1. Introduction

Since proposed by T. Tajima and J. M. Dawson [1] in 1979, laser wakefield accelerators (LWFA) have made tremendous progress in electron acceleration and x-ray light sources. Electrons can be trapped and accelerated in the wakefield driven by an intense femtosecond (fs) laser with typical accelerating electric field of 100 GV/m [2,3], possessing the inherent fs time duration and tens of pC charge, thus reaching multi-kA peak beam current. Electrons with GeV energy [4,5], ultra-high brightness [6], and kA beam current [7,8] have been obtained experimentally.

LWFA can also be used to generate betatron and undulator radiation. Despite the longitudinal acceleration, electrons can simultaneously undergo transverse betatron oscillations in the wakefield which lead to the so called betatron radiation. Betatron x-ray pulses usually have broadband spectrum [9–11] since the transverse wakefield is so strong that acts as a wiggler. Undulator radiation combing permanent magnet and LWFA has also been demonstrated for visible light [12] and soft x-ray [13]. In such experiments, electrons accelerated by LWFA were injected into a permanent magnet undulator placing tens of centimeters downstream the gas jet. For this type of undulator, the typical values for period and magnetic field are of 1cm and 1 Tesla, respectively [14]. However, owing to the large divergence of LWFA electron beams (~mrad), the electron flux would be reduced by few orders of magnitude after propagating through the long gap between the accelerator and undulator. Thus, the x-ray brightness is strongly limited, e.g., 1.3 × 10¹⁷photons/s/mrad²/mm²/0.1%BW as reported in the soft x-ray region [13], and a plasma undulator was proposed to obtain higher photon brightness by placing a nanowire array a few millimeters downstream the LWFA [15].

Shaped quasi-static strong magnetic field can be generated by laser driven capacitor-coil target [16–18], which is comprised of two parallel metal plates connected with a shaped coil, as schematically shown in Fig. 1. The two plates form a capacitor and drive a quasi-static (time scale of a few nanoseconds) current through the coil when one is irradiated by a high-energy nanosecond (ns) laser propagating through a hole on another plate, which collects the hot electrons emitted from the laser-produced plasma. Magnetic field exceeding hundreds of Tesla and lasting tens of ns is typical for such experiments [17,19,20].

 

Fig. 1 Schematic diagram of the bifilar elliptical undulator and the coupled x-ray source. (a) Intense femtosecond laser is focused into a gas jet to drive a wakefield and accelerate electrons. Capacitor-Coil target is formed by a bifilar coil and two copper plates. High energy nanosecond laser is focused on one plate through a hole on another. Electrons accelerated in the wakefield present elliptical orbits after injecting into the undulator along the coil axis. Ultra-bright x-ray will be emitted. The coil pitch defines the undulator period λ_u while the undulator strength is determined by the magnetic field amplitude on axis. The gap between the LWFA and the capacitor in x can be as short as millimeter-scale. (b) Three-view drawing of the bifilar coil.

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In this paper, we present a helical undulator based on a bifilar shaped capacitor-coil. A bifilar coil [21] containing DNA-like double helical windings presents a helical magnetic distribution on axis when currents with equivalent amplitude flowing in opposite directions in those two wingdings. When synchronized LWFA electron beam is generated upstream the coil, the strong quasi-static magnetic field acts as an elliptical undulator and will bent the trajectories of electrons injecting along the coil axis thus produce synchrotron radiation. Significantly, the period of this laser driven elliptical undulator can be as short as a few hundred microns and the amplitude of the magnetic field as high as tens of Tesla, while the gap between LWFA and the undulator can be as short as a few millimeters. Combing this short period high strength undulator with LWFA will make better use of the high electron flux of LWFA and lead to a much higher x-ray brightness.

2. Results

2.1. Principle of the short period high strength undulator

The proposed scheme of the undulator and coupled x-ray source is presented in Fig. 1. We consider a laser with 1 ns pulse duration, 1 kJ energy and 1.06 μm wavelength is focused onto one of the copper plates (anode) with a focal spot size of 50 μm. Another copper plate (cathode), which collects the expanding hot electrons, is separated 600 μm from the first one, and the diameters of the two plates both are 3.6 mm, similar to that used in current experiments [17,20]. One can expect discharge currents with opposite directions in the two helices since the charge separation will build up an electrical potential between the two plates and the capacitor acts as a voltage source [22]. The LWFA is coaxially placed upstream the coil with 1 mm gap, and the initial point of the electron beam is defined as origin (x = 0, y = 0, z = 0) of the whole system. The helix pitch defines the undulator period λ_u while the undulator strength is determined by the magnetic field amplitude on axis. For simplicity, in the following we consider a 20 turn bifilar coil with 500 μm internal diameter, 500 μm helix pitch and 50 μm wire diameter.

The temporal evolution of the coil current can be obtained since the capacitor coil can be described as a RLC electric circuit which is governed by the laser and target geometry [23,24]. For the capacitor coil considered above, the resistance, self-inductance, mutual inductance and the capacitance are R = 0.158 Ω, L = 6.0 nH, M₁₂ = M₂₁ = −0.5 nH and C = 0.184 pF, respectively. It is known that plasma dynamics in the capacitor is essential but difficult for the coil current modeling, and we estimate the coil current according to the model developed by V.T.Tikhonchuk [24]. This model, accounting for the space charge neutralization and plasma magnetization between the capacitor plates, shows good validity when compared with former experimental data. For our case, this model indicates a current with 36.0 kA peak amplitude, ≤ 1 ns raising time and τ_re = L/R~38.0 ns relaxation time can be generated. In principle, we can realize the variation of undulator period and strength, respectively, by changing the coil pitch and the time delay between the fs laser and ns laser.

The spatial field pattern is essential for a magnetic undulator. For an ideal bifilar coil, near the coil axis (y = 0, z = 0), the axial magnetic field B_x vanishes while the transverse component B_y, B_z are sinusoidal with the period equals to the coil pitch λ_u. The static 3D field distribution of the bifilar coil is calculated numerically and the whole coil geometry, including the connection wires between the coil and copper plates, has been taken into account with the undulator length L_u = 12.0 mm. The calculation is based on the fact that the time interval (τ_e = L_u/c~40.0 ps) of relativistic electrons propagating through the whole undulator is negligible small compared with relaxation time τ_re of the coil current (38.0 ns), thus the current amplitude can be regarded as a constant and static magnetic field distribution is assumed. The field map in Fig. 2 shows the magnetic field in the horizontal plane (y = 0) under a 30 kA coil current. Figures 2(a), 2(b) and 2(c) represents B_x, B_y, B_z, respectively. The map apparently shows an alternating polarity magnetic field distribution with 500 μm period determined by the coil pitch, and, along the axis, magnetic field presents an elliptical distribution. The magnetic field experienced by one typical electron distinctly confirms the elliptical field pattern, as illustrated by the red dotted line in Fig. 2. It should be noted that taking the whole coil geometry into account is vital for accurately modeling the undulator field. For a symmetric bifilar coil, the field amplitude B_y0, B_z0 should be exactly equal with each other. For our case, however, coil asymmetry originating from connection wires results in the average field amplitude B_y0 = 16.3 T and B_z0 = 14.8 T, respectively. Further, the magnetic field decreases to zero at 1 mm away from the coil (x = 0 mm and x = 12 mm), thus the gap between LWFA and undulator can be set to as short as 1 mm without net effect on each other. This will allow making better use of high electron flux of LWFA and eventually lead to an increase of x-ray brightness [15].

 

Fig. 2 Magnetic field distribution with 30 kA coil current. (a), (b), (c) represents B_x, B_y, B_z with the same scale, respectively, and the coordinate is shown in Fig. 1. The field map shows alternating-polarity magnetic field distribution with the period equivalent with coil pitch. The red dotted line represents the magnetic field experienced by a typical electron and shows an elliptical field pattern. The axial part B_x equals to zero while B_y, B_z are sinusoidal.

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We will show the electron dynamics inside the undulator, which is also critical for the operation of the coupled x-ray source. Figure 3 shows transverse motions of a typical electron inside the undulator and transverse phase space distribution of an electron beam. LWFA electrons are injected into the undulator along the coil axis. Electron presents an elliptical orbit with a period of 500 μm, as shown in Figs. 3(b) and 3(c). It should be mentioned that the sinusoidal electron motion is combined with small transverse deflection and drift as the result of field inhomogeneity, i.e. electron position in z direction drifts 0.57 μm at the exit (x = 12 mm) of the undulator as illustrated in Fig. 2(a). This drift is consistent with the value 0.62 μm given by the second field integral [14]:$z=\raisebox{1ex}{$e$}\!\left/ \!\raisebox{-1ex}{$\gamma m_{0}c$}\right.I{I}_y=\raisebox{1ex}{$e$}\!\left/ \!\raisebox{-1ex}{$\gamma m_{0}c$}\right.\underset{-\infty }{\overset{+\infty }{{\displaystyle \int }}}\underset{-\infty }{\overset{s}{{\displaystyle \int }}}{B}_y\left(s^{\prime }\right)d{s}^{\prime }ds$, where e, m₀, γ and c are the elementary charge, the electron rest mass, the Lorentz factor of the electron and the speed of light in vacuum, respectively. For lower electron energy, this drift will increase but still negligible (< 5 μm) when compared with the coil diameter for electrons considered in this paper. Electron beams will maintain this small drift and defocus due to the initial divergence when moving in the undulator. The corresponding electron beam phase space distribution at the undulator entrance and exit are shown in Figs. 3(d)–3(g). After propagating through the undulator, the beam radius (root mean square, rms) in y and z direction grow up from 3.0 μm to 5.19 μm and 5.30 μm, respectively, while the divergence is consistent with the initial one. It should be noted that this growth in beam size is acceptable for the purpose of high brightness undulator radiation, while, however, toward x-ray light sources where extensive undulator periods are required, i.e. x-ray free electron lasers (X-FEL), magnet devices for carefully manipulating electron phase space distribution are needed.

 

Fig. 3 Transverse motions and phase space distribution of electrons. (a), (b), (c) represent z, y^’ = p_y/p_x, z^’ = p_z/p_x for a typical electron, respectively, where z is the electron position in one transverse direction and p_i is the electron momentum in corresponding direction. (d), (e) Electrons are Gaussian distributed at the undulator entrance ($x=0$) with the rms beam radius of 3 μm and divergence of 0.6 mrad in y and z. (f), (g) After propogation through the undulator, electrons present a defocusing elliptical distribution with the rms beam radius and divergence of 5.3μm and 0.6 mrad, respectively. For this calculation, electron parameters are consistent with the UTEXAS experiment result listed in Table. 1.

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2.2. X-ray radiation coupling the undulator with LWFA

To get further insight of the features of the x-ray source, we modelled the coupled system numerically by synchrotron radiation code WAVE [25]. Insertion devices can be characterized by the strength parameter, K = eB₀λ_u/2πm_ec, where B₀ = (B_y² + B_z²)^1/2 is the on axis magnetic field amplitude for elliptical undulator and λ_u is undulator period as mentioned above. For the 30 kA coil current with λ_u = 500 μm, B₀ = 22.02 T thus K equals to 1.07. The strength paramater differs undulator (K$\le$1) and wiggler (K$\gg$1) regime in terms of photon energy and yield [26]. In the undulator regime (K$\le$1), electrons radiate around the fundamental wavelength λ_γ = λ_u(1 + K²/2)/2γ² on axis with the number of photons N_γ = 2παK²/3 emmited per magnet period, α ≈1/137 is the fine structure constant and K is the strength parameter. To verify the validity of the proposed x-ray source, undulator radiation coupling this undulator and state-of-the-art LWFA is calculated numerically. Magnetic field inside a 12 × 0.5 × 0.5 mm box with grid resolution (λ_u/50, λ_u/25, λ_u/25) in x, y, z direction, respectively, was loaded into WAVE for electron beam tracking and spectrum calculation. In order to base our studies on experimentally verified data, we take state-of-the-art electron beam parameters close to that obtained by three groups [4–6] into account and summarize the electron pamameters in Table 1. In the case of LBNL electron beam, as shown in Fig. 4, x-ray pulse peaked at 252.7 keV with 59.7% (FWHM) relative energy spread (RES) and a peak brightness of 3.13 × 10²⁵ photons/s/mrad²/mm²/0.1%BW can be generated. This brightness is comparable and even higher than that of the third generation sychrotron radiation based on energy recovery linear accelerators but at a much higher photon energy. The Stokes vector at peak photon energy is (1, −0.07, 0.007, 1), which suggest the x-ray is almost pure right-hand circularly polarizd. This relatively large x-ray RES mainly comes from the large RES (6%) and divergence of the electron beam since they contribute to the spectrum broadening as [27] (∆λ_γ/λ_γ)² = (2∆γ/γ)² + (γ²ε²/σ²)² + (1/N)², where ε and σ are the emmitance and transverse beam radius of the electron beam. And x-rays with RES down to 15.1% (FWHM) can also be generated by electron beam of SIOM case thanks to the small electron RES (< 1%). For those three cases, energy conversion efficiencies from femtosecond laser to x-rays are on the order of 10⁻⁶ to 10⁻⁵, which is comparable to that of betatron radiation from LWFAs [9,11,28].

Table 1. Undulator and electron beam parameters

View Table

 

Fig. 4 Undulator radiation spectrum by state-of-the-art LWFA. Electron beams are injected into the undulator at x = 0 with the initial parameters sampling from Gaussian distribution. The blue solid, yellow dashed and red dot-dashed lines denote electron beams with parameters close to that obtained by SIOM, UTEXAS, and LBNL, respectively.

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On the other hand, tunabilities in photon energy and brightness are of key importance for applications of x-ray light source. In synchrotron radiation science, a commen way of tuning the photon energy is variation of the magnetic field amplitude B₀ since the variation of electron energy is rather complicated and even impossible due to the focuing and deflecting devices. For the undulator proposed, variation of B₀ can be obtained by tuning the coil current. This can be realized either by varying the time delay between the fs and ns laser or by variation of the energy of the ns laser [29]. Magnetic field amplitude B₀ under various coil current and corresponding K are shown in Fig. 5(a). The field amplitude increses linearly with the coil current, and as a rule of thumb 1.0 kA equals to 0.74 T for our capacitor coil geometry. The strength parameter K can be varied from 0.34 to 1.07 while the coil current increses from 10 kA to 30 kA. And the corresponding peak photon energy can be varied from 17.3 keV to 12.2 keV when we take 1 GeV electrons as an example, as illustrated in Fig. 6(b). The variation of electron energy presents a more efficient way since the coupled x-ray source is free of other beam manipulating devices and the dependence of photon energy and total power radiated on electron energy scales as γ². Peak x-ray photon energy increse from 4.2 keV to 59.2 keV while the electron energy ranging from 0.6 GeV to 2.2 GeV with 30 kA coil current, blue dot-dashed line in Fig. 6(b). Higher photon energy can also be obtained with lower coil current on the cost of lower x-ray brightness, i.e., 6.3 keV to 84.9 keV under 10 kA coil current with the same electron energy range above, red line in Fig. 6.

 

Fig. 5 Tunability of the coupled x-ray source. (a) Amplitude for B₀ under different coil current. The strength of the undulator can be tuned by variation of the coil current, e.g. by varying the time delay between the fs and ns laser. (b) Peak energy of x-ray photon emitted by electrons with different energy or coil current. The period of the undulator is fixed to 500 μm.

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Fig. 6 FEL pierce parameter and 1D FEL gain length for the proposed undulator under the variation of field strength B₀ and period λ_u. The stars denote ρ and L_g1d for the undulator discussed above. The energy, initial beam radius, duration and charge assumed for the LWFA electron beam are 300 MeV, 3 μm, 10 fs and 60 pC, respectively, corresponding to 1 keV x-ray photons with λ_u = 500 μm and B₀ = 22 T undulator.

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3. Discussion

So far we have shown the validity of the undulator and performance of the coupled x-ray sources. Owing to its intrinsic ultra-high acceleration gradient and kA electron beam current, LWFA is also regarded as a promising candidate for compact X-FEL. However, the foremost challenge for realizing FEL gain driven by LWFA is the percent level energy spread. Since for the FEL instability to occur, the RES of electron beam has to be smaller than FEL pierce parameter ρ = (2γ)⁻¹(I/I_A)^1/3(A_uλ_u/2πσ)^2/3, of which for keV-level resonant x-ray photon energy is usually 10⁻⁴-10⁻³, where I is the electron beam current, I_A = 4πε₀mc³/e ≈17 kA is the nonrelativistic Alfvén current, σ is the rms transverse electron beam radius and A_u = K[J₀(x)-J₁(x)]/$\sqrt{2}$with x = K²/ (4 + 2K²), J₀ and J₁ are Bessel functions and for helical undulator we should take A_u = K/$\sqrt{2}$. Several possible solutions have been proposed by FEL community either by longitudinally decompressing the electron beam or using a transverse gradient undulator to increase the acceptance for energy spread [30,31] and by LWFA community by controlling the spread in injection time to decrease energy spread [32,33].

It is interesting to take one step further by considering the possibilities of applying this undulator to compact X-FEL driven by LWFA. We show the dependency of ρ and 1D power gain length L_g1d = λ_u/(4π$\sqrt{3}$ρ), defined by P∝exp(x/L_g1d), on the field strength B₀ and undulator period λ_u. For simplicity, we assume an electron beam of 300 MeV average energy which corresponding to 1 keV resonant x-ray energy under λ_u = 500 μm and B₀ = 22 T undulator as discussed above. As shown by Fig. 6(a), for a fixed undulator period λ_u = 500 μm, ρ increase with the field strength B₀ while L_g1d decrease. On the other hand, for fixed field strength B₀ = 22 T, larger undulator period will also increase the RES acceptance and shorten 1D gain length, as illustrated in Fig. 6(b). For the proposed system, stars in Fig. 6, ρ = 4.43 × 10⁻³ and L_g1d = 5.18 mm, respectively. This value of FEL pierce parameter ρ is close to electron beam RES of state-of-the-art LWFA while the gain length is only half of the undulator length. For self-amplified spontaneous emission (SASE) FEL, the radiation power typically saturates at about 20 L_g1d when taking 3D effects such as diffraction and emittance into account [27]. For the undulator proposed, 20 L_g1d is only 10.36 cm, corresponding to N_s = 11 segments of the undulator. This superficial but heuristic discussion suggests that high field strength undulator will be beneficial, simultaneously, for increasing the RES acceptance [30] and making the resultant X-FEL more compact. However, usually, it is hard to increase the field strength and undulator period at the same time. We may seek for a balance between larger B₀ and larger L_g1d under certain laser conditions and start-to-end FEL simulations are needed to quantitatively evaluate the performance of the undulator.

4. Conclusions

In summary, we have presented a short period high strength elliptical undulator using a bifilar shaped capacitor coil target. With currently available laser parameters, we show that an elliptical undulator with hundreds of microns in period and tens of Tesla in magnetic field amplitude can be generated. Undulator radiation by coupling this undulator with laser plasma accelerators was also presented. We demonstrate that right-hand circular polarized x-ray with tunable energy ranging 5-250 keV and high brightness of the order of 10²⁵ photons/s/mrad²/mm²/0.1%BW can be produced. For this coupled system, FEL pierce parameter can be increased close to the RES of state-of-the-art LWFA electron beam while the 1D gain length is short down to a few millimeters. This concept can be further improved, for example, by using micro-fabricated capacitor coil with tens of microns period thus requiring less energetic lasers. This may lead to an ultra-compact synchrotron radiation and even X-FEL, not only the accelerator and undulator parts but the footprint of the whole system.