Optical Accelerator: Scaling Laws and Figures of Merit

Transcription

1 SLAC-PUB Otical Accelerator: Scaling Laws and Figures of Merit Levi Schächter (1), obert L. Byer () and obert H. Siemann (3) (1) lectrical ngineering Deartment, Technion IIT, Haifa 3000, ISAL () Deartment of Alied Physics, Stanford University- MC4085, Stanford CA 94305, USA (3) SLAC, MS-7, Stanford University, P.O. Box 4349, Stanford, CA 94309, USA Abstract. Indications that solid-state lasers will reach wall-lug to light efficiencies of 30% or more make a laser-driven vacuum-accelerator increasingly aealing. Since at the wavelength of relevant lasers, dielectrics may sustain significantly higher electric field and transmit ower with reduced loss comaring to metals, the basic assumtion is that laser accelerator structures will be dielectrics. For structures that have tyical dimensions of a few microns, resent manufacturing constraints entail lanar structures that in turn, require re-evaluation of many of the scaling laws that were develoed for azimuthally symmetric structures. Moreover, structures that oerate at a wavelength of a few centimeters are machined today with an accuracy of microns. In future it will not be ossible to maintain 4-5 orders of magnitude difference between oerating wavelength and achievable tolerance. An additional difference is, that contrary to resent accelerators where the number of electrons in a micro-bunch is of the order of a 10 10, in an otical structure this number dros to a few thousands. Consequently, the relative imact of individual electrons may be significantly larger. Acceleration structures with higher degree of symmetry, similar to otical fibers, have also some inherent advantages however thermal gradients as well as heat dissiation may become critical imediments. The imact of all these factors on the erformance of a laser accelerator structure needs to be determined. fficiency, wakes and emittance scaling laws that have been develoed recently will be resented. It will be shown that there are some inherent advantages in combining the accelerator structure and the laser cavity in one system. INTODUCTION Indications that solid-state lasers will reach wall-lug to light efficiencies of 30% or more make a laser-driven vacuum-accelerator increasingly aealing. Since at the wavelength of relevant lasers, dielectrics may sustain significantly higher electric field and transmit ower with reduced loss comaring to metals, the basic assumtion is that laser accelerator structures will be made of dielectrics. Oen, quasi-otical structures and near-field accelerators with dimensions comarable to the wavelength are both being considered. xamles of these two are: a) the LAP[1] crossed laser beam accelerator where the interaction between the crossed laser beams and the articles is limited by slits to satisfy the Lawson-Woodward theorem[-3], and b) the hotonic band-ga (PBG) concet where a laser ulse is guided in a dielectric structure with a vacuum tunnel bored in its center [4]. Lithograhy, which would 10th Advanced Accelerator Concets Worksho (AAC 00), 3-8 June 00, Mandalay Beach, California Work suorted in art by Deartment of nergy contract D-AC0-76SF00515

2 result in lanar structures, and otical fiber drawing are manufacturing techniques that seem well suited for laser driven structures that have tyical dimensions of a few microns. Structures that oerate at a wavelength of a few centimeters, are machined today with an accuracy of microns. In future, it will not be ossible to maintain 4-5 orders of magnitude difference between oerating wavelength and the achievable tolerance since this would entail engineering of a surface at the atomic level. As a result, the size of irregularities may be of the same order of magnitude as the micro-bunches, and they may generate wakes that in turn, may alter the dynamics of electrons. Fortunately, the electromagnetic roerties of materials at wavelengths that are significantly smaller than 0.1µm do not differ dramatically from these of the vacuum. This fact is exected to reduce somewhat the sensitivity to manufacturing tolerances mentioned above. For all these reasons, re-evaluation of many of the scaling laws that were develoed for near-field, F-driven, azimuthally symmetric structures is necessary. In this note the toics are organized as follows: in the next section it is shown that for a given laser ower there is an otimal number of electrons which should be accelerated in order to get imum efficiency. Since the otimal number of electrons that can be accelerated is roortional to the square root of the laser ower, it is natural to determine the imum ower sustained by the structure before breakdown occurs. This is exressed in terms of the energy er unit surface (fluence) that the material can sustain Section 3. At otical wavelengths, dielectric materials can tyically sustain higher electric fields therefore, they become natural constituents of acceleration structures. In the fourth section it is shown that since dielectrics have a frequency deendent coefficient, the number of modes excited (wake field) is by orders of magnitude smaller than in the existing S-band accelerator. In the fifth section, we consider the emittance of an asymmetric acceleration structure dictated by the manufacturing constraints and in the last section some thermal considerations are discussed. FFICINCY One of the major questions that will determine the fate of any future accelerator is the efficiency since it is natural to require acceleration of the largest number of electrons at the minimal inut ower. In other words, imum efficiency. Its existence may be readily conceived for a given inut ower, acceleration of a small number of electrons entails low efficiency. At the other extreme, increasing the number of electrons increases the beam loading, and as a result, the gradient is reduced and again the efficiency is relatively low. In between these two extremes, imum efficiency must occur and the question is what is its value for a given inut ower. Denoting the laser ower injected in the structure by and the resulting gradient at the location of the electrons by imedance as acc P laser, it is ossible to define the interaction

3 int accλ (1) P laser which characterizes any acceleration structure; in this exression λ is the wavelength in the vacuum. This laser ulse accelerates a bunch (Q) that as it moves in an arbitrary acceleration structure generates an electromagnetic wake. Associated with this wake there is decelerating electric field comonent ( dec ) which by virtue of the linearity of Maxwell s equations, must be roortional to Q. Therefore, without loss of generality it is ossible to assume that there is an effective transverse dimension denoted by, determined by the details of the structure and the bunch, such that eff = Q ; () dec 4πε 0 eff it is tacitly assumed here that the bunch is significantly smaller than the transverse dimension where the bunch roagates see Aendix. Three generic cases have been investigated in the ast and have direct relevance to the resent analysis: (i) for a relativistic (γ ) bunch moving in a vacuum tunnel of radius bored in a dielectric medium [5], it may be shown that eff =. A similar result is obtained if the dielectric is of finite width or if the dielectric structure is relaced by an arbitrary dielectric eriodic structure confined by a metallic surface ( eff = 1.14 )[6]. (ii) Analysis of a similar bunch moving at a height h above a dielectric half-sace [7] reveals an effective length eff = h and a similar result is exected for a dielectric layer. Moreover, when the bunch moves arallel to a dielectric cylinder [8] of radius at a distance h from axis, eff = ( h ). (iii) For a wide bunch ( ) moving at a height h above a grating of arbitrary geometry, summarized in Figure 1. = h. These cases are eff A first figure of merit may be develoed by comaring the imedance linked to the acceleration mode and that seen by the wake. For this urose it is imortant to bear in mind that contrary to acc, which is a very narrow-band signal, the sectrum of dec may be very broad. However, since it is ossible to establish the ower associated with this field, P = Q, then the imedance definition as in q.(.1) entails wake v dec λ = 1 (3) dec wake 0 Pwake π ( eff / λ)

4 Configuration eff eff = h = eff h h eff = ( h ) eff = 1.14 h = h eff FIGU 1. The effective radius ( eff ) for different configurations.

5 Vacuum tunnel Dielectric FIGU. Schematic of the honey-comb hotonic band ga structure analyzed in ef. 4. wherein 0 = µ / 0 ε 0 is the vacuum imedance. For examle, the interaction imedance of the hotonic band-ga structure analyzed in ef. 4, see Figure, has an interaction imedance of int = 19.5Ω at 1µm and for an accelerating gradient of 1GV/m the total ower required is about 50kW. Furthermore, since the radius of the vacuum tunnel is = 0.678λ and it was demonstrated [ef. 9] that eff 1.3, then wake 86Ω. With these two imedances, it is ossible to determine the effective (loaded) gradient as 1 Qc wake eff acc = dec Plaserint λ λ (4) therefore, the gain in the kinetic energy of the bunch in structure of length L is U Q L whereas the total electromagnetic energy stored in the structure is U kin eff P τ = P em laser f laser L 1 βgr c β gr where τ f is the fill-u time and cβ gr is the grou velocity. The exression for the fill-u time tacitly assumes full overla of the electron bunch during the entire transit time of the electrons. Moreover, it is assumed that the laser ulse duration is shorter than the time it takes the ulse to traverse the structure, in fact, the difference between the two is taken to be equal to the bunch transit time τ L L tr c c τ L 1 βgr = imlying that the minimal ulse duration is τ β τf =. c β gr Based on these two energy definitions, the efficiency of the acceleration rocess is gr

6 wherein Q laser n wake c gr η P λ 1 βgr and β U ( Q) QQ kin 0 = (5) Uem Qn Q λ P laser int 0 is the charge for which the cwake effective gradient vanishes see q. (4). In other words, all the energy gained by the bunch from the laser field is lost to Cerenkov radiation. Clearly, imum efficiency occurs for an otimal value of the charge given by Q 1 λ P laser int ot Q0 = (6) cwake in which case the value of this efficiency is η 1 Q 1 βgr Qn 4 1 βgr 0 = = int wake (7) imlying that the imum efficiency is determined by the ratio of the imedances and the grou velocity. For the PBG structure mentioned above [4], the grou velocity is 0.58c therefore the imum ossible efficiency is 7.86%. It is likely that considerations of energy sread will lead to charges and efficiencies below the otimum values. BAK-DOWN LIMIT The otimal number of electrons to be accelerated is determined by the laser intensity and the latter is limited rimarily by the surface threshold limit exressed in terms of energy iminging er unit surface i.e. the threshold damage fluence F. Based on ef. 10 there is a relatively simle model for this quantity namely, 1/ 1.44 τ [sec] τ[sec] >10 1/4 F[J/cm ].51 τ [sec] 0.4 < τ[sec] < 10 (8) τ [sec] < 0.4 However, while it was relatively simle to determine the amount of energy in a structure based on the ower and the fill-u time, it is difficult to rovide a general relation between the imum field ( ) and the accelerating gradient acc. Nevertheless, by virtue of the linearity of Maxwell s equations, similar to q.(1), it is ossible to determine an imedance

7 λ (9) P laser which is a characteristic of the structure. Based on this definition and on q.(1) it is evident that int =. (10) acc Furthermore, by calculating the imum field for a given gradient and knowing the interaction imedance in the structure, the imedance, is set. One ste further is achieved by realizing that the imum amount of energy iminging uon a unit surface (fluence) of the structure is roughly determined by the energy density in the region of the imal field, the grou velocity and the ulse duration or exlicitly, 1 = F ε 0ε cβgr τ. (11) Note that, for examle, the radius of curvature i.e. the smallest geometric arameter of the structure is tacitly considered to determine the imum field. Consequently, based on qs. (8-9) and using q.(11) the imum laser ower is given by () 0 acc laser εβgrτ int P λ F =. (1) Accordingly, based on q.(6) Q = λ ε F. (13) () 0 acc 0 ot ε wake βgrcτ For the PBG structure it was found that =.1acc. The main relations so far are summarized in the next table.

8 TABL 1. The arameters of the hotonic band ga structure are: rod sacing a / λ = 1. 3, radius of the central cylinder / λ = 0.678, the interaction imedance = 19.5 Ω, grou velocity β = 0.58, the int gr effective radius / =.1. acc = 1.3 eff and the ratio of the accelerating and imum surface field Quantity General xression PBG (ef. 4) τ = 00sec Number of electrons Laser ower Maximum gradient Q λ ε F () = 0 acc 0 ot N 4 el = ε wake βgrcτ () 0 acc laser εβgrτ int P λ F = () P laser = 15.8[kW] 1 () = P () λ = 0.55[GV/m] () acc laser int Q Decelerating field = dec 4πε Maximum efficiency η 0 eff gr wake acc dec = 0.7[GV/m] 1 βgr int = 41 β η = 7.86% FQUNCY DPNDNC OF ε One of the major differences between a future (otical) accelerator based on a dielectric material and the current systems based on metallic structures is the fact that the latter suorts excitation and roagation of many thousands of modes other than the main accelerating mode whereas the former, may suort only a few modes before the material becomes transarent. In order to quantify this statement we shall consider the simlest model ossible: a oint-charge moving at a constant velocity (v) in an azimuthally symmetric waveguide filled with a dielectric material (ε). Proagating modes develo rovided the oles are real quantities namely, the zeros of the function ω + c s β 1 ε( ω) β = 0 (14) are real; here is the radius of the waveguide and s reresents the zeros of the zeroorder Bessel function of the first kind s=1,,. Assuming that ε r ω < ωc εω ( ) = 1 ω > ωc (15)

9 then for a relativistic article the roagating modes are limited to s<s c where ω ε 1 c s c Integer r 1. (16) c π Higher modes do not contribute to radiation and as such, all the electromagnetic (evanescent) field corresonding to these higher modes is attached to the bunch. In order to better quantify this statement let us consider two examles; in both cases the cut-off of the material is at λ c =00nm and ε r =. In an S-band structure the tyical radius is of the order of centimeters (=1cm) therefore s c =10 5 whereas for an otical system the tyical radius is =1µm thus s c = 10!! Clearly, exression (16) constitutes a quantitative measure for the number of modes in a tyical structure whose transverse dimension is and ω c is the cut-off angular frequency of the dielectric material. MITTANC One of the imortant obstacles in the rocess of imlementing an otical vacuum accelerator is the constraint imosed by the emittance therefore, a few comments regarding the scaling laws linked to the emittance in a non-azimuthally symmetric structure are in lace. In order to have the correct orders of magnitude, one should bear in mind that in an azimuthally symmetric structure, the ratio of the transverse force to the longitudinal force is virtually negligible since it is roortional to F F z 1 ω b ; (17) 4γ c 4 for examle, tyical values are γ 10, ωb / c 1wherein b is the radius of the bunch. On the other hand, in a non-symmetric structure of a tyical transverse dimension a, see Figure 3 for a tyical configuration, the ratio of the two forces is F ω a F c z 1. (18) 3D numerical simulations of a bunch of 30GeV electrons indicate that the relative emittance change is drastically affected by the transverse dimension ε (out) (in) υ ε b (in) ε (19) a

10 wherein it was assumed that the bunch exeriences a gradient of 1GV/m along a 10cm long structure and the initial bunch length was 90 o of otical hase. The arameters b and ν deend on the radius of the beam and the corresonding emittance: x : b= ν = b y: b= 10.5 ν = b b b b b therefore in case b = 3µm the coefficients for εx are b x =15.4 and ν x =4.4; a, b and b are exressed in microns.. Bunch length ( χ ) also affects the emittance. A similar analysis shows that if o χ < 90, the emittance change is quadratic in χ ε ε ε (out) (in) (in) 1.1 = χ 180 o. (0) here b =3µm and a=0µm. According to the emittance growth in a single stage, two drawbacks are evident: in order to kee the emittance growth moderate it is necessary to have a relatively large transverse dimension (tyically a few vacuum wavelengths) imlying oeration in a multi-mode regime. And even then, according to simulations, the increase in the emittance is almost 10% in a 10cm long segment. In order to overcome this imediment due to the asymmetry of the structure it is quite natural to symmetrized the structure. This is done by constructing each module by sets of four non-symmetric segments each one rotated by 90 o. In this way, the average value of the transverse field comonents is zero. However, there remain two questions to be addressed: (i) is the zero average transverse field condition sufficient to maintain the low emittance over extended length? (ii) What would be a reasonable division in a tyical acceleration module? a y x FIGU 3.: Schematic of one ossible configuration of a non-symmetric acceleration structure. mittance variation along such a segment is strongly deendent on a.

11 In order to address these questions a one meter long module was simulated. The two lates are a=0µm aart, the initial radius of the bunch is b =µm, its average energy is 30GeV with an energy sread of 1%, its length being 3.6 o on the scale of a 1.06µm laser wavelength which in turn generates at the electrons location a gradient of 1GV/m. For these arameters, the emittance at the outut aroaches the limit of a symmetric structure when using a large number of modules, secifically, ε ε (in) ε (out) (in) 4.14 (1) N 1 st + 15 where N st is the number of segments. For 400 segments the emittance increase in a 1m long acceleration structure is 0.6%, entailing a imum segment length of.5mm. THMAL CONSIDATIONS Although dielectric material are known to have low ohm loss there still is an imortant asect of temerature gradients and heat dissiation. lectromagnetic field variations in the cross section of the structure are inevitable and consequently, so are temerature gradients since electromagnetic ower is artially absorbed by the dielectrics. eduction of these gradients is ossible if: (i) the material used has good thermal conductivity and (ii) heat dissiation at the outer boundary is roerly designed. In order to have some order of magnitude, consider a 5cm long module of an outer circumference of 40µm in which we inject 15kW of laser ower, the loss of the material is tanδ =10-4. For this articular set of arameters the total amount of ower that needs to be dissiated is of the order of 15W/cm which is two orders of magnitude lower that what we may anticiate to be able to dissiate I D 1.5kW/cm. However, secial attention is required here since heat dissiation engineering, at the micron-scale favors lanar configuration. DISCUSSION The fate of any future linear collider will be determined by the energy efficiency of the entire system. Tyical collider designs (500GeV) redict 8MW of average beam ower assuming electrons/sec. At an acceleration efficiency of 7.86% as calculated in Section for the PBG, this entails 0.1GW of average laser ower and assuming a wall-lug to light efficiency of 30%, the total amount of average ower is about 0.34GW. For accelerating electrons/sec the bunch reetition rate of the system would need to be about 1GHz since the otimal number of electrons er bunch was demonstrated to be 10 5.

12 All laser based schemes rely on the fact that a relatively small fraction of the energy stored in the laser cavity is extracted and used in the acceleration structure. Concetually, it seems ossible to take advantage of the high intensity electromagnetic field that develos in the cavity and incororate the acceleration structure in the laser cavity as illustrated in Figure 4. Active Medium Active Medium Acceleration Structure Acceleration Structure Acceleration Structure Acceleration Structure Active Active Medium Medium FIGU 4.: Schematic of 4 segments of combined acceleration structure and laser cavity. By choosing the round-tri time of the laser cavity to be equal to the reetition rate of the system, it is ossible to use the wake generated by the macro-bunch ν, amlify it in the active medium and further inject it in the acceleration structure for accelerating the macro-bunch ν+1. Nominally, this may lead to a significantly higher overall efficiency since energy is re-cycled. ACKNOWLDMNTS Τhis study was suorted by the United States Deartment of nergy and the Israel Science Foundation. FNCS 1. Huang Y. C., heng D., Tulloch WM. and Byer.L., Al. Phys. Lett. 68, (1996).. Lawson J. D., I Trans. Nucl. Sci. NS-6, (1979). 3. Woodward P. M., J. I 93, 1554 (1947) 4. Xintian ddie Lin, Phys. eview Secial Toics Accelerators and Beams, 4, (001). 5. Schächter L. and Schieber D., NIM A, 388, 8-16 (1997). 6. Schächter L. and Schieber D., Physics Letters A., 93, 17- (00). 7. Schieber D. and Schächter L., Physical eview., 57, (1998). 8. Schieber D. and L. Schächter L., Physical eview, 64, (001). 9. Schächter L. Wake-Field in a Photonic Band Ga Structure, Private communication 10. Stuart B.C., Feit M.D., ubenchick A.M., Shore B.W. and Perry M.D.,Phys. ev. Lett., 74(1), (1995).

13 APPNDIX Based on the analysis in ef. 5 the longitudinal electric field acting on a relativistic oint-charge (Q) as it moves along the axis of a vacuum channel of radius bored in a dielectric medium (ε) is given by Q 1 ξ jxγβcτ / z ( τ = t z/ v) = dx e 4 πε π I ( x ) + ξ I ( x ) whereinξ γ ε 1/ ε ; note that the radial deendence is omitted here since it was found that for a relativistic bunch ( γ 1) the transverse dimension has a negligible effect. In case of a bunch of duration τ, the average electric field that acts on the bunch is τ / τ / Q ξ = dτ dτ dx e 4 πε τ τ π I ( x ) + ξ I ( x ) 0 τ / τ / 0 1 jxγβc( τ τ )/ At the limit of γ the contribution to the first integral is from the region since ξ γ therefore x 1 τ / τ / Q e = 4 dτ dτ du 4πε τ τ π ju γ c( τ τ )/ ξ. 0 / 1 u τ τ / + The last integral can be evaluated analytically ju χ 1 e 1 du = e + π 1 u and it is convenient to define τ τγ c/ ξenabling to calculate analytically the other two integrals χ Q = 1 e τ 4 sinh( τ ). πε 0 τ τ Two comments are in lace at this stage: firstly, the average decelerating field is indeendent of γ since τ = cτ ε / ε 1. Secondly, within an excellent aroximation this relation may be exressed by

14 Q πε + τ If the ulse duration is significantly shorter than the transverse dimension of the channel, τ 1, whereas at the other extreme( τ 1), Q 4πε 0 Q ε. πε τ ε 4 0 c 1