1. Introduction

X-ray phase-contrast imaging (PCI) gives great contrast compared to X-ray absorption imaging in lowly absorbing samples, such as soft tissue or small objects and has previously been used in biomedical applications such as in vivo mammography [1,2], imaging human joints [3] as well as material science, among others. This method relies on measuring the phase-shift induced by the sample, instead of absorption, which traditional radiography relies on. The phase shift is measured as an intensity modulation at a detector and is obtained by applying a phase-retrieval algorithm. Further post-processing allows for calculations of the projected thickness, making the images suitable for standard tomography to obtain a 3D reconstruction of the sample.

back projIn-line, or propagation based, phase-contrast imaging (PB-PCI) requires no optical components, which simplifies the experimental set-up. With appropriate propagation distances, the phase-shift will by itself evolve into a measurable intensity modulation. The drawback of PB-PCI is that it requires a more complex phase-retrieval method compared to other techniques, such as grating based PCI (where the phase-shift may be obtained directly) [4].

Several factors impact the image quality, such as contrast, signal-to-noise ratio (SNR) and resolution. These quantities are all associated, forcing a compromise between them, and several reports discuss the process of optimizing PB-PCI [5–8]. For example, greater contrast results in lower SNR. Soft X-rays (energy less than 5 keV) yield greater contrast (as the phase contrast is proportional to the phase-shift which is proportional to the wavelength) compared to hard X-rays; this will also increase the average absorption in the object, resulting in lower SNR [5]. For very small/thin samples, the absorption is also small as it scales exponentially with the optical depth according to the Beer-Lambert law. In such cases, soft X-ray PB-PCI may be advantageous over hard X-ray PB-PCI due to its increase in contrast without significantly degrading the SNR. By choosing geometrical parameters that maximize SNR while maintaining the highest possible contrast, a soft X-ray source may be optimal for PB-PCI of thin samples.

Lower energetic X-rays are preferentially absorbed over higher energies in the sample, resulting in the X-ray beam having a higher mean energy after the sample compared to before, this is referred to as beam hardening. This can lead to artifacts in the tomographic reconstruction when using a polychromatic source. These artifacts appear as either dark streaks going through brighter areas (streaking artifacts) or an increased brightness at the edges of an object (cupping artifacts). Beam hardening becomes less prominent for smaller samples as there is less absorption. On the other hand, it increases for a softer polychromatic X-ray source as this leads to more absorption in the sample.

The X-ray source size will determine the optimal magnification and therefore influences the overall resolution [6]. In general, a smaller source size performs better but to properly determine the experimental parameters, this size has to be known and a source size measurement is needed.

Although small commercial X-ray sources with a few μm in size are readily available in microfocus X-ray tubes, they have a limited flux. Furthermore, they produce a continuous beam which limits the possibilities for temporally resolved studies. The X-ray source generated by laser wakefield acceleration (LWFA) can be very small, and was recently shown to be suitable for PB-PCI [9–12]. LWFA give high flux in relation to the source size and the X-rays are generated in very short pulses (fs), allowing for time-resolved studies of ultra-fast phenomena.

In LWFA, the production of X-rays is accomplished by focusing a high-power laser pulse onto a gas target. As the intensity at the focus is very high (on the order of 10¹⁸ W/cm²), a plasma will be generated. The main part of the pulse is, due to the ponderomotive force, pushing electrons out from high intensity regions which creates a void of electrons inside the plasma. Within this electron void, referred to as a “plasma bubble”, a very strong electrostatic potential gradient is formed (on the order of 1 TV/m) [13–21] and some of the electrons trapped within may be accelerated to hundreds of MeV. Electrons not located on the optical axis in the plasma bubble will, due to the focusing force of the plasma, start to oscillate. These oscillations of electrons lead to the generation of X-rays, and the size of the X-ray source which is a few μm, is smaller than the plasma bubble diameter.

Synchrotron radiation is produced whenever a charged particle is accelerated perpendicular to its velocity and, for relativistic electrons the radiation spectrum is in the X-ray range. For a single relativistic particle, this radiation becomes highly directed, having an opening angle that is inversely proportional to the Lorentz factor, as ϕ = 2/γ. In LWFA, a group of particles (an electron bunch), is oscillating within an envelope, and the X-ray beam divergence depends on the maximum envelope amplitude r_β, plasma wavelength λ_p and electron energy as $\theta \propto \sqrt{(2/\gamma )}\cdot r_{\beta }/{\lambda }_p$ [22]. The generated X-ray spectrum is synchrotron-like [23–25] and, as such, the spectrum is broadband and may be characterized by its critical energy, E_c, i.e. the median energy for which 50 % of the total number of photons have an energy below E_c. In a LWFA, E_c ∝ n_eγ²r_β, where, n_e is the electron density in the plasma.

This is to our knowledge the first time a laser-plasma accelerator using ionization injection has been used for PB-PCI. Previous works have been done using self-injection [26] which relies on the self-evolution of the laser pulse to break the plasma wave and thereby inject electrons into the plasma bubble. Ionization injection [27] allows for a more controlled injection mechanism by using a gas mixture, e.g. helium and nitrogen. Electrons are continuously injected as long as the peak intensity of the laser is above a certain threshold or until the bubble is fully loaded with charge. This leads to a broad spectrum and it has been shown that ionization injection increases the accelerated charge and the X-ray yield compared to self-injection [28,29].

The injection method will impact the characteristics of the X-ray beam and, ionization injection has shown to be less sensitive to laser energy and beam-size variations as well as producing an X-ray beam with a higher, more stable flux. It also shows a more stable divergence and critical energy compared to self-injection [28,30]. Since the electrons are injected close to the peak of the laser pulse, they also have a preferred direction of oscillation, resulting in an X-ray beam with a polarization ratio up to 80 % and an asymmetric divergence [29].

2. Experimental method

The experiment was performed at the Lund Laser Center using a chirped-pulse-amplification-based 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 × 10¹⁸ W/cm² in vacuum, corresponding to a peak normalized vector potential of a₀ = 1.7.

 

Fig. 1 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.

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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 N⁵⁺ 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 a₀ 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.

 

Fig. 2 (a) A typical electron spectrum for a backing pressure of 230 mbar. The spectrum is broad and decays exponentially towards higher energies. The mean (gray dashed line) and maximum energy (solid blue line) are shown in b), the error bars represent the standard deviation over 10 different spectra. The collected charge (dotted orange line) shows a maximum at a backing pressure of 230 mbar.

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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 × 10¹¹ ph/sr, corresponding to an electron number density of 1 × 10¹⁹ 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.

 

Fig. 3 (a) Critical energy and X-ray photon yield obtained from the Ross filter measurements. Maximal photon yield is obtained at a backing pressure of 230 mbar, corresponding to an electron number density of 1 × 10¹⁹ cm⁻³. At this pressure, a critical energy of 2.4 keV is obtained. Error bars represent the standard deviation within the average of 10 X-ray pulses. The backing pressure used during the tomographic scan resulted in a peak photon yield of approximately 1.9 × 10¹¹ ph/sr, a divergence of 48 × 67 mrad² in the vertical and horizontal direction respectively and a critical energy of 2.4 keV. (b) knife edge measurement of the source size, using a 25 μm tungsten wire. The obtained data (dots in b)) was compared to simulated values (shown as a shadow), resulting in a vertical source size of 2.6 ± 0.2 μm and a horizontal source size of 3.6 ± 0.2 μm.

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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]

To determine optimal r₁ and r₂ 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 r₁ = 0.6 m and r₂ = 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 r₁, and the final distances were r₁ = 0.3 m and r₂ = 1.8 m.

The detected image, I₀, 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 = (I₀ − 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).

 

Fig. 4 (a) shows a raw PCI of a 100 μm thick CH-line with some visible edge enhancement due to the phase-contrast effect. The projected thickness of the same CH-line, obtained by using Eq. (3), is shown in (b). (c) shows a PCI of the Chrysopa specimen accompanied by three tomographic slices, the location of these at the specimen is indicated by the red dashed arrows.

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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]

As the X-ray source is polychromatic, the effective refractive index decrement was calculated as the spectrally weighted average δ_eff = ∫ w_λδ_λdλ/∫w_λdλ and the effective attenuation coefficient μ_eff was calculated in the same way. The projected thickness was calculate by Eq. (3) using δ_eff and μ_eff.

Two different samples were used; a 100 μm thick plastic line (CH-line) tied in a reef-knot and 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 [41–46], 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.

 

Fig. 5 The volume rendering of the Chrysopa specimen is shown here, a) shows the upper part. (b) shows the head where some finer details are resolved, such as the follicles. These are located within the red circle and a line-out plot is shown of two such follicles, showing them to be on the order of 30 μm in width and 10 μm in height. (c) shows a section of the head, which has hollow chambers and (d) shows a close up of the tarsus (leg) where two hooks can be seen at the tip.

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3. 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.

4. 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 pump-probe 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.

Author contributions statement

K.S., I.G.G., O.L and M.H planned and performed the experiments. I.G.G., K.S., A.P., J.B.S. and H.E. prepared the experimental setup. K.S., I.G.G and J.B.S. analyzed the data and K.S and M.H wrote the manuscript. K.S., I.G.G., M.H. and O.L. discussed and interpreted the results. All authors reviewed the manuscript.