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Free electron lasers with static and dynamic plasma wigglers
Nuclear Instruments
*H &.__
and Methods in Physics Research A 358 (1995) 437-440
NUCLEAR INSTRUMENTS 8 METHODS IN PHVSICS RESEARCH
__ I!!!! ELSEVIER
Secrion A
Free electron lasers with static and dynamic plasma wigglers N.I. Karbushev Moscow Radiotechnical
Institute, Varshacskoye
7 113519 Moscow, Russian Federation shone 13_,
Abstract In free electron lasers, plasma wigglers of static and dynamic types may be used. In the case of a static plasma wiggler a spatially modulated plasma density results in a periodic dielectric permittivity, but pumping electromagnetic fields are absent. A dynamic plasma wiggler arises when an electromagnetic wave propagating in a plasma is scattered by an electron beam. The wave fields act on beam electrons but modulate plasma particles also. Two mechanisms are responsible for the interaction between an electron beam and electromagnetic waves. One of them is analogous to that in usual FELs and exists only in dynamic plasma wigglers. Another mechanism is conditioned by space-temporal modulation of the density and velocity of plasma particles and exist in both static and dynamic plasma wigglers. The contributions of the two mechanisms into the interaction process may be comparable. Both types of magnetized plasma wigglers are considered in a linear approximation.
1. Introduction Different types of plasma wigglers may be used in free electron lasers (FELs) for amplification and generation of electromagnetic waves. In comparison with conventional FELs, a number of advantages can be realized in such wigglers. Plasma parameters can be easily changed and so parameters of plasma wigglers also can be changed easily. Plasma compensates the space charge, and more intense electron beams without decreasing their quality may be used. Two types of plasma wigglers may exist: static and dynamic ones. The first type of plasma wigglers is made when its density is modulated in space and this modulation is stationary in time. The second type of plasma wigglers arises in the case of propagation of an intense electromagnetic wave in the plasma. This wave is a pumping wave for an electron beam and a modulator for plasma density and velocity. In static plasma wigglers effective interaction of electron beams with fast electromagnetic waves is possible due to slow space harmonics of the wave which arise in a dielectric medium with periodic permittivity. Such a mechanism is like that in periodic slow-wave structures. In dynamic plasma wigglers two mechanisms of interaction exist. One of them is analogous to that in usual FELs and is conditioned by wave electromagnetic fields acting on beam electrons. Another mechanism is the result of spacetemporal plasma modulation by a pumping wave and is the same as in static plasma wigglers. There are several different branches of electromagnetic waves in plasma [l-.5] and they can all be used as pumping waves. Some of them are slow waves with phase 0168-9002/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0168-9002(94)01263-6
velocities less than the speed of light but there are also fast waves. The process of interaction of electron beams with amplified or generated electromagnetic waves in static and dynamic plasma wigglers can be described in a general way [6]. A static wiggler will correspond to the particular case of zero velocity of the plasma particles and zero frequency of the pumping wave. In the present paper, the dispersion equation describing the amplification and generation of electromagnetic waves by an electron beam in an unbounded plasma wiggler is derived and analyzed. An electron beam and plasma are proposed to be strongly magnetized so that motion of their electrons is one-dimensional only in the longitudinal direction.
2. Basic equations The system considered is described by the equation for the axial component of the electric field EZ and the equations of motion and continuity for electrons:
a2
a2
ax’+az’,. aLIp h L
at
+ [!p.b
&)~+(-$&)4_i;=o, aL’, h 1= az
eEz
(m, m?)
’
an,, a
L at + z(np,bL'p.b) = ‘1 where 1, z and x are the time and axial and transverse coordinates, ~~~ b and nP ,b are the axial velocity and the
X. THEORY/CONJECTURE/SPECULATION
NJ. Karbushet) / Nucl. instr. and Meth. in Phys. Res. A 358 (I 99.5) 437-440
438
density of plasma and beam electrons
respectively, e and and c is the speed of light. The density of total current j, is directed along the z axis and is equal to the sum of plasma and beam currents: m are the charge and mass of the electron,
j, =j,
+jh = e(npLlp + nbLsb).
(2)
When the pumping wave with the axial component electric field EZ, = Re[ E,, exp( - iw,,t + ikoz -I- iK”.x)],
of
(3)
where E, is its amplitude, wC, is the frequency, and k, and K~ are the axial and transverse components of the wave vector, propagates in a beam-plasma system, the plasma and beam become modulated in space and time. Amplitudes of velocities and densities of the electrons can be find from linearized Eqs. (1):
analog sums, and the amplitudes of all perturbed values are coupled. In the case of a small enough pumping wave amplitude, we may restrict consideration to only three main harmonics in sum (6) with numbers n = 0, + I, because the amplitudes of other harmonics are much smaller. Let us define them with the signs “s”, ” +” and .‘ - “. Further, when the density of a beam is not too great, so that wz K w:, it is sufficient to take into account an electron beam only in terms containing the small values w’+ or w’_ in the denominators (equations w’?= 0 mean the synchronism of a corresponding wave harmonic with an electron beam). With such assumptions, the following expressions for the amplitudes of the perturbed values can be obtained in a linear approximation:
i eE,, LJpo= y-&- >
0
l’b0
z-
nbO=
i eE, rnr3uh
’
k
2ii,r~,o, “;I
and u are average values of where o’0 = w 0 - k”“; ‘,b electrons densities and the beam velocity. The following dispersion relation for the pumping wave,
(8)
2
DC”,,
K,’ =_"b
is valid, where wp,b = (4nez7i,,,/m)‘/2 are the Langmuir frequencies of the plasma and beam, and y = (1 U”/c2)- l/7- is the relativistic factor.
VbS” -
i eE,
my306
3. Calculation of perturbed beam electrons
values
for plasma
’
and
If the electromagnetic wave propagates in a beamplasma system together with a pumping wave, the axial component of its electric field should be written down in the form of the sum of space-temporal harmonics rr. E;,= c Re{E,,exp[-i(w,+nwo)f+i(k,+nk,)z n= --r +i(Ks+“K,&]},
(6)
where ES,,, wS, k, and K, are the amplitudes of harmonics with the numbers n and frequency and components of a wave vector of an electromagnetic wave. Perturbations of velocities and densities of electrons can be presented as
The amplitudes of the plasma and beam current densities can be written down as Ip.hb =
enp,bs
ws/ks)
(11)
NJ. Karbusher /Nucl.
4. Dispersion
equation and discussion
Substituting expressions for the current densities (11) with those for the velocities and the densities of electrons (7)~(IO) into Eq. (1) for the electric field, one can find the correlation between the amplitudes of harmonics E,,, and the dispersion equation. It follows that the ratio of amplitudes of harmonics is equal to
It is convenient the form
to write down the dispersion
k,)
equation in
- wi;
4 -
- p
4w,“v’
D(ws.
k,)=O,
D(w*,
k,)=O,
I[
w+, k,)
where the function D( w +, k +) is the analog introduced in Eq. (5) and _ -
kc, ks
+-+--.
(16)
which must be fulfilled simultaneously. In the case of induced scattering the solution to the dispersion equation (13) demonstrates an instability without the electron beam. If the plasma-wave system is stable, the resonance conditions (16) correspond to the parametric coupling of electromagnetic and plasma waves accompanied by effective transformation of waves of different types. In the last case the presence of an electron beam and fulfillment of synchronism conditions (15) can lead to a strong beam-plasma instability where one may write down the dispersion equation (13) in form
w;y3
(13)
1 _=-
The dispersion equation (13) also describes induced (parametric) scattering of electromagnetic waves by the plasma without the electron beam. The conditions of such process are determined by two approximate equations
@us, D( y3aJgZ 1 Lo71 L’J
7
D( w,,
439
Ins&. and Meth. in Phys. Rex A 358 (1995) 437-440
to that
(17)
It follows from Eq. (17) that the amplitudes of fast and slow waves increase with the same growth rate [8,9]. For static plasma wigglers, the frequency we and the amplitude of the electric field E;,, above, must be replaced with zero but the inhomogeneous perturbation of plasma is finite. So, Eqs. (12)-(14) density II~,~= k,,Tz,r,,,/q, can be simplified. Only the second term in the expression for (Y* in Eqs. (14) is non-zero which points to the fact that only one interaction mechanism is present in such cases. In static plasma wigglers, the conditions of synchronism (15) can be fulfilled only with the negative sign, and the resonance conditions (16) for parametric induced scattering or parametric coupling of waves may not be reached, i.e. D(w+, k,)+O.
k+ 5. Conclusion
When D(w+, k *) # 0 the dispersion equation (13) is equivalent to that of ordinary FELs [6,7]. However, the expression for LYf in Eqs. (14) contains two terms of the same order conditioned by two mechanisms of interaction. The first term is determined by direct modulation of an electron beam by a pumping wave and the second is the result of plasma modulation. These two mechanisms can amplify or compensate each other and in general must be taken into account together. An electron beam interacts with those electromagnetic waves for which w, f w
D(w,,
k,) =o.
The interaction of electromagnetic waves with an electron beam in static and dynamic plasma wigglers can be described in a general way. The dispersion equation in such cases is similar to that for usual free electron lasers. There are two mechanisms which are responsible for the amplification and generation of waves by an electron beam in plasma wigglers: that coupled with direct effects of pumping wave fields on beam electrons and one determined by space-time modulation of the density and velocity of plasma electrons. Both these mechanisms must be taken into account.
(15)
Because of the variety of dispersion branches for waves in plasma both synchronism conditions with positive and negative signs in Eqs. (15) can be realized while in usual vacuum FELs only that with negative sign is possible.
References [I] Ya.6. Faynberg, Fizika Plasmy 11 (1985). [2] V.A. Buts, V.I. Miroshnichenko and V.V. Ognivenko, Techn. Fiz. 50 (1980) 2257.
X. THEORY/CONJECTURE/SPECULATION
Zhurn.
[3] V.A. Balakircv, Izv. Vuzov. Radiofizika 15 (I’X?) 11X-L [4] N.I. Karbushcv and VS. Rogov, Abstracts of the 9th All-Union Conf. on Linacs of Charged Particles (1985) p. 9.5. [5] V.A. Balakirev, V.I. Miroshnichenko and Ya.B. Faynberg. Fizika Plazmy 12 (1986)983. [h] NJ. Karbushev. V.P. Poponin and A.A. Rukhadzc. in: Generatory Kogerentnogo lzlucheniya na Svohodnykh Elektronakh, ed. A.A. Rukhadze (Mir. Moscow. 1983) p. 214.
[7] NJ. Karbushev and A.D. Shatkus. Nucl. Instr. and Meth. A 304 (1991) 559. [x] G.I. Batskikh. K.G. Gurcev. N.I. Karbushev and A.I. Lisitsyn, Proc. 8th Int. Conf. on High-Power Particlc Beams. Vol. 2 (1990) p. 1173. [9] N.I. Karbushev, Izv. Vuzov. Radiofizika 34 (1991) X3.
*H &.__
and Methods in Physics Research A 358 (1995) 437-440
NUCLEAR INSTRUMENTS 8 METHODS IN PHVSICS RESEARCH
__ I!!!! ELSEVIER
Secrion A
Free electron lasers with static and dynamic plasma wigglers N.I. Karbushev Moscow Radiotechnical
Institute, Varshacskoye
7 113519 Moscow, Russian Federation shone 13_,
Abstract In free electron lasers, plasma wigglers of static and dynamic types may be used. In the case of a static plasma wiggler a spatially modulated plasma density results in a periodic dielectric permittivity, but pumping electromagnetic fields are absent. A dynamic plasma wiggler arises when an electromagnetic wave propagating in a plasma is scattered by an electron beam. The wave fields act on beam electrons but modulate plasma particles also. Two mechanisms are responsible for the interaction between an electron beam and electromagnetic waves. One of them is analogous to that in usual FELs and exists only in dynamic plasma wigglers. Another mechanism is conditioned by space-temporal modulation of the density and velocity of plasma particles and exist in both static and dynamic plasma wigglers. The contributions of the two mechanisms into the interaction process may be comparable. Both types of magnetized plasma wigglers are considered in a linear approximation.
1. Introduction Different types of plasma wigglers may be used in free electron lasers (FELs) for amplification and generation of electromagnetic waves. In comparison with conventional FELs, a number of advantages can be realized in such wigglers. Plasma parameters can be easily changed and so parameters of plasma wigglers also can be changed easily. Plasma compensates the space charge, and more intense electron beams without decreasing their quality may be used. Two types of plasma wigglers may exist: static and dynamic ones. The first type of plasma wigglers is made when its density is modulated in space and this modulation is stationary in time. The second type of plasma wigglers arises in the case of propagation of an intense electromagnetic wave in the plasma. This wave is a pumping wave for an electron beam and a modulator for plasma density and velocity. In static plasma wigglers effective interaction of electron beams with fast electromagnetic waves is possible due to slow space harmonics of the wave which arise in a dielectric medium with periodic permittivity. Such a mechanism is like that in periodic slow-wave structures. In dynamic plasma wigglers two mechanisms of interaction exist. One of them is analogous to that in usual FELs and is conditioned by wave electromagnetic fields acting on beam electrons. Another mechanism is the result of spacetemporal plasma modulation by a pumping wave and is the same as in static plasma wigglers. There are several different branches of electromagnetic waves in plasma [l-.5] and they can all be used as pumping waves. Some of them are slow waves with phase 0168-9002/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0168-9002(94)01263-6
velocities less than the speed of light but there are also fast waves. The process of interaction of electron beams with amplified or generated electromagnetic waves in static and dynamic plasma wigglers can be described in a general way [6]. A static wiggler will correspond to the particular case of zero velocity of the plasma particles and zero frequency of the pumping wave. In the present paper, the dispersion equation describing the amplification and generation of electromagnetic waves by an electron beam in an unbounded plasma wiggler is derived and analyzed. An electron beam and plasma are proposed to be strongly magnetized so that motion of their electrons is one-dimensional only in the longitudinal direction.
2. Basic equations The system considered is described by the equation for the axial component of the electric field EZ and the equations of motion and continuity for electrons:
a2
a2
ax’+az’,. aLIp h L
at
+ [!p.b
&)~+(-$&)4_i;=o, aL’, h 1= az
eEz
(m, m?)
’
an,, a
L at + z(np,bL'p.b) = ‘1 where 1, z and x are the time and axial and transverse coordinates, ~~~ b and nP ,b are the axial velocity and the
X. THEORY/CONJECTURE/SPECULATION
NJ. Karbushet) / Nucl. instr. and Meth. in Phys. Res. A 358 (I 99.5) 437-440
438
density of plasma and beam electrons
respectively, e and and c is the speed of light. The density of total current j, is directed along the z axis and is equal to the sum of plasma and beam currents: m are the charge and mass of the electron,
j, =j,
+jh = e(npLlp + nbLsb).
(2)
When the pumping wave with the axial component electric field EZ, = Re[ E,, exp( - iw,,t + ikoz -I- iK”.x)],
of
(3)
where E, is its amplitude, wC, is the frequency, and k, and K~ are the axial and transverse components of the wave vector, propagates in a beam-plasma system, the plasma and beam become modulated in space and time. Amplitudes of velocities and densities of the electrons can be find from linearized Eqs. (1):
analog sums, and the amplitudes of all perturbed values are coupled. In the case of a small enough pumping wave amplitude, we may restrict consideration to only three main harmonics in sum (6) with numbers n = 0, + I, because the amplitudes of other harmonics are much smaller. Let us define them with the signs “s”, ” +” and .‘ - “. Further, when the density of a beam is not too great, so that wz K w:, it is sufficient to take into account an electron beam only in terms containing the small values w’+ or w’_ in the denominators (equations w’?= 0 mean the synchronism of a corresponding wave harmonic with an electron beam). With such assumptions, the following expressions for the amplitudes of the perturbed values can be obtained in a linear approximation:
i eE,, LJpo= y-&- >
0
l’b0
z-
nbO=
i eE, rnr3uh
’
k
2ii,r~,o, “;I
and u are average values of where o’0 = w 0 - k”“; ‘,b electrons densities and the beam velocity. The following dispersion relation for the pumping wave,
(8)
2
DC”,,
K,’ =_"b
is valid, where wp,b = (4nez7i,,,/m)‘/2 are the Langmuir frequencies of the plasma and beam, and y = (1 U”/c2)- l/7- is the relativistic factor.
VbS” -
i eE,
my306
3. Calculation of perturbed beam electrons
values
for plasma
’
and
If the electromagnetic wave propagates in a beamplasma system together with a pumping wave, the axial component of its electric field should be written down in the form of the sum of space-temporal harmonics rr. E;,= c Re{E,,exp[-i(w,+nwo)f+i(k,+nk,)z n= --r +i(Ks+“K,&]},
(6)
where ES,,, wS, k, and K, are the amplitudes of harmonics with the numbers n and frequency and components of a wave vector of an electromagnetic wave. Perturbations of velocities and densities of electrons can be presented as
The amplitudes of the plasma and beam current densities can be written down as Ip.hb =
enp,bs
ws/ks)
(11)
NJ. Karbusher /Nucl.
4. Dispersion
equation and discussion
Substituting expressions for the current densities (11) with those for the velocities and the densities of electrons (7)~(IO) into Eq. (1) for the electric field, one can find the correlation between the amplitudes of harmonics E,,, and the dispersion equation. It follows that the ratio of amplitudes of harmonics is equal to
It is convenient the form
to write down the dispersion
k,)
equation in
- wi;
4 -
- p
4w,“v’
D(ws.
k,)=O,
D(w*,
k,)=O,
I[
w+, k,)
where the function D( w +, k +) is the analog introduced in Eq. (5) and _ -
kc, ks
+-+--.
(16)
which must be fulfilled simultaneously. In the case of induced scattering the solution to the dispersion equation (13) demonstrates an instability without the electron beam. If the plasma-wave system is stable, the resonance conditions (16) correspond to the parametric coupling of electromagnetic and plasma waves accompanied by effective transformation of waves of different types. In the last case the presence of an electron beam and fulfillment of synchronism conditions (15) can lead to a strong beam-plasma instability where one may write down the dispersion equation (13) in form
w;y3
(13)
1 _=-
The dispersion equation (13) also describes induced (parametric) scattering of electromagnetic waves by the plasma without the electron beam. The conditions of such process are determined by two approximate equations
@us, D( y3aJgZ 1 Lo71 L’J
7
D( w,,
439
Ins&. and Meth. in Phys. Rex A 358 (1995) 437-440
to that
(17)
It follows from Eq. (17) that the amplitudes of fast and slow waves increase with the same growth rate [8,9]. For static plasma wigglers, the frequency we and the amplitude of the electric field E;,, above, must be replaced with zero but the inhomogeneous perturbation of plasma is finite. So, Eqs. (12)-(14) density II~,~= k,,Tz,r,,,/q, can be simplified. Only the second term in the expression for (Y* in Eqs. (14) is non-zero which points to the fact that only one interaction mechanism is present in such cases. In static plasma wigglers, the conditions of synchronism (15) can be fulfilled only with the negative sign, and the resonance conditions (16) for parametric induced scattering or parametric coupling of waves may not be reached, i.e. D(w+, k,)+O.
k+ 5. Conclusion
When D(w+, k *) # 0 the dispersion equation (13) is equivalent to that of ordinary FELs [6,7]. However, the expression for LYf in Eqs. (14) contains two terms of the same order conditioned by two mechanisms of interaction. The first term is determined by direct modulation of an electron beam by a pumping wave and the second is the result of plasma modulation. These two mechanisms can amplify or compensate each other and in general must be taken into account together. An electron beam interacts with those electromagnetic waves for which w, f w
D(w,,
k,) =o.
The interaction of electromagnetic waves with an electron beam in static and dynamic plasma wigglers can be described in a general way. The dispersion equation in such cases is similar to that for usual free electron lasers. There are two mechanisms which are responsible for the amplification and generation of waves by an electron beam in plasma wigglers: that coupled with direct effects of pumping wave fields on beam electrons and one determined by space-time modulation of the density and velocity of plasma electrons. Both these mechanisms must be taken into account.
(15)
Because of the variety of dispersion branches for waves in plasma both synchronism conditions with positive and negative signs in Eqs. (15) can be realized while in usual vacuum FELs only that with negative sign is possible.
References [I] Ya.6. Faynberg, Fizika Plasmy 11 (1985). [2] V.A. Buts, V.I. Miroshnichenko and V.V. Ognivenko, Techn. Fiz. 50 (1980) 2257.
X. THEORY/CONJECTURE/SPECULATION
Zhurn.
[3] V.A. Balakircv, Izv. Vuzov. Radiofizika 15 (I’X?) 11X-L [4] N.I. Karbushcv and VS. Rogov, Abstracts of the 9th All-Union Conf. on Linacs of Charged Particles (1985) p. 9.5. [5] V.A. Balakirev, V.I. Miroshnichenko and Ya.B. Faynberg. Fizika Plazmy 12 (1986)983. [h] NJ. Karbushev. V.P. Poponin and A.A. Rukhadzc. in: Generatory Kogerentnogo lzlucheniya na Svohodnykh Elektronakh, ed. A.A. Rukhadze (Mir. Moscow. 1983) p. 214.
[7] NJ. Karbushev and A.D. Shatkus. Nucl. Instr. and Meth. A 304 (1991) 559. [x] G.I. Batskikh. K.G. Gurcev. N.I. Karbushev and A.I. Lisitsyn, Proc. 8th Int. Conf. on High-Power Particlc Beams. Vol. 2 (1990) p. 1173. [9] N.I. Karbushev, Izv. Vuzov. Radiofizika 34 (1991) X3.