- Email: [email protected]
Pulse-shaping in short-pulse FEL oscillators using multiple resonators
Nuclear Instruments and Methods in Physics Research A 393 (1997) 237-241
NUCLEAR INSTRUMENTS & METHODS IN PHVSBCS RESEARCH Section A
ELSEVIER
Pulse-shaping in short-pulse FEL oscillators using multiple resonators G. Shvet?*,
J.S. Wurteleb
‘Princeton Plasma Phvsics Laboratoty Princeton, NJ 08543. USA b Department ?f Physics, University qf California, Berkeley. CA 94720. USA
Abstract A novel method is proposed for the efficient production of optical pulses that are shorter concept uses short electron bunches and a multiple cavity FEL oscillator configuration.
1. Introduction Most free-electron laser user facilities operate as low gain oscillators and the majority are driven by radiofrequency (rf) linacs. The electron beam consists of very short (picosecond or sub-picosecond) electron bunches, spaced by a multiple of the rf wavelength. The radiation pulse length is determined by the bunch length, L,,, if 3 CCL,, where A =N,,,,I is the slippage length, i, is the radiation wavelength, and N, is the number of wiggler periods. For electron bunches much shorter than the slippage distance, the pulse length is (roughly) determined by the slippage length, A. To make more precise statement about the duration and structure of radiation pulses in a short-bunch oscillators requires the inclusion of the cavity desynchronism and losses. The supermode theory of small-gain FEL oscillators in the short bunch limit was recently developed [l]. Many of the analytical conclusions of Ref. [l] parallel those of the pioneering numerical work of Dattoli et al. [Z] on the linear radiation supermodes in oscillators driven by electron bunches with an arbitrary temporal profile. One of the useful aspects of the formalism presented in Ref. [I] is that the equations describing the temporal pulse evolution of a short-bunch oscillator are very similar to those describing backward wave oscillators (BWO) (see, for example, Ref. [3]). This implies that boundary conditions are specified at the heat and the tail of the radiation pulse
*Corresponding
author. Tel.: + 1 609 243 2609: fax: + 1 609
233 2662; e-mail: [email protected]. 016%9092/97/$17.00 Copyright PII SOI 68-9002(97)00483-X
than a slippage length. The
and is essential to our analysis of the multi-cavity pulseshaping concept described below. Our previous work showed [l] that an FEL oscillator driven by very short electron bunches (shorter than the slippage length) can generate radiation pulses shorter than the slippage length. These very short radiation pulses, predicted by the linear theory of Ref. [ 11. can only be obtained in the operating regime where cavity desynchronism is very small. Unfortunately, it is precisely in this regime that nonlinear multi-mode effects, such as limit-cycle saturation and transition to chaos, dominate the FEL dynamics. This results in nonrepetetiveness of radiation pulses from pass to pass. Here we propose a novel scheme which circumvents this limitation and enables the steady-state generation of ultra-short radiation pulses with high efficiency (i.e., larger than the typical value of +N,). Our scheme avoids the complications of operating the oscillator with many unstable supermodes, yet achieves efficiences characteristic of that regime. The key idea is to split an FEL oscillator into three sections: a prebuncher, a drift space. and a pulse shaper. The latter section is where most of the radiation is produced. Both the prebuncher and the pulse shaper are operated as oscillators with high-Q optical cavities. The main difference between these two sections is that the prebuncher operates with a relatively large cavity desynchronism, which ensures that (i) there is only one linear[~ unstable radiation supermode and (ii) this supermode saturates quickly and at low intensity. The pulse shaper is operated in an ouerdamped mode, i.e. there are no linearly unstable radiation supermodes in the pulse-shaping cavity. Instead, the pulse shaper is driven by a bunched electron beam.
1(1#1997 Elsevier Science B.V. All rights
reserved IV. LONG-WAVELENGTH
FELs
238
G. Shvets, J.S. Wurtele /Nucl. Instr. and Meth. in Phys. Rex A 393 (i99.7) 237-241
and
2. Theoretical model and basic formalism
Our model is one-dimensional, low gain and assumes an electron bunch length much less than a slippage length. The evolution of laser pulser is then controlled [l] by three parameters: the reduced Colson parameter, r=jcL,/A, the cavity desynchronism SY, and the cavity losses LYE, where
(1) Moreover, by an appropriate normalization of “time” (a continuous equivalent of pass number), cavity losses, and desynchronism
is was shown that the evolution of an FEL oscillator is governed by only the normalized cavity desynchronism, v, and losses u. The independent variables are t and 5 =(ct--)/A, the normalized distance from the head of the radiation pulse (which is assumed to move with the speed of light, c). The field is described by a slowly varying complex amplitude, A, normalized so that IAl2=4nN,j,P(z, t)/P,, where P(z, t) is the intra-cavity optical power and PC= mc2y,(l,/e) is the electron beam power. The FEL oscillator can be modelled with a field equation for the radiation and a Vlasov equation for electrons:
e-‘~-Qli/2”’
f,(P, 5) =
(P - Q?
,ieBl
+
Bl
In Eqs. (5) and (6) B, = (e-“), P, =(pe-“‘), Q=(p), and C? = ((p - Q)2) are t-dependent moments of the particle distribution function f: bunching, momentum bunching, average momentum, and momentum spread, respectively. Since fresh electrons are entering the oscillator on each pass, distribution function f depends on normalized pass number t parametrically through optical field. In a storage ring, where the memory of a FEL interaction is preserved in the electron distribution from pass to pass, Eq. (3) would be modified. The particular choice of the distribution function in Eqs. (5) and (6) corresponds to a Gaussian closure. This truncated description of the electrons by four global moments was shown [1,4], to accurately describe such complex FEL phenomena as nonlinear saturation of FEL amplifiers, synchrotron oscillations, nonlinear superradiance, limit cycles in FEL oscillators, etc. The explicit expression for the distribution functionfi, presented here for the first time, will be later used to describe the effect of the drift section. The output distribution function at the end of the prebuncher is evolved through the drift section to obtain the input distribution function at the entrance of the pulse-shaper, which determines the output of the novel FEL configuration we propose in this paper. Substituting Eqs. (4H6) into Eqs. (2) and (3) yields the set of closed equations [l]: aA aA CY --v~+~A=~B,,
aT
as
-aB, = _iP 13 al
ah
= -A - iSBl - 2iQPi + 2iQ2B,, a<
af+ pae af @
af
= (Ae” + c.c.)--,
aQ = - [ABT + CC.]) at
aP
-
where q(l)= 1 if 0 < i < 1 and q(c)=0 otherwise. The variables p=&8 and B are canonically conjugate. This description can be recast into a form similar to that of Ref. [il. As was first demonstrated in Ref. [4], the essential FEL physics can be captured by assuming a particular form of the electron distribution function:
fb, 0,5) =fo(p, 5) + e”fl(p, 5) + c.c.,
(4)
where
(5)
as
C = -2[APT
+ CL],
(9)
(10)
(11)
where S = C? + Q2 = (p’). Eqs. (f+(l 1) are integrated between 5 =0 and 5 = 1. Physically, 5 =0 corresponds to the radiation slice which overlaps the electron bunch at the entrance into the wiggler while 5 = 1 corresponds to the radiation slice which overlaps the electron bunch at the exit of the wiggler. Radiation which has already slipped ahead of the electron bunch due to finite cavity desynchronism (< =O) decays exponentially as exp(&/2u). The boundary condition for the radiation amplitude A is set at < = 1 (one slippage length behind the pulse),
G. Shvets. J.S. Wurtele/Nucl.
Instr. and Meth. in Phw. Rrs. A 393 (1997) 237-241
and will be assumed vanishing in this paper. Boundary conditions for particle variables are fixed at the wiggler entrance, i.e. at X=0. Setting boundary conditions at different boundaries for particles and fields is reminiscent of backward wave oscillators [3], where particles are initialized at the entrance into the interaction region while the microwave fields (which have group velocity in the opposite direction of their phase velocity due to the slow-wave structures) are initialized at the exit of the interaction region. Initializing particles and fields at different locations is intuitively clear for a backward wave oscillator. but is less intuitive for an FEL oscillators, where the “interaction window” O<< < 1 itself moves with the speed of light, and is equal to one slippage length. Eqs. (7)-(11) can be linearized by assuming Q and S constant (that is, by discarding Eqs. (10) and (11)). Also assume that electron bunches are entering the wiggler with the same degree of prebunching, momentum detuning, and momentum spread from pass to pass, that is, B,(< = 0,t) = B,,,
Pl(il = 0, r) =
P,o,
S(<= o,T)=y;+ fJ&
Q(<= 0,~) = yo,
I A(;,T) = A,(g)+
1
W,$le"",
infrared applications), a separate oscillator is needed to prebunch the beam. This oscillator has to be linearl.r unstable to grow from (presumably small) density noise on the electron bunches. Details of this double-oscillator configuration are given in the next section. Here we assume that electron bunches are somehow prebunched at the desired radiation wavelength and concentrate on the physics of the overdamped synchronized cavity (OSC) configuration for pulse shaping. An OSC configuration is defined by the experimentally controlled FEL oscillator parameters, normalized cavity desyncronism 1’ and normalized cavity losses ‘x, under which (i) oscillator is operated very close to synchronism, r<>~J?T(v/~)“~. Hence. after 1 ix steady state is achieved. Eqs. (7)--( 11). with atimer= d/dr=O, can then be used to model the steady-state operation. As we will see below, ultra-short radiation pulses can only be generated in the linear regime, where Q and S can be assumed constant. The linearized steadystate equations can then be recast as
(12)
The solutions of the linearized equations can be written as a sum of a stationary solution with inhomogeneous boundary conditions (12) at i;’=0 (A( t = 1) =0 is assumed throughout the paper), and a series of time-dependent solutions with homogeneous (i.e. vanishing) boundary conditions. For example, (13)
1=1 where r/j, is the Ith supermode with a complex gain coefficient i,, [l]. Typically, the linearly unstable supermodes $1, which saturate nonlinearly by depleting the electron energy and inducing electron energy spread, are assumed to dominate any steady-state solution. This is not necessarily the case. We demonstrate below that a high-Q cavity operating near synchronism, which has no linearly unstable supermodes, can serve as a pulse-shaper, provided that the entering electron bunches are slightly prebunched.
3. Overdamped synchronized cavity for pulse shaping In this section we discuss a novel FEL oscillator configuration for generating ultra-short radiation pulses. This scheme utilizes a weakly pre-bunched electron beam. If a low-power long-pulse radiation source at the desired wavelength is available (as it may be the case for microwave or sub-millimeter frequency range), prebunching can be achieved in a separate bunching structure which is powered by the external source. If external source is not available (as will likely be the case for
139
(14) and
a’ + a-2
( 1 a2
(BleiY~~‘)
=
iAeiYG,
(15)
i
where we assumed Q = )jo, S =J$ + &. As Eq. (15) indicates, momentum spread cr enters the particle equation in a way reminiscent of Raman FELs. Eq. (14) describes the steady state of radiation emitted by a prebunched electron beam and deserves a careful physical interpretation. Even though Eq. (14) is time-independent, it contains the physics associated with the advance of he superradiantly emitted radiation at 5 = 1 towards the head of the electron bunch due to cavity desyncronism. In fact, in the vicinity of 4 = 1 cavity losses are relatively unimportant, as radiation is emitted by a highly bunched electron beam at the exit from the wiggler. The buildup of IAl from zero at < = 1 to its peak value is mainly governed by an interplay of the strong bunching (the RHS of Eq. (14)) and the cavity desyncronism. On the contrary, far from <= 1, cavity losses tend to dominate over the effects of cavity desyncronism. Eq. (14) is a typical example of a boundary layer problem, where a term with the highest (first) derivative is multiplied by a small parameter 1’. Hence, Eqs. (14) and (15) can be solved between < = 0 and t=<,, .e 1 neglecting the effect of cavity desyncronism. and then between 4 = &,,,, and 5 = 1 to insure the boundary condition A(( = 1) =O. This approach will be justified by the results of calculation and the assumption of strongly damped cavity. Note that lengthening the cavity (v < 0) would be counterproductive to generation of ultrashort pulses. This is because the peak value of 1.41would
IV. LONG-WAVELENGTH
FELs
240
G. Shuets, J.S. Wurtele JNucl. Instr. and Meth. in Phys. Res. A 393 (1997) 237-241
be achieved at 5 = 1, and a significant pedestal will be formed for {> 1 due to the high Q of the cavity. In the proposed scheme the optical resonator is shortened (v>O). Since, as we later show, the peak value of IAl greatly exceeds lAl(t =0), pedestal formation for 5 ~0 is of no consequence. To illustrate the approach to this boundary layer problem, assume, for simplicity, ye =O. An approximate solution to Eqs. (14) and (15) can then be expressed as A(t) = A,(t) - A,(( = l)e”(t-l)‘“,
(16)
where A,(t) and B,(t) are solutions far away from the boundary layer at 5 = 1, related by &l(5) = 2&(5)/~ >
(17)
and obtained by solving a linear equation 2 a2 + r.3 BOeiyor= i-Be, c( a52
( )
with boundary conditions B,(5=O)=b, and dB,/d<(l=O) =O. The largest amplitudes of the radiation field are expected for a=O. As this is also the most amenable to analytical calculation case, we assume here that a=O. More realistic cases with cr # 0 and y, # 0 will be treated numerically.
Solving Eq. (18) yields A,(()
=bO[e(G+ivGX +
e-CJiii+iJi7i)ij,
a
As Eqs. (19) and (16) indicate, the half-width of the laser pulse is of order o- =& for 5 -=zI&,, and c+ = v/a for r ==C &ax and G+ = v/a for t -Ct,,,. By choosing a =0.05 and v=O.Ol the total pulse width of about a fifth of slippage length can be achieved. Results of a time-dependent numerical simulation of nonlinear equations (7)-( 1l), for y, = 5.0, be = 0.01, and CJ= 0 are presented in Fig. 1. An important point to have in mind is that narrow single-spike solutions, exemplified by Fig. 1, can only be obtained in the linear regime, where overbunching of the electrons does not lead to nonlinear saturation. To insure that the pulse-shaper operates in a linear regime, initial prebunching has to be small (1% in the numerical example shown in Fig. 1). If a nonlinearly saturated FEL oscillator is used as a prebuncher, electrons exiting from the prebuncher may be overbunched. The role of a drift space is to debunch the electron beam to an acceptable level which would insure that the electron dynamics in a pulse shaper is linear. Physically, electrons acquire energy spread inside the prebuncher. This energy spread leads to debunching (in phase and momentum) after
20.0
1.0
0.8 15.0
0.6 s
10.0
0.4
5.0 0.2
0.2
0.4
0.6
0.8
-I 0.0 1.0
<=(ct-z)/A Fig. 1. Normalized field amplitude and bunching in a pulse shaper (one slippage window) for GI=0.05, Y= 0.001, y0 = 5.0. and CJ=O.O: solid line - field amplitude; dashed line ~ bunching.
G. Shvets, J.S. Wurtele/Nucl.
Instr. and Meth. in Phys. Rex A 393 (1997) 237-241
a sufficiently long free drift. As Eq. (3) indicates. propagating a distance &, through drift space is equivalent to multiplying fi by a factor exp( -iPtd). It is straightforward to demonstrate that electrons are debunched by a factor d x exp( -a2ri). Hence, the drifting section is a tool for controling relative and absolute amounts of phase and momentum bunching of a beam, entering the pulse-shaper. Energy spread, introduced in the prebunching section, will not severely affect the FEL performance in the pulse shaper because the radiation pulses in the pulse-shaper are much narrower than those in the prebuncher, making the former insensitive to energy spread. Numerical examples demonstrating an operation of a multi-cavity FEL oscillator will be presented in the forthcoming publications. In conclusion, we suggested a novel scheme of an FEL oscillator. overdamped synchronized cavity, capable of providing a steady-state train of ultra-short optical pulses with high efficiency. The operation of this device depends critically on availability of a weakly prebunched electron beam. A multi-cavity (prebuncher +drift space) configuration for producing such a beam in described.
241
Acknowledgements This work was supported by the US DOE Division of High Energy Physics. One of us (GS) acknowledges the support of the US DOE Postdoctoral Fellowship.
References [l]
N. Piovella, P. Chaix, G. Shvets. D.A. Jaroszynski, Phys. Rev. E 52 (1995) 5470; N. Piovella, P. Chaix, D.A.
Jaroszynski, G. Shvets. Nucl. Instr. and Meth. A 375 (1996) 156. [Z] G. Dattoli, A. Marino, F. Romanelli. Optics Commun. 35 (1980) 407; G. Dattoli, A. Marino, A. Renieri, F. Romanelli, IEEE J. Quantum Electron. 17 (1981) 1371. [3] N.S. Ginzburg, S.P. Kuznetsov, T.N. Fedoseeva. Radio Phys. Quant. Electron. (1971) 728. [4] R. Bonifacio, F. Casagrande, L. De Salvo Souza, Phys. Rev. A 33 (1986) 2836; R. Bonifacio, L. De Salvo Souza, P. Pierini. N. Piovella, Nucl. Instr. and Meth. A 296 f 1990) 358.
IV. LONG-WAVELENGTH
FELs
NUCLEAR INSTRUMENTS & METHODS IN PHVSBCS RESEARCH Section A
ELSEVIER
Pulse-shaping in short-pulse FEL oscillators using multiple resonators G. Shvet?*,
J.S. Wurteleb
‘Princeton Plasma Phvsics Laboratoty Princeton, NJ 08543. USA b Department ?f Physics, University qf California, Berkeley. CA 94720. USA
Abstract A novel method is proposed for the efficient production of optical pulses that are shorter concept uses short electron bunches and a multiple cavity FEL oscillator configuration.
1. Introduction Most free-electron laser user facilities operate as low gain oscillators and the majority are driven by radiofrequency (rf) linacs. The electron beam consists of very short (picosecond or sub-picosecond) electron bunches, spaced by a multiple of the rf wavelength. The radiation pulse length is determined by the bunch length, L,,, if 3 CCL,, where A =N,,,,I is the slippage length, i, is the radiation wavelength, and N, is the number of wiggler periods. For electron bunches much shorter than the slippage distance, the pulse length is (roughly) determined by the slippage length, A. To make more precise statement about the duration and structure of radiation pulses in a short-bunch oscillators requires the inclusion of the cavity desynchronism and losses. The supermode theory of small-gain FEL oscillators in the short bunch limit was recently developed [l]. Many of the analytical conclusions of Ref. [l] parallel those of the pioneering numerical work of Dattoli et al. [Z] on the linear radiation supermodes in oscillators driven by electron bunches with an arbitrary temporal profile. One of the useful aspects of the formalism presented in Ref. [I] is that the equations describing the temporal pulse evolution of a short-bunch oscillator are very similar to those describing backward wave oscillators (BWO) (see, for example, Ref. [3]). This implies that boundary conditions are specified at the heat and the tail of the radiation pulse
*Corresponding
author. Tel.: + 1 609 243 2609: fax: + 1 609
233 2662; e-mail: [email protected]. 016%9092/97/$17.00 Copyright PII SOI 68-9002(97)00483-X
than a slippage length. The
and is essential to our analysis of the multi-cavity pulseshaping concept described below. Our previous work showed [l] that an FEL oscillator driven by very short electron bunches (shorter than the slippage length) can generate radiation pulses shorter than the slippage length. These very short radiation pulses, predicted by the linear theory of Ref. [ 11. can only be obtained in the operating regime where cavity desynchronism is very small. Unfortunately, it is precisely in this regime that nonlinear multi-mode effects, such as limit-cycle saturation and transition to chaos, dominate the FEL dynamics. This results in nonrepetetiveness of radiation pulses from pass to pass. Here we propose a novel scheme which circumvents this limitation and enables the steady-state generation of ultra-short radiation pulses with high efficiency (i.e., larger than the typical value of +N,). Our scheme avoids the complications of operating the oscillator with many unstable supermodes, yet achieves efficiences characteristic of that regime. The key idea is to split an FEL oscillator into three sections: a prebuncher, a drift space. and a pulse shaper. The latter section is where most of the radiation is produced. Both the prebuncher and the pulse shaper are operated as oscillators with high-Q optical cavities. The main difference between these two sections is that the prebuncher operates with a relatively large cavity desynchronism, which ensures that (i) there is only one linear[~ unstable radiation supermode and (ii) this supermode saturates quickly and at low intensity. The pulse shaper is operated in an ouerdamped mode, i.e. there are no linearly unstable radiation supermodes in the pulse-shaping cavity. Instead, the pulse shaper is driven by a bunched electron beam.
1(1#1997 Elsevier Science B.V. All rights
reserved IV. LONG-WAVELENGTH
FELs
238
G. Shvets, J.S. Wurtele /Nucl. Instr. and Meth. in Phys. Rex A 393 (i99.7) 237-241
and
2. Theoretical model and basic formalism
Our model is one-dimensional, low gain and assumes an electron bunch length much less than a slippage length. The evolution of laser pulser is then controlled [l] by three parameters: the reduced Colson parameter, r=jcL,/A, the cavity desynchronism SY, and the cavity losses LYE, where
(1) Moreover, by an appropriate normalization of “time” (a continuous equivalent of pass number), cavity losses, and desynchronism
is was shown that the evolution of an FEL oscillator is governed by only the normalized cavity desynchronism, v, and losses u. The independent variables are t and 5 =(ct--)/A, the normalized distance from the head of the radiation pulse (which is assumed to move with the speed of light, c). The field is described by a slowly varying complex amplitude, A, normalized so that IAl2=4nN,j,P(z, t)/P,, where P(z, t) is the intra-cavity optical power and PC= mc2y,(l,/e) is the electron beam power. The FEL oscillator can be modelled with a field equation for the radiation and a Vlasov equation for electrons:
e-‘~-Qli/2”’
f,(P, 5) =
(P - Q?
,ieBl
+
Bl
In Eqs. (5) and (6) B, = (e-“), P, =(pe-“‘), Q=(p), and C? = ((p - Q)2) are t-dependent moments of the particle distribution function f: bunching, momentum bunching, average momentum, and momentum spread, respectively. Since fresh electrons are entering the oscillator on each pass, distribution function f depends on normalized pass number t parametrically through optical field. In a storage ring, where the memory of a FEL interaction is preserved in the electron distribution from pass to pass, Eq. (3) would be modified. The particular choice of the distribution function in Eqs. (5) and (6) corresponds to a Gaussian closure. This truncated description of the electrons by four global moments was shown [1,4], to accurately describe such complex FEL phenomena as nonlinear saturation of FEL amplifiers, synchrotron oscillations, nonlinear superradiance, limit cycles in FEL oscillators, etc. The explicit expression for the distribution functionfi, presented here for the first time, will be later used to describe the effect of the drift section. The output distribution function at the end of the prebuncher is evolved through the drift section to obtain the input distribution function at the entrance of the pulse-shaper, which determines the output of the novel FEL configuration we propose in this paper. Substituting Eqs. (4H6) into Eqs. (2) and (3) yields the set of closed equations [l]: aA aA CY --v~+~A=~B,,
aT
as
-aB, = _iP 13 al
ah
= -A - iSBl - 2iQPi + 2iQ2B,, a<
af+ pae af @
af
= (Ae” + c.c.)--,
aQ = - [ABT + CC.]) at
aP
-
where q(l)= 1 if 0 < i < 1 and q(c)=0 otherwise. The variables p=&8 and B are canonically conjugate. This description can be recast into a form similar to that of Ref. [il. As was first demonstrated in Ref. [4], the essential FEL physics can be captured by assuming a particular form of the electron distribution function:
fb, 0,5) =fo(p, 5) + e”fl(p, 5) + c.c.,
(4)
where
(5)
as
C = -2[APT
+ CL],
(9)
(10)
(11)
where S = C? + Q2 = (p’). Eqs. (f+(l 1) are integrated between 5 =0 and 5 = 1. Physically, 5 =0 corresponds to the radiation slice which overlaps the electron bunch at the entrance into the wiggler while 5 = 1 corresponds to the radiation slice which overlaps the electron bunch at the exit of the wiggler. Radiation which has already slipped ahead of the electron bunch due to finite cavity desynchronism (< =O) decays exponentially as exp(&/2u). The boundary condition for the radiation amplitude A is set at < = 1 (one slippage length behind the pulse),
G. Shvets. J.S. Wurtele/Nucl.
Instr. and Meth. in Phw. Rrs. A 393 (1997) 237-241
and will be assumed vanishing in this paper. Boundary conditions for particle variables are fixed at the wiggler entrance, i.e. at X=0. Setting boundary conditions at different boundaries for particles and fields is reminiscent of backward wave oscillators [3], where particles are initialized at the entrance into the interaction region while the microwave fields (which have group velocity in the opposite direction of their phase velocity due to the slow-wave structures) are initialized at the exit of the interaction region. Initializing particles and fields at different locations is intuitively clear for a backward wave oscillator. but is less intuitive for an FEL oscillators, where the “interaction window” O<< < 1 itself moves with the speed of light, and is equal to one slippage length. Eqs. (7)-(11) can be linearized by assuming Q and S constant (that is, by discarding Eqs. (10) and (11)). Also assume that electron bunches are entering the wiggler with the same degree of prebunching, momentum detuning, and momentum spread from pass to pass, that is, B,(< = 0,t) = B,,,
Pl(il = 0, r) =
P,o,
S(<= o,T)=y;+ fJ&
Q(<= 0,~) = yo,
I A(;,T) = A,(g)+
1
W,$le"",
infrared applications), a separate oscillator is needed to prebunch the beam. This oscillator has to be linearl.r unstable to grow from (presumably small) density noise on the electron bunches. Details of this double-oscillator configuration are given in the next section. Here we assume that electron bunches are somehow prebunched at the desired radiation wavelength and concentrate on the physics of the overdamped synchronized cavity (OSC) configuration for pulse shaping. An OSC configuration is defined by the experimentally controlled FEL oscillator parameters, normalized cavity desyncronism 1’ and normalized cavity losses ‘x, under which (i) oscillator is operated very close to synchronism, r<
(12)
The solutions of the linearized equations can be written as a sum of a stationary solution with inhomogeneous boundary conditions (12) at i;’=0 (A( t = 1) =0 is assumed throughout the paper), and a series of time-dependent solutions with homogeneous (i.e. vanishing) boundary conditions. For example, (13)
1=1 where r/j, is the Ith supermode with a complex gain coefficient i,, [l]. Typically, the linearly unstable supermodes $1, which saturate nonlinearly by depleting the electron energy and inducing electron energy spread, are assumed to dominate any steady-state solution. This is not necessarily the case. We demonstrate below that a high-Q cavity operating near synchronism, which has no linearly unstable supermodes, can serve as a pulse-shaper, provided that the entering electron bunches are slightly prebunched.
3. Overdamped synchronized cavity for pulse shaping In this section we discuss a novel FEL oscillator configuration for generating ultra-short radiation pulses. This scheme utilizes a weakly pre-bunched electron beam. If a low-power long-pulse radiation source at the desired wavelength is available (as it may be the case for microwave or sub-millimeter frequency range), prebunching can be achieved in a separate bunching structure which is powered by the external source. If external source is not available (as will likely be the case for
139
(14) and
a’ + a-2
( 1 a2
(BleiY~~‘)
=
iAeiYG,
(15)
i
where we assumed Q = )jo, S =J$ + &. As Eq. (15) indicates, momentum spread cr enters the particle equation in a way reminiscent of Raman FELs. Eq. (14) describes the steady state of radiation emitted by a prebunched electron beam and deserves a careful physical interpretation. Even though Eq. (14) is time-independent, it contains the physics associated with the advance of he superradiantly emitted radiation at 5 = 1 towards the head of the electron bunch due to cavity desyncronism. In fact, in the vicinity of 4 = 1 cavity losses are relatively unimportant, as radiation is emitted by a highly bunched electron beam at the exit from the wiggler. The buildup of IAl from zero at < = 1 to its peak value is mainly governed by an interplay of the strong bunching (the RHS of Eq. (14)) and the cavity desyncronism. On the contrary, far from <= 1, cavity losses tend to dominate over the effects of cavity desyncronism. Eq. (14) is a typical example of a boundary layer problem, where a term with the highest (first) derivative is multiplied by a small parameter 1’. Hence, Eqs. (14) and (15) can be solved between < = 0 and t=<,, .e 1 neglecting the effect of cavity desyncronism. and then between 4 = &,,,, and 5 = 1 to insure the boundary condition A(( = 1) =O. This approach will be justified by the results of calculation and the assumption of strongly damped cavity. Note that lengthening the cavity (v < 0) would be counterproductive to generation of ultrashort pulses. This is because the peak value of 1.41would
IV. LONG-WAVELENGTH
FELs
240
G. Shuets, J.S. Wurtele JNucl. Instr. and Meth. in Phys. Res. A 393 (1997) 237-241
be achieved at 5 = 1, and a significant pedestal will be formed for {> 1 due to the high Q of the cavity. In the proposed scheme the optical resonator is shortened (v>O). Since, as we later show, the peak value of IAl greatly exceeds lAl(t =0), pedestal formation for 5 ~0 is of no consequence. To illustrate the approach to this boundary layer problem, assume, for simplicity, ye =O. An approximate solution to Eqs. (14) and (15) can then be expressed as A(t) = A,(t) - A,(( = l)e”(t-l)‘“,
(16)
where A,(t) and B,(t) are solutions far away from the boundary layer at 5 = 1, related by &l(5) = 2&(5)/~ >
(17)
and obtained by solving a linear equation 2 a2 + r.3 BOeiyor= i-Be, c( a52
( )
with boundary conditions B,(5=O)=b, and dB,/d<(l=O) =O. The largest amplitudes of the radiation field are expected for a=O. As this is also the most amenable to analytical calculation case, we assume here that a=O. More realistic cases with cr # 0 and y, # 0 will be treated numerically.
Solving Eq. (18) yields A,(()
=bO[e(G+ivGX +
e-CJiii+iJi7i)ij,
a
As Eqs. (19) and (16) indicate, the half-width of the laser pulse is of order o- =& for 5 -=zI&,, and c+ = v/a for r ==C &ax and G+ = v/a for t -Ct,,,. By choosing a =0.05 and v=O.Ol the total pulse width of about a fifth of slippage length can be achieved. Results of a time-dependent numerical simulation of nonlinear equations (7)-( 1l), for y, = 5.0, be = 0.01, and CJ= 0 are presented in Fig. 1. An important point to have in mind is that narrow single-spike solutions, exemplified by Fig. 1, can only be obtained in the linear regime, where overbunching of the electrons does not lead to nonlinear saturation. To insure that the pulse-shaper operates in a linear regime, initial prebunching has to be small (1% in the numerical example shown in Fig. 1). If a nonlinearly saturated FEL oscillator is used as a prebuncher, electrons exiting from the prebuncher may be overbunched. The role of a drift space is to debunch the electron beam to an acceptable level which would insure that the electron dynamics in a pulse shaper is linear. Physically, electrons acquire energy spread inside the prebuncher. This energy spread leads to debunching (in phase and momentum) after
20.0
1.0
0.8 15.0
0.6 s
10.0
0.4
5.0 0.2
0.2
0.4
0.6
0.8
-I 0.0 1.0
<=(ct-z)/A Fig. 1. Normalized field amplitude and bunching in a pulse shaper (one slippage window) for GI=0.05, Y= 0.001, y0 = 5.0. and CJ=O.O: solid line - field amplitude; dashed line ~ bunching.
G. Shvets, J.S. Wurtele/Nucl.
Instr. and Meth. in Phys. Rex A 393 (1997) 237-241
a sufficiently long free drift. As Eq. (3) indicates. propagating a distance &, through drift space is equivalent to multiplying fi by a factor exp( -iPtd). It is straightforward to demonstrate that electrons are debunched by a factor d x exp( -a2ri). Hence, the drifting section is a tool for controling relative and absolute amounts of phase and momentum bunching of a beam, entering the pulse-shaper. Energy spread, introduced in the prebunching section, will not severely affect the FEL performance in the pulse shaper because the radiation pulses in the pulse-shaper are much narrower than those in the prebuncher, making the former insensitive to energy spread. Numerical examples demonstrating an operation of a multi-cavity FEL oscillator will be presented in the forthcoming publications. In conclusion, we suggested a novel scheme of an FEL oscillator. overdamped synchronized cavity, capable of providing a steady-state train of ultra-short optical pulses with high efficiency. The operation of this device depends critically on availability of a weakly prebunched electron beam. A multi-cavity (prebuncher +drift space) configuration for producing such a beam in described.
241
Acknowledgements This work was supported by the US DOE Division of High Energy Physics. One of us (GS) acknowledges the support of the US DOE Postdoctoral Fellowship.
References [l]
N. Piovella, P. Chaix, G. Shvets. D.A. Jaroszynski, Phys. Rev. E 52 (1995) 5470; N. Piovella, P. Chaix, D.A.
Jaroszynski, G. Shvets. Nucl. Instr. and Meth. A 375 (1996) 156. [Z] G. Dattoli, A. Marino, F. Romanelli. Optics Commun. 35 (1980) 407; G. Dattoli, A. Marino, A. Renieri, F. Romanelli, IEEE J. Quantum Electron. 17 (1981) 1371. [3] N.S. Ginzburg, S.P. Kuznetsov, T.N. Fedoseeva. Radio Phys. Quant. Electron. (1971) 728. [4] R. Bonifacio, F. Casagrande, L. De Salvo Souza, Phys. Rev. A 33 (1986) 2836; R. Bonifacio, L. De Salvo Souza, P. Pierini. N. Piovella, Nucl. Instr. and Meth. A 296 f 1990) 358.
IV. LONG-WAVELENGTH
FELs