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1196 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-17, NO. 7, JULY 1981

A Unified Theory of Magnetic Electrostatic Bremsstrahlun

Scattering, and Cerenkov- Free-Electro

Abstract-This paper discusses in a comparative way the main operating parameters of various free-electron lasers (FEL’s), providing a useful tool for laser design and a comparative evaluation of the various lasers. We show that the various kinds of FEL’s satisfy the same gaindispersion relation and differ only in a single coupling parame - ter K. The different gain regimes which are common to all FEL’s are delineated. We find the small signal gain in all the gain regimes (warm and cold beam, low- or high-gain, single electron, collective or strong coupling interaction). The laser gain parameter, radiation extraction efficiency, maximum power generation, and spectral width are given and compared in the various kinds of FEL’s and gain regimes. The maximum power generation of all FEL‘s (except Compton-Raman scattering) is shown to be limited by an interaction region width parameter. This parameter and, consequently, the laser power are larger in the highly relativistic limit by a factor -yo in all bremsstrahlung FEL’s, i n comparison to Cerenkov-Smith-PurceU FEL’s. Some expressions which were derived earlier for the magnetic bremsstrahlung FEL, like the expression for gain in the low-gain regime with the space charge effect correction and the low-gain expression for efficiency, are shown to be special cases of more general expressions.

I . INTRODUCTION E presently have a number of detailed theoretical analyses of free-electron lasers (FEL‘s) of various kinds:

magnetic bremsstrahlung [ 1 ]-[IO], electrostatic bremsstrahlung 11 11 , [ 121 , stimulated Compton-Ramon scattering [ 131 - [ 161 , and Cerenkov-Smith-Purcell [ 171 - [2 11 . However, it would be desirable at this point to have a simple unified model which describes simultaneously all the kinds of FEL’s, allowing easy comparison among the various lasers and providing simple expressions for the various operating parameters required for laser design.

As is shown in the following sections, such a unified analysis is possible because the different kinds of FEL‘s all satisfy to a good approximation similar dispersion and gain relations. The origin of the similarity of the various FEL’s is that they all involve longitudinal coupling between single electrons or electron plasma waves (space charge waves) and an electromagnetic

Manuscript received July 8, 1980; revised August 19, 1980. The work of A. Gover was sponsored in part by the Air Force Office of Scientific Research under Grant AFOSR-80-0073.

A. Gover is with the Faculty of Engineering, Tel-Aviv University, Tel-Aviv, Israel.

P. Sprangle is with the Plasma Physics Division, Naval Research Labo- ratory, Washington, DC 20375.

I x x x x x X X I

------------- x x x x x x x x

2 .0 2 4 2.Q

Fig. 1 . A schematic representation of all kinds of FEL amplifier structures.

wave. This coupling is carried out via different mechanisms in the different FEL’s discussed.

The qualitative distinction among the different kinds of FEL mechanisms was discussed in detail in [ 17 ] . The basic difference between bremsstrahlung FEL‘s and Cerenkov-Smith- Purcell FEL’s is that, in the first case, a periodic (magnetostatic or electrostatic) force operates on the electron beam, allowing phase matching (synchronism) with the electromagnetic wave by providing to the single electron or the space charge waves a negative crystal momentum k , :

where L is the periodicity of the periodic field. On the other hand, in the Cerenkov-Smith-Purcell FEL‘s the synchronism is obtained by increasing the momentum (wavenumber) of the electromagnetic wave in a slow wave structure (a periodic waveguide or a dielectric waveguide).

The stimulated Compton-Raman scattering problem is very similar to that of the bremsstrahlung FEE except that, instead of a static periodic force, the electron beam is modulated by an intense electromagnetic (pump) wave which propagates in a counter direction to the electron beam and which facilitates coupling between the electron plasma waves and a forward- going scattered wave of higher frequency. The bremsstrahlung FEL is sometimes regarded as a special case of Compton- Raman scattering with zero frequency pump.

Fig. 1 illustrates schematically the general structure of all the FEL‘s discussed in the present article. They are all composed of an electron beam of uniform cross section which propagates at an average velocity uoz through an electromagnetic waveguide and parallel to its axis (2-direction). The crossed areas symbolically represent the source of the interaction agent

0018-9197/81/0700-1196$00.75 0 1981 IEEE

GOVER AND SPRANGLE: FREE-ELECTRON LASERS 1197

(pump) which allows the interaction between the electromagnetic wave and the electron beam. This can be, in the different FEL’s, coil windings, periodic magnets, periodic electrodes, helixes, corrugated walls, dielectric walls, etc. The figure shows a schematic laser amplifier structure in which the input radiative power P(0) is amplified along an interaction length I , producing an output power P(1) > P(0). A free-electron laser oscillator structure will consist of similar elements but will also need the means for a feedback mechanism (for example, a Fabry-Perot resonator).

radiation) is derived for all FEL’s. In Section I11 the common dispersion-gain relation of FEL’s is discussed. The different k2

gain regimes, which evolve from the dispersion relation (and PL apply to all FEL‘s), are delineated in Section IV. Expressions for the maximum gain, efficiency, power, and spectral width of the various FEL’s are derived in Sections V-VI11 and discussed in a comparative way.

Throughout the paper we kept a full relativistic analysis. In most of the recent work on FEL’s the electron beam is highly kz

relativistic. Therefore we gave the extreme relativistic limits A

of the relevant expressions (given in brackets). But in some (d) COMPTON--RAMaN

FEL structures, like the Smith-Purcell experiment [22], the Orotron [23], [24], the Ledatron [25], or the Ubitron [26], the electron beam is nonrelativistic or moderately relativistic. For this reason the derivation was made for all FEL‘s in a general way without using the somewhat simplifying assumption that the beam is highly relativistic.

(a) CERENKOV

In the next section the radiation condition (wavelength of @I) SMITH-PURCELL

(C) BREMSSTRAHLUNG

w

The system of units used.in this paper is MKS. b

k 11. THE RADIATION CONDITION The radiation condition of the various free-electron lasers

can be derived from the kinematics of the interaction scheme, without requiring involved analysis,

In all the FEL structures, shown schematically in Fig. 1, an electron beam propagates inside the FEL structure together with a waveguided electromagnetic wave along the same direction (z-axis). A necessary condition for interaction is close synchronism (phase matching) between the interacting electromagnetic wave and the electron (plasma) waves. When such “near synchronism” is obtained, energy can be transferred from the electron beam to the radiation field (amplification) or vice versa (electron acceleration). This “synchronism” condition results in the radiation relation which determines the wavelengths at which amplification should be expected.

Fig. 2 describes schematically the interaction schemes of the various free-electron lasers discussed. The approximate wavenumbers of the interacting waves or wave components (space harmonics) are listed in the first two columns of Table I. Equating the wavenumbers (phase matching or momentum conservation) yields the radiation condition (third column in Table I). The expressions in brackets correspond to the highly relativistic limit yo, >> 1 [yo, is defined later in (7)].

In Table I and Fig. 1, k,, represents the wavenumber of the interacting electromagnetic wave which is, in general, a waveguide mode for which kzO < k = w/c. For a plane wave propagating in the z-direction, or for a low-order mode in a wide waveguide, one has k,, = k = w/c. In a dielectric wave-

Fig. 2. Dispersion diagrams of the interacting waves illustrating the synchronism (phase matching) condition for the different FEL schemes.

guide (Cerenkov FEL) one may have k,, > k. uo is the mean velocity of the electron beam which is propagating in the z-direction. In the case of the transverse pump FEL’s (two last rows in Table I), the electron beam also has at each point a transverse velocity, and the parameter which is used in the synchronism condition is uoz, the longitudinal component of the average beam velocity. For the longitudinal FEL‘s (first three rows in Table I) uoz = uo (boz = bo).

The dispersion curve of the electron waves is represented symbolically in Fig. 1 by the thick line of slope (w/kz) = uoz . A single electron which is forced to oscillate at frequency w and at the same time propagates in the z-direction with velocity u, is generating a traveling current wave with wavenumber w/u,. In the case of a warm electron beam there is some spread in the wavenumbers w/u, due to the variance vZth in the velocities. When the beam is cold enough, the eigenmodes of the beam plasma will be the longitudinal space charge waves with wavenumber (w ? wL)/u,, where w; is the modified plasma frequency of the electron beam. Either case is represented approximately by the thick dotted line of slope u,,, assuming Uzth << uoz and wL << w.

The dispersion curve of the electromagnetic wave, in the case of the Cerenkov FEL [Fig. 2(a)], is the well-known dispersion


TABLE I THE WAVENUMBERS OF THE SYNCHRONOUS COMPONENTS OF THE INTERACTING WAVES APSD THE RADIATION CONDITION FOR VARIOUS FEL‘S

EM Component Plasma Component FEL Wavenumber Wavenumber Radiation Condition

Cerenkov kzo - vo w h

w n-’(h) = po cos @

Smith-hrcell kzo + ko - -‘pol - C O S @

p 3 O -COS@

VO L

Longitudinal Bremsstrahlung kzo - -ko w -1

V o

Transverse Bremsstrahlung kzo

Compton-Raman k

_ - w k0

vo z

curve of a dielectric waveguide mode which at high enough frequency tends to a slope (phase velocity) c/n (n is the high- frequency index of refraction). If the beam which is propagated in close proximity to the dielectric waveguide has a mean velocity voz >c/n, then synchronism between the electron beam and an electromagnetic mode may be possible around the curve’s crossing region:

- N w - k,, . (2)

“02

This condition is equivalent to a condition for synchronism between the wave phase velocity and the beam velocity w/kzo uoz. The radiation condition in Table I (row 1) is derived from (2) using the definition of the “mode zig-zag angle” 4:

where k o l e . In the case of the Smith-Purcell FEL [Fig. l(b)], the

phase matching condition between the electron-beam waves and the electromagnetic mode is obtained by adding “crystal momentum ko” to the mode wavenumber kzo. To be more concrete, we may say that, in the periodic waveguide of which the Smith-Purcell FEL is composed, the electromagnetic eigenmodes are Floquet-Bloch modes and the smooth waveguide dispersion diagram is modified into a Brillouin diagram with period ko (which for clarity is only partly drawn as a curve section parallel to the smooth waveguide dispersion curve and displaced by k , to the right). The synchronism condition is obtained when the dispersion curve of the first- order space harmonic (kzo t k,) matches the curve of the electron-beam wave:

_- w - k,, + k,. (4)

vo z

This can also be interpreted in the velocity space as a synchronism condition between the wave phase velocity and the beam velocity: w/(kzo + k,) = uoz. The radiation condition of the Smith-Purceg effect [22] is derived in Table I from (4) using the definition for the “mode zig-zag angle” @:

c o s 4 = -. kzo k

The scheme for the bremsstrahlung FEL [Fig. l(c)] is similar to that of the Smith-Purcell FEL and results in the same synchronism condition (4), and consequently, the same radiation condition (Table I), The difference is that in this case a periodic magnetic or electric force operates on the electron beam and endows its waves with a negative crystal momentum -ko . Thus, the electromagnetic waveguide mode interacts with the - 1 order space harmonic of the electron-beam wave.

In the case of the most familiar transverse bremsstrahlung FEL (magnetic or electric), the interaction can take place with a transverse electromagnetic (TEM) wave for whch kzo = k(@ = 0). Hence the radiation condition for this case, which is given in Table I (row 4), is derived by simply substituting 4 = 0 in the previous result (row 3). In this case and also in the transverse electrostatic and Compton-Raman FEL‘s, the electron beam has a transverse component of the average beam velocity due to the transverse force applied by the pump, and voz Z vo . We defined for these cases

In the case of Compton-Raman scattering [Fig. l(d)], the periodic electrostatic or magnetostatic pump of the transverse bremsstrahlung FEL is replaced by an intense TEM wave with a frequency wo and wavenumber ko = 277th propagating in a counter direction to the electron beam. The combined effect of the scattering and scattered electromagnetic waves generates in the beam an electron-current wave which oscillates with the difference frequency w - wo and propagates with wavenumber (w - wo)/voz - ko . In wavenumber space the phase matching condition between the electron wave and the amplified electromagnetic wave is

~- w - wo

uoz ko N k (8)

which results in the radiation condition of Table I (row 5 ) . The phase matching condition (8) can be also explained in a


different way. The combined effect of the scattering and scattered waves generates in a parametric process a current with frequency w - wo and wavenumber k - k,. The scattering process is resonant when the phase velocity of the current is synchronous with the electron beam

w - w, k + ko - 'OZ - N

which is, of course, equivalent to (8).

111. THE DISPERSION-GAIN RELATION We consider the interaction between an electromagnetic

waveguide mode

and an electron beam, both propagating in the tz-direction in a free-electron laser structure, as shown in Fig. 1. The waveguide can be either a uniform or a periodic waveguide.

Due to the interaction of the electromagnetic wave with the electron beam, its amplitude grows gradually at a spatial growth rate assumed to be small compared to the wavelength. With this assumption, and an assumption that only one mode exists in the waveguide, the Maxwell equations may be reduced into two simple one-dimensional equations for the amplitude of the electromagnetic mode [27], [20], [21] :

(1 1)

EZ(X,Y'Z) =a(.) Gz(x,y,z)+-Jz(x,y,z) (12) 1

I W f

where a (z), the amplitude of the electromagnetic mode, is defined by

E,(x,.Y,z) = a(z) Gt(X,Y, z). (13)

Et(x,y,z) is the transverse field component of the uncoupled electromagnetic mode (10) and P is the Poynting vector power of this uncoupled mode :

P E ; ! ~ 1: Re [&,(x,y,z)xK~(x,y,z)] *c dxdy.

(14)

This formulation is useful for most kinds of waveguides of relevance to free-electron laser structures, including periodic waveguides. In the particular case of a uniform cross section waveguide (which is of relevance for all the FEL's except the Smith-Purcell type), the uncoupled mode field is indeDendent ofz: &.(x,y,z)=&(x,y). IntheSmith-PurcellFEL,&(x,y,z) is a periodic function of z with the period of the periodic waveguide - L .

The current J(x , y , z) is the alternating current which is induced in the electron beam by the interaction mechanism of any kind of free-electron laser scheme. When J = 0 (no interaction) the solution of (1 1) is

a (z) = a (0) eikzo (15)

giving back [considering also (12) and (13)] the uncoupled mode (1 0).

In order to complete the analysis of any particular free- electron laser scheme, the alternating current J ( x , y , z) which is induced by the electromagnetic wave in the electron beam should be calculated by solving the electron equations (the force equation, the Vlasov equation, the Dirac equation, or other) in the particular structure considered. In the linear regime this will usually lead to a linear relation between the current and the electromagnetic field. Equipped with such a relation, it would usually be straightforward to solve the linear (1 l), (12) by a Laplace transform technique.

This procedure was used successfully on both Cerenkov- Smith-Purcell [20], [21] and longitudinal electrostatic bremsstrahlung [ 111 FEL's. The Laplace transformed amplitude of the electromagnetic wave Z(s) E sow e-" a(z) dz was found in either case to be given by the same expression:

1 t ~ ( L J , s t ik,)/E (s - ikzo) [1 + x(o, s t ik,)/~] - iK x(w, s t iko)/e

Z(S) = . a (0)

(1 6) where

X ( w , s t i k , ) G ( l t a 2 ) X p ( w , s t i k o ) . (1 7)

xp (w, s) is the well-known longitudinal plasma susceptibility of an electron-beam plasma propagating in free space in the z-direction. I t is defined by

-00

(1 8)

g. (O) is the canonical momentum distribution function of the I rlectron beam entering the interaction region; u, = pz / (ym) is the electron longitudinal velocity component.

The parameter a2 in (17) is equal to zero for all FEL's discussed except the longitudinal electrostatic bremsstrahlung FEL, for which it was found to be [ 111

where 'Po = Eo /ko is the amplitude of the periodic electrostatic potential in this laser.

In (16) k , is the periodicity parameter which is given for all FEL's except the Cerenkov FEL by (1). In the Cerenkov FEL, which does not utilize a periodic structure, one uses k , = 0.

The coupling parameter K is listed in Table I1 for the various kinds of FEL's. A , is the electron-beam cross section area. gZl (x ,y ) , which appears in row 2 of the Table, is the longitudinal electric field profile of the first-order space harmonic of the electromagnetic mode in the Smith-Purcell FEL periodic waveguide:

I t is instructive to find out that the Cerenkov, Smith-


TABLE I1

ARE GIVEN IN BRACKETS) THE COUPLING PARAMETER K OF VARIOUS FEL‘S (THE HIGHLY RELATIVISTIC LIMIT EXPRESSIOKS

FEL K

Cerenkov Ae n

P h .-

71 Smith-Purcell ._ P h

71 Longitudinal Electrostatic or2 . _ P h

Purcell, and longitudinal electrostatic bremsstrahlung FEL‘s all have similar expressions for the coupling coefficient K . This stems from the fact that they all involve direct longitudinal interaction of an electromagnetic wave component with a synchronous electron plasma wave component. In the Ceren- kov scheme the z component of the total electromagnetic mode field &(x, y ) can be synchronized and coupled to electron plasma waves. In the Smith-Purcell (or traveling wave tube) type scheme only the electromagnetic mode first-order space harmonic & (x, y ) is synchronous with the electron plasma waves. In the longitudinal electrostatic bremsstrahlung scheme the total electromagnetic mode &(x, y ) is synchronous with only the -1 order space harmonic of the electron plasma wave which has an amplitude a [see (19)]. In all three cases the coupling coefficient is proportional to the radiation wavenumber k = 2r/h and to the “relative power” factors of the interacting wave components.

The dispersion-gain relation (16) applies to a good approximation also to the case of the magnetic bremsstrahlung FEL. Assuming operation near synchronism and introducing a “relative power” factor (filling factor) Ae/Ag (Ag is the effective cross section area of the electromagnetic mode), we were able to reduce the dispersion equations developed in [4], [lo] for the magnetic bremsstrahlung laser to a form identical with (16). The coupling coefficient K which results for this case is listed in Table I1 (row 5). Bo in this expression is the amplitude of the helical magnetic field modulation.

The transverse electrostatic bremsstrahlung FEL scheme is essentially equivalent to the magnetic bremsstrahlung FEL except that a periodic transverse electrostatic force is applied on the electrons by periodic alternating electrodes replacing the periodic magnetic (Lorentz) force which is applied in the

magnetic bremsstrahlung FEL by means of static magnets or coils. It was also suggested that high-amplitude electrostatic field modulation can be obtained due to the change in the space charge field of an electron beam transversing through a periodically rippled waveguide, and that this can be utilized for a free-electron laser scheme [ 121, [28].

By a simple extension of the linear analysis of magnetic bremsstrahlung FEL’s [4], [lo] it is possible to show [43] that the general gain-dispersion relation (16) applies, to a good approximation, to the transverse electrostatic bremsstrahlung FEL as well, with appropriate modification of the coupling coefficient K (given in Table 11, row 4).

In the Compton-Raman FEL the electron beam is pumped (wiggled) by both the transverse electric and magnetic fields of a time-varying electromagnetic wave propagating in counter direction to the electron beam. The dispersion relation which results from the analysis of this interaction [16] can be reduced to the general form (16), with K given in Table I1 (row 6), and a substitution o -+ o - wo made in (16), (18). The parameter K is expressed in the table for this case also in terms of the Poynting vector power density of the electromagnetic pump So (which for a circularly polarized wave is given by So = $ d m * E:).

It should be noted that, in all the transverse modulation FEL’s (transverse magnetic, electrostatic, and Compton- Raman scattering), the electron mechanical momentum distribution fo(&, i?,,,j72) inside the interaction region (in the absence of the amplified electromagnetic wave) has an average transverse momentum (quiver momentum) amplitude p o l # 0. This transverse momentum is endowed to the electron beam by the periodic (helical) transverse force of the “wiggler” field, and is vanishing (poi = 0) in the case of the longitudinal


FEL's. However, the distribution function g(O)(p,, p y , p , ) which appears in (1 8) is generally defined for all FEL's in terms of the transverse momentum variables pl = p i - pol where Fl is the electron total transverse mechanical momentum at the entrance to the interaction region and pol is the average quiver momentum. These variables are the canonical momentum components in the case of magnetic bremsstrahlung and Compton-Raman scattering FEL's, and are therefore constants of the motion in the wiggler field (approximately so also for the transverse electrostatic bremsstrahlung FEL). The amplitude of the quiver momentum pol is given for the different FEL's by

for magnetic bremsstrahlung

for electrostatic bremsstrahlung (21)

p for Compton-Raman. 0 0

If the wiggler field is turned on adiabatically, the dependence of g(O) on the transverse "canonical" momentum variables p x , p y is the same for all FEL's and is equal to the initial transverse momentum distribution of the electron beam far away from the interaction region. The dependence of g(O) on the axial momentum variable p z is shifted in the case of the transverse modulation FEL's and centered in the wiggler field around an average axial momentum

Poz = 4F-z where p o is the average total mechanical momentum. In the longitudinal modulation FEL's poz = po.

The axial average velocity is calculated from (22) using the relation uoz =poz/(yom). The parameter yoz can be then calculated from (7), or directly from (2 l ) , using the formula

We have shown that the different FEL's-both the longitudinal and transverse modulation schemes-satisfy the same gain-dispersion relation (16). As mentioned earlier, this common relation stems from the fact that the basic interacting waves in all interaction schemes are the same electromagnetic mode and the longitudinal single electron or space charge waves, which are coupled in either case by a different mechanism. We now turn to normalize the common relation (1 6) and derive the FEL gain expressions.

The common gain-dispersion relation (16) would yield the power gain for any FEL at any wavelength when Z(s) is inverse Laplace transformed and substituted in

The operating wavelength is determined from the solution of the dispersion equation which is the condition for the vanishing of the denominator in (16):

(s - ikzo) [l t x(o, s + iko)/e] - ~ K X ( W , s + iko)/e = 0.

(25) For a weakly coupled system (K is small), the eigenmodes of

the system have wavenumbers close to the eigenmodes of the uncoupled system (K = 0), which are the electromagnetic wave (s = ikzo) and the electron-beam plasma waves (solutions of the plasma dispersion equation: 1 t x / e = 0). The approximate wavenumbers of these waves (more precisely, the wavenumbers of the space harmonics which participate in the interaction) are listed in Table I, giving rise to the radiation conditions listed in the third column of the table. When the wavenumbers of the interacting waves match (synchronism), the dispersion equation (25) vanishes and the appearance of a pole in (16) indicates strong coupling of the electron plasma and electromagnetic waves. We should point out that (16) and the parameters in Table I1 were all derived with the assumption of operation near synchronism.

The susceptibility function x(w,s) [see (17), (18)] can be expressed in terms of familiar functions and parameters. The distribution function g(')((p) may be substituted in terms of a normalized distribution function of a single variable:

~ ~ ~ ~ ~ ~ z ~ . ~ ~ ~ ~ o ' ~ P x . P ~ . p , ) d p , d p y -00

where no is the electron-beam density, p z t h is the longitudinal momentum spread of the electron-beam distribution, and poz is the average electron-beam momentum in the longitudinal (2)

direction. In terms of the normalized function 9, the plasma susceptibility (1 8) can be written as

where

f = I

iw/s - uoz

U z t h

Ezth is the longitudinal kinetic energy spread of the electron beam. In stimulated Compton-Raman scattering, substitute 0'0- 0 0 .

Often the electron distribution is approximated by a shifted Maxwellian distribution. For this case


1 g(x) = - fi

and

(33)

is the so-called plasma dispersion function which is tabulated in [29].

Before we go to the next section where the solutions of (16), (25), and the laser gain regimes are discussed, let us examine (25) in the limit of a cold beam (pz th -+ 0). In this limit we get from (29), (28) It I .+ 00 and GI({) -+ 1/s2. Substitution in (27) and (1 6) gives

. -12

x(w, s) = - E W P

(w + iu,zs)2

Z(s) = r 2 .a(o> (s - ikzo) [(s + ik, - i xy t $1 - i K

uoz uoz 00 z

and the dispersion relation (25) is 12

(s - ikz0) [(s t ik, - i Ey t $1- iK 2 - - 0. (37) uoz uoz uo z

This equation is similar to the conventional traveling wave tube dispersion equation [30]. Its physical significance is seen when we take the limit K = 0 (no interaction). We see that the uncoupled eigenmodes of the system in the cold-beam limit are the electromagnetic wave

s = ik,, (38)

and the slow and fast plasma waves (correspondingly)

s+iko =i(x+ G) s+ik , =i(e- 2).

uoz Do2 (39)

The dispersion relation (37) can be further simplified into the compact form

6 k ( 6 k - B - B p ) ( 6 k - B + 8 , ) t Q = O (41)

where we defined the complex wavenumber modification due to coupling4 k :

s ik,, t i6 k , (42)

the synchronism (detuning) parameter:

B E - ko - kzO uo z

(43)

(w -+ w - wo in stimulated Compton-Raman scattering), the space charge parameter:

Fig. 3. Numerically calculated gain curves of the FEL for various values of the normalized thermal spread parameter eth(K = 1, g p = 1).

and the gain parameter:

IV. FREE-ELECTRON LASER GAIN REGIMES

(44)

In principle, the calculation of the gain of any FEL at arbi- trary operating conditions is straightforward, requiring only an inverse Laplace transform of (16), then using (24) and the appropriate coupling parameter from Table 11. In practice the execution of an inverse Laplace transform may be somewhat difficult in the general case (Pzth # 0) where the exact plasma dispersion function (28) or (34) must De used in (27). In this general case it may be most useful to evaluate the inverse transform

by numerical integration in the complex field. A computer program for performing this integral was developed [21]. Representative gain curves calculated by this program are shown in Fig. 3 for the case of a Maxwellian distribution (33). The gain curves are given in terms of the interaction length normalized synchronism parameter 3

(47)

and various values of the normalized thermal spread parameter

In the Compton-Raman FEL, w + w - wo , and consequently, (48) changes into


(a) WARM BEAM

kzo - 28ih 4 1

Y w/v02 i t h / &

@) LOW GAIN TENUOUS BEAM kzo ko -9 zrr/e +-+- -

w IV02 * z .6 /ev

kzo .. k0

w/voz . ep

b o ko

(c) LOW GAIN SPACE-CHARGE DOMINATED , 27/Q - Y

(d) HIGH GAIN COLLECTIVE , z m L

w/voz 8,

(e) HIGH GAIN STRONG COUPLING kz0 f8k ko p d 3 I -

W/VOZ 1 / 2 ~ 1 / 3 ’ Fig. 4. “Phase matching” (momentum conservation) diagrams at the

different gain regimes assuming operation at the maximum gain point. The section at the end of each diagram indicates the width of the gain curve in wavenumber space. There i s gain as long as the mismatch between the arrows is within this section.

The thermal spread parameter eth has the interpretation as the spectral width (evaluated in wavenumber space) of electron wavenumbers A(w/u,) in an electron beam with an average longitudinal velocity uoz and a velocity spread u i t h . As we will see later in this section, the beam is considered cold or warm if gfh << 1 or eth >> 1 , respectively. Fig. 3 displays ex- amples in both limits as well as in the intermediate cases. The normalized coupling and space charge parameters is and ep, which appear in Fig. 3, are defined by

- _ K =KI (50)

In many practical limits, exact calculation of the gain curve is not necessary and analytical expressions for the gain may be derived with certain approximations. These different limits (gain regimes) were delineated by a number of authors [4], [7], [ lo], [ 171, [IS], [20] for various kinds of FEL‘s. ‘Indeed they are common to all of them, since they result from the same gain-dispersion relation (1 6). We will briefly describe the gain characteristics in these regimes.

Fig. 4 displays “phase matching” (momentum conservation) diagrams which help to understand the various gain regimes discussed. It is essentially a “blow up” of the phase matching diagram in Fig. 2(b), looking at the details of the phase matching near the electron wave’s dispersion curve at the various gain regimes. In the bremsstrahlung FEL‘s, ko in Fig. 4 is reversed in direction. Also, in the stimulated Compton-Raman scatter-

Fig. 5. The normalized low-gain curves F(F, 8,)(55) for various values of the normalized space charge parameter g,.

(pzth -+ 0) the dispersion equation (25) reduces to (37) or (41), which are simple third-degree“polynomia1 equations with a finite number of roots (three). If the roots are found, then the evaluation of the inverse Laplace transform (46) may be straightforwardly calculated by use of the residuum-method. This gives

where Ai are the residues of (36) at si. The low-gain regime occur‘s at the limit when the normalized

gain parameter g i s very small: e << 1 (a sufficient condition):

Q~zi j ; = ~ 1 3 . (5 3)

In this limit the roots of (37) and (41) will be close to the roots of the uncoupled waves (Q = 0)’ [see (38)-(40)] and may be expanded to first order in K or Q around these zero-order values. After a lengthy mathematical calculation [21] , substituting (52) in (24), and calculating the power output to first order in K (or Q), one gets

(54)

where IAPI IP(1) - P(0) I << P(O), and

The function F(& is shown in Fig. 5 for various values of the parameter &, (51). Since F($, ep) < 1, the parameter gindicates an upper limit on the power gain available.

Equations (54) and (55) give the low-gain characteristics of free-electron lasers including the space charge effect. For a tenuous beam or a short interaction length, << n, and then (55) reduce to

ing, ko = wo/c is reversed and w/uoz + (o - wo)/uoz. In the d sin @/2) Cerenkov FEL, ko = 0. The other details remain the same in F(@, 0) = -z- [ ’ el2 ] all FEL’s.

which is the familiar single electron gain function which ap- Cold-Beam Low-Gain Regime pears in the analyses of the various free-electron lasers when

As was shown in the previous section, in the cold-beam limit the space charge effect is neglected [ 2 ] -[5] , [lo] , [ 191 , [20] .

de (5 6)


Notice also that the gain curve for this case (Fig. 5, ep = 0) resembles the computer calculated curve of Fig. 3 for e,, = 0.1 << 1 (cold beam). It attains its maximum value at

2.6 w 2.6 1 uoz 1

e=---- or -+-=kzo + k o . (57)

This situation is illustrated in Fig. 4(b). It is interesting to point out that the expression which was

derived by Lousiell et al. [31] for the magnetic bremsstrahlung FEL gain in the low-gain regime including the space charge effect is actually the Taylor expansion of (55) to first order in 3; :

(58)

This expansion is correct, of course, only for 3, << n. The general space charge limited expression (55) has the

interesticg physical interpretation as the result of the interference of the electromagnetic wave (38) with the slow space charge waye (39) on one hand and with the fast space charge wave (40) on the other hand. Whenever the electromagnetic wave is synchronous with the slow space charge wave (0 N -0, or k,, t ko w/uo, t e,), the interference yields maximum net gain. When the electromagnetic wave is synchronous with the fast space charge wave (0 Bp or kzo t ko w/uoz - e,), maximum attenuation is obtained. The phase matching diagram in the space charge dominated low-gain regime (gp >> n) at the maximum gain point

6 = -ep or - t ep = kzo t ko w

U O z (59)

is shown in Fig. 4(c). The gain expression (54) may be of practical importance in

situations when low-gain operation is expected (as in laser oscillators) and a weak coupling parameter (K) is unavoidable (as in the case of short wavelength operation). In this situation one may try to increase the gain parameter i5 (53) by increasing the electron density and the interaction length I , thus arriving at the regime e, >> n. For example, with beam parameters no = lo1’ ~ m - ~ , yo = yoz = 10, uoz c , and I = 2 m, one gets gp 2: 5 and it is necessary to use (55) to describe the gain.

In the extreme space charge dominated limit e, >> n we get that, at the maximum gain point (59), the normalized gain function (55) attains a value F(- g,, 3,) = 1/(2Pp). Thus, the maximum gain [(54), (55)] is given by

We notice that in this low-gain space charge dominated regime the gain grows proportionally to 6, while in the tenuous beam limit (56) where collective effects were negligible, the gain was proportional to no (single electron interaction). Notice also that, since gP >> n was assumed, the parameter

gives again an upper limit on the gain available. It should be pointed out here that as the electron density is

increased and the space charge effects become important, the

saturation (trapping) effect would start at a lower input power level of the electromagnetic radiation. In this case the linear analysis which led to (54) may fail to describe the situation with practical power levels and a complete nonlinear analysis should be used [32], [33].

Cold-Beam High-Gain Collective (Raman) Regime For weak enough coupling (Q<< 1) the roots of (41) are

all real and small, thus the roots si [see (42)] are all imaginary and close to the uncoupled wavenumbers (38)-(40). They then give rise to the “interferential” gain expression (54), (55). As e is increased, the polynomial (41) starts having one real and two complex (conjugate) solutions. Consequently, one of the roots si [(42)] must have a positive real part (corresponding to gain) and the other two have a negative and a vanishing real part (corresponding to loss and a constant amplitude, respectively). In this limit-“the high-gain limit-” we can neglect the interference between the different roots in (52), (24) and keep only the exponentially growing wave. Then, from (24)

In - - ‘(’) - 2 In IAi I t 2 (Re si) 1 = 2 (Re si)Z = -2 (Im 6k)l P(0)

(61) where si is the exponentially growing root.

synchronous with the slow space charge wave In the particular case when the electromagnetic wave is

e = -ep or - t ep = kzo + ko w

(62) UO z

and assuming 16k I << O P , (41) may be approximated by a simple second-degree equation:

Q 2%

(6k)2 = --.

The root of this equation, which corresponds to a growing wave, is 6k = -idQm and, substituting in (61), it gives

This gain regime is often termed as the stimulated Raman regime because it involves stimulated scattering of the electromagnetic wave by the slow space charge plasma wave [4], [ lo] , [34] , [35], [36]. The phase matching diagram o f these waves [see (62)] is shown in Fig. 4(d).

In this gain regime the gain grows with electron density proportionally to &I4. Also, we point out that the derivation of (64) required the constraints >? IIm 6kl I >> 1 ; thus we see from (64) that the parameter &gives an upper limit of the gain available in this regime.

Cold-Beam High-Gain Strong Coupling Regime In the limit of high gam and strong coupling (“strong pump”)

the FEL parameters satisfy Q‘l’ >> 0, (or, equivalently, K >> 0,). Near synchronism

8 - 0 or -=kzo t k o uoz

w


6 and 6, are negligible relative to 16kl in (41), which reduces then into a straightforwardly soluble third-degree polynomial equation:

(13k)~ = -Q. (66)

The root of this equation, which corresponds to a growing wave, is

6k = - i6 2

and, using (61), it gives [4] , [7] , [ 101 , [ IS ] , [20]

Hence, in this gain regime the parameter solely determines the available power gain. The gain depends on the electron- beam density in proportion to

Equation (67) indicates that the real part of the wavenuniber k,, changes appreciably due to the interaction (by Re 6k). Instead of (65) it would be perceptive to draw the phase matching diagram at the maximum gain point [Fig. 4(e)] in terms of the modified wavenumber k,, t Re 6k:

Re 6k = - Q1I3 or - t - &'I3 = (k,, -b Re 6k) t ko . 1 w 1 2 voz 2

Warm-Beam High-Gain Regime In deriving the cold-beam dispersion ind gain relations (36),

(37), we used an asymptotic expansion of (28) which is valid only for I{ I >> 1. If this condition is not satisfied, one has to go back and solve (25) with the plasma susceptibility given by (27) and (28) and not by (35).

Using (42) in (29), the parameter { may be written around s = ik,, in terms of the detuning parameter 6 [(43)] :

0 - 6k {z-

n CI t k

where etk is defined by (48). If 0 t h >> IIm 6kl (which later yields 6:h >> Q) and etk >> 6, (which is equivalent to k,, t ko >> kb-the space charge wavenumber is much shorter than the Debye length), then it follows from (70) that the require- ment I t1 = 1(6 - Re 6 k)' t (Im 6 k)' 11/2/6tk >> 1 cannot be satisfied at any of the synchronism conditions required for the gain regimes previously discussed. In these conditions we are bound to look for gain in the regime 15 1 5 1 and solve (25) with x(w, s) given by (27) and (28).

With a small enough coupling coefficient IC, it is possible again to solve the dispersion equation (25) by means of a first- order expansion of the roots in terms of ;. In the conditions stated above it is possible to show that, apart from the electromagnetic-like root ( 3 8 ) , all the other zero-order solutions of the dispersion equation are complex, corresponding to plasma waves which decay strongly by Landau damping. It is sufficient then to consider only the isolated root s = ikzo [(38)] which would give an exponentially growing wave.

The first-order expansion of (25) around s = ik,, gives

e - Re6k e {r E - --

th t k

From (28); we have

(75)

Because of our initial assumptions = lF$il/6tk << 1, the Lorentzian inside the integral (75) is much narrower than the function g'(i): Hence

Im G'(fr) N ng'(tr)- (76)

Using the previously assumed inequality k b << k,, + ko, (71) can be written in the form

In terms of the detuning parameter 6 , the gain is given by [4] , ~ 0 1 , ~ 7 1 [181 , POI

(78)

For a Maxwellian electron distribution (33), the imaginary part of the plasma dispersion function (76) is

Im G f ( f r ) = - 2 f i r r e . -s: (79)

This function is shown in Fig. 6. We see that it attains its maximum value when

{,.=-I/&?' or

e = - 6 t h / f i or

The diagram of this phase matching condition is shown in Fig. 4(a).

At the maximum gain point (80) Im G'(- l / d ) N 1.5 and the maximum gain is

Since we assumed initially >> 1, again the parameter indicates an upper limit on the power gain available.


ImG’ 1

-:v -1.5

Fig. 6. The gain curve function Im G’(cY) for a Maxwellian distribution (79).

The warm-beam regime is a regime of single electron interaction [17] and therefore the gain turned out to be proportional to the electron-beam density no.

Warm-Beam Low-Gain Regime Assuming in the low-gain limit that K + 0 and also that

the space charge effect can be neglected (eth >>ep or k,, t ko >> kbj , (1 6) reduces to

Z(s) = [-- 1 t lim iKX(0, S t & , ) / E

s - ik,, 8-0 (s - ikzo)(s - ik,, -t 2.6)

By substituting (27)-(29) in (82) we get the explicit dependence of Z(s) on s. The inverse Laplace transform (46) can be carried out now through the integral over x (or u,) in the expression for x [(27), (28)] and evaluated by means of the residuum method. The result can be then substituted in (24), resulting in, after some tedious mathematical expansion (to first order in K),

This expression can be interpreted as a convolution between the warm-beam gain curve g‘(O/e,,) [(78)] and a spectral line shape function sin2 (8/2)/(8/2)2 which can be attributed to the wavenumber uncertainty due to the finite interaction length. When integration by parts is applied to (83), the resulting expression will involve convolution between the cold-beam gain curve (dld8) [sin2 (8/2)/(8/2)2] [see (56)] and a line shape function g(O/O,,), which corresponds to line broadening due to the electron velocity spread. Notice that the linear convolution relation (83) is applicable only in the limit of single electron interaction (k,, + ko >> kD) and low gain (APIP(0) << 1). However, in the derivation of (83), there was no restriction on g t h , and it would apply to both cold and warm beams as well as to the intermediate regime.

In the limit et, >> 1 (warm beam) the line shape function sin2 (8/2)/ [2n(8/2)’ ] reduces to a “delta” function and (83) reduces to

This expression is consistent with the high-gain expression (78) which gives an identical result when we approximate In [P(Z)/P(O)] = AP/P(O). The high-gain expression (78) thus applies in the low-gain limit as well. The reason is that, in the case we had, where the dispersion equation (25) had only a single significant root (s N ik,,), the high-gain approximation (61) is correct at any gain level.

In the limit of a cold beam (gth << l), (83) reproduces the low-gain cold tenuous beam limit expression (54), (56).

v. THE GAIN PARAMETER OF FEL’s From the discussion above it appears that the normalized

gain parameter r? [(53)] is a good figure of merit to characterize the gain of any free-electron laser in all gain regimes [(54), (60), (64), (68), (78), (83)]. While the maximum gain obtainable depends also on parameters which characterize the beam (gp , st,), the dependence of the gain on the kind of FEL considered enters in all these equations only through the parameter e. Furthermore, the parameter Dindicates an upper limit of the gain in all regimes. As we increase g b y either increasing the coupling coefficient or the electron-beam density, this upper limit grows at a declining rate from roughly e, through

(if we go through the high-gain collective regime), and finally, as Q 1 I 3 . A simple necessary condition for obtaining appreciable gain in any FEL structure and any gain regime is

In comparing the gain of the various kinds of free-electron lasers discussed in this chapter, it is best to compare their normalized gain parameter e. This was calculated by substituting K from Table 11 in (53) and is presented in Table 111 for the various kinds of FEL‘s.

Since in many cases the current density Jo is a limiting factor, we expressed in Table 111 the modified plasma frequency in terms of the current density 0: = eJo/ (m~ou, ,~~z-yo) . The parameter re which appears in the table is the classical electron radius:

e> 1.

e“ re z = 2.818 X m. (85)

The expression given for the longitudinal electrostatic bremsstrahlung FEL (row 3) is taken from [ 111 . The expression in brackets corresponds to the highly relativistic limit, assuming operation at an angle Gm = (‘ )-I at which the gain parameter is maximal.

When one considers the dependence of the parameter e on the operating parameters of the various FEL’s, it should be noticed that some of these parameters are dependent on each other. For example, the wavelength h, period L , velocity uo , and radiation angle @ depend on each other through the radiation relations which are given for the various kinds of FEL‘s in Table I. In comparing various types of FEL‘s and examin- ing the dependence of their gain on the operating parameters, one should specify which are the independent parameters. In Table 111 the independent parameters are assumed to be X, uo ,

4 7r€o mC


Smith-hrcell

Transverse Mag. ”

Jo 1.9 Ae (1 + P O Z ) ~ YOZ 2

e c2 mc2 A, Poz YO Transverse Electrostatic 2n--.-u ._.

4 3 .-

Compton-Raman

and @. The period L (or in the Cerenkov FEL-the index n) should be determined from the relations of Table I. This choice is most convenient for the purpose of comparison among the different FEL’s.

The factor 3 ~ ~ J J A e I E z ( x , y ) I 2 dx dx/P which appears in the first three rows of Table I can be interpreted as a relative power factor or generalized “filling factor,” which indicates which fraction of the electromagnetic wave power participates in the interaction. In Table I11 (row 3) the calculated expression for the filling factor of the electrostatic bremsstrahlung FEL [l l] was substituted in. Numerical calculation of this factor for a Cerenkov FEL was done in [21] . In the case of a Smith-Purcell laser structure, the calculation of the filling factor requires the full Floquet mode solution of the electromagnetic wave in the periodic waveguide. While such an involved calculation would be avoided in the present work, we may point out that the filling factors of the Cerenkov and Smith-hrcell FEL‘s are only a function of the waveguide and electron-beam geometrical dimensions normalized to the radiation wavelength. Therefore, if one scales the geometry of the waveguide and the beam in proportion to the operation wavelength, it appears from the radiation conditions of Table I that over a wide range of wavelengths the “filling factor” is only a function of the independent parameters uo and 4, and does not depend directly on the wavelength X. Consequently, Table I11 is an explicit representation of the gain dependence on X for the various FEL‘s, assuming that the limiting factor is the current density Jo, and that the waveguide geometrical dimensions of the laser amplifier are optimized for the operating wavelength. It is instructive to note that the wavelength dependence of the gain parameter e is a proportionality relation for all FEL’s except the Cerenkov-Smith-Purcell type, for which it is an inverse proportion.

The dependence of the gain parameter on the electron-beam energy is not explicit in Table I11 for the Cerenkov-Smith-

Purcell FEL‘s, since the filling factors are a function of the electron velocity. However, it is probably correct to conclude that the gain parameter Q goes down as the energy is increased in all kinds of FEL’s, but that in the longitudinal interaction FEL‘s (first three rows in Table 111) it drops at a faster rate than in the transverse interaction FEL’s (last three rows).

We should point out that, in particular cases like the magnetic bremsstrahlung FEL operated with a high-energy linear accelerator [42], it may be more practical to use different independent parameters to describe the scaling laws of the gain parameter. Since the beam energy can be varied to quite a large extent with such accelerators, it is useful to express yoz in terms of the wavelength X and magnet period L. If we use the radiation relation of Table I (row 4) in the extreme relativistic limit, and (23), the expression for the magnetic bremsstrahlung gain parameter QMMB can be written as

This indicates that, for a constant magnet period L , the gain parameter scales with the wavelength like h3I2, as argued in [44]. It should also be appreciated that the limits on obtainable current density Jo may be wavelength dependent. As total current is increased and beam area A , decreased (keeping the filling factor Ae/Ag constant), higher gain is obtainable; but if the electron beam has a finite emittance (defined as the product of beam area and angular spread), then the maximum current density may be limited by the maximum allowable beam angular spread, which is consistent with proper operation of the FEL in the particular (low-gain) regime that it is designed to operate in. This limit is then wavelength dependent and requires more detailed consideration for particular cases [44].

In comparing magnetic and electrostatic bremsstrahlung


TABLE IV THE EFFICIENCY COEFFICIENT 3 USED FOR CALCULATING THE EFFICIENCY

(88) AT DIFFERENT GAIN REGIMES

Gain Regime - rl

Low-gain tenuous beam 2.611 Collective (low or high gain) 9 P High-gain strong coupling 4 Q 113 = f 113

Warm-beam high-gain

Warm-beam low-gain

et al. [lo] . For a magnetic bremsstrahlung FEL with a cold beam, this estimate has been shown to be in excellent agreement with the more rigorous self-consistent nonlinear formulation of [32] . Also, it agrees in the low-gain regime with the result of [38], which was obtained with a more elaborate analysis.

Consider any kind of FEL operating at any of the cold-beam gain regimes discussed before (the warm-beam regime must be discussed separately). The combined excitation of the coupled electromagnetic wave and the electron-beam space charge waves has a component which propagates with wavenumber k,, t ko t Re 6k and phase velocity

FEL‘s, it is necessary to point out the basic advantage of the first kind, which stems from the fact that the force applied on a relativistic electron by a magnetic field is usually much greater than the force applied by an electrostatic field, considering the state of the art of laboratory-available periodic magnetic and electric fields (the latter are limited mostly by breakdown effects). However, if one also considers free- electron lasers in the nonrelativistic regime, it is seen from Table I11 that the gain of electrostatic FEL’s grows higher at low Po, values.

As a last point in the comparative discussion of free-electron laser gain, we should indicate the advantage of transverse interaction FEL‘s compared to the longitudinal interaction FEL‘s in the consideration of waveguide losses of the electromagnetic wave. The effect of waveguide losses were not included in the present model. However, for practical FEL design, it may be in some cases an important consideration which cannot be ignored. To a good approximation it may be argued that the net gain of a free-electron laser is equal to the gain of a lossless FEL structure substracted by the waveguide losses of the electromagnetic mode. Since transverse electric or transverse electromagnetic modes which are used in transverse interaction FEL‘s suffer usually less waveguide loss than the transverse magnetic modes used in longitudinal interaction FEL’s, this provides a substantial advantage to the first kind when net gain is being compared.

VI. FREE-ELECTRON LASER EFFICIENCY As discussed in a number of articles [32] , [33] , [37] , high

efficiencies of the order of tens of percents are potentially achieveable in free-electron lasers using various efficiency enhancement techniques like tapering the pump period and intensity [32], [33], [45] or using energy-retrieval schemes like the storage ring [46] or depressed collector [2], [37] (which is currently used in conventional traveling wave tubes). In this section we will consider only the basic efficiency qo of the various FEL‘s, noting, however, that this efficiency may be increased appreciably when efficiency enhancement techniques are used.

Since the efficiency of radiative energy extraction from the electron beam is limited by saturation effects, a full nonlinear analysis is necessary to derive the FEL efficiency qo. Never- theless, it is possible to estimate its value from the linear analysis by using a simple consideration presented by Sprangle

0

= kzo t ko t Re 6k ’

The estimate of FEL efficiency is based on the argument that the saturation mechanism is electron trapping in the periodic potential of the excitation, which propagates with phase velocity uph (86). It can be shown that the maximum deceleration in the electron velocity during the interaction up to full trapping is 2 6u,, where Su, is the initial axial velocity difference between the electron beam and the phase velocity of the trapping potential:

Thus the maximum radiative energy extraction efficiency can be calculated as the relative change in the electron-beam kinetic energy when its axial velocity decreases by 6v,:

[vo = ,,,;ip] where

The expression in brackets corresponds to the highly relativistic limit.

The efficiency can now be calculated for any of the cold- beam gain regimes described in Section IV by simply substituting the appropriate 0 and Re 6 k in (88), (89). The values of these parameters will usually be assumed to correspond to the maximum initial gain condition. The efficiency coefficient G, in the case of a cold beam, is exactly the wavenumber mismatch drawn for various gain regimes in diagrams (b)-(e) of Fig. 4. It is listed in Table IV for the different gain regimes. Notice that in the low-gain and collective gain regimes !Re 6k( << 10 I and i j -0 and in the high-gain strong coupling regime (Re 6k( >> 101 and i j N Re 6k.

The estimate of the laser efficiency in the warm-beam high- and low-gain limits requires a somewhat different consideration. As is indicated from (75) and (83), in these gain regimes only a small fraction of the electrons in the electron-beam distribution participate in the interaction. In the high-gain case


(75) these are the electrons with normalized axial velocities which lie in the regime Ix - tp I = Iu, - uph I/&h 5 Iti I = 16 ki I /e th or whose velocities lie within the spectral width of the wave due to its growth Iw/u, - krl 5 6ki. From either of these relations we find that the class of electron velocities, which takes place in the interaction, is

During the interaction the trapped electrons reverse their velocity relative to the phase velocity of the potential wave Uph, until at saturation a local plateau is generated and g'((uph - uOZ)/uith) = o (see Fig. 7).

In the warm-beam low-gain case the saturation process is similar, except that the class of electrons that participate in the interaction is determined by the width of the spectral line shape function sin2 (1!?/2)/(@/2)~ which appears in the convolution (83): 18' - el = I w/u, - uph I1 5 TT. The wavenumbers of these electrons lie within the wavenumber spectral width determined by the finite interaction length I (ob,) - uph I 5 TT/Z. These relations can be written as

The radiation extraction efficiency of the free-electron laser in the warm-beam regimes is equal to the relative change in the kinetic energy due to the interaction, where the calculation of the change in beam kinetic energy before the interaction and after saturation is most conveniently done in momentum space:

AEKE - 1 P O =-----

EKE YO - 1 no s'"" pzph - + 6 p z 6 p z

* (Y - 1) k ' O ' ( P Z ) - g'oTPzph)l dPz (92) where pzph = yphmuph and 6pz E yo yizm6 u,. First-order expansion of both y and g(')(pz) around pz =pzph and some mathematical steps give

(93)

Using (go), (91), and (77), we finally find that in both high- and low-gain warm-beam regimes the efficiency is given again by (88) with the parameter .fi listed in Table IV (rows 4 and 5) for the high- and low-gain limits, respectively. The expressions in parentheses were calculated for an electron beam with a Maxwellian distribution (33), assuming initial tuning to the maximum gain point 5, = - I/*.

The radiation extraction efficiency for a12 FEL 's and all gain regimes is estimated by the simple expression (88) with the efficiency coefficient < given in Table IV for the different gain regimes. This expression applies to FEL amplifier structures,

vph voz VZ

Fig. 7. The electron distribution function in the warm-beam gain regime drawn in velocity space before (continuous line) and after saturation (broken line).

assuming saturation is reached within the interaction length I. For oscillator structures (in which saturation is always obtained within the interaction length) it applies with the assumption of single-mode operation.

The efficiency depends on the interaction length 2 only in the low-gain tenuous beam regimes (both cold and warm beam). Notice, however, that the expressions in Table IV (rows 1 and 5) apply with the assumption that saturation is reached right at the interaction length 1. In the cold-beam low-gain and collective regimes (rows 1 and 2 in Table IV), the efficiency is proportional to X and depends only on the beam parameters (velocity and density) and the interaction length. It is thus the same for all FEL's considered. Also, in the warm-beam low- gain regime (Table IV, row 5) the efficiency is independent of the kind of FEL considered and depends only on the parameters of the beam and the interaction length. The efficiency in this limit is proportional to h3.

In the high-gain strong coupling and high-gain warm-beam limits (rows 3 and 4 in Table IV) the efficiency depends on the kind of FEL considered through the parameter Q. Its dependence on wavelength and beam velocity varies for different FEL's and can be found by substituting in (88) the corresponding : parameter, the parameter 0th (48), and the values of the parameter Q, as listed in Table IV for the different FEL's. In all cases the efficiency q0 grows with the wavelength X. This dependence is particularly strong in the high-gain warm-beam case, where it goes like X6 in the Cerenkov-Smith-Purcell FEL's and X' in the bremsstrahlung FEL's.

In the case when beam energy-retrieval schemes are used, the total efficiency can be calculated from the expression for the basic efficiency (88) by multiplying it by an efficiency enhancement factor M. This factor is equal in the case of a pulsed mode storage ring to the number of times that the electron beam can be recycled before its spread becomes intoler- able, or in the case of steady-state operation, to the ratio between the electron energy and the acceleration gap energy. In the case of a depressed collector scheme, it is equal to the ratio between the electron-beam acceleration and collection voltages. In the case of efficiency enhancement by pump tapering, we cannot use at all the estimate (88) and a full nonlinear analysis is necessary [32], [33], [45].

In the conclusion of this section we will briefly discuss the revelations of the different saturation characteristics in the two separately treated limits of warm and cold beams. We saw that while in the warm-beam case only a class of electrons (90)


or (91) participates in the interaction and gets trapped at saturation, in the cold-beam limit all the electrons participate in the interaction and trapping process. As pointed out before in [17] [39], the local plateau formation (Fig. 6) is analogous to the saturation and diminishing of population inversion in a class of atoms in Doppler broadened gas lasers. In terms of laser theory, the saturation nature of a warm-beam FEL corresponds to inhomogeneous broadening and that of a cold beam t o homogeneous broadening [40].

As in gas lasers, there is a “hole burning” effect in the gain curve of the warm-beam FEL. However, there is still an important difference between the two: in contrast to the homo- geneously broadened gas laser, in FEL‘s the interaction mechanism itself changes the population of velocity classes in the vicin- ity of the interacting class of electrons. The gain curve of the saturated FEL (the derivative of the saturated distribution curve in Fig. 6) displays from the point of saturation on an effect of “hole burning” at u, uph. But it also shows a new effect of “Ml heaping” at the sides of the “hole” (vz uph Su,) where the slope of the distribution function is high.

These different saturation behaviors will have explicit expression in FEL oscillation characteristics [47]. In the homogeneous broadening cold-beam regime there will be “mode competition” between the modes of a long cavity laser, all “attempting” to extract power from the same electrons. This will tend to depress multimode operation in the laser. In the mhomogeneous broadening warm-beam regime, there will be at saturation an interesting new effect of “mode cooperation” which will tend to increase the number of oscillating modes in a laser cavity which has a sufficiently dense spectrum ofmodes. The effect of mode cooperation will tend to wash out any bumps on the electron distribution function and spread it out and will produce wide-band radiation. Notice that this “super inhomogeneous” saturation behavior is different from that of inhomogeneously broadened gas lasers where there is no cooperation between the modes.

VII. FREE-ELECTRON LASER POWER One of the most attractive qualities of free-electron lasers

seems to be the potential capability of high-power operation. Since the interaction is done in vacuum and the active medium is not a material, problems of material damage and thermal dis- tortion of the wavefront are avoided. Since the radiation extraction efficiency of FEL‘s can be made quite high, the laser power can be quite high, too, and limited essentially by the electron-beam power which can be drawn into the interaction region.

The radiative power which is generated by the free-electron laser is

A P = VIVO (94)

where qo is the basic radiation extraction efficiency (88), I is the total beam current, and V is the beam acceleration voltage:

V = (yo - 1) mc2/e. (9 5)

For a given efficiency and beam acceleration energy, the radiative power generation (94) will depend only on the amount

(C) INTERACTION RANGE

l / 2 q“ R

Fig. 8. Cross section diagrams of various FEL structures: (a) sheet beam in a planar waveguide structure, (b) annular beam in a cylindrical waveguide structure, (c) solid beam in a cylindrical waveguide structure. The cross-marked regions symbolize the source of periodic static field in bremsstrahlung FEL’s and the slow wave structure in Cerenkov-Smith-Purcell FEL’s.

of electron current I which can be made to interact with the electromaghetic wave in the free-electron laser. In many cir- cumstances there will be a limit on the current density Jo which can be generated at the FEL input and propagated along the interaction length, hence, the limiting amount of interacting current will be given by

I = A , J , . (96)

Can the electron-beam cross section A, be increased indefi- nitely? Apart from various technical limitations, we should be aware that the transverse dimensions of the electron beam should be limited by the range over which the periodic static fields and the electromagnetic mode field have an appreciable value, so that an appreciable coupling coefficient (Table 11) exists to carry out the interaction. This limitation stems from the fact that the static periodic fields and the electromagnetic modes should satisfy correspondingly the Laplace equation

(v‘{E,(x,~) sin koz ; ~,(x,y) sin koz} = O (97)

in the case of bremsstrahlung FEL’s, and the wave equation

for the Cerenkov-Smith-Purcell type (in the Cerenkov case

The solution of (97), (98) in the case of a planar waveguide structure [Fig. S(a)] is straightforward, giving, for the bremsstrahlung FEL‘s,

ko = 0 , G I -+ G o ) .

{EO (x7 Y) ; ’0 (x, .?)I Ore -qx (99)

where

q = n o =L 271

and, for the Cerenkov-Smith-Purcell FEL‘s,


TABLE V INTERACTION ~ N G E AND UPPER BOUND OF EXTRACTABLE POWER IN FEL‘s

Interaction Rage Max. Power

FEL ’i(9 2q 4n

Transverse Bremsstrahlung

P 4 z l

r 2 4 1 Longitudinal Bremsstrahlung PO P;Y:

1 - po cos @ 1 - po cos @ r 4 d l

Cerenkov-Smith-Purcell Po70 [YO1 r,Y; I

G1(x,y)ae-qx (101)

where

4 = (kzo + k d 2 - 2 * [ a21 ‘I2 (102)

Thus, both the static pump field in the case of bremsstrahlung FEL‘s and the slow electromagnetic wave component in the Cerenkov-Smith-Purcell FEL’s decay exponentially away from the source of the periodic static field (magnets, coils, or electrodes) or from the slow wave structure (periodic waveguide, dielectric waveguide). In order to obtain an appreciable coupling coefficient ( K ) , the electron beam should be passed roughly within the range of one exponential e-folding distance of the squared field intensity 1/(2q). This distance, which we will term “the interaction range,” can be found in terms of X, poz, and cp by substituting in (loo), (102) the synchronism condition (4) and the radiation conditions of Table I. The interaction range for the various FEL‘s are listed in the second column of Table V. The expressions in brackets correspond to the special case of (yo >> 1) where, for the longitudinal bremsstrahlung case, we substitute in this limit 4 = cpm =

In all FEL’s the interaction range (1/2q) is proportional to the wavelength X and only when the beam is highly relativistic ( T ~ >> 1) can the interaction range be appreciably larger than a wavelength. This explains why at short wavelengths it was necessary in recent FEL research (as compared to the old microwave tubes art) to go to relativistic electron beams in order to obtain appreciable radiation. We see from Table V that, in the highly relativistic limit, bremsstrahlung FEL‘s have an advantage of roughly a factor yo in the interaction range compared to Cerenkov-Smith-Purcell FEL‘s. There is no difference between them in the nonrelativistic limit (poz << 1).

For other waveguide cross section geometries, like a cylindrical waveguide [Fig. 8(b), (c)], the solution of (97), (98) gives other function solutions for the transverse variation of the fields instead of (99), (101). For the case of a cylindrical waveguide, the fields depend on the transverse coordinates, like modified Bessel functions of the argument qr ( y is the distance from the cylinder axis). When r >> l /q, the modified Bessel functions behave near the waveguide walls like an exponential function -eq‘. In a crude approximation we may extend the notion of interaction range to waveguides with

( 6 Y O 1 .

general cross sections and claim that the interaction region, at which the coupling coefficient K has an appreciable value, ex- tends along the periphery of the waveguide within one interaction range (1/24) away from the walls [see Fig. 8(a), (b)]. Out of this region the coupling coefficient drops down exponentially.

It is evident that for obtaining strong interaction it is usually preferable to pass the electron beam within the interaction region where k: has its maximum value. So, in the case of a planar geometry [Fig. 8(a)], we will prefer to use a sheet beam of dimensions we X he where we is the waveguide width and he = 1 /(2 4). In the case of a cylindrical waveguide; it is best to use an annular beam of thickness he N 1/(2q) and periphery we = 2 nR [Fig. 8(b)]. In case the experimental conditions require the use of a solid cylindrical beam, it is preferable (for the sake of obtaining appreciable gain) to keep the waveguide radius not much bigger than R 2 1/(2q). The maximum beam cross section is then approximately A, = n/(2q)2.

It should be pointed out that at the present time there is little theoretical analysis for operation of various kinds of FEL’s in regions where either the electromagnetic wave mode or the pump field (in bremsstrahlung FEL’s) varies transversely [48], [49]. Magnetic bremsstrahlung FEL experiments which were carried out in the low-gain regime [42] used a thin solid beam which was propagated along the axis of a superconduct- ing magnet [Fig. s(~)] where the periodic magnetic field can be considered (to first-order approximation) transversely uniform. On the other hand, experiments which were carried out in the high-gain regime with intense relativistic electron beams (for example, [36] ) usually use an annular electron beam [Fig. 8(b)] where there is a certain transverse gradient in the periodic magnetic field intensity across the beam cross section. A transverse gradient in the magnetic field intensity may impose a limitation on propagating the electron beam in a confined cross section along a long enough interaction length. It will also have the effect of increasing the effective longitudinal momentum spread of the beam, as is evident from inspection of (21), (22), (99) (this may be partly overcome by the “gain expansion” mechanism in particular configurations [48]). These effects will impose limits on the width of the electron beam which can be used in particular configurations of bremsstrahlung FEL‘s, depending on the electron-beam energy and the gain regime of operation. This, as well as technical limitation on propagating the electron beam too close to


the waveguide walls, would usually limit the electronbeam cross section area A, to less than the maximum interaction region cross section area we X l(24).

However, it is generally true that in either case the electron- beam cross section area cannot be made much larger than the interaction region area if appreciable gain is to be attained; hence, for the goal of estimating the upper bound of radiative power generation in free-electron lasers, the estimate that the useful electron-beam cross section area is equal to the interaction r,g' p ion area

A, we X 1/(24) (103)

is an appropriate assumption.

sion for the maximum available power: Using (95), (96), (88), and (103) in (94), we get the expres-

This is listed in the .third column of Table V for the different kinds of FIEL's. The efficiency coefficient i j is listed in Table IV for the different gain regimes. The expressions in brackets in Table V correspond to the highly relativistic limit (yo >> 1) where, in the case of the longitudinal electrostatic bremsstrahlung FEL, operation at the maximum radiation angle Grn = (&yo j-' was assumed.

In practice, the limitations discussed above will not allow realization of the optimal case where the whole interaction region cross section is filled with electrons. Consequently, the practically extractable power falls short of the upper bound for power extraction (104) by a factor Io/(Johe X 1/24), which is the ratio between the actual drawn current and the maximum current which would ideally be passed through the whole interaction region cross section. This factor will be usually closer to 1 for Cerenkov-Smith-Purcell FEE'S where there are less limitations for filling up the interaction region cross section.

Table 61 and (104) apply to all FEL's except the Compton- Raman scattering FEL. In this case there is no fundamental limit on the interaction region cross sectional area and it is simply determined by the electron-beam cross sectional area (assuming it is smaller than the pump electromagnetic-beam cross section). The maximum radiative power extractable from the electron beam should be then expressed in terms of the electron-beam cross section A, by substituting (103) into (104):

Notice that this is the power extracted from the electron beam only. In the case of a nonrelativistic electron beam, a significant amount of extra power may be extracted from the electromagnetic pump as well. The estimation of this power will require a separate discussion.

The lack of an interaction width limitation in the case of the Compton-Raman FEL means that the extractable power can be increased in t h s case by simply increasing the beam cross section area A, (assuming the current density Jo is a limiting

factor). In practice this increase may be limited by limitations on the total beam current and by the amount of realizable pump wave power [recall that for (105) to be valid the pump power density and the corresponding laser gain should be high enough to lead to saturation within the interaction length I]. The lack of an interaction width limitation in the Compton- Raman FEL should be especially advantageous for short wavelength FEL operation and finite available beam energy. In this case the interaction width of other FEL's diminishes (Table V, column 2) and the Compton-Raman scattering process has an advantage. A viable option for producing radiation in this limit may be the composition of a two-step FEL, in which one kind of FEL produces high-power radiation at a relatively long wavelength, and this radiation operates as a pump in an adjacent Compton-Raman laser which produces radiation at a short wavelength. Simultaneous operation of a bremsstrahlung and a Compton-Raman FEL, operating as spontaneous emission amplifiers, was demonstrated in [41].

In the cold-beam low-gain regime (which may, in particular, be a useful operation regime in FEL oscillators) and in the collective regime ( i j= 2.6/1 and i j= e,, respectively) the maximum power is proportional to h2 for all FEL's, and to h for Compton-Raman scattering. In both cases it is independent of the FEL gain parameter (or coupling coefficient). Neverthe- less, it cannot be concluded that laser gain is immaterial in determining the laser power. It should be recalled that the assumption is, in all cases, that the laser saturates within the interaction length 1. In the case of a very-low-gain laser this assumption would require an impracticable input power (in the case of an amplifier) or an impracticable resonator quality factor (in the case of a laser oscillator). Thus, even in the low- gain regime, the laser gain will indirectly affect the maximum available power through the choice of 1.

In the warm-beam low-gain regime, the substitution of Gth

[see (49)j and ;i (Table N, row 5) results in (104) a h5 wavelength dependence of the maximum power available in all FEL's except the Compton-Raman FEL (105) for which there is a h4 dependence.

In the high-gain, strong coupling, and warm-beam regimes there is an explicit dependence of the maximum power on the gain parameter Q. From Tables 111, rV, and V we find that the maximum power generation of all FEE'S still grows with h for all FEL's considered. There is a particularly strong wavelength dependence in the high-gain warm-beam case. This dependence goes like h7 in the Cerenkov-Smith-Purcell FEL's, like h13 in the bremsstrahlung FEL's, and like h12 in the Compton- Raman FEL.

It is instructive to point out that the maximum power generation grows very strongly with beam energy in all FEL's and all cold-beam regimes. We may say that, apart from the possibility of generating a short wavelength radiation at high electron-beam energies, the high-power generation level at this limit is what most remarkably distinguishes modern free- electron laser development from classical microwave tubes.

We see from Table V (column 3) that in the highly relativistic limit bremsstrahlung FEL's have an advantage in maximum power generation by a factor of about yo compared to the Cerenkov-Smith-Purcell FEL's (at least in the low-gain and

GOVER AND WRANGLE: FREE-ELECTRON LASERS 1213

collective regimes). This stems from the fact that the interac- TABLE VI tion width of the first kind is larger than that of the second kind by the same factor (Table V, column 2). However, in the nonrelativistic limit (Po << 1) the maximum power generation drops down in these regimes at the same fast rate (like 0;) for both kinds of FEL's. It should also be recalled that the upper bound of FEL power given by Table V can be approached with fewer difficulties in the case of the Cerenkov-Smith- Purcell FEL's because in the bremsstrahlung FEL there are stronger limitations on the electron-beam cross section area, which result from the transverse gradient of the pump field.

As in the discussion of the gain parameter in Section V, we point out here that the operation condition and the independent parameters should be clearly identified when the scaling laws are derived. Table V and (104) give the upper bound on extractable power in terms of the independent parameters h,cp, and "/oz. The period L or the index of refraction n (in the Cerenkov FEL) is determined from the radiation condition (Table I). In cases like the magnetic bremsstrahlung FEL operated with a high-energy linear accelerator [42] the beam energy may be varied to quite a large extent, and it is thus more useful to express yoz in terms of h and L using the radiation condition of Table I (row 4). For the case of a highly relativistic beam with a solid-beam cross section operating in the cold-beam low-gain regime [42], we substitute into (104) yoz = (L/2h)'12, we = n X 1/(2q) = $,h/(2n), and 6= 2.6/1. Using (23), we get for this particular case

- ,

The upper bound on the power extraction seems to scale like It should be noted that the limit on current density Jo

and the factor by which the realizable power falls short of the upper bound of power extraction (106) may depend on h and the other independent parameters. This dependence requires more detailed consideration [44].

We point out that the expression of maximum power generation derived in this section applies without change also for FEL's with energy-retrieval schemes, but of course fails com- pletely in efficiency enhancement schemes like tapering. How- ever, if the total efficiency 17 in the latter case is known, the maximum power AP, can be still calculated from (94), (95), (96), and (103) with qo substituted by 17 and l/(2q) given in Table V.

As an example, consider the case of the transverse bremsstrahlung FEL operating in the cold-beam low-gain regime with parameter values similar to those of the Stanford experiment [42]: yo = 48, L = 3.2 cm, yoz = 39 ( B = 0.24 T), no = 2 X lo9 cm-3 (Jo = 9.6 A/cm2), h = 10.6 pm, and I = 5.2 m. Equation (106) gives AP, = 84 kW. In the actual experiment the basic and technical limitations on the electron-beam diameter discussed before did not allow the use of an optimal size beam which fdls the whole interaction region Jo X n(l/24)' = 2 A [as assumed in (106)] and only a current of 70 mA was passed in the center of the waveguide. Hence, the expected power generation falls short of the calculated upper bound of power generation by a reduction factor Io/ [(Jon X

THE DETUNING PARAMETER VALUE AT MAXIMUM GAIN e,, AND THE DETUNING PARAMETER RANGE OF GAIN A9

Regime ern A0

Low-gain tenuous beam -2.61l 2nll Low-gain space charge -0P 2nll High-gain collective -0P 2-

High-gain strong pump 0 2~ lI3

(1/2q)' ] = 0.07 A/2 A, resulting in AP = 3 kW, in agreement with the measured value [42].

VIII. FREE-ELECTRON LASER SPECTRAL WIDTH In all gain regimes considered in Section IV the gain is a

function of the detuning parameter 8 = (w/vo,) - k,, - ko (43). In each gain regime there is an optimal value of the detuning parameter 0, (which is always negative) for which the gain is maximal. The values of 8, at the different gain regimes are given in Table VI. The phase matching diagrams corresponding to the maximum gain condition are drawn in Fig. 4 for the different gain regimes.

When we keep all the laser operating parameters constant and only change the radiation wavelength X, the detuning parameter 0 changes from its maximum gain value 8,. The differentiation of (43) gives

where

is the group velocity of the electromagnetic mode in the waveguide.

Equation (107) applies to all FEL's. In the Cerenkov FEL the evaluation of the group velocity of the mode ug may require the complete solution of the uncoupled mode dispersion relation in the dielectric waveguide. The group velocity in this case may depend on the index of refraction dispersion of the dielectric dn/do as well as the waveguide dispersion. In all other FEL structures the group velocity can be evaluated straightforwardly by differentiation of (5):

ug = c cos q4 (109)

and substitution in (107) gives

In the case of low-order modes in wide cross section waveguides and in the transverse FEL's, we have q4 = 0 and, using the radiation condition of Table I,


If A0 is the range of detuning parameter value around 0,) over which there is still gain, then Aw or Ah are the laser spectral widths in the frequency or wavelength domains, respectively.

The parameter A0 can be evaluated in the different gain regimes by simple inspection of the gain dependence on 0 [(78), Fig. 6 in the warm-beam limit, and (55), Fig. 5 in the low-gain cold-beam limits] and by some algebraic investigation of the dispersion relation (41) in the cold-beam high-gain regimes (collective and strong pump). The results are given in Table VI. The wavenumber mismatch ranges, over which the laser has gain, are also illustrated in Fig. 4 for the various gain regimes.

For all FEL‘s the spectral width is given by substituting the appropriate parameter A0 in (1 lo), (1 1 l), or (107). Only in the high-gain regimes (rows 3 and 4 in Table VI) does the spectral width Aw depend on the particular kind of laser or on the wavelength X through the coupling parameter K or the gain parameter Q, which are given in Tables I1 and 111, respectively.

From the validity conditions of the different gain regimes it comes out that the parameter A0 and, consequently the spectral width, grow up the lower the gain regime is listed in Table VI. Hence, the lower limit on the spectral width occurs in the low-gain regimes (rows 1 and 2). In (1 11) it gives

A h L h I ’

- -_ -

This is usually quite a wide spectral width, which grows with the wavelength and beam energy (as can be verified by substituting the radiation conditions from Table I). In a typical example like the Stanford experiment [42] L = 3.2 cm, 1 = 5.2 m, and (Ah/h) = 0.6 percent.

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Avraham Gover (S’72-MY75) was born in 1945 in Poland. He received the B.S. degree in physics in 1968 and the M.S. degree in solid state physics in 1971, both from Tel-Aviv University, Tel-Aviv, Israel, and the Ph.D. in applied physics in 1975 from the California Institute of Tech- nology, Pasadena.

He worked for a year as a Research Fellow at the California Institute of Technology and a Consultant at Meret Co. and Spectrolab and then joined the Tel-Aviv University School of Engi-

neering, Tel-Aviv, Israel, in 1977 as a Senior Lecturer. He is conducting research in the field of quantum electronics at Tel-Aviv University and is a consultant to JAYCOR on contract with the Naval Research Lab.

Dr. Gover is a member of the Optical Society of America.

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