The ion channel free-electron laser with varying betatron amplitude

The ion-channel laser (ICL) is an ultra-compact version of the free-electron laser (FEL), with the undulator replaced by an ion channel. Previous studies of the ICL assumed transverse momentum amplitudes which were unrealistically small for experiments. Here we show that this restriction can be removed by correctly taking into account the dependence of the resonance between oscillations and emitted field on the betatron amplitude, which must be treated as variable. The ICL model with this essential addition is described using the well-known formalism for the FEL. Analysis of the resulting scaled equations shows a realistic prospect of building a compact ICL source for fundamental wavelengths down to UV, and harmonics potentially extending to x-rays. The gain parameter ρ can attain values as high as 0.03, which permits driving an ICL with electron bunches with realistic emittance.


Introduction
The free-electron laser (FEL) [1,2] produces highly coherent, ultra-short duration light pulses with extremely high peak brilliance, and photon energies extending to above 10 keV. FELs are very useful for ultrafast time-resolved studies of the structure of matter but require high energy electron beams and long undulators, which makes them large and expensive. In spite of the high cost, several large national and international x-ray FELs [3] have been, or are being, built because of their potential for delivering new science and applications.
FELs are based on the collective interaction of high energy electrons that are periodically deflected by an undulator. The combined undulator and radiation fields give rise to a ponderomotive force that bunches the electrons on a wavelength scale and results in intense coherent emission. The self-amplified spontaneous emission FEL [2] produces coherent radiation by amplifying incoherent synchrotron radiation spontaneously emitted by the initially uncorrelated electron beam.
However, magnetostatic undulators are not the only means of providing a periodic transverse force. Whittum et al [4] suggested in 1990 that an ion-channel laser (ICL) could use the 'betatron' motion of electrons in an ion-channel to emulate an undulator, resulting in a very compact device.
An important difference between the FEL and ICL is the spatial periodicity of the transverse oscillations. In the FEL, this is fixed by the undulator, whereas in the ICL it depends on the ion density, and both the electron energy and oscillation amplitude. Due to this latter dependence, maintaining resonance with the emitted field in an ICL requires a small amplitude spread, unless the transverse momentum is very small. Only the latter case with very small amplitudes was treated in [4] and [5]. However, this is constrained to very low emittances that are very difficult to achieve in practice.
In this paper, we consider the more general and realistic case of high transverse momentum, which requires the betatron amplitude to be treated as variable. We derive a set of equations for the ICL describing, on a slow timescale, the complex amplitude of the amplified wave, and the axial momenta, betatron amplitudes, and ponderomotive phases of the oscillating electrons. We assume ultra-relativistic axial and high transverse momenta, but non-relativistic transverse velocity; we study the steady-state regime by neglecting slippage between electrons and wave, but include space-charge effects.
The form of the equations allows one to apply the well-known scaling procedure for the FEL, with an analogous fundamental coupling parameter ρ [2]. We present analytical and numerical results showing that for small ρ the evolution of field amplitude, phase bunching, and axial momentum in the ICL is virtually identical to the FEL.
We investigate how the growth of the radiation field depends on the initial spreads of axial electron momentum and betatron amplitude. Sufficiently low betatron amplitude spreads (compared to the mean oscillation amplitude) can be achieved by injecting the electrons off-axis and/or under an angle, as shown schematically, for just two electrons, in figure 1. The admissible betatron amplitude spreads lead to a small source size for the emitted radiation, which necessitates guiding to avoid diffraction. Small overlap between the radiating electrons and the guided mode makes space-charge effects relatively much more important than in the FEL. We find that at large values of ρ electron beams with realistic amplitude spreads and emittance can be used to drive the ICL and show that by removing the restriction of small amplitude, an UV ICL with high efficiency should be feasible.
The next section sets out our model for the ICL. In section 3, we apply a formalism similar to that for the FEL, before presenting numerical solutions in section 4. We discuss the results in section 5, and finally draw conclusions.

Hamiltonian
We consider test electrons in a cylindrical channel otherwise void of electrons, along the z-axis, with homogeneous stationary ion background. For motion in the y-z-plane, the electron energy is where ω ε = e n m [ ( )] p 2 0 0 1 2 is the plasma frequency for the background density, n 0 (with e the elementary charge, m the electron mass, and ε 0 the permittivity of free space).

Transverse electron motion
The test electron will perform betatron oscillations cos , Figure 1. Schematic of electron injection into the ion channel. An offset in position (y (0)) and/or momentum ( p (0)) from the channel axis (z-axis) leads to betatron oscillations (red trajectory) in the parabolic potential V(y), with amplitude β r . The orange trajectory is for an electron with slightly different initial conditions, but equal β r . An electron bunch with suitable initial distribution in phase space can have a small betatron amplitude spread.
represents a linearly polarized propagating wave, with phase ϕ ω = − t kz, where ω ≈ ck, and slowly varying complex amplitude a 0 .

Ponderomotive force
The corresponding axial ponderomotive force is is the ponderomotive phase, and the dot designates the total time derivative, where J 0,1 are Bessel functions, and accounts for the modulation of the axial velocity [6]:

Betatron amplitude evolution
; the betatron energy thus evolves as As β W depends on p z only through γ 0 , the first term on the rhs can be written as . The minus sign is due to p y , z, and t (and thus + p eA y rather than v y ) being kept fixed in the partial derivative. Similarly, as the only dependence on z and t is through A, the two remaining terms Hence, on a slow scale, the betatron amplitude evolves according to Betatron oscillations and wave are in resonance when the ponderomotive phase is stationary. At the position (0)¯z of the electron, with time-averaged velocity v z , the phase evolves as

Space-charge
In an electron bunch, space-charge forces [6] contribute to the slow longitudinal force; thus ⎡

Radiation emission
The radiation from these electrons adds to the wave amplitude: Here η h , defined above, accounts for the reduced emission at the fundamental frequency of the harmonic spectrum [6]. For a planar source, the amplitude of the ℓth harmonic evolves as ). The resulting spectrum has a synchrotron-like envelope, with critical frequency ω ω ≈ β a 3 8 c 3 . In resonance, 1. η m accounts for the spatial overlap of current density and radiation mode, which will be discussed below.

Steady-state
In the following, we neglect slippage between electrons and wave, as in the steady-state FEL regime [2], thus Combining equations (2) and (6) then yields a r r 0 1 2 2 is conserved if correlations between electron energy and ponderomotive phase can be neglected. Furthermore, equations (5) and (6) may be combined to express energy conservation:

FEL formalism
We express the energies γ j 0, and betatron amplitudes β r j , of individual electrons in terms of their initial averages and relative deviations q j and s j , respectively: Neglecting slippage, equations (2), (3) and (5), for each electron, and (6) form a closed set of equations, which are similar to the FEL equations [2]. In analogy to the FEL-parameter, we define  where the prime denotes the derivative with respect to τ, is the average detuning from resonance, and is the bunching factor. Equations (10) and (11)   is similar to that for the FEL [2]. For δ¯below a threshold value, there is an unstable solution with amplitude growing exponentially at a rate Γ ρ ω κ = β | Im ( )|, which gives the gain of the ICL. For small δ¯and ρ,

Numerical results
We have numerically solved the set of equations (8)- (11) for different values of ρ and δ, with small initial field, = − a |¯|(0) 10 0 3 , and vanishing initial bunching, ). We varied the initial spreads of momenta, σ q (¯(0)), and of betatron amplitudes, σ s ) to explore their effect on the interaction and determine the threshold conditions for a realizable ICL. Figure 2 shows the spreads of momentum σ q (¯), and betatron amplitude σ s (¯) as functions of τ for varying initial values, together with the corresponding field intensities, for ρ = 0.01 and δ = 2.0, which is optimized for fastest growth. For small initial spreads, the evolution of intensity a |¯| 0 2 , bunching b | |, and average 〈 〉 q |¯| and spread σ q (¯) of the momentum deviations is similar to the conventional FEL, with stages of lethargy, exponential growth, and saturation, where each of the scaled variables is of order unity and oscillates quasi- periodically [7]. If σ q (¯) initially is close to its saturation value, ∼2.0, it remains approximately constant, and the growth of the field is suppressed. Interestingly, the amplitude spread σ s (¯), which does not play a role in the FEL but affects the resonance in the ICL, evolves in an analogous way to σ q (¯). However, the threshold for σ s (¯(0)) to suppress the growth of a |¯| 0 2 is ρ ∼0.7 ; the contribution from σ s (¯) to the relevant spread σ P (¯) is scaled with ρ η h , cf. equation (13).
Varying ρ from 0 to 0.05, while maintaining optimized detuning, space-charge effects increase the scaled saturation intensity by about one third, and reduce the scaled gain coefficient by one half. The linearization yielding equations (12)- (14) is valid for ρ < 0.003. Figure 3 shows the dependence of the growth rate on initial momentum and amplitude spreads. These plots (and similar ones for the saturation amplitude) yield an approximate condition for amplification in the ICL: i.e., the relative spread, between different electrons, in the variable P, equation (9), must be less than ρ ∼ . These admissible spreads imply optimum detuning δ. The condition Δγ γ ρ < z z in [8], referring to variations of the 'axial energy' within a cycle, does not apply, since these are taken into account by the emission efficiency η h for the fundamental frequency of the harmonic spectrum.

Discussion
Whittumʼs original proposal for the ICL [4] would be very difficult to realize experimentally, at least for high γ˜0, due to the restriction to very small transverse momenta, ≪ β a 1, which would also lead to very low gain and low efficiency and thus unfeasibly long devices. However, we have shown here that large transverse momenta can realistically be used, by explicitly taking into account the effect of the betatron amplitude on the resonance. This allows the coupled radiation-matter equations to be cast in a form similar to that of the conventional FEL.
An ion channel can be realized experimentally by focusing a laser pulse with relativistic amplitude ω > E m c e L L (where ω L is the laser frequency) into plasma. Its ponderomotive force displaces the electrons from its path and a 'bubble' structure is formed, which provides the required transverse field in addition to a longitudinal wakefield [9]. To minimize the variation of γ˜0, the electron bunch should be close to dephasing, at the centre of the 'bubble'; as their velocities are different this limits the useful propagation length. Figure 4 shows possible initial phase-space distributions resulting in low amplitude spread. Small σ β β r R ( ) , ρ ∼0.7 , can be achieved by injecting electrons either off-axis at a distance β R , with σ ρ ≈ β y R ( ) 0.5 ( figure 4(a)) and a range of betatron phases σ φ and σ ρ ≈ β y R ( ) 0.8 ( figure 4(b)). An electron bunch trapped in the 'bubble' could be offset from the axis by perturbing the propagation direction of the laser and thus of the 'bubble' [10].
The normalized emittance, 0.8˜6.6 10 10 cm m˜. In the x-direction, perpendicular to the polarization, the matched bunch width for ϵ ϵ =  ). An analysis similar to [11] shows that the plasma channel can guide high-frequency modes if its radius exceeds ω c p . The overlap factor of the lowest order mode with an electron bunch, oscillating with amplitude matched to the channel radius, is η ≈ 0.01 m for ω ω ≫ p . Electron bunches can be accelerated in laser wakefields to γ = − 200 300 0 with relative energy spread as small as σ γ γ ∼ ( (0))˜0.01 0 0 [12], and normalized emittance ϵ ∼ − π 10 m . The current in these cases is ≈ I 600 A, and the efficiency of converting kinetic electron energy into radiated energy, ρ ≈ a |¯| 2% ; the current is 10 kA, the peak power emitted at λ, 50 GW, and the photon rate × . While in the present study we consider planar electron motion and linear polarization of the radiation field, the theory can be extended to describe betatron oscillations in both transverse directions, leading to elliptical polarization. The simplest case, equal amplitudes in both directions, with a phase difference of π 2, results in circular polarization. In this case, the distance from the axis and the magnitude of the transverse momentum are slowly-varying; as a consequence, there will be no synchrotron-like spectrum. Experimentally, these cases can be realized by combining off-axis and oblique injection out of plane.

Conclusions
In conclusion, we have developed a comprehensive model for the ICL by studying the collective interaction of electrons moving in an ion channel with a propagating wave. Including the effect of variable betatron amplitude on the resonance between field and electrons is essential to correctly describe experimentally accessible regimes. We show that the ICL can be described in a similar way to the FEL and define an analogous ρ-parameter, which can realistically reach 0.03, which is high for an FEL. However, space charge effects are relatively much more important here, with a corresponding new coupling parameter ρ ρ ∼ 400 . Numerical solutions with varying initial spreads in axial momentum and betatron amplitude confirm the condition, known from the FEL, that the relative momentum spread must be less than ρ ∼ , and give an analogous condition for the amplitude spread. A numerical example indicates the requirements for operating an ICL down to UV fundamental wavelengths, with harmonics potentially extending to x-rays. A high value of ρ is essential for reconciling the requirement of low betatron amplitude for emission at short wavelength with experimentally accessible amplitude spread. While the present study covers the steady-state case and neglects longitudinal plasma fields, the model can be readily extended to the timedependent regime where superradiant pulses will evolve [15][16][17][18][19]. The resonance with a propagating wave also plays an important role in laser-driven betatron oscillations [20,21].