Laser wakefield accelerator based light sources: potential applications and requirements

Laser wakefield acceleration is a scheme that uses a high intensity laser pulse propagating through a plasma (figure 1), such that the laser pulse length cτ_L is shorter than the plasma wavelength (λ_p = 2πc/ω_p, where $\omega_{\rm p}^2=e^2n_{\rm e0}/m_{\rm e}\epsilon_0$ and n_e0 is the initial electron number density of the plasma). A relativistic electron plasma wave can be generated by the ponderomotive force (essentially the gradient of the field energy density) of the pulse. This wave is known as a plasma wakefield, and can be used to accelerate trapped electrons with velocities close to the phase velocity of the wave. Trapped electrons, which remain in the accelerating phase of the wave, can reach energies determined by the electric field in the plasma wave and the length over which the acceleration takes place.

Zoom In Zoom Out Reset image size

For a sufficiently high energy laser pulse, phase matching of an electron beam to the wakefield is maintained over a length known as the dephasing length. Since the laser driver group velocity is less than the speed of light, due to dispersion by the plasma, and the electron beam can approach the speed of light to an arbitrarily small degree as it gains energy, eventually the electron beam will outrun the accelerating fields of the wake. The maximum energy transferred from wake to electron beam is at this distance, and can be estimated by considering the velocity of an electron to be close to the speed of light and the wakefield velocity to be given by the linear group velocity of the laser in plasma, $v_{\rm g}=c\sqrt{1-\omega_{\rm p}^2/\omega_0^2}$, where ω₀ is the laser frequency. A more substantial analysis has been given by a mixture of particle-in-cell simulations and scaling laws developed in [50, 51]. By these scalings, the maximum energy gain, that can be realistically expected, scales as

$$\begin{equation} \Delta E/m_{\rm e}c^2\approx \left(\frac{\omega_0^2}{\omega_{\rm p}^2}\right) \left(\frac{\lambda_0}{1\;\mu{\rm m}}\right) \sqrt{\frac{I}{10^{18}\;{\rm W\,cm}^{-2}}}\;, \end{equation} \tag{ 1 }$$

where λ₀ is the laser wavelength and I is the focused intensity. It is important to note that this formula assumes a matched spot-size and a matched beam length. For the scalings of Lu et al [51] the pulse duration would be cτ_FWHM = 2r_b/3, the spot-size $k_{\rm p} w_0 = 2 \sqrt{a_0} = r_{\rm b}$, where r_b is the blowout radius, and a₀ the laser peak normalized vector potential.

In the three-dimensional (3D), highly non-linear LWFA regime, when a short laser pulse with an intensity I > 10¹⁸ W cm⁻² is focused inside a plasma, the laser ponderomotive force completely expels the plasma electrons away from the strong intensity regions to form an ion bubble in the wake of the pulse [51]. Electrons trapped at the back of this structure are accelerated and wiggled by the focusing force of the more massive and immobile ions to produce broadband, synchrotron-like radiation in the keV energy range (figure 2). One of the many exciting prospects of LWFA sources is that the electron bunch durations produced in such interactions have been demonstrated to be of fs duration [52, 53]. The radiation pulse generated by the electron beam will have equivalent duration, and hence fs x-ray pulses are likely to be generated by such interactions. This has very exciting implications for time-resolved pump–probe experiments using such laser-generated x-ray pulses.

Zoom In Zoom Out Reset image size

The spectrum of plasma betatron radiation is characterized by a betatron strength parameter a_β = γk_βr_β, where k_β is the wavenumber of the betatron oscillation and r_β is the radius of the oscillation [33, 54]. For a_β ≪ 1, the spectrum is a Doppler shifted peak at 2γ²ω_β corresponding to the betatron frequency ω_β. For a_β ≫ 1, the on axis spectrum is equivalent to the characteristic synchrotron spectrum [33, 55]. In this case the spectrum is broad (synchrotron-like) and extends up to a critical energy,

$$\begin{equation} E_{\rm crit}=3\gamma^3\omega_\beta. \end{equation} \tag{ 2 }$$

The average number of photons radiated by a single electron is [33]:

$$\begin{equation} \overline{N} = \frac{\pi}{3} \frac{e^2}{4\pi\epsilon_0c\hbar}\left(1+\frac{\alpha_\beta^2}{2}\right)\alpha_\beta^2N_\beta/\overline{n}, \end{equation} \tag{ 3 }$$

where N_β is the number of betatron oscillations, and $\overline{n}$ is the average harmonic number, which for synchrotron-like emission (large α_β), is $\overline{n} = E_{\rm crit}/\hbar\omega_\beta$. Taking into account the acceleration of the electrons results in a more complicated interaction, but the spectrum is still broadband with a peak energy lower than that predicted by the critical energy corresponding to the highest energy the electrons gain [36, 43, 56].

Another mechanism for radiation generation is Compton scattering of the electron beam from electromagnetic radiation. An electron, initially at rest, oscillating in a laser field experiencing non-relativistic motion emits radiation at the laser frequency. If the electron is initiated with a relativistic momentum counter-propagating with respect to the laser pulse, then it gains a Doppler upshift. For a very relativistic electron with Lorentz factor γ₀, and a lower laser intensity, the upshift in frequency results in emission in a spectral peak at a frequency $\omega_1=4\gamma_0^2\omega_0$.

As the laser intensity increases, the Lorentz force due to the magnetic field begins to become significant, and hence the motion of the electron becomes more complicated. The radiation spectrum starts to pick up higher harmonics of the laser frequency, which gives rise to 'non-linear' Compton scattering. As the intensity increases further, the relativistic motion of the electron in the direction of the laser propagation results in a Doppler-shift of the fundamental frequency, in addition to increasing the spectral power in the harmonics of the down-shifted frequency. For a higher laser intensity, there is a slight down-shift of the up-shifted frequency, as the laser accelerates the electron beam against its motion. However, the normalized laser field strength parameter, a₀ = eE₀/m_ecω₀, where E₀ is the peak electric field, and the normalized betatron (wiggler) parameter, a_β, are almost interchangeable in the description of Compton scattering for a relativistic electron colliding with a laser pulse [57]. Hence, as the strength parameter increases, the photon spectrum tends towards a synchrotron-like broad spectrum, extending to much higher photon energies than the shifted fundamental.

The emission of photons in such processes clearly indicates that a force is applied to the electron to conserve momentum. This radiation force has a classical form, which is self-consistent within the limits that the acceleration timescale is much larger than τ₀ = 2e²/3mc³ = 6.3 × 10⁻²³ s [58], which is principally a damping of motion due to loss of momentum to the radiation. One of the interesting phenomena arising from this laser-electron interaction is that the radiation damping is theoretically predicted to be so extreme that for a sufficiently intense laser, the electron beam may lose almost all its energy in the interaction time [59–61]. This means that the radiation force is comparable to the accelerating force, which has the implication that the spectrum of the radiation should be strongly modified.

XPCI records the modifications of the phase of an x-ray beam as it passes through a material, as opposed to its amplitude recorded with conventional x-ray radiography techniques. When x-rays pass through matter, elastic scattering causes a phase shift of the wave passing through the object of interest. The cross-section for elastic scattering of x-rays in low-Z elements is usually much greater than for absorption [62]. The total phase shift induced on an x-ray wave when it travels a distance z through a sample with complex index of refraction n = 1 − δ + iβ is due to the real part of the index and calculated with the relation:

$$\begin{equation} \Phi(z)=\frac{2\pi}{\lambda}\int_{0}^{z}{\delta(x){\rmd}x}, \end{equation} \tag{ 4 }$$

where λ is the x-ray wavelength. For two distinct low-Z elements, the difference in the real part of the complex index of refraction is much larger than the difference in the imaginary part. It means that for quasi transparent objects such as biological samples or tissues, this technique is more sensitive to small density variations, and offers better contrast than conventional radiography. For the past decade, XPCI has been a very active topic of research for medical, biological, and industrial applications. Consequently, several XPCI techniques have been developed based on interferometry [62], gratings [63] and free space propagation [64]. In combination with these techniques, XPCI has been done with various x-ray sources. Examples includes images of a small fish recorded with a standard x-ray tube and gratings [65], images of a bee obtained with a Mo K-alpha laser-based source [66] and phase contrast radiography using x-pinch radiation [67]. Even though, as suggested by equation (4), it is suitable to use a monochromatic x-ray source for XPCI, polychromatic sources with high spatial coherence can also be used [68, 69]. In this case, the scheme is much simpler and does not require using complex and expensive x-ray optics. Much of the sources currently used for XPCI do not have a high temporal resolution desirable to take snapshots of laser-driven shocks or other phenomena. XPCI measurements of shocks done at synchrotrons were limited to a temporal resolution of ∼100 ps [70]; betatron x-ray radiation, where the source size is less than a few micrometers [38], has the potential to offer three orders of magnitude better time resolution. For a source size of 2 µm and a critical energy of 8 keV, the transverse coherence length of betatron radiation was measured at L_trans = 3 µm 5 cm away from the source, which is sufficient to observe Fresnel diffraction fringes [37]. Using free space propagation techniques, proof-of-principle XPCI measurements of biological samples have recently been done [9, 10] with betatron radiation. These promising results have led to an extension of this technique to higher x-ray energies [71], with an example shown in figure 3.

Zoom In Zoom Out Reset image size

To generate a single-shot image, a large photon number is required. As an approximate threshold, a megapixel (1024 × 1024 pixels) is a reasonable number of elements to make an image. The relative fluctuations from Poisson statistics will scale as $1/\sqrt{\overline{N}_{ij}}$, where $\overline{N}_{ij}$ is the average number of detected photons per pixel. Therefore, for a low noise image the number of photons per shot should be N ≫ 10⁶, assuming the x-rays uniformly fill the detector and are detected. In practice N ≫ 10⁸ is more realistic, given non-uniformities, overfill and detection efficiency. Generally speaking, the ideal source for XPCI should have an average brightness superior than 10¹² photons/mm²/mrad²/s/0.1% BW, be monochromatic, and be easily tunable from 10 to 150 keV. For standard medical projection imaging, photon energies in the range 10–30 keV are used for soft tissue absorption radiography such as mammography and 50–150 keV for radiography of hard tissue like bone. Phase contrast imaging is approximately a thousand times more sensitive than absorption contrast, but the advantage over absorption contrast will be more prominent in the hard x-ray region [72]. Although it is not monochromatic, betatron radiation already achieves performances sufficient for XPCI. In order to improve the quality of XPCI experiments with betatron radiation, future directions include a repetition rate increase up to 30 Hz for in vivo imaging [73] and a better field of view, currently limited to <50 mrad. Having a 30 Hz repetition rate allows motion freezing and time lapse imaging of biological phenomena. It corresponds to data acquisition rates of XPCI beam lines at synchrotrons and would allow real-time visualization of internal physiological mechanisms such as the respiration and circulatory systems, or the beating of a heart. One of the applications of XPCI in the medical field is to look at breast cancer. Current systems, as well as synchrotrons and grating-based systems have a field of view of ∼10 cm, whereas current betatron x-ray imaging experiments have a field of view of ∼1 cm, depending on magnification. X-ray microtomography experiments are currently limited to ∼5 µm resolution, which is comparable to what betatron radiation can do. One particular suggested niche area for XPCI with betatron radiation could be ultrafast x-ray imaging with femtosecond resolution [74]. Except XFEL sources, betatron radiation offers the best time resolution ever achieved for XPCI.

Extremely powerful x-ray absorption techniques, such as extended x-ray absorption fine structure (EXAFS) and x-ray absorption near edge structure (XANES), can reveal electron–ion equilibration mechanisms, when extended to the sub-picosecond time scale [75, 76]. XANES and EXAFS are diagnostic tools providing direct information on valence and core electronic structure, as well as on local atomic order. It is therefore suitable to characterize changes of structures and phase transitions. In standard conditions (no heating, solid density, and room temperature), the absorption spectrum of a material exhibits a sharp edge that reveals the clear separation between occupied and unoccupied states in the conduction band at the Fermi energy. The edge is followed by EXAFS modulations expected from the lattice structures of a solid, which disappear after the heating, indicating a rapid loss of order while the electronic and ionic temperatures increase, and evidencing a fast phase transition. In addition, the broadening of the edge slope reveals the expected broadening of the Fermi level. These techniques have been developed on OMEGA to diagnose iron up to 560 GPa [77], and at LULI-2000 to diagnose Mott nonmetal transitions [78] and to investigate the electronic structure of highly compressed Al [79]. XANES has been used on tabletop systems to characterize the phase transitions of Al up to a few eV [80]. However, these techniques require further development. Models still need improvements to describe in detail the changes in the XANES and EXAFS spectra, mainly because the time resolution of these experiments was intrinsically limited by the x-ray probe duration. Ultrafast x-ray absorption experiments done at the advanced light source synchrotron radiation facility have unraveled the electronic structure of warm dense copper [81], but they required specific slicing techniques to reduce the synchrotron pulse duration that can be realized at the expense of x-ray flux.

Due to its broad, continuous spectrum, femtosecond pulse duration and synchronization with the drive laser, there is wide recognition that betatron radiation has the potential to offer a significant alternative in this domain, but several improvements have to be made to the source in order to achieve this. A typical absorption spectrum of a material exhibits sharp edges, followed by EXAFS oscillations that are generally separated by a few eV and extend up to ∼200 eV after the edge. In general, the oscillation amplitude of the EXAFS signal is on the order of a few % of the total absorption signal (the edge step). Ideally, the random statistical noise, ${\rm SN}=1/\sqrt{N_{\rm Ph}}$, where N_Ph is the number of x-ray photons in the energy band of interest, should be 1/1000 of the EXAFS signal. This means that the condition N_Ph > 10⁶ eV⁻¹ must be fulfilled to realize an EXAFS experiment with good statistics. Currently, state-of-the-art betatron radiation sources produce on the order of 10⁸ photons (in the full spectrum) [37], and numbers of 10⁴–10⁵ photons eV⁻¹ have been reported around the 1.56 keV Al K-edge [41]. While these numbers are encouraging, progress needs to be made to improve the source flux and shot-to-shot stability. The source of variation is likely the variation on (i) the electron oscillation amplitude (due to the fact that self-trapping is a highly non-linear mechanism), and (ii) the electron bunch charge, which can be as high as 50% rms. The variation on the oscillation amplitude can be improved by deliberately exciting off-axis [82].

3.3.1. Nuclear resonance fluorescence.

While x-rays interact with the inner-shell electrons of atoms, gamma-rays, on the other hand, interact with the nuclei of atoms and are a powerful tool to reveal isotope-specific properties of certain materials. An application of particular interest is nuclear resonance fluorescence (NRF). NRF is an isotope-specific process, in which a gamma-ray photon with a specific energy (typically a few MeV) is absorbed by a nucleus, which in turns relaxes back to the equilibrium by emitting a photon at slightly lower energy (due to recoil) [83]. Similarly to the fact that K-shell or L-shell fluorescence lines in atoms reveal properties of the spin and oxidation states of a chemical element, NRF lines reveal specific properties of a nucleus, like excitation energies, spin and parities [84]. Since NRF is an isotope-specific process, it can be particularly useful for isotopic assay and detection of materials, an application that spans the domains of non-destructive evaluation, homeland security, nuclear waste assay, stockpile stewardship or mining. The only practical drawback of NRF lines is the fact that their relative spectral width is extremely small (ΔE/E ∼ 10⁻⁶). Although it is possible to detect NRF lines with broadband gamma-ray sources produced from bremsstrahlung radiation, it is more desirable to excite them with a narrow-band gamma-ray source. Compton scattering, a process in which laser photons are scattered off a relativistic beam of electrons to produce bright sources of x-rays and gamma-rays, is an ideal candidate to enable efficient NRF detection.

Despite Doppler-broadening, the NRF lines exhibit a relative energy width of ΔE/E ∼ 10⁻⁶, which is well below the energy resolution of standard germanium-based detectors. Therefore it is desirable to use a gamma-ray source with a narrow energy linewidth to efficiently excite and detect NRF lines. One could argue that the best solution is to excite the NRF line of a given isotope with the same isotope as the source of exciting radiation. However, recoil energy losses upon emission and absorption prevent us from doing this. Indeed, the recoil yields a shift toward lower energies and the line emitted is off resonance by:

$$\begin{equation} \Delta E_{\rm R}=E^2/Mc^2. \end{equation} \tag{ 5 }$$

In the case of ⁷Li, which has an NRF line at 0.478 MeV and a nucleus of atomic weight 7, ΔE_R = 35 eV. It is larger than the natural linewidth and than the Doppler width (1.33 eV) at room temperature. Within this context, it has been shown that Compton scattering sources produced with a conventional accelerator can efficiently excite NRF lines [11]. As discussed in the next section, the bandwidth of LWFA-based Compton scattering sources is currently on the order of 50%, and therefore not ideal for NRF detection applications.

3.3.2. Other photonuclear reactions.