FREE ELECTRON LASER: OPERATING PRINCIPLES

Transcription

1 FREE ELECTRON LASER: OPERATING PRINCIPLES Emilio Giovenale ENEA - Frasati; INN/FIS/LAC V. E. Fermi 7-44 Frasati (Italy). Introdtion Free Eletron Laser (FEL) is one of the most reent among oherent radiation sores: in 977 J.M.J Madey and oworkers at Stanford University obtained first lasing from an FEL operating at λ 3.47 µm, with λ 8 nm bandwidth and an average power of 36 mw []. The FEL is really different from onventional lasers, bease, instead of exploiting the stimlated emission from atomi or molelar systems, it makes se of the radiation emitted from a relativisti aelerated eletron beam to obtain radiation amplifiation, throgh the interation of the e-beam with a spatially periodi stati magneti field. The first qestion that we mst answer to is related to the possibility for a free eletron to emit radiation withot violating onservation priniples for energy and momentm. It is easy to notie that every time a free eletron emits radiation, there mst be the interation with an external field, that allows the flfillment of onservation rles. We an give some example of sh fields: Synhrotron radiation emission: here it is the magneti field of the bending magnets that allows onservation []. Bremsstrahlng: the external field is the Colomb field of the atomi nlei [] Compton sattering: in this ase the E.M. field of the inident wave allows onservation [] Smith-Prell radiation: is the radiation generated by a harge passing lose to a metal grid; in this ase the field reqired for onservation is generated by the harges inded on the grid srfae [3]. Cherenkov radiation: is the radiation emitted by a harge moving in a medim with a veloity greater than the veloity of light in that medim; the field is generated by the asymmetri polariation inded in the dieletri [4]

2 . FREE ELECTRON DEVICES There are many devies sed to prode radiation starting from Free Eletron emission. Probably the most known Free Eletron Devie is the Klystron [5]. It is able to generate high power entimeter wavelength radiation. It was developed in the 3s by W.W. Hansen and oworkers and it was able to overome the limitations of eletroni tbes, whose ability to prode radiation was limited at high freqenies, when the triode dimensions were omparable to the wavelength and the athode-anode flight time was no more negligible respet to the radiation osillation period. Fig.. : The klystron In the Klystron (fig..) an aelerated eletron beam is injeted into a ondtive avity, whose dimensions are of the order of magnitde of the reqired emission wavelength. Eletrons will find inside the avity a Radio Freqeny eletri field, and its longitdinal omponent will prode a veloity modlation in the eletron beam. After a drift spae this veloity modlation will reslt in a harge density modlation, ths proding the bnhing of the eletron beam. The bnhed beam is then injeted in another avity, where it exites an E.M. wave. In this onfigration the Klystron ats as an amplifier, bt if we ople the seond avity to the first one, so that part of the generated radiation is given bak to the first avity to bnh the beam, we have realied an osillator. The operating priniple of the Klystron, based pon the seqene veloity modlation Æ bnhing Æ oherent emission, is very important, bease it is ommon to many free eletron devies. There are many other Free Eletron devies, like the Magnetron [5] and the Travelling Wave Tbe (TWT) [5,6]. Other Free Eletron Devies, realied in the 5s, are the Orotron [7], where something like a stimlated Smith-Prell effet is sed, the Ubitron [8], that an be onsidered a non-relativisti FEL, and the Gyrotron [9]. The limit of sh devies is related to the shorter ahievable wavelength, whih is in the mm-wave region. Only with the FEL was possible to overome this limitation, exploiting the relativisti effets of the high energy eletron beam.

3 . LORENTZ TRANSFORMATIONS In order to better nderstand the FEL physis, it is neessary to onsider the relativisti effets de to the high energy of the eletron beam. We will then reall briefly the main eqations that are sed when onsidering a relativisti e-beam. Let s onsider two referene frames. We will all x, y and the oordinates of the first system and with x', y' and ' the oordinates of the seond system. Time in the two systems will be denoted by t and t' respetively. Let s onsider the seond system moving respet to the first system at niform veloity v along the x diretion, as indiated in fig.. y y' ' x v Fig..: Inertial referene frames x' The oordinate transformation between the two systems are rled by the Lorent Transformations: x' γ(x-vt) y' y ' t' γ(t-vx/ ) (.) where γ is the so-alled relativisti fator : γ v β (.) Starting from these eqations it is possible to desribe the behavior of relativisti partiles, when β v/ is lose to.

4 .3 DOPPLER EFFECT If we want to derive the eqations that desribe the relation between the freqeny ν in the first referene frame and the freqeny ν' in the seond referene frame, it is onvenient to onsider the light as a olletion of photons. Energy and momentm of eah photon an be expressed in terms of the freqeny aording to the following formlas: & hν & p n E hν (.3) where n is the diretion of propagation of the light wave and h erg s is the Plank onstant. It is now possible to introde in or system the qadrivetor formalism, i.e. an extension of the three-dimensional spae oordinate vetor, where a forth oordinate is added in order to take into aont the time transformations when hanging referene frame in relativisti systems. R (x, y,, t) t was made dimensionally homogeneos with the spatial vetor omponents by mltiplying it by the onstant (veloity of the light). We an then define the qadri-momentm as a qadrivetor with the sal p x, p y e p spae momentm omponent and the qantity E/ as time omponent. P (p x, p y, p, E/) If we now apply the Lorent transformations to the 4 omponents of the qadrimomentm: ( ) E p' x γ px vt γ p v x p' y p y (.4) p' p E' E px γ v E' γ ( E v p ) x

5 If we remind the expressions for E and p x it is easy to obtain: hν hν ' γ hν v osθ (.5) The angle θ is indiated in fig..3 n θ v x Fig..3: oordinate system sed for the freqeny transformation So there will be a relativisti Doppler shift, given by ν ' γν ( β osθ ) (.6) if the propagation diretion is along the x axis (θ ) we have: ν ' γν ( β osθ ) ν ( β ) β ν β β β + β ν + β β + β (.7) Using these formlas it is easy to nderstand why relativisti effets are important in a Free Eletron Laser in order to derease the emission wavelength. In Free eletron devies the emission wavelength is sally omparable to the physial dimensions of the strtre proding the emission. In the FEL emission is generated inside a magneti ndlator, i.e., in a spatially periodi magneti field, that drives the eletron beam into a spatial osillation. The system is skethed in fig..4 Fig..4: oordinate system for eletrons travelling inside a magneti ndlator []

6 If the eletron beam is propagating along the diretion, de to the Lorent Fore eletrons osillate along the x diretion, with a spatial period eqal to the ndlator period λ. The assoiated freqeny of osillation is ths πv /λ ~π/λ for highly relativisti eletron beams. If we onsider a referene frame moving together with the eletron at a longitdinal veloity v, in this system it will be possible to see the eletron osillate along the x diretion, emitting at a freqeny that will be pshifted de to Lorent time transformation: t t' γ ' γ v where v <v > / (.8) In this referene frame the eletron osillate emitting like an antenna at freqeny all over the solid angle [], and this freqeny is pshifted by a fator γ. Moreover this is the freqeny of emission in the referene frame moving with the eletron, bt we will see the radiation from the laboratory rest frame. So oming bak to the rest frame, the emission will be ompressed in a one of apertre θ /γ [] and the freqeny will experiene a relativisti doppler shift given by: + β v where β β v / (.9) Sbstitting the expression for and performing allations we obtain: γ k where k π/λ (.) ( + K ) that expressed in terms of wavelength gives: λ λ ( K + γ ) (.) It is easy to notie that the dimension sale λ is reded by a fator γ exploiting the relativisti effets.

7 . Synhrotron emission It is well known that a harge emits radiation when aelerated. This behavior was already predited in the XIX entry: Larmor in 897 derived, sing lassial eletrodynamis, the formla expressing the power P irradiated by a non relativisti aelerated harge of mass m and harge e : e P 3 m 3 d p & dt (.) where p is the partile momentm and the veloity of the light in vam. This emission reslts to be important in irlar partile aelerators: a relativisti eletron of energy E m γ, moving at onstant veloity vβ in a onstant magneti field B, de to the Lorent Fore will move along a irlar trajetory of radis ρ : Eβ ρ Be E Be (for relativisti eletrons β ~ ) (.) The harge will then emit radiation and the irradiated power is: P 3 e 4 ( m ) E B 4 (.3) This formla was derived in 944 by Iwanenko e Pomeranhk, and desribes the power irradiated by a relativisti harge of energy E moving in a onstant magneti field B. It is easy to notie that this power is proportional to the sqare of the energy, bt dereases as the 4 th power of the rest mass of the partile m. This means that we expet to notie a onsiderable amont of radiation emitted from low mass and high energy harges, i.e. from high energy eletrons or positrons. When bilding the first partile aelerators, this synhrotron emission was onsidered an annoying problem, bease it was neessary to deliver ontinosly energy to the eletron beam, in order to ompensate the energy lost in irradiation. Nevertheless the peliar harateristis of sh radiation appeared to be sefl for many appliations. The harateristis that make synhrotron radiation appealing are: Wide emission band, p to UV, x-ray e γ radiation tnability high intensity high polariation degree The emission spetrm of synhrotron radiation is indiated in fig..

8 The so-alled ritial freqeny is expressed by: 3 E ρ m 3 3 ρ γ 3 (.4) where ρ is the bending radis expressed by (.) and γ is the relativisti fator : γ E m (.5) Synhrotron radiation / ~ Fig.. Synhrotron radiation spetrm / Looking at the figre it is possible to notie that a onsiderable amont of power is available p to freqeny of abot. The vale of depends on the 3 rd power of the energy, so that inreasing the eletron energy, also the maximm freqeny emitted with onsiderable power inreases. As an example we an allate the vale of for 3 different kind of eletron aelerators: ρ E (GeV) (H) λ (m) Small aelerator (Mirotron). 3 6 µm Typial Synhrotron (Grenoble - Trieste) nm Great aelerator (CERN) nm Table. : Critial freqenies allated for different eletron aelerators

9 In the first example the highest sefl freqeny is in the infrared region, while it is evident that the typial synhrotron is designed in order to obtain strong emission p to the x-ray region. The se of a mh bigger mahine allows strong emission at higher freqeny, p to γ rays, bt the se of sh a mahine is not sggested de to high osts and experimental diffilties. Another interesting propriety of the synhrotron radiation is related to it diretionality: de to the relativisti effets, radiation will not be emitted all arond the solid angle, bt it will be ompressed in the forward diretion, in a one of apertre θ []. The light one amplitde is expressed by: m θ E γ (.6) For highly relativisti eletrons the vale of θ an be very small, giving rise to a highly diretional emission. Let s allate the vale of θ for the previosly examined mahines: E (GeV) θ Small aelerator (Mirotron). 5 mrad Typial Synhrotron (Grenoble - Trieste) 5 µrad Great aelerator (CERN) 5 µrad Table. : Emission one apertre allated for different eletron aelerators Smmariing, inreasing the eletron energy, emitted power inreases and onentrates along the motion diretion. The emission diretionality an be sed to explain the spetral harateristis of the synhrotron radiation: if we onsider the light emitted by an eletron moving along a irlar trajetory of radis ρ, an hypothetial observer (fig..) will see the radiation emitted by the eletron only when it s position is inside the light one, that is only along the ar l e θρ. ρ θ l l θ e p Implso di le θ δt Osservatore Fig..: Geometrial explanation of the spetral harateristis of synhrotron radiation

10 The plse dration of the light plse, as seen from the observer, will be eqal to the different between the transit time of the eletrons along the ar l e and of the photon along the line l p ρ sinθ. This differene an ths be written as: l l e p δ t v (.7) where v is the eletron veloity, that an be expressed in terms of it s energy as: m v E γ (.8a) Ths we obtain δ t ϑρ ρ sinθ ρ θ ρ θ sinθ sinθ γ γ (.8b) θ if we remind that θ /γ. Expanding in power series p to first order in θ, i.e. + θ + θ 3 5 o( θ ) and sinθ θ θ + o( θ ) 6, we obtain ρ ρ ρ ρ m δ t θ + θ θ θ θ γ E 3 (.8) Aording to Forier analysis the bandwidth of a plse of length δt is of the order of π/δt, so that we obtain: π 3 π γ δ t 4 ρ 3 (.9) that, exept for a nmerial fator π, is eqal to the definition of the ritial freqeny, that is ths proportional to the bandwidth. The small vale of the θ angle for high energy eletrons (θ /γ) will ase the emission of very short light plses, that in trn will reslt in a wide emission band. Sh a wide band is very sefl, bease it is possible to selet the desired freqeny in a wide range of possibilities, from the infrared region p to the x-ray region, giving rise to a wide tnability of the system. Nevertheless there is a drawbak for sh a sitation: all the emitted power is distribted over a wide spetral range, so that power per nit

11 freqeny reslts to be small. If or appliation reqires high power per nit freqeny, it is neessary to find a way to rede the spetral bandwidth, i.e. to obtain longer light plses. This an be done hanging the interation sheme for the eletron, sing the soalled "Mot Sheme [], sing a magneti ndlator. 3. Undlator Emission We have seen that synhrotron radiation, de to the wide spetral emission band, an provide a limited amont of power per nit freqeny. In order to overome this limit Mot proposed, in 95, a new interation sheme, designed to lengthen the dration of the light plse emitted by an eletron moving in a magneti field. In the Mot Sheme the eletron moves inside a spatially periodi magneti field, generated by a magneti ndlator. Sheme is skethed in fig. 3.: nder the effet of the almost sinsoidal magneti field, the eletron ndergoes an osillatory trajetory along the ndlator axis. RELATIVISTIC ELECTRON BEAM EMITTED RADIATION N S N S N S N S N S N S N S N S S N S N S N S N S N S N S N S UNDULATOR N Fig. 3.: Trajetory of eletrons inside a magneti ndlator If we hoose the magneti field intensity, the ndlator period λ and the eletron energy in sh a way that the mean deviation angle from the ndlator axis is smaller that the light emission one θ, an hypothetial observer, plaed along the ndlator axis, wold see the eletron radiate all along the trajetory inside the ndlator, ths generating a longer light plse, that in trn means a narrower emission band. When all these onditions are flfilled, we an laim we are in the ndlator regime In order to derive the onditions for the ndlator regime we an start from the motion eqation for an eletron inside a periodi stati magneti field. Using the referene frame in fig..4 it is possible to write the expression for the Lorent Fore ating on an eletron moving along the axis: & d p e & & v B (3.) dt where B & π, B os,, λ is the ndlator period and we have assmed a λ perfetly sinsoidal magneti field (with only the y omponent different from ero).

12 x In the vetor prodt & y & v B v By o ero: & & π v B v B os x λ only the x omponent will be different from (3.) This means that, de to the Lorent Fore, the eletron will be aelerated along the transverse diretion x, and that the aeleration will be desribed by: dp dt x e d π B os dt λ (3.3) performing a time integration: dpx eb d π ebλ p dt x os dt dt dt λ π π sin λ (3.4) The mean deviation angle from the eletron trajetory and the ndlator axis is (with the reasonable approximation p ~ p ): ( px ) ( p ) θ ebλ p sin π π λ (3.5a) if we remind that E ~ p, ths p ~ E/ and that <sin (x)> ½ we obtain: θ λ eb π E (3.5b) If we want to obtain a long light plse, this angle mst be smaller than the emission one apertre, that is expressed by /γ; this an be written as follows: θ ebλ m π E γ E π writing the magneti field as B B os we obtain λ (3.6) B B B B, ths, to flfil the ndlator regime we mst satisfy:

13 e B λ π m (3.7a) The qantity on the left is alled ndlator parameter, and is denoted by K: K e B λ π m (3.8) ths, we are in the ndlator regime if: K (3.7b) When the ndlator ondition is flfilled a long light plse will be emitted, narrowing the spetral band, giving ths rise to emission harateried by higher power per nit freqeny. This inrement an be easily allated sing a proedre similar to that sed for synhrotron radiation emission. The time dration of the light plse is given by the differene in transit time between the photons and the eletrons in the ndlator: L δ t v L (3.9) where L is the ndlator length and v is the mean eletron veloity along the ndlator axis; in or referene frame this an be written as: v To obtain the vale for v v v we remind that: x v. v γ (3.) ebλ p sin π p m v x e γ π λ ths it is easy to obtain eb v m sin K λ π π x sin (3.) π γ λ γ λ then K v v vx sin π K sin π + (3.) γ γ λ γ λ averaging along an ndlator period, and reminding that <sin (x)> ½ we obtain:

14 v v + K γ ( ) (3.3) it is now possible to allate δt: δ t (3.4) L v L L L ( + K ) γ x and that for γ >>, then x γ It is known that + x + o( x ) L L δ t / + + / γ γ + ( K ) ( K ) (3.5) The bandwidth expression is then: π π γ (3.6) δ t L + K It is possible to notie that for ndlator emission the bandwidth is proportional to the sqare of the eletron energy γ, while for irlar synhrotrons it was proportional to the 3 rd power of the energy γ 3 ; ths, for ndlator emission, bandwidth is γ times narrower then for irlar synhrotrons. When we deal with high energy eletron, with γ >>, Mot Sheme allows a onsiderable narrowing of the emission bandwidth, and a orresponding inrease of the power per nit freqeny. We already derived in hapter the expression for the ndlator emission; we repeat here, with some more details the derivation of the emission entral freqeny for a magneti ndlator. If a relativisti eletron beam is propagating along the diretion (ndlator axis), de to the Lorent Fore eletrons osillate along the x diretion, with a spatial period eqal to the ndlator period λ. The assoiated freqeny of osillation is ths πv /λ ; for highly relativisti eletron beams v ~, ths: π π v λ λ (3.7) If we onsider a referene frame moving together with the eletron at a longitdinal veloity v, in this system it will be possible to see the eletron osillate along the x

15 diretion, emitting at a freqeny that will be pshifted de to Lorent time transformation: t t' ' γ (3.8) γ v where v <v > / In this referene frame the eletron osillates emitting like an antenna at freqeny all over the solid angle [], and this freqeny is pshifted by a fator γ. Moreover this is the freqeny of emission in the referene frame moving with the eletron, bt we will see the radiation from the laboratory rest frame. So oming bak to the rest frame, the emission will be ompressed in a one of apertre θ /γ and the freqeny will experiene a Relativisti Doppler Shift given by: v where β + β β ' v / ; ths (3.9a) reminding that ' + β β + β + β β β + β ' ' β ' (3.9b) + β we obtain: + β β β β β + β β (3.) from 3.3 it is evident that β ( + K ) γ γ ( + K ) ( + K ) ( ) where k π/λ That expressed in terms of wavelength gives: γ γ β ( + K ), then π γ λ + K k (3.) π λ + K λ λ π ( + K π γ γ ) (3.)

16 It is easy to notie that the dimension sale λ is reded by a fator γ exploiting the relativisti effets. The emission bandshape an be easily allated: radiation is emitted as a plse train omposed by ' t' N' periods. The spetrm of sh a strtre is the well known π sin fntion []: ( ϑ / ) ( ϑ / ) sin f ( ) (3.3) where ϑ π N is the so-alled detning parameter. The bandshape for the ndlator emission is shown in fig. 3.. Spontaneos emission. //Ν θ Fig. 3.: Undlator emission bandshape The emission of the ndlator is not omposed by a single line, at the entral freqeny, bt is omposed by a finite width band. This an be interpreted as a line broadening effet: in this ase we are looking at the homogeneos broadening effet, de to the dynami of the emission proess, i.e. the finite transit time of the eletron inside the ndlator, that is eqal for all eletrons in the eletron beam, independently on their physial properties, like energy, momentm and trajetory. This is analogos to the line broadening ased by finite lifetime in atomi systems. In the eletron beam the differene in energy, momentm and trajetory of the different eletrons will ase a frther broadening of the emission line, that an be lassified as inhomogeneos. The latter an be ompared to inhomogeneos broadening in a gas emission de to doppler shift of the single atom emission.

17 4. Synhrotron radiation stimlated emission Up to now we have seen what happens when an eletron passes throgh a magneti ndlator. In sh a proess only a field is present: the stati magneti field of the ndlator (that in the eletron referene frame beomes an eletromagneti field). Let s now onsider what happens when other EM modes are present dring the interation: we will observe the emission properties of sh a system and the variations of the modes intensity dring the proess. Let s onsider an EM mode opropagating with the eletrons inside the ndlator. We want to allate the rate of energy exhange between the eletrons and the EM field. Eletrons osillate inside the ndlator in the transverse plane x with period λ. In order to obtain energy exhange between the eletrons and the EM field it is neessary to have synhronism between the transverse osillations of the eletrons and the osillations of the Eletri field of the opropagating EM wave. This will happens if the eletron, after one ndlator period, will find the eletri field with the same phase. If we remind that the eletron veloity v <, it is evident that this ondition an be flfilled if the mean longitdinal eletron veloity v is hosen in sh a way that the eletron performs a omplete osillation in the time needed for the light to over an ndlator period pls a wavelength. This ondition an be expressed as follows: Being t e the time needed for the eletron to over an ndlator period λ Being t p the time needed for the EM wave to over the distane λ + λ the synhronism ondition is expressed by the eqation t e t p ; if we remind that te λ λ + λ and that t p, where vf is the phase veloity of the EM wave v f, v v k we obtain: f λ λ + λ v v f (4.) if we define k π/λ eq. 4. an be rewritten as: v k k k + k k k f v f λ v (λ +λ) v + v v k v ( k + k ) k f taking in mind that v f and v β we obtain: k β ( k + k ) (4.) this eqation, that defines the so alled beam line, desribes the points of the ( k, /) spae where the synhronism ondition is flfilled

18 It is now neessary to onsider the dispersion relation of the strtre where the interation takes plae: /f(). The intersetion between the rve representing the dispersion relation and the beam line in the (k, /) spae gives the emission freqeny for the given ndlator at the given beam energy. If the interation ors in vam, the dispersion relation is linear: /k and the emission freqeny an be easily deded from the intersetion of this line with the beam line, as shown in fig. 4.. Fig. 4. : representation of the beam line and of the vam dispersion relation The analyti soltion is easily deded solving the linear system: k k ( β ) kβ then, if we remind that γ β ( k + k ) β ( + β ) β β β β β γ + k k k k k + β β + β β ( β ) (4.3) Eq. 4.3 for relativisti eletrons (β ) beomes: γ k (4.4) We immediately notie that this formla is idential to eq. 3., giving the entral spontaneos emission freqeny for an ndlator. In terms of wavelength we have (like in eq. 3.) λ λ ( + K ) (4.5) γ

19 It is easy to verify that synhronism ondition is flfilled also if in the time t e the EM wave slips over the eletron by a qantity eqal to or more wavelengths. The more general relation is then: t p λ + nλ v f with n integer, greater than ero (4.6) Following the same allation performed for the ase n it is possible to derive the general formlation for the beam line: β ( k + nk ) (4.7) n is the so alled harmoni nmber and the intersetion of the different beam lines harateried by different vales of n with the dispersion relation, gives the emission freqenies for the fndamental (n ) and the higher harmonis (n > ), as shown in Fig. 4.. Fig. 4. : representation of the harmonis beam lines and of the vam dispersion relation Then, it is possible to modify the emission freqeny by hanging: The eletron energy The ndlator period The K parameter, proportional to the ndlator magneti field.

20 4. GAIN In order to allate the ahievable gain in the interation between the eletrons and the EM wave, we start allating the energy variation of the eletron: given γ E the m energy variation an be expressed by: dγ e dt m E & v & (4.8) Only the x and omponents of the eletron veloity are different from ero. In the EM wave the eletri field is transverse, so that E & v & E v ; moreover we have: T x K π vx sin γ λ ET E os t k + ( φ ) where φ is the EM wave phase (4.9) with a simple sbstittion we obtain: dγ e dt m K π sin E os t k + γ λ eek sin k k t + m γ ( ) os( φ ) ( φ ) sing the simple trigonometri expression sinα sinβ sin( α + β ) + sin( α β ) have: dγ eek dt m γ eek m γ eek m γ { sin[ ( k + k) t + φ ] + sin[ ( k k) + t φ ]} { sin[ ( k + k ) t + φ ] sin[ ( k k ) t + φ ]} + ( sinψ sinψ ) with ( # ) ψ ± sin k k t + φ (4.) ( ) we (4.) both terms are rapidly osillating with t, bt if is hosen lose to the resonane freqeny it is possible to neglet the seond term, bease it will osillate more

21 rapidly, giving then a negligible ontribtion after averaging over time. Ths, approximating ψ ψ we an write: dγ dt eek sinψ m γ π with ψ k + + φ λ t (4.) (4.3) It is now possible to write the time derivative of the fntion ψ: dψ π d + k + dt λ dt d ψ π d k + dt λ dt (4.4) d let s remind that v vx + dt γ the eqation expressing dγ /dt mst now be opled with the expression for d/dt, that an be simplified averaging over the longitdinal osillation, that is negligible when ompared to the longitdinal veloity. The average over of the sin(k ) reslts in a / K fator ½, so that eqation (4.9) for v x beomes v x, then: /γ d K dt + γ γ γ ( K ) (4.5) sing the series expansion ( ) x x o x we obtain: d dt K + + K γ γ (4.6) performing the time derivative: d dt dγ dt ( + K ) (4.7) γ 3

22 ebλ sbstitting the vale (4.) for dγ /dt (4.) and the expression of K we π m obtain: d e EBλ K sin dt 4 + ψ (4.8) 4πγ m ( ) ( ) let s remind that ( k k ) d ψ d +, so that: dt dt d ψ e EBλ ( k + k ) 4 ( + K ) sinψ (4.9) dt 4πγ m if we now define : ee K + + γ m ( k k ) ( K ) Ω 4 (4.) It is possible to write the famos pendlm eqation that desribes the FEL dynamis [3, 4]: d ψ Ω sinψ dt (4.) The Gain per single pass is defined as the relative energy variation of the radiation, i.e.: W G W p p (4.) it is easy to allate the vale of W p, bease the energy inrease of the EM mode orresponds to the energy loss of the eletrons, ths: Wp m γ (4.3) we an then write: G m γ Wp The variation of γ an be expressed in terms of ψ; if γ << γ we have: It is valid if γ onstant over the single pass, i.e. in low Gain onditions

23 ( + K ) d dt γ dψ + dt 3 ( k k ) γ d dt (4.4) we obtain then: 3 γ dψ γ (4.5) K k k dt ( + )( + ) and gain an be written as: ( + )( + ) 3 γ dψ G m (4.6) K k k dt W If in the eletron beam there are N eletrons, orresponding to a beam rrent I, p N IL eβ (4.7) onsidering N times the mean energy exhanged by eah eletron we have: IL G eβ m γ ( + K )( k + k ) 3 p dψ (4.8) dt W To obtain the vale for d ψ it is possible to perform a pertrbative analysis of the dt pendlm eqation, that is beyond the task of this paper. Here we report only the final reslts of the gain allations: ( + ) ( λ ) 5 ( β γ ) Σ L K K N I G π I 3 ( k + k ) d sin dθ θ θ (4.9) where θ π - n is the so-alled detning parameter, I is the Alfven rrent, e given by I 7 ka and Σ L is the laser mode setion. In vam Σ L is the setion r

24 of the gassian mode into the resonator, while inside a wavegide, for a generi TE mn mode, is expressed by: Σ L m n + a b n b ab 4 σ (4.3) where a and b are the wavegide transverse dimensions and σ if both m and n are different from ero, σ otherwise. It is easy to notie that the gain depends on many parameters. In order to simplify the expression f the gain, we an perform some approximations:. k << k λ. operation at resonane: λ ( + K ) γ With these approximations the expression for the gain is: K G π 5 β γ 3 λ Σ N L 3 I I d sin dθ θ θ 4π K 3 λ N 5 3 β γ I I d sin dθ θ θ (4.3) Gain is proportional to the beam rrent intensity, to the be of the nmber of ndlator periods, to the sqare of the ndlator parameter K, that is in trn proportional to the ndlator magneti field intensity, while it is inversely proportional to the be of the energy γ. One the mahine is bilt, it is sally impossible to hange the vales of γ and N, so that the only parameters that an be exploited to inrease the gain while operating are the beam rrent I and the ndlator parameter K, that an be modified by varying the ndlator poles distane or, in eletromagneti ndlators, by varying the rrent generating the magneti field. The eletron beam rrent is then a ritial parameter for the FEL operation, and we need to obtain high rrents to ahieve vales of the gain ompatibles with laser emission. The dependene on /γ 3 is one of the reasons that makes so diffilt to bild short wavelength FELs (λ is proportional to / γ ).

25 4. EFFICIENCY Let s now onsider one more eqation 4.3 for the small signal gain, reported in fig. 4.3: G Fig. 4.3 : Small signal gain as a fntion of the detning parameter θ If de to the interation the mean energy of the eletrons dereases so mh that the detning parameter θ πn is no more in the positive branh of the gain rve, the FEL emission proess stops. This happens when θ > π, then the maximm vale of θ ompatible with FEL emission is: θ θ π (4.3) whih orresponds to a maximm energy variation of the eletron beam E, that an be deded as follows: the FEL emission freqeny is proportional to the sqare of the eletron beam energy: E E E then E E θ 4πN N (4.33) This is the maximm mean energy variation of the eletron beam allowed in order to ontine emitting, i.e. the maximm theoretial effiieny for the FEL emission proess:

26 η N (4.34) This eqation is known as Renieri limit for the FEL effiieny []. It is possible to notie that while gain inreases like N 3, the effiieny dereases with N. This an be nderstood if we remind the spetral harateristis of the ndlator emission: if the nmber of periods (i.e. the length) of the ndlator is inreased, an hypothetial observer plaed along the ndlator axis will see a longer light plse, bease the time dration of the plse is the time needed to the light to pass throgh the longer ndlator, and the emission will take plae on a narrower spetral band. Being longer the interation time, also the gain will be higher, bt being the gain rve proportional to the derivative of the emission band, with a narrower gain rve the eletrons losing energy will go ot of tne more rapidly, ths reding the effiieny of the proess. One possible soltion to the Renieri effiieny limit onsists in trying to follow the eletron energy loss, smoothly hanging the parameters of the FEL along the ndlator, in order to hange the interation freqeny while the eletron loses energy. This is the so-alled ndlator tapering, i.e. the variation of the ndlator gap along the axis, in order to hange the K parameter. This way the entral emission freqeny will move together with the eletron energy motion nder the Gain rve, inreasing the total effiieny Another possibility is related to energy reovery: it is possible to ollet the already nsefl eletrons after the interation and reover their energy, that an be sed to aelerate new eletrons. 4.3 FEL LINE BROADENING The expression of the gain has been obtained onsidering a monoenergeti eletron beam, with ero displaement and ero anglar divergene from the ndlator axis. In a real beam it is not possible to neglet the effets of energy spread and beam divergene. These deviations from the ideal onditions will ase a broadening of the emission band, mh alike an inhomogeneos broadening, that ases the redtion of the gain and the displaement of the entral emission freqeny. The latter an be expressed by [5]: 4πγ λ ( + K + α γ ) (4.35) The orretion term α γ, is de to off-axis eletrons (α is the deviation angle from the ndlator axis). It is possible to write the single ontribtions of energy and anglar dispersion to inhomogeneos broadening []:

27 i ε + x + y / (4.36) If we define σ ε, the standard deviation of the energy distribtion of the eletron, it an be proved that: ε σ ε (4.37) Moreover, if we onsider the ndlator field being not perfetly sinsoidal, with the presene of sextpole terms h x and h y, and an eletron beam harateried by transverse emittanes ε x and ε y, the broadening terms dependent on the oordinates an be written as: γε x, y hx, y x, y λ + K K (4.38) In order to evalate the effet of the inhomogeneos broadening, it is neessary to ompare it with the homogeneos broadening term, de to the finite transit time of the eletrons inside the ndlator. It is then possible to define the so alled inhomogeneos broadening normalied parameters: e, x, y µ ε, x, y (4.39) If µ ε,x,y <<, the FEL is operating in homogeneos broadening regime: TOT [ + µ + µ µ ] ε x + y (4.4) where the term µ ε +µ x +µ y desribes the deviation from the homogeneos regime. The effets of the inhomogeneos broadening an be seen in fig. 4.4:

28 Fig. 4.4 Emission band (- - - ) and gain ( ) in the homogeneos broadening regime (a), inhomogeneos broadening ased by energy spread µ ε (b), inhomogeneos broadening ased by anglar dispersion along x µ x (), inhomogeneos broadening ased by energy spread and anglar dispersion µ ε µ x µ y (d) [] It is possible to perform a nmerial analysis of the effets of inhomogeneos broadening on the peak vale of the gain. Sh an analysis gives: G max G ( +.7µ ε ) + ( + µ ) + ( + µ ) x y (4.4) where G is the vale of the gain allated taking into aont homogeneos broadening only. It is possible to verify that inreasing the nmber of ndlator periods N, also the vale of the parameters µ i inreases, inreasing ths the inhomogeneos broadening. When the FEL operates with a radiofreqeny driven eletron aelerator, a frther sore of inhomogeneos broadening appears: the bnh strtre of the eletron beam is refleted in an analogos strtre in the emitted light, and the laser plse is then omposed by a train of light plses. In order to obtain the interation between the eletron bnhes and the light plses propagating bak and forth inside the optial resonator, there mst be sperposition between eletrons and light, i.e. the light plse shold find, after a rond trip, another eletron bnh entering the avity. This means that the time distane between two (or more than two) eletron bnhes mst be eqal to the rond trip time for the light plse in the resonator. This distane, for a RF aelerator, is the radiofreqeny period T, so that the mathing ondition an be written as:

29 L nt (4.4) where n is a positive integer; n means that the light plse generated by the i-th eletron bnh, after a rond trip will not find the (i+)-th eletron bnh, bt the (i+)-th (there is a jmp of bnh), and so on for n3,4, Anyway, also if this ondition is flfilled, we mst be aware that the light veloity in vam is always greater that the eletron veloity, ths along the ndlator period the light plse slips over the eletron bnh by the amont: Νλ the so alled "slippage length" (4.43) This relation follows immediately if we remind that the synhronism ondition for FEL operation reqires that the light plse passes over the eletron bnh by the a wavelength λ after an ndlator period λ, therefore, after N periods we obtain the expression for. The existene of the slippage prodes some effets:. eletrons and photons are not perfetly sperimposed over the whole interation time. de to the fat that the forward part of the plse overomes the eletron bnh, the rear part of the plse will interat for a longer time, experiening an higher gain. This asymmetry reslts in the shift of the enter of the plse, that seems to be slowed (fig. 4.5). This effet, alled lethargy, prodes as an effet the fat that in order to obtain the mathing between eletron bnhes and light plses, the avity length mst be reded of a small length dl. a) b) Fig. 4.5 Lethargy mehanism: (a) when the interation starts the eletron bnh (. ) and the light plse are sperimposed, and their interation prodes radiation ( ); (b) the light plse (---) begins overoming the eletron bnh (. ) the interation generates new radiation ( ) that appears to be retarded respet to the original plse position. A possible way to avoid the slippage onsists in slowing down the light plse, making se of a wavegide resonator. De to dispersion relation of the wavegide, the light veloity inside the wavegide an be slower then, and it is possible to make it eqal to the mean longitdinal veloity of the eletrons. The effet of the slippage an be evalated introding the parameter:

30 Nλ µ (4.44) σ where σ is the normalied standard deviation of the longitdinal distribtion of the eletrons. µ is alled "normalied slippage", and, like for the other µ parameters, an expression relates the maximm gain to the vale of µ : G max G µ + 3 (4.45) So there will be a gain redtion de to the non perfet sperimposition of the eletron plse and the light plse. This onfirms the advantages of sing a wavegide resonator when dealing with long wavelength emission, bease it is possible to slow down the light plse in the wavegide and operate at ero slippage ondition. 5. Free Eletron Laser omponents In the typial Free Eletron Laser eletrons are aelerated by an eletron aelerator, then, by means of a transport hannel, they are injeted inside the ndlator, where the FEL interation ors inside an optial resonator. The emitted light is then extrated from the resonator and delivered to the detetion setion. The typial layot for an FEL is reported in Fig. 5. Fig. 5.: Typial FEL layot

31 The eletron aelerator is probably the most important omponent of the FEL: most of the performane of the laser depends on the harateristis of the eletron beam. The laser emission wavelength depends on the eletron energy γ : The gain strongly depends on the rrent in the e-beam The gain is strongly affeted by the e-beam qality: we have seen that energy spread E and anglar deviation in the beam indes gain redtion. In order to better define the e-beam qality it is possible to introde the so-alled beam emittane : l et s onsider a single transverse oordinate, say x, and let s have a look to the phase spae (x,x ). The ideal partile, with ero displaement and deviation from the axis wold lay in the (,) point. In a real beam eah eletron will have its own position and divergene, and will be represented by a point in the phase spae x-x. We an define the beam emittane ε x the area of the ellipse ontaining 95% of the partiles, as shown in fig. 5. x' x Fig. 5.: transverse phase spae and definition of the emittane The parameters E, representing the eletrons energy spread, together with the emittanes ε x and ε y define the beam qality, determining the entral freqeny shift, the broadening of the emission and the redtion of the gain The hoie of the aelerator depends on the reqired emission spetral region: looking at the FEL emission wavelength it is easy to notie that tnability an be exploited hanging the vale of K, bt the emission spetral region is mainly determined by the eletron energy γ, that is a design parameter and sally annot be hanged dring operation. The hoie of the aelerator depends also on the reqired harateristis of the otpt radiation: the temporal strtre of the otpt radiation depends on the temporal strtre of the e-beam, so if ontinos emission is needed we mst se an eletrostati aelerator, like a Van der Graaf generator, a Cokroft-Walton aelerator or a Tandem aelerator. These aelerators an prode a ontinos eletron beam sing harge reovery tehniqe, i.e. the e-beam is reirlated from the otpt of the ndlator bak into the aelerator. Pratial limitations arise from the sie of sh devies, and the maximm

32 eletron energy is limited to a few MeV, orresponding to an emission in the region between the mm-wave and the FIR. If a plsed radiation strtre is aeptable, Radio Freqeny Linas are probably the best hoie. In sh a devie an eletromagneti Radio Freqeny field is sed to aelerate the eletrons p to high energy. The eletrons proded by sh a devie are omposed of a seqene of maroplses, with a time dration of the order of µs, eah one omposed by a series of miroplses, of time dration of a few ps, separated by the RF period (fig. 5.3). µs T r f ps Fig. 5.3 : Time strtre of a radiofreqeny aelerator eletron emission The next omponent of the system is the eletron transport hannel. It is sally omposed by a metal pipe, in high vam onditions, with magneti elements to steer and fos the eletron beam. Simple dipole oils are sed to deflet the beam in order to perform position and angle orretions. This allow to injet eletrons in the ndlator perfetly on-axis. In order to avoid the natral tendeny of the eletron bnh to expand, magneti qadrpoles are generally sed. Qadrpoles at on the e-beam exatly in the same way a lens ats on a light beam. The matrix formalism sed for magneti lenses an be applied to standard optis. Other magneti elements an be sed to perform varios operations on the e-beam. The next element of the system is the magneti ndlator. The ndlator an be bilt sing permanent magnets or sing eletromagneti devies. There are two main geometries: the planar ndlator and the helial ndlator. In bilding a planar ndlator the task is to obtain a field as lose as possible to a prely sinsoidal field. In order to obtain sh a field onfigration it is generally sed the soalled Halbah onfigration, that makes se of 4 magnets per pole per period (fig. 5.4).

33 Fig. 5.4: Halbah onfigration for ndlator magnets Single magnets field shold be measred with are, and then the magnets mst be sorted by means of a dediated software. When the ndlator is bilt, the field map an be measred and, if neessary, it is possible to insert some thin ferromagneti foils in the right position in the ndlator, in order to orret the small deviations from the sinsoidal behavior (shimming tehniqe). The optial resonator ats exatly like in a onventional laser. One more onstraint appears when sing a RF aelerator as eletron sore: the resonator length mst be hosen in order to allow sperposition of the light plse with sbseqent eletron bnhes. This means that the rond trip transit time for the radiation mst be an integer L mltiple of the RF period: in vam this an be expressed by eq. 4.4: nt The radiation is finally extrated from the resonator and sent to the detetion area, where performanes of the FEL are measred and experiments an be performed sing the FEL radiation. The harateristis of the detetors depend on the FEL parameters. Usally FELs are able to deliver high peak power, so that detetors mst be hosen taking in mind this harateristi of the otpt radiation. 6. Conlsions Priniples of operation of the Free Eletron Laser have been presented. The present tehnology allows the design and realiation of FELs emitting in a wide spetral range, from the mm-wave region p to the X-ray region. Being the FEL a omplex and expensive mahine, it is not onvenient to se it in a region already overed by onventional lasers. From the above analysis it appears logial to limit the interest to the extreme ones of the spetrm, i.e. the mm-wave/fir region, beyond the possibilities of onventional free eletron devies, where interesting appliations an be fond in biology, solid state and molelar physis, and where the osts related to the FEL realiation are not too high. At the other end of the spetrm we find the x-ray region,

34 where a high brightness oherent sore, emitting very short plses wold be of enormos importane to many appliation in different researh fields, and old jstify the very high osts and the tehnial diffilties related to the realiation of sh a devie. At present there are mahines nder onstrtion worldwide, and a third one is being designed in Italy. The realiation of sh a mahine will improve dramatially the experimental possibilities respet to the atal sitation, that makes se of synhrotron radiation in the X-ray region. Referenes: [] D.A.G. Deaon, L.R. Elias, J.M.J. Madey, G.J. Ramiam, H.A. Shwettman, T.I. Smith (977), Phys. Rev. Lett. 38, 89 [] J.D. Jakson (975), Classial Eletrodynamis, Wiley, New York [3] S.D. Smith, E.M. Prell (953), Phys. Rev. 9, 69 [4] J.V. Jelley (958), Cerenkov Radiation, Pergamon, London [5] J.C. Slater (963), Mirowave Eletronis, Van Nostrand, Prineton [6] J. Piere (95), Travelling Wave Tbe, Van Nostrand, New York [7] S.F. Rsin, G.D. Bogomolov (966), JETP Lett., 4, 6 [8] R.M. Phillips (96), IRE Trans Eletron Devies ED-7, 3 [9] J. Shneider (959), Phys. Rev. Lett.,, 54 [] G. Dattoli, A. Renieri and referenes therein, Laser Handbook vol. IV, North Holland, Amsterdam [] H. Mot (95), J. Appl. Phys., 57 [] S.G. Lipson, H. Lipson (969), Optial Physis, Cambridge University Press, Cambridge [3] W.B. Colson (977), Phys. Lett. 64A, 9 [4] A. Doria, G.P. Gallerano, A. Renieri (99), Opt. Comm., 8, 48 [5] A. Renieri, in Developments in High-Power Lasers and Their Appliations (98), C. Pellegrini ed., North-Holland, Amsterdam, p. 44