Principles of Gyrotron Powered Electromagnetic Wieelers for Free

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-23, NO. 1, JANUARY 1987 103 Principles of Gyrotron Powered Electromagnetic

Wieelersvu for Free-Electron Lasers

B.G. DANLY, G. BEKEFI, R. C. DAVIDSON, R J. TEMKIN, T. M.TRAN, AND J. S. WURTELE

Abstract-The operation of free-electron lasers (FEL’s) with axial An alternate concept to the conventional FEL which has electron beams and high-power electromagnetic wiggler fields such as received both considerable experimental and theoretical those produced by high-power gyrotrons is discussed. The use of short attention recently is the use of an electromagnetic wave wavelength electromagnetic wigglers in waveguides and resonant cavities can significantly reduce required electron beam voltages, resulting as the wiggler rather than the field from a magnetostatic in compact FEL’s. Gain calculations in the low- and high-gain Comp- wiggler [11]-[16]. In this case, if thewiggler wave is ton regime are presented, including the effects of emittance, transverse propagating in freespace, the resonance condition be- wiggler gradient, and electron temperature. Optimized scaling laws comes k, = 4y2k(l + a:,,)-’, where k is now the wave- for the FEL gain and the required electromagnetic wiggler field power number of the electromagnetic wave. The ability to use are discussed. Several possible configurations for FEL’s with electromagnetic wigglers powered by millimeter wavelength gyrotrons are short wavelength electromagnetic waves as the wiggler presented. Gyrotron powered wigglers appear promising for operation field would allow a substantial reduction of the beam en- of compact FEL’s in the infrared regime using moderate energy (<10 ergy necessary for obtaining any given laser frequency. MeV) electron beams. FEL operation at lowerbeam energy has many major technological implications,including the availability of alternate accelerator technologies, reduced activation and I. INTRODUCTION radiation hazard,reduced system size, and reduced HE free-electron laser (FEL) has been shown to be a shielding requirements. Tpromising source of radiation over a broad range of Recently, novel short period electromagnetic wigglers frequencies from the optical to the millimeter range [1]- [ 171, [18] have also been proposed. Both these magneto- [ 101. These FEL’s have all employedmagnetostatic wig- static wigglers and the magnetostatic wiggler discussed glers made from either permanent magnets or electromag- here can make a significant impact on FEL technology. nets. This paper is concerned with the principles of elec- There are, however, significant differences between electromagnetic (time-varying) wigglers [ 111-[161. We tromagnetic wigglers and short period magnetostatic wig- consider electromagnetic wigglers powered by high-power glers. Although theshort period magnetostaticwiggler sources of millimeter wave radiation such as the gyrotron may be more straightforward to construct,.the electro- or cyclotron autoresonance maser. magnetic (EM) wiggler has several advantages. For relativistic electron beams, the resonancecondition The EM wiggler can have an appreciably larger phase for the conventional FEL with a magnetostatic wiggler space acceptance. Because the transverse wigglerfield in- can be expressedas = 2y2k,(l + ai))-’,where k,(k,) homogeniety can be varied independently of the wiggler is the optical (wiggler) wavenumber, y is the relativistic periodicity, the transverse variation can be made small; factor, y = (1 - /3: - /3;)-”*, and a, is the normalized this allows the use of larger radius and larger emittance vector potential of the wiggler magnetostatic field, B,(a, beams. = eB,/mc2k,). Because the wavelength of magnetostatic One advantage of the magnetostatic (MS) wiggler is that wiggler fields is usually limited to X, = 27r/k, 1 2 cm, the period is well determined and reproducible. The source the conventional FEL requires high-energy electron beams of the electromagnetic radiation for theEM wiggler must typically in excess of 40 MeV to reach the visible- and have a high degree of frequency stability. Gyrotrons have near-IR portions of the spectrum. such a high degree of stability; they are thus well suited as an EM wiggler power source. There exists little oper- Manuscript received May 9, 1986; revised August 7, 1986. This work ational experience with the cyclotron autoresonance maser was supported in part by the Department of Energy, the National Science to date, but it may also be attractive after further devel- Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research. The work of T. M. Tran was also supported by the Swiss opment. National Science Foundation. The advantages of FEL operation with electromagnetic B. G. Danly, G. Bekefi, R. C. Davidson, R. J. Temkin, and J. S. Wur- pump (wiggler) waves has led to the study of several novel tele are with the Plasma Fusion Center and the Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139. configurations. One important and promising concept is T. M. Tran is with CRPP, Association Euratom-Confederation Suisse, that of the two-stage FEL [11]-[15] in which a conven- CH-1007, Lausanne, Switzerland, on leave at the Plasma Fusion Center tional magnetostatic wiggler produces an electromagnetic andthe Department of Physics, Massachusetts Institute of Technology, , Cambridge, MA 02139. wave in the first stage, and this wave is then used as a IEEE Log Number 861 1214. pump wave utilizing the same electron beam in a second

0018-9197/87/0100-0103$01.00 O 1987 IEEE 104 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-23, NO. 1. JANUARY 1987 stage. In this paper, we present an analysis of an FEL fects of e-beam emittance and transverse wiggler field in- configuration which also utilizes two stages and an elec- homogeneity are discussed. The acceptance of the FEL tromagnetic wiggler.However, the concept [16] dis- with a TE1, mode electromagnetic wiggler is derived. The cussed here differs from the concept discussed above in gain scaling with beam brightness, number of wiggler pe- that separate electron beams are used for each stage, and riods, and current are discussed, and sample gain calcu- a cyclotron resonance (gyrotron) interaction mechanism lations are presented. produces the wiggler electromagnetic field. The genera- In Section 111, the wiggler field strength is calculated tion of an electromagnetic pump wave by a gyrotron re- for two configurations, traveling wave TE,, waveguide quires only a low to moderate voltage electron beam, from modes, and standing wave TEIn1cavity modes. The de- fifty to a few hundred kilovolts, depending on the desired pendence of the field strength on source power, mode, electromagnetic field strength. The electromagnetic wave cavity configuration, wiggler wavelength and number of produced by the gyrotron is then stored as an electromag- wiggler periods is discussed. In Section IV, several con- netic standing wave acting as the wiggler. Although the figurations for an FEL with a gyrotron powered EM wig- gyrotron has been selected here as the prototype device gler are proposed and discussed, and conclusions are pre- for producing the electromagnetic wiggler field, most of sented. All of this study deals with thesingle particle the results for FEL gain obtained in this paper would also Compton regime. apply to EM wigglers produced by other high-power sources of millimeter wave radiation. 11. ELECTROMAGNETICWIGGLER FEL GAIN The use of the gyrotron interaction for the first stage A. Low-Gain Compton Regime would benefit from recent progress in gyrotron oscilla- The gain for an FEL with an electromagnetic wiggler tors. Gyrotrons are capable of routinely producing high- operating in the Compton regime isoptimized in this sec- power at high frequency(millimeter and submillimeter tion. The FEL resonance condition for a relativistic elec- bands) with high efficiency [ 191, [20]. A one megawatt, tron beam with energy ymc2 (U[MeV] = 0.51 l(y - 1)) 140 GHz gyrotron is currentlyunder development at propagating in anelectromagnetic wiggler fieldof fre- 2 112 . M.I.T. [19], and a 2 MW, 3 mm waveiength pulsed gy- quency w = ck = 27rc/X = c(ki + k I) 1s given by rotron has been demonstrated in the Soviet Union [20]. Alternatively, one could use relativistic cyclotron res- k, = 2y2k,(l + atv)-’ (1) onance devices, such as the relativistic gyrotron or the where k, = k + k,, and a, is the normalized vector po- cyclotron autoresonance maser (also known as the wig- tential of thewiggler field, defined here by = eB,J gler-free FEL) [21] to produce very high-power electro- mc2kll.Here k = (ki + k:)”2 is the wavenumber of the magnetic pulses in the millimeter or submillimeter band. wiggler field with kll and ki as its axial and transverse The power would propagate through the second stage FEL components. The inclusion of kL is necessary when con- interaction region as a traveling wave in a waveguide; the sidering electromagnetic wiggler fields in waveguides and electromagnetic wiggler is then a traveling wave rather resonant cavities, as is the case in this paper. The FEL than a standing wave in a high Q-cavity . The relativistic gain is a strong function of a,(G cc ut,); the calculation gyrotron has received considerable attention and several of reasonable values for a, is deferred until the next sec- experimentshave achieved impressiveresults, with 20 tion. MW at 35 GHz and 8.5 percent efficiency achieved in one In the cold beam low-gain Compton regime, the single [22], and 23 MW at 40 GHz with 5 percent efficiency pass gain expression [23], [24] is given by achieved inanother [23]. Similarly, a cyclotron autoresonance maser has been operated at 2.4 mm with 10 W)- P(0) Gcold > MW of power and 2 percent efficiency [24]. P(0) The use of high Q-resonators for the wiggler field is xy2~;5/21~3~2,( 1 + a:) -3’2 appropriate for use with gyrotrons capable of long pulse G~~,~= 1.77 X (2) operation. Gyrotrons with long pulse or CW capability Ab employ thermionic cathodes. The high Q-resonator is re- where X, is the wavelength of the electron wiggle motion, quired to reach wiggler field strengths high enough for given by X, = X/(1 + kll/k) = X(l + P,’>-’, where Pph reasonable FEL gain in the second stage. Alternatively, = o/ckl,is the phase velocity of the wiggler electromag- relativistic gyrotrons and cyclotron autoresonance masers netic wave, and where I is the beam current in amperes. produce significantly more peak power than do gyrotrons, In (2), Ab is the cross-sectional area of the electron beam. but these sources are in a less developed state. For these Eq. (2) applies to a beam with zero emittance, with zero sources, propagation of the wiggler field as a traveling transverse wiggler field inhomogeniety and filling factor wave may be more appropriate. In this paper, both trav- of one (e-beam cross-sectional area = radiation beam eling wave and standing wave resonator fields are consid- cross-sectional area); hencewe refer to it as thecold beam ered as possible electromagnetic wigglers. gain, Gcold.In order to include finite temperature effects This paper is organized as follows. In Section 11, the we consider below the effect of finite emittance and trans- gain for an FEL with an electromagnetic wiggler in the verse wiggler field inhomogeniety on the FEL resonance low- and high-gain Compton regimes is presented; the ef- condition. DANLY etELECTROMAGNETIC al.:POWERED GYROTRON WIGGLERS FOR FEL's 105

An electromagnetic wiggler produced from a gyrotron where the subscripts on $ refer to electrons with different or cyclotron autoresonance maser is assumed to be in a orbits. The equation of motion describing the evolution TEI, mode of a cylindrical waveguide or cavity. We con- of $ for a single electron can be shown to be [28] sider a circularly polarized traveling wiggler wave; the field dependence is given by the vector potential

A",= 2A,r- l(kLr)[cos (kilz + ut) cos 4 where k, = k + kli for the electromagnetic wiggler. The kJ. r brackets ( ) denote an average overthe wiggle period 2nl - sin (kllz + ut) sin 41 C, - J;(kLr) [cos (kl,z+ ut) k,. Eq. (5) can then be rewritten as sin 4 + sin (kllz+ ut) cos +IC6 3 (3) where A, = B,/kll, and B, is the peak magnetic field on axis. Jl(x) is a Bessel function and kL = vl,/a, where vln is the nth nonzero root of J;(x) = 0, and a is the waveguide radius. Wear the axis, the vector potential can be At this point we consider electron orbits for the case of approximated by an electron beam for which the minimum beamradius occurs at the center of the interaction region, but for which IA",l =A,(, - y). there is no additional focusing within the FEL interaction. Axial guide fields [29] and quadrupole fields can allow The wiggler field is thus slightly defocusing as a result of beam transport over longer distances, but are not consid- the minus sign in (4). This is in contrast to the case of a ered here. The addition of focusing fields can have both magnetostatic wiggler in which the field varies in the beneficial and deleterious effects on the FEL resonance transversedirection as either a hyperbolic cosine or a [30]. ' modified Bessel function and is focusing. Although the We consider one electron to have an orbit (orbit 1) de- electromagnetic wiggler is defocusing, the defocusing will scribed by be very weak because the field strength a, = eA,/mc2 will x(z) = ro cos (k,z) + r generally be weak (a: << 1). However, for large a, this defocusing effect could be compensated by external fo- y (z> = ro sin (k,z) cusing. where An important difference between the waveguide electromagnetic (EM) wiggler and the conventional magnetostatic (MS) wiggler is that the transverse wiggler field variation depends on kl for the EM wiggler, whereas in the magnetostatic wiggler it depends on k,. Thus, in the is theequilibrium radius of the helical wiggle motion. The EM wiggler, one can vary the transverse inhomogeniety second particle is assumed to pass through the interaction independently of the wiggler periodicity. Since k, can be region at an angle given by the emittance E = rr' ( = (1/ made small by making thewaveguide radius relatively a) X area in phase space) of the electron beam. For this large, the transverse inhomogeniety can in principle be particle, the orbit (orbit 2) is described by small. This flexibility is not available in the case of mag- E netostatic wigglers. This has dramatic effects on the gain x(z) = __ (1 - 7r$-z2) cos (k,,,z) + - z achievable with the two systems as will be discussed later. Ykw r The effects of a transverse wiggler field inhomogeniety k2 c2 k, Z 0 and a finite emittance cause electrons in a non- y(z) = - (1 - -A-4r2 z2) sin (k,z). ideal electron beam to undergo different phase shifts in YkW the FEL interaction region [27]. Because the energy trans- Using = (dx/dz)2 + (dy/dz)', we can then evalua.te fer between the electron and optical mode is proportional (-y2P; ) for both orbits. The result is to the vector dot product of the electron velocity and the electric field of the optical wave, theslowly varying phase for the FEL interaction is $ = k,z + k,z - wst. In the low-gain Compton regime, net energy transfer over an interaction length L requires that the relative variation of $ for any two electrons over the length L be less than some fraction of 2a radians. Choosing a/2 as the allowable rel- (8b) ative phase variation [27], we can write where b = k:~~/4ki,r~. It is usually the case that k2,r2/4 << 1 and that the last term of the second expression is negligible. Then, (8) be- 106 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-23, NO. I, JANUARY 1987 comes Similarly, the transverse wiggler field inhomogeniety contributes to a detuning spread - 222 = aNu,k,rb. (12) 22 y2E2 (y P.>2 = 4, + - -The total detuning spread is then given by = 8th,l + r2 . Ob. Substitution of these expressions into(7) together with the Degradation of the cold beam gain due to finite k, and FEL resonance condition (1) then result in a condition on E, can be calculated numerically as afunction of eth.How- the allowable emittance and wiggler inhomogeniety for ever, for an electronbeam in the intermediate cold-warm tenuous beam,low-gain regime, Jerby and Gover [26] the FEL: have shown that the numerically computed gain can be 12Natk:r’ + closely approximated by the expression

I where N = IA,,, is the number of electron oscillations in thewiggler, and where wehave assumed k, = 2y2k,, a: << 1, where the cold beam gain is given by (2). If (9) is written in terms of r and yr’ , one obtains the The gain as given by (13) can be optimized for fixed I, equation of an ellipse in (r, yr’) phase space. The nor- N, and brightness B, = I/a2e; with respect to the electron malized acceptance of theelectromagnetic wiggler can beam radius rb. The resulting optimum radius is given by then be defined as theproduct of the majorand minor axes of the ellipse:

A, = yrr’ = (2&Nu,k,)-’. (10) or, for the case48(N~,a,k,)~ << 1, which usually holds, Provided the phase space of all the electrons in abeam is bounded by this ellipse at the center of the wiggler, all particles will suffer less than a 7r/2 relative phase shift. This isto be compared with theacceptance of a helical magnetostatic wigglergiven by [27]; A? = The optimum electron beam radius decreases as the number of periods and the current decrease and increases as (2&Nu,k,)-’ in the limit of ut << 1. An important difference between theelectromagnetic the beam brightness decreases. A lower limit onrb results wiggler and the magnetostatic wiggler is apparent from from the diffraction of the optical beam and the concom- comparison of the acceptance in the two cases. Fur short itant reduction in FEL coupling to the optical mode [32]. Thislimit is ari,,pt 1 X,L/2 the minimum optical wavelength wigglers and comparable a,,., the acceptance h, of the electromagnetic wiggler can be made significantly mode area. When this limit is violatedby (14) or (15),we larger than the acceptance of a corresponding magneto- take static wiggler, because in the case of the EM wiggler, k, can bedecreased significantly without significantly changing k, = k + k,,. The effect of finite emittance and a transverse wiggler Two different gain scalings result from the two different field inhomogeniety on the gain can becalculated by expressions for the beam radius. For low currents treating each as a thermal spread in the FEL detuning pa- 7fX,X,B, rameter. Many authors have treated these effects in such z

B=240.N=200

Y -

B = 2 40, N=100

5 10 20 50 100 IO 20 50 IO0 CURRENT [AI CURRENT CAI

Fig. 1. FEL gain at X, = 10 pm for 30 GHz (X = 1 cm) electromagnetic Fig. 2. FEL gain at X, = 10 pm for 60 GHz (X = 5 mm) electromagnetic wiggler (a, = 0.05). Different brightnesses, B,(kA/cm2 rad’), and num- wiggler (a, = 0.05). Different brightnesses, B,(kA/cm2 rad2), and number of wiggler periods N are shown. ber of wiggler periods N are shown. from (17) that as Z approaches Il, the gain is saturated by harmonics has been proposed [34]; such a system is po- the finite emittance as described by gth,1. tentially capable of very high brightness. The useof pho- For Z > I,, the optimum beam radius is given by (14), tocathodes may also bring about large increases in bright- and the gain is given by ness. The gain is also calculated assuming a brightness of 5 X lo3 kA/cm’ rad2. = 1.77 G x lop3 The calculated FEL gain is shown in Fig. 1 for the case of a 30 GHz, X = 1 cmgyrotron powered electromagnetic wiggler and a X, = 10 pm FEL output wavelength. A 7.8 MeV electron beam would be required for 10 pm opera- (184 tion. The wiggler electromagnetic wave phase velocity is When (14) can be approximated by (15), the gain can be assumed to be p?’ = k,,/k = 0.9 forall the following approximated by examples, The gam for twodifferent brightnesses and for N = 100and N = 200wiggler periods is shown.The 3/2 1/2 2 2 G = 1.77 - X, X, N awBn (1 8b) brightnesses are quoted in units of kA/cm2 rad’. For the 4 lo3 kA/cm2 rad’ brightness, the gainscaling changes from where B, is in kA/cm2 rad2. For currents above ZI,the a linear dependence oncurrent described by (17) to a con- small signal gain is independent of current and depends stant gain independent of current for Z > II = 57 A. For only on brightness. The gain described by (lab) is ap- higher brightness the transition between scaling laws oc- proximately half the cold beam gain at r = rb,opt. curs at higher currents. For both regimes, the gain scales The small signal gain has been calculated as a function as N2. of current for several different values of the electromag- The calculated gain for a 60 GHz gyrotron powered netic wiggler wavelength and beam brightness. A wiggler electromagnetic wiggler (X = 5 mm) is shown in Fig. 2 amplitude of a, = 0.05 is assumed; this corresponds to a for three different brightness beams. The transition be- traveling wave with a magneticfield of approximately 0.5, tween the different gain scalings can be seen for the 240 1, and 2.5 kG for X = 1, 0.5, and 0.2 cm, respectively. kA/cm2 rad2 brightness, and the beginning of the transi- Such a wiggler amplitude requires high-power sources of tion can be seen for the B, = 240 kA/cm2 rad’, N = 100 electromagnetic radiation such as gyrotrons or other cy- cases. In the region where gain scales linearly with cur- clotron resonance devices. Adiscussion of powerrequire- rent, different gain for different brightness beams at the ments and possible configurations is deferred to the fol- same current results from the finite emittance corrections lowing sections. to the gain (13). It is clear from Fig.2 that for reasonable Figs. 1-3 show the gain versus current for several beam brightness, the gain for a h = 5 mm wiggler with a, = brightnesses. Brightnesses of 240 kA/cm2 rad2 and 1 X 0.05, X, = 10 pm, (5.35 MeV beam) is sufficient for FEL lo3 kA/cm2 rad2 are possible with present RF linacs and operation. electrostatic accelerators [33]. A 2.5 MeV injector em- The gain for X = 2 mm electromagnetic wiggleris ploying accelerator cells excited at the fundamental and shown in Fig. 3. At this wiggler wavelength, only a 3.2 108 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-23, NO. 1, JANUARY 1987

B = 240, N = 200

-1 / I

5 10 20 50 100 CURRENT CAI

Fig. 3. FEL gain at X, = 10 pn for 150 GHz (X = 2 mm) electromagnetic wiggler (a,= 0.05). Different brightnesses, B,(kAlcm* rad2), and number of wiggler periods N are shown

- MeV beam is required for X, = 10 pm operation.The pressed as Oth = 2rNAyIy [26]. For a single frequency decrease in gain with increase in current for theB, = 240 RFaccelerator, Ayly = 6:/8, and, with a reasonable kA/cm2 rad2, N = 100 and N = 200 cases is due to the limit- on this contribution to the detuning spread parame- current dependence in (18a). ter, Oth I n12, this condition can be written 6: I2lN. It is clear from these figures that no appreciable im- The slippage problem worsens as 6, decreases, but the en- provement in gain is achieved by using electron beams ergy spread problem worsens as 6, increases. An estimate which exceed the limiting current II.Thus, with all other of the limit to N due to these two effects can be obtained parameters fixed, higher FEL gain requires higher quality by combining these two conditions: beams with low emittance. Because of the gain dependence on NU,)^ for both (17) and (18), it appears attractive to increase N and decrease a, by the same factor. Although the efficiency of the electromagnetic wiggler would decrease (efficiency - 112 N), For an RF linac withfRF - 500 MHz, so - 0.2, and X, the FEL gain based on (17) or (18) would remain un- = 10 pm,Nlim = 200 periods. Operation with higher fre- changed. The advantagewould be a reduced a, and hence quency optical radiation reduces the slippage problem, but a reduced electromagnetic power density required for rea- the energy spread requirements for any accelerator tech- sonable FEL gain.For certain accelerator technologies nology become more severe as N increases. this may prove practical. However, two additional effects Although EM wiggler FEL operation with conventional could reduce this potential advantage. Pulse slippage and RF linear accelerators may be limited to wigglers of - 200 periods, RF linearaccelerators employing higher order the finite energy spread contribution to 8th will set an upper limit on the degreeto which this trade-off can be made mode, harmonically excited cells [34] could lessen both for different accelerator technologies. these constraints on the maximum N. The operation of an EM wiggler FEL with an electrostatic accelerator would A slippage parameter s = N X,Icrp can be defined for a given wiggler (N) and electron pulse temporal width rp. avoid the slippage problem,but the constraint dueto finite When s approaches unity, initially overlapping electron energy spread remains. and optical pulses separate completely in transit through One final contribution to the detuning spread parameter the wiggler, resulting in a considerable degradationof the which may become significant for very low voltage beams gain (laser lethargy). For electrostatic and induction ac- is due to space charge. Thiscontribution is approximately celerators, T~ is generally large and slippage is not an is- 1261 sue. For RF linacs, we require s Iso, some maximum allowable value. For rp = 6,TRF/2w,where 6, is the phase angle of the electron bunch in radians, and TRF = 1/f& is the RF accelerator period, the condition on s can be writ- where I is in amperes. For N - 200, I - 100 A, and y ten 6: > ~~Nx,~~~Ics~. -- 10, sc - n14, andthe space charge contribution to The contribution to Bth from energy spread can be ex- 6th may result in a small decrease in FEL gain. For larger DANLY et al.:POWERED GYROTRON ELECTROMAGNETIC WIGGLERS FOR FELs 109

N and smaller y, this correction becomes more signifi- expressing G(z,p,, t) = Go(P,) -t 6G(z,p,, and solving cant. the linearized Vlasov-Maxwell equations for 6G(z, pz:t), 6a,(z, t), and 6a,(z, t). As an equilibrium constraint it is B. High-Gain Compton Regime found that the wiggler frequency w and wavenumber k are The calculations discussed in Section 11-A refer to related self-consistently by FEL’s with weak electromagnetic wiggler fields (a, << 1) where the gains are low (G < 1). However, the high- gain regime may well be reached using anintense electromagnetic wiggler. The computations in this section are where i)$, = 4nribe2/m is the nonrelativistic plasma fre- appropriate to this mode of operation; the free-electron quency-squared, Go(p,) is the equilibrium distribution laser stability properties have also been calculated in the function, and yo = yT(6a, = 0, 6a, = 0) is the relativistic high-gain Compton regime using a model based on the mass factor defined by Vlasov-Maxwellequations. Weconsider a relativistic electron beampropagating in the positive z-direction through thebackward-traveling, circularly-polarized electromagnetic wiggler with vector potential (k,, = k in this We parallel the kinetic stability analysis given in I351 and discussion) originally carried out for a static helical wiggler. Making mcL the Compton-regime approximation and assuming elec- A,@, t) = -a, [cos (kz + d)Z, - sin (kz + wt)2,] e tromagnetic perturbations with (nearly) right-circular po- larization, the linearized Vlasov-Maxwell equations give (19) the dispersion relation = &2 - c2&2 - 2 where a, eB,/rnc2k = const is the normalized wiggler pb amplitude. For present purposes, transverse spatial vari- ations are neglected (alax = 0 = Way), and the electron beam is assumed to be sufficiently tenuous that longitu- dinal perturbations canbe neglected (Compton-regime approximation with 64 = 0). Furthermore, we consider Here, u, = pz/yomis the axial velocity,k is the wavenum- the class of self-consistent beam distribution functions of ber of the perturbation, andLJ is the (complex) oscillation the form [35], [36] frequencywith Im LJ > 0 corresponding to temporal growth. Moreover, the normalization constant & in (25) h(Z, p, t) = fibB(Px) &Py) Gk,Pz, t> (20) is defined by where P, and Py are the exact transverse canonical mo- menta defined by P, = p, - (e/c)A,,(z, t) - (e/c) 6Ax(z, t) and P, = py - (e/c) Ayw(z,t) - (e/c) 6A,(z, t). Her;, p = ymO is themechanical momentum, y = (1 + p / and we have assumed that the beamelectrons are moving m2c2)1’2is the relativistic mass factor, iib = const is the in the positive z-direction with average electron density, and 6A, and &A, are the transverse components of the perturbed vector potential. For Go(P,) = WPJ G$(yo). (27) the class of distribution functions in (20), note that the In (27), U(p,) is the heaviside function defined by U(p,) electrons move on surfaces with P, = 0 = Py correspond- = 1 for pz > 0 and U(p,) = 0 for pz < 0. Forfuture ing to zero transverse emittance. Moreover, it is readily reference, it is straightforward to show from (24) that the shown that the one-dimensional distribution function G(z, axial velocity pZc = u, can be expressed as p,, t) evolves according to the nonlinear Vlasov equation 1351

The Compton-regime dispersion relation (25) can be used to investigate detailed kinetic stability properties over a wide range of system parameters and choice of distribution functions Gl(yo).In the regime of practical inter- est, we assume that the electron beam is sufficiently tenuous that

A A2 + [-a, sin (IQ, + ut) + 6a,,l2 . aupb << C2g2. (29) r (22) Here, 6ax = (e/mc2)6A,and 6ay = (e/mc2)6A,, and lon- In addition, it is assumed that the distribution of beam gitudinal perturbations have been neglected 64 2: 0. electrons Gof(yo)is strongly peaked about average energy A detailed kinetic stability analysis [35] proceeds by yo = yb with (small) energy spread Ayo satisfying 110 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-23, NO. I, JANUARY 1987

,I Vr vb (b) Fig. 4. Distribution function G: relative to the ponderomotive velocity u, in the (a) monoenergetic limit, and (b) the weak resonant growth regime. V, is the beam velocity.

AyO << Yb. (30) Substituting (34) into the dispersion relation (25) readily gives The FEL growth rate generally exhibits a sensitive dependence on the detailed form of GX(yo)even when the inequality in (30) is satisfied. 1) FEL Resonance Conditions: Within the context of (29) and (30), the dispersion relation (25) exhibits strong FEL coupling for frequency and wavenumber (2,k) = (us,k,) satisfying the simultaneous resonance conditions w, = ck,y where vb = c[1 - (1 aL)/yi]"2.The dispersion relation W, = w + (k,y+ k)Vb. (31) + (33, valid for monoenergetic electrons, can be solved for Here, the average beam velocity vb = &,e is defined by the growth rate Im d over a wide range of system param- cpb = [ 1 - (1 + at,)/y$]"2c.Solving (3 1) for k,7 gives eters.For present purposes,we assume a$$, << c2,t?, ck, = (1 - V,/C>~'(O+ kVb), or equivalently, and express 4 = w,~+ and k = k, + 6k,, where w, and k, solve (3 1). Retaining the dominant contribution 7; k, = ~ 2 (1 + Ph)(w/kc + bb)k. (32) proportional to [($ - w) - (E + k)Vh]-' on the right- 1 + a%" hand-side of (33, the characteristic maximum growth rate If further it is assumed that c2k2 >> 2$, jr dp,Gof/yo, is determined from then w = ck is a good approximationto (23), andthe characteristic wavenumber k, of the FEL radiation can be expressed as for 6k, = 0 and I6w,/ << w,. Wemake use of (33) to (33) eliminate k, + k in (36) in favor of k. Solving for Im then gives the characteristic growth rate

For relativistic electrons, Pb 2: 1 and (33) is equivalent to (1). (37) 2) Strong FELInstability-Monoenergetic Electrons: As a first application of the kinetic dispersion relation Because an ideal electron beam (monoenergetic electrons) (25), we consider the case where the energy spread Ayo has been assumed, the growth rate in (37) corresponds to is sufficiently small that I k + kl Au, << I 2 - w 1. Here, the largest growth rate achievable in the high-gain Comp- Au, is the axial velocity spread characteristic of G:(yo), ton-regime. and Au, is related to the energy spread Ay0 by Au, = c[(l 3) Weak ResonantFEL Growth: Kinetic stability + a~,)/y~](c/Vb)Ayo.For sufficiently small Ayo, it is valid properties are altered significantly when the energy spread to approximate G;(yo) by [Fig. 4(a)] in the beam electrons is sufficiently large that 112 + kl Au, >> /Im $1 [Fig. (4b)l. In this case, resonant electrons vb Gl(yO) = 6(y0 - Yb) (34) play the controlling role indetermining the instability growth rate. For ad$ << c2gand IIm &I << 112 + ~(Au,, where the normalization condition is Srdp,G,'(yo) = 1. it is readily shown from Eq. (35) that the growth rate is DANLYFEL’s etELECTROMAGNETIC al.:FORPOWERED GYROTRONWIGGLERS 111 given by is also possible, and perhaps desirable, to use an electromagnetic standing wave in a high Q-resonator. High values of a, may be obtained by storage of the gyrotron power in asingle high Q-cavity mode.The resulting Here, the resonant energy T,.is defined by +,= (1 + &/ standing wave wiggler field can be written as the sum of m2c2 + ai,)”2, where 0,. = TrmOr,and the resonant axial two oppositely directed traveling waves of equal ampli- velocity 0, is defined by tude. Near the z-axis, the field can be written as wr-w &(z, t) = Aw(gx cos (kllz + ut) - ZY sin (kllz+ ut) u, = n (39) k+k’ + Ex cos (k,,z - wt) + EY sin (kllz - ut)). Combining (39) with the definition of 9,.also gives For an electron beam propagating in the gZ direction, the backward travelingwave component produces the fast wiggle motion of the electron beam with a wavenumber of k: = k/PII + kll.In a standing wave field, the forward wave component of the wiggler field also produces awig- In (39) and (40), w, = Re & = ck for &hib<< c2i2.Note from (38) that the growth rate Im & depends on the de- gle motion on the electron beam with a wavenumber of k, = k/fill kll.The equilibrium radius of the fast time tailed form of G,f(yo) in the resonant region where u, = - scale helical motion due to the backward wiggler wave is ur = (w, - w)/(k + k). As a particular example, we consider the case where r = a,/yk,f. The radius of the electron guiding centerhel- the electrons have a Gaussian distribution in energy with ical motion due to the forward waveis given by G;(yo) = const X exp [-(yo - ~~)~/2(Ay~)~],where yb = const is the average energy, and Ayo = const is the 1 - (kll/k)I1 + 1/2y2 ’ energy spread. It is readily shown from (38) that the maximum growth rate occurs for y, - yb = -Ayo with For kll --f k, R + uwy/k, and the guiding center helical motion of a single electronmay be comparable to the electron beam radius. In thiscase, the detuning parameter spread may be significantly modified, and the deleterious effect of finite k, will be more severe. For kll/k < 1, the Here, (nle)”2 = 1.075and 6; = 1 - (1 + a:,)/$. For sufficiently large energy spread, as shown in [37] for the radius for the guiding centerhelical orbit of a single elec- case of a static magnetic wiggler, the growth rate in (38) tron becomes much smaller than typical electron beam ra- can be substantially less [38] than the ideal growth rate dii, so that the preceeding treatment for a circularly po- for monoenergetic electrons in (37). That is,beam quality larized traveling wave adequately describes the circularly (as measured by Aya/yb) can have a larger influence on polarized standing wave wiggler. detailed FEL stability behavior in the high-gain Compton Operation with a standing wave electromagnetic wig- regime. gler results in twoseparate FEL resonances with reso- The critical energy spread (Ayo/yb)c, for the transition nance conditions k, = 2y2(k k kll)(1 + 2~:)- ’. These two to the weak resonant FEL instability is readily calculated resonances correspond to the two separate components of the motion due to the forward and backward waves. In from (Im &)MAX= (k, + k)Au,, where Au, = c[(l + a:)/ y:](c/Vb)Ayo. Making use of (41), we find the design of the high Q wiggler cavity and the laser optical cavity, care must be taken to prevent oscillation at the low frequency FEL resonance. A detailed analysis of the gain for the standing wave electromagnetic wiggler and the phenomenon of detrapping in the limit k,,/k << 1 For a narrow energy spread with Ayo/yb << (Ayo/yb)cr, have been investigated and will be presented elsewhere. the dispersion relation (33, derived for monoenergetic electrons, gives an excellent description of FEL stability properties. On the other hand, for sufficiently large en- D. Pump Depletion ergy spread that Ayo/yb >> (byolyb)cr,the (weak) growth The amount of depletion of wiggleror pump wave rate is given by the resonant expression in (38). For pa- power by the generated optical wave can be estimated by rameters a, - 0.5, Z/A, = 3.3 kA/cm2, yb - 10, and w/ considering thequantum picture of electronsCompton 2a - 60 GHz, (42) gives (Ayo/yb)cr- 0.005. Because scattering pump photons into the optical wave. The ratio the energy spread for typical accelerators [33] is likely to of the power lossby the wiggler, P,, to the optical power be comparable to or larger than (Ayo/yb)cr, growth the rate produced by the FEL interaction, P,7 is PJP, = w/u, 2: in the high-gain regime should be calculated from (38). 1/4y2. Formoderate y,the average pump power depletion is a small fraction of the average optical power generated. C. Effect of Wiggler Field Forward Wave Furthermore, gyrotrons are capable of high average power In the preceding discussionwe have considered an operation, so pump depletion is unlikely to pose a signif- electromagnetic traveling wave (3) as the wiggler field. It icant constraint on FEL performance. Pump depletionwill 112 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-23, NO. 1, JANUARY 1987 be more important for FEL operation in the collective re- nance masers (P - 10 - 100 MW) are thus capable of gime with high current beams. producing reasonable a, values in circular TEl, wave- The gain for gyrotron powered electromagnetic wig- guide modes. As an example, consider a power P = 20 glers has been shown to be sufficient for FEL oscillator MW at X = 5 mm propagating in a TE12 mode (u12 = operation with modest current electron beams. The elec- 5.33) in waveguide of radius a = 0.5cm. Then kl = tromagnetic power required to produce the wiggler field 10.66 cm-’, k,, = 6.65 cm-’, and the magnetic field and strengths sufficient for high gain is discussed in the fol- vector potential are B, = 0.7 kG and a, = 0.062. For lowing section. power in the range of 10-100 MW, the vector potential ranges from a, - 0.04 - 0.14. 111. GYROTRONPOWERED EM WIGGLERFIELD Alternatively, the wiggler field can be stored in a high STRENGTH Q-cavity. For a TEInlcavity mode in a right circular cyl- The feasibility of a gyrotron powered electromagnetic inder, the time averaged stored energy, W, can be written wiggler for FEL applications will depend not only on the as available beam current, brightness, and number of wig- glerperiods, but also on theattainable wiggler field strength, a,. The FEL gain scales as a:; it is therefore importantto determine the scaling of a, with source where B, = A,/kll is the magnetic field strength of one of power,frequency, cavity Q,and wiggler cavity mode. the traveling wave components of the standing wave and This scaling is addressed in this section for the cases of where d is the length of the cavity (d 1 L, the length of traveling wave propagation in waveguides and standing the FEL interaction). The equilibrium fields in the cavity waves in high-Q resonators. can be obtained by equating the supplied power from the For a traveling wave electromagnetic wiggler, we con- gyrotron, Pg, to the total time average dissipated power, sider a TE1, cylindrical waveguide mode. The TEl, modes Pd. The total Q of the system can be written QT = wW/ have fields on the axis that can be approximated by plane Pd. At equilibrium, Pg = Pd, the field can be expressed waves. Such modes are well suited for use as wigglers as with high voltage axialelectron beams. Furthermore, B, = 4.47 X io3 [cVf, - ~)J~V~,)I-~/~ high-power gyrotrons can produce significant power in these modes. For a power P flowing in a circular waveguide of radius a, the peak magnetic field on axis, B,, can be related to P by [39] where Pg is in (MW), lengths arein (cm) and B is in (kG). The normalized vector potential can then be expressed as

a, = 2.62 X 103 - 1)~:(~~,)]-~~~

where kL = vl,/a, and vln is the nth nonzero root of J;(x) = 0. For P in (MW) and wavenumbers in (cm-’), the (47) field in (kG) can be expressed as The use of high Q-cavities can allow high a, for moderate B, = 3.65 x - ~)J:(V~,)]-”~ki ak [(vy, fi power. The total Q isrelated to the cavity ohmic Q, QOHM, for an arbitrary TE,, mode. For propagation in a TEIl and any external or diffractive Q, Q,,byQ,’ = mode, the Bessel function factor [*I-”* = 1.11, and B, QGAM + eo1.The upper limit on QT is QOHM. Further- = 4.06 X lo-’ k, akfi. For fixed power and fixed more, Pg is the power supplied to the wiggler cavity and k, , B,v decreases slowly with higher order modes TE,,, n not necessarily the total power produced by the source. > 1. The normalized vector potential for the circularly The degree to which the source power equals Pg and QT polarized traveling wave is then related to power by = QoHM will depend in a detailedway on the system configuration. a, = 2.14 X lo-’ [(v:, - l>J~(~l,)]-”~ As an example of typical cavity parameters, we consider a high Q cylindrical cavity with QT = QOHM/2and *k’JFp@i. (44) a X = 5 mm electromagnetic wiggler (kll/k = 1). For a TE13 cavity mode with N = 200 periods in a cavity of The normalized vector potential increases with P and radius a = 1.56 cm the resulting field strength is a, = kl . The FEL cold beam gain is thus proportional to Pk2,. 4.2 X lop2G. For a dissipated power of Pg = 1 However, (10) indicates thatthe acceptance decreases MW, a, = 0.042, and B, = 0.9 kG. with increasing kl . Thus the gain given by (13) will have Several practical considerations may provide an upper an optimum with respect to k,. If this optimum occurs limit to the attainable B, and a, for an EM wiggler. Sur- too close tocutoff (k,, << k), other problems such as large face breakdown does not appear to be a limitation at the ohmic loss may arise. wiggler field strengths necessary for reasonable FEL gain High-power pulsed gyrotrons and cyclotron autoreso- [40]. In high average power systems, ohmic heating of DANLY et al.: GYROTRONPOWERED ELECTROMAGNETIC WIGGLERS FOR FEL’s 113

2 .o I

I I 1- 6 8 10 ulMeVl

0.I 0.2 0.5 I 2 X, Ccml

Fig. 5. Comparison of gain for short wavelength magnetostatic (MS) hybrid wiggler and electromagnetic (EM) wiggler. The following parameters are assumed: X, = 10 p.m. I = 100 A, B, = 240 kAicm’ rad’, and N = 200. The voltage (U)required is also shown. the wiggler resonator walls may require active cooling, lated from the full expression [(17), (18)], including the but in FEL’s with modest pulse repetition rates this does effects of finite emittance and wiggler field inhomogene- not appear to be a problem. One problem that may prove ity. The gain for the magnetostatic wiggler has been cal- important in high Q-cavities is conversion of the stored culated assuming the optimization of reference [27], and power into unwanted modes by imperfections or tapers in with 8, = n/2, as used in [27]. The gain is shown as a the cavity. This may require mode supression techniques. dashed line for gains above unity because the low-gain Such mode conversion is well understood in gyrotrons and expressions were used to calculate the gain. overmoded resonators,and the application of existing For short wavelength wigglers, X, < 1.5 cm, the elec- techniques can minimize or eliminate this mode conver- tromagnetic wiggler produces substantially more gain than sion. However, this problem deserves further study and the conventional hybrid undulator. At wiggler periodici- is not addressed in this paper. Phase jitter in an EM wig- ties larger than 1.5 cm for these parameters, the magne- gler system could also be a problem and should be further tostatic wiggler will have higher gain and will be more investigated. attractive. Smaller gaps for the magnetostatic undulator will shift the MS gain curve in Fig. 5 to the left, but the IV. DISCUSSIONAND CONCLUSIONS gyrotron powered EM wiggler will maintain higher gain As an illustration of the advantages of an electromag- at short wavelengths. Of course, marked improvements in netic (EM)wiggler over a conventional magnetostatic permanent magnet materials or the development of other (MS) wiggler for short wigglerwavelength and low beam short wavelength wigglers [ 171, [ 181 could produce fields voltage operation, the gain forboth types of wigglers with comparable to those of EM wave wigglers. Nevertheless, comparable parameters is depicted in Fig. 5. An output millimeter wavelength electromagnetic wigglers powered wavelength of X, = 10 pm is assumed, and the brightness,by gyrotrons or other cyclotron resonance devices appear current, and number of wiggler periods aretaken to be B, promising. = 240 kA/cm2 rad’, Z = 100 A, and N = 200 for both Microwave undulators have been used for the genera- cases. The beam voltage U required to produce X, = 10 tion of synchrotron radiation in the visible region of the pm radiation is also shown. The EMwiggler field strength spectrum using high voltage electrom beams [42]; in this is assumed to be constant with a, = 0.05, and we have paper we areinterested in the application of high fre- assumed kll/k = 0.9 (X, = X/1.9). The magnetostatic quency (millimeter wave) electromagnetic wigglers to wiggler is assumed to have a gap fixed at 7 mm, and the free-electron lasers. There are several possible configu- fieldis calculated from the usual expression [41] for a rations for FEL’s with gyrotron powered electromagnetic hybrid undulator. For the EM wiggler the gain is calcu- wigglers. The wiggler fields may be stored in a high-Q 114 IEEEJOURNAL OF QUANTUMELECTRONICS, VOL. QE-23,NO. 1, JANUARY 1987

Gyrotron Resonator

Fig. 6. One possible GEM wiggler FEL configuration. cavity or they may be traveling waves in waveguide or pling problems are nonexistent. The temporal duration of free space. In the case of a standing wave EM wiggler in the wiggler pulse constitutes a limitation of this configu- a cavity, several possibilities exist. The power source may ration. If field emission cathodes are used, the wiggler be separated from the wiggler cavity, with external cou- electromagnetic pulse will be limited by gap closure. The pling intothe cavity. High-power microwave or milli- FEL gain must then be sufficiently high to reach satura- meter wave circulators or isolators would be necessary to tion during the wiggler pulse. For the high-Q cavity con- prevent reflection of power back to the source. Although figuration with a thermionic cathode, much longer dura- these components may be feasible forlower frequency EM tion (ps-CW) wiggler fields could be produced. wigglers (X - 0.5 - 1 cm),higher frequency wigglers An infrared FEL system could be made extremely com- are likely to require an alternate concept. pact if the gyrotron which powered the electromagnetic One possible configuration is shown in Fig. 6. The gy- wiggler was also used to power the electron accelerator. rotron interaction and the FEL interaction take place in Part of the gyrotron output could be used to power a cy- the same closed cavity. The gyrotron interaction occurs clotron resonance accelerator [46]-[48]. The resulting in the region of the cavity where k, 2: k and where there system could be quite compact with one high-power gy- is the axial magnetic field necessary for the interaction. rotron providing both the accelerating and wiggler fields. The radiation produced in the gyrotron section is trapped The use of high current (>1 kA) high brightness beams inside the high-Q cavity. TheFEL interaction then occurs with short wavelength electromagneticwigglers is also in a region of the high-Q cavity in which ki < k. The possible. Larger EM wiggler powers may be realistic in same mode produced in the gyrotron section of the cavity this case because the required EM wiggler power would acts as the wiggler. The high voltage beam for the FEL still be small compared to the electron beam power. interaction can be brought into the cavity through tapered Operation of an FEL with a gyrotron powered EM wig- openings in the cavity which are cutoff to the trapped mode gler does not preclude efficiency enhancement schemes. or through slots in the resonator walls. The operation of As in the case of a magnetostatic wiggler, the EM wiggler gyrotrons ,in TE1, modes with slotted resonators can be can be tapered. By varying the waveguide wall radius, advantageous in terms of mode descrimination in the gy- a(z) = v,,/k,(z), the wiggler wave number k,(z) = k + rotron interaction region. Alternatively, the high voltage kli(z)and the field strength on axis may be tapered. For beam may exit through a holein the electrom gun provid- free space electromagnetic wigglers, no such controlled ing the beam for the gyrotron interaction; gyrotron elec- tapering is possible. tron guns are axisymmetric with the emitter strip located Other novel efficiency enhancement schemes are also off axis by several centimeters. The optical resonator for possible. A compound wiggler consisting of a relatively the radiation produced by the FEL interaction may be ax- long wavelength untapered EM wiggler and a strongly ta- isymmetric and located as shown in Fig. 6. pered magnetostatic wiggler could allow high efficiency The generation of a few megawatts of power by the gy- FEL operation. Initially, at the start of the high voltage rotron in Fig. 6 could produce wiggler field amplitudes electron beam macropulse, the EM wiggler would pro- sufficientfor FEL oscillatoroperation. The high field vide high gain. As the optical field approaches saturation, strengths in the gyrotron interaction region would require the gyrotron powering the EM wiggler could be turned operation of the gyrotron with short interaction lengths off, allowing the EM wiggler field to decay. The only re- [19], [43]; this is typical for high power gyrotrons. maining wiggler field is then due to the tapered magne- A quasioptical gyrotron [44] or gyroklystron operating tostatic wiggler. The extraction efficiency of the magne- with a ring resonator could also be employed. A cyclotron tostatic wiggler in the presence of a strong optical field autoresonance maser (CARM) 'could operate as the wig- can then be very high. gler power source in either an opticalresonator or a Bragg In conclusion, the recent considerable progress made in resonator [45]. high power gyrotrons and other cyclotron resonance de- In analternate configuration foran electromagnetic vices can have important implications for free-electron la- wiggler, the wiggler field could be a traveling wave pro- sers. The operation of an FEL with short wavelength gy- duced by a gyrotron or CARM. Higherpeak powers would rotron powered electromagnetic wigglers can yield a be required than in the case of high-Q cavity, but thecou- significant reduction in the electron beamenergy required DANLYFEL’s et ELECTROMAGNETIC al.:POWEREDFOR GYROTRONWIGGLERS 115 to produce a given optical frequency. Such a considerable tron laser using low energy electron beams,” Phys. Rev. Lett., vol. 42, pp. 977-980, 1979. reduction in beam energy would result in more compact [13] V. L. Bratman, G. G. Denisov, N. S. Ginzburg, A. V. Smorgonsky, FEL systems with reduced activation and radiation shield- S. D. Korovin, S. D. Polevin, V. V. Rostov,and M. I. Yalandin, ing problems. Advantages of the gyrotron powered EM “Stimulated scattering of waves in microwave generators with high- wiggler over the two-stage FEL include the high stability current relativistic electron beams: Simulationof two-stage free-electron lasers,” Int. J. Electron., vol. 59, pp. 247-289, 1985. of gyrotron operation and efficient generation of the wig- [14] S. Von Laven, S. B. Segall, and J. F. Ward, “A low loss quasioptical gler field by a separate low voltage electron beam. cavity for a two stage free electron laser (FEL),”in Proc. Free Elec- tron Generators of Coherent Radiation, June 26-July 1, 1983, Proc. The gain in the low- andhigh-gain Compton regime has SPIE, VO~.453, pp. 244-254, 1983. been presented, and nonideal effects such as finite emit- [15] H. R. Hiddleston, S. B. Segall, and G. C. Catella, “Gain enhanced tance, finite wiggler transverse gradient, energy spread, free electron laser with an electromagnetic pump field,” Physics of and pulse slippage have been discussed. Reasonablegains Quantum Electronics, vol. 9. Reading, MA: Addison-Wesley, 1982, pp. 849-865. (- 20-80 percent) are found for electron beams with a [16] J. S. Wurtele, G. Bekefi,B. G. Danly,R. C. Davidson, and R. J. brightness obtainable from present day accelerators. Re- Temkin, “Gyrotron electromagnetic wiggler for FEL applications,” Bull. Am. Phys. SOC., vol. 30, paper 655, p. 1540, 1985. quirements on gyrotron power were obtained, and several [17] V.L. Granatstein, W. W. Destler,and I. D.Mayergoyz, “Small possible gyrotron powered EM wiggler FEL configura- period electromagnet wigglers for free electron lasers,” Appl. Phys. tions were suggested. The gyrotron powered electromag- Lett., vol. 47, pp. 643-645, 1985. [18] L. Elias, X. Kimel, and G. Ramian, “Free electron lasers and magnetic wiggler appears promising for compact infrared free- netic materials,” J. Magnetism, Mag. Materials, vol. 54, pp. 1645- electron lasers. 1646,1986. [19] K. E. Kreischer, B. G. Danly, J. B. Schutkeker, and R. J. Temkin, “The design of megawatt gyrotrons,” IEEE Trans. Plasma Sci.,vol. ACKNOWLEDGMENT PS-13, pp. 364-373, 1985. The authors are grateful to G. Johnston, K. Kreischer, [20] A. Sh. Fix, V. A. Flyagin, A. L. Goldenberg, V. I. Khiynyak, S. A. Malygin, Sh. E. Tsimring, V. E. Zaperalov, Znt. J. Electron., vol. and N. Tishby for helpful discussions. 57, pp. 821-826, 1984. [21] A. Fruchtman and L. 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