Calculation and optimization of laser acceleration in vacuum

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1 PHYICAL REVIEW PECIAL TOPIC - ACCELERATOR AND BEAM, VOLUME 7, 3 () Clcultion nd otimiztion of lser ccelertion in vcuum Z. Hung nd G. tukov tnford Liner Accelertor Center, tnford, Cliforni 939, UA M. Zolotorev Center for Bem Physics, Lwrence Berkeley Ntionl Lbortory, Berkeley, Cliforni 97, UA (Received 5 June 3; ublished 3 Jnury ) Extrordinrily high fields generted by focused lsers re envisioned to ccelerte rticles to high energies. In this er, we develo new method to clculte lser ccelertion in vcuum bsed on the energy exchnge rising from the interference of the lser field with the rdition field of the rticle. We ly this method to simle ccelerting structure, erfectly conducting screen with round hole, nd show how to otimize the energy gin with resect to the hole rdius, lser ngle, nd sot size, s well s the trnsverse rofile of the lser. Limittions nd energy scling of this ccelertion method re lso discussed. DOI:.3/PhysRevTAB.7.3 PAC numbers:.75.jv,.6. m I. INTRODUCTION Accelertion of chrged rticles by lser fields in vcuum cn be clculted s Z U cc e E vdt; () where U cc is the energy gin, e is the chrge, E is the electric field, v is the rticle s velocity, nd the time integrl is tken long the rticle s th. In stright trjectory roximtion, when v in Eq. () is considered s constnt unerturbed velocity, ccording to the Lwson-Woodwrd theorem [], lser ccelertion in vcuum is ossible only in close roximity to mteril boundries. The ccelertion occurs becuse currents nd chrges induced by the lser field in the mteril distort the incident electromgnetic field in wy which gives nonzero vlue for the integrl. A direct clcultion of the integrl in Eq. () requires solving Mxwell s equtions in the vicinity of the mteril boundries. In most cses, this leds to formidble electromgnetic roblem nd requires extensive numericl comuttions. Only very simle geometries llow n nlyticl clcultion of the energy gin directly from Eq. () (see, e.g., [,3]). In this er we develo new method to clculte the energy gin U cc. It is bsed on the energy blnce eqution for the electromgnetic field energy nd the rticle s energy, nd only requires knowledge of the rdition field in the fr zone. In its most generl formultion, it is not limited to vcuum nd stright trjectories. It cn lso be used for ccelertion in medium (e.g., inverse Cherenkov ccelertion), nd curviliner trjectories (such s in inverse free-electron lser (FEL) ccelertion). To demonstrte dvntges of the new method, we ly it to reltively simle roblem: lser ccelertion of rticle ssing through round hole in erfectly conducting metl screen. The ssumtion of erfect conductivity of the metl is vlid if the lser frequency is smller thn the lsm frequency for the metl. Two different lser illumintions re considered: first with higher-order lser mode, nd second with two crossed Gussin lser bems. Note tht crossed Gussin lsers re used in the LEAP exeriment t tnford University [] nd in the roosed E-63 exeriment t LAC [5]. In the limit when the hole rdius tends to zero, we show tht our result grees with direct clcultion of the integrl (). Tking into ccount the effect of dmge threshold for mterils, we show how our clcultions lso llow otimiztion of the energy gin for given lser rmeters nd find the limits of this ccelertion method. II. RELATION BETWEEN RADIATION FIELD AND ENERGY GAIN Consider bunch ssing through hole in erfectly conducting metl screen, s shown in Fig.. The hole my hve n rbitrry she, lthough in subsequent sections we ssume tht it is round, with rdius. At the time of ssge, the bunch is irrdited by lser ulse, nd due to the interction with the lser light, rticles in the bunch re ccelerted or decelerted deending on the hse of the lser wve. We introduce surfce of lrge rdius R enclosing volume V which includes the ccelertion re. Eventully, we tke the limit R!. Initilly, t t!, rticle in the bunch nd the lser ulse re locted outside of the surfce. After the interction, when t!, they leve the volume V. We use the energy blnce eqution for the electromgnetic field (see, e.g., [6,7]): 3-98-==7()=3(8)$. The Americn Physicl ociety 3-

2 PRT-AB 7 CALCULATION AND OPTIMIZATION OF LAER... 3 () V FIG.. (Color) Lyout of vcuum lser ccelertion exeriment. A erfectly conducting screen with round hole of rdius is locted t z. Initilly, rticle nd lser ulse re t osition outside of the volume Z dv E H V 8 ZV dvj E nd; where the integrtion goes over the volume V with the surfce boundry, n is unit vector in the outwrd direction norml to the surfce, j is the current density, is the Poynting vector, c=e H, ndh is the mgnetic field. Integrting this eqution over time, from t to t, nd tking into ccount tht t t there is no electromgnetic field inside the volume V, we find Z Z dt dvj E dt nd: V The current density j includes the current in the metl nd the current ssocited with the moving oint chrge, j evxt, where Xt is the rticle s trjectory. The former, R however, does not contribute to the integrl dvj E, becuse this integrl is equl to the energy deosited inside the metl due to the Ohmic heting, which is neglected in the limit of erfect conductivity. Hence the integrl reduces to e R v Edt tken long the trjectory. It is equl to the energy gin (or loss, if negtive) U of the rticle due to the interction with the field. Hence Z U dt nd: () Note tht this formul is exct nd is vlid for rbitrry curviliner motion of the rticle under the influence of n externl field. z In the clcultion of fields, it is convenient to use the Fourier trnsform, which we define s E! Z Et dte H! i!t : (3) Ht Using Prsevl s theorem we find U c Z d! Re E!H! nd c Z d! Re E!E!d; () where the sterisk denotes comlex conjugte, nd we use the reltions H n E nd n E, vlid in the fr zone. The field entering Eq. () is suerosition of the lser field, E L, nd the rticle s field, E P : E E L E P ; (5) where the letter in the suerscrit indictes tht these re the fields in the resence of the screen. ubstituting Eq. (5) intoreq. () nd emloying the nottion UE ; E c d! R Re E!E!d we obtin severl terms. The term UE L ; E L corresonds to the integrted energy flow of the lser light through the surfce without the bem. This term vnishes becuse we ssume tht there re no losses in the screen nd hence the incoming lser energy is equl to the outgoing one. The term UE P ; E P describes the energy rdited by the rticle ssing through the hole in the screen when there is no lser field. This term scles s squre of the rticle s chrge nd is not relevnt to the ccelertion of single rticle considered in this er. Only the cross term, Z c d! Re E L E P d; (6) is resonsible for ccelertion of the rticle. In wht follows, we need nottion for the lser field without the screen, E L, nd the bem field without the screen, E P.We define the rdition fields E LR nd E PR s difference between the field with the screen nd the field in free sce: E LR E L E L, E PR E P E P. The rdition fields re generted by currents flowing in the screen. The fields E L nd E P cn be considered s suerosition of the rdition fields nd the fields without the screen, i.e., E L E L E LR ; E P E P E PR : (7) For clcultion, it is convenient to cst Eq. (6) into different form. Using the second exression in Eq. (7) we reresent Eq. (6) s sum of two terms. The first one involves the rticle s field without the screen: Z c d! Re E L E P d: 3-3-

3 PRT-AB 7 Z. HUANG, G. TUPAKOV, AND M. ZOLOTOREV 3 () This term describes interference of the chrge s Coulomb field in vcuum with the lser field. In the limit R!, this term vnishes becuse the Coulomb field moves with the chrge with velocity v<c, nd the lser light rogtes with the seed of light c. ince we ssume tht the lser ulse overls with the rticle in the vicinity of the hole, t lrge distnce from the hole these two fields re serted in sce. Hence, the rticle s ccelertion is given by the second term: Z U cc c d! Re E L E PR d; (8) for which we use the nottion U cc. Notice tht the resence of the field E PR in this eqution indictes tht rticle cn be ccelerted only if it rdites. Although in the bove derivtion we refer to the lyout of the ccelertion exeriment outlined in Fig., our result is not limited by this secific rrngement. With slight modifiction, it cn lso be used for clcultion of the energy gin for other ccelertion schemes, such s, e.g., inverse FEL or inverse Cerenkov ccelertion. The close reltionshi between ccelertion nd rdition hs been exlored in Refs. [,8]. Recently, Eq. (8) is lso derived by Xie [9]. In our clcultions of the rdition field below we ssume tht the rticle moves with constnt velocity. Hence, we neglect the effect of the lser field on the rticle s trjectory, s well s the effect of rdition rection. uch n roximtion describes liner ccelertion roortionl to the lser electric field. III. DIFFRACTION RADIATION ON A ROUND HOLE Following the roch develoed in the revious section, we first clculte the rdition field E PR of the rticle. We now ssume tht the hole in the screen is round, with rdius, nd consider reltivistic rticle moving long the xis of the screen with constnt velocity v close to the seed of light. In the limit of the lrge Lorentz fctor,, the rdil electric nd zimuthl mgnetic fields of the rticle re E P r r; z; t H P er r; z; t ; (9) r z vt 3= where r x y is the rdil distnce. To clculte the rdition field in the fr zone we use diffrction formuls [7,,]. This roch is vlid if the reduced wvelength of the rdition, =, is much smller thn the rdius of the hole, nd the diffrction ngle is smll. According to the diffrction theory [7], the field behind the screen, E P, t lrge distnce R!nd in the region z>, cn be clculted by integrtion of the incident field E P on the screen t z : E P eikr i R Zhole k e ikr n E P d; () where r x; y is the two-dimensionl vector in the lne of the hole, k is the wve number vector in the direction of the rdition, k jkj!=c, nd n is the unit vector erendiculr to the surfce of the hole. The integrtion in Eq. () goes over the cross section of the hole. Eqution () is derived in [7] for the cse when the incident wve rogtes in free sce. In our roblem the incident field is the Coulomb field crried by the rticle. In this cse, Eq. () gives the totl field behind the screen including the field of the rticle, nd to find the rdition field, we need to subtrct the Coulomb field of the electron. The ltter cn be clculted s the sme integrl in Eq. () in the limit!, tht is when the screen is removed. The result of such subtrction is n integrl, with the sign oosite to tht in Eq. (), in which the integrtion goes over the screen surfce, rther thn the hole []: E PR E P E P eikr i R Zscreen k e ikr n E P d: () A more rigorous roof of this eqution cn be found in Ref. []. The rticle s field on the screen is given by E r r; ;t nd H r; ;t in Eq. (9). Fourier trnsformtion of these fields defined by Eq. (3) gives E P r r;! H P ke r;! c K kr ; () where K n (n ; ; ;...) is the modified Bessel function, nd we hve used v c in the bove exression. In the limit of lrge, the ngle of the rdition reltive to the z xis,, is smll,. ubstituting Eq. () into Eq. () nd neglecting higher-order terms in, we find tht E PR hs the rdil comonent only, E PR r k eikr R ek c e ikr R rdre r r;!j kr rdrk kr J kr; (3) where J n (n ; ; ;...) is the Bessel function. The integrtion in the lst formul cn be crried out nlyticlly [], E PR r with A!; e k c A!; eikr R ; () k J kk J kk k :

4 PRT-AB 7 CALCULATION AND OPTIMIZATION OF LAER... 3 () This formul grees with the rigorous solution of the diffrction rdition roblem obtined in Ref. [3], if one tkes the limit, k of their result. In the limit (but is still much less thn ) we hve [] A!; e c J k; (5) which in smll-ngle roximtion yields A!; e c : (6) ince the hole rdius dros out from the lst eqution, it is lso vlid in the limit!, when there is no hole in the screen. In this limit, it is usully clled the trnsition rdition. Putting this exression into Eq. () yields E L r z>e!! L eikr R k w Z drr J kre r =w ; () where, s ws defined in ec. II, the suerscrit L stnds for the diffrcted lser field. Note tht for!, Eq. () cn be integrted to yield E L r z>;!e!! L eikz z ex k w k w 3 ; () IV. ACCELERATION BY A HIGHER-ORDER LAER MODE For the lser field, s in Ref. [3], we first consider rdilly olrized TEM mode with the trnsverse field: E L? r;z;te e ik Lzi! L t w w r w ex r w i k Lr f i ; (7) where! L is the lser frequency, k L! L =c = L, the lser wist with trnsverse size w is ssumed to be locted t the screen, nd hence w w z ; z R k Lw z R f z z R ; rctn z ; z z R : (8) The choice of this higher-order mode is motivted in rt by the fct tht it mtches the rdil olriztion of the diffrction rdition in Eq. () nd is exected to roduce better ccelertion for the sme lser energy. Eqution () (with the suerscrit P substituted for L ) enbles us to clculte the diffrction of the lser field through the round hole. First, we Fourier trnsform Eq. (7): E L r;z;! dte i!t E L r r;z;t E!! L e ikzw w r w ex U cc ce k L w r w ikr f i d ek L c d ek L c Using the orthogonlity of Bessel functions, : (9) kl r rdrk rdrk kl r which is consistent with Eq. (9) in the limit z!. As ointed out in ec. II, it is convenient to reresent the diffrcted lser field s sum of the originl lser field (when the screen is bsent) nd the field E LR due to the rdition of the currents in the screen, E L E L E LR. For the rdition field, we hve E LR r E!! L eikr R k drr e r =w J w kr: () This field ws clculted in the region z>. However, due to the symmetry of the screen, it is symmetric bout the oint z. In the region z<, is then tken to be the ngle reltive to the z xis; here E LR r reresents the reflected wves rogting in the direction oosite to the incident lser bem. We now clculte the ccelertion of this lser mode using the energy blnce Eq. (8). Ignoring ure hse fctor, nd noting tht both fields E L nd E PR hve rdil olriztion, the energy gin of the rticle is Z U cc c d! de L r E PR r : (3) In the region z<, only the reflected field E LR cn interfere with the rdition field since they rogte in the sme direction (to the left of the screen in Fig. ). Writing R d R R d R R d nd inserting Eqs. (3), (), nd (), we hve J k L r Z J k L r dr r e r =w J k L r dr r e r =w J k L r : () 3-3-

5 PRT-AB 7 Z. HUANG, G. TUPAKOV, AND M. ZOLOTOREV 3 () we obtin dj krj kr kr kr ; (5) kr U cc ee Z k L w r dre r =w kl r K : (6) For n ultrreltivistic rticle we hve k L =w nd we my use the roximtion K x =x. Introducing the lser focusing ngle f =k L w,the condition for the roximtion cn lso be written s f. Eqution (6) then yields r U cc ee w e =w P e L c ex w ; (7) where the verge ower crried by this mode is P L c Z de L r Hr L 8 z lne c Z 8 E rdr r w ex r w c 3 E w : (8) Eqution (7) shows n imortnt result: in order to ccelerte rticle the lser bem should lso irrdite the mteril wll of the screen. If the focl size of the lser light is so smll tht it does not touch the metl, w, the ccelertion diminishes exonentilly. For otiml ccelertion, we should hve <w with the mximum energy gin in units of mc : mx s P L ; (9) where P m c 5 =e 8:7GW. For TW lser, we find mx 6. In generl cse of rbitrry reltion between k L nd =w, Eq. (6) cn be rewritten s cc s P L GA; B; (3) where GA; B B A P P dxx e x K Bx; A w ; B k Lw : (3) f The mximum of G is when A B. When B, GA; e A, which is the roximtion used in Eq. (7). When A, G;B B ex B ; B ; (3) where ;Z R Z dte t =t is the incomlete Gmm function. As shown in Aendix A, Eq. (3) grees with the direct integrtion of Eq. () in the bsence of hole, confirming the vlidity of this roch. V. ACCELERATION BY TWO CROED LAER BEAM Another lser ccelertion scheme emloys ir of linerly olrized lser bems with the Gussin fundmentl mode focused to the screen nd crossed t smll ngle to the z xis. If the two identicl lsers re out of hse by, the trnsverse comonents cncel while the longitudinl comonents dd. In the bsence of bemssge erture, the ccelertion hs been directly clculted by integrting the longitudinl field long the bem trjectory [3]. Here we clculte the energy gin from the energy blnce Eq. (8). It is sufficient to consider one tilted lser since the totl energy gin of two crossed lser bems t roer reltive hse is twice s lrge. First we clculte the lser field in the resence of the screen, following closely the derivtion of ec. IV. The Gussin fundmentl mode for smll tilt ngle t the screen loction, z, is E L x r;z ;!E!! L e ikx sin ex The diffrction integrl cn be evluted s [7] Z r w : (33) E L x z>e!! L ikeikr rdr R ex r w J kr; (3) where cos =,nd is the zimuthl ngle of the wve vector k with resect to the z xis. In the region z<, the totl lser field is the incident field nd the reflected field given by E LR x E!! L ikeikr R rdr ex r w J kr: (35) To comute Eq. (8), we note tht je L E B j je L x E BR r cosj nd mke use of the Bessel function exnsion [] J kr U cc ee k L ee w X m J m krj m kre im : (36) Integrtion over icks u only m terms. Then following the integrtion stes of ec. IV, we find Z ex r J k L r rdrk kl r =w dxe x J f x w ; (37)

6 PRT-AB 7 CALCULATION AND OPTIMIZATION OF LAER... 3 () αot / αf /w FIG.. The otiml tilt ngle s function of the hole rdius for the tilted Gussin lser bem. where the extr fctor of on the right-hnd side tkes into ccount two crossed lser bems, nd we hve ssumed tht k L =w to use K x =x for the roximte exression. For vnishing hole s!, we hve U cc ee w f ex f ; (38) in greement with Ref. [3] when the injection oint is t z I nd the extrction oint is t z F. Atthe otiml tilt ngle ot : f, the mximum energy gin is :3eE w. For n rbitrry, Eq. (37) cn be used to obtin the otiml tilt ngle nd the mximum energy gin (see Figs. nd 3). As shown in Fig. 3, the mximum energy gin in units of mc cn be roximted by s P mx 3:6 L P ex w : (39) Here P L ce w =8 is the totl lser ower for the two Gussin bems. Comring with Eq. (7), the energy gin of the crossed lsers hs essentilly the sme exonentil deendence on the rdius of the hole. For the sme lser ower, the rdilly olrized TEM mode is more effective for ccelertion (by bout fctor of.6) becuse it mtches the olriztion of the diffrction rdition in this ccelerting structure (see ec. VI B for more discussions). VI. DICUION A. Limittions due to mteril dmge Results of revious sections suggest tht the lser should irrdite the ccelertor structure, which is subject to mteril dmge t certin threshold lser fluence. Considering the cse of the higher-order lser mode in ec. IV, we rewrite Eq. (3) s cc U =w L w P GA; B t L FL =L BGA; B; () P t L where U L is the lser flsh energy, t L is the lser ulse durtion, nd F L :U L =w is the mximum lser fluence t r :7w for this higher-order mode lser. We hve lso ssumed the hole rdius <:7w for effective ccelertion nd used the revious nottions A =w nd B k L w = = f from Eq. (3). ince the lser fluence t the mteril dmge threshold is knowntobef th J=cm for sub-s lser ulses [], we ssume tht the lser oertes t the dmge threshold [i.e., by tking F L F th in Eq. ()] nd otimize the lser sot size or the focusing ngle in order to obtin the mximum energy gin. Figure shows the otiml focusing ngle tht mximizes BGA; B in Eq. (). For w, the otiml lser focusing ngle f ot nd the otiml sot size w ot L =. γ mx (PL /P) / 3 (γα f )ot /w FIG. 3. The mximum energy gin of two crossed lser bems evluted t the otiml tilt ngle from Eq. (37) (solid line), nd comred with the roximte Eq. (39) (dshed line) /w FIG.. The otiml lser focusing ngle s function of the hole rdius t the mteril dmge threshold

7 PRT-AB 7 Z. HUANG, G. TUPAKOV, AND M. ZOLOTOREV 3 () The scling f ot hs simle hysicl exlntion. As it follows from Eq. (7), in the limit of lrge ngles f, the energy gin does not deend on the sot size w nd scles with the lser ower s P = L.This hens becuse rticle intercts with the lser on the Ryleigh length, nd lthough, for given P L, incresing w mkes the mlitude of the lser field smller, corresonding increse in the Ryleigh length comenstes for the smller field nd mkes the energy gin indeendent on w. Further incresing w, however, mkes the ngle f smller thn.inthisregime, the interction length becomes shorter thn the Ryleigh length it is determined by the hse slige due to the difference between the rticle s velocity nd the hse velocity of the lser light. As it follows from Eq. (3), the energy gin in this regime (for given P L ) decreses with w. For given fluence (lser ower er unit re) the otiml vlue of f turns out to be t the boundry between those two regimes. At the otiml focusing ngle f ot, we hve B = f ot in Eq. () nd B GA; B :8 for w from Fig. 5. If we tke tyicl short-ulse lser with t L fs nd L m, the mximum frctionl energy gin limited by the fluence dmge threshold is roximtely mx s F :6 th L 7:5 3 : () P t L ince the interction length is bout equl to the Ryleigh length z R w ot= L L = t the otiml sot size, the effective ccelertion grdient is U cc 7:5 3 mc z R L = GeV=m: () For 5 MeV electron (i.e., ), the energy gin is bout 375 kev from Eq. (), nd the ccelertion grdient B G(/w,B) /w FIG. 5. The function B GA; B in Eq. () evluted t the otiml sot size or focusing ngle. is bout MeV=m ccording to Eq. (), in greement with the exected erformnce of the E-63 roosl [5]. Finlly, suose tht the lser is oerted t the dmge threshold fluence, the otiml lser energy t the otiml sot size for mximum energy gin is U L F thw ot : 8: nj: (3) For fs lser ulse intercting with 5 MeVelectron bem, the otiml lser ower is P L :8 GW. A lrger lser ower t the given ulse durtion requires lrger lser sot to void the mteril dmge nd hence smller longitudinl field for ccelertion. ince the interction length is still L = limited by the hse slige of the rticle in the lser field, the totl energy gin is ctully smller if the lser ower is lrger thn this otiml vlue. B. Otiml lser rofile As ointed out in ec. V, the rdilly olrized TEM mode is more effective for lser ccelertion thn the tilted Gussin fundmentl mode becuse it mtches better with the diffrction rdition ttern. For otiml ccelertion, one might consider shing the lser trnsverse rofile in such wy tht chieves mximum ccelertion for given lser ower. It is esy to see from clcultions in ec. IV tht for the otimum ccelertion the ngulr distribution of the reflected lser light E LR must mtch exctly the ngulr distribution of the rticle s rdition. In the cse w, this mens [see Eq. (6)] E LR r E w!! L eikz z : () The corresonding lser ower is P L lnce w =, for. Integrting the energy blnce Eq. (8) then yields mx lnee w s P ln L : (5) We see tht the otiml lser rofile (with the ngulr distribution of the trnsition rdition) only imroves the mximum energy gin by smll fctor ln even for n ultrreltivistic rticle. VII. CONCLUION In summry, liner ccelertion by lser field in vcuum is ossible only if rticle rdites in ssing the ccelerting structure. In this er, we exress the energy gin by the rticle s n interference integrl of P

8 PRT-AB 7 CALCULATION AND OPTIMIZATION OF LAER... 3 () the lser field nd the rdition field in the fr zone nd hence void clcultion of ny ner field tht ccelertes the rticle. We ly this new method to study lser ccelertion in simle ccelerting structure ( conducting screen with bem-ssing hole) nd to otimize gin for given lser rmeters. We show tht for otiml ccelertion, the lser should irrdite on the ccelerting structure (i.e., the dimension of the hole should be less thn the lser sot size), nd the lser focusing ngle (s well s the crossing ngle in the cse of the two crossed lser bems) should be comrble to the rdition oening ngle. Limited by the dmge threshold fluence, the mximum energy gin in this ccelerting structure is roortionl to the electron energy, but the ccelertion grdient scles s. ACKNOWLEDGMENT The uthors would like to thnk P. Emm for creful reding of the mnuscrit. This work ws suorted by the Dertment of Energy Contrcts No. DE-AC3-76F55 nd No. DE-AC3-78F98. APPENDIX: COMPARION WITH DIRECT CALCULATION OF ACCELERATION In the cse with no hole,, when the screen stos the lser bem from rogting to the region z> the energy gin in lser field cn be clculted directly. Prticles re ccelerted by the lser bem in the region z< nd sto intercting with the lser bem fter ssing through the screen. (In the bsence of the screen, the lser would decelerte electrons in the region z>so tht the net energy gin is zero.) For the lser mode given by Eq. (7), the longitudinl electric field cn be found from the Mxwell eqution r E L nd is roximted by E L z i r k? E L? L k L r@r rel r E e ik Lzi! L t w i r r w k L w w wf ex r w ik Lr f i : (A) Consider reltivistic rticle moving in the z direction long the xis of the system, r nd z vt. The energy gin cn be obtined by integrting the longitudinl lser field long the rticle s trjectory from z to z (the loction of the screen): Z U cc dzee L z r ;tz=v i ee k L w Z ee w B ex dz iz=z R ex ik L z c v B ; B ; (A) where B k L w =, nd the squre brcket term describes the gin reduction due to reltive slige of the rticle in the lser field. This exression is identicl to Eq. (3) derived using the energy blnce Eq. (8). An roximte exression of Eq. (A) is given in Ref. [3]. [] R. B. Plmer, in Advnced Accelertor Concets, edited by P. choessow, AIP Conf. Proc. No. 335 (AIP, New York, 995),. 9. [] F. Csers nd E. Jensen, CERN Technicl Reort No. CERN/P 89-69, 989. [3] E. Esrey, P. rngle, nd J. Krll, Phys. Rev. E 5, 53 (995). [] R. Byer, T. Plettner, C. Brnes, E. Colby, B. Cown, R. iemnn, nd J. encer, in Proceedings of the Prticle Accelertor Conference (IEEE, Pisctwy, NJ, ). [5] C. Brnes, E. Colby, B. Cown, R. Noble, D. Plmer, R. iemnn, J. encer, nd D. Wlz, LAC Technicl Reort No. E-63,. [6] L. D. Lndu nd E. M. Lifshitz, The Clssicl Theory of Fields, Course of Theoreticl Physics Vol. (Pergmon, London, 979), th ed. (trnslted from the Russin). [7] J. D. Jckson, Clssicl Electrodynmics (Wiley, New York, 975), nd ed. [8] M. Zolotorev,. Chttodhyy, nd K. McDonld, htt://vier.he.rinceton.edu/~mcdonld/ccel/vcuumccel.df [9] M. Xie, in Proceedings of the 3 Prticle Accelertor Conference (IEEE, Pisctwy, NJ, 3). [] M. Ter-Mikelin, High-Energy Electromgentic Processes in Condensed Medi (Wiley-Interscience, London, 97). [] B. M. Bolotovskii nd E. A. Glst yn, Phys. Us. 3, 755 (). [] I. Grdshteyn nd I. Ryzhik, Tble of Integrls, eries, nd Products (Acdemic Press, New York, ), 6th ed. [3] G. Dome, E. Ginfelice, L. Plumbo, V. G. Vccro, nd L. Verolino, Nuovo Cimento oc. Itl. Fis.A, (99). [] B. turt, M. Feit, A. Rubenchik, B.W. hore, nd M. Perry, Phys. Rev. Lett. 7, 8 (995)