Electron Beam Guiding by a Laser Bessel Beam

PHYSICAL REVIEW ACCELERATORS AND BEAMS 23, 081301 (2020)

Electron beam guiding by a laser Bessel beam

Levi Schächter 1,* and W. D. Kimura 2 1Technion–Israel Institute of Technology, Haifa 32000, Israel 2STI Optronics, Inc., Bellevue, Washington 98005, USA

(Received 7 April 2020; accepted 15 June 2020; published 3 August 2020)

We investigate the dynamics of electrons counterpropagating along a radially polarized optical Bessel beam (OBB). (i) It is shown that a significant fraction of the electrons can be transversally trapped by the OBB even in the case of “unmatched” injection. Moreover, (ii) these transversally trapped particles (TTPs) can be transported without loss along many thousands of wavelengths. As long as there is full longitudinal overlap between the electrons and laser pulse, this transport distance is limited only by the length of the OBB region. (iii) The unique profile of the transverse field components facilitates guiding either azimuthally symmetric pencil beams or annular beams. Space charge tends to totally suppress the annular beams, and it reduces the amount of charge trapped on axis for pencil beams. (iv) Assessment of the emittance of the TTPs alone reveals typical values of 10–50 pm. In fact, our simulations indicate if we trace the emittance of those particles that are trapped from the input to the output of the OBB, we find that this emittance is conserved. (v) We developed an analytic model whereby we average over the fast oscillation associated with the counterpropagating electrons and OBB. The resulting Hamiltonian has a Bessel 2 potential J1ðuÞ, which, when operated in the linear regime near equilibrium, causes rotation of the phase space. A Kapchinskij-Vladimirskij beam-envelope equation is derived including space-charge and emittance effects. Relying on conservation of the longitudinal canonical momentum, the energy spread in the interaction region is determined in terms of the OBB intensity and the electron energy.

DOI: 10.1103/PhysRevAccelBeams.23.081301

I. INTRODUCTION beam applications where conventional magnetic or electrostatic focusing cannot be used. Guiding charged particles in vacuum along significant The notion of using optical beams to manipulate the distances plays a pivotal role in many systems such as trajectory of electrons is certainly not new. For example, radiation sources, accelerators, and electron microscopes. there are various ways high-energy electrons can be Essentially, there are three types of transport systems: focused with optical beams that rely on nearby structures electrostatic systems are used primarily in electron micro- [1,2]. The fields created within two colliding laser pulse can scopes and magnetostatic elements are used in high-energy change the energy of subrelativistic electrons [3]. Optical accelerators. In a small, but by no means less important micromanipulation of micron-sized particles using Bessel fraction, radio-frequency quadrupoles are used to both optical beams has been demonstrated [4]. What distin- transport and accelerate either electrons or ions. In the guishes our 8scheme from these other methods are (i) it present study, we demonstrate that a laser Bessel beam can occurs in free space with no nearby structures, (ii) it is transport efficiently a counterpropagating electron beam in designed to guide relativistic electrons, and (iii) it can be vacuum. This may provide a highly useful alternative easily scaled to long interaction lengths, e.g., many meters. means for guiding electrons, especially for situations where Before we investigate the dynamics of the transport by an it may be undesirable or impractical to have relatively large extended focus Bessel beam, it is useful to briefly review structures, such as magnets or metal components, near the the essentials of acceleration of electrons by a focused electron beam. An example might be ultracompact electron beam in vacuum. The first hint regarding electron acceleration by a focused laser beam is in the work of Boivin and Wolf [5], where they investigated the electromagnetic field *Corresponding author. (Gaussian beam) in the vicinity of the focal point and found [email protected] that a significant longitudinal electric field develops. In their words, “It seems plausible that such strong longi- Published by the American Physical Society under the terms of tudinal fields could be used for accelerating charged the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to particles. However, because of the considerable complexity the author(s) and the published article’s title, journal citation, of the field in the focal region, the practical feasibility of and DOI. such a proposal must await a more detailed study.”

2469-9888=20=23(8)=081301(17) 081301-1 Published by the American Physical Society LEVI SCHÄCHTER and W. D. KIMURA PHYS. REV. ACCEL. BEAMS 23, 081301 (2020)

The next important step in our context was the configu- facilitated by the finite extent of the interaction and the ration suggested by Steinhauer and Kimura [6]. Its essence momentary capture of the electron over this extended was threefold: (i) a radially polarized laser beam; (ii) an length. Implied in this work, as it pertains to this paper, aligned axicon focus that generates an extended focal region is that CAS is enabled by the fact that diffraction of the (Bessel beam); and (iii) the approximate matching of focused laser beam leads to a significant component of its particle and laser phase velocities as both propagate spectrum having a phase velocity equal to or slower than c. in the same direction. They showed that acceleration is To conclude this brief overview, at the high laser feasible provided the interaction region is terminated intensities involved, radiation-reaction effects on electron a ≫ 1 abruptly at both ends. A similar conclusion for a beams were considered [12]. Provided and the initial γ ≫ 1 Gaussian beam was reached by Esarey, Sprangle, and relativistic factor ini , the dynamics divides into three Krall [7]. A couple of years later, the same group published regimes: (i) For relatively low-energy electrons, radiation another study [8] that is closely related to that presented by damping effects are negligible. (ii) At higher electron 2γ a Because of the apparent similarity between these two (iii) For ini , the radiation-reaction induces longi- studies, it is important to emphasize from the very beginning tudinal electron trapping. This trapping process is stable the differences between these studies and the work presented with respect to the spatial properties of the electron beam in this paper. The primary interest of these two studies was and results in a significant energy loss of the electrons. In our case, we aim to transversally confine the beam with the on creating a longitudinal accelerating force; thus, they both lowest possible laser power; therefore, we assume that the assumed copropagating electron and laser beams. radiation reaction is negligible. Our focus is on counterpropagating beams, where we are Our idea stems from a previous concept [13] of gen- not interested in utilizing the longitudinal force, but rather erating x-ray radiation similar to channeling radiation in we are using the fact that the particles will experience a crystals, but in vacuum, by counterpropagating a beam of rapidly oscillating transverse force. As we will show, this electrons through an optical Bessel beam (OBB). For the effective transverse force can guide the electron beam. proof of principle analysis, we initially imposed some very Contrary to Ref. [8], which employs the paraxial approxi- stringent constraints on the model in order to facilitate the mation, we employ the exact solutions of TM01 in vacuum, analytic solution. Recently, after we alleviated most of the although the finite cross section of the electron beam constraints, rather than investigating the spontaneous radi- (e-beam) implies that only a small number of transverse “ ” ation emitted, we decided to examine more closely the Bessel periods are used in practice. In other words, we trajectories of the test particles ignoring the spontaneous employ an ideal Bessel beam. radiation. The essence of this investigation is presented A different approach was adopted by Hora et al. [9]. here: We found the OBB is able to transport electrons over They considered the feasibility of acceleration of individual its entire extent (many thousands of laser wavelengths). electrons by Hermit-Gaussian laser beams, and it was The diffraction-free character of Bessel beams [14] means shown that the transition from elastic to inelastic scattering that Rayleigh range limits no longer apply and the length of occurs for a laser normalized amplitude a ¼ eE0λ0= 2 the Bessel beam can be made, in principle, as long as 2πmc > 0.1 and very small incident angle θ ≪ π=4, desired by simply increasing the diameter of the laser beam where e and m represent the charge and the rest mass, being focused by the axicon. We also found that typical respectively, of an electron, c is the speed of light in normalized transverse emittances of less than 100 pm were vacuum, and λ0 and E represent the wavelength and the calculated for the transversally trapped particles (TTPs). In amplitude, respectively, of the laser. Moreover, later it was fact, we found that their emittance is preserved along the demonstrated in simulations [10] that, for an initial electron interaction length. energy of 26 MeV and some stringent initial conditions To thoroughly investigate the processes involved, we (among them a ≥ 100), emerging 1.5 GeV electrons are developed an analytic model whereby we average over feasible. This is in comparison with 32 MeV electrons for the fast oscillation associated with the counterpropagating the case of inelastic scattering. electron and Bessel laser beam. The resulting Hamiltonian 2 Further support of this result was provided 3 years later has a radial Bessel potential J1ðuÞ, which, when operated in by Wang et al. [11], whereby the characteristics and the linear regime near equilibrium, causes rotation of the essential conditions under which an electron in a vacuum phase space. This allows us to show the analytic conditions and a laser beam can undergo a capture and acceleration needed for the emittance to be conserved. A Kapchinskij- scenario (CAS) were shown. They confirmed that, if Vladimirskij beam-envelope equation is derived including a ≃ 100, the electron can be captured, primarily longitu- space-charge and emittance effects. In the OBB, the beam dinally, and violently accelerated to energies of the order of radius at equilibrium is reconfigured according to the 1 GeV, corresponding to an effective acceleration gradient intensity of the OBB, electron energy, and space charge. 1 TV=m. As in the case of Ref. [2], the net energy gain is Relying on conservation of the longitudinal canonical

081301-2 ELECTRON BEAM GUIDING BY A LASER … PHYS. REV. ACCEL. BEAMS 23, 081301 (2020) momentum, the energy spread in the interaction region is determined in terms of the OBB intensity and the electron energy. We show that there is an inherent advantage to using a counterpropagating electron beam.

II. DESCRIPTION OF ANALYSIS CONFIGURATION Let us first generate a clear picture of the configuration we intend to investigate. We conceive a radially “ ” polarized annular plane wave, which extends between FIG. 1. An axicon lens converts a radially polarized annular R ≤ r ≤ R , and is focused by an axicon lens such that R L=c namely, i max max; i is the radial coordinate of the p and, further, that the axicon lens is exposed to ith particle. Another aspect that is ignored is associated F F P τ = a fluence max below its damage threshold [15] ¼ in p with the axicon lens, which has a small central hole in order π R2 − R2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 τ ¼ ct=λ0; x¯i ¼ xi=λ0; y¯i ¼ yi=λ0; z¯i ¼ zi=λ0; ψ i ¼ 2π x¯i þ y¯i sin θ0; ¯ ¯ ¯ ¯ ¯ 2 Ui ¼ πEJcðψ iÞ sinðχiÞ; Vi ¼ EJ0ðψ iÞ cosðχiÞ; E ¼ eE0λ0=mc ;JcðuÞ ≡ 2J1ðuÞ=u:

081301-3 LEVI SCHÄCHTER and W. D. KIMURA PHYS. REV. ACCEL. BEAMS 23, 081301 (2020)

The first three equations represent the particles’ momen- affected by the intensity of the OBB. The conservation of tum dynamics, the fourth the individual’s particle energy canonical longitudinal momentum, as reflected in Eq. (6), dynamics, and the fifth the phase variation as experienced by indicates that, for a narrow pencil beam such that the particle along the wave. In the equations of motion as J0ð2πr¯i sin θ0Þ ≃ 1, the energy of the particle can be solved formulated in Eq. (2), we omitted the fact that the force terms analytically. In Appendix B, we show that the initial energy Δγ2 are exerted only in the region where the OBB exists, spread ini increases in the interaction region according to 0

081301-4 ELECTRON BEAM GUIDING BY A LASER … PHYS. REV. ACCEL. BEAMS 23, 081301 (2020)

2πr¯ θ p ; eq;s sin 0 ¼ 1;s ð11Þ TABLE I. The default values of the parameters used in the various simulations. where p1;s are the relevant zeros of the first-order Bessel ¯ −5 function J1ðp1;sÞ ≡ 0 and s is an integer s ¼ 0; 1; 2… with λ0 ¼ 1.064 ½μm E ¼ 1.2 × 10 γ0 ¼ 20 _ −4 its limit determined by the geometric radius of the electron Er;0 ¼ 33 ½MV=m L ≃ 1.0 ½m jγið0Þx¯ið0Þj ≤ 10 S 147 4 = 2 −3 beam and the angle of the laser OBB, i.e., p1;0 ¼ 0, z;0 ¼ . ½MW cm F ≃ 1.0 ½J=cm jγi − γ0;ij ≤ 10 γ0;i θ 10 π=180 R ≃ 5λ N 360 p1;1 ¼ 3.832, p1;2 ¼ 7.016, p1;3 ¼ 10.173…. Note that 0 ¼ ð Þ b 0 mp ¼ in the longitudinal direction, according to the conservation of the canonical longitudinal momentum in the z the process repeats itself. It is for this reason that we direction, the electron oscillates. Furthermore, we can N consider one period of the exerted field that contains mp assume that a small transverse oscillation ðδr¯s;iÞ around microparticles [19–20]. As a typical example, we can the equilibrium does occur, namely, r¯i ¼ r¯ ;s þ δr¯s;i; eq conceive a 3-mm-long pulse of 109 electrons, and the therefore, ignoring the longitudinal oscillation as well as exerted force has a 3 μm period; therefore, in one period, the energy modulation, γiðτÞ¼γið0Þ ≡ γi;0, for relativistic there are 106 electrons. These can be divided into 1000 particles ðp¯z;i; γi ≫ 1Þ, the radial equation of motion simplifies to read microparticles, each one containing 1000 electrons. Throughout all simulations that follow, the relative error, d2 defined based on Eq. (8), did not exceed 10−9%. α χ u δr¯ 0; du2 þ s;i sinð 0;i þ Þ s;i ¼ ð12Þ In Fig. 2(a), the blue circles show how the transverse phase space evolved after half a meter. Several facts are evident: ¯ 2 −1 where αs;i ≡ EJ0ðp1;sÞ½4πγi;0cos ðθ0=2Þ and u ≡ 2 4πcos ðθ0=2Þct=λ0. This is the Mathieu equation that can have periodic (thus stable) solutions. In Appendix C, 1 x 10-3 we show that a stable solution is possible for s ¼ 1 if α1;i ¼ (a) z = L / 2 ¯ 2 −1 ½J0ðp1;0ÞE½γ0;i4πcos ðθ0=2Þ ≤ 0.454 or, explicitly, 5 x 10-4 eE λ γ ≥ 0 175 0 0 : 0 . mc2 ð13Þ 0 For most practical regimes of interest, the right-hand side of Eq. (12) is much smaller than one; therefore, for relativistic electrons, this constraint is always satisfied. -5 x 10-4 This is the first important result of this study, since it indicates that, within the framework of a linear approxi- Normalized Transverse Momentum mation, electrons moving in an OBB can follow stable -1 x 10-3 s 0 -10 -5 0 5 10 trajectories which can form a pencil beam ð ¼ Þ or x / λ annular beam ðs>0Þ. 1 x 10-3 V. DYNAMICS WITHOUT SPACE CHARGE (b) z = L

The important result mentioned above and analyzed sub- 5 x 10-4 ject to a linear approximation is next investigated numeri- cally relying on the exact set of equations in Eq. (2) and employing the set of parameters specified in Table I. 0 Figure 2 shows the transverse phase space when the space-charge effect is ignored. As reference, we show -4 the initial (z ¼ 0) phase space represented by red circles in -5 x 10

both frames. Each circle represents a microparticle that Normalized Transverse Momentum contains many electrons that are “glued together” all along -3 -1 x 10 the interaction length. It is tacitly assumed that, by splitting -10 -5 0 5 10 N x / λ the entire beam into mp microparticles, the characteristic features of the entire ensemble will be preserved. It should FIG. 2. Phase space in the space-charge-free case. The trapping be made crystal clear that, while we focus here on the of particles in the vicinity of the equilibrium points is evident: x¯ ∼ 0 x¯ ∼ 3 512 ’ transverse dynamics, the latter is solved self-consistently j eqj and j eqj . . The initial beam s phase space does x¯ ∼ 6 43 x¯ < 5 with the longitudinal dynamics. And in the longitudinal not cover the third equilibrium ðj eqj . Þ since j j ; direction we consider one period of the exerted field, therefore, electrons are not trapped in this state. (a) At since at a given location, during the next period of time, z ¼ L=2. (b) At z ¼ L.

081301-5 LEVI SCHÄCHTER and W. D. KIMURA PHYS. REV. ACCEL. BEAMS 23, 081301 (2020)

Already at this stage, part (20%) of the microparticles shown) the phase space at the input to be jx¯ij ≤ 2, then no J x¯ ≡ 0 (representing the electrons) are trapped ½ 1ð eqÞ on axis particles are trapped in the vicinity of the second equilib- x¯ ≃ 0 “ ” eq forming a pencil beam. Another fraction (29%) is rium point. This is the second important result of this study, trapped in the vicinity of the second zero of the Bessel since it indicates that the OBB supports the generation of x¯ ≃ 3 512 function corresponding to j eqj . , indicating the gen- either a pencil or an annular beam for rather large eration of an “annular beam.” (The parameters for this amplitudes. Thus, the stable trajectories are not limited particular examplewere chosen to illustrate how it is possible to the linear regime. for both a pencil and an annular beam to be guided simulta- A third important result is associated with the emittance. z L ε ≃ neously by the OBB. Although actual applications would Its value for the pencil beam at the output ð ¼ Þ is n 53 2 typicallyguideonlyonetypeofbeam,thisparticularexample . pm. Tracing backwards to those electrons that made it will be useful in understanding the effects of space charge as to the end of the OBB and calculating their emittance at z L=2 ε ≃ 53 5 discussedshortly.)Athirdgroupisnottrapped,butratherthey ¼ , we find that the emittance is n . pm, which slowly “diffuse” away from the center. Not shown is a fourth clearly shows that the emittance of the TTPs is conserved group of electrons that were scattered in the first few for at least the second half meter. The slight deviation can centimeters of the interaction. be attributed to the usage of a relatively small number of In Fig. 2(b), only the trapped electrons (49%) are left, microparticles (70) affecting the statistics. and this is a representative picture for the last quarter of the interaction region. Note that there are no trapped micro- VI. DYNAMICS WITH SPACE CHARGE x¯ ≃ 6 43 particles in the third equilibrium point j eqj . . In fact, For guiding a significant number of monoenergetic only microparticles that are initially in the vicinity of the electrons, we need to confine them as much as possible equilibrium are trapped. For example, if we reduce (not on the scale of the OBB wavelength with the limit being the

1 x 10-3 1 x 10-3 (b) z = L / 2 (a) z = L / 2 -5 κ = 4.2x10-6 κ = 4.2x10 sc -4 sc 5 x 10-4 5 x 10

0 0

-4 -5 x 10-4 -5 x 10 Normalized Transverse Momentum Normalized Transverse Momentum

-3 -3 -1 x 10 -1 x 10 -10 -5 0 5 10 -10 -5 0 5 10 x / λ x / λ

1 x 10-3 1 x 10-3 z = L (c) z = L (d) κ = 4.2x10-6 κ = 4.2x10-5 sc sc 5 x 10-4 5 x 10-4

0 0

-5 x 10-4 -5 x 10-4 Normalized Transverse Momentum Normalized Transverse Momentum

-1 x 10-3 -1 x 10-3 -10 -5 0 5 10 -10 -5 0 5 10 x / λ x / λ

−6 −6 FIG. 3. (a) κSC ¼ 4.2 × 10 , z ¼ L=2. (c) κSC ¼ 4.2 × 10 , z ¼ L. Space charge tends to reduce the trapping when comparing κ 4 2 10−6 κ 0 κ 4 2 10−5 z L=2 κ 4 2 10−5 sc ¼ . × with the zero space-charge case sc ¼ as revealed in Fig. 2. (b) SC ¼ . × , ¼ . (d) SC ¼ . × , z ¼ L. Elevating the charge by one order of magnitude suppresses entirely the annular beam.

081301-6 ELECTRON BEAM GUIDING BY A LASER … PHYS. REV. ACCEL. BEAMS 23, 081301 (2020) repelling force the electrons exert on each other. When this now is twofold: expand the equations of motion to account force is greater than or equal to the radial force exerted by for the space-charge effect and assess the effect of the latter. the OBB, the electron beam will tend to blow up. Our goal The former is explained in detail in Appendix D, and in

1 x 10-3 1 x 10-3 z = L / 2 (a) z = L / 2 (b) x = 2 λ x = 1 λ max max -4 -4 5 x 10 κ = 4.2 x 10-6 5 x 10 κ = 4.2 x 10 -6 sc sc p = 10-4 p = 4 x 10 -4 x,max x,max 0 0

-5 x 10-4 -5 x 10-4 Normalized Transverse Momentum Normalized Transverse Momentum

-1 x 10-3 -1 x 10-3 -10 -5 0 5 10 -10 -5 0 5 10

x / λ x / λ

1 x 10-3 1 x 10-3 (c) z = L z = L (d) x = 2 λ x = 1 λ max max -4 -4 5 x 10 κ = 4.2 x 10-6 5 x 10 κ = 4.2 x 10-6 sc sc p = 10-4 p = 4 x 10-4 x,max x,max 0 0

-5 x 10-4 -5 x 10-4 Normalized Transverse Momentum Normalized Transverse Momentum

-1 x 10-3 -1 x 10-3 -10 -5 0 5 10 -10 -5 0 5 10

x / λ x / λ

1 x 10-4 4 x 10-4 (e) (f)

5 x 10-5 2 x 10-4

0 0

-5 x 10-5 -2 x 10-4 Normalized transverse momentum Normalized transverse momentum -1 x 10-4 -4 x 10-4 -2.0 -1.0 0.0 1.0 2.0 -1.0 -0.5 0.0 0.5 1.0 x / λ x / λ

−6 −4 −6 FIG. 4. Guiding of a pencil beam of electrons by OBB. (a) κSC ¼ 4.2 × 10 , z ¼ L=2, px;max ¼ 10 . (c) κSC ¼ 4.2 × 10 , z ¼ L, p 10−4 κ 4 2 10−6 x;max ¼ . The low space-charge ( sc ¼ . × ) beam is significantly squeezed spatially, but the spread of the transverse momentum increases by almost one order of magnitude. The initial emittance is 0.1 nm. (e) Phase space at the input is marked with red, whereas in green are labeled particles at the input that eventually become trapped and make it to the output. A well-defined region ðjx¯ij ≤ 1Þ in the phase space is occupied by only those electrons that will become trapped, implying that the emittance of the trapped ¯ ¯ −4 electrons is within a good approximation constant. (b),(d),(f) The same parameters as (a),(c),(e) except xmax ¼ 1.0, px;max ¼ 4 × 10 . Note that many fewer electrons are trapped, since the selection is more stringent, but the emittance of the trapped particles (0.2 nm) is significantly lower than that of the entire ensemble at the input (1 nm).

081301-7 LEVI SCHÄCHTER and W. D. KIMURA PHYS. REV. ACCEL. BEAMS 23, 081301 (2020) what follows we discuss the main findings from our TABLE II. Emittance in the various regimes considered. In numerical investigation. In the framework of the approxi- brackets is the percentage of the TTPs near the axis. mation specified in Appendix D, space charge can be κ ≡ 2r N ε εpencil ½pm @ εpencil ½pm @ represented by a single parameter sc el Δ, where ½pm@ trapped trapped −15 κ z 0 z L z L=2 r 2 8 10 sc ¼ ¼ ¼ el ¼ . × ½m is the classical radius of the electron and NΔ represents the number of electrons per unit length 0 35.5 53.2 (19.4%) 53.5 (19.4%) in the direction of motion. In zero order, for stability, we 4.2 × 10−6 35.5 45.1 (19.2%) 48.5 (19.2%) require that the transverse force exerted by the OBB is 4.2 × 10−5 35.5 17.4 (11.1%) 21.1 (11.1%) dominant, which is shown in Appendix D to be equivalent 4.2 × 10−6 14.4 34.4 (39.7%) 29.2 (39.7%) 2 4 2 10−6 to πE0 ≫ eNΔ=4πε0λ0γ and is readily satisfied even at . × 28.5 10 (13.1%) 10 (13.9%) moderately relativistic energies, e.g., γ ≃ 20. The first observation we make is that space-charge momentum coordinate its spread is almost one order of effects tend to suppress the annular beams: Figs. 3(a) magnitude larger—yet the electrons remain trapped. and 3(c) correspond to a small space-charge parameter In Fig. 4(e), we see the transverse phase space at the κ 4 2 10−6 z L=2 z L sc ¼ . × at ¼ and ¼ , respectively. We input marked with red, whereas in green we labeled those observe that fewer electrons (36% compared to 49% in the particles at the input that eventually became trapped and space-charge-less case in Fig. 2) are trapped and those made it all the way through to the output. Clearly, there is a trapped have a slightly larger spatial spread. Elevating well-defined region ðjx¯ij ≤ 1Þ in the phase space occupied the number of electrons by one order of magnitude, by only those electrons that will become trapped. This κ 4 2 10−5 sc ¼ . × , as evident in Figs. 3(b) and 3(d), the brings us to the fourth important result of this study that the annular beam is absent even in the middle of the interaction emittance is not reduced, but rather it is conserved; in other length. For the pencil beam, 11% of the particles are words, the emittance of the trapped electrons at the input is trapped and are transported over the 1-m-long interaction within a good approximation constant along at least the z L region. The emittance at the output ð ¼ Þ of the pencil second half of the interaction region. ε ≃ 45 1 beam is n . pm; whereas, tracing backwards, the The transverse phase space at the output of Fig. 4(c) ε ≃ 48 5 trapped electrons had an emittance of n . pm at leads us to examine the case when the initial transverse z L=2. This is for the low space-charge case. For the ¼ phase space is squeezed spatially ðjx¯ij ≤ 1Þ but enlarged high space-charge case, the emittance is a factor of 2.5 −4 along the transverse momentum p¯x;i ≤ 4 × 10 axis— ε z L ≃ 17 4 ε z L=2 ≃ 21 1 j j lower, nð ¼ Þ . pm and nð ¼ Þ . pm. see Figs. 4(b) and 4(d). In the middle of the interaction Although to a lesser extent, in this case, too, the emittance region, there are very few (15%) microparticles that of the TTPs may be considered essentially constant in the remained untrapped [Fig. 4(b)], and this number diminishes second half of the interaction region. We should note that at the end of the guiding process [Fig. 4(d)]. In this case, the the effects of possible radiation loss have not been included emittance of the TTPs at the output is 10 pm. Figure 4(f) in this analysis. Furthermore, there may be a self-filtering reveals that the relatively large transverse momentum process occurring in which low emittance electrons tend to spread as well as the space-charge effect makes the origin remain well guided, whereas higher emittance electrons of the trapped particles more complex compared to the case tend to be lost during transit along the first half of the 1 m in Fig. 4(e), but the emittance remains unchanged (10 pm). 106 OBB length ( wavelengths). Again, in the framework of our numerical analysis, the Next, we focus our attention to a regime where only emittance of the trapped particles seems to be conserved. In a pencil beam is guided. For this purpose, we limit the Table II, we summarize the normalized emittances in the x¯ ≤ 2 initial spatial spread of the electrons to j ij and various regimes. we consider the case of a weak space-charge effect κ 4 2 10−6 p¯ ≤ ð sc ¼ . × Þ; the initial momentum spread is j x;ij 1 10−4 VII. ANALYTIC MODEL: SLOWLY × . Figures 4(a) and 4(c) show the phase-space VARYING AMPLITUDE distribution at z ¼ L=2 and z ¼ L, respectively. These results reveal that, despite space-charge effects, there is A deeper insight into the preceding numerical results may no major change in the transverse phase space along be obtained by developing a Hamiltonian that accounts for most of the last half-meter guidance “tunnel” formed by averaging over the fast oscillations associated with the the OBB [see Figs. 4(a) and 4(c)] and the TTPs are electron and the wave propagating in opposite directions. transported without loss (40% of the initial particles are InAppendixE,weshowthattheHamiltonianassociatedwith trapped). It is interesting to note that the electrons, which the radial motion ρi ¼ αiJ1ðCiÞ sinðχi;0 þ ψÞþCi ≃ Ci are guided along the second half of the interaction, have an with the normalized time variable ψ ¼ 2πðct=λ0Þð1 þ emittance comparable with their value in the previous cases cos θ0Þ and the normalized radius ρi ¼ 2πðri=λ0Þ sin θ0, [εnðz ¼ LÞ ≃ 34 pm and εnðz ¼ L=2Þ ≃ 29.2 pm]. While as well as the normalized coupling coefficient 2 −1 clearly the phase space is squeezed spatially, along the αi ≡ ðeE0λ0=mc Þ½2πγi;0ð1 þ cos θ0Þ ≪ 1, reads

081301-9 LEVI SCHÄCHTER and W. D. KIMURA PHYS. REV. ACCEL. BEAMS 23, 081301 (2020) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ε =K ε ≫ ε; 2 1 2 2 sc sc trapped from input to output, we find, within an error R ¼ ε þ ε þ ε ≃ ð21Þ −1=2 b;eq K sc sc ε=K ε ≪ ε; ∝ N sc ð mp Þ associated with a relatively small number of microparticles (70–40), that this emittance is conserved. which indicates that, if the equilibrium is space-charge To thoroughly investigate the processes involved, we ε ≫ ε dominated ð sc Þ, the beam radius is independent of developed an analytic model whereby we averaged over the R2 ∝ ε the emittance, whereas, in the opposite case, b;eq . fast oscillation associated with the counterpropagating electron and Bessel laser beam. The resulting 2 IX. CONCLUSIONS Hamiltonian has a Bessel (radial) potential J1ðuÞ which, when operated in the linear regime and near equilibrium, In conclusion, we considered the trajectories of electrons causes rotation of the phase space, and using this we counterpropagating along a radially polarized optical 2 2 determined the analytic condition ðeE0λ0=mc γ¯Þ ¼ Bessel beam. It was shown that a significant fraction of 2 x_2 x2 −1 for the emittance to be conserved. the electrons can be transversally trapped by the OBB even h i ih i i Finally, our choice of counterpropagating electrons is not in the case of “unmatched” injection, i.e., an arbitrary phase with respect to the optical field. Moreover, these trans- arbitrary. In Appendix G, we show that, while the stability versally trapped particles can be transported without loss is identical in both cases, namely, the oscillation follows over more than 105 laser wavelengths (1 μm). The distance the same frequency, the transverse confinement (kick) is θ =2 2 is limited only by the length of the OBB as long as there is ð 0 Þ weaker in the copropagating case. full longitudinal overlap between the electrons and laser pulse. The unique profile ½J1ðuÞ of the transverse field ACKNOWLEDGMENTS components supports the propagation of annular beams in addition to an azimuthally symmetric pencil beam in the This study was supported by the Israel Science vicinity of the zeros of the Bessel function, namely, Foundation (ISF) and STI Optronics, Inc. Internal Research and Development. J1ðuÞ¼0. Qualitatively, space charge tends to suppress annular beams, and it reduces the amount of charge trapped on axis. APPENDIX A: EQUATIONS OF MOTION Assessment of the emittance of the TTPs alone reveals (see Table II) typical values of 10–50 pm for the parameters Fromtheexplicitexpressionsofthepotentials[Eq.(1)]that specified in Table I. In fact, our simulations indicate that, if satisfy the Lorentz gauge, we may derive the three nonzero we trace the emittance of those particles that are eventually field components in the cylindrical coordinates system:

∂ ðOBBÞ E0 ω0 z μ0Hϕ ¼ − Az ¼ − J1 r sin θ0 sin ω0 t þ cos θ0 ; ∂r c sin θ0 c c ∂ E θ ω z ðOBBÞ 0 cos 0 0 Er ¼ − Φ ¼ J1 r sin θ0 sin ω0 t þ cos θ0 ; ∂r sin θ0 c c ∂ ∂ ω z E − AðOBBÞ − ΦðOBBÞ E J 0 r θ ω t θ : z ¼ ∂t z ∂z ¼ 0 0 c sin 0 cos 0 þ c cos 0 ðA1Þ

With these field components, we may write the equation of motion of the ith electron in the Cartesian coordinate system:

d e γ β − E − cμ β H ; dt ð i x;iÞ¼ mc ð x;i 0 z;i y;iÞ d e γ β − E cμ β H ; dt ð i y;iÞ¼ mc ð y;i þ 0 z;i x;iÞ d e γ β − E cμ β H − cμ β H ; dt ð i z;iÞ¼ mc ð z;i þ 0 x;i y;i 0 y;i x;iÞ dγ e i − cβ E cβ E cβ E ; dt ¼ mc2 ð x;i x;i þ y;i y;i þ z zÞ z χ ω t i θ : i ¼ 0 þ c cos 0 ðA2Þ

081301-10 ELECTRON BEAM GUIDING BY A LASER … PHYS. REV. ACCEL. BEAMS 23, 081301 (2020)

The last expression determines the phase experienced by the ith particle. Keeping in mind that in cylindrical coordinates Ex ¼ Er cos ϕ and x ¼ r cos ϕ, we get Ex ¼ðx=rÞEr. Similarly, Ey ¼ðy=rÞEr, Hx ¼ −Hϕ sinϕ ¼ −ðy=rÞHϕ, and Hy ¼ Hϕ cos ϕ ¼ðx=rÞHϕ. Thus, we have

d e xi ðγiβx;iÞ¼− ðEr;i − η0βz;iHϕ;iÞ ; dt mc ri d e y γ β − E − η β H i ; dt ð i y;iÞ¼ mc ð r;i 0 z;i ϕ;iÞ r i d e x y γ β − E η β i H η β i H ; dt ð i z;iÞ¼ mc z;i þ 0 x;i r ϕ;i þ 0 y;i r ϕ;i i i d e xi yi γi ¼ − βx;i Er;i þ βy;i Er;i þ βzEz ; dt mc ri ri d χ ω 1 β θ : dt i ¼ 0ð þ z;i cos 0Þ ðA3Þ

At this stage, we may substitute the explicit expressions for the field components derived in Eq. (A2), and, defining ω0 ψ i ¼ c ri sin θ0 as the argument of the Bessel functions, we find

d e E0 xi ðγiβx;iÞ¼− ðcos θ0 þ βz;iÞ J1ðψ iÞ sin χi ; dt mc sin θ0 ri d e E0 yi ðγiβy;iÞ¼− ðcos θ0 þ βz;iÞ J1ðψ iÞ sin χi ; dt mc sin θ0 ri d e xi yi E0 ðγiβz;iÞ¼− E0J0ðψ iÞ cos χi − βx;i þ βy;i J1ðψ iÞ sin χi ; dt mc ri ri sin θ0 d e xi yi E0 cos θ0 γi ¼ − βx;i þ βy;i J1ðψ iÞ sin χi þ βz;iE0J0ðψ iÞ cos χi ; dt mc ri ri sin θ0 d χ ω 1 β θ : dt i ¼ 0ð þ z;i cos 0Þ ðA4Þ

Defining the normalized quantities as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 τ ¼ ct=λ0; x¯i ¼ xi=λ0; y¯i ¼ yi=λ0; z¯i ¼ zi=λ0; ψ i ¼ 2π x¯i þ y¯i sin θ0 ; ¯ ¯ ¯ ¯ ¯ 2 Ui ¼ πEJcðψ iÞ sinðχiÞ; Vi ¼ EJ0ðψ iÞ cosðχiÞ; E ¼ eE0λ0=mc ;JcðuÞ ≡ 2J1ðuÞ=u;

wed finally get APPENDIX B: CANONICAL LONGITUDINAL γ x¯_ − θ z¯_ x¯ U¯ ; MOMENTUM AND ENERGY SPREAD dτ ð i iÞ¼ ðcos 0 þ iÞ i i d The conservation of the longitudinal canonical momen- γ y¯_ − θ z¯_ y¯ U¯ ; γ dτ ð i iÞ¼ ðcos 0 þ iÞ i i tum [Eq. (6)] implies that for relativistic particles, i (and χ d 1 d i) can be solved analytically: _ 2 2 ¯ ¯ ðγiz¯iÞ¼ ðx¯ þ y¯ Þ Ui − Vi; ¯ dτ 2 dτ i i E sin½χið0Þþ2πð1 þ cos θ0Þτ γiðτÞþ dγ 1 d 2πð1 þ cos θ0Þ i 2 2 ¯ _ ¯ ¼ − cos θ0 ðx¯i þ y¯i Þ Ui − z¯iVi; ¯ dτ 2 dτ E sin½χið0Þ ¼ γið0Þþ : ðB1Þ dχ 2πð1 þ cos θ0Þ i _ ¼ 2πð1 þ z¯i cos θ0Þ; ðA5Þ dτ From this last expression, we may conclude that, if the initial energy and phase are statistically independent, the which constitutes the fundamental set of equations intro- average energy is independent of the OBB, i.e., hγiðτÞi ¼ duced in Eq. (2) and used throughout this study. hγið0Þi, whereas the second moment is given by

081301-11 LEVI SCHÄCHTER and W. D. KIMURA PHYS. REV. ACCEL. BEAMS 23, 081301 (2020)

0.5 3.79

0.4 3.78

α ()k 0.3 α α 2()k 3.77 app()k 0.2

3.76 0.1

0 3.75 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 k k (a) (b)

FIG. 5. Dispersion curves corresponding to solution of Eq. (C2). (a) In red is the first passband and in blue its polynomial approximation. (b) The second passband is relatively far away from the first one, and it is one order of magnitude narrower. The eigenvalues are independent of the angle χ0. ¯ 2 χ 2 2 E 0, and within a reasonable approximation the first passband hγi ðτÞii ¼hγi ð0Þii þ 2πð1 þ cos θ0Þ in the first Brillouin zone can be approximated by α k ≃ 4α k − k2 0 ≤ α k ≤ α ≡ α k ð Þ maxj j,where ð Þ max ð ¼ × 1 − cos 2π 1 cos θ0 τ : B2 f ½ ð þ Þ g ð Þ 0.5Þ ≃ 0.454. The red curve in Fig. 5(a) shows the numerical solution for jnj ≤ 5, whereas in blue we plot the approximation Averaging in time over one oscillation period, we from the above. After a wide forbidden gap (corresponding to conclude that the initial energy spread Δγ2 increases ini unconfinedtrajectories),thesecondbranchα2ðkÞisoneorderof according to α 3 79 α 3 757 magnitude narrower, i.e., max ¼ . , min ¼ . .See eE λ =mc2 2 Fig. 5(b). Δγ2 Δγ2 0 0 : ¼ ini þ 2π 1 θ ðB3Þ ð þ cos 0Þ APPENDIX D: SPACE CHARGE This last result represents the electrons’ energy spread In free space, an azimuthally symmetric pencil beam of increase when the beam enters the OBB region. Because of uniform density tends to diverge due to the net repelling ðscÞ 2 2 2 conservation of longitudinal canonical momentum, once force Fr ¼ re ðN=πRbΔzÞ=ð2γ ε0Þ within the volume of the e-beam exits the OBB, the electrons’ energy spread the beam. N is the number of electrons in the volume 2 drops back to its original value. πRbΔz, and the beam radius is Rb. Actually, as we show subsequently, the quantity that matters is the number of electrons N per unit length Δz; thus, we define ’ APPENDIX C: MATHIEU S EQUATION NΔ ≡ N=Δz. In practice, the radial distribution varies in In this Appendix, we determine the constraints on α that time, and, therefore, the space-charge force (after averaging ensure a periodic (stable) solution to Mathieu’s equation over the longitudinal dimension) is e2N 1 1 FðscÞ r; t Δ h r − r t : d2 r ð Þ¼ 2 h ½ ið Þii ðD1Þ α u χ Ψ u 0; 2πε0 γ r 2 þ sinð þ 0Þ ð Þ¼ ðC1Þ du The earlier expression for the radial force is readily P retrieved if we notice that for a uniform distribution ∞ ¯ R namely, ΨðuÞ¼expð−jkuÞ n −∞ Ψn expð−jnuÞ,where 2 Rb 2 2 ¼ ð2=RbÞ 0 dρρhðr − ρÞ¼r =Rb if r ≤ Rb and 1 other- u ¼ð−π; πÞ and jkj < 0.5 is the projection on the first wise. Consequently, by analogy to the latterZ and the fact Brillouinzone.TheMathieuequationnowhasanalgebraicform 2 Rb r2 dρρ3 that for a uniform distribution h iu ¼ 2 ¼ X∞ Rb 0 M − α−1δ Ψ˜ −jnu 0; 1 2 ½ n;n0 n;n0 n0 expð Þ¼ ðC2Þ 2 Rb, we consider the following approximation for the n;n0¼−∞ space-charge term: M ≡ 1 k n0 −2 δ −jχ − δ ( r2 with n;n0 2j ð þ Þ ½ nþ1;n0 expð 0Þ n−1;n0 × r2 ≤ 2 r2 ; 2 h i ii ˜ 0 2 ¯ h r − r t ≃ 2 r : expðþjχ0Þ, Ψn0 ≡ ðk þ n Þ Ψn0 .Equation(C1) is an eigen- h ½ ið Þii h i ii ðD2Þ α−1 2 2 value problem, or, explicitly, represents the eigenvalues of 1 r > 2hri ii the matrix M. In the framework of a periodic solution, it can be shown that the eigenvalues are independent of the initial angle

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r¯2 2 r¯2 r 2 8 10−15 This is obviously an approximation that becomes exact for wherein b ¼ h i i, and el ¼ . × ½m is the the case of a uniform radial distribution, but it obviously fails classical radius of the electron. For an assessment of the for large deviations. Therefore, we can expect it to describe value of the space-charge parameter, let us consider first an properly a pencil-beam-like configuration but be less accu- example density of 109 electrons confined by a ð100 μmÞ3 21 −3 rate when an annular beam is involved. In the latter case, at cube; thus, nb ≃ 10 ½m . A beam of similar density but 11 −1 bestthismodelprovidesuswithageneraltrend.Despitethese radius Rb ¼5½μm contains NΔ ≃ 0.75 × 10 ½m ; thus, κ ≡2r N ≃4 2 10−4 limitations, this approach greatly speeds up the numerical sc el Δ . × . For stability, we require that the processing time while still providing rough assessments. transverse force exerted by the OBB is dominant: Subject to the assumptions above, the transverse space- θ z¯_ U¯ ≫κ γ−2 ⇒2πeE λ =mc2 ≫2r N γ−2 jðcos 0 þ iÞ ijmax sc i 0 0 el Δ i . charge forces can be readily incorporated in the equations For the parameters specified above, this is readily satisfied. of motion [Eq. (2)], which then read d 1 1 γ x¯_ − θ z¯_ U¯ − h x¯ ; dτ ð i iÞ¼ ðcos 0 þ iÞ i i γ2 r¯2 i APPENDIX E: EFFECTIVE HAMILTONIAN i i d 1 1 Our purpose is to develop an analytic approach to _ _ ¯ γiy¯i − θ0 z¯i Ui − hi y¯i; simplify the equation of motion taking advantage of the dτ ð Þ¼ ðcos þ Þ γ2 r¯2 i i fact that the electron and the laser are counterpropagating; d 1 dr¯2 1 γ z¯_ i U¯ h z¯_ − V¯ ; thus, the absolute phase varies rapidly. We further neglect dτ ð i iÞ¼ 2 dτ i þ r¯2 i i i the energy variations and consider zero angular momentum i dγ 1 dr¯2 1 in Eq. (9); thus, i − i θ U¯ − h − z¯_ V¯ ; ¼ cos 0 i i 2 i i dτ 2 dτ r¯i d2ρ i ≃ −α J ρ χ ψ ; dχ 2 i 1ð iÞ sinð i;0 þ Þ ðE1Þ i 2π 1 z¯_ θ ; dψ dτ ¼ ð þ i cos 0Þ r¯2 wherein ψ ¼ 2πτð1 þ cos θ0Þ, ρi ¼ 2πr¯i sin θ0, and i r¯2 ≤ r¯2; ¯ −1 2 i b αi ¼ E½2πγi;0ð1 þ cos θ0Þ . The radial location of the hi ¼ 2r NΔ r¯b : ðD3Þ |fflfflffl{zfflfflffl}el electron has a slowly varying component and a fast ≡κ 1 r¯2 > r¯2; sc i b varying one:

ρi ¼ Ai cosðχi;0 þ ψÞþBi sinðχi;0 þ ψÞþCi; dρ ρ_ ≡ i A_ B χ ψ B_ − A χ ψ C_ ; i dψ ¼ð i þ iÞ cosð i;0 þ Þþð i iÞ sinð i;0 þ Þþ i ̈ _ ̈ _ ̈ ρ̈i ¼ðAi − Ai þ 2BiÞ cosðχi;0 þ ψÞþðBi − 2Ai − BiÞ sinðχi;0 þ ψÞþCi; ðE2Þ for substituting in Eq. (E2), we present also the first and second derivative of ρi: ̈ _ ̈ _ ̈ ðAi − Ai þ 2BiÞ cosðχi;0 þ ψÞþðBi − 2Ai − BiÞ sinðχi;0 þ ψÞþCi

≃ −αiJ1½Ai cosðχi;0 þ ψÞþBi sinðχi;0 þ ψÞþCi sinðχi;0 þ ψÞ; and using the orthogonality of the trigonometric functions we get Z 1 2π C̈ ≃ −α dψJ A ψ B ψ C ψ ; i i 2π 1½ i cosð Þþ i sinð Þþ i sinð Þ Z0 1 2π Ä − A 2B_ ≃ −α dψJ A ψ B ψ C 2ψ ; i i þ i i 2π 1½ i cosð Þþ i sinð Þþ i sinð Þ Z0 1 2π ̈ _ 2 Bi − 2Ai − Bi ≃ −2αi dψJ1½Ai cosðψÞþBi sinðψÞþCisin ðψÞ: ðE3Þ 2π 0

In the framework of the slowly varying amplitude approximation,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the derivatives of A and B are assumed to be negligible. 2 2 Furthermore, we keep in mind that, by defining ai ¼ Ai þ Bi , θi ≡ arctanðBi=AiÞ, the argument of the Bessel function reads Ci þ ai cosðψ − θiÞ; therefore, we get

081301-13 LEVI SCHÄCHTER and W. D. KIMURA PHYS. REV. ACCEL. BEAMS 23, 081301 (2020) Z 2π ̈ 1 1 dJ1ðχÞ Ci ≃ −αi dψJ1½Ci þ ai cosðψÞ sinðψ þ θiÞ ≃ −αi Bi; 2π 0 2 dχ Z χ¼Ci 1 2π 1 A ≃ 2α dψJ C a ψ 2ψ 2θ ≃ O a2 ; i i 2π 1½ i þ i cosð Þ 2 sinð þ iÞ ð Þ Z0 2π 1 1 2 Bi ≃ 2αi dψJ1½Ci þ ai cosðψÞ ½1 − cosð2ψ þ 2θiÞ ≃ αiJ1ðCiÞþOða Þ; ðE4Þ 2π 0 2

2 3 ω¯ ψ where we again employed the slowly varying amplitude δC ψ sinð s Þ δC 0 s;ið Þ 6 cosðω¯ sψÞ 7 s;ið Þ 4 ω¯ 5 ; approximation; namely, we assume that the amplitudes and _ ¼ s _ δCs;iðψÞ δCs;ið0Þ the phase vary slowly compared to trigonometric functions. −ω¯ s sinðω¯ sψÞ cosðω¯ sψÞ Based on the last three expressions, we have Ai ≃ 0, F2 Bi ≃ αiJ1ðCiÞ, and ð Þ that has a determinant of unity corresponding to a rotation d2 1 d C ≃ − α2 J2 C : of the phase space. In the framework of the present 2 i i 1ð iÞ ðE5Þ dψ 4 dCi notation, the emittance is given by

So we need to solve only the latter, since once Ci is −2 εðψÞ¼λ0sin ðθ0=2Þ known, the radial coordinate is known: ρi ¼ αiJ1ðCiÞ × qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi χ ψ C 2 _ 2 _ 2 sinð i;0 þ Þþ i, wherein again the first term is the fast × hδCi ðψÞihδCi ðψÞi −hδCiðψÞδCiðψÞi : ðF3Þ oscillatory term of the radial location, whereas the second is its slowly varying counterpart. Consequently, after the transients have decayed, ω¯ sψ ≫ π, Based on this equation of motion, we can deduce the the emittance is given by Hamiltonian associated with the radial motion, as we qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ignored the change in the total kinetic energy. For this ε ≡ ε ψ → ∞ λ −2 θ =2 δC2 δC_ 2 dC =dψ ∞ ð Þ¼ 0sin ð 0 Þ h s;ii∞h s;ii∞ purpose, we multiply the equation of motion by i : 1 1 2 −2 2 _ 2 d 1 dC 1 ≃ λ0sin ðθ0=2Þ ω¯ shδC ð0Þi þ hδCs;ið0Þi ; i α2J2 C 2 s;i ω¯ dψ 2 dψ þ 4 i 1ð iÞ s 2 ðF4Þ 1 dCi 1 2 2 ¼ 0 ⇒ H ¼ þ αi J1ðCiÞ; ðE6Þ Z 2 dψ 4 1 uþπ F2 u ≡ dxF2 x where we define ð Þ 2π ð Þ. The best case wherein H is the normalized Hamiltonian and obviously it u−π scenario dε∞=dω¯ s ¼ 0 is achieved by matching the poten- is positively defined, i.e., H ≥ 0. tial, represented by ω¯ s, to the initial phase space: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi APPENDIX F: EMITTANCE OF THE eE λ ω¯ δC_ 2 0 δC2 0 −1 ⇒ 0 0 TRANSPORTED BEAM s ¼ h s;ið Þih s;ið Þi mc2γ¯ pffiffiffi In this Appendix, we examine the emittance as electrons 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p dδx=dτ 2 δx2 −1 : are oscillating near one of the zeros of the potential 1;s: ¼ hð Þs;iiτ¼0h s;iiτ¼0 ðF5Þ jJ0ðp1;sÞj J1ðp1;sÞ ≡ 0, s ¼ 0; 1; 2; …. Consider a linearized solution C ≃ p δC of the form i 1;s þ s;i, and then we may write Subject to this condition, the emittance is conserved along 2 the entire interaction length: d 1 dδCs;i 1 2 2 2 þ α J ðp1;sÞðδCs;iÞ dψ¯ 2 dψ 4 i 0 ε ε ψ 0 : ∞ ¼ ð ¼ Þ ðF6Þ d2 1 0 ⇒ α2J2 p δC 0: ¼ dψ 2 þ 2 i 0ð 1;sÞ s;i ¼ ðF1Þ APPENDIX G: CO- AND 2 COUNTERPROPAGATING BEAMS Defining the normalized oscillating frequency ωs;i ≡ 2 2 2 2 −2 ð1=2Þαi J0ðp1;sÞ ≡ ω¯ s γ¯ γi and keeping in mind that, for The goal in Appendix G is to compare the dynamics of most practical acceleration applications, the energy spread co- and counterpropagating beams. To obtain the equations −1 is of the order ∼0.1%, we can assume γγ¯ i ∼ 1, wherein that describe the copropagating motion, we switch the γ¯ ¼hγii. As a result, the transport matrix that describes the trajectory of the particles in the z direction: ziðtÞ → −ziðtÞ. trajectory is Therefore, instead of Eq. (2), the system is described by

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d γ x¯_ − θ − z¯_ x¯ πE¯ χ ; dτ ð i iÞ¼ ðcos 0 iÞ i sinð iÞ d γ y¯_ − θ − z¯_ y¯ πE¯ χ ; dτ ð i iÞ¼ ðcos 0 iÞ i sinð iÞ d 1 d γ z¯_ − x¯2 y¯2 πE¯ χ E¯ χ ; dτ ð i iÞ¼ 2 dτ ð i þ i Þ sinð iÞþ cosð iÞ dγ 1 d i − θ x¯2 y¯2 πE¯ χ z¯_ E¯ χ ; dτ ¼ cos 0 2 dτ ð i þ i Þ sinð iÞþ i cosð iÞ dχ i 2π 1 − z¯_ θ ; dτ ¼ ð i cos 0Þ ðG1Þ where the various normalizations are identical to those in the counterpropagating case. We focus on the transverse motion and ultrarelativistic case γθ0 ≫ 1, and for comparison we present the corresponding equations for the counterpropagating case:

ðG2Þ

2 It is evident that, for otherwise identical conditions, the confining force is ðθ0=2Þ weaker in the copropagating case. This is the main reason for our choice of using a counterpropagating OBB. Next, we focus our attention on the radial stability of the beam for both cases. We show for both cases that the slow varying component of the radial motion is stable. Subject to the previous assumptions, we may further assume that the electrons energy is constant; thus, Copropagating Counter propagating d2x¯ 1 E¯ d2x¯ 1 E¯ i ≃ ux¯ ; i ≃ − ux¯ ; 2 2 sin i 2 sin i ðG3Þ du 2πθ0 γ du 8π γ 2 u ¼ χð0Þþπθ0τ;u≃ χð0Þþ4πτ: Note that the normalized time ðuÞ has a different (natural) definition. Clearly, the latter varies much slower in the copropagating case. In cylindrical coordinates, the motion may be explicitly written as

x¯i ¼ ρi cos ϕi; d x¯_ ≡ x¯ ρ_ ϕ − ϕ_ ρ ϕ ; i du i ¼ i cos i i i sin i ̈ _ ̈ _2 x¯i ¼ ρ̈cos ϕi − 2ϕiρ_i sin ϕi − ϕiρi sin ϕi − ϕi ρi cos ϕi; ðG4Þ and, assuming vanishing angular momentum (which is conserved), we get Copropagating Counterpropagating d2ρ 1 E¯ d2ρ 1 E¯ G5 i ≃ uρ ; i ≃ − uρ : ð Þ 2 2 sin i 2 sin i du 2πθ0 γ du 8π γ The radial location of the electron has a fast varying component and a slowly varying one:

ρi ¼ Ai cos u þ Bi sin u þ Ci; ðG6Þ

081301-15 LEVI SCHÄCHTER and W. D. KIMURA PHYS. REV. ACCEL. BEAMS 23, 081301 (2020) where the three unknown quantities (A, B, and C) are slowly varying functions of u; thus, d ρ_ ≡ ρ A_ B u B_ − A u C_ ; i du i ¼ð i þ iÞ cos þð i iÞ sin þ i ̈ _ ̈ _ ̈ ρ̈i ¼ðAi − Ai þ 2BiÞ cos u þðBi − 2Ai − BiÞ sin u þ Ci:

Substituting in Eq. (G6) and using the orthogonality of f0ðuÞ¼1, fcðuÞ¼cos u, and fsðuÞ¼sin u for the copropagating case,

E¯ f u ∶ C̈ B ; 0ð Þ i ¼ 2 i 4πθ0γi _ fcðuÞ∶ − Ai þ 2Bi ¼ 0 ⇒ Ai ≃ 0; 1 E¯ 1 E¯ f u ∶ − 2A_ − B ≃ C ⇒ B ≃ − C : sð Þ i i 2 i i 2 i ðG7Þ 2 πθ0γi 2 πθ0γi

Combining the three results and repeating the procedure for the counterpropagating case, we obtain

ðG8Þ

Consequently, employing regular time normalization ðτ ¼ ct=λ0Þ for both cases, we conclude that the radial motion of a narrow pencil beam is stable for both copropagating and counterpropagating beams, i.e.,

d2 1 E¯ 2 C ≃ − C : 2 i i ðG9Þ dτ 8 γi

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