Optimization of soft X-ray phase-contrast tomography using a laser wakefield accelerator

: X-ray phase-contrast imaging allows for non-invasive analysis in low-absorbing materials, such as soft tissue. Its application in medical or materials science has yet to be realized on a wider scale due to the requirements on the X-ray source, demanding high flux and small source size. Laser wakefield accelerators generate betatron X-rays fulfilling these criteria and can be suitable sources for phase-contrast imaging. In this work, we present the first phase-contrast images obtained by using ionization injection-based laser wakefield acceleration, which results in a higher photon yield and smoother X-ray beam profile compared to s e l f-injection. A peak photon yield of 1 . 9 × 10 11 ph / sr and a source size of 3 µ m were estimated. Furthermore, the current laser parameters produce an X-ray spectrum mainly in the soft X-ray range, in which laser-plasma based phase-contrast imaging had yet to be studied. The phase-contrast images of a Chrysopa lacewing resolve features on the order of 4 µ m. These images are further used for a tomographic reconstruction and a volume rendering, showing details on the order of tens of µ m.

. Schematic of the experimental setup showing the most relevant components. The laser pulse propagates from left to right and is focused on the entrance of the gas cell. Electrons are accelerated and generate X-ray radiation which co-propagates with the electrons and the laser pulse. The wire grid is used to determine the X-ray source size and is not present during the tomographic image acquisition. The dipole magnet disperses the electron beam onto a scintillating screen to monitor the energy. The X-rays propagate to the sample, mounted on a rotational stage. The X-rays are then allowed to propagate a large distance and are finally detected by the CCD. The coordinates (x s , y s ), (x, y) and (x d , y d ) refer to the transverse plane of the X-ray source, sample and detector respectively. with a polarization ratio up to 80 % and an asymmetric divergence [29].

Experimental method
The experiment was performed at the Lund Laser Center using a chirped-pulse-amplificationbased multi-terawatt laser system with titanium-doped sapphire as the amplifying medium. In this experiment the laser was set to deliver pulses with an energy of 730 mJ on target with a duration (FWHM) of 37 fs, as measured using an intensity autocorrelator, at a central wavelength of 800 nm. The final compression is done in a grating-based compressor in vacuum and is fine-tuned using an acousto-optic programmable dispersive filter (Fastlite Dazzler) in the front-end of the laser system. After temporal compression the pulses are sent through vacuum tubes to the main experimental vacuum chamber.
The main parts of the experimental setup are illustrated in Fig. 1. The temporally compressed laser pulses are focused using an off-axis parabolic mirror with an effective focal length of 775 mm onto the entrance of a 6-mm-long gas cell. The diameter of the laser pulse in the focal plane is approximately 13 µm (FWHM), leading to an estimated peak intensity of 6.5 × 10 18 W/cm 2 in vacuum, corresponding to a peak normalized vector potential of a 0 = 1.7.
The gas cell is filled with a mixture of helium and nitrogen with a ratio of 1:99 a few tens of milliseconds before the arrival of each laser pulse. This mixture is chosen to allow for a substantial ionization injection of electrons to be accelerated [31]. The backing pressure used for filling the gas cell is set using an electrically controlled gas regulator. Previous studies on similar gas cells showed a filling factor of 90%. The repetition rate of the laser system is 10 Hz but the effective repetition rate is closer to 0.1 Hz, as the residual gas needs to be evacuated prior to the next arriving laser pulse. For the peak normalized vector potential reached in vacuum, the helium atoms are fully ionized and the nitrogen atoms ionized to N 5+ long before the arrival of the peak of the laser pulse. The released electrons, together with the ions, constitute the background plasma which the main part of the laser pulse interacts with.
The ratio between the peak power and the critical power for self-focusing [32,33] in the linear regime is approximately 9, and thus the laser pulse is expected to be guided over a much longer distance than the Rayleigh range and at the same time reaching much higher a 0 than the estimated value in vacuum. Thus, the threshold for ionization of nitrogen to the second highest level is expected to be reached after some propagation in the plasma and only close to the peak of the laser pulse. The electrons released here are more easily trapped and accelerated in the following plasma wave driven by the laser pulse.
The electrons accelerated in the interaction exit the cell along the optical axis and are dispersed according to their energy using a 20 cm long dipole magnet with a peak magnetic field of 0.8 T, before impacting on a scintillating screen (Kodak Lanex Regular). The scintillating screen is imaged from the back side using a 16-bit scientific CMOS camera (Andor Zyla 4.2 Plus). By numerically tracing electrons of different energies through a measured map of the magnetic field, the relation between electron energy and position on the scintillating screen along the dispersed direction is determined. Furthermore, the response of the full imaging system is calibrated and used together with previously published calibration factors for the scintillating screen [34] to determine the amount of charge detected at each position. This enables the reconstruction of the energy spectrum of the accelerated electrons. Figure 2(a) shows a typical spectrum for a backing pressure of 230 mbar. The mean electron energy remains fairly constant over the range of scanned pressure, see Fig. 2(b). The maximum electron energy, defined here as the energy at 10% of the peak charge-per-energy, dQ/dE, shows a stronger trend to decline towards higher pressure and the maximum of the collected charge coincides with the maximum photon yield.
The dipole magnet is used to separate the electrons from the laser beam and the beam of X-rays generated in the acceleration process. The X-ray beam exits the experimental chamber through a 250 µm thick beryllium window and enters the vacuum chamber housing the CCD-chip of the X-ray camera (Andor iKon-L SO), sensitive to X-ray up to 20 keV, through a beryllium window of the same thickness. The X-ray CCD has 2048 × 2048 square pixels with a size of 13.5 µm. This detector is positioned 2.1 m from the front of the gas cell. The separation of the two beryllium windows is 6 cm over which the X-ray beam propagates in air.
To characterize the flux distribution and energy distribution of the X-ray pulse, a Ross-filter array [35,36] was inserted into the X-ray beam close to the exit of the experimental chamber. The filter array used was composed of intersecting strips of the elements Cu, Sn, Zr, Fe, Ni, and V, of thickness 25, 3, 3, 3, 5 and 3 µm, respectively. The average signal values in the image behind each strip provide samples of the X-ray spectrum to which an assumed synchrotron-like spectrum is fitted and allows for determination of the critical energy.
This way, the critical energy was determined while varying the background electron number density by changing the backing pressure to the gas cell. The result is shown in Fig. 3(a), together with the peak photon yield and mean photon yield. The error bars indicate the standard deviation within the average of 10 different X-ray pulses. The photon yield has a broad maximum centered around 230 mbar with a maximum value of 1.9 × 10 11 ph/sr, corresponding to an electron number density of 1 × 10 19 cm −3 . The critical energy at these pressures is approximately 2.4 keV. The X-ray beam divergence was estimated by fitting a Gaussian function to the X-ray intensity profile on the CCD camera. It was estimated to be 48 (vertical) and 67 mrad (horizontal) FWHM at 230 mbar. The divergence remained unchanged up until 400 mbar.

Results
A source size measurement was performed by placing a grid of 25 µm thick tungsten wires in the X-ray beam, 10 mm from the exit of the gas cell. As there was some concern for the durability of the 25 µm wires, the thicker 50 µm wires were also included in the grid. Having wires in both the vertical and horizontal plane allowed for a size measurement in both planes. The resulting diffraction patterns were fitted to simulated data that was obtained by the use of the equation [37] where x s and x d are the coordinates in the source plane and detector plane respectively. r 1 is the source-sample distance and r 2 is the sample-detector distance. B(x s ) is the spatial distribution of the source which was assumed to be Gaussian andĪ(x d ) is the spectrally weighted average of where x is the object plane coordinates. The total propagation distance is r = r 1 + r 2 . The wire's transmission function q is given by where R is the wire radius and µ(λ) the attenuation coefficient. The weights w λ used for the spectral average was determined by the spectral probability distribution by considering the synchrotron-like spectrum of the source at E c = 2.4 keV, the quantum efficiency of the X-ray CCD, the photon absorption through the two beryllium windows and the air gap to the detector. The intensity distribution I(x d , λ) is Figure  3(b) shows the best fit for the horizontal and vertical source size respectively for the 25 µm thick tungsten wire. This results in a vertical size of approximately 3.6 ± 0.2 µm and a horizontal size of 2.6 ± 0.2 µm (FWHM).
To determine optimal r 1 and r 2 that result in the best SNR for PB-PCI, the findings by Ya. I. Nesterets et al. [5] were implemented. The procedure maximizes SNR with respect to the optimal magnification for a symmetrical Gaussian feature of a homogeneous object, wavelength, source size and detector resolution. For a source size of 2.6 µm, pixel size of 13.5 µm and a magnification that yields the best detector resolution results in r 1 = 0.6 m and r 2 = 1.7 m. This was further investigated by performing Fresnel-Kirchoff diffraction simulations and analyzing the contrast using different distances. The simulations gave better SNR with a magnification larger than what was obtained via the optimization procedure developed by Nesterets et al. This, in combination with the limited space inside the experimental chamber, motivated the use of a smaller r 1 , and the final distances were r 1 = 0.3 m and r 2 = 1.8 m.
The detected image, I 0 , is processed before any further calculations by subtracting a dark field image I d and normalizing to a flat field image, I f , as I = (I 0 − I d )/(I f − I d ). This flat and dark field correction constitutes an issue as the flat field changes from shot-to-shot, is non-uniform, and it is not possible to simultaneously acquire a flat field and a corresponding sample image. By taking the average pixel value at several positions in the image and generating an cubical interpolated mesh, I g , one obtains an approximated image background gradient, which results in a more representative image. This was implemented for the phase-contrast images of the Chrysopa specimen, Fig. 4(c).
All single-shot phase retrieval algorithms make some assumption on absorption and A. Burvall et al. gives a good overview on these [38]. The soft X-ray spectrum is subject to some absorption in the sample which limits the choice to Paganins single-material algorithm [39], since it does not require absorption close to zero. Instead, one assumes the absorption to be proportional to the refractive index decrement, δ. For a monochromatic source of wavelength λ, the projected thickness of the sample is [39] where ì r ⊥ is the position vector perpendicular to propagation, F denotes the 2D Fourier transform, ì k ⊥ is the spatial frequency vector, ì k ⊥ = (u, v), so that | ì k ⊥ | 2 = u 2 + v 2 . The spatial frequencies, u, v, were calculated as M · F(−1/2+(n i −1)/(n−1)), where M is the magnification, F the number of pixels per unit length, n i the pixel number and n total number of pixels in the corresponding dimension [40]. I( ì r ⊥ , r 2 ) λ is the normalized and background subtracted PCI. As the X-ray source is polychromatic, the effective refractive index decrement was calculated as the spectrally weighted average δ e f f = ∫ w λ δ λ dλ/ ∫ w λ dλ and the effective attenuation coefficient µ e f f was calculated in the same way. The projected thickness was calculate by Eq.  a small green lacewing of the Chrysopa genus. For each of these samples, 900 images were taken, divided into 5 images at each angle over 180 degrees in 1 degree increments. Each set of 5 images were averaged to increase the SNR. The last set of images obtained at 180 degrees are not necessary for the tomographic reconstruction but were used to find the center of rotation by overlapping with the set of images at 0 degrees. As some shot-to-shot pointing fluctuation is present in the laser system and, a general pointing drift, i.e. the laser focus drifts toward a general direction over time, the image will move on the detector. Taking an average of several images would diffuse any details and decrease the contrast. To account for this, a method of aligning the images before averaging was adapted by using a stationary reference object positioned in the plane of the sample. By performing edge-detection and cross-correlation computations the images were translated to cancel the pointing drift. The alignment can be sub-pixel accurate by interpolating either the images or the cross-correlation matrix.
Applying the procedure for background correction described earlier and calculating the projected thickness using Eq. (3) on a sample of known thickness, the 100 µm thick CH-line, showed good agreement. Figure 4(b) shows the projected thickness of this CH-line. This thickness varied along the line but this was believed to be due to the manufacturing process or possibly the process of tying the knot, which may have deformed the wire to some degree.
The tomographic images were reconstructed by the filtered backprojection from sinograms, which in turn were generated from the projected thickness data set. The tomographic reconstruction assumed parallel beam geometry and using the open-source software 3D Slicer [41][42][43][44][45][46], a volume rendering was made of the sample from the tomographic images.
Here we present a single PCI of the Chrysopa specimen sample, Fig. 4(c), along with tomographic images. Location of the cross-section is indicated by a red dashed arrow. In the PCI, some very fine details are visible such as the hairs, estimated to be 4 µm thick. The tomographic images were reconstructed by averaging and aligning 5 shots for each angle, calculating the projected thickness using Paganin's algorithm, Eq. (3), and applying the filtered backprojection formula to the sinograms. The finer details are lost when calculating the projected thickness as Eq. (3) also acts as a low-pass filter. Finally, we present the volume rendering, shown in Fig. 5. In these images, the finest details visible are the small follicles at the neck. These are roughly 30 µm in diameter and 13 µm in height.

Discussion
The X-ray yield has a broad maximum centered around a backing pressure of 230 mbar which coincides with the maximum collected charge in the electron beam. The electron spectra exhibits a Maxwellian-like distribution which is most likely due to the continuous injection and the fact that the length of the gas cell is much longer than the dephasing length. To be able to continuously operate at the optimal electron density, the plasma density needs to be consistent. However, at a constant gas cell backing pressure, the plasma density might be decreased due to the decrease of the filling factor after a few hundred laser pulses due to ablation of the entrance and exit holes in the gas cell. This might be overcome by using gas cells with an entrance hole for the laser pulse that does not ablate easily. The peak photon yield and mean photon yield are close in magnitude, indicating a smooth beam profile. The X-ray beam divergence was observed to be fairly constant over the range of 150 − 400 mbar along with the critical energy.
The source was measured to be (2.6 ± 0.2) × (3.6 ± 0.2) µm, with the horizontal size being the larger of the two. This asymmetry was expected and the size is larger in the direction of the polarization of the laser pulse (horizontal polarization). It has previously been observed that the accelerated electrons, when overlapping with the laser pulse tail, oscillate with larger amplitudes along the laser polarization [31,47]. Consequentially, this creates a larger X-ray source in this direction. The influence of the laser beam on the electron beam is reported to decrease as the laser pulse is made shorter [48]. Thus, a shorter pulse would be preferable to decrease the source size, assuming the photon flux does not decrease.
The SNR could be improved by introducing additional filters and/or increasing the number of shots averaged. An increase in the laser and gas injection repetition rate would facilitate this. The calculated thickness of the sample depends strongly on the ability to detect diffraction fringes in the X-ray beam, making the detector resolution a crucial aspect of the setup [5]. To further improve, a detector with higher pixel density would be mandatory, this may however decrease the SNR due to the smaller pixel size.
The tomographic reconstruction could be further improved as parallel beam geometry was assumed. This was not the case as the beam is divergent (estimated to about 60 mrad). This would result in a rhombus distortion in the sinogram, introducing some error in the reconstruction if not taken into account. The beam divergence is still relatively small, hence the error will not be significant, but the reconstruction would still be improved if assuming a cone beam geometry.

Conclusions
It has been shown that ionization injection-based LWFA can provide sufficient X-ray flux and beam quality for high-resolution PCI by using ionization injection at a suitable plasma electron density. This allows for rapid data collection at high-repetition rate systems and the sample is only illuminated during data acquisition, minimizing the applied radiation dose. The phase contrast images resolve features on the order of 4 µm and the final 3D volume rendering shown in Fig. 5, resolves details on the 10 µm scale such as the follicles, indicated by the red arrows. Despite the sample having a considerable thickness and absorption at the photon energies used, soft X-ray PB-PCI provides good resolution and seems to be limited primarily by the source size. To further decrease the source size, the oscillation amplitude of the accelerated electrons would need to be decreased. This would also decrease the critical energy of the emitted X-ray spectrum which is proportional to the oscillation amplitude. As such, this may be a possibility to gain a tunable X-ray source by using clever injection methods that allow for injection at a certain transverse position, thereby controlling the amplitude.
Using lower energetic X-rays improves the contrast but it also decreases the SNR due to the increased absorption. By using a thinner sample, the increase in contrast remains but as the absorption decreases, the SNR improves, and in the extreme case for a pure phase-object (no absorption) softer X-rays will always be more beneficial over hard X-rays. There is some loss in detail as the projected thickness is calculated, which is needed to perform the tomographic reconstruction, but one may view this as complementary data to the PCI. The volume rendering creates a full 3D object which can be manipulated and analyzed by a number of different tools and software packages, providing a valuable workspace.
This technique is intriguing since it allows for X-ray imaging of small objects that generally have low absorption, along with the possibility to fully 3D render the object. The advantage of using laser-plasma produced X-rays is the compactness of the system compared to a synchrotron source, its small source size and relatively high flux. An additional advantage is the automatic synchronization. By extracting part of the laser beam, one may easily construct a pumpprobe experiment with accurate timing (assuming the laser beam has enough energy to spare). Furthermore, the ultra-short X-ray pulse duration, which is on the order of fs, allows for temporally resolved experiments. The X-ray beam divergence allows for a compact setup when imaging mm-sized objects as the distances needed for the beam to expand and cover the full sample are relatively small. Smaller samples, especially ones exhibiting ultra-fast dynamics (such as chemical reactions or molecular vibrations) might require some focusing optics in order to achieve a satisfactory X-ray flux on the sample. Such small systems would on the other hand benefit from the increase in contrast due to using soft X-rays and this method may have a large impact in other fields such as micro-and nano-structures, cell biology and aerosol studies.