PFC/JA-86-27
On the Linear Theory of the
Circular Free Electron Laser
H, Saito and J. S. Wurtele
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, MA
02139
July 1986
This work was supported by the Japan Society for the Promotion of Science and the Office of Naval
Research .
By acceptance of this article, the publisher and/or recipient acknowledges the U.S. Government's
right to retain a nonexclusive royalty-free licence in and to any copyright covering this paper.
On the Linear Theory of the Circular Free Electron Laser
H. Saito*and J.S. Wurtele
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, MA 02139
A small signal theory of a Free Electron Laser with a rotating electron beam in a uniform
axial magnetic field and in an azimuthal wiggler field (the "circular" FEL) is developed.
The analysis includes the low and high gain regimes and the influence of longitudinal space
charge forces. It is found that the circular FEL instability has two regimes: a strong pump
regime and a negative mass dominated regime. The negative mass regime replaces the
weak pump (Raman) regime found for the usual FEL geometry in which the electron beam
propagates in the axial direction ( the "linear" FEL). The dispersion relation is evaluated,
and the resulting growth rates are compared with the those of the linear FEL. For a cold
beam, at fixed output frequency, the growth rate in the strong pump region is larger, by
a factor of 12/3, in the circular FEL. The negative mass instability is shown to increase
the growth rate and modify the bandwidth of the circular FEL in the strong pump region.
However, the circular FEL performance is found to be more sensitive to the energy spread
than the linear FEL.
*Present address: Institute of Space and Astronautical Science, Tokyo, Japan
1
I.
Introduction
A new type of free-electron laser, the "circular" Free Electron Laser, has been theoreti-
cally1 3 and experimentally 4 5 investigated. As originally proposed by Bekefi, 1 the magnetic
field configuration consists of a uniform axial field, B 0 _, and an azimuthally periodic wiggler
field, B.(r,0). The relativistic electron beam rotates azimuthally and wiggles axially, as
shown in Figure 1. Such a wiggler field is generated by an assembly of magnets placed behind two concentric metal cylinders. Near the center of the gap, the wiggler magnetic field
is primarily radial and is transverse to the electron flow velocity. The metallic boundaries
act as a coaxial waveguide for the radiation. The conventional FEL geometry, in which the
electron beam propagates axially and wiggles in the transverse plane, will be referred to
here as a "linear" FEL.
Yin and Bekefi3 derived the dispersion relation for the circular FEL in the strong pump,
high gain region. Their analysis used a fluid model, and did not include longitudinal space
charge or energy spread.
Here we extend their work by developing a one dimensional
nonlinear model which includes space charge, energy spread and the transverse profile of
co-axial waveguide modes. Our nonlinear model is similar to the description of the linear
FEL presented by Kroll, Morton and Rosenbluth and others. &8
This paper analytically solves for the gain characteristics of the circular FEL in the low
gain limit and in the high gain strong and weak pump limits. Linearization of the equations
of motion for the circular FEL, in the small signal regime, results in a cubic dispersion
relation. This cubic is then compared to the cubic resulting from the small signal analysis
of the linear FEL.9'
0
At fixed frequency, in the cold beam, strong pump regime, growth
rate of the circular FEL is larger, by a factor of -y 3 . than that of the linear FEL. On the
otherhand, the circular FEL has a greater sensitivity to energy spread than does the linear
2
FEL. We find that the negative mass instability can significantly influence the growth rate
of the circular FEL. In particular, there is no equivalent to the Raman regime in the circular
FEL. Instead, as the beam current is increased the negative mass instability dominates and
the growth rate becomes independent of the pump field strength.
In Sec. II a nonlinear, one dimensional model for the circular FEL is derived. In Sec.
III the low gain limit is examined, and in Sec. IV the exponential gain dispersion relation is
derived and studied. We briefly discuss optimization in Sec. V. and Sec. VI contains some
conclusions.
II.
Theoretical Model
Wiggler and Electromagnetic Fields
A.
The configuration of the circular free electron laser is shown in Fig. 1.
A uniform
magnetic field Jo = Boi is applied in the axial (z) direction. The corresponding vector
potential is given by 4 0 = (Bor/2)6. The wiggler field B, is given by3
=-i$w~r0)
BrN
sin(NO)[(
+N-({
b
+11
(a! )(N2-1)2N4
(j-;2
(1)
+ -
B)N-i_ ,blN+lla'
cos(N)() )
2
a
r
(N
2
-1)/2N
b
where a' and b' denote the radii of the outer and inner magnet surface, N is the number
of wiggler periods and b, is the amplitude of the wiggler field.
and
We adjust the radii a'
b' so that the azimuthal component of 6,, vanishes at the electron beam radius ro =
(aiNb-1bN+1)1/2N.
The wiggler field can then be approximated by
3
k' = -
r sinNO
(2)
near r = ro. The corresponding vector potential of the wiggler field is
cos NOW
N
(3)
The radiation mode of the circular FEL is a coaxial waveguide mode which is a travelling
wave in the azimuthal direction. Since the electrons are assumed to have no axial streaming
velocity, k. = 0 corresponds to the case where the growth rate of the FEL instability is the
largest.3 The coupling between the axial electric field (E.) and the oscillatory axial electron
velocity (v.) due to the wiggler field results in the FEL interaction. The radiation mode must
be transverse magnetic (TM), since, for k. = 0, only TM modes have axial components of
the electric field. However, the ponderomotive potential of FEL causes bunching of electron
beam in the azimuthal direction and thereby drives a longitudinal space charge wave.
The vector potential
of the TM mode radiation is
A.(F, t) = -A,(r, t) cos(pO - u.,t + $4i,
(4)
and the axial electric field E. corresponding to ', is
E- (F,t) = -wA,(r,t) sin(pO - wt +4)
(5)
We assume dA.,/dt < w,. The radial dependence A,(r,t) is given by
A,(r,t) = M(t)X,(kr),
where M denotes the time dependent field amplitude. The function X, is
4
(6)
XP(kr) = Y(ka)Jp(kr) - Jp(ka)Y,(kr),
where J, and Y, are the first and the second kind of Bessel function, respectively."
(7)
In
Eq. (7), p is the mode number in the azimuthal direction. The radial wave number k, is
determined by solving X,(kb) = 0. The coaxial waveguide modes are labelled TMN,, where
the index q refers to the q'th zero of X,(kb) = 0. At cutoff, k. = 0 and k,. = w,/c.
The energy density 11 and the circulating power P (Poynting power in the azimuthal
direction) of the TM~,4 mode radiation per a unit axial length are given by
U = (r/2)eoM
2
{a2X'(kra)2 _ 2X'kr)2
2
kr)2 - 2-1EjXVk)
P =(coc/4k,) M2 [XP,
,o (k,r)
X,,, (k,.r)
(8)
(9)
v=O,.
where Xp,,(krr) is defined as
Xp,,(krr) = Y(ka)J,,(kr) - Jp(ka)Y,(kr).
B.
(10)
Nonlinear Equations
An annular relativistic electron beam with infinitely small thickness is introduced in
the gap at r = ro, where the azimuthal component of the wiggler field B. vanishes. The
electrons are subject to the cyclotron motion which is slightly perturbed by the wiggler and
the radiation. We have the simple relation
ro = co
5
(11)
where "c = eBo/m and B0 is the amplitude of the axial uniform magnetic field. Here
3 = v/c and -y= (1 -
02)-1/2,
where v is the electron velocity.
Neither the wiggler field nor the cutoff radiation mode vary in the axial (z) direction.
Therefore, the z component of the canonical momentum is conserved and the axial velocity
v: is given by
V. = (c/'y) v'{a. cos N6 -
IaIcos(pO - wt +
)},
(12)
where a, = (EroB5v,/'mcN), and la. = (e/V2_mc)A,(ro,t).
For the electrons which are subject to the concentric motion in the uniform and the wiggler field, the radial component of the canonical momentum is conserved to order (a/y).
The radial variation of the wiggler field and waveguide mode is neglected. This approximation requires an annular electron beam, and places constraints on the beam energy spread
which are discussed in Sec. IV.D. Under the further assumption that the wiggler field only
slightly perturbs the electron motion, (a,,/'y)2 < 1, the radial velocity can be assumed to
vanish:
V, = 0.
(13)
Note that there is no radial component to the vector potential. To obtain the azimuthal
component, vq, of the electron velocity, we use Eqs. (12) and (13) and the definition of y.
Thus
=
c#1 -
{a, cos NO - Ia, 1cos(po
We now define the electron phase C as
6
Wt+4)}2].
(14)
= (p+ N)O - w~t.
(15)
Using (11) and (14). the electron angular velocity defined by dO/dt = ve/ro is
d
a
2-
[1
2Ia.acos((+
-
4)
(16)
,
where we averaged dO/dt over a wiggler period and used 0 2: 1, and Ia.I1< a,,.
The electron phase C evolves according to Eqs. (15) and (16):
[(p + N)
I -
_ 2a,aue') + C.C.
(
-
-W,,
(17)
where the index j has been introduced as an electron label and the complex presentation
a, = Ia,| e
is used. Here c.c. denotes complex conjugate. By setting dC/dz = 0 for an
electron with the average beam energy, yo, the FEL resonant frequency is found to be
W4
wo = (p+ N)-
YO
I
a2
_ 2'
2
.
(18)
(8
The electron energy evolves according to
dyj
dt
_1[
-
2i'as,-
2
)J
1 [
+
+ [
'iw2
i (p +N)wc/gsy
(e i)e'C + c.c.,
(19)
where the symbol ( ) means an average over the electron energy and phase. In Eq.(19), w,
is the plasma frequency defined by
=
p
r.
meor(a2 - b 2 )(
(20)
Here N, is the number of electrons per unit axial length. The first term on the right hand
side in Eq. (19) is the ponderomotive force produced by the wiggler and radiation fields,
7
9 10 For the circular
and the second term comes from the longitudinal space-charge force. '
FEL, when the geometric factor
f,
(described below, Eq. (21)) is positive, the system
has a negative mass instability. 1 2-15 This instability results from the azimuthal bunching
induced by the FEL interaction. The coupling factor fi, which includes the influence of the
14
waveguide on the negative mass instability, is given by
f 1 = g(p + N)
a2
-
b2
2b.
(21)
4ro
where
Ar
(b ++b_
g= t
+(p+N)--
+
2
ro
-1
.
(22)
The quantity Ar is the half width of the electron beam and the b± denote the wave admittance, and are discussed in Appendix A. As will be shown in Section IV.B, a small signal
analysis of Eqs. (17) and (19) yields, in the absence of the wiggler field, the well-known
growth rate for the negative mass instability.1
The evolution of the slowly varying amplitude, a,(t), is found by inserting Eq. (4) in the
0
wave equation. Analogously to the linear FEL in a waveguide,1 the wave equation is next
multiplied by the mode factor rXp(krr) (Eq. (7)) and integrated over the radial coordinate
(b< r < a). Averaging over a wiggle period then yields the desired equation
da,da
m (ea
ed ).
i
&2w,
(23)
The couping factor for the mode, f2, is found, using well-known orthogonality relations
of Bessel functions," to be
8
f
(a2 - b2 )X,(k,.ro)
a X|(k,.a) - b2X' (k,b)'
2
We recall that k,. is the radial wave number of the TMJQ mode and
f2
depends on the mode,
the waveguide-geometry and the beam position. When the azimuthal mode number p is
small enough, the radial profile X,(kr) of the field is approximately sinusoidal and
f2
is
then
f2 =
7.0
sin k,(a - ro),
(25)
where k,. ~ 7r4/(a - b)/2. When the electron beam is near the center of the gap, ro ~
(a + b)/2, Eq. (25) reduces to f2 ~ 2 for odd q, and
~ 0 for even q.
f2
In summary, we have a complete set of the equations which describe the circular FEL.
Equations (17), (19) govern the electron motions and Eq.(23) is the self-consistent wave
equation.
C.
Comparison with the Linear FEL
Before analyzing the small signal theory of the circular FEL, we compare the equations
of evolution of the circular FEL with those of the conventional linear FEL. We consider
a linear FEL with the ideal wiggler field
=
-,
sin
i, kz, and with no guiding axial
magnetic field. The velocity dz/dt of the electron is given by
cd = -
(1 + a2 - (aaeC
For the linear FEL, the electron equations of motion are
9
+ c.c.)).
(26)
+k
dk
d
2+
-y
c.c. +
dt
+
( 1 + a2 - 2a,a-e.
P
-itk + k,}c
(c~)e
.
-W,,
+ c.c. ,
(27)
(28)
where k and km, denote the wavenumber of the radiation and the wiggler field, and the
electron phase is defied by ( = (k + k, )z - w~t. The resonance condition for the linear
FEL is
Wo=c(k + kw) 1 -
2-y(1+ a2)}.
(29)
The wave equation of the linear FEL is identical to that of the circular FEL, Eq. (23),
except that the spatial coupling factor f2 is replaced by the corresponding coupling factor
f2L. The factor f2L depends on transverse geometry and the overlap between the electron
beam and the radiation mode. The appropriate coupling factors for the longitudinal space
charge field (fiL) and the electromagnetic mode (f2L) have been discussed elsewhere.10
The wavenumber of the radiation and the wiggler fields for the circular FEL are defined
as
k = p/ro,
(30)
k, = N/ro.
(31)
From Eq. (11) and the definitions of k and k,, we find
(p+ N)-
= c(k + k.) (1 -
10
,
(32)
where y2 > 1 is used. Equation (32) can be used to show that Eqs. (17), (19) and (18) for
the circular FEL are "formally" identical to those Eqs. (27), (28) and (29) for the linear
FEL.
The synchronous phase. C, evolves accordin5 to Eq. (17) for the circular FEL and Eq.
(27) for the linear FEL. The different energy dependence in the Eqs. (17) and (27) results
from using dO/dt (dz/dt) in the definition of C for the circular (linear) FEL. Among the
consequences of this are differences in the growth rates and efficiencies of the FELs.
The vacuum waveguide dispersion relation in a linear FEL is
w2 = c2 k 2 + W2,,
(33)
where k is the axial wavenumber and wet the cutoff frequency. For the coaxial waveguide
of the circular FEL, with (a- b)/a << 1, the vacuum dispersion relation is approximately"
W
(wp)2 + (w2 q2
/4),
-
(34)
where w+ = 2c/(a + b) and w- = 7rc/(a - b). If the electron beam is near the midpoint of
the gap, (ro ~ (a + b)/2), then wop ~ ck, where k = p/ro is the wavenumber. Equation
(34) is then in the same form as Eq. (33), with (w2_q
2
_
/4)1/2
considered the "cutoff
frequency" in the azimuthal direction.
III.
The Low Gain Regime
This section examines the circular FEL in the low gain region. We assume a cold,
tenuous electron beam and therefore neglect the space charge term in Eq. (19).
Each
electron has the same initial energy yo and the different initial phase C. We drop the label
11
j in this section. Since the gain is low, the radiation can be considered constant. Equations
(17) and (19) become
-
-
-(p + N)-,,
dt
d
(35)
I
w, I a,
1
sin C.
(36)
di
Linearizing Eqs. (35) and (36), with 6- =
-
o, and -yo the initial beam energy, yields
= Aw + Qby,
d-y =
di
(37)
Esin (,
(38)
where e = w.la, a,yo, the frequency detuning Aw = wo - w., wo = (p+ N)(w./yo)(1 -
aw,/2yo), and
Q
= -(p + N)(wo/ys)(1 - 3a2 /2'y).
For the linear FEL, a similar analysis of Eqs. (27) and (28) yields equations of the
same form as Eqs. (37) and (38), except that the coefficients are now wo = c(k + k,)[1 (1/2-y)(1 + a2,)], and
Q
= c(k + kw)-y- 3 (1 + a2,.
Using the wavenumber k, and k,. of the circular FEL defined in Sec. Il.C, we find that
e, Aw and wo are identical for the circular FEL and the linear FEL. Only
different:
(circular FEL)
Wo,
Q
Q is
-0oi10( 1 + a2,),
(linear FEL)
(39)
With Aw = 0, the Hamiltonian H for Eqs. (37) and (38) is
H(6y,() = (1/2)Q(6y)
12
2
-
6
cos(,
(40)
where e = w;,a,aw/yo both for the circular and the linear FEL, and
(39). Equation (40) allows to interpret
Q as
Q is
defined in Eq.
the inverse of the mass of a particle trapped
by the potential well. In this analogy, the circular FEL has a "negative mass", the absolute
value of which is "lighter" than that of the linear FEL by a factor y .
The height, 6y,, = 2(e/IQ )1/2, of the ponderomotive potential is given by
(2
(circular FEL)
Ia, a.,
2 -to
The synchrotron frequency 1 = (CIQI)1/
lalaw,
I + a,
2
(linear FEL)
in the ponderomotive potential is
((wo/yo)V$1|al,
(circular FEL)
(wo/)Vaa(1+ a,2,,),
(linear FEL)
(42)
Equations (37) and (38) can be used to solve for the gain in the same way as in the
linear FEL.7 We define the energy gain G as the ratio between the loss of electron energy
and the field energy of the radiation (Eq. (8)). The gain of the circular FEL as a function
of time is
G(t)
a .(t/2) f 2 h(Awt/2),
(43)
where h denotes the spectrum function h(x) = d/d:(sinX/z) 2 , and the spatial coupling
factor
f2
is given in Eq. (24).
The well-known formula for the corresponding gain of the linear FEL in the low gain
region is shown in Table 1. The gain is proportional to
Q, and hence the gain of the circular
FEL is larger than that of the linear FEL by a factor y2 . Also, the spectrum shape is
inverted with respect to Awo = 0. A similar "inverted spectrum shape" is found16 in a
particular parameter region of a linear FEL with a helical wiggler and axial magnetic field.
13
IV.
High Gain Regime
A.
Dispersion Equation
This section examines the high gain regime, in which the radiation field undergoes
exponential growth. The complex presentation, in which a.(t) = Ja,(t)je'('), is used.
Energy spread and space charge are examined in this section.
We begin the linearization of Eqs. (17), (19) and (23) by defining the perturbed quantities 6b-),
b(j, and 6a,:
(i
= Co + Awit +
63
(44)
73 = i'jo + &yj,
a, = ba.,
where the subscript "jO" refers to the initial values of the jth electron. The frequency
detuning for the jth electron is
- a
Aus = (p + N)±
±
-
w.
(45)
The time dependence of the perturbations is given as :
bcj
6
1;exp
[
{i (Cjo + Ao.t + Ft)} + c.c.]
1
[[bj exp
{i ((jo + Awjt +
rt)} + c.c.j,
ba, = 6d, exp (irt),
where -ImP is the growth rate wi of the radiation mode.
14
(46)
Substituting Eqs. (44) and (46) in Eqs. (17), (19) and (23) yields the linearized equations for b(j, 6-y, and 6a,:
2
- (p + N)
-3a
2
i.(o+Awt+rt) + c.c
+ c.c
[(p+ N)4ae,co+Awt+rt)6a,
+
(47)
(i( Aw + r)beo+Aw~t+rt) + c.c.)
iw~aaueiCio+Awi t+rt)6a,
=
2
+
cc
-Y O
2 (wp+ N)we/, o
[( 6 j)ei'(o+Awt+rt)+ c.c
T6,=-f2 O( ei +)
4w, vJ0
'Vjo
An initial distribution that is independent of the electron phase ( was assumed. Using the
k and k, defined in Sec. II.C, Eqs. (47) of the circular FEL become identical to the similar
small signal equations of the linear FEL, except for the underlined coefficient of &bi. This
coefficient is exactly the factor Q defined in Sec. III.
A straight forward algebraic calculation gives the dispersion equation from Eq. (47):
f2w aj
3
(p + N)w
N
04,
-
a. 0
(48)
+
(1 +X )W, {
k
(p + N)we
3#, -
where
15
a" 04
Ck3 -
W,
#3-
a f
a=
((Aw + r)~
vj',
(49)
=((Aw,
+ rr
j;
and -I denotes the electron susceptibility defined by
XI = fixw {1,
The susceptibility X1 is proportional to
Q.
- (3a./2)33.
(50)
Note that the electron xII for the circular
FEL is positive, while the space charge wave in the linear FEL geometry has the negative
susceptibility
X = -f1Lwp/33(1 + a').
(linear FEL)
(51)
This difference is a consequence of the negative mass effect in the circular FEL, where the
electrons are subject to the cyclotron motion slightly perturbed by the wiggler field.
Next we introduce energy spread. The unperturbed distribution function is assumed to
be uniform in the range of
For A-y/-yo
Vv
- yol < Ay, where the energy spread is small, A-y/yo
< 1.
< 1, the frequency detuning Eq. (45) for the jth electron in the circular FEL
is approximately given by
Aw, ~ Awo - wo(Ay, /yo),
where A
=j
(52)
- yo, Awo = wo - w, and wo is defined in Eq. (18). For comparison, the
linear FEL has the frequency detuning
16
Aw, : Awo + wo(Ayj/y3),
(linear FEL)
(53)
where wo is defined by Eq. (29) and Awo = wo - w,. We note that the frequency detuning
Aw3 of the circular FEL is more sensitive-to the energy spread than-that-of the linear FEL
by a factor of -yo.
This results in a greater sensitivity of the growth rate to the energy
spread in the circular FEL. as shown in 4-3.
The full dispersion relation is found using
=
+ 2AwOr + AwO
O" {I
(54)
(woAyyo)2',
-
Together with 0,,, cm appears in Eq. (48). However, the ratio of the two terms is an order
of Awo/w, and we can neglect am.
With these approximations, the cubic dispersion equation of the circular FEL can be
obtained using Eqs. (48) and (54):
r 3+
fA
+ 2a
B.
+
2
2
2D
I~
fw
2)2+
(~a.
WO
tI
\
(3
2+-
2-yO
YO
f
f 2 w woa,
=
5.
4703
(55)
Comparison with the Linear FEL
To understand the dispersion relation of the circular FEL, it is useful to recall the
dispersion equation of the linear FEL with no guiding axial magnetic field 9 '10:
.,
r 3 + 2Awol' 2 +
.
Aw5 -
1
g \2
fLi
(
'
-
.,
+ a,
-
f2Lw woat,
(+ a)
. (56)
The dispersion relations (55) and (56). for the circular FEL and the linear FEL, have
three terms which differ. First, from the factor Q defined in Eq. (39), the coupling terms
17
(the right hand side) differ by -y.
This increases the growth rate and the efficiency of the
circular FEL as we see in Section IV.C. Secondly, the space-charge terms differ by --y'. This
term is exactly the electron susceptibility Eqs. (50) and (51). The positive susceptibility of
the circular FEL causes the negative mass instability. The third difference is in the energyspread term in the linear coefficient of F. The circular FEL is more sensitive to the energy
spread than the linear FEL, as shown in Section IV.D.
The circular FEL is characterized by the coupling between the positive energy electromagnetic wave and the negative energy space-charge wave, which is itself unstable due
to the negative mass effect. On the other hand, the conventional linear FEL couples the
electromagnetic wave with the stable beam space-charge mode. Putting a,,, = 0 in the
equations for the circular FEL, Eqs. (17), (19) and (23), the radiation field terms disappear
and the equations describe only the space-charge wave in the 0-direction. The frequency
F + Awo of these space-charge waves can be obtained by setting a. = 0 in Eqs. (55) and
(56):
1/2 W
±if
F
+ AWO =
1/
P,
10
(circular FEL)
,1I&(linear FEL)
(57)
where a cold beam, Ay = 0, approximation has been made. The stabilization of the negative
mass instability by the waveguide wall or by energy spread is discussed in Appendix A.
Another case in which the FEL instability couples with an unstable space-charge wave
in the beam is in the linear FEL with guide axial magnetic field. The space-charge wave
is driven unstable by the presence of the wiggler and the axial magnetic field in particular
parameter region.1617 As is the case here, the FEL electromagnetic wave couples with the
unstable space-charge wave through the FEL interaction.
18
C.
Cold Beam Limit
The dispersion relation of the circular FEL in the cold beam, strong pump limit is
obtained by neglecting the space charge wave and setting Ay = 0 in Eq. (55):
t+ 2Awr
2
+ zwlr
=
f2
woa/4 0.
(58)
The peak growth rate of the strong pump regime of the circular FEL occurs for Awo = 0:
2 1/3
V3= 25 /gYo (f 2 w woa24 .
(59)
The efficiency 77 is easily obtained by using well-known techniques18 :
'
n'
(f
2
2 w2
1/3
.
(60)
In Table 1 are the growth rates and efficiencies for the circular and linear FEL. In the strong
pump, cold beam parameter region the growth rate and the efficiency of the circular FEL
are seen to be larger than those of the linear FEL by a factor of ^0/
The circular and linear geometries behave quite differently in the weak pump (Raman)
regime. In the linear FEL the weak pump regime begins when
a,,
j
<
Crit,L
=
25 /2 tfAL ) 1/4 /1'2\"2L
1/4(P12,
(61)
(1)
where -yji = yo/(1 + a,)2.
In the circular FEL, however, the susceptibility Eq.
(50) is positive and the term
proportional to wer in Eq. (55) corresponds to an instability. Thus, instead of the FEL
instability, the negative mass instability dominates in the weak pump regime of the circular
FEL. Its growth rate is found, from Eq. (57), to be
19
=
fi
1
2-
.
(negative mass instability)
(62)
7Yo
This result agrees with the well-known growth rate of the negative mass instability 1 ' (see
Appendix A). The condition for the strong pump regime in circular FEL is that the growth
rate Eq. (59) of the FEL is much larger than Eq. (62) of the negative mass instability. This
corresponds to requiring that 1 > 3 crit for the strong pump regime of the circular FEL,
where
-
5/2 ( fS
(3
1/4
(63)
-
Ocrit -
We summarize the above results in Table 1.
In Figs. 2 and 3, the cold beam growth rates of the circular (Eq. (55)) and linear (Eq.
(56)) FEL are solved for numerically. The normalized growth rate w1 /wo as functions of the
normalized detuning parameter Awo/wo, are plotted for the circular FEL in Fig. 2 and the
linear FEL in Fig. 3. For both Figs. 2 and 3, -jo = 5, a,,, = 2, and the plasma frequency
varies. For the sake of simplicity, we assume f, = fiL = 1 for the cases plotted with solid
lines in Figs. 2 and 3.
In the circular FEL, the instability near the peak Awo 2 0 is a largely electromagnetic
FEL interaction which agrees with the analytic expression of the growth rate Eq. (59).
These peak growth rates are larger than those of the linear FEL by about
2-y
. As the
frequency detunes, the character of the instability becomes increasingly electrostatic. This
negative mass instability is very wideband and its asymptotic growth rate is independent
of wiggler field strength (Eq. (62)). As the plasma frequency increases, the negative mass
instability become more dominant because of the increase in Ocrit defined in Eq. (63). To
explicitly show the effect of the negative mass instability, we put
20
f, =
0 for the case plotted
-by the broken line in Fig. 2 (w,/wo = 2 x 10-2). With fi = 0 the bandwidth of the FEL
instability becomes narrow and the peak growth rate decreases.
In Fig. 4, we plot the peak growth rate of both FELs as functions of the pump field
strength j3, for the case of yo = 5, w,/wo = 2 x 10-2, and 5 x 10-3. In the strong pump
regime, the circular FEL has the larger growth rate with the same scaling
/3 as the linear
FEL. However, in the weak pump regime, the negative mass instability dominates and the
growth rate is independent of the pump field strength
D.
#
3
.
Energy Spread
The effect of energy spread is examined by numerically solving the cubic dispersion Eqs.
(55) and (56). The normalized growth rate wi/wo from the dispersion equations (55) and
(56) are shown as functions of the normalized detuning parameter Awo/wo, for the circular
FEL (Fig. 5) and the linear FEL (Fig. 6) with Yo = 5, a,, = 2, w,/wo = 2 x 10-2, and
the varying values of the energy spread Ay /-yo. The growth rate of the circular FEL starts
decreasing at a smaller energy spread than the linear FEL. The negative mass contribution
to the growth rate is effectively suppressed by the energy spread.
Figure 7 shows the normalized growth rate wi/wo as functions of the energy spread
&yf/yo, for circular and linear FELs with yo = 5, a,, = 2, and varying values of the
plasma frequency. The growth rate of the circular FEL starts degrading at the smaller
energy spread than the linear FEL. This critical value of the velocity spread increases with
increasing plasma frequency. As the energy spread begins to substantially reduce the gain
the square distribution function used to derive the cubic dispersion relations (55) and (56)
should be replaced by a more realistic model.
If the electrons have an energy spread Ay, the electron beam has a non-zero beam width
21
Ar. Equation (11) gives the simple relation between Aj and Ar.
Ar = Ar0
'YO
(64)
where 30 n I is assumed. Note Ar and Ay are defined as the half width.
The energy spread not. only affects the gain through the linear term in the cubic dispersion relation, but also through the beam-mode overlap. As will be shown in Sec. V.A,
optimum coupling requires maximization of the spatial overlap factor
f2
by adjusting the
electron beam position ro. For a beam with zero radial width, the optimal coupling occurs
when the electron beam radius is located at the peak of the mode profile. With a beam
of non-zero width, the coupling factor
f2
is reduced. Using the definition of
and Taylor expanding Bessel functions in Ar/ro, we find Af 2 /f
2 a- p
2
f2
(Eq. (24)),
(Ar/ro)2 . Here Af 2
is the reduction in coupling for a beam at radius r = ro + Ar. Equation (64) can then be
used to obtain the desired criterion on the energy spread:
< 1/p.
(65)
This condition implies that TM,, modes with high p (or high frequency) can be coupled
to efficiently by a beam with small energy spread. For example, we need A-y/-yo < 10-2 to
couple effectively to a TMp mode with p = 100.
Another constraint on the energy spread is given by the azimuthal component of the wiggler field. The full wiggler field is given by Eq. (1) and includes an azimuthal component. By
choosing the radii a' and b' of the outer and inner magnets, such that ro = (a'N-IN+1)1/2N,
the azimuthal component vanishes at the electron beam position ro. However, the azimuthal
component exists at radii other than r = ro. This field component, as well as gradients
in the radial field. influence the orbits of electrons with r
22
#
ro. While the detailed orbit
calculations are left for future investigations, a criteria similar to ker, < 1 for the linear
FEL can be derived. Let b.& and fl
be spatial amplitudes of the azimuthal and the radial
component of the wiggler. Using Eq. (1), we find 5 ./B.,. ~ N(Ar/ro). Thus neglecting
the azimuthal field component requires
(66)
-Y <
Yo
N
Since N < p, the spatial coupling criteria Eq. (65) is more difficult to satisfy than the
criteria Eq. (66) for the one dimensional approximation of the wiggler field.
V.
Discussion
A.
Optimization of Beam Position
To optimize the gain of a given mode, it is necessary to maximize the spatial coupling
factor f2 by choice of the electron beam position. As is seen in Eq. (24),
f2
is proportional
to the square of electric field amplitude at the beam radius.
Even if the beam is optimized for coupling to one specific mode, other nearby modes
may also be amplified. Using Eq. (34) for low-aspect-ratio waveguide (a - b)/a < 1, the
azimuthal mode number of the radiation with radial mode number q is given by
=
where o+ = w+ /(wyo
assume that
a2
>
1
o>a.
[N IN2 0,2 -q 2a! (o-
1)1
(67)
),a- = w/(wcy1), w+ = 2c/(a + b), w- = 1c/(a - b) and we
These two solutions correspond to the high and low frequency
branches of the FEL instability. Negative values of p correspond to modes travelling in the
-6 direction.
23
The number of modes which can couple varies according to whether a+ ' 1. It is found
from Eq. (11) that a+ 1 occurs when the electron beam position ro "'o(a+b)/2. If a+ > 1,
then only a finite number of radial modes with (1 < q
p. (q); here qa, = N(o /a_ )(
of radial modes (q
=
- 1)-1/2.
qh)
have two solutions p+(q) and
On the other hand, for a+< 1, an infinite number
1.2,.. .) have two solutions p+(q), p-(q). To avoid possible multi-mode
oscillations the number of possible modes which may oscillate should be reduced. This can
be achieved by setting the electron beam outer half of the gap (ro > '0o(a + b)/2) and
decreasing qh.
In most cases of interest, the radiation field has an azimuthal index p
> 1. Higher-
azimuthal modes of coaxial waveguides are characterized by whispering-gallery modes, in
which the field energy is localized near the outer electrode. The location of the peak of the
field shifts outward as p increases. Figure 8 illustrates this. The solid lines of Figure 8 show
the position r, (the left vertical axis) of the peak amplitude of the electric field of TM,
mode for q = 1 as a function of p, for a coaxial waveguide with a = 6.51cm, b = 5.40cm. The
broken line is the possible oscillating mode number p+, p- for -yo = 5.1, N = 12, and q = I
as a function of the axial magnetic field Bo (the right vertical axis). The axial magnetic
field also determines the electron cyclotron radius ro. The intersection of the solid line (the
position of the peak amplitude of the TMh,
mode) and the broken line (the electron beam
radius) gives the optimum condition (Bo or ro) for the maximum coupling between the field
and the electron beam.
Figure 9 shows the result of optimizing for each radial mode by adjusting the axial
magnetic field.
The frequency (Fig. 9(a)) and the growth rate (Fig. 9(b)) are shown
for -yo = 5.1, Ay = 0, N = 12, a = 6.51cm, b = 5.40cm, w, = 1.78 x 10
1
sec-', and
._ = 6.72kG as functions of the axial magnetic field or the electron-beam radius. The
24
growth rate is calculated from Eq. (59), neglecting the negative mass instability. The solid
and the broken lines are, respectivelythe high and low frequency branches.
The region
ro < 6o(a + b)/2 = 5.84 cm corresponds to a+ < 1 and there are an infinite number of
possible modes with different radial mode numbers q. The higher modes q > 5 are omitted
from Fig. 9 for the sake of simplicity. The region ro > 5.84cm corresponds to 0'
> 1
and there are a finite number of modes with q :; qh. When the electron beam is near the
midpoint of the gap, the high-frequency branch has an extremely high frequency. The high
p values place constraints on beam energy spread which seem difficult to satisfy (see Section
IV.D); therefore the growth rate of such modes is not plotted in Fig. 9(b). These modes
occur for radii between 5.76cm and 5.90cm. While the axial magnetic field varies from 1.35
kG to 1.45kG, each of the modes with q = 1, 2, 3, and 4 has a maximum growth rate at
the optimum point for each mode. These optimum parameters can be obtained by means
of the optimizing scheme in Figure 8. The peak of the higher radial mode corresponds to
the high p number.
B.
Bandwidth
Schuetz and coworkers'
pointed out that the circular FEL be used as a wideband
amplifier. In this section, we evaluate the bandwidth of the circular FEL and find a similar
result.
As the radiation frequency w, departs from the center frequency wo, Awo become finite.
Solving the dispersion equation (55), a typical width Awo of the detuning can be estimated.
The corresponding radiation-frequency width bw, depends on the slopes of the dispersion
curve of the waveguide and the beam mode. Using Eq. (18) and the dispersion relation of
the waveguide, we get the radiation-frequency width as
25
6
ow,
AwWi
0
w
( dw.
-y
dp
(68)
-
where dL,/dp is the derivative of the waveguide frequency with respect top. If (a- b)/a < 1,
differentiation of Eq. (34) yields cdL/dp
(w
1w,
)p. The 6oaxialwaveuide size (orw+),
the axial magnetic field, and the electron energy can be chosen such that dw,/dp : wc/y
This choice means that the two dispersion curves are grazing.
condition is identical to cr
in Sec.
o.
Note that this grazing
= 1 for a coaxial waveguide with small gap, where o+ is defined
V.A. The circular FEL may be very wideband when the dispersion curves are
grazing.
Another factor which determines the bandwidth is the spatial coupling between the
electron beam and the radiation field. As Fig. 8 shows, the peak position of the TM,
mode shifts outward as p increases. The typical radial width of the TM,, mode is roughly
~ ro/p(p
> 1) (see Section IV.D). Assuming that the peak position of the TM,, mode
coincides with the electron beam position ro, modes from TMN to TM,+bp,
may couple
efficiently with the electron beam. The peak position of the TM,+b,q mode is at ro + ro/p.
This 6 p is approximately given by
p :- ro ( d,
(69)
where r, is the location of the radial maximum of the TMN. mode. As an example, we
can evaluate drp/dp of the coaxial waveguide of Fig. 9, using Fig. 8. The corresponding
bandwidth is given by 6w., = (d,/dp)6p.
In summary, two factors to limit the bandwidth: the dispersion and the spatial coupling.
As a typical value, we find bw,/w, ;: 0.3 for the parameters of Fig. 9, with BO = 1.4kG,
26
q = 2. This shows that the circular FEL may work as a very wideband amplifier.
VI.
Conclusions
We have examined 4he linear theory of the circular FEL in a one dimensional model.
This model includes the longitudinal space charge, transverse coaxial waveguide modes, and
energy spread. Linearization of the equations of motion yields a cubic dispersion relation.
We study this cubic dispersion relation for the circular FEL and compare it with that
of the linear FEL. Analytic expressions for the gain in the low gain regime, and the growth
rate in the strong pump and negative mass dominated regimes are derived. These results
are summarized in Table I. The coupling term between the space charge wave and electromagnetic mode is found to be larger, for the circular FEL, by a factor of Y.
The circular
FEL, on the other hand, is more sensitive to energy spread than the linear FEL.
Another new result found here is the influence of the negative mass instability on the
FEL growth rate. Unlike the linear FEL, the circular FEL does not have a weak pump
(Raman) regime. Instead, the FEL growth rate becomes identical to that of the unstable
space charge wave. In the strong pump (Compton) regime, the FEL peak growth rate is
enhanced by the negative mass instability and the bandwidth is increased.
VII.
Acknowlegement
We thank George Bekefi, Ronald Davidson and Han Uhm for useful discussions. This
work was supported in part (H.S.) by the Japan Society for the Promotion of Science and
in part (J. W.) by the Office of Naval Research.
27
Appendix A.
Derivation of f, in Eq. (21)
The negative mass instability has been studied by many authors.'2.' 5 Uhm and Davidson1 4 analyzed the negative mass instability of an intense relativistic electron beam utilizing
a kinetic description which includes the equilibrium self-field, the transverse magnetic perturbations, the inner and outer cylindrical conductor, and the spread in canonical angular
momentum (AP). The growth rate wi is given in Eq. (92) of reference [14] under the
assumption
(Al)
1< t < -n
Ar'
where t is the azimuthal harmonic number and t = p + N for the negative mass instability
in the circular FEL. The condition (Al) is equivalent to the spatial coupling criteria Eq.
(65) in the text. Therefore, (Al) should be satisfied in the circular FEL. The spread in
canonical angular momentum (A) in Ref. [14] can be written as AP = mcro
0 A,
where
A-y is the energy spread A-y in our notation. The growth rate of Eq. (92) in Ref. [14] is
expressed in our notation as
2-
W2 = ig
4ro
2
2
W
-yo
-
,
(A2)
'to
The second term of Eq. (A2) shows the stabilizing influence of beam energy spread. The
first term is geometric and g is defined in Eq. (22) of the text. The quantities b± in Eq.
(22) are the wave admittances at the inner and outer boundaries of the electron beam and
are given by
b+ = -i
.E,
-Es,,,,
28
,(A3)
E,
where ro :t
r are the radii of the inner and outer boundaries of the beam. For coaxial
waveguide, b± are
tc
b
+= ~
b-
c
row.
Yj(k,.a)J(kro) - J'(ka)Yt(krro)
Y('(ka)Jj(krro) - Jt(ka)Yf'(kro)'
Yj(k,b)Jt(krro) - J{(kb)Yt(k,-ro)
Yj(kb)Jt(k,.ro) - J\(kb)Yj'(kro)"
(A5)
(M)
In Fig. 10, we plot (b+ + b )/2 for the parameters of Fig. 9 (a = 6.51cm, b = 5.40cm,
yo = 5.1) and varying electron beam radii. Note that Ic/row, in Eqs. (A5,6) reduces to
t/0o and b+, b_ are functions of I, yo, a/ro and b/rO. The negative mass instability has
a significant impact on the peak value and the spectrum shape of the growth rate in the
circular FEL, as is shown in Sec. IV.C. The growth rate of the negative mass instability
depends on fi and it is stabilized when fi < 0. When the width of the electron beam is
not too large (see Eq. (22)), the wave admittance alone determines the value (and sign)
of fi.
Generally, (b+ + b )/2 oscillates from the negative to the positive value with an
almost constant period of I and gradually tends to -Yo. An inductive impedamce at the
beam corresponds to b+ + b- < 0 and the negative mass instability is then stabilized by
the waveguide. In the case of an inductive impedance, the stabilizing influence of the outer
conducting wall dominates over that of the inner wall. The boundary condition Ee = 0 at
the outer conducting wall may reduce the azimuthal component of the electric field at the
electron beam, which is essential in the negative mass instability. As is shown in Fig. 10,
this stabilization effect is enhanced as the beam is close to the outer conducting wall.15
29
In our analysis in the text the growth rate of the negative mass instability is derived by
putting a,,, = 0 in the dispersion equation (55), yielding
fi
-
"Yo - (..
(A7)
-to
This result is analogous to the result of Eq. (A2) by Uhm and Davidson."
two equations yields
f, as defined
Comparing the
in Eq. (21) of the text. More recently we have examined
the circular FEL utilizing a fluid model and derived
f, from first principles.
In Fig. 9, the growth rate has its peak for q = 2 radial mode, p = 229 at ro = 6.08cm.
We fnd b+ = 6.65, b- = 4.29 (close to yo = 5.1) and
30
f, =
3.94 from Eq. (21).
References
IG. Bekefi, Appl. Phys. Lett., 40, 578 (1982).
2
R.C. Davidson, W.A. McMullin, and K. Tsang, Phys. Fluids, 27, 233 (1984).
3 Y.Z.
4 G.
Yin and G. Bekefi, Phys. Fluids, 28, 1186 (1985).
Bekefi, R.E. Shefer, and W.W. Destler, Appl. Phys. Lett., 44, 280 (1984).
sW.W. Destler, F.M. Aghmir, D.A. Boyd, G. Bekefi, R.E. Shefer, and Y.Z. Yin, Phys.
Fluids, 28, 1962 (1985).
6 N.M.
Kroll, P.L. Morton, and M.N. Rosenbluth, IEEE J. Quantum Elect., QE-17, 1436
(1981).
7 W.B.
Colson, IEEE J. Quantum Elect., QE-17, 1417 (1981).
8D. Prosnitz, A. Szoke, and V. K. Neil, Phys. Rev. A 24, 1436 (1981).
'R. Bonifacio, C. Pelligrini, and L.M. Narducci, Opt. Commun., 50, 373 (1984).
10 T. J. Orzechowski, E. T. Scharlemann, B. Anderson, V. K. Neil, W. M. Fawley, D.
Prosnitz, S. Yarema, D.B. Hopkins, A. C. Paul, A. M. Sessler, and J. S. Wurtele, IEEE
J. Quantum Elect. QE-21, 831 (1985).
"R.A. Waldron, Theory of Guided Electromagnetic Waves, Van Nostrand Reinhold,
London, 1969.
12C.
E. Nielson, A. M. Sessler, and K. R. Symon, in Proceedings of the Internat. Conf. on
Accelemtors, CERN, Geneva, 1959.
' 3 R.J. Briggs and V.K. Neil, Plasma Phys., 9, 209 (1967).
31
"H.S. Uhm and R.C. Davidson. Phys. Fluids, 20, 771 (1977).
"sD. Chernin and Y.Y. Lau, Phys. Fluids, 27, 2319 (1984).
"6 H.P. Freund, P. Sprangle, D. Dillenburg, E.H. da Jornada, R.S. Schneider, and B. Liberman, Phys. Ref. A26, 2004 (1982).
17
H.P. Freund, and P. Sprangle, Phys. Rev. A28, 1835 (1983).
"8 P. Sprangle, R.A. Smith, and V.L. Granatstein, Infrared and Millimeter Waves, vol. 1,
pp. 279-327, Academic Press, New York, 1979.
19 L.S.
Schuetz, E. Ott, and T.M. Antonson, Bull. Am. Phys. Soc., 30, 1549 (1985).
32
Figures
FIG. 1. Configuration of the circular FEL.
FIG. 2. The growth rate ;i /wo of the circular FEL as a function of detuning Awo/wo, for
-yo = 5, a, = 2, A-y/yo
0, f2 = 1, and varying plasma frequency.
f, =
1 for the solid
lines. In broken line, the space charge effect is neglected (fi = 0).
FIG. 3. The growth rate wi /wo of the linear FEL as a function of detuning Awo/wo. Same
parameters as in Fig. 2.
FIG. 4. The growth rate wi/wo as a function of the pump field strength #_. The parameters
are yo = 5, Ay/yo = 0, fi = f2 = 1, wp/wo = 2 x 10-2 and 5 x 10-3.
FIG. 5. The growth rate wi/wo of the circular FEL as a function of detuning Awo/wo, for
yo = 5, wp/wo = 2 x 10-2, a,, = 2,
f, =
1 and varying energy spread.
FIG. 6. The growth rate wi /wo of the linear FEL as a function of detuning Awo/wo. Same
parameters as in Fig. 5.
FIG. 7. The growth rate w/wo as a function of the energy spread. The parameters are
A y/yo, for -yo = 5, a. = 2, f, = f2 = 1 and varying plasma frequency.
FIG. 8. The diagram for the spatial coupling optimization. The solid lines are positions
r, for the peak field of the TMN modes as functions of p for q = 1. The broken lines
are p_, p+, the azimuthal mode numbers, as functions of the axial magnetic field BO or
electron-beam radius ro for N = 12, q = 1, yo = 5.1, a = 6.51cm, b = 5.40cm.
FIG. 9. Example of gain optimization by adjusting axial magnetic field. (a) frequency w/27r,
and (b) growth rate wi as functions of axial magnetic field BO or electron-beam radius ro.
33
The solid and dashed lines correspond to the high frequency and low frequency branches
of the interactionrespectively. Higher modes q > 5 are omitted from the figures. The
parameters are N = 12, yo = 5.1,we
b
1
= 1.78 x 10 10 sec , B. = 6.72kG, a = 6.51cm and
5.40cm.
FIG. 10. The wave admittance (b+ + b-) /2 as a function of azimuthal mode number 1, for
a = 6.51cm, b = 5.40cm, -yo = 5.1, and varying electron beam radii.
34
-S
U
aN
39
C
30
0
N
N
N
N
0
-J
w
3
of
*0
0
-
in
0
0
N
*I
---S
-J
30
.9-S
N
3
CL
N
0
In
-j
N
0'
-J
0
0
C
CL
-S
.2
39
U
o
C
0
b..
0
U
N
0
N
0,3
.4-.
.4-
h.
Nj
0
0
-S
-J
31
0
IC)
In
U
0
0'
N
5to
N
N
*0
C
0
.4-
S
-S
U-
w
IL
M
N
-S
C
0
N
In
I.)
.4--
o
U
0
0
-C
C-)
0.
0
--
N
O.
.c3
cm
N
3
.C
JIA
.
N
-
-
E
z
A
N
A
o
0L
-
0 L
In
cv
CIc
-
V
.cx
0
C"V
0
C
ROTATING
ELECTRON BEAM
CONDUCTING
WALLS
PERMANENT
MAGNETS
Figure 1
xl0-2
I
I
I
I
Circular FEL
ya =
1.5
A
'O
0
=0
1.0
W p
3
-
-=2xl0
I=2x10p-
0.5
I
-
2
1
I
I
0
I
-1.5
x 10-
-1.0
-0.5
5 x10-3
I
0
Aw/w0
Figure 2
I
0.5
1.0
x 101
4
I
,
xlo-3
s
I
Linear FEL
I
=5
YO
aw=2
3
=p
2xIC-2
_0
0
3
3
2
-
xl-3
1
i
-'.5
-1.0
-0.5
Aw /
Figure 3
5x 10
0
w
0.5
1.0
x 10-2
-y =5 , Ay/y=0
Circular FEL
:23=
I -2
-. WMN
iegOTive moss
. Instability
:3
0-3
-
-w
&
I
-3
0
a
aI
ml
-2
I0
5x10in Linear FEL
5
x IC-3
I I aIaaaaI
I
I
I
I
I
IALt
~
-I
l0
Figure 4
1 2 aa
1
IL~
xI-2
Circular FEL
-t.5
wp/we
= 2 x 10-2
LAy
-=0
aw=
0
:3
N
0
1.0
3m
5 x
0.5
10-3
1x 10-2
5 x
10-3
2 x 10-2
0
-1.5
- 1.0
-0.5
0
A w0 1w 0
Figure 5
0.5
1.0 x 10-1
Linear FEL
4
3
:3
WP /wo = 2 x10-2
aw = 2
_
2
-y
=0
2
5 x
I
~
10-2
1x lo0~
0
-1.5
-1.0
-0.5
0
1w 0
Figure 6
0.5
1.0 x 10~2
N
Nl~
I
0
AO
--
b
I
0
I
0
N
-
gms
II I
v
0
Nh
0
.
'1
Lim
.j
w
LLm
,,
r')I
0
0,
I sIIl&LJ
o0
I
0-)
10
Om /
!m
LC)
I
i
d
2
Q
d
I
I
0
I
I
El
I
-o
(pD
LCS
(p
I
I
I
I
i
I
I
I
I
8
Nlo
I
/
I
I
/
2fwo
0O
I
/
/
I
I
0
0(\Jc
/
0
0
-op
-
IT
a
-
-
-
-
I
Q
(p
-
-
-
r
T
(p0
LC)
-6
(a )
400
-
___________________
U.
T,=5.1
N=12
N
T
I
I
I
I
300 -9=
4
(NJ
I
I
I
I
3
3
200
I
/
.:
:4/
3 /I
OOO'
100
ow
w-p
-00--
so* 000
Noo--
Io-
I
-
O
-
-
I
5.8
5.6
I
I
I
I
I
I
5>
I
G2
6.0
I
1.5
I
1.4
Figure 9a
I
I
I
I
-
-
-
-
-------
GA
I
Q
(c m)
Bo
(KG)
(b)
y =5.1
'
9
0
-
N=I 2
0
.
132
q=4
1-0
'C
00 I
2
1'1
0
a)
(I,
:3
Ill\
9
10
\
4
\
\
\
3
I
2
,I
10
b
,
I
5.6
I
,
I
5.8
I
6.0
6.2
1
1.5
-1
1.4
Figure 9b
6.4 c (cm)
B,
(KG)
ro =
5.80cm
10
c\J
0
I
+
+
-D
rr
-10
6.08cm
6.25cm
5.80
cm
-20
_-
0
100
Figure 10
-
- A-
200
300