Optimized operation of dielectric laser accelerators: Single bunch

We introduce a general approach to determine the optimal charge, efficiency and gradient for laser driven accelerators in a self-consistent way. We propose a way to enhance the operational gradient of dielectric laser accelerators by leverage of beam-loading effect. While the latter may be detrimental from the perspective of the effective gradient experienced by the particles, it can be beneficial as the effective field experienced by the accelerating structure, is weaker. As a result, the constraint imposed by the damage threshold fluence is accordingly weakened and our self-consistent approach predicts permissible gradients of ∼10 GV=m, one order of magnitude higher than previously reported experimental results—with unbunched pulse of electrons. Our approach leads to maximum efficiency to occur for higher gradients as compared with a scenario in which the beam-loading effect on the material is ignored. In any case, maximum gradient does not occur for the same conditions that maximum efficiency does—a trade-off set of parameters is suggested.


I. INTRODUCTION
Today's electron accelerators are predominantly driven by rf sources. Being structure-based systems, operating with the lowest electromagnetic mode, the accelerating gradient is limited by the breakdown in the metal's surface [1]. However, in the optical regime it has been experimentally shown [2,3] that dielectric materials held higher fields before breakdown. Therefore, it indicates what should be the general trend, namely operating at sub-mm or optical wavelengths, as this allows higher accelerating gradients. For example, the Stanford Linear Collider's gradient is of the order of 20 MeV=m [4], while in an optical accelerator 50 times this value is anticipated [5].
Another profound difference between a laser driven accelerator as compared with its microwave counterpart is their material. At optical frequencies Ohm loss makes metals prohibitively lossy. Thus, low loss dielectric materials are virtually the only alternative for an accelerating structure, regardless of whether the latter is used as an optical electron collider [6], a possible light source [7], or as a module for medical devices [8]. Throughout the years several dielectric structures have been proposed [9][10][11][12] and more recently experimental results were reported [13][14][15]. In all these configurations fluence damage [16] is a limiting factor, whereas in rf machines, breakdown at the metalvacuum interface is a critical impediment [17].
Both rf and laser accelerators have an important phenomenon in common, this is the beam loading. It results from the wakefield generated by each particle [18], thus reducing the effective gradient experienced by the same or trailing particles. As shown subsequently, the field reduction may be beneficial, since the structure is exposed to a weaker electromagnetic field.
In the framework of this paper we present a general approach relevant to any guided-mode or resonant Floquet harmonic in a dielectric laser-driven acceleration structure, which aims to achieve a self-consistent analysis of the optimal charge, gradient, and efficiency. Contrary to the approach in Ref. [19], we take into account the short range wakefield only once. But in addition, we account for its effect on the dielectric material too. Proper design of the operation parameters, indicates that beam-loading although reduces the effective gradient experienced by the particles, it also enlarges the gap between the maximum field experienced by the material and the damage threshold fluence. As a result, the laser power may be increased and so is the amount of accelerated charge. The optimization process developed predicts an unloaded gradient of ∼10 GV=m. This is one order of magnitude higher than previous experiments demonstrated [14,15].

II. SYSTEM DESCRIPTION
Let us now introduce the general approach. Common to all various structures is the vacuum channel, where the electrons propagate, and the single TM 01 mode that accelerates them. Although preliminary results were previously presented for planar Bragg structures [20], for the numerical examples presented subsequently in this study, we adopt an azimuthally symmetric dielectric-loaded waveguide [21], as shown in Fig. 1, since it is possible to evaluate all the quantities of interest analytically. However, while the group velocity, interaction impedance, and wake parameters may vary from one configuration to another, the general trend is expected to be the same. We anticipate the same quantities to control the interaction whether it is a guided mode or synchronous harmonic (open structure).
Specifically, the fields components of a TM 01 mode which propagates at a phase velocity equals to the speed of light in a cylindrical vacuum channel are where G 0 is the accelerating unloaded gradient, ω L is laser's angular frequency, and η 0 is the wave impedance in vacuum. The idealized acceleration structure consists of a dielectric (ε r ) loaded waveguide, whereby for a given dielectric (fused Silica) and vacuum channel radius (R int ), the external radius (R ext ) is set by imposing single mode (TM 01 ) operation and phase velocity equal to the speed of light in vacuum [see Eq. (B3)]. Moreover, note that for a given dielectric material ε r , imposing the group velocity sets R int and vice versa. Additionally, more assumptions are at the foundation of our model: (i) the microbunch is a point-charge, and remains so all along the interaction region. (ii) The space-charge effect inside each microbunch is ignored. (iii) The laser pulse has a sharp rise/fall time relative to one laser period, and (iv) the conversion from a propagating laser mode to an accelerating TM 01 mode (coupling process) is considered to be ideal. Each one of these idealizations may reduce the optimal charge, the gradient experienced by the electrons and of course the overall efficiency of the system, however these are beyond the scope of this study. (v) Fluence damage threshold is the only concern here and we ignore nonlinear optical effects (Kerr [23], Brillouin, Stokes, etc.). (vi) Since the focus of this study is the behavior of the field near the vacuum-dielectric interface, we ignore the fluence damage threshold of the confining metal. We wish to reiterate that the metallic boundary is not intrinsic to the operation, but rather allows us to readily determine the interaction impedance, wakefield and group velocity, for the sake of showcasing an analytical example for our general formulation. (vii) In case of single bunch operation, both the laser pulse and the wake leave the accelerating module before the next laser pulse fills in.
Previous studies [6,19,20] formulated the efficiency of accelerating single bunch (η 1 ) with a charge q in terms of three quantities: (i) the accelerating unloaded gradient generated by the laser ðE acc ¼ G 0 Þ; (ii) the wake coefficient (κ) which by virtue of linearity of Maxwell's equations relates the decelerating electric field that acts on the bunch ðE wake ¼ κqÞ, and (iii) κ 1 is the projection of κ on the fundamental (accelerating) mode. In these studies it was tacitly assumed that the maximum gradient applicable is limited by the damage threshold fluence-ignoring beamreduction of the field experienced by the dielectric. In the next paragraph we explain how we suggest to consider this effect, which would be referred to as the reduced field.
Virtually in all relevant acceleration structures the energy flux ðS z ¼ E r H Ã ϕ =2Þ reaches its maximum at the vacuumdielectric interface where ε r is the dielectric coefficient adjacent to the vacuum channel, and R int is the radius of the latter. We wish to emphasize that this is the flux of the TM 01 mode in the dielectric in terms of the accelerating gradient (G 0 ), in the absence of the electron beam. However, since we examine the effect of the electrons on the fundamental mode, we shall consider an effective electric field (E eff ). Note that the effective electric field in the material near the dielectricvacuum interface is . Given a single mode operation ω L R int =ð2ε r cÞ < 1, the maximum effective field in the material is approximately E eff ≃ E z .
Since the phase velocity is c, the latter is G 0 reduced by the projection (κ 1 ) of the wakefield solely on the accelerating mode, ignoring the contribution of all the other modes. Consequently, the maximum energy flux in the dielectric is assumed to be given by or explicitly, the effective field in the reduced case (superscript R) is E ðRÞ eff ≡ G 0 − κ 1 q. Note that the latter quantity is defined in the dielectric. Evidently, if the reduction κ 1 q is FIG. 1. Schematics of the basic principle and envisaged geometry. Single relativistic microbunches (blue) is accelerated by TM 01 laser mode (red) which propagates at cβ gr group velocity, in a vacuum channel of a dielectric (ε r ) loaded cylinder. The wake (green), generated by the bunch, propagates at its velocity [22] along the interaction region L geo . ignored, namely E ðURÞ eff ≡ G 0 , in what follows we refer to this incidence as the unreduced case (superscript UR).
It is important to emphasize that the projection of the wake on the fundamental mode is proportional to the value 2hðτÞ whereby hðτÞ is the Heaviside step function. In this study we consider a conservative scenario whereby the typical value corresponds to the front of the wake rather than the main trailing wake. If we consider the latter, the reduction leads to a better optimization but we prefer the conservative route rather than the best case scenario.
Given the flattop laser pulse duration τ p , the maximum Poynting vector and the fluence are related S max ðR int þ 0Þ ¼ Fðτ p Þ=τ p . Therefore substituting the latter to Eq. (3), the effective field experienced by the dielectric is Furthermore, we assume that the fluence dependence on the pulse duration is known and determined by Fðτ p Þ; explicitly, in the examples presented in this study, we adopt the expression in Ref. [24] which is a parametrization of the measured data presented in Ref. [16] Note that although this fluence formula is based on experimental results of laser wavelength λ L ¼ 1 μm, we adopt it also for λ L ¼ 2 μm, due to lack of experimental data with the latter. Next, we introduce one more condition: full overlap between the laser pulse that propagates at cβ gr and a relativistic single microbunch along the interaction length, L geo . This condition is satisfied by setting the pulse duration, which is equal to the delay time τ D [6], to be While the group velocity is an electromagnetic property of the structure, the interaction length is determined by two parameters: the energy gain required Δγ acc and the effective loaded gradient that acts on the charge G Loaded ¼ G 0 − κq. Therefore, the geometric length is given by where m e and e are the electron's mass and charge respectively. It is important to emphasize the difference between G loaded and E eff , the former is the electric field acting on the particle whereas the latter is the electric field that the structure is exposed to at the vacuum-dielectric interface. At this stage, considering Eqs. (4)- (7), the unloaded gradient G 0 and charge q are yet unknown. However, the inter-dependence between the parameters in Eqs. (4)- (7) results in a transcendental equation for G 0 and q. Given the five free parameters: normalized energy gain (Δγ acc ), the vacuum channel's radius ðR int Þ, the group velocity (β gr ), the laser wavelength (λ L ), and the geometrical length of the structure ðL geo Þ, we can determine the gradient given the charge or vice versa, we can calculate the charge given the gradient. In either one of the options the solution is selfconsistent. Given the solution for the gradient and charge, the efficiency ðη 1 ¼ 4κ 1 qG −1 0 ð1 − κqG −1 0 ÞÞ can be evaluated. Moreover, we show in what follows that optimal values exist for the efficiency as well gradient.

III. SELF-CONSISTENT ANALYSIS
A flavor of the general trend of such a self-consistent solution, for a single bunch in the reduced case, is given in Fig. 2 for the parameters in Table I and a range of interaction length values (L geo ). While a detailed comparison between the unreduced and reduced cases is given in Sec. IV, for the sake of simplicity, in what follows we focus on the properties of the latter. Figure 2 reveals a monotonic variation of the charge (dashed blue), whereas both the efficiency (red) and unloaded gradient (dotted green) reach maximum. However, the conditions for maximum efficiency and maximum unloaded gradient cannot be satisfied simultaneously. Moreover, as shown in Fig. 2, it is advantageous to operate in short structures, where higher gradients are available. For example, an unloaded gradient of G 0 ≃ 9 GV=m and loaded gradient of G Loaded ≃ 3 GV=m are FIG. 2. Reduced case quantities for single bunch as a function of geometrical length. Left vertical axis: normalized charge q Ã ¼ κq=G 0 (dashed blue), efficiency normalized to its maximum η 1;max (red)-see derivation in Appendix B. Right vertical axis: unloaded gradient (turquoise) and loaded gradient (dotted green). While the charge varies monotonically, both the efficiency and unloaded gradient reach maximum. Notably, maximum efficiency and maximum gradient (G 0 ) occur for different geometrical lengths, and therefore cannot be satisfied together. feasible in a 0.3 mm long structure with an efficiency of 0.58η 1;max . However, for a relatively long interaction length (L geo ¼ 6 mm), the maximum unloaded gradient and charge are G 0 ¼ 4.7 GV=m, q ¼ 0.25 pC respectively, therefore the loaded gradient is G Loaded ¼ 0.145 GV=m and the efficiency is 0.08η 1;max .
As could be inferred from Eq. (6), when the bunch enters the structure the laser pulse might be half a way in the structure, while no wake has been generated in this region to reduce the exposure of the dielectric to the intense optical field. During this time period ðτ D Þ, the field at the dielectric interface may be higher than the damage threshold fluence (DTF). Therefore, the latter would set an upper limit on the unloaded gradient values (G 0 ). Figure 3 reveals the DTF gradient ðG DTF Þ dependence on the delay time ðτ D Þ, and the corresponding fluence values (F) presented in Eq. (5). Therefore, the fact that the delay time might be of the order of picoseconds, would set an upper limit on the unloaded gradient values. Consequently, it is advantageous to reduce the delay time by either shortening the structure or operating with a higher group velocity.
Before we proceed, we would like to note that since dielectric breakdown is nonrecoverable, a failure at the injector may cause the accelerator structure to be destroyed by high-intensity laser fields. Therefore, it might be critical to build a control system that would block the laser field in case the electron bunch does not arrive at the structure, due to some upstream fault, similar to control systems of electron damping rings.

IV. DISCUSSION
At this stage, we may compare between the reduced case, in which the beam loading effect on the dielectric is considered, to the unreduced case, in which the effect is ignored. The four constraints formulated in Eqs. (4)- (7) reduce the number of free parameters to five: the radius of the vacuum channel R int , the structure's geometric length L geo , the group velocity β gr , the required energy gain Δγ acc , and the driving laser's frequency ω L . The solution, expressed in terms of the charge in a microbunch for the unreduced case, is given by where the pulse duration is τ p ¼ L geo ðβ −1 gr − 1Þ=c, and Fðτ p Þ is the fluence. Both the wake coefficient κ, and its TABLE I. Parameters of the laser and the envisaged structure shown in Fig. 1. (*) Average laser power formula is given by first mode's weight W 1 ≡ κ 1 =κ depend on the vacuum channel radius and the group velocity β gr . According to the set of equations for the reduced case, we can show that the charge in a microbunch is Next, given the charge, the gradient G 0 is calculated based on Eq. (4) or Eq. (7). Therefore, as shown in Appendix C, in both cases (un-reduced and reduced) the loaded gradient is the same (e.g. G Loaded ¼ G ðRÞ Loaded Þ, although occurring for different geometric length. Figure 4 shows the efficiency as a function of the loaded gradient for two vacuum channel radii -0.75λ L (blue) and 0.5λ L (red), and two cases for each radii -reduced (solid curves) and unreduced (dotted curves). First, it is evident that higher maximum efficiency (81% vs 67%) occurs for wider vacuum channel (0.75λ L vs 0.5λ L ). Second, maximum efficiency occurs for higher loaded gradient in the reduced case as compared with the unreduced case. For example, for R int ¼ 0.5λ L , 67% maximum efficiency occurs for loaded gradient of 5 GV=m in the reduced case and only 2 GV=m in the un-reduced case.
In what follows we show that it is advantageous to operate with short (sub mm) structures since the loaded gradient is anticipated to be higher. Table II presents three scenarios (corresponding to different structure lengths), each of which regards both the reduced and unreduced cases. Explicitly, for short structures (scenarios I and II wherein L geo ≤ 0.2 mm), the loaded gradient is one order of magnitude higher than the long structure (scenario III wherein L geo ¼ 5.2 mm). Also, for long structures a variety of other problematic effects arise such as laser defocusing.
Although the loaded gradient is the same for both the reduced and unreduced cases, the former is preferable in terms of microbunch charge and efficiency. The microbunch charge is nearly three times higher in the reduced case for all three scenarios, implying that the unloaded gradient is higher as well. For instance, in a 0.2 mm long structure, q ¼ 240 fC and G 0 ¼ 8.6 GV=m in the reduced case as compared with q ¼ 80 fC and G 0 ¼ 5.7 GV=m in the un-reduced case. Moreover, for the sub mm structures, the reduced case efficiency is higher than the unreduced case. For example, in a 0.2 mm long structure, the reduce case efficiency is 25% higher than the unreduced case. Such high efficiency leads to high charge, a total of 1.5 × 10 6 electrons in a bunch. In order to employ the single bunch theory developed in the current paper, in Ref. [25] we present three different regimes of operation for high energy physics applications: maximum efficiency, maximum charge, and maximum loaded gradient. We demonstrated the tradeoffs between the regimes, that result in loaded gradients of 1 to 6 [GV/m], efficiencies of 20% to 80%, and electrons flux of 10 14 [el/sec], without significant concerns regarding damage threshold fluence.

V. CONCLUSION
In conclusion, we solved a self-consistent set of nonlinear constraints, taking into account the beam loading reduction on the material at the dielectric-vacuum interface. This case, which was referred to as the reduced case, presents unloaded gradient of ∼10 GV=m. While the loaded gradient is anticipated to be the same in both cases (reduced and un-reduced), the maximum efficiency (and higher amount of charge) occur for a higher loaded gradient in the reduced case (5 GV=m) as compared with the unreduced case (2 GV=m).
Furthermore, we showed that it is advantageous to operate in sub millimeter structures, where higher gradients are available and are facilitated by shorter laser pulse and consequently by higher damage threshold fluence gradient (G DTF ).
In addition, maximum gradient does not occur for the same conditions that maximum efficiency occurs. However, the latter, together with the laser field, determine the optimal charge to be accelerated, a total of ∼10 6 electrons in a bunch.

ACKNOWLEDGMENTS
This study was supported by Israel Science Foundation and Rothschild Caesarea Foundation.

APPENDIX A: ENERGY GAIN DEPENDENCE ON THE GROUP VELOCITY
In this Appendix we investigate the energy gain dependence on the group velocity, fluence, and structure's length for the reduced case. Let us assume that the wake's first mode is dominant (κ ≃ κ 1 ), and the pulse duration is τ p ¼ τ D . The latter condition occurs for either a single bunch or for a short train of microbunches duration as compared with the delay time (τ B ≪ τ D ). Therefore the energy gain required is By substituting Eq. (5) and the parameters in Table I, we get the typical values of which are presented in Table III for a single bunch, assuming the wake's first mode is dominant (κ ≃ κ 1 ). The corresponding loaded gradient values are shown in Table III for three cases-each case corresponds to a different regime of the fluence. These results indicate that, due to the fluence dependence on pulse duration, high energy gain is available only for long structures, whereas the corresponding loaded gradient permissible is low. Figure 5(a) shows the energy gain for the three fluence regimes in Eq. (5) as a function of group velocity. Although the energy gain is smaller as the structure is shorter, it is independent of the group velocity. The latter's dependence on the structure's internal radius is shown in Appendix B. Figure 5(b) shows the energy gain for three values of group velocity, as a function of fluence. While the energy gain is a square root of the fluence [Eq. (A1)], the self consistent analysis indicates virtually linear dependence for F > 4.4 J=cm 2 . Moreover, the energy gain discontinuity pertain to the discontinuity in the fluence (shown in Fig. 3).

APPENDIX B: WAKE IN A PARTIALLY LOADED METALLIC CYLINDER
In this Appendix we calculate the wake's weight functions W s of the envisaged configuration, shown in Fig. 1, by solving the nonhomogeneous dispersion relation in the presence of charge distribution in the structure.
The current density of a beam with a total charge q and radius R b , distributed at r i , z i , moving at a velocity v is Let us assume a TM 01 laser mode is copropagating with the beam in the z-direction with a phase velocity β ph equal to the beam's velocity. The longitudinal electric field component of the mode is where , K 0 and I 0 are the modified Bessel function of the first and second kind respectively, T 0 ðΛrÞ ≡ J 0 ðΛrÞY 0 ðΛR ext Þ − J 0 ðΛR ext ÞY 0 ðΛrÞ, and J n and Y n are the n-th order Bessel function of the first and second kind respectively.
In the absence of charge, the homogeneous dispersion relation is where T 1 ðΛrÞ ≡ −J 1 ðΛrÞY 0 ðΛR ext Þ þ J 0 ðΛR ext ÞY 1 ðΛrÞ. The coefficients C 1 , C 2 in Eq. (B2) can be derived from the boundary conditions at the internal radius In the time-domain, the averaged (over r) longitudinal electric field on the beam's radius R b is where XðωÞ≡2ε r T 1 ðΛR int ÞI c ðΓR b Þ=I c ðΓR int Þ, and I c ðxÞ ¼ 2I 1 ðxÞ=x for x > 0 and 1 for x ¼ 0. The first term represents the self-field of the particle, whereas the second term is the contribution of the structure. Using Cauchy's residue theorem for the second term, the wakefield is where hðtÞ is the Heaviside step function, κ ¼ 1=ð2πε 0 R 2 int Þ is the wake coefficient, and the wake's weight functions for the sth mode are wherein the angular frequency ω s is the solution of the homogeneous dispersion relation in Eq. (B3). Please note that P s W s ¼ 1, and that the first fundamental mode is ω L ¼ ω 1 .
As an example, Fig. 6(a) shows the wake's weight functions for 60 modes and their corresponding angular frequencies for λ L ¼ 2 μm, R int ¼ 0.5λ L , ε r ¼ 2.1 and the relativistic particle beam (v ¼ c). For this case, the phase velocity of the laser mode is β ph ¼ 1. As a result the weight functions are independent of the beam radius.
Please note that the projection of the wake on the fundamental (accelerating) mode ðκ 1 ¼ κW 1 Þ represents the maximum efficiency of the structure for a single bunch For the self-consistent formulation, the group velocity dependence on the radii must be presented. Despite the fact that in this section we develop the nonhomogeneous solution of the wake, it warrants to point out that this solution contains information about the homogeneous solution, specifically about the group velocity in the system, Figure 6(b) shows the dependence of the group velocity on the internal radius. For each internal radius, the external radius is calculated via the dispersion relation, and then the group velocity is derived.

APPENDIX C: LOADED GRADIENT IN THE REDUCED AND UNREDUCED CASES
The set of Eqs. (4)- (7), describing the bunch's charge q in the reduced case, could be written in terms of the charge in the un-reduced case [Eq. (9)], namely Keeping in mind that the effective field in the dielectric [Eq. (3)] implying that the loaded gradients are the same for the reduced and unreduced cases. Therefore, the unloaded gradient in the reduced case is expected to be higher, and so does the amount of charge in the single bunch-as was shown in Table II.