1. Introduction

Soft x-ray radiation with keV or lower photon energy has broad applications in nanolithography, biomedical imaging, material characterization, and attosecond sciences [1]. Generation of bright soft x-ray radiation is challenging and still under extensive research. Laser produced plasma (LPP) soft x-ray sources emit incoherent radiation over 4π solid angle, requiring delicately fabricated high-precision x-ray collection optics [2,3]. X-ray free electron lasers (XFEL) provide collimated, ultrafast, tunable, coherent x-ray radiation with an extremely high photon flux and high brightness [4], providing advanced light sources for sciences. However, the size and cost of the particle accelerator based light source facilities limit their accessibility to global researchers with increasing demands to advanced x-ray sources.

Table-top x-ray sources based on high harmonic generation [5] or laser wakefield acceleration [6,7] provides complementary and less expensive solutions to situations that the photon flux requirement is moderate. Thus, though the photon flux of table-top x-ray sources is lowered by orders of magnitude, they find applications in x-ray absorption fine structure spectroscopy [8], angular-resolved photoemission spectroscopy [9], coherent diffractive imaging [10] and so on. Thanks to recent progresses on the ultrafast laser technology [11–13], high harmonic generation of keV soft x-ray radiation has been demonstrated with a photon flux of ∼10⁶ photons per shot using mid-infrared laser pulses [14], because the cut-off photon energy increases linearly with the laser ponderomotive potential which is proportional to the square of the laser wavelength [5]. However, to further increase the photon energy is technically challenging, because the conversion efficiency drops dramatically as the laser wavelength increases [15] and a stringent phase-matching condition is required [16,17].

Laser wakefield acceleration based femtosecond x-ray radiation is another option to compact table-top x-ray source [18], via mechanisms such as inverse Compton scattering [19], betatron radiation [20,21], and plasma acceleration followed by a conventional undulator [22]. During the process of laser wakefield acceleration [6,7], electrons are injected into the plasma wave or “bubble” [23], and the longitudinal electric field in the bubble, or the wakefield, accelerates electrons to GeV energy [24–27]. Meanwhile, the transverse electric restoring field drives radially displaced electrons back to the laser propagation axis, thus the wiggling electrons emit high-energy betatron photons [28,29].

Most betatron radiation experiments are based on laser wakefield acceleration of >100 MeV electron, using large-scale laser systems with hundreds of terawatt or petawatt peak power [30–32]. The typical betatron photon energy is tens of keV, in the hard x-ray regime. In order to increase the photon yield, the betatron amplitude is enlarged by tailoring the plasma profile [33] or inserting a titled shock front [34]. The plasma bubble can also be driven to wiggle transversely in an oscillating plasma channel, so the betatron spectral range and the photon number are enhanced [35]. However, research on betatron radiation in the soft x-ray regime is rarely reported so far, because MeV low energy electrons accelerated by a sub-terawatt laser pulse are needed for soft x-ray betatron radiation [36–38]. However, in this case with a high ambient plasma density, the trajectory of a MeV low energy electron wiggling in a small-size plasma bubble determines a low betatron photon number in the soft x-ray regime.

In order to enhance the betatron emission with a soft x-ray spectrum, we have proposed a two-color laser wakefield acceleration scheme, using MeV electrons accelerated by a sub-terawatt laser-driven plasma wakefield. In this scheme, a near-infrared (e.g., 800 nm wavelength), millijoule-level laser pulse excites a nonlinear plasma wakefield in the plasma bubble, accelerating electrons to a broadband energy spectrum with a maximum energy of ∼60 MeV. Another weak, mid-infrared, control laser field co-propagates with the 800 nm driving laser pulse. The wavelength of the mid-infrared control laser field is comparable to the plasma bubble longitudinal size, introducing the transverse oscillation of the plasma bubble. The transverse oscillation of the bubble shakes accelerated electrons trapped inside, enhancing the betatron radiation with tunable photon energy in the soft x-ray regime.

2. Theoretical analysis of betatron enhancement in a two-color laser wakefield

Electrons injected and accelerated in the plasma bubble radiate electromagnetic waves, and the feature of the radiation is determined by the dimensionless strength parameter $K = \gamma {k_\beta }{r_\beta }$, where $\gamma $ is the relativistic factor of accelerated electrons, ${k_\beta }$ is the betatron motion wave number proportional to the betatron motion frequency ${k_\beta } \sim {{{\omega _\beta }} / c}$, and ${r_\beta }$ is the amplitude of the betatron motion. According to the classical electrodynamic theory [39], electrons oscillate in the “undulator” regime and emit radiation at the fundamental frequency $\omega = 2{\gamma ^2}{\omega _\beta }$ with a narrow bandwidth for $K \ll 1$. While for $K \gg 1$, electrons oscillate in the “wiggler” regime and radiate closely spaced harmonics of the fundamental frequency of ${\omega _1} = {{2{\gamma ^2}\;\;{\omega _\beta }} /{(1 + {{{K^2}}/2})}}$ up to the critical frequency ${\omega _c} = {{3K{\gamma ^2}{\omega _\beta }} /(1 + {{K^2}}/2}$. The total emitted photon number is proportional to ${N_0}{N_e}K$, where ${N_e}$ is the number of oscillating electrons and ${N_0}$ is the averaged betatron oscillation period.

 

Fig. 1. The schematic diagram of betatron enhancement in the two-color laser wakefield acceleration scheme. The orange ellipse represents the 800 nm driving laser pulse, and the sine-shape red solid line represents the mid-infrared control laser field. The blue circle represents the plasma bubble. The bubble is moving downwards due to the transient positive mid-infrared control field. The injected electrons thus experience a transverse force downwards from the restoring field of the bubble. Due to the superluminal phase velocity of the mid-infrared control laser field, the bubble and injected electrons experience high frequency transverse oscillation when the control laser field rapidly changes its sign.

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For electrons accelerated in a laser plasma wake, the betatron motion frequency is ${\omega _\beta } = {{{\omega _p}} / {\sqrt {2\gamma } }}$ where ${\omega _p}$ is the plasma frequency. The oscillation amplitude is limited by the bubble size, which is inversely proportional to ${\omega _p}$ [28]. In the case of GeV or hundreds of MeV electron acceleration, the electron relativistic factor is large $\gamma \sim {10^2} - {10^3}$), thus the strength parameter $K \gg 1$ and the betatron radiation is in the wiggler regime. However, for the case of MeV electron acceleration by sub-terawatt laser pulses, the electron relativistic factor is at the order of tens, so the electron oscillation and radiation are at the boundary between the “undulator” and “wiggler” regimes. The emitted photon is in the soft x-ray spectral region, and the photon number is reduced as the injected electron number ${N_e}$ and the strength parameter $K$ decrease. Given the accelerated electron energy, increasing the betatron frequency is equivalent to increasing the strength parameter $K$ and the averaged betatron oscillation periods ${N_0}$, thus enhancing the betatron photon number.

We propose to increase the betatron motion frequency by applying a sub-relativistic, mid-infrared control laser field co-propagating with the intense near-infrared laser pulse driving the plasma wakefield (Fig. 1). The weak mid-infrared laser field controls the motion of the plasma bubble in two aspects.

First, the wavelength of the mid-infrared laser field is comparable to the longitudinal size of the plasma bubble, applying an equivalent “direct current” (DC) force on sheath electrons forming the bubble and introducing transverse offset of the bubble from the driving laser axis. This infrared laser field induced asymmetric transverse bubble structure has also been observed in the case of a few-cycle laser pulse driven plasma wake or bubble, which experiences a significant transverse oscillation in phase with the red-shifted driving laser field [40].

Second, the optical frequency of the mid-infrared control field ${\omega _{IR}}$ is slightly larger than the plasma frequency ${\omega _p}$, so the waveform of the control field propagates at a superluminal phase velocity ${v_{IR}} = {c / {\sqrt {1 - {{\omega_p^2} / {\omega_{IR}^2}}} \gg c}}$. On the other hand, the propagation velocity of the plasma bubble is close to the light speed, determined by the near-infrared driving laser pulse group velocity and the etching velocity of the driving laser leading edge [41]. The superluminal infrared laser field keeps surpassing and shaking the sheath electrons of the plasma bubble, causing transverse oscillation of the bubble at a high frequency

Thus, electrons injected and accelerated in the plasma bubble experience an additional oscillating transverse electrostatic field due to the bubble oscillation, and wiggle at the oscillation frequency ${\omega _m}$ as being accelerated to relativistic energy.

Given the ambient plasma density optimized for electron acceleration, the infrared laser frequency can be finely tuned to regulate the enforced wiggling frequency of accelerated electrons. For example, one can simply tune the mid-infrared output central frequency of an optical parametric amplification system [42]. The infrared laser should be transparent in the plasma so the central frequency ${\omega _{IR}}$ is always larger than the plasma frequency ${\omega _p}$. As the infrared laser field frequency decreases and gets close to the plasma frequency, the electron enforced oscillation frequency increases monotonically and approaches the plasma frequency at the maximum. Compared to betatron oscillation frequency ${\omega _\beta } = {{\omega _p}}/{\sqrt {2\gamma}}$ without the control field, the enforced oscillation frequency is significantly enhanced by $\sqrt {{\rm{2}}\gamma } $, and the number of oscillation periods of each electron increases as well. Thus, both the emitted photon energy and photon number increase. Moreover, the enforced electron transverse oscillation is decoupled from the near-infrared driving laser pulse and the electron acceleration dynamics in the longitudinal direction, suggesting an opportunity for a tunable betatron x-ray source.

Due to the lower MeV energy of accelerated electrons and the smaller bubble size than the cases of multi-terawatt or petawatt laser driven wakefield acceleration, the enhanced betatron radiation is approximately in the undulator regime. The emission spectrum is centered on ${\omega _c} = 2{\gamma ^2}{\omega _m}$. The total photon number is $\sum\nolimits_{{N_{\rm{0}}}} {\sum\nolimits_{{N_{\rm{e}}}} {1 /137} ({2\pi/9){K_{{N_0},{N_e}}}}}$ where ${N_0}$ is the number of the betatron period of the electron beam, ${N_e}$ is the number of oscillating electrons, and ${K_{{N_0},{N_e}}}$ is the strength parameter for a specific electron oscillating at a betatron motion period [20]. For example, it is possible to accelerate electrons (11.7 pC total charge, 45 MeV averaged electron energy) using millijoule laser pulses. If electrons experiencing 7 periods of enforced transverse oscillation with even small amplitude of 15 nm in a micron-size bubble, the strength parameter K is around 0.5, emitting ∼1.3×10⁶ photons with 1.25 keV photon energy in a single shot. With a high repetition rate driving laser system, e.g., 1kHz, 10⁹ soft x-ray photons are obtained per second, promising for various applications.

3. Electron acceleration and one-electron betatron radiation

To quantitatively investigate the betatron radiation enhancement by the mid-infrared control laser field, we have performed particle-in-cell (PIC) simulations of MeV electron acceleration in laser plasma wakefield using the Osiris code and calculated the radiation from the accelerated relativistic electrons based on the PIC simulation output of electron trajectories. The PIC simulations are two dimensional, including the longitudinal direction of laser propagation and one of transverse directions. The simulation window size is 127 µm in both longitudinal and transverse directions, so the long mid-infrared control laser pulse can be accommodated in the moving window. The longitudinal grid size is ${{{\lambda _0}}/{8\pi }}$ where ${\lambda _0} = 800nm$ is the driving laser wavelength, and 8 particles are initially placed in a unit cell.

The driving laser pulse (pulse duration 13 fs, pulse energy 35 mJ, ${1 /{e^2}}$ focal spot radius 3.6 µm, corresponding to ${a_0} = \sqrt {46.5P({\rm{TW}})} \lambda {\rm{/}}{w_0}{\rm{ = }}2.5$ [43]) is initially centered at z = −109 µm and focused at the entrance (z = 0 µm) of a 178 µm thick homogeneous gas medium including neutral hydrogen (1.65×10¹⁹ cm⁻³) and nitrogen (1.74×10¹⁷ cm⁻³) gases, allowing both self-injection and ionization injection [44]. The mid-infrared control pulse with 6.25 µm central wavelength (${\omega _{IR}} = 1.28{\omega _p}$) propagates collinearly with the driving laser pulse. The control pulse duration is 424 fs, ${1/{{e^2}}}$ focal spot radius is 31.75 µm, and the trailing edge of the control pulse overlaps with the driving pulse and the plasma wake at the gas medium entrance, so during the propagation, the plasma bubble keeps longitudinally overlapping with the control pulse, whose group velocity is much less than c. The control pulse energy is 21 mJ so its normalized vector potential is sub-relativistic (${a_{IR}}$ ∼ 0.3). The control field polarization axis x is perpendicular to the driving field polarization axis y, so the bubble transverse oscillation along the control field polarization direction is not affected by the driving laser field.

Figure 2(a) shows the energy spectrum of accelerated electrons in the 2D PIC simulation, when both the driving and the control laser fields are present (red solid and dashed lines). The electron energy spans from 5 to 55 MeV. We have also labeled electrons ionized from hydrogen, L-shell and K-shell of nitrogen, so electrons injected via the mechanisms of ionization injection (solid line) and self-injection (dashed line) can be differentiated from each other. 97% of total accelerated electrons are injected via ionization injection during which electrons are close to the laser propagation axis. Due to the mid-infrared control laser field, the plasma bubble experiences high frequency enforced transverse oscillation. After propagation for 7.6 µm, the transverse position of the first plasma bubble bucket moves from the positive x-direction (Fig. 2(b)) to the negative direction (Fig. 2(c)), meanwhile the sign of the control laser field on the axis of the driving laser is flipped (black solid lines in Fig. 2(c)). In comparison, the bubble is symmetric around the optical axis of the driving laser pulse when the control field is absent (Fig. 2(d)). The averaged transverse position of sheath electrons can be calculated from electron density distribution at different times [40]. The transverse bubble centroid position shows an oscillation behavior at a frequency of ${\rm{0}}{\rm{.049}}{\omega _{\rm{0}}}$ (Fig. 2(e)), consistent with the theoretical enforced bubble oscillation frequency ${\omega _m} = {\rm{0}}{\rm{.048}}{\omega _{\rm{0}}}$ according to Eq. (1). This consistency confirms that the transverse bubble oscillation can be actively controlled by the mid-infrared control laser field.

 

Fig. 2. Simulation results of transverse bubble oscillation induced by the control laser field. (a) The energy spectrum of accelerated electrons in 2D simulations without the control light (blue lines), 2D simulations with the control light (red lines), and 3D simulations without the control light (black lines) respectively. Dashed lines are for ionization-injected electrons and solid lines for self-injected electrons. (b)(c) The snapshots of the bubble electron density spatial distributions at ∼136 and 143 µm propagation positions respectively, showing a transverse motion from the positive to the negative x-directions. The black lines denote the transverse electric field E_transverse on axis multiplied by 20 (containing both static electric field and control light electric field), and the redlines show the driving pulse electric field E₀ on axis respectively. z_w represents the propagation length of the simulation window. (d) The bubble electron density spatial distribution at 136 µm propagation positions without the control pulse, compared to (b). (e) The averaged centroid transverse position of the oscillating bubble sheath electrons at each propagation position.

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We have also estimated the total charge of electron acceleration by performing additional 3D PIC simulations. Without the slow group velocity, 424 fs long control laser pulse, a simulation with a compact longitudinal window size is sufficient to study the electron acceleration dynamics. As shown in Fig. 2(a) black line, the electron energy spectrum is from 5 to 55 MeV, and the total charge is 16.0 pC, including 90.2% ionization-injected electrons (Fig. 2(a), black solid line). Both 2D and 3D simulation results confirm that ionization injection is the dominant injection mechanism and only a small portion of electrons are self-injected either with or without the control field. In this case, we can neglect the contribution from self-injected electrons in our radiation analysis while preserving the realistic radiation property. By assuming the total charge of ionization-injected electrons in the 2D case without the control field equals to the total charge of ionization-injected electrons in the 3D simulation without the control field, we can calculate the charges of injected electrons in all 2D cases. Thus in the 2D simulation with the control field, the ionization-injected electrons’ charge is 11.7 pC, which is 97.4% of all injected electrons. We can further calculate the betatron radiation using the charge number.

The single-electron trajectory of each accelerated electron is analyzed to study the influence of the enforced bubble transverse oscillation. We have found the initial electron transverse position at the moment of injection into the bubble is critical to the high-frequency enforced oscillation dynamics. For electrons initially placed close to the laser axis, the transverse force acted by the static electromagnetic field of the plasma bubble is small. Instead, the force acted on the electron oscillates in phase with the bubble oscillation at a frequency of ${\omega _m}$ (Fig. 3(a), red solid line), indicating that the electron transverse oscillation is dominated by the mid-infrared laser field. The transverse velocity and position at different longitudinal positions are also calculated in Figs. 3(b) and (c) (red solid lines), showing a significant component of transverse oscillation at ${\omega _m}$.

 

Fig. 3. Enforced transverse oscillation dynamics of a single electron and its betatron radiation. (a) Transverse force along the x direction acted on electrons injected near the axis (red solid line) and far away from the axis (green solid lines) when the mid-infrared control laser field is present, compared to the case without the control field (blue solid line). The dashed red line shows the growth of the electron Lorentz factor. (b) Transverse velocity of corresponding electrons in (a), with the inlet graph zooming into the range from 40 to 160 µm. (c) Transverse positions along the x direction of the three analyzed electrons. (d) The single-electron radiation calculated using Eq. (2) for three analyzed electrons. The enhancement of photon energy and radiation power is obvious (red solid line compared to blue solid line).

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For electrons initially injected at a transverse position far from the laser axis, the electrostatic field of the plasma bubble is dominant to drive the electron transverse oscillation, and the effect of the control field is to implement the electron transverse motion with another weak high-frequency mode at ${\omega _m}$. The transverse velocity and position of the off-axial injected electron (Figs. 3(b) and (c), green solid lines) predominantly oscillate at a low frequency, similar to the case without the control field (Figs. 3(b) and (c), blue solid lines).

The single-electron electromagnetic radiation from test electrons in Figs. 3(a)-(c) has been calculated using the Liénard-Wiechert potential [39]

As shown in Fig. 3(d), both the on-axial (red solid) and off-axial (green solid) injected electrons emit much more x-ray radiation than the electron in the case without the control field. The peak photon energy is boosted from less than 100 eV to 1 keV, in the soft x-ray spectral region. In addition, the electron injected on the laser axis shows more significant radiation power and photon energy enhancement, because the control field induced enforced transverse oscillation is dominant. As an estimate of photon energy, the electron injected on the laser axis experiences transverse oscillation of angular frequency 0.047ω₀, close to the enforced bubble oscillation frequency 0.049ω₀ shown in Fig. 2(d). The maximum electron energy is $\gamma \sim 90$ (Fig. 3(a) dotted red line), thus the maximum photon energy is around $\omega = 2{\gamma ^2}{\omega _\beta }\sim 1.{\rm{1keV}}$, consistent with simulation results (Fig. 3(d)). Moreover, the betatron radiation is insensitive to the time delay between the driving and the mid-infrared control field within a reasonable range.

It is also necessary to clarify whether the enforced transverse oscillation of injected electrons is because of the static electromagnetic field of the oscillating plasma bubble or the control laser field itself. Our analysis shows that the static field by the oscillating bubble is the driver of the enforced high-frequency oscillation of accelerated electrons. First of all, the mid-infrared control laser field is sub-relativistic (${a_{IR}}$ ∼ 0.3) and much larger than the static field by the plasma bubble, however, the forces acted on the relativistic electron by the electric and magnetic field components cancel each other. In contrast, the electric and magnetic components of the bubble static field are in the same direction. Second, the spatial profile of the control laser field inside the bubble is approximately homogeneous, so the laser field experienced by the on-axial and off-axial electrons should be similar. However, as shown in Fig. 3(a), the transverse oscillating force acted on the on-axial electron is much larger than that on the off-axial electron. This difference can be explained that only the on-axial electron can observe the sign flip of the static field inside an oscillating bubble. Finally, the force acted on the on-axial electron is in phase with the bubble oscillation, rather than the external control laser field. The transverse infrared control laser field has little direct effect on the accelerated electron bunches.

4. Incoherent summation of betatron emission from all accelerated electrons and the wavelength tunability

To calculate the total radiation from accelerated electrons, we have randomly selected 2000 from 176616 ionization-injected particles from the PIC simulation results, and calculated one-electron radiation from each particle using Eq. (2). The particle density distribution and motion in the electron bunch is random, so the total radiation should be obtained by incoherently adding up one-electron radiation from all particles. Figure 4(a) shows the total betatron radiation spectrum along the laser propagation direction with (red solid line) and without (blue solid line) the mid-infrared control laser field. When the control field is present, the peak photon energy is shifted from ∼40 eV to 1 keV, in the soft x-ray regime. By integrating the radiation intensity spectrum over photon energy, the radiation brightness along the laser propagation direction is enhanced by over 10 times. We have also calculated the total photon number over the photon spectrum at different radiation angles, yielding the angular distribution of the photon number (Fig. 4(b)). The angular distribution shows a circular profile, and the full width at half maximum (FWHM) of the angular profile is 20 mrad.

 

Fig. 4. Properties of the total betatron radiation from all accelerated electrons. (a) The incoherent summation of radiation spectrum on axis. Red and blue denote the case with and without control field respectively. (b) The angular distribution of the total photon number integrated over the whole photon energy spectrum. (c) The photon energy spectrum of all photons emitted within the azimuthal angle of 42mrad. (d) The on-axis photon energy spectra for different control laser frequencies at 1.15ω_p (red), 1.28ω_p (green), 1.41ω_p (blue), and 3.32ω_p (magenta) respectively. The black line labels the case without the control field. (e) The photon number angular distributions for different control field frequencies, 1.41ω_p (left) and 1.15ω_p (right).

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We have further integrated the total betatron radiation over different radiation directions, as shown in Fig. 4(c). The maximum photon energy is up to 2 keV, significantly enhanced from <100 eV in the case without the control field. The total betatron radiation pulse energy has also increased by 8 times, including 1.87×10⁶ high energy (∼keV) soft x-ray photons. One can expect that using laser wakefield acceleration driven by a kHz femtosecond laser system with tens of millijoule pulse energy, 10⁹ soft x-ray photons with keV photon energy can be obtained for various applications.

The relation between the control field frequency and the enhanced betatron photon energy has been investigated. The betatron peak photon energy increases from 846 eV to 1.35 keV, as the control field frequency decreases from $1.41{\omega _p}$ to $1.15{\omega _p}$ (Fig. 4(d)). As the control field frequency is getting close to the plasma frequency, the enforced bubble oscillation frequency and electron betatron motion frequency increase dramatically, further enhancing the photon energy. We have also noticed the decrease of the peak value of radiation as frequency drops down, and it is because of the reduced number of ionization-injected electrons from 14.6 pC to 8.9 pC. The photon number angular distributions with ${\omega _{IR}} = 1.15{\omega _p}$ and ${\omega _{IR}} = 1.41{\omega _p}$ are also shown in Fig. 4(e), and no significant difference in the betatron spatial profile is observed.

We have further investigated the case of resonantly matching the electron enforced oscillation frequency ω_m with the intrinsic betatron oscillation frequency ω_β (Fig. 4(d) magenta line). However, neither transverse oscillation of the plasma bubble nor the betatron radiation enhancement is observed. Thus, the increase of the transverse oscillation frequency of accelerated electrons plays a more important role than increasing the oscillation amplitude in enhancing the betatron radiation. In addition, the fact that no enhancement is observed at the resonant control field frequency has further justified that the plasma bubble oscillation is responsible for the oscillation of accelerated electrons and the enhancement of the betatron radiation.

5. Conclusion

In summary, we have proposed a novel scheme to enhance betatron radiation driven by millijoule laser wakefield acceleration. By applying a tunable mid-infrared control laser field, the laser wavelength is almost resonantly matched with the bubble longitudinal size, introducing a high-frequency transverse oscillation of the plasma bubble. The oscillating plasma bubble further shakes the injected relativistic electrons and emits highly collimated, soft x-ray pulses with almost tens of times of betatron photon energy and total radiation enhancement. Complimentary to hard x-ray betatron radiation emitted by electrons accelerated in multi-terawatt or petawatt laser wakefield, this two-color field enhanced betatron radiation scheme has further broadened the betatron radiation spectral range and demonstrated an opportunity of controlling collective dynamics of a plasma wave using laser field with tailored waveform.

Table-top femtosecond x-ray sources have made great processes, manifested by ∼1 keV high harmonic generation using mid-infrared laser pulses [14] and tens of keV photon energy betatron radiation using multi-terawatt lasers [21]. Extending the betatron photon energy to several keV, especially ∼1 keV, helps fill the soft x-ray photon energy gap between these two techniques. As laser technology progresses, e.g., high-frequency femtosecond laser amplification and optical parametric amplification, reliable electron acceleration at high repetition rate by millijoule laser wakefield can be expected in near future. Mid-infrared laser field enhanced betatron radiation based on high repetition rate, millijoule laser wakefield acceleration may provide an alternative option to table-top, collimated, ultrafast keV soft x-ray light source for real applications.