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Broadband instability in free electron lasers
Volume 79, number 1,2
OPTICS COMMUNICATIONS
1 October 1990
Broadband instability in free electron lasers Ju.B. V i k t o r o v , a, A.B. D r a g a n o v b, A.K. K a m i n s k i y a, N.Ya. K o t s a r e n k o b, S.B. R u b i n a, V.P. S a r a n t s e v a, A.P. Sergeev a a n d A.A. Silivra b " United Institute o f Nuclear Research, Moscow, USSR b Department ofTkeoretical Pkysics, Pkysical Faculty, Kiev State University, Kiev, 252022 USSR
Received 9 January 1990; revised manuscript received 10 May 1990
Experimental and theoretical investigation of some instabilities developing in the relativistic electron beam in the periodical magnetostatic pump is presented. It is shown, in particular, that under cyclotron resonance condition the generation of the electromagnetic waves is possible in a broadband regime. This broadband regime is due to development of cyclotron mode broadband instability not investigated earlier.
I. Introduction The possibility of generation or amplification of electromagnetic waves in relativistic electron beams via parametric interaction (FEL) attracts the interest o f scientists. As a rule, the p u m p wave in these devices is represented by a periodical magnetostatic field created by means of an assembly o f magnets or electric current. Instability developed in such a system and provided by parametrical coupling o f the electromagnetic wave with space charge waves of the relativistic electron beam is investigated in detail (see, for example, refs. [ 1-3] and list of references in ref. [ 1 ] ). The frequency when the amplification (or generation) of electromagnetic wave is possible, equals to ~
k w v l f f ( 1 - Vll/C ) ,
( 1)
where v, is the longitudinal component o f the electron speed; 2 n / k w is the spatial period of the p u m p wave. But under experimental conditions the electron beam is focused by relatively strong magnetic field. So when the time of electron passing o f one period o f p u m p wave approximately equals to the time of cyclotron revolution the transverse speed o f electrons increases, and consequently the growth rate of electromagnetic wave increases too, while the frequency of growing wave decreases and, therefore, expression ( 1 ) should be modified. But a theoretical investigation of this cyclotron resonance regime is usually fulfilled under a great number of simplifying assumptions. This leads to the fact that results of such investigation are applicable in a highly small region of possible experimental parameter values and, moreover, are not applicable close enough to the cyclotron resonance because new instabilities can occur in this case [4,5].
2. Experiment The scheme of an experimental set-up is shown in fig. I. The linear induction accelerator l produces an electron beam with the current I = 2 0 0 A; the electron energy ~= 1.5 MeV (relativistic factor 7 = 4 ) ; the energy spread A~/~= 2%; the time duration of impulse z = 200 ns; the bean radius rg=0.3 cm. The current impulse shape is close to right-angular, the typical oscillogram is shown in fig. 2 curve I. 0030-4018/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
81
Volume 79, number 1,2
OPTICS COMMUNICATIONS
{5
3
4
8'
7
_
1 October 1990
~_
./
\
# I
"\j 0
,/
g
! 300
Fig. 1. The experimental set-up.
P
~, ns
5
Fig. 2. The experimental oscillograms, h the current impulse; 2: the microwave power impulse in the first regime; 3: the microwave power impulse in the second regime.
The interaction region is waveguide 4 being a thin-wall stainless tube with inner diameter 2.9 cm. The magnetic field in the interaction region consists of the longitudinal magnetic field Ho < 10 kG created by solenoid 2 and transverse right hand helical field of wiggler 3 with value Hw< 5 kG. One period of helix equals to 7.2 cm. The wiggler field slowly increases from 0 to m a x i m u m value on the first five helix periods and then is constant on the interaction region length. In the longitudinal direction the interaction region is restricted by two diaphragms 5, 6 with the distance between them of 0 < L < 250 cm. In the interaction region entrance the relativistic electron beam current is controlled by Rogovskiy belt 7 and in the exit by Faraday cup 6. The radiation spectrum is determined by the assembly of semiconductor detectors with waveguides having different cut-offwavelengths within the limits of 3.4 m m < 202 < 11.9 mm. To decrease measured value spread connected with nonstability o f feeder voltages and beam current, detector 8 results are compared with the main detector 9 results capable to register radiation with the wavelength smaller than 11.9 mm. This results in measured value spread of < 10% being provided in independent experiments. Two strongly different regimes have been registered. When the longitudinal field Ho is directed opposite to the electron longitudinal speed direction, the radiation impulses being 50...80 ns lagged behind the beginning of current impulse, with time duration o f 100... 120 ns and the power o f approximately 2 M W are registered. The characteristic radiation impulse shape is shown in fig. 2, curve 2; the characteristic radiation spectrum shape is shown in fig. 3, curve 1. The electron beam radiation has been observed under the longitudinal field value o f Ho = 2.7 kG and in wide region of wiggler field strength o f 1.5 k G < H w < 3.0 kG with m a x i m u m of radiation when wiggler field strength is equal to 2.2 kG. Another oscillating regime has been observed when the longitudinal magnetic field direction coincided with the direction of longitudinal motion of electrons. As is shown in fig. 2, curve 3, radiation impulse shape prac-
dP d~
m /
-~/
J
3.4
82
/
\
\
\,"
"f2 0
Fig. 3. The typical view of the radiation spectrum in the narrowband ( 1) and broadband (2) regimes.
Volume 79, number 1,2
OPTICS COMMUNICATIONS
1 October 1990
tically repeats current impulse shape in this regime. But the spectrum bandwidth is much greater then in the previous case and is practically constant in the whole measured band (the characteristic radiation spectrum shape is shown in fig. 3 curve 2). The radiation power resonantly depends on both wiggler and longitudinal magnetic field strength. For example increasing or decreasing the wiggler field by 50...80 G from an optimum value leads to fatal decreasing of oscillation power, but corresponding change of longitudinal field allows to restore the oscillation. In such manner it is possible to reach the oscillating regime with the same level of power of the order of 2 MW in the wide band of wiggler field values from 150 G to 1000 G and focusing field values from 6 kG to 7.5 kG.
3. Theory An elementary theory of the processes observed can be built on the analysis of the interaction of plane electromagnetic waves in a relativistic electron beam. The electron beam is assumed to be compensated in both static space charge and direct current and homogeneous in transverse direction. Let us assume the external magnetic field is given by H = Hw( ex cos kwz + ey sin k,,,z ) + Hoez ,
(2)
where Hw is the amplitude of the wiggler field; Ho is the amplitude of the longitudinal magnetic field. It is well known that the electron can move in the field of such configuration with the helical trajectories [2,6] provided adiabatic condition of electron injection in such a field: Vxo=V± coskwz,
Vyo=v± s i n k ~ z ,
V~o=V,,
(3)
where v±, v, are the roots of the following system of equations, V2 + V ~ I = C 2 ( 1 - - y - 2 ) ,
V± =Vllcowl(con-yk~Vll
) ,
(4)
where cou= e H o / m c is the cyclotron frequency; COw= eHw/mC. The typical dependence of the longitudinal electron velocity for constant level of wiggler field is plotted in fig. 4 versus x,oH/ykwc. It should be noted that for negative value of the parameter x (that corresponds to a backward direction of the longitudinal magnetic field in regard to direction of the longitudinal electron velocity) there is only one branch of curve (4). The numerical simulation and analysis of stability point out that only first and second regions are experimentally realizable (see fig. 4). Linearizing of Maxwell equations and relativistic equation of motion and substituting solution in form of E_+ ~ exp [ i ( COt- kz + kwZ) ] for transverse waves and Ez ~ exp [ i ( COt- kz) ] for longitudinal waves results in an algebraic system of equations. To have a nontrivial solution of this system we should equate the determinant of this system to zero. Then the dispersion equation for waves propagating in such system takes the form
.S~ll "fO
Vlli/IIIII/II~
Fig. 4. The dependence of the normalized longitudinal velocity on the normalized longitudinal magnetic field, l, 2: stable trajectories; 3: unstable trajectories; shaded area: the region of the broadband instability existence.
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[n 2- (o92D) ( 1 -/3~, )l { (o92-c2k~+)(o9~-c 2k2 )(n+ n _ + J 2) -- ( 0 ) 2 / 7 ) [if2+ ((A)2 - c2k % ) ( o 9 - k _
+
(o92/32/27) [o9(g2+
71--(0`)4/729 [ ( o ) - k +
vii ) q-•2_ (0`) 2 - c Z k 2_ ) ( o 9 - k + L'Ij) ]
+ d ) ( ( 0 2 - c 2 k 2 ) nt- (.o(Q_ - z ~ ) ( o 9 2 - c2]¢2 )]
uii ) ( ( o - k
uii ) _ / 3 2 0,)•] }
+ [ - (o9w/27) (o92-c2k2+) + (og~,fl~./27) (ck+ -ogflll) ]{ ( 092-c2k2- ) [ -ckfl± (-Q+ £ 2 ~
A) +12~2w(.O_ +A) ]
1 + (og~/7)[ (ck/3±~+ -f2~2w)(og-k v,,) + flufl±~_(oge-c2k2 )+ ~f12 og(ckfl± (~ +.O+)+2~2.Qw) ]
- (o94/7~)/3,/3± ( o 9 - k _ ~',,)I H- [ (o9w/27) ( o 9 2 - c 2 k { ) H- (602/32 / 2 7 ) (ck - fll,og) 1{ (ogZ-c2k%) [ -ckfl~ (D+ £2_ - D _ J ) -~2~2w(£2+ - d ) ] + (0`)2/7) [ (ck/31 ~
+g2f2w)(o9-k+ vu) +13,1/3±~+ (ogZ-c2k2) + ½/3~o9(ck/3±(.O+ - ~
- (o9U72 )/3~/31 (o9-k+ v~) } = O,
) - 2s'2£2w) ]
(5)
where
k+ =k-T-kw,
~+=f2++_J,
£2=o9-ku11, .62+=og-k+vllT-ogH/7+_J, /3±=v±/c,
d=/3~ 7(/3~ ogH-/311o9w)/2,
£2w=ckw/3± +ogw/Y+fl± 7(//± OgH--/311Ogw) , O92=4nnbe2/m.
In assuming 0.)8/3± 2 2 /~(A)H 2 <( 1 this equation takes the form [ (O)--kuii )2--0`)2/73 ] [ (0.92-- c2k 2 ) (0,)_ k+ uii --(DH/7) - ( . 0 2 ( o , ) - k +
X [ ( o 9 2 - c 2 k 2_ ) ( o 9 - k
Ull q-OgH/7) --0)2( o 9 - k
Pll ) / 7 ] --~0 .
Ull ) / ~ ] (6)
In the latter equation the first term evidently describes space charge waves of the relativistic electron beam, and the second and the third terms describe consequently right- and left-hand polarized pure electromagnetic mode coupled passively with fast and slow cyclotron modes of the relativistic electron beam. Note that only transverse waves have the parametrical shift of dispersion on + k.~, moreover, the direction of this shift is determined by the polarization of these waves. The dispersion curve scheme is shown in fig. 5. It should be noted that reversing of longitudinal magnetic field results in the transverse wave dispersion curves exchanging places with each other. The three groups of wave mentioned above are eigenmodes of the system considered, therefore, the points of crossing of different branches of curves are the points of synchronism. In the synchronism points' vicinity the instability can occur. Usually the instability developing under parametrical coupling of electromagnetic wave with space charge waves of the relativistic electron beam is investigated. For relatively small amplitude of transverse electron velocity the solution of (5) gives the real part of synchronism frequency close to value ( 1 ) and the imaginary part of it is
% 84
Fig. 5. The scheme of the dispersion curves for the waves propagating in the considered system. 1: direct and counter left-hand em wave; 2: slow cyclotron mode; 3: direct and counter righthand em wave; 4: fast cyclotron mode; 5" fast and slow space charge modes.
Volume 79, number 1,2
-Imo=
O 473/4
w
OPTICS COMMUNICATIONS
~
1 October 1990
(7)
IOH/7--kwUll I
The criterion of this regime when the slow space charge wave takes part in the interaction separately (the so called Raman regime) is
o2kwvJ32 (OH~7--k,~vll)2 < <
(.0b/~3/2
.
The so called Compton regime, i.e. regime when one cannot separate fast and slow space charge waves, has the instability rate -Imo=
"v/3 2~x / 2 ~ " ("
4
7
ObOw
,2/3
\OH/7--kwvbl/
(8)
and the criterion of this regime is
33/202wkwPll / 162( o . / 7 _ ] G , Pll )2 :>> 0b/73/2. It should be noted that the considered instability occurs with any direction of longitudinal magnetic field but the difference between these two cases is connected not only with the corresponding sign of ot~ in eqs. (5), (6) but also with correct consideration of the longitudinal velocity vii dependence on OH (see fig. 4). The synchronism point of the left-hand electromagnetic mode with slow cyclotron mode (of right-hand polarization) can have even higher frequency value o~
2kw v, - oH/7
(9)
1- v , / c
The temporal rate of this instability ~.~2 b ~, R3/2
-Imo~
Ob
.....
t1~,
7 3/2 IO.)H/7--kwUII 13
(10)
grows substantially near the cyclotron resonance. Nevertheless for usual experimental conditions the temporal rate is smaller than the one of instability of the space charge mode (7), (8). In contrast to the previous cases this point of synchronism and, consequently, this instability disappears for backward direction of the longitudinal magnetic field.
4. T h e cyclotron resonance case
Now let us consider the cyclotron resonance case. Earlier in the number of papers (see, for example, refs. [7-9] ) it has been pointed out that near the cyclotron resonance kwv, ,~ o , / y
( 11 )
the instability rate grows. Expressions (7), (8), (10) evidently imply this fact, but, as can be seen from eq. (6), under condition ( 11 ) the physical picture of wave interaction in an relativistic electron beam proves to be more complex and rich, because new instability of the cyclotron modes, noninvestigated earlier, arises. As is seen from fig. 5 under condition ( 11 ) the dispersion curves of the different cyclotron modes draw together and, since asymptotically they are parallel lines, they have no point of crossing (point of synchronism) but coincide and, hence, the developing instability becomes broadband. Mathematically it means that the corresponding system of equations to determine the synchronism frequency
o--kVll =OH/7--kwVll , o--kVll = --OH/7+ kwVll , 85
Volume 79, number 1,2
_ Z'~
OPTICS COMMUNICATIONS
1 October 1990
l
_2¸
/f
Fig. 6. The dependence of the broadband instability temporal growth rate on frequency (numerical solution). 7=4, Oh= 1.76× 1010S [,h=5.46kG, 2w=7.2cm, Hw=340G.
becomes degenerated. Within the limit of sufficiently high frequency (co >> con/)', kwc) one can find this instability rate in form - - I m c o ~ [ cowcoHfl± ( ~ L
27fl,i
/4co2fll~'~((l~fll~)74+ l-l)]
\ Nl
I/2
(12)
cowconfl±
The typical dependence of the instability rate obtained via numerical solution of (5) is shown in fig. 6. It is seen that really for sufficiently high frequencies the instability rate does not depend on frequency. It should be noted that similar dependence of the instability rate is obtained in ref. [2]. Although the frequency of this broadband instability is not restricted in such a model, it is evident that on the frequency of order co~7211mcol (Ae/e)-~ (where Ae/e is the thermal energy spread of the relativistic electron beam) this instability cannot develop. Note that besides the instabilities considered in this paper another relatively low frequency instabilities on the backward electromagnetic modes can take places in free electron lasers.
5. Discussion
For the instability developing in the experiment one passing enhancement of the single is to exceed the losses in waveguide wall and losses of reflecting of part of the signal to create the positive feedback. This situation has been evidently realized under two regimes in our experiments. The feature of the first regime is that the instability has been developed under relatively high wiggler field amplitude. This is connected with the fact that the transverse electron speed is relatively small under the backward direction of the longitudinal magnetic field (see fig. 4 for x < 0). That is why in order to reach the threshold instability rate it is necessary to increase the wiggler field. But with further increasing of the wiggler field the longitudinal electron speed decreases, and consequently the frequency of the instability according to ( 1 ) decreases. This leads to the fact that the radiation frequency becomes lower than the cut-off frequency of the waveguide leading to detector, and therefore the radiation cannot be registered. The second experimental regime similar to those described in ref. [ 10] corresponds apparently to the development of the wideband instability. The numerical solution of dispersion eq. (5) under condition ( 11 ) shows that this instability is possible within the definite interval of the longitudinal magnetic field values inside of which the instability rate has maximum. This interval is marked in fig. 4. Moreover the whole band of the wiggler field value corresponds to used longitudinal magnetic field value. Thus it is possible to "compensate" within the definite limits the alteration of one field by means of another magnetic field alteration as the experiment shows. Under the experimental conditions in the second regime the longitudinal electron velocity varied within the limits of 0.8 < fill< 0.94, but the generated radiation spectrum did not vary practically. Moreover, in the both experimental regimes the longitudinal electron speed values were equal (fl± = 0.23; fl, = 0.94). The great dif86
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ference of radiation spectrum confirms the principle difference of physical p h e n o m e n a in these regimes. At the same time the presence of short-wave c o m p o n e n t with 2 < 7 m m in the radiation spectrum of the second regime when ~li = 0.94 contradicts expression ( 1 ) a n d can be explained by the b r o a d b a n d instability of the cyclotron modes described above. One has to notice that the regime similar to latter one was registered in ref. [ 10 ].
6. Conclusion Thus two principally different regimes can occur u n d e r parametrical interaction of electromagnetic waves in relativistic electron b e a m depending on the longitudinal and wiggler magnetic field values. If the direction of the longitudinal magnetic field is backward to the longitudinal direction of electron speed, the instability of the electromagnetic wave parametrically interacting with space charge modes of the relativistic electron beam develops near the frequency ( 1 ). The experimentally registered radiation spectrum is narrowb a n d in this case. If the direction of the longitudinal magnetic field coincides with the longitudinal electron speed direction and its value is close enough to the cyclotron resonance condition ( 1 1 ), the instability of the cyclotron modes occurs in a broad frequency band. Practically flat radiation spectrum in the b a n d from 4.5 m m to 1 1.9 m m has been registered experimentally in this case.
References [ 1] T. Marshall, Free electron lasers (McMillan, 1985). [2] I.B. Bernstein and L. Friedland, Phys. Rev. A 23 ( 1981 ) 816. [ 3 ] J.A. Pasour and S.H. Gold, IEEE J. Quantum Electron. QE-21 (1985) 845. [4 ] N.Ya. Kotsarenko and A.A. Silivra, Izv. VUZov Radiofizika (in Russian) 30 ( 1987 ) 1279. [5] A.B. Draganov, N.Ya. Kotsarenko and A.A. Silivra, Soy. Phys. JTP (in Russian) 58 (1988) 928. [6 ] H.P. Freund, P. Sprangle, D. Dillenburg, E,H. da Jornada, B. Liberman and R.S. Shneider, Phys. Rev. A 24 ( 1981 ) 1965. [7] P. Sprangleand V.L. Granatstein, Appl. Phys. Let. 25 (1974) 377. [8] T. Shiozava, J. Appl. Phys. 54 (1983) 3712. [9] J. Fajan, NIMPR 250A (1986) 342. [ 10] K.L. Felch, L. Vallier,J.M. Buzzi, P. Drossart, H. Boehmer, H.J. Doucet, B. Etlicher, H. Lamain and C. Roville, IEEE J. Quantum Electron. QE-17 (1981) 1354.
87
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Broadband instability in free electron lasers Ju.B. V i k t o r o v , a, A.B. D r a g a n o v b, A.K. K a m i n s k i y a, N.Ya. K o t s a r e n k o b, S.B. R u b i n a, V.P. S a r a n t s e v a, A.P. Sergeev a a n d A.A. Silivra b " United Institute o f Nuclear Research, Moscow, USSR b Department ofTkeoretical Pkysics, Pkysical Faculty, Kiev State University, Kiev, 252022 USSR
Received 9 January 1990; revised manuscript received 10 May 1990
Experimental and theoretical investigation of some instabilities developing in the relativistic electron beam in the periodical magnetostatic pump is presented. It is shown, in particular, that under cyclotron resonance condition the generation of the electromagnetic waves is possible in a broadband regime. This broadband regime is due to development of cyclotron mode broadband instability not investigated earlier.
I. Introduction The possibility of generation or amplification of electromagnetic waves in relativistic electron beams via parametric interaction (FEL) attracts the interest o f scientists. As a rule, the p u m p wave in these devices is represented by a periodical magnetostatic field created by means of an assembly o f magnets or electric current. Instability developed in such a system and provided by parametrical coupling o f the electromagnetic wave with space charge waves of the relativistic electron beam is investigated in detail (see, for example, refs. [ 1-3] and list of references in ref. [ 1 ] ). The frequency when the amplification (or generation) of electromagnetic wave is possible, equals to ~
k w v l f f ( 1 - Vll/C ) ,
( 1)
where v, is the longitudinal component o f the electron speed; 2 n / k w is the spatial period of the p u m p wave. But under experimental conditions the electron beam is focused by relatively strong magnetic field. So when the time of electron passing o f one period o f p u m p wave approximately equals to the time of cyclotron revolution the transverse speed o f electrons increases, and consequently the growth rate of electromagnetic wave increases too, while the frequency of growing wave decreases and, therefore, expression ( 1 ) should be modified. But a theoretical investigation of this cyclotron resonance regime is usually fulfilled under a great number of simplifying assumptions. This leads to the fact that results of such investigation are applicable in a highly small region of possible experimental parameter values and, moreover, are not applicable close enough to the cyclotron resonance because new instabilities can occur in this case [4,5].
2. Experiment The scheme of an experimental set-up is shown in fig. I. The linear induction accelerator l produces an electron beam with the current I = 2 0 0 A; the electron energy ~= 1.5 MeV (relativistic factor 7 = 4 ) ; the energy spread A~/~= 2%; the time duration of impulse z = 200 ns; the bean radius rg=0.3 cm. The current impulse shape is close to right-angular, the typical oscillogram is shown in fig. 2 curve I. 0030-4018/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
81
Volume 79, number 1,2
OPTICS COMMUNICATIONS
{5
3
4
8'
7
_
1 October 1990
~_
./
\
# I
"\j 0
,/
g
! 300
Fig. 1. The experimental set-up.
P
~, ns
5
Fig. 2. The experimental oscillograms, h the current impulse; 2: the microwave power impulse in the first regime; 3: the microwave power impulse in the second regime.
The interaction region is waveguide 4 being a thin-wall stainless tube with inner diameter 2.9 cm. The magnetic field in the interaction region consists of the longitudinal magnetic field Ho < 10 kG created by solenoid 2 and transverse right hand helical field of wiggler 3 with value Hw< 5 kG. One period of helix equals to 7.2 cm. The wiggler field slowly increases from 0 to m a x i m u m value on the first five helix periods and then is constant on the interaction region length. In the longitudinal direction the interaction region is restricted by two diaphragms 5, 6 with the distance between them of 0 < L < 250 cm. In the interaction region entrance the relativistic electron beam current is controlled by Rogovskiy belt 7 and in the exit by Faraday cup 6. The radiation spectrum is determined by the assembly of semiconductor detectors with waveguides having different cut-offwavelengths within the limits of 3.4 m m < 202 < 11.9 mm. To decrease measured value spread connected with nonstability o f feeder voltages and beam current, detector 8 results are compared with the main detector 9 results capable to register radiation with the wavelength smaller than 11.9 mm. This results in measured value spread of < 10% being provided in independent experiments. Two strongly different regimes have been registered. When the longitudinal field Ho is directed opposite to the electron longitudinal speed direction, the radiation impulses being 50...80 ns lagged behind the beginning of current impulse, with time duration o f 100... 120 ns and the power o f approximately 2 M W are registered. The characteristic radiation impulse shape is shown in fig. 2, curve 2; the characteristic radiation spectrum shape is shown in fig. 3, curve 1. The electron beam radiation has been observed under the longitudinal field value o f Ho = 2.7 kG and in wide region of wiggler field strength o f 1.5 k G < H w < 3.0 kG with m a x i m u m of radiation when wiggler field strength is equal to 2.2 kG. Another oscillating regime has been observed when the longitudinal magnetic field direction coincided with the direction of longitudinal motion of electrons. As is shown in fig. 2, curve 3, radiation impulse shape prac-
dP d~
m /
-~/
J
3.4
82
/
\
\
\,"
"f2 0
Fig. 3. The typical view of the radiation spectrum in the narrowband ( 1) and broadband (2) regimes.
Volume 79, number 1,2
OPTICS COMMUNICATIONS
1 October 1990
tically repeats current impulse shape in this regime. But the spectrum bandwidth is much greater then in the previous case and is practically constant in the whole measured band (the characteristic radiation spectrum shape is shown in fig. 3 curve 2). The radiation power resonantly depends on both wiggler and longitudinal magnetic field strength. For example increasing or decreasing the wiggler field by 50...80 G from an optimum value leads to fatal decreasing of oscillation power, but corresponding change of longitudinal field allows to restore the oscillation. In such manner it is possible to reach the oscillating regime with the same level of power of the order of 2 MW in the wide band of wiggler field values from 150 G to 1000 G and focusing field values from 6 kG to 7.5 kG.
3. Theory An elementary theory of the processes observed can be built on the analysis of the interaction of plane electromagnetic waves in a relativistic electron beam. The electron beam is assumed to be compensated in both static space charge and direct current and homogeneous in transverse direction. Let us assume the external magnetic field is given by H = Hw( ex cos kwz + ey sin k,,,z ) + Hoez ,
(2)
where Hw is the amplitude of the wiggler field; Ho is the amplitude of the longitudinal magnetic field. It is well known that the electron can move in the field of such configuration with the helical trajectories [2,6] provided adiabatic condition of electron injection in such a field: Vxo=V± coskwz,
Vyo=v± s i n k ~ z ,
V~o=V,,
(3)
where v±, v, are the roots of the following system of equations, V2 + V ~ I = C 2 ( 1 - - y - 2 ) ,
V± =Vllcowl(con-yk~Vll
) ,
(4)
where cou= e H o / m c is the cyclotron frequency; COw= eHw/mC. The typical dependence of the longitudinal electron velocity for constant level of wiggler field is plotted in fig. 4 versus x,oH/ykwc. It should be noted that for negative value of the parameter x (that corresponds to a backward direction of the longitudinal magnetic field in regard to direction of the longitudinal electron velocity) there is only one branch of curve (4). The numerical simulation and analysis of stability point out that only first and second regions are experimentally realizable (see fig. 4). Linearizing of Maxwell equations and relativistic equation of motion and substituting solution in form of E_+ ~ exp [ i ( COt- kz + kwZ) ] for transverse waves and Ez ~ exp [ i ( COt- kz) ] for longitudinal waves results in an algebraic system of equations. To have a nontrivial solution of this system we should equate the determinant of this system to zero. Then the dispersion equation for waves propagating in such system takes the form
.S~ll "fO
Vlli/IIIII/II~
Fig. 4. The dependence of the normalized longitudinal velocity on the normalized longitudinal magnetic field, l, 2: stable trajectories; 3: unstable trajectories; shaded area: the region of the broadband instability existence.
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Volume 79, number 1,2
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[n 2- (o92D) ( 1 -/3~, )l { (o92-c2k~+)(o9~-c 2k2 )(n+ n _ + J 2) -- ( 0 ) 2 / 7 ) [if2+ ((A)2 - c2k % ) ( o 9 - k _
+
(o92/32/27) [o9(g2+
71--(0`)4/729 [ ( o ) - k +
vii ) q-•2_ (0`) 2 - c Z k 2_ ) ( o 9 - k + L'Ij) ]
+ d ) ( ( 0 2 - c 2 k 2 ) nt- (.o(Q_ - z ~ ) ( o 9 2 - c2]¢2 )]
uii ) ( ( o - k
uii ) _ / 3 2 0,)•] }
+ [ - (o9w/27) (o92-c2k2+) + (og~,fl~./27) (ck+ -ogflll) ]{ ( 092-c2k2- ) [ -ckfl± (-Q+ £ 2 ~
A) +12~2w(.O_ +A) ]
1 + (og~/7)[ (ck/3±~+ -f2~2w)(og-k v,,) + flufl±~_(oge-c2k2 )+ ~f12 og(ckfl± (~ +.O+)+2~2.Qw) ]
- (o94/7~)/3,/3± ( o 9 - k _ ~',,)I H- [ (o9w/27) ( o 9 2 - c 2 k { ) H- (602/32 / 2 7 ) (ck - fll,og) 1{ (ogZ-c2k%) [ -ckfl~ (D+ £2_ - D _ J ) -~2~2w(£2+ - d ) ] + (0`)2/7) [ (ck/31 ~
+g2f2w)(o9-k+ vu) +13,1/3±~+ (ogZ-c2k2) + ½/3~o9(ck/3±(.O+ - ~
- (o9U72 )/3~/31 (o9-k+ v~) } = O,
) - 2s'2£2w) ]
(5)
where
k+ =k-T-kw,
~+=f2++_J,
£2=o9-ku11, .62+=og-k+vllT-ogH/7+_J, /3±=v±/c,
d=/3~ 7(/3~ ogH-/311o9w)/2,
£2w=ckw/3± +ogw/Y+fl± 7(//± OgH--/311Ogw) , O92=4nnbe2/m.
In assuming 0.)8/3± 2 2 /~(A)H 2 <( 1 this equation takes the form [ (O)--kuii )2--0`)2/73 ] [ (0.92-- c2k 2 ) (0,)_ k+ uii --(DH/7) - ( . 0 2 ( o , ) - k +
X [ ( o 9 2 - c 2 k 2_ ) ( o 9 - k
Ull q-OgH/7) --0)2( o 9 - k
Pll ) / 7 ] --~0 .
Ull ) / ~ ] (6)
In the latter equation the first term evidently describes space charge waves of the relativistic electron beam, and the second and the third terms describe consequently right- and left-hand polarized pure electromagnetic mode coupled passively with fast and slow cyclotron modes of the relativistic electron beam. Note that only transverse waves have the parametrical shift of dispersion on + k.~, moreover, the direction of this shift is determined by the polarization of these waves. The dispersion curve scheme is shown in fig. 5. It should be noted that reversing of longitudinal magnetic field results in the transverse wave dispersion curves exchanging places with each other. The three groups of wave mentioned above are eigenmodes of the system considered, therefore, the points of crossing of different branches of curves are the points of synchronism. In the synchronism points' vicinity the instability can occur. Usually the instability developing under parametrical coupling of electromagnetic wave with space charge waves of the relativistic electron beam is investigated. For relatively small amplitude of transverse electron velocity the solution of (5) gives the real part of synchronism frequency close to value ( 1 ) and the imaginary part of it is
% 84
Fig. 5. The scheme of the dispersion curves for the waves propagating in the considered system. 1: direct and counter left-hand em wave; 2: slow cyclotron mode; 3: direct and counter righthand em wave; 4: fast cyclotron mode; 5" fast and slow space charge modes.
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-Imo=
O 473/4
w
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1 October 1990
(7)
IOH/7--kwUll I
The criterion of this regime when the slow space charge wave takes part in the interaction separately (the so called Raman regime) is
o2kwvJ32 (OH~7--k,~vll)2 < <
(.0b/~3/2
.
The so called Compton regime, i.e. regime when one cannot separate fast and slow space charge waves, has the instability rate -Imo=
"v/3 2~x / 2 ~ " ("
4
7
ObOw
,2/3
\OH/7--kwvbl/
(8)
and the criterion of this regime is
33/202wkwPll / 162( o . / 7 _ ] G , Pll )2 :>> 0b/73/2. It should be noted that the considered instability occurs with any direction of longitudinal magnetic field but the difference between these two cases is connected not only with the corresponding sign of ot~ in eqs. (5), (6) but also with correct consideration of the longitudinal velocity vii dependence on OH (see fig. 4). The synchronism point of the left-hand electromagnetic mode with slow cyclotron mode (of right-hand polarization) can have even higher frequency value o~
2kw v, - oH/7
(9)
1- v , / c
The temporal rate of this instability ~.~2 b ~, R3/2
-Imo~
Ob
.....
t1~,
7 3/2 IO.)H/7--kwUII 13
(10)
grows substantially near the cyclotron resonance. Nevertheless for usual experimental conditions the temporal rate is smaller than the one of instability of the space charge mode (7), (8). In contrast to the previous cases this point of synchronism and, consequently, this instability disappears for backward direction of the longitudinal magnetic field.
4. T h e cyclotron resonance case
Now let us consider the cyclotron resonance case. Earlier in the number of papers (see, for example, refs. [7-9] ) it has been pointed out that near the cyclotron resonance kwv, ,~ o , / y
( 11 )
the instability rate grows. Expressions (7), (8), (10) evidently imply this fact, but, as can be seen from eq. (6), under condition ( 11 ) the physical picture of wave interaction in an relativistic electron beam proves to be more complex and rich, because new instability of the cyclotron modes, noninvestigated earlier, arises. As is seen from fig. 5 under condition ( 11 ) the dispersion curves of the different cyclotron modes draw together and, since asymptotically they are parallel lines, they have no point of crossing (point of synchronism) but coincide and, hence, the developing instability becomes broadband. Mathematically it means that the corresponding system of equations to determine the synchronism frequency
o--kVll =OH/7--kwVll , o--kVll = --OH/7+ kwVll , 85
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l
_2¸
/f
Fig. 6. The dependence of the broadband instability temporal growth rate on frequency (numerical solution). 7=4, Oh= 1.76× 1010S [,h=5.46kG, 2w=7.2cm, Hw=340G.
becomes degenerated. Within the limit of sufficiently high frequency (co >> con/)', kwc) one can find this instability rate in form - - I m c o ~ [ cowcoHfl± ( ~ L
27fl,i
/4co2fll~'~((l~fll~)74+ l-l)]
\ Nl
I/2
(12)
cowconfl±
The typical dependence of the instability rate obtained via numerical solution of (5) is shown in fig. 6. It is seen that really for sufficiently high frequencies the instability rate does not depend on frequency. It should be noted that similar dependence of the instability rate is obtained in ref. [2]. Although the frequency of this broadband instability is not restricted in such a model, it is evident that on the frequency of order co~7211mcol (Ae/e)-~ (where Ae/e is the thermal energy spread of the relativistic electron beam) this instability cannot develop. Note that besides the instabilities considered in this paper another relatively low frequency instabilities on the backward electromagnetic modes can take places in free electron lasers.
5. Discussion
For the instability developing in the experiment one passing enhancement of the single is to exceed the losses in waveguide wall and losses of reflecting of part of the signal to create the positive feedback. This situation has been evidently realized under two regimes in our experiments. The feature of the first regime is that the instability has been developed under relatively high wiggler field amplitude. This is connected with the fact that the transverse electron speed is relatively small under the backward direction of the longitudinal magnetic field (see fig. 4 for x < 0). That is why in order to reach the threshold instability rate it is necessary to increase the wiggler field. But with further increasing of the wiggler field the longitudinal electron speed decreases, and consequently the frequency of the instability according to ( 1 ) decreases. This leads to the fact that the radiation frequency becomes lower than the cut-off frequency of the waveguide leading to detector, and therefore the radiation cannot be registered. The second experimental regime similar to those described in ref. [ 10] corresponds apparently to the development of the wideband instability. The numerical solution of dispersion eq. (5) under condition ( 11 ) shows that this instability is possible within the definite interval of the longitudinal magnetic field values inside of which the instability rate has maximum. This interval is marked in fig. 4. Moreover the whole band of the wiggler field value corresponds to used longitudinal magnetic field value. Thus it is possible to "compensate" within the definite limits the alteration of one field by means of another magnetic field alteration as the experiment shows. Under the experimental conditions in the second regime the longitudinal electron velocity varied within the limits of 0.8 < fill< 0.94, but the generated radiation spectrum did not vary practically. Moreover, in the both experimental regimes the longitudinal electron speed values were equal (fl± = 0.23; fl, = 0.94). The great dif86
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ference of radiation spectrum confirms the principle difference of physical p h e n o m e n a in these regimes. At the same time the presence of short-wave c o m p o n e n t with 2 < 7 m m in the radiation spectrum of the second regime when ~li = 0.94 contradicts expression ( 1 ) a n d can be explained by the b r o a d b a n d instability of the cyclotron modes described above. One has to notice that the regime similar to latter one was registered in ref. [ 10 ].
6. Conclusion Thus two principally different regimes can occur u n d e r parametrical interaction of electromagnetic waves in relativistic electron b e a m depending on the longitudinal and wiggler magnetic field values. If the direction of the longitudinal magnetic field is backward to the longitudinal direction of electron speed, the instability of the electromagnetic wave parametrically interacting with space charge modes of the relativistic electron beam develops near the frequency ( 1 ). The experimentally registered radiation spectrum is narrowb a n d in this case. If the direction of the longitudinal magnetic field coincides with the longitudinal electron speed direction and its value is close enough to the cyclotron resonance condition ( 1 1 ), the instability of the cyclotron modes occurs in a broad frequency band. Practically flat radiation spectrum in the b a n d from 4.5 m m to 1 1.9 m m has been registered experimentally in this case.
References [ 1] T. Marshall, Free electron lasers (McMillan, 1985). [2] I.B. Bernstein and L. Friedland, Phys. Rev. A 23 ( 1981 ) 816. [ 3 ] J.A. Pasour and S.H. Gold, IEEE J. Quantum Electron. QE-21 (1985) 845. [4 ] N.Ya. Kotsarenko and A.A. Silivra, Izv. VUZov Radiofizika (in Russian) 30 ( 1987 ) 1279. [5] A.B. Draganov, N.Ya. Kotsarenko and A.A. Silivra, Soy. Phys. JTP (in Russian) 58 (1988) 928. [6 ] H.P. Freund, P. Sprangle, D. Dillenburg, E,H. da Jornada, B. Liberman and R.S. Shneider, Phys. Rev. A 24 ( 1981 ) 1965. [7] P. Sprangleand V.L. Granatstein, Appl. Phys. Let. 25 (1974) 377. [8] T. Shiozava, J. Appl. Phys. 54 (1983) 3712. [9] J. Fajan, NIMPR 250A (1986) 342. [ 10] K.L. Felch, L. Vallier,J.M. Buzzi, P. Drossart, H. Boehmer, H.J. Doucet, B. Etlicher, H. Lamain and C. Roville, IEEE J. Quantum Electron. QE-17 (1981) 1354.
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