Currently, high electron energies of tens of GeV are achieved with radio-frequency (RF) linear accelerators that operate in the GHz frequency range with a typical length of a few kilometers^[1]. In idealized conditions, breakdown^[2] limits the accelerating electric field to the order of a few hundreds of . In practice, gradients reach values of when operating at room temperature^[1] and in their superconductive counterpart^[3].

In the past two decades, with the immense progress in laser technology, laser plasma accelerators have become able generate hundreds of ^[4–6]. In this scheme, high intensity focused laser pulses with lengths of the order of the plasma wavelength generate an intense wake. The plasma wake, which trails behind the laser pulse with the same group velocity, can accelerate electrons from the plasma itself. Work is in progress to accelerate electrons that do not originate in the plasma.

Based on the chirped pulse amplification (CPA) technique^[7], pulsed laser technology facilitates focus on plasma target pulses with intensities as high as and duration of the order of femtoseconds ( s) for a laser wavelength of that is optimized to a plasma density of . In a series of experiments reported in 2004^[8–10], quasi-monoenergetic e-beams with energies of the order of 100 MeV have been demonstrated. More recently, several groups have demonstrated quasi-monoenergetic e-beams with energies of up to 2 GeV^[11].

An intense wake may also develop by replacing the laser pulse with an energetic e-beam in plasma, and it is shown experimentally^[12] that an initial bunch of 40 GeV can generate an intense wake of the order of , which results in acceleration of a trailing bunch to an energy of about 80 GeV with about 16% energy spread. The total number of injected electrons in the bunch is and the spot size is , whereas the number of electrons accelerated to 80 GeV is about . For comparison, in a dielectric loaded waveguide, a 60 MeV bunch of electrons can generate Cerenkov gradients of ^[13]. The total number of injected electrons in the bunch is and the spot size is , whereas the number of accelerated electrons is about .

In the paradigm analyzed here, the gradients are more modest (order of ) and it is conceptually closer to a conventional two-beam acceleration scheme. It relies on transferring energy stored from the active medium to a train of electron bunches – see Refs. [14, 15]. In contrast to previously mentioned schemes, where the energy for the acceleration comes from either energetic laser pulses or an energetic electron bunch, in this scheme the energy comes from excited atoms. In the first approach of this scheme, an electron bunch injected into an active medium generates a wake comprised of a broadband spectrum of evanescent waves. Since the active medium is a resonant medium, only the fraction of the wake spectrum that is close to the resonance frequency of the medium will be amplified. Thus, it is proposed to inject a spatially modulated bunch with periodicity equal to the resonance wavelength so that a large portion of the wake spectrum lies in the vicinity of the resonance frequency of the medium. As a result, the amplified wake accelerates directly the injected bunch. This approach has been demonstrated in an experiment performed at Brookhaven National Laboratory Accelerator Test Facility (BNL-ATF)^[16], in which a density modulated bunch with an energy of 45 MeV gained energy of 200 keV from an active gas mixture.

For a description of the paradigm, we start with a general description of the system followed by a simplified model used to investigate the essential phenomena involved. A trigger bunch propagates in a vacuum channel surrounded by a low loss dielectric layer which is thick enough to sustain the 8–10 atm pressure of the mixture consisting of the active medium. The latter in turn is confined by a Bragg waveguide which facilitates excitation of the active medium on the one hand and allows full confinement of a resonant Cherenkov wake. One of the eigenmodes is amplified by the active medium and many wavelengths behind the trigger bunch the former accelerates a trailing bunch. For the sake of simplicity, the analysis that follows relies on a metallic waveguide that contains the gaseous active medium.

Initially, a triggering bunch of electrons generates a Cerenkov wake with electric field in the longitudinal direction, which in turn is amplified by the active medium - stimulated emission process. As the wake field is amplified, the population inversion in the active medium is reduced and, as a result, the spatial gain is also reduced. When the spatial gain is zero, the wake reaches saturation and it can accelerate a second bunch of electrons trailing behind the triggering bunch.

Previous study of this scheme used a linear model^{[15, 17–19]} and a simplified nonlinear model^[20]. In the linear model, where a constant population inversion density (PID) is assumed, the wake is exponentially amplified. Since the propagating wake in the cylindrical waveguide propagates in an oblique angle, it reflects multiple times from the boundaries. As a result, the effective propagation path is longer than if the wake were propagating in a boundless medium. For a structure of a given length, the effective gain of the wake is enhanced. The simplified nonlinear model assumes the propagation of one electromagnetic (EM) mode^[20]; it does not account for the multiple reflections of the wake from the waveguide boundary and only the dynamics of the polarization field is considered.

In this paper we extend the previous linear model^{[18, 19]} and include the nonlinear dynamics of the active medium. The extended model can describe the wake saturation level and the interval of time in which the wake reaches saturation. While the previously mentioned approaches^[18–20] predict the saturation process qualitatively, the present approach is far more quantitative.

This paper is organized as follows. Section 2 presents the dynamics equations that describe the Cerenkov wake amplification by the active medium. It also describes the dynamics of the polarization and the population inversion density. In Section 3 the single-mode resonance condition, the value of the saturated wake and the saturation length are calculated analytically. In addition, we calculate for a single mode the width or the spot size of the longitudinal wake. In Section 4 we show numerically for a modulated trigger bunch the dynamics of the wake and the population inversion density for a gas mixture. We conclude (Section 5) with discussion and conclusions.

The structure of interest consists of a cylindrical metallic waveguide of radius filled with an active medium (see Figure 1). Far from the resonance of the active medium it is assumed that the dielectric coefficient of the medium is and it is frequency independent. A bunch of electrons, the trigger bunch, is injected into the structure with velocity larger than the Cerenkov velocity , where is the speed of light in vacuum. Assuming azimuthal symmetry, the bunch excites an entire manifold of transverse magnetic (TM) modes which propagate behind^[21]. Each mode consists of longitudinal and radial electric field components () as well as an azimuthal magnetic field ().

This superposition of TM modes, also known as the wake field, travels at the same speed as the trigger bunch. Among this infinite manifold of TM modes only those with frequency equal to or close enough to the resonance frequency of the active medium, , will be amplified through the stimulated emission process. Thus, the EM field has the following generic form

 (2.1)

Similarly to the EM fields, the current density of the trigger bunch is assumed to be of the form

 (2.2)

Having in mind that the growth rate is much smaller than the resonance frequency, the dynamics of the fields can be separated into two major time scales: the ‘fast’ and the ‘slow’ time scales. The ‘fast’ time scale is the medium resonance period of time whereas the ‘slow’ time scale associated with the growth rate of the field envelopes, , is , where is the ‘plasma frequency’ of the active medium. Here, is the initial PID, is the average dipole moment and is the dipole moment. The parameter is the reduced Planck constant and is the vacuum permittivity. Thus, the dynamics of the wake and the active medium can be described by the slowly varying envelope approximation. Consequently, the dynamics of the normalized EM fields derived from the Ampere and Faraday laws read

 (2.3)

 (2.4)

 (2.5)

The expression is the normalized bunch current, where , and describes the electron bunch profile in the longitudinal direction. In this study, the bunch injected at with a length of has a profile of , and zero otherwise, where .

The active medium is modeled semi-classically as a two-level system within the framework of the dipole approximation^{[22, 23]}. In addition, it is assumed that only stimulated emission can reduce the population inversion density and collisions of the second kind are neglected here. The response of the active medium to the wake is through the normalized polarization fields and ,

 (2.6)

 (2.7)

The dynamics of the PID, , reads

 (2.8)

Finally, the set of equations introduced in Equations (2.3)–(2.8) conserves energy,

 (2.9)

 (2.10)

In this section we determine analytically the single-mode resonance condition, the value of the saturated wake and the saturation length. In addition, we calculate for a single mode the spot size of the longitudinal wake.

In the considered structure only the modes with frequencies adjacent to the resonance frequency will be amplified. In our model it is possible to find the modes that will be amplified from the dynamics of as follows. Substitution of and from Equations (2.4) and (2.5), respectively, into Equation (2.3) results in

 (3.1)

 (3.2)

The dynamics of the wake can be divided into three parts. In the first part of the amplification, where the PID is weakly depleted i.e., , known also as the linear regime, the solution of Equations (2.3)–(2.7) assuming a single-mode longitudinal wake is

 (3.3)

 (3.4)

 (3.5)

 (3.6)

 (3.7)

In the second part of the amplification the PID is significantly depleted () as a result of the stimulated emission (right-hand side of Equation (2.8)), thus reducing the effective gain of the medium. In this nonlinear regime of amplification, where the PID is getting depleted and the wake is intense, both the amplified wake and the PID can experience Rabi oscillations^[24] before reaching the third regime of deep saturation. In the latter case the PID is completely depleted () and as a result the effective gain is zero. Hence, the medium is transparent to the propagating wake and the interaction reaches full saturation.

The value of the saturated wake can be found from the energy conservation (Equation (2.9)). Assuming that and at most of the energy is stored in the active medium rather than the trigger bunch () then Equation (2.9) becomes

 (3.8)

 (3.9)

 (3.10)

 (3.11)

To determine the saturation time we first evaluate the time interval of the quasi-linear regime and then the nonlinear regime. We define the time interval of the quasi-linear regime to be from the point where the trigger bunch ends () until the point where the PID reaches its first time zero, . Motivated by the linear regime (see Equations (3.4) and (3.7)), where the amplitude during satisfies

 (3.12)

 (3.13)

 (3.14)

 (3.15)

Since , the Rabi frequency in physical units is , where is in the same form as the Rabi frequency in a homogeneous medium.

For a relativistic bunch that travels close to the Cerenkov velocity ( and ) the Rabi frequency is . Hence, in the nonlinear regime where the wake oscillates at

 (3.16)

 (3.17)

 (3.18)

 (3.19)

 (3.20)

Finally, the theoretical spot size, , of the longitudinal wake can be calculated through . Since the first zero of is and from the single-resonance condition , we obtain that the spot size of the longitudinal wake is

 (3.21)

In this section we show the wake dynamics for an active gas mixture with the set of parameters that is given in Table 1.

Since the bunch profile is continuously changing except for a finite number of points where it can have first-order discontinuity, the initial conditions of all the envelopes are set to zero but the initial PID is .

Figure 2 shows the wake and the medium dynamics on the waveguide axis (). In this example the trigger bunch (Figures 2(a) and (b) dashed curves) which appears at and ends at (corresponding to a bunch length of ) generates the initial wake. As seen in Figure 2(a), the wake (solid curve) amplification begins in the linear regime, where the PID (dashed–dotted curve) is weakly depleted (). In this regime the growth rate of the medium is constant, which results in exponential amplification of the wake (Figure 2(b)).

As the wake is amplified the PID is depleted. In this nonlinear regime of amplification, where the PID is significantly reduced and the wake is intense, Figure 2(a) shows that the amplified wake and the PID experience Rabi oscillations. Finally, when the PID relaxes to zero, the effective gain of the medium is zero and the wake reaches deep saturation.

Figure 2(a) shows that the normalized wake reaches a saturation value similar to the theoretical calculation (Equation (3.10) with ) of . Also, the normalized saturation time in this example is , where the depletion time is and the relaxation time is . These time parameters are in good agreement with the theoretical formulas (Equations (3.13) and (3.18)) of , and .

Figure 2(b) shows in a logarithmic scale the nonlinear wake amplification in physical units of . Clearly, in the linear regime the wake (solid curve) grows exponentially and it saturates about 6 cm behind the trigger bunch to a value of . In addition, Figure 2(b) shows in the linear regime good agreement between the nonlinear wake dynamics (solid curve) and the linear wake dynamics (dots) which is calculated for .

Similarly to the linear regime approach^{[18, 19]}, the waveguide radius is chosen such that only a single TM mode will be in resonance with the active medium. Indeed, Figure 3 shows at that among the first 500 modes of that are calculated in the simulation only is the dominant mode.

Figure 4 shows the two-dimensional plot of the longitudinal wake. At , corresponding to , Figures 4(b) and (c) show that the radius of the spot is about , which is smaller than the waveguide radius of . This spot size radius is in good agreement with our theoretical expression (Equation (3.21)) of . This means that such a wake can accelerate a second bunch of electrons with a beam radius of less than a hundred microns.

The energy balance of the structure shown in Figure 5 reveals that initially most of the energy comes from the active medium (Figure 5(c)) and only a small fraction of it comes from the trigger bunch (Figure 5(b)). In the steady state of the amplification all the stored energy from the active medium is transferred to EM energy (Figure 5(a)) such that the energy of the longitudinal wake, , is the same as that of the transverse wake, . In addition, the stored energy of level 1 in the active medium, , is equal to that of level 2, (Figure 5(c)), which means that in the steady state the total stored energy in the active medium is zero, . The deviation from energy conservation is shown in Figure 5(d) and can be used for the numerical simulation error which in our example is about .

As mentioned previously, the shape of the single-mode wake is sensitive to the active medium bandwidth, , and the Cerenkov parameter, . These two parameters affect the Rabi frequency and the relaxation time . Figure 6 shows the dynamics of the wake for different Cerenkov and bandwidth parameters. The solid curve corresponds to the same wake dynamics as in Figure 6(a) with and . As seen for a Cerenkov parameter five times larger than our former example (dashed curve) the number of Rabi oscillations is significantly lowered. However, the saturation length, , is greatly increased as a result of increased —see Equation (3.13). It may be possible to suppress the Rabi oscillation when the medium bandwidth is enlarged. However, increasing the bandwidth can result in multi-mode interaction if . Figure 6(a) shows (dotted curve) that for a bandwidth that is five times larger the shape of the wake in the steady-state regime has a steady oscillation which results from multi-mode coupling. In this example, , whereas for the first single-mode propagation we have , where . Figure 6(b) shows the mode spectrum of the wake. As seen for large bandwidth (dotted curve), the adjacent modes ( and 361) participate in the wake dynamics shown in Figure 6(a) (dotted curve).

The two major parameters that characterize our scheme are the value of the wake at the saturation and the saturation length. The sensitivities of these two quantities to the other parameters of the scheme are revealed in Figure 7. Specifically, Figure 7 shows the dependence of the value of the saturated wake (Figure 7(a)) and the saturation time (Figure 7(b)) on the waveguide radius, the dielectric constant, the active medium bandwidth, the bunch energy and the bunch radius. Clearly, the saturation value of the wake is significantly dependent on the waveguide radius, , the medium bandwidth, , and the parameter , which is proportional to the Cerenkov parameter .

For a relativistic beam the scaling law of the saturated wake on the structure parameters can be explained by our analytical calculations (Equation (3.11)) and the single-resonance condition of (see Equation (3.2)),

 (4.1)

Similarly, the scaling law of the saturation time on the structure parameters can be explained by our analytical calculations of the saturation time (Equations (3.14) and (3.20)) and the single resonance condition of (see Equation (3.2)),

 (4.2)

At this point the saturation time is significantly large because the deviation from satisfying the resonance condition is larger than for a larger waveguide radius . Specifically, for the deviation from the nearest resonance mode is , whereas for the deviation is . When the waveguide radius is close to satisfying a single resonance, then there is a small dependence on the saturation time, as confirmed by the logarithmic dependence in Equation (4.2). Otherwise the wake growth rate is significantly reduced.

In this paper we present the nonlinear aspects of a new scheme of electron acceleration by an active medium. In contrast to plasma-based accelerators, where the energy for generating the intense wake originates in the intense laser pulse, in our scheme the energy source is stored in excited atoms/molecules. Most importantly, we have shown that a centimeter-sized cylindrical waveguide filled with a gas mixture can generate gradients of the order of traveling at the trigger bunch velocity.

We analyzed the dynamics of the wake and the active medium both analytically and numerically. Our numerical simulations indicate that the dynamics of the wake can be divided into three regimes. In the first regime, known as the linear regime, the PID is nearly constant (). Hence, the gain of the medium is constant which results in exponential growth of the wake. The second regime starts when the amplified wake reaches high intensity, and the PID is significantly reduced () by the stimulated emission effect. In this regime both the PID and the wake can experience Rabi oscillations. The Rabi frequency in our confined structure can be significantly larger than in a boundless medium. More specifically, this frequency is inversely proportional to the square root of both the Cerenkov parameter and the mode number. Finally, in the third regime, when the Rabi oscillations are relaxed, the PID reaches complete depletion () and the wake reaches deep saturation.

Energy conservation proves that most of the initial energy originates from the active medium, and in the deep saturation regime half of the initial stored energy is transferred to the longitudinal EM field component and the other half is transferred to the transverse EM field components.

To obtain maximum performance from our studied structure we should fulfill the following constraints. First, it is desired to design the structure for single-resonance-mode operation to obtain a constant value of the wake in the saturation regime. In order to avoid multi-mode propagation of adjacent frequencies, the active medium bandwidth should be narrow enough. Second, the spot size of the wake should be large enough in order to accelerate most of the trailing train bunch. We found that for single-mode propagation a large spot size is achieved for a small Cerenkov parameter. The last requirement is obviously the generation of a high gradient in a short saturation length.

On one hand, we have found that for a relativistic trigger bunch the saturation value is strongly dependent on the waveguide radius and the medium bandwidth but weakly dependent on the Cerenkov parameter. On the other hand, we have found that for single-mode propagation and a relativistic trigger bunch the saturation length is strongly dependent on the Cerenkov parameter and the medium bandwidth but weakly dependent on the waveguide and electron beam radius. Consequently, optimal performance may be achieved with a large waveguide radius filled with a high density of excited atoms and a trigger bunch that travels at a velocity slightly above the Cerenkov velocity. Interestingly, the short saturation length of the wake may be used as a seed pulse for backward Raman amplification in plasma^[25].

For a proper perspective, it is important to comment that the high value of the gradient is limited by self-focusing and ionization effects^[23], which are not analyzed here, but which may be overcome by using a vacuum channel, as previously mentioned in Section 1.

In spite of the relatively modest gradient, compared with laser plasma accelerators, our paradigm may benefit from a few aspects. First, our scheme can support staging^{[26, 27]} in a fairly natural way. Second, our setup, in principle, does not seem to suffer from instabilities that may evolve in plasma at a high repetition rate. A typical repetition rate for laser plasma accelerators is less than 1 Hz for 100 J laser pulses with fs duration in a laser plasma accelerator^[6]. The various laser wake acceleration schemes in plasma take advantage of the fact that on the fs time scale all ion instabilities are far below threshold; at high repetition rate these instabilities may develop^[28]. Their suppression may require a dramatic reduction of the laser power.

Regarding our paradigm as a new source of coherent radiation excited by a trigger bunch, it possesses important features that distinguish it from a conventional laser driven by spontaneous emission. The most remarkable ones are the excitation of a specific mode number and wake propagation at the same phase and group velocity as the trigger bunch.