- Email: [email protected]
Electron beam bunching with nonsinusoidal modulation field
15 November
1997
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
143 (1997) 301-307
Full length article
Electron beam bunching with nonsinusoidal modulation field Michael G. Kong a.‘, Trevor M. Benson b, Christos Christopoulos ” Department h Department
of Electrical of Electrical
Engineering and Electronic
Received
und Electronics, Erqgineering.
Unirersity
Unioersifl
qf Liverpool.
of Nottingham,
LiL’erpool Nottingham
b
L69 3GJ. UK NG7 2RD. UK
17 March 1997; accepted 3 June 1997
Abstract Bunching electrons by means of nonsinusoidal modulation is considered for linear electron beam devices including microwave klystrons and free electron lasers. A simple analytical formulation is developed to study the electron bunching in a nonsinusoidal bunching field for low power devices where space charge effects are not important. It is found that in general the current of the bunched electron beam depends on the derivative of the bunching field rather than the bunching field itself. This finding bears some important new implications for the consequent performance of electron beam devices. The analytical formulation is then verified with a numerical simulation. Space charge effects are also considered as a modification to the new formulation which leads to interesting insights to electron bunching with nonsinusoidal modulation. 0 1997 Elsevier Science B.V.
1. Introduction The mechanism of electron beam bunching is one of the most important aspects of the field-particle interaction in many linear beam devices operating in the spectrum from microwave to visible [l-l 11. For instance, an effective operation of many O-type microwave tubes relies, to a large extent, upon the quality of the electron beam bunching [ l,2]. The principle of prebunching electrons for an improved system performance of these linear beam devices may be best understood in the context of conventional klystron cavities where the bunching process is essentially one-dimensional and thus the current of a bunched electron beam can be formulated analytically [l-3]. As a result, a large variety of complex bunching problems, including those where fringe fields and transit effects are important, can be accurately analysed with a single analytical formulation [12,13]. Although prebunching electrons is a mature technique for conventional microwave tubes, its possible application to more modern electron beam devices is less well under-
’ E-mail: [email protected].
stood and this is a subject of considerable interest [4-l I]. For instance, the principle of prebunching electrons has been recently explored for relativistic microwave tubes [4,5], gyrotrons [6], and free electron lasers [8-l 11. The idea of prebunching electrons over a distance of many radiation wavelengths has also been considered [7-l I]. and this distributed bunching technique should be compared with the conventional localised bunching method which employs microwave cavities [1,2]. Note that with a distributed bunching mechanism, it is now possible to optimise the electron bunching by altering the bunching signal along the whole length of the bunching system. One example is to alter gradually the amplitude of the local magnetic field and/or the wiggler periodicity of a freeelectron laser [14], and such a modification has shown to be capable of producing stronger electromagnetic radiation than what is possible from the usual sinusoidal bunching field [ 15,161. In other words, the performance of a linear beam device may be improved if the bunching field has some appropriate nonsinusoidal waveform. It is therefore important to understand the electron bunching mechanism for a bunching field of nonsinusoidal waveform so that alternative bunching techniques may be derived. In Section 2, a simple analysis of the electron beam
0030-4018/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII s0030-4018(97)00304-0
302
M.G. Kong et al./Optics
bunching is presented I( free electron lasers. In th the resulting formulatio current of the bunched derivative of the bunchit field itself. This finding cations and thus leads to native bunching techniqt oped field-particle code line modelling (TLM) m the new analytical formt gridless buncher cavity. I the numerical calculatior the results of our analyt current. Space charge eff tion 4. Finally in Section are summarised.
2. Analytical
Communications
. both microwave klystrons and absence of space charge effects, suggests that, in general, the electron beam depends on the g field rather than the bunching ears some important new impliinteresting suggestions for alters. In Section 3. a newly devel17,181 based on the transmission thod [19] is employed to verify !ation through an example of a the small bunching signal limit, is in excellent agreement with cal formulation of the electron cts are then considered in Sec5. the conclusions of this study
formulation
Suppose that there is n electric field oscillation in an electron bunching regior This electric field could be generated by means of ex ,iting a conventional microwave buncher cavity [4,9,20], OI the establishment of a ponderomotive field in a distribut :d bunching system [ l4-16,211. If an electron beam wi I an initially uniform density distribution is fed into thr electron bunching region along the direction of the electi c field, it undergoes a velocity modulation. This velocity nodulation then converts into a density modulation after the modulated electron beam travels a sufficiently long distance into a subsequent drift region. Depending upon the actual applications, spacecharge effects can be eithr significant [4,8,22], or negligible [9,14,16,20]. To empt Isize the voltage derivative dependence of the electron beam current however, space charge effects are assumec to be negligibly small and thus our analysis in this sectio applies to low current devices such as compact free elect on lasers [9,20,21]. In the absence of space charge effects. the dynamics of an electron beam may I e approximated by that of a reference electron. This is :overned by ymc’)
&
= qE. u,
(1)
where E is the electric fir Id, q, u, and Y are the charge, the velocity, and the rel; tivistic factor of the electron, respectively. In the one-dii tensional limit, the above equation becomes
dy
dt-
-
L( ml?
;,t)r(
:)
143 (1997) 301-307
/ derived from y = I / \i I - (o/c)‘, dr
ZJ).
- --( d,- - y’nn
dt
y3c d/l
_-- cZ
y’? =-_ dt c’
(4)
As indicated in the above equation, the electron changes its velocity due to the presence of the electric field as well as the change in its kinetic energy (through the change in its y and c). However to the first order of the electric field, the contribution of the electron kinetic energy change is negligible and thus the electron velocity at the exit of the electron bunching region may be approximated as 1’t tr)
= r,,+ -
q
I
(5)
+&.t,)dz. y&n1: 0 -x.
Here y,, is the electron’s initial relativistic factor, and time t, is a reference to represent the time when the electron is in the bunching region. For convenience, we choose our reference time at the instant, to, when the electron arrives at the entrance of the bunching region. If we further introduce
V(r,) = -/+k(:.t,)dz,
(6)
--x
VII= -(y,- l)rnLZ/Y,
(7)
to denote the effective bunching voltage of the bunching field and the kinetic energy in eV of the electron, respectively, Eq. (5) becomes I
W,)
L’(to) = (‘tl I + Yo(Y0 + 1)
[
v,,
t-t,=4
-I
I
F(t())
Yo(Yo+ 1)
vo
(3)
I
(9)
’
which, in the small bunching signal limit of f ==KV,, may be approximated to ”
t-to=-
I
I
l-
1’0
Yo(Y0
@o) + 1)
“0
1
(10)
Suppose now we consider two adjacent electrons in the electron beam which arrive at the entrance of the electron bunching region at toI and t02, respectively. After passing through the bunching region and advancing into the drift region, the time interval between them becomes
f’(to2)- C(to1)
Z/O0
d:
-’
I+
1’0
f2 -
dl
(8) I
To travel a distance ,- from the exit of the electron bunching region into the drift region, an electron requires a time period of
Using the relationship dr
Eq. (2) is reduced to
t1 = (to2 - to,) -
YOCY” +
1)
v,,
. (11)
Consequently from the charge conservation l,,At,, the bunching current becomes !
-=
IO
I
1
I-
Yo(Yo+‘)
dP
--
z/co
-’
I
“0 dto
law /At =
’
where I, and I are the electron beam current at the entrance of the electron bunching region and at the exit of the drift region, respectively. In the case of a conventional klystron, the bunching voltage is of sinusoidal waveform and may be expressed as p(r,) = - ~,coswtO with w being the angular resonant frequency of the klystron cavity. Also the electron beam is usually not relativistic and therefore Y,) = I. Under these conditions, Eq. (12) becomes zw v, - = 1 - 2, -sinwt, I0 ,o v,, i I
-1
1
(13)
and this is identical to the expression in the literature [ 1,2]. Comparison of Eq. (12) with Eq. (13) suggests that in general the current of a bunched electron beam depends on the derivative of the bunching voltage, even though it is usually perceived to be dependent on the bunching voltage itself for the case of a conventional klystron as indicated in Eq. (13). This difference bears many important implications. For instance, Eq. (13) suggests that a long drift distance is required in order to achieve the maximum instantaneous current (when the denominator of Eq. (13) becomes zero) if the bunching signal is small. A long drift distance is not desirable as it leads to a large system size and a significant emittance growth [23]. However the required drift distance can be reduced considerably if a nonsinusoidal bunching signal is employed. The voltage derivative dependence of the electron current in Eq. (12) suggests that the maximum instantaneous current can be alternatively achieved over a shorter drift distance if the bunching signal increases or decreases in time faster than the usual sinusoidal variation. Possible choices of suitable waveform include a square wave, which theoretically has an infinitely large dc/dt at the edges of the bunching voltage and as such it requires a very short drift distance to achieve the maximum instantaneous electron current. Practical realisation of a square wave bunching voltage may be achieved by means of a spatially confined plasma slab [24] as a localised bunching system. For the case of a distributed bunching system, the concept of a step wiggler magnet [25] appears to be a feasible option to construct a practical square wave bunching signal. It is also of interest to note that the current of a bunched electron beam formulated in Eq. ( 13) always has a time dependence. This suggests a discrimination among different electrons in the beam, and its implication is that there will always be some electrons detached from those already tightly bunched. In other words, there will be some electrons wondering between adjacent bunches with a sinu-
soidal bunching signal. Increasing the bunching voltage OI the drift distance cannot sweep these detached rlecrronx into bunches since the time dependence of the electron current is independent of the values of V,, and ;. .As :I result. the electron beam could never be optimiscd into :I series of tight bunches. With a nonsinusoidal hunching voltage, on the other hand, it is possible to remove the time dependence of the electron current by making dc/dt a constant in Eq. (12). For instance, a bunching voltage 01‘ sawtooth waveform leads to a constant d?,/dr and thus results in an electron current that is independent of time. This suggests that electrons in the electron beam are now indistinguishable, indicating that they are now grouped completely into tight bunches. An idealised electron bunching is very desirable since it leads to the maximum electronic efficiency in many linear beam devices. Therefore it is important to explore techniques that may lead to the realisation of a sawtooth bunching signal. Practical implementation of a sawtooth bunching signal may be better achieved in a distributed bunching system whose characteristics can be altered gradually along the direction of electron passage, and one possible solution is to employ a multiharmonic wiggler magnet [ 161. Another interesting deduction from Eq. (12) is that a much longer drift distance is required for relativistic electrons because of the yo(yo + I ) dependence of the electron current in Eq. (12). Therefore to achieve the required electron bunching over a short drift distance, it is advantageous to prebunch electrons at a non-relativistic velocity first and then post-accelerate those already prebunched electrons to the required beam voltage. This technique is useful to minimise the size of linear beam systems [20,26].
3. Numerical verification To validate the voltage derivative dependence of the electron current. a newly developed 2D field-particle code [17.18] is employed to study the bunching process in a rectangular buncher cavity. The code is based on the
k-----b_
Cavity ridge Gap or gridded area
Fig. 1, A rectangular klystron a’ = 3 cm, and b’ = 0.45 cm.
cavity
with a = 4 cm, b = 2 cm,
M. G. Kong et al. / Optics Communications
transmission line modelling method [19], which employs voltage and current pulse ; to simulate the electromagnetic fields in the problem spate. The geometry of the example cavity is illustrated in Fig. I. A two-dimensional mesh of 30 X 80 nodes is used to I eplace the cross sectional area of the cavity with appropriat: boundary conditions. The resonant frequency of the cavity is calculated first in the absence of the electron berm, and it is then used to specify the frequency of a sinustidal signal for the excitation of the cavity. After a stable field oscillation is established in the cavity, a stream of electrons with a nominal accelerating voltage of V,, = 50 kV is launched into the cavity along the axial line of its gap opening from a location far
143
f 1997)
301-307
to the left of the cavity. Space charge effects are removed in the TLM simulation by choosing an electron beam whose plasma frequency is much smaller than the cavity resonant
frequency.
Instantaneous
current
of the electron
beam is recorded at various spatial locations in the drift region by calculating numerically the time interval of two adjacent electrons in the beam, At, and by using the charge conservation law I/I,, = A t,/A t. To emphasize the bunching property, the ac component of the normalised bunching current ia, = I/I, - I is chosen for comparison of the analytical formulae, Eqs. (12) and (I 3) with the TLM simulation. It is of interest to compare the numerical calculation
150 100
1.0 0.5 Bunching Parameter
1.0 0.5 Bunching Parameter Fig. 2. Normal&d numerical (~:ircles) and analytical and (b) a gridless model model.
(solid line) bunching current (I/I,)
with the latter based on (a) a gridded cavity model,
with Eq. (13) first, which essentially assumes a sinusoidal bunching voltage. To this end, the amplitude of the bunching voltage V,,, needs to be calculated. V,,, may be formulated by assuming the electric field distribution to be that at the cavity gap of a gridded cavity. With this idealised cavity field distribution, V, = VP,, = E,,, b’ with E,,, being the peak electric field across the cavity gap. Alternatively, a gridless cavity model may be assumed for the formulation of V,,, and this gives v, = VP, =
+XE,;,,( /-%
z,r,)dz,
(14)
where t, is the instant when the electric field integral reaches its maximum. For a sinusoidal signal, this corresponds to sinwt, = 1 in Eq. (13). Consequently, the analytical ac normalised current of the electron beam is 1 IX = 1 - (zw/2rs,)
(v&)(/v,,)
(15)
’
1 IX = 1 - [( 3JVpo)/~Wo)]
(vp,/vp,)
*
(16)
for the gridded cavity model and the gridless cavity model, respectively. Eqs. (15) and (16) are the instantaneous measurement of the electron beam current, and they are formulated for the electron which arrives at the entrance of the bunching region when sin wt, = 1. Numerically, the condition of sinmt, = 1 is found by choosing two electrons that yield a maximum instantaneous electron current. Fig. 2 shows a comparison of the numerical instantaneous i,, with that calculated from Eq. (13) (through Eqs. (15) and (16)). both plotted against the bunching parameter
(‘7)
For the example studied, VP0= 9.52 V, VP, = 15.90 V, and 0/27r = 2.207 GHz. The TLM simulation and analytical computation using Eq. (13) show a large surge of the electron current occurring when the two electrons catch up each other. However the surge locations differ from each other for both the gridded cavity model and the gridless cavity model. To explain the observed disagreement, we consider the electron transit time. For our example, an electron of 50 kV requires a transit time equal to 6.7% of the cavity oscillation period to pass through the ridged area of 0.45 cm in Fig. 1. This corresponds to a transit time factor of sin( wD/c,,)/( oD/P,) = 0.9895. If this transit time factor is used to correct the analytical curve in Fig. 2a. the latter moves towards larger values of x along the bunching parameter axis, leading to a greater excursion of the analytical curve from the numerical one. On the other hand, the calculation of the transit time for the gridless cavity model requires a knowledge of the axial distance over which the cavity field is finite. For our example, this distance is approximately 2 cm and the corresponding transit time is found to be 33% of the period of the cavity oscillation. However the usual correction using the transit time factor may not be easily applied to gridless cavities since it is formulated by assuming a square wave distribution of the cavity field in the cavity ridge area. Therefore although a correction of Eq. (16) for a finite transit factor is needed, its implementation to a gridless cavity does not appear to be straightforward in the framework of conventional klystron theory. One possible way to account for the finite transit time in gridless cavities is to consider the variation of the cavity field experienced by the reference electron. It is of interest to note that although the bunching signal is sinusoidal, the electric field experienced by the electron during its passage
1.0 0.5 Bunching Parameter Fig. 3. Comparison
of numerical (circles) bunching current with new analytical (solid line) formula.
E(z,t,) = E,,,,, cos cot,, the it the arrival time of the electron at the entrance of the bu ‘thing region. In other words, the electric field experiences by the electron is not exactly sinusoidal and therefore II q. (13) is no longer accurate. For this reason. we needx tc employ Eq. (12) instead of Eq. (13). The field integral of 3q. (6) is computed numerically for two electrons which. at the entrance of the bunching region. experience the l:.rgest difference in the electric field integral. For these I’VOelectrons, the voltage derivative is found to be 2.0597 X IO” V/s and this reduces Eq. (12) to cannot be approxitnatecl by bunching field measured
iac=
1 1-
1.5602(~0/2,‘,)(v,,/v,)
’
(18)
where w/2n= 2.201 Gt[z and VP0= 9.52 V have been used. The above equation is then used to compare with the TLM simulation in Fig. 3 where excellent agreement is observed. The validation of Eq. (I 2) is therefore confirmed.
where the deceleration field, a,, is given by a, = leE,,I/(
of the electron
rim).
(20)
Eq. (20) can be shown to have the following
The above relationship is then used interval of two adjacent electrons entrance of the bunching region at tively. In the small bunching signal after some algebra, t2-1,
1 -M<-
=(fo2-lo,)
where the modification
- = [J( &))( f - tO) - a,( t -- 1”)?/2,
c
7
du(t,)
t’.
dt,
,
(22)
I
factor, M, is given by
(23)
2a,z/u~
retarding field
= (2:leEZol)/(r,‘~,‘m).
Once again, the charge conservation
(‘9)
to consider the time which arrive at the I,, and fo2, respeclimit, this leads to,
h4=&+1). x=
The analytical formula:ion in the preceding sections is based on the assumption I hat space charge effects are not important, and thus its apl lication is limited to low current electron beam devices SW 1as compact free electron lasers [9,16,20]. However for many high power devices, space charge effects are likely t 3 be significant and their treatment usually requires a :areful account of kinetic and potential energy partition cf electrons, the nonlinear propagation characteristics of tt e electron beam, as well as the geometrical details of the bunching system [27]. Such a comprehensive treatment is likely to be device-dependent [27] and thus it does nr t appear to be appropriate to attempt a generalised fcllmulation of the space charge effects on the electron bun :hing by nonsinusoidal modulation. Instead, a simple model of space charge effects is employed to incorporate tile space charge effects into our analytical formulation so 11Iat some insights may be gained into the bunching procesr in the presence of the space charge. Our treatment is II )t self-consistent but it portrays what is conjectured to be llappening in experiments. In the presence of the sl lace charge, electrons approaching their bunch experienct a retarding field, E:,, arising from the Coulomb repulsion forces. For simplicity, we assume this retarding fielcl is constant. Therefore for an electron leaving the bunchi lg region at a speed u(jO) at I”, its arrival time, t, at a distance z into the drift region is now such that
solution
(2’)
with X being the normalised
4. Space charge effects
in the retarding
(24) law is used to give
IO
I=
(25)
1 -M(~/u;)(du(r,)/dt,)
Note that from Eq. (8), ddjo)
-=
dr0
1 df(r,) --
1’0
~o(yo+
1)
Vo
(26)
dro
Therefore Eq. (25) becomes 1
z/l:oMdf(ro) -dr, Yo(Y0 + 1) vo
-’
(27)
I
Fig. 4 plots the modification factor, M, as a function of the normalised retarding field, X. It is clear that the modification factor is always greater than 1 and that as the retarding field increases it becomes larger. This suggests that the presence of the retarding field reduces the drift distance needed for the electrons to get bunched. In other words, the repulsion forces of the electrons’ Coulomb interaction speed up the electron bunching. The physical reason is that the same amount of deceleration results in a larger percentage reduction in velocity for a slow electron than that for a fast electron. Therefore it takes a shorter time for the initially retarded fast electron to catch up the slow electron than that needed when the deceleration is absent. Our assumption of a constant retarding field is of course idealised since the retarding field varies along the electron beam. It is interesting to note that the slow
should mental Raman effects
be useful to conjecture what happens in the experioperation of linear electron beam devices such as free electron lasers [8,26] where the space charge play an important role.
Acknowledgements
0:. 0.0
The authors wish to thank the UK Engineering and Physical Science Research Council (EPSRC) for their financial support of this work under Grant GR/J 13106. 0.2
0.4
0.6
Normalised Retarding Field
0.6
1.0 References
Fig. 4. Modification retarding field, X.
factor,
M, as a function
of the normalised
electron experiences a stronger retarding field for being closer to the electron bunch which exerts the Coulomb repulsion force on it. Therefore its velocity is further reduced by this larger retarding field, resulting in an even shorter time spell before the fast electron catches up. However it should be stressed that this bunching length reduction does not indicate whether or not the electrons are more tightly grouped in the presence of the space charge. In fact, the space charge leads to a sizeable amount debunching even though the electron bunching is more rapidly achieved.
5. Concluding
remarks
Electron beam bunching by means of nonsinusoidal modulation was analysed with a simple formulation of the electron current. It was found that in general the current of the bunched electron beam depends on the derivative of the bunching field rather than the bunching field itself as what has been understood for conventional klystron-like devices. Some important new implications of this finding were discussed, from which possible techniques were identified to either shorten the required drift distance or to group more electrons into their bunches. Possible implementation of these techniques in experiments were considered for both localised bunching systems and distributed bunching systems. These proposed techniques should directly lead to an enhanced system performance of low power linear beam devices. For applications with more powerful beam devices, space charge effects were considered with a simple account of the Coulomb repulsion forces. It was found that the presence of the space charge speeds up the bunching process and thus group electrons into bunches over a shorter drift distance. The resultant modified formulation
of Microwave [ll M. Chodorow, C. Susskind, Fundamentals Electronics (McGraw-Hill. New York, 1964). H.A. Watson, Principles of Electronic Dl J.W. Gewartowski, Tubes (McGraw-Hill, New York, 1965). [31 J.R. Pierce, Traveling-wave tubes (Van Nostrand. New York, 1950). 141 V. Serlin, M. Friedman, IEEE Trans. Plasma Sci. 22 (1994) 692. [51 E. Kuang et al., IEEE Trans. Plasma Sci. 22 (1994) 511. [61 H.W. Mathews et al., IEEE Trans. Plasma Sci. 22 (1994) 861. 171 W.B. Colson, in: Laser Handbook, Vol. VI, Eds. W.B. Colson, C. Pellegrini, A. Renieri (North-Holland, Amsterdam, 1990). [XI J.S. Wurtele et al., Phys. Fluids B 2 (1990) 401. [91 F. Ciocci et al.. Phys. Rev. Lett. 70 (1993) 928. t101 Cl. Dattoli et al., J. Opt. Sot. Am. 10 (1993) 2136. [ill F. Ciocci et al.. IEEE J. Quantum Electron. 31 (1995) 1242. iI21 J.R. Pierce, W.G. Shepherd, Bell Sys. Tech. J. 26 (1947) 663. [131 G.M. Branch Jr, IRE Trans. Electron Devices 8 (1961) 193. [141 N.A. Vinokurov, A.N. Skrinsky, Preprint 77-59 of the Institute of Nuclear Physics, Novosibirsk, 1979. 1151 N.M. Kroll, P.L. Morton, M.N. Rosenbluth, IEEE J. Quantum Electron. 17 (1981) 1436. [I61 M.G. Kong, Phys. Rev. E 52 (1995) 3060. [I71 G. Kong, T.M. Benson, C. Christopoulos, Electron. Lett. 30 (1994) 1657. [I81 M.M. Al-Asadi, T.M. Benson, C. Christopoulos. Int. J. Numer. Model. 9 (1996) 201. Modelling Method: [I91 C. Christopoulos, The Transmission-Line TLM (IEEE-OUP, London, 1995). 1201 G. Dearden et al., Nucl. Instrum. Meth. A 341 (1994) 80. [21] R.A. Stuart, M.G. Kong, Optics Comm. 129 (1996) 284. [22] K. Saito et al., Nucl. Instrum. Meth. A 375 (1996) 237. [23] S. Hendrickson, J.R. Cary, IEEE Trans. Plasma Sci. 24 (1996) 439. 1241 T.C. Katsouleas et al., IEEE Trans. Plasma Sci. 24 (1996) 443. [25] D. Jaroszynski et al., Phys. Rev. Lett. 74 (1995) 2224. [26] M. Cohen et al., Phys. Rev. Len. 74 (1995) 3812. [27] M. Friedman et al., J. Appl. Phys. 64 (1988) 3353.
1997
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
143 (1997) 301-307
Full length article
Electron beam bunching with nonsinusoidal modulation field Michael G. Kong a.‘, Trevor M. Benson b, Christos Christopoulos ” Department h Department
of Electrical of Electrical
Engineering and Electronic
Received
und Electronics, Erqgineering.
Unirersity
Unioersifl
qf Liverpool.
of Nottingham,
LiL’erpool Nottingham
b
L69 3GJ. UK NG7 2RD. UK
17 March 1997; accepted 3 June 1997
Abstract Bunching electrons by means of nonsinusoidal modulation is considered for linear electron beam devices including microwave klystrons and free electron lasers. A simple analytical formulation is developed to study the electron bunching in a nonsinusoidal bunching field for low power devices where space charge effects are not important. It is found that in general the current of the bunched electron beam depends on the derivative of the bunching field rather than the bunching field itself. This finding bears some important new implications for the consequent performance of electron beam devices. The analytical formulation is then verified with a numerical simulation. Space charge effects are also considered as a modification to the new formulation which leads to interesting insights to electron bunching with nonsinusoidal modulation. 0 1997 Elsevier Science B.V.
1. Introduction The mechanism of electron beam bunching is one of the most important aspects of the field-particle interaction in many linear beam devices operating in the spectrum from microwave to visible [l-l 11. For instance, an effective operation of many O-type microwave tubes relies, to a large extent, upon the quality of the electron beam bunching [ l,2]. The principle of prebunching electrons for an improved system performance of these linear beam devices may be best understood in the context of conventional klystron cavities where the bunching process is essentially one-dimensional and thus the current of a bunched electron beam can be formulated analytically [l-3]. As a result, a large variety of complex bunching problems, including those where fringe fields and transit effects are important, can be accurately analysed with a single analytical formulation [12,13]. Although prebunching electrons is a mature technique for conventional microwave tubes, its possible application to more modern electron beam devices is less well under-
’ E-mail: [email protected].
stood and this is a subject of considerable interest [4-l I]. For instance, the principle of prebunching electrons has been recently explored for relativistic microwave tubes [4,5], gyrotrons [6], and free electron lasers [8-l 11. The idea of prebunching electrons over a distance of many radiation wavelengths has also been considered [7-l I]. and this distributed bunching technique should be compared with the conventional localised bunching method which employs microwave cavities [1,2]. Note that with a distributed bunching mechanism, it is now possible to optimise the electron bunching by altering the bunching signal along the whole length of the bunching system. One example is to alter gradually the amplitude of the local magnetic field and/or the wiggler periodicity of a freeelectron laser [14], and such a modification has shown to be capable of producing stronger electromagnetic radiation than what is possible from the usual sinusoidal bunching field [ 15,161. In other words, the performance of a linear beam device may be improved if the bunching field has some appropriate nonsinusoidal waveform. It is therefore important to understand the electron bunching mechanism for a bunching field of nonsinusoidal waveform so that alternative bunching techniques may be derived. In Section 2, a simple analysis of the electron beam
0030-4018/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII s0030-4018(97)00304-0
302
M.G. Kong et al./Optics
bunching is presented I( free electron lasers. In th the resulting formulatio current of the bunched derivative of the bunchit field itself. This finding cations and thus leads to native bunching techniqt oped field-particle code line modelling (TLM) m the new analytical formt gridless buncher cavity. I the numerical calculatior the results of our analyt current. Space charge eff tion 4. Finally in Section are summarised.
2. Analytical
Communications
. both microwave klystrons and absence of space charge effects, suggests that, in general, the electron beam depends on the g field rather than the bunching ears some important new impliinteresting suggestions for alters. In Section 3. a newly devel17,181 based on the transmission thod [19] is employed to verify !ation through an example of a the small bunching signal limit, is in excellent agreement with cal formulation of the electron cts are then considered in Sec5. the conclusions of this study
formulation
Suppose that there is n electric field oscillation in an electron bunching regior This electric field could be generated by means of ex ,iting a conventional microwave buncher cavity [4,9,20], OI the establishment of a ponderomotive field in a distribut :d bunching system [ l4-16,211. If an electron beam wi I an initially uniform density distribution is fed into thr electron bunching region along the direction of the electi c field, it undergoes a velocity modulation. This velocity nodulation then converts into a density modulation after the modulated electron beam travels a sufficiently long distance into a subsequent drift region. Depending upon the actual applications, spacecharge effects can be eithr significant [4,8,22], or negligible [9,14,16,20]. To empt Isize the voltage derivative dependence of the electron beam current however, space charge effects are assumec to be negligibly small and thus our analysis in this sectio applies to low current devices such as compact free elect on lasers [9,20,21]. In the absence of space charge effects. the dynamics of an electron beam may I e approximated by that of a reference electron. This is :overned by ymc’)
&
= qE. u,
(1)
where E is the electric fir Id, q, u, and Y are the charge, the velocity, and the rel; tivistic factor of the electron, respectively. In the one-dii tensional limit, the above equation becomes
dy
dt-
-
L( ml?
;,t)r(
:)
143 (1997) 301-307
/ derived from y = I / \i I - (o/c)‘, dr
ZJ).
- --( d,- - y’nn
dt
y3c d/l
_-- cZ
y’? =-_ dt c’
(4)
As indicated in the above equation, the electron changes its velocity due to the presence of the electric field as well as the change in its kinetic energy (through the change in its y and c). However to the first order of the electric field, the contribution of the electron kinetic energy change is negligible and thus the electron velocity at the exit of the electron bunching region may be approximated as 1’t tr)
= r,,+ -
q
I
(5)
+&.t,)dz. y&n1: 0 -x.
Here y,, is the electron’s initial relativistic factor, and time t, is a reference to represent the time when the electron is in the bunching region. For convenience, we choose our reference time at the instant, to, when the electron arrives at the entrance of the bunching region. If we further introduce
V(r,) = -/+k(:.t,)dz,
(6)
--x
VII= -(y,- l)rnLZ/Y,
(7)
to denote the effective bunching voltage of the bunching field and the kinetic energy in eV of the electron, respectively, Eq. (5) becomes I
W,)
L’(to) = (‘tl I + Yo(Y0 + 1)
[
v,,
t-t,=4
-I
I
F(t())
Yo(Yo+ 1)
vo
(3)
I
(9)
’
which, in the small bunching signal limit of f ==KV,, may be approximated to ”
t-to=-
I
I
l-
1’0
Yo(Y0
@o) + 1)
“0
1
(10)
Suppose now we consider two adjacent electrons in the electron beam which arrive at the entrance of the electron bunching region at toI and t02, respectively. After passing through the bunching region and advancing into the drift region, the time interval between them becomes
f’(to2)- C(to1)
Z/O0
d:
-’
I+
1’0
f2 -
dl
(8) I
To travel a distance ,- from the exit of the electron bunching region into the drift region, an electron requires a time period of
Using the relationship dr
Eq. (2) is reduced to
t1 = (to2 - to,) -
YOCY” +
1)
v,,
. (11)
Consequently from the charge conservation l,,At,, the bunching current becomes !
-=
IO
I
1
I-
Yo(Yo+‘)
dP
--
z/co
-’
I
“0 dto
law /At =
’
where I, and I are the electron beam current at the entrance of the electron bunching region and at the exit of the drift region, respectively. In the case of a conventional klystron, the bunching voltage is of sinusoidal waveform and may be expressed as p(r,) = - ~,coswtO with w being the angular resonant frequency of the klystron cavity. Also the electron beam is usually not relativistic and therefore Y,) = I. Under these conditions, Eq. (12) becomes zw v, - = 1 - 2, -sinwt, I0 ,o v,, i I
-1
1
(13)
and this is identical to the expression in the literature [ 1,2]. Comparison of Eq. (12) with Eq. (13) suggests that in general the current of a bunched electron beam depends on the derivative of the bunching voltage, even though it is usually perceived to be dependent on the bunching voltage itself for the case of a conventional klystron as indicated in Eq. (13). This difference bears many important implications. For instance, Eq. (13) suggests that a long drift distance is required in order to achieve the maximum instantaneous current (when the denominator of Eq. (13) becomes zero) if the bunching signal is small. A long drift distance is not desirable as it leads to a large system size and a significant emittance growth [23]. However the required drift distance can be reduced considerably if a nonsinusoidal bunching signal is employed. The voltage derivative dependence of the electron current in Eq. (12) suggests that the maximum instantaneous current can be alternatively achieved over a shorter drift distance if the bunching signal increases or decreases in time faster than the usual sinusoidal variation. Possible choices of suitable waveform include a square wave, which theoretically has an infinitely large dc/dt at the edges of the bunching voltage and as such it requires a very short drift distance to achieve the maximum instantaneous electron current. Practical realisation of a square wave bunching voltage may be achieved by means of a spatially confined plasma slab [24] as a localised bunching system. For the case of a distributed bunching system, the concept of a step wiggler magnet [25] appears to be a feasible option to construct a practical square wave bunching signal. It is also of interest to note that the current of a bunched electron beam formulated in Eq. ( 13) always has a time dependence. This suggests a discrimination among different electrons in the beam, and its implication is that there will always be some electrons detached from those already tightly bunched. In other words, there will be some electrons wondering between adjacent bunches with a sinu-
soidal bunching signal. Increasing the bunching voltage OI the drift distance cannot sweep these detached rlecrronx into bunches since the time dependence of the electron current is independent of the values of V,, and ;. .As :I result. the electron beam could never be optimiscd into :I series of tight bunches. With a nonsinusoidal hunching voltage, on the other hand, it is possible to remove the time dependence of the electron current by making dc/dt a constant in Eq. (12). For instance, a bunching voltage 01‘ sawtooth waveform leads to a constant d?,/dr and thus results in an electron current that is independent of time. This suggests that electrons in the electron beam are now indistinguishable, indicating that they are now grouped completely into tight bunches. An idealised electron bunching is very desirable since it leads to the maximum electronic efficiency in many linear beam devices. Therefore it is important to explore techniques that may lead to the realisation of a sawtooth bunching signal. Practical implementation of a sawtooth bunching signal may be better achieved in a distributed bunching system whose characteristics can be altered gradually along the direction of electron passage, and one possible solution is to employ a multiharmonic wiggler magnet [ 161. Another interesting deduction from Eq. (12) is that a much longer drift distance is required for relativistic electrons because of the yo(yo + I ) dependence of the electron current in Eq. (12). Therefore to achieve the required electron bunching over a short drift distance, it is advantageous to prebunch electrons at a non-relativistic velocity first and then post-accelerate those already prebunched electrons to the required beam voltage. This technique is useful to minimise the size of linear beam systems [20,26].
3. Numerical verification To validate the voltage derivative dependence of the electron current. a newly developed 2D field-particle code [17.18] is employed to study the bunching process in a rectangular buncher cavity. The code is based on the
k-----b_
Cavity ridge Gap or gridded area
Fig. 1, A rectangular klystron a’ = 3 cm, and b’ = 0.45 cm.
cavity
with a = 4 cm, b = 2 cm,
M. G. Kong et al. / Optics Communications
transmission line modelling method [19], which employs voltage and current pulse ; to simulate the electromagnetic fields in the problem spate. The geometry of the example cavity is illustrated in Fig. I. A two-dimensional mesh of 30 X 80 nodes is used to I eplace the cross sectional area of the cavity with appropriat: boundary conditions. The resonant frequency of the cavity is calculated first in the absence of the electron berm, and it is then used to specify the frequency of a sinustidal signal for the excitation of the cavity. After a stable field oscillation is established in the cavity, a stream of electrons with a nominal accelerating voltage of V,, = 50 kV is launched into the cavity along the axial line of its gap opening from a location far
143
f 1997)
301-307
to the left of the cavity. Space charge effects are removed in the TLM simulation by choosing an electron beam whose plasma frequency is much smaller than the cavity resonant
frequency.
Instantaneous
current
of the electron
beam is recorded at various spatial locations in the drift region by calculating numerically the time interval of two adjacent electrons in the beam, At, and by using the charge conservation law I/I,, = A t,/A t. To emphasize the bunching property, the ac component of the normalised bunching current ia, = I/I, - I is chosen for comparison of the analytical formulae, Eqs. (12) and (I 3) with the TLM simulation. It is of interest to compare the numerical calculation
150 100
1.0 0.5 Bunching Parameter
1.0 0.5 Bunching Parameter Fig. 2. Normal&d numerical (~:ircles) and analytical and (b) a gridless model model.
(solid line) bunching current (I/I,)
with the latter based on (a) a gridded cavity model,
with Eq. (13) first, which essentially assumes a sinusoidal bunching voltage. To this end, the amplitude of the bunching voltage V,,, needs to be calculated. V,,, may be formulated by assuming the electric field distribution to be that at the cavity gap of a gridded cavity. With this idealised cavity field distribution, V, = VP,, = E,,, b’ with E,,, being the peak electric field across the cavity gap. Alternatively, a gridless cavity model may be assumed for the formulation of V,,, and this gives v, = VP, =
+XE,;,,( /-%
z,r,)dz,
(14)
where t, is the instant when the electric field integral reaches its maximum. For a sinusoidal signal, this corresponds to sinwt, = 1 in Eq. (13). Consequently, the analytical ac normalised current of the electron beam is 1 IX = 1 - (zw/2rs,)
(v&)(/v,,)
(15)
’
1 IX = 1 - [( 3JVpo)/~Wo)]
(vp,/vp,)
*
(16)
for the gridded cavity model and the gridless cavity model, respectively. Eqs. (15) and (16) are the instantaneous measurement of the electron beam current, and they are formulated for the electron which arrives at the entrance of the bunching region when sin wt, = 1. Numerically, the condition of sinmt, = 1 is found by choosing two electrons that yield a maximum instantaneous electron current. Fig. 2 shows a comparison of the numerical instantaneous i,, with that calculated from Eq. (13) (through Eqs. (15) and (16)). both plotted against the bunching parameter
(‘7)
For the example studied, VP0= 9.52 V, VP, = 15.90 V, and 0/27r = 2.207 GHz. The TLM simulation and analytical computation using Eq. (13) show a large surge of the electron current occurring when the two electrons catch up each other. However the surge locations differ from each other for both the gridded cavity model and the gridless cavity model. To explain the observed disagreement, we consider the electron transit time. For our example, an electron of 50 kV requires a transit time equal to 6.7% of the cavity oscillation period to pass through the ridged area of 0.45 cm in Fig. 1. This corresponds to a transit time factor of sin( wD/c,,)/( oD/P,) = 0.9895. If this transit time factor is used to correct the analytical curve in Fig. 2a. the latter moves towards larger values of x along the bunching parameter axis, leading to a greater excursion of the analytical curve from the numerical one. On the other hand, the calculation of the transit time for the gridless cavity model requires a knowledge of the axial distance over which the cavity field is finite. For our example, this distance is approximately 2 cm and the corresponding transit time is found to be 33% of the period of the cavity oscillation. However the usual correction using the transit time factor may not be easily applied to gridless cavities since it is formulated by assuming a square wave distribution of the cavity field in the cavity ridge area. Therefore although a correction of Eq. (16) for a finite transit factor is needed, its implementation to a gridless cavity does not appear to be straightforward in the framework of conventional klystron theory. One possible way to account for the finite transit time in gridless cavities is to consider the variation of the cavity field experienced by the reference electron. It is of interest to note that although the bunching signal is sinusoidal, the electric field experienced by the electron during its passage
1.0 0.5 Bunching Parameter Fig. 3. Comparison
of numerical (circles) bunching current with new analytical (solid line) formula.
E(z,t,) = E,,,,, cos cot,, the it the arrival time of the electron at the entrance of the bu ‘thing region. In other words, the electric field experiences by the electron is not exactly sinusoidal and therefore II q. (13) is no longer accurate. For this reason. we needx tc employ Eq. (12) instead of Eq. (13). The field integral of 3q. (6) is computed numerically for two electrons which. at the entrance of the bunching region. experience the l:.rgest difference in the electric field integral. For these I’VOelectrons, the voltage derivative is found to be 2.0597 X IO” V/s and this reduces Eq. (12) to cannot be approxitnatecl by bunching field measured
iac=
1 1-
1.5602(~0/2,‘,)(v,,/v,)
’
(18)
where w/2n= 2.201 Gt[z and VP0= 9.52 V have been used. The above equation is then used to compare with the TLM simulation in Fig. 3 where excellent agreement is observed. The validation of Eq. (I 2) is therefore confirmed.
where the deceleration field, a,, is given by a, = leE,,I/(
of the electron
rim).
(20)
Eq. (20) can be shown to have the following
The above relationship is then used interval of two adjacent electrons entrance of the bunching region at tively. In the small bunching signal after some algebra, t2-1,
1 -M<-
=(fo2-lo,)
where the modification
- = [J( &))( f - tO) - a,( t -- 1”)?/2,
c
7
du(t,)
t’.
dt,
,
(22)
I
factor, M, is given by
(23)
2a,z/u~
retarding field
= (2:leEZol)/(r,‘~,‘m).
Once again, the charge conservation
(‘9)
to consider the time which arrive at the I,, and fo2, respeclimit, this leads to,
h4=&+1). x=
The analytical formula:ion in the preceding sections is based on the assumption I hat space charge effects are not important, and thus its apl lication is limited to low current electron beam devices SW 1as compact free electron lasers [9,16,20]. However for many high power devices, space charge effects are likely t 3 be significant and their treatment usually requires a :areful account of kinetic and potential energy partition cf electrons, the nonlinear propagation characteristics of tt e electron beam, as well as the geometrical details of the bunching system [27]. Such a comprehensive treatment is likely to be device-dependent [27] and thus it does nr t appear to be appropriate to attempt a generalised fcllmulation of the space charge effects on the electron bun :hing by nonsinusoidal modulation. Instead, a simple model of space charge effects is employed to incorporate tile space charge effects into our analytical formulation so 11Iat some insights may be gained into the bunching procesr in the presence of the space charge. Our treatment is II )t self-consistent but it portrays what is conjectured to be llappening in experiments. In the presence of the sl lace charge, electrons approaching their bunch experienct a retarding field, E:,, arising from the Coulomb repulsion forces. For simplicity, we assume this retarding fielcl is constant. Therefore for an electron leaving the bunchi lg region at a speed u(jO) at I”, its arrival time, t, at a distance z into the drift region is now such that
solution
(2’)
with X being the normalised
4. Space charge effects
in the retarding
(24) law is used to give
IO
I=
(25)
1 -M(~/u;)(du(r,)/dt,)
Note that from Eq. (8), ddjo)
-=
dr0
1 df(r,) --
1’0
~o(yo+
1)
Vo
(26)
dro
Therefore Eq. (25) becomes 1
z/l:oMdf(ro) -dr, Yo(Y0 + 1) vo
-’
(27)
I
Fig. 4 plots the modification factor, M, as a function of the normalised retarding field, X. It is clear that the modification factor is always greater than 1 and that as the retarding field increases it becomes larger. This suggests that the presence of the retarding field reduces the drift distance needed for the electrons to get bunched. In other words, the repulsion forces of the electrons’ Coulomb interaction speed up the electron bunching. The physical reason is that the same amount of deceleration results in a larger percentage reduction in velocity for a slow electron than that for a fast electron. Therefore it takes a shorter time for the initially retarded fast electron to catch up the slow electron than that needed when the deceleration is absent. Our assumption of a constant retarding field is of course idealised since the retarding field varies along the electron beam. It is interesting to note that the slow
should mental Raman effects
be useful to conjecture what happens in the experioperation of linear electron beam devices such as free electron lasers [8,26] where the space charge play an important role.
Acknowledgements
0:. 0.0
The authors wish to thank the UK Engineering and Physical Science Research Council (EPSRC) for their financial support of this work under Grant GR/J 13106. 0.2
0.4
0.6
Normalised Retarding Field
0.6
1.0 References
Fig. 4. Modification retarding field, X.
factor,
M, as a function
of the normalised
electron experiences a stronger retarding field for being closer to the electron bunch which exerts the Coulomb repulsion force on it. Therefore its velocity is further reduced by this larger retarding field, resulting in an even shorter time spell before the fast electron catches up. However it should be stressed that this bunching length reduction does not indicate whether or not the electrons are more tightly grouped in the presence of the space charge. In fact, the space charge leads to a sizeable amount debunching even though the electron bunching is more rapidly achieved.
5. Concluding
remarks
Electron beam bunching by means of nonsinusoidal modulation was analysed with a simple formulation of the electron current. It was found that in general the current of the bunched electron beam depends on the derivative of the bunching field rather than the bunching field itself as what has been understood for conventional klystron-like devices. Some important new implications of this finding were discussed, from which possible techniques were identified to either shorten the required drift distance or to group more electrons into their bunches. Possible implementation of these techniques in experiments were considered for both localised bunching systems and distributed bunching systems. These proposed techniques should directly lead to an enhanced system performance of low power linear beam devices. For applications with more powerful beam devices, space charge effects were considered with a simple account of the Coulomb repulsion forces. It was found that the presence of the space charge speeds up the bunching process and thus group electrons into bunches over a shorter drift distance. The resultant modified formulation
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