shih and A. Yariv, Tenth International Quantum
Electronics Conference, Atlanta, Georgia, May 29,
1978.
7. F. A. Hopf, P. Meystre, G. T. Moore, and M. 0. Scully, in
Physics of Quantum Electronics, S. F. Jacobs, M. Sar-
gent, and M. 0. Scully, eds. (Addison Wesley, Reading,
Mass., 1978), Vol. 5.
8. We have considered using a magnet system in the drift
space. However, it is found that the velocity dependence
of the magnet is a second-order effect that is too small to
affect the first-order bunching effect.
9. D. L. Webster, J. Appl. Phys. 10, 501 (1939).
10. J. M. J. Madey, Nuovo Cim. 50B, 64 (1979).
11. If 0ra. is defined as the maximum allowable width of
electron distribution, where the gain drops by a certain
factor, it is found that 0tmax does not depend on parame-
ters strongly near zeros of A.
Efficient higher-Stokes-order Raman conversion in molecular
gases: errata
H. Komine and E. A. Stappaerts
Northrop Research and Technology Center, I Research Park, Palos Verdes Peninsula, California 90274
Received December 3,1979
The following references Awere inadvertently omitted
during the editorial process:
Reference 6 should also have included works by V.
Wilke and W. Schmidt, "Tunable coherent radiation
source covering a spectral range from 185 to 880 nm,"
Appl. Phys. 18, 177-181 (1979); J. A. Paisner and R. S.
Hargrove, "Tunable, efficient VUV and IR generation
using high order stimulated Raman scattering in H2,"
presented at OSA/IEEE Conference on Laser Engi-
neering and Applications, May 30-June 1, 1979, paper
II-4.
Reference 8 should also have included a paper by W.
R. Trutna, Jr., Y. K. Park, and R. L. Byer, "The de-
pendence of Raman gain on pump laser bandwidth,"
IEEE J. Quantum Electron. QE-15, 648-655 (1979).


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