## Abstract

Compact accelerators of the future will require enormous accelerating gradients that can only be generated using high power laser beams. Two novel techniques of laser particle acceleration are discussed. The first scheme is based on a solid-state accelerating structure powered by a short pulse CO_{2} laser. The planar structure consists of two SiC films, separated by a vacuum gap, grown on Si wafers. Particle acceleration takes place inside the gap by a surface electromagnetic wave excited at the vacuum/SiC interface. Laser coupling is accomplished through the properly designed Si grating. This structure can be inexpensively manufactured using standard microfabrication techniques and can support accelerating fields well in excess of 1 GeV m^{−1} without breakdown. The second scheme utilizes a laser beatwave to excite a high-amplitude plasma wave, which accelerates relativistic particles. The novel aspect of this technique is that it takes advantage of the nonlinear bi-stability of the relativistic plasma wave to drive it close to the wavebreaking.

## 1. Introduction

The ability of a laser beam to concentrate large amounts of electromagnetic energy in a small volume over a short period of time makes laser-driven acceleration techniques promising for future compact accelerators. Extremely high accelerating gradients delivered by lasers may result in a significant reduction of the accelerator size. Because the electric field of a laser is predominantly transverse, it has to be rotated using an accelerating structure in order to impart energy to a charged particle. The structure can be based either on a solid material, or on a plasma. While breakdown, single-pulse heating and the resulting cyclic stress are not important for a plasma-based structure, those issues are of the utmost importance to solid structures (Whittum 1999). Examples of the solid laser-driven structures include an open grating in an inverse Smith–Purcell accelerator (Palmer & Giordano 1985), a dielectric-loaded accelerator (Rosenzweig *et al*. 1995), an overmoded microchannel metal waveguide (Steinhauer & Kimura 2003), or a photonic bandgap accelerator (Smith *et al*. 1994; Shapiro *et al*. 2001; Cowan 2003).

All these schemes have important limitations. The dielectric-loaded accelerator needs a metallic casing to confine radiation. Metals are lossy at infrared (IR) frequencies and susceptible to breakdown for high field amplitudes. Because the electric field at the metallic wall is of the same order as the peak accelerating field in the vacuum gap, breakdown and pulsed heating of the metallic casing is a serious issue. The overmoded microchannel metallic waveguide suffers from the same limitation as the dielectric accelerator (metal at high field), and, in addition, has a low shunt impedance because the electric field in an overmoded guide is primarily transverse. To overcome the above limitations, a novel accelerating structure has been recently suggested (Shvets & Kalmykov 2004): the Surface Wave Accelerator Based on SiC (SWABSiC). It has no metallic parts and has a narrow accelerating gap (to maximize *E*_{x}/*E*_{⊥}, where *E*_{x} is the accelerating field component).

The accelerating structure consists of two standard (400 μm thick) double-side polished silicon wafers. At the bottom of each wafer a few micrometre thick SiC film is epitaxially grown using atmospheric pressure chemical vapour deposition (APCVD) (Zorman *et al*. 1995). On the top of each wafer a diffraction grating is etched with a period of order 10 μm. The grating couples the incident laser radiation into the structure. The resulting structure shown in figure 1 forms a sandwich with a few micrometre vacuum gap between the SiC films. Charged particles are accelerated by the surface electromagnetic waves localized near the vacuum/SiC interface. The structure can be powered by a widely available and fairly efficient carbon dioxide (CO_{2}) laser with a wavelength μm. The operating wavelength is dictated by the frequency-dependent dielectric permittivity of SiC which is negative for the frequencies in the CO_{2} tunability range.

The resulting setup can support accelerating fields well in excess of 1 GeV m^{−1} without a breakdown, and provide the path to compact and inexpensive particle accelerators. SWABSiC is a robust structure, because the working surface of the accelerator (SiC film) is attached to a large Si wafer capable of handling thermal and mechanical stress. Here we report on our first experiments aimed at exciting and diagnosing surface waves at the SiC/air interface. Excitation of surface waves is deduced from laser transmission through and reflection from the structure. A number of diagnostic challenges arise when laser energy is coupled to the surface waves through a large wafer. For example, dips in transmission and reflection are sometimes associated with the excitation of wafer-guided modes instead of the surface waves. Modelling must be used, as described below, to identify and quantify surface waves.

As the second advanced acceleration technique based on plasmas we consider a plasma beatwave accelerator (PBWA). Beatwave excitation mechanism is realized when the driver intensity (laser or particle beam) is modulated with the temporal periodicity of the plasma wave. PBWA is the oldest plasma-based acceleration concept dating back to the original idea of using plasmas as the accelerating structure (Tajima & Dawson 1979). Despite being the oldest of the experimentally realized acceleration concepts (Clayton *et al*. 1993), it continues attracting significant experimental and theoretical attention (Tochitsky *et al*. 2004; Lindberg *et al*. 2004). It was recently realized (Shvets 2004) that one can improve a PBWA by using a pair of *long* laser pulses detuned from each other by a frequency (where is the electron plasma frequency, *n* is the plasma density, −*e* and *m* are the electron charge and mass, respectively) whose beatwave amplitude exceeds a certain threshold . The electric field amplitudes of the laser pulses are *E*_{1} and *E*_{2} and their corresponding frequencies and . The resulting plasma wake is essentially independent of the beatwave amplitude and duration. This phenomenon, referred to as Dynamical Bi-Stability (DBS), is caused by the relativistic nonlinearity of a high-amplitude plasma wave. Here we develop a Hamiltonian description of a strongly driven plasma wave. Using the standard nonlinear dynamics concepts such as the separatrix crossing (Timofeev 1978; Tennyson *et al*. 1986), we analyse the evolution of the plasma wave and explain how a long laser beatwave pulse of duration can leave behind a relativistic wake.

## 2. Design and electromagnetic properties of a SWABSiC

We have carried out the cold tests of the SWABSiC by doing proof-of-principle experiments aimed at excitation and diagnostics of surface waves at the SiC/air interface. Before describing the details of the experimental measurements and the supporting simulations, we review the basic fabrication steps of SWABSiC and the rationale for using SiC. The structure of the accelerating surface wave in the gap region between the two films of SiC was described earlier (Shvets & Kalmykov 2004). The surface wave in a SWABSiC is a *double-interface* wave, because it is supported by two interfaces: SiC/air and air/SiC. The purpose of the experiments described below is to demonstrate that the surface wave at the SiC/air interface can be excited by a CO_{2} laser. Excitation is done using a grating etched in a Si wafer on which the SiC film is grown as shown in figure 1. After the laser beam is refracted by the grating into the silicon, it propagates through the entire thickness of the wafer (400 μm) before reaching the SiC film. One of the experimental challenges is to diagnose surface waves in the presence of other waveguide (Fabry–Perot) resonances. To simplify the problem, we have not bonded the two halves of the accelerator together. Therefore, *single-interface* surface waves are excited and diagnosed in this series of experiments.

### (a) Material properties of silicon carbide and structure fabrication

To build a testable prototype of a SWABSiC, we have used the cubic 3C–SiC polytype of silicon carbide because it can be grown on Si wafers (Zorman *et al*. 1995). SiC is a wide-bandgap (2.3 eV) semiconductor, which exhibits a very high electrical breakdown field (300 MV m^{−1} DC). SiC is rapidly becoming the material of choice for electronics and sensor devices that operate in hostile and high-temperature environments (over 1000 °C), where conventional silicon-based electronics (limited to 350 °C) cannot function. The ability of SiC devices to function at high temperatures and high radiation fluxes makes this material very promising for particle acceleration applications. Those must operate under high accelerating gradients (large electric and magnetic fields), significant heating due to the resistive dissipation of the accelerating fields and occasional beam losses and radiation by the accelerating particles. The high thermal conductivity, approximately 3.8 W cm^{−1} K^{−1}, is important as well. The real part of frequency-dependent dielectric permittivity of SiC is negative in a frequency band (Spitzer *et al*. 1959; Shchegrov *et al*. 2000):(2.1)where *ω*_{L}=972 cm^{−1} (*λ*_{L}=1/*ω*_{L}=10.29 μm) and *ω*_{T}=796 cm^{−1} (*λ*_{T}=1/*ω*_{T}=12.56 μm) are the normalized frequencies of the longitudinal and transverse optical phonons, respectively. The high-frequency dielectric constant for the 3C–SiC crystal is *ϵ*_{∞}=6.5. The real part of the dielectric permittivity (2.1) is negative near *λ*=10.6 μm produced by a CO_{2} laser. Thus, a common and efficient power source for the SiC-based accelerating structures is available. The damping constant *γ*=5 cm^{−1} is relatively small. Silicon has no IR resonances and does not introduce any additional damping at *λ*≈10.6 μm. This is validated by our transmission/reflection measurements from a pure Si wafer (no grating or SiC film), which demonstrated that less than 5% of the incident energy is scattered or lost to heating. In the mid-IR, the dielectric permittivity of Si is almost frequency-independent and equal to *ϵ*_{Si}=12.

We have fabricated one-half of the SWABSiC accelerating structure. Using standard microfabrication methods, which will be described elsewhere, the accelerating cavity at the bottom of a doubly polished Si wafer was etched out using selective reactive ion etching. The cavity dimensions are 2000×400 μm^{2}, its depth is 2.7 μm. The rest of the bottom side was protected by a 1.5 μm thick SiO_{2} layer. Using the same techniques, we have etched the grating on the topside of the Si wafer. The grating pitch is *λ*_{g}=16 μm, its width is 8 μm and its depth is 1.5 μm. After the above processing steps, the wafer ended up protected by an oxide film everywhere except inside the cavity. It was sent to Case Western Reserve University where a 1.5 μm-thick SiC film was grown using APCVD (Zorman *et al*. 1995). Crystalline SiC film grows only on Si but not on the amorphous SiO_{2}. Thus, we were able to peel off the amorphous (and weakly adherent) SiC grown on the oxide without damaging the crystalline oxide grown inside the cavity. The wafer was then cleaned from the remaining oxide.

The total area covered by the grating is 2×2 mm^{2}. It is located directly above the cavity and entirely covers it. This arrangement enables a convenient measurement of reflection/transmission coefficients with and without the SiC film. To measure energy losses with the grating only, the CO_{2} laser beam is pointed at the portion of the grating that exists outside of the accelerating cavity. This measurement is very important, because the grating itself can present a significant source of energy losses by coupling the laser beam into the guided modes of the wafer. Those guided modes can propagate through the wafer outside of the grating area and, therefore, be lost from the reflection/transmission energy balance. Numerical simulations of the guided modes are presented in §2*c*. The SiC film introduces a new source of losses that are particularly enhanced when a surface wave is excited. Therefore, energy losses due to the grating alone is an important ‘null’ test when the energy loss to the surface wave is estimated.

### (b) Experimental tests of the surface wave excitation

The grating on the top of the Si wafer couples the incident radiation with different modes of the multi-layer waveguide. The most important for our purposes is the surface wave supported by the SiC/air interface. The electric field associated with this mode peaks at the interface and is relatively low inside the wafer because it exponentially decays into the SiC film. Diagnosing the surface wave is possible by detecting a drop in the total collected energy that consists of the reflected and transmitted waves. It is reasonable to assume that this energy deficit is deposited into the surface wave and dissipated due to the finite phonon lifetime.

For cold tests (measurements of reflection and transmission coefficients as functions of incident angle and frequency), we use a line-tuneable CO_{2} continuous wave (CW) laser with ^{13}C isotope fill (Access Laser Company Merit-G). No more than 200 mW of average power was used. The beam from the CO_{2} laser was aligned with a HeNe laser using a dichroic beam splitter. Red beam from HeNe laser was used as a pointer. A ZnSe lens with a focal length *f*=15 cm and *f*/10 numerical aperture was used to focus the mid-IR beam. A set of calibrated pinholes was used to establish the position of the focal point and estimate the diameter of the focused beam. It was found that the diameter of the focused beam was approximately 200 μm, so it completely fits inside the accelerating cavity footprint. For each wavelength in the 10.67<*λ*<11.31 μm range, the ratio of the reflected and transmitted power to the incident laser power is measured as a function of the angle of incidence *α*. Laser radiation is linearly polarized: magnetic field is parallel to the grating lines. Transmitted and reflected radiation is collected in all propagating diffractive orders: *n*=0, −1, −2. The sample experimental data are shown in figure 2*a*: with and without the SiC film.

First, we measured the total energy balance *E*_{gr} with the grating but without the SiC film, and discovered an energy deficit of up to 50% for some incidence angles (diamonds in figure 2*a*). We speculate that these launching losses are due to the strong coupling to the wafer-guided modes that propagate away from the grating region. Our simulations reveal that, even though the grating is rather shallow, high diffractive orders (up to the sixth) can contribute to the excitation of the wafer-guided modes. In the presence of a SiC film, resistive losses in SiC occur in addition to the launching losses. That is why the total losses *E*_{SiC} in the case with the film (squares in figure 2*a*) are consistently higher than without the film. To get a quantitative measure of the losses in the SiC, we have subtracted the *E*_{gr} from *E*_{SiC} and plotted the results in figure 2*b*. The resulting curve is peaked at the *α*_{m}=35°, with the half-width of Δ*α*=8°. This agrees well with the theoretical prediction of(2.2)which yields *α*_{th}=34.1° for *λ*=10.85 μm. Equation (2.2) is derived under the assumption that the surface wave is excited at the interface between two semi-infinite media: SiC and air. Given that the film is not semi-infinite, the agreement between *α*_{m} (which is, presumably, the incidence angle at which the surface wave is excited) and *α*_{th} is adequate. However, the large launching losses observed in the experiment are vexing from both the conceptual (needs to be explained) and practical (needs to be overcome) standpoints. In §2*c*, we present the results of the numerical modelling in support of our hypothesis that the launching loss is caused by the strong coupling to the wafer-guided modes.

### (c) Numerical simulations of the SWABSiC

A series of numerical simulations using a finite elements electromagnetic code FEMLAB was performed. Frequency-dependent *ϵ*_{SiC} given by equation (2.1) was assumed (including losses), and *ϵ*_{Si}=12. In the simulation, we approximate the laser amplitude with a plane transverse magnetic (TM) electromagnetic wave with the wavelength *λ*=10.8 μm emitted by a flat antenna placed above the wafer parallel to the wafer. At the upper and lower borders of the simulation box totally absorbing boundary conditions are imposed. Periodic boundary conditions are applied to the side borders at *X*=±8 μm (see figure 4). Therefore, we model the events that may occur near the centre of the laser focal spot within distances smaller than the focal spot size (less than 1 mm). Owing to the software limitations, we took the wafer thickness to be *D*=14.8 μm, which is much less than that in the real-scale experiment; the grooves are 1.6 μm high and the SiC film is 1.6 μm thick, as in the experiment. The results of such simulations can guide us in interpreting experimental results.

The simulated curves of transmission, reflection and losses are shown in figure 3. Energy losses grow in the broad angular range 34°<*α*<38° (compare with figure 2*b*). Everywhere in this range the surface wave has a large amplitude (up to a factor of 5 larger than in vacuum in the absence of the dielectric structure). The curves of reflection and absorption exhibit anomalies in the vicinity of *α*≈37°. We associate these anomalies with the excitation of the guided modes of the structure. To confirm this point of view, we show the results of two sample simulations in figure 4 that can be interpreted as follows. The grating transforms the incident beam into a set of modes corresponding to different diffraction orders. These modes can propagate inside the structure (and near the SiC/vacuum interface). The longitudinal electric field can be expanded as , where the longitudinal wavenumber differs from the fundamental by integer multiples of the grating wavenumber, , where *λ*_{g} is the grating period. The near-luminous surface wave characterized by a longitudinal component of phase velocity close to the speed of light in vacuum () can be excited in the *n*th diffractive order. The required angle of incidence *α* is such that the wavenumber *k*_{xn} satisfies the dispersion relation for the surface wave at the SiC/vacuum interface: . This wave is almost longitudinal at the border between the vacuum and SiC. The near-luminous wave can be effectively excited in the *first* diffractive order if the incidence angle satisfies equation (2.2). For *λ*_{g}=16 μm and *λ*=10.8 μm, we find and *α*_{th}≈37.58°. It is clear from figure 4 (where both plots are given for *α*≈*α*_{th}) that the electric field is almost entirely longitudinal at the SiC/vacuum interface. This surface wave is powered by the field associated with the first diffractive order and has a phase velocity approaching the speed of light. Although the phase velocity of a single-interface surface wave is always smaller than the speed of light, a double-interface surface mode can be luminous (Shvets & Kalmykov 2004), and, therefore, appropriate for accelerating ultra-relativistic particles.

The principal difference between the top and bottom plots in figure 4 is that the former illustrates excitation of the predominantly surface mode at the incidence angle *α*=35°, sufficiently removed from the sharp reflectivity spikes in figure 3. The bottom plot in figure 4 corresponds to *α*=37.3°, for which the guided mode is excited in the *minus sixth* diffractive order. The electric field inside the silicon is at least twice as large compared to the electric field in the SiC film; the Poynting flux is directed alongside the wafer and is reversed against the energy flux in the incident electromagnetic wave. The modal analysis of the two-layer waveguide without grating performed by Otwinowski (2005) shows that the minus sixth diffraction order is indeed an eigenmode of the Si wafer. Figure 3 (bottom) illustrates that a shallow grating can strongly couple to high diffractive orders. Therefore, a laser beam with the finite spot size (and, therefore, finite angular width) can excite multiple-guided modes, resulting in a significant launching loss. While the experimentally measured energy balance shown in figure 2 does not show the very sharp spikes as the simulated figure 3, it is most likely due to the finite angular divergence of the laser beam which is focused to the 200 μm diameter.

## 3. Beatwave excitation of plasma waves in the bi-stability regime

Research in the field of advanced plasma-based accelerators has to address a very different set of challenges. Plasma structures are not designed to last: new plasmas can be recreated whenever necessary. Plasma accelerators, however, face a different challenge: how to convert transversely polarized laser pulses into longitudinally polarized plasma waves capable of accelerating particles. The solution was proposed early on (Tajima & Dawson 1979): to use a nonlinear ponderomotive force of a laser pulse to ‘rectify’ its field and excite relativistic plasma waves. Any nonlinear mechanism of driving accelerating waves is somewhat inefficient as long as the excited field is much smaller than the driving field. Because plasma is *not* a very nonlinear medium, extremely high laser intensities are needed to approach the limit of *E*_{p}∼*E*_{laser}, where *E*_{p} and *E*_{laser} are the peak values of the plasma and laser fields, respectively. This can be accomplished in the PBWA where plasma waves are driven close to the plasma resonance. In a PBWA, plasma waves can approach the cold wavebreaking limit of . One of the limitations of a PBWA is that the plasma wave amplitude is sensitive to the pulse duration and amplitude.

It was recently suggested (Shvets & Kalmykov 2004) that this sensitivity could be reduced by using the nonlinear phenomenon of relativistic bi-stability. Below we consistently derive the simplified equations describing the bi-stability and introduce a Hamiltonian description of a single plasma wave in terms of a one-dimensional particle moving in the time-varying external potential. The role of the potential is played by the time-dependent laser beatwave. Particle motion in the phase space will be described using the standard concepts of the nonlinear dynamics such as action conservation and separatrix crossing. We will demonstrate that the conservation of the Hamiltonian explains how a strong plasma wave is left behind a long beatwave. The amplitude of the plasma wave is *independent* of the pulse duration and amplitude.

We assume that the beatwave generated by a pair of non-evolving frequency-detuned by *ω*_{B} laser beams is moving with the speed close to the speed of light *c*, and, therefore, all beatwave quantities are functions of the co-moving coordinate . This quasi-static approximation assumes that the laser beatwave evolves on a time-scale slower than the beatwave duration *τ*_{L}. For a pair of linearly polarized laser pulses with electric field amplitudes *E*_{1} and *E*_{2} and the corresponding frequencies *ω*_{1} and *ω*_{2}=*ω*_{1}−*ω*_{B}, the normalized beatwave amplitude *a* can be introduced as . The electrostatic plasma wave characterized by the electrostatic potential *ϕ* driven by the laser beatwave satisfies the well-known quasi-static equation:(3.1)where and *ω*=*ω*_{B}/*ω*_{p}.

To analyse equation (3.1) in the limit of a small driver *a*≪1, it is convenient to make an asymptotic expansion of the above equation in slow variables using the method of Krylov–Bogoliubov–Mitropolsky (KBM). To facilitate the implementation of the KBM method, equation (3.1) is expanded around *Φ*=0 and rewritten in the form of a nonlinear driven oscillator equation:(3.2)where the small parameter *ϵ*≪1 is introduced for ordering of the terms of equation (3.2). The following hierarchy of the small terms will be assumed: , and .

The KBM expansion of the solution of equation (3.1) in slow-varying amplitude *u*(*τ*) and phase *θ*(*τ*) ordered in powers of *ϵ* is then given by(3.3)(3.4)(3.5)where the slowly varying phase is the difference between the phases of the plasma wave and the beatwave driver.

Substituting the expressions (3.3)–(3.5) into equation (3.2), equating the terms proportional to the same powers of *ϵ*, and taking into account that leads to the following set of equalities.

To the order of *ϵ*^{2} (the lowest power),Eliminating secular terms gives *A*_{1}=0, *F*_{1}=0 and .

To the order of *ϵ*^{3}, a similar procedure yields , and .

To the order of *ϵ*^{4}, elimination of the secular terms yields *A*_{3}=0 and *F*_{3}=−*a*/2.

Combining all terms, up to the order *O*(*ϵ*^{4}), the equations for the slow amplitude and phase take the final form(3.6)(3.7)

This dynamical system is very similar to the one obtained earlier (Shvets 2004), except that an additional term in the phase equation responsible for the relativistic shift of the plasma frequency due to the laser field is included. The total nonlinear (relativistic) frequency shift of the driven plasma wave consists of the relativistic *γ* correction due to the transverse laser jitter (the *a*/2 term) and due to the longitudinal plasma oscillation (the 3/16*u*^{2} term). Both terms are added to the *ω*−1 term describing the detuning of the beatwave from the exact plasma resonance. For the parameters of relevance to high amplitude plasma wave excitation, the *a*/2 frequency detuning term is extremely small, and will be neglected below. Although similar to equations (3.6) and (3.7) descriptions of the nonlinear plasma wave excitation have been derived in the past (Noble 1985; Mori 1987), the response to a realistic time-varying (turned on and off) beatwave pulse has never been accurately explored. Below we develop a Hamiltonian description of the plasma wave and demonstrate how a finite plasma wake can be left behind the beatwave pulse due to the phenomenon of DBS. The adiabatically driven nonlinear dynamical system described by equations (3.6) and (3.7) exhibits a bi-stable behaviour if its phase portrait changes topology and the system goes through a bifurcation while the laser beatwave slowly changes its amplitude. Specifically, depending on the beatwave duration and the initial conditions of the plasma wave , , the solution of equations (3.6) and (3.7) can have two possible oscillatory solutions for large times (after the driver is turned off): either with the final amplitude , or with the larger amplitude .

Hamiltonian description of this dynamical system is obtained by introducing the action-angle variables and *θ*. It is easy to show that the equations of motion for *I* and *θ* can be derived from the time-dependent Hamiltonian *H* as , , where(3.8)

Thus, the plasma wave is described as an equivalent particle whose one-dimensional motion is characterized by action-angle variables (*I*, *θ*) and Hamiltonian *H*. As the beatwave amplitude *a*(*τ*) turns on and off, the equivalent particle traces a path in the (*I*, *θ*) phase space. Assuming that *a*(*τ*) varies slowly, we explain the particle's motion in the phase space using the standard concepts of Hamiltonian dynamics: conservation of the adiabatic invariant (equal to the phase space area) and separatrix crossing.

If the driver amplitude *a* is time-independent, then the Hamiltonian is conserved and the particle moves along the curves *H*=*H*_{0}, with *H*_{0} determined by the initial conditions. If *a*(*τ*) is changing slowly with time, with , where *T*_{osc} is the period of nonlinear oscillations, then *H*_{0} also adiabatically changes with time, i.e. is parametrically dependent on *τ* through its dependence on *a*. For time periods Δ*t* satisfying particle trajectory follows the *H*=*H*_{0}(*a*) curves in the (*I*, *θ*) phase space. To preserve the adiabatic invariant , the *H*=*H*_{0}(*a*) curves are such that the phase space area inside them is a constant. Therefore, *H*_{0}(*a*, *J*_{0}) is a function of both the initial action *J*_{0} and the beatwave amplitude *a*.

Adiabatic passage of the representative particle through the phase space can be visualized by following figure 5, which describes the particle phase space portraits for various values of the beatwave amplitude *a*. The large dot representing the particle moves along the curves that are slowly evolving with *a*. Note that for , the solution of *H*=*H*_{0} has two branches as illustrated by figure 5*a*,*b*. The precise value of *a*_{crit} is slightly lower than *a*_{0} for a finite *u*_{0}. The lower branch (small solid loop in figure 5*b*) is confined inside the separatrix while the upper branch (solid curve in figure 5*b*) is outside (above) the separatrix. The two branches merge with the separatrix for *a*=*a*_{crit} as shown in figure 5*c*,*f*. At that moment the area inside the inner loop of the separatrix is equal to . For , the solution of *H*=*H*_{0} has only one branch that lies below the separatrix as shown in figure 5*d*. The separatrix disappears for *a*>*a*_{0}.

Therefore, at the time *τ*_{crit}, such that , particle necessarily crosses the separatrix. No matter how slowly *a*(*τ*) is changing, adiabatic invariant *J* is changed at the separatrix becoming . Here, *S* is the total phase space area under the upper branches of the separatrix (including the inner loop of area *S*_{0}). Separatrix crossing is illustrated in figure 5*d*. The Hamiltonian *H* is unchanged during the crossing for long beatwave pulses. Note that the separatrix crossing occurs with the change of the adiabatic invariant, but without the change of the Hamiltonian. As the beatwave amplitude further increases, the particle continues its nonlinear oscillations along the *H*=*H*_{0}(*a*, *J*) curves while preserving the new adiabatic invariant *J*=*J*_{1}. When *a* starts decreasing and eventually returns to *a*_{crit}, a bifurcation occurs. The term ‘bifurcation’ in this case implies that the particle can proceed along one of the two possible trajectories. Depending on the beatwave duration *τ*_{L} and the initial particle phase *θ*_{0}, a particle can follow two possible paths: (i) return back inside the inner separatrix loop and reclaim its original adiabatic invariant *J*_{0} or (ii) move outside of the separatrix and acquire a new adiabatic invariant . In the former case, after the beatwave is turned off. In the latter case, . For the simulation parameters used for making figure 5 (, , , ), we have an example of the possibility (i) being realized for the initial plasma wave phase *θ*_{0}=0. In that case the time sequence of the particle's phase portraits is (*a*)–(*b*)–(*c*)–(*d*)–(*e*)–(*d*)–(*c*)–(*b*)–(*a*). A very small-amplitude plasma wave is left behind the long beatwave driver when the possibility (i) is realized. A different initial phase *θ*_{0}=10, however, presents an example of the possibility (ii). Figure 5 shows the corresponding phase space sequence (*a*)–(*h*) and illustrates by the panel (*h*) that .

A bifurcation (particle remaining on the inside or moving to the outside of the separatrix) occurring at *τ*=*τ*_{crit} determines whether or not a significant plasma wave is left behind the laser beatwave. We refer to this phenomenon, however, as a DBS because of the remarkable theorem we have empirically observed in the numerical simulations: whether or not the particle ends up inside or outside the separatrix, its Hamiltonian is the *same* for a given laser beatwave amplitude. In other words, for any value of *a*, *J*_{0} and *J*_{2}(*J*_{0}). Here, *J*_{2} refers to the new adiabatic invariant acquired by the particle with the initial adiabatic invariant *J*_{0}. The accuracy of this theorem improves with the pulse adiabaticity, i.e. with the increasing pulse duration *τ*_{L}. Analytically proving this theorem would take us beyond the scope of this paper, but it can indeed be proven in the adiabatic limit using the approach similar to the one used (Kotel'nikov & Stupakov 1990) in the context of the cyclotron plasma heating. In the context of plasma wake generation, the important consequence of the theorem is that for *a*<*a*_{crit}, there are two solutions to the *H*=*H*_{0}(*a*, *J*) equation corresponding to the two different *J*'s (*J*=*J*_{0} and *J*=*J*_{2}). Typically, one of them corresponds to a much larger plasma wake. We refer to the discovered phenomenon as the DBS: dynamical because it requires time-dependent separatrix crossing, and Bi-Stability because of the two solutions of the *H*=*H*_{0}(*a*, *J*) equation.

The presented examples are given in terms of dimensionless parameters: beatwave strength *a*, normalized beatwave frequency , normalized pulse duration , initial normalized plasma amplitude *u*_{0}. These parameters can be achieved for a variety of plasma densities, laser intensities and frequencies. As an example, we use *ω*_{1}/*ω*_{p}=10, 2*πc*/*ω*_{1}=1 μm, *n*=10^{19} cm^{−3} and . In that case, the amplitude of the wake left behind the laser beatwave is *E*_{p}≈200 GeV m^{−1}. Note that *E*_{p}/*E*_{1}≈0.35, so the electric field of the plasma wave is comparable to that of the incident laser.

## 4. Conclusions

In conclusion, we have reviewed two fundamentally different approaches to laser-particle acceleration. The first approach utilizes a solid structure powered by a short pulse laser. As an example we considered a SWABSiC that uses a thin film of SiC epitaxially grown on a Si wafer. Accelerating surface waves at the air/SiC interface are excited by a CO_{2} laser through the grating etched in the Si wafer. Because the accelerating fields are linear in the incident laser field, and can significantly exceed it if the resonant excitation of the surface waves is accessed, moderate laser powers are needed to produce accelerating gradients of the order of a GeV m^{−1}. Preliminary experimental results verifying the existence of the surface waves are presented.

The second approach, PBWA, utilizes an underdense plasma. Accelerating waves are excited by an intense two-colour laser beam, and the frequency detuning between the two colours is assumed to be slightly smaller than the plasma frequency. When the laser beam intensity exceeds a detuning-dependent threshold, the phenomenon of DBS occurs: a long beatwave leaves behind a relativistic wake. DBS was explained in terms of a representative Hamiltonian particle describing the plasma wave, which undergoes a complicated nonlinear dynamics in the phase space. This dynamics includes separatrix crossings and phase space bifurcations. Whether the wake is excited or not is determined by the pulse duration and the initial (if any) plasma wave amplitude and phase. Examples of the wakefield comparable in amplitude to the electric field of the laser field are given.

## Acknowledgments

We gratefully acknowledge C. Zorman for growing silicon carbide films. This work is supported by the US Department of Energy under Contracts Nos. DE-FG02-04ER54763, DE-FG02-04ER41321, W-31-109-ENG-38, and by the National Science Foundation grant PHY-0114336 administered by the FOCUS Center at the University of Michigan, Ann Arbor.

## Footnotes

One contribution of 15 to a Discussion Meeting Issue ‘Laser-driven particle accelerators: new sources of energetic particles and radiation’.

- © 2006 The Royal Society