1. Introduction

High power ultrashort-pulse laser facilities across the world start to operate PW class laser systems and many planed experimental projects are related to the generation of synchrotron X-ray emission during the LWFA of electrons. Although this process is well known, there is a lack of experimental reference data and experimental scaling law to guide programs at the PW level. Here we describe the parameters affecting LWFA-based X-ray synchrotron generation and we present the scaling law obtained for this synchrotron X-ray emission with smaller scale experiments realized with a 20 fs laser system operated with a maximum power of 200 TW on gas jet target. We emphasize that the primary concern for these experiments was to optimize the synchrotron X-ray emission for imaging applications.

High intensity lasers can produce femtosecond X-ray pulses extending up to the 10’s keV hard X-ray range called Betatron X-ray beams. They are generated when an intense femtosecond laser pulse is focused onto a gas jet target. This laser pulse produces an under-dense plasma and excites a wakefield wave in which electrons are trapped and accelerated to high energies over mm range distances. These trapped electrons incur Betatron oscillations across the propagation axis and emit X-ray photons [1–3]. The emission properties for these Betatron oscillations are depending on the strength parameter $K=r_\beta k_p\sqrt {\gamma /2}$, which in this case occurs in the wiggler regime: $K\gg 1$. In this definition, $\gamma$ is the relativistic factor of the electron, $r_\beta$ is the Betatron transverse amplitude of motion, and $k_p=\omega _p/c$ ; where $\omega _p$ is the plasma frequency, and $c$ the speed of light in vacuum. The integrated radiation spectrum $dI/d\omega$ is similar to that produced by synchrotrons, characterized by its critical energy $E_c=\hbar \omega _c$, where $\omega _c=3K\gamma ^2\omega _\beta /2$ and $\omega _\beta$ is the Betatron frequency. These synchrotron X-ray beams are broadband, collimated within ten’s of mrad, and with femtosecond duration. The main interest for X-ray imaging is the micrometer range source size, which provides radiation with high spatial coherence and permits phase contrast imaging and tomography [4–7]. These X-ray beams characteristics are described in several works [8,9]. To trap electrons in the wakefield and accelerate them, low-Z gases are generally used as the interaction medium [10,11]. In this case, electrons are mostly trapped by transverse self-injection which requires a normalized vector potential amplitude $a_o \sim 4$ [12]. Thus, the transverse self-injection step usually follows laser pulse self-focusing which permits to increase the laser intensity. Several articles have introduced ionization trapping by mixing controlled amount of higher Z gas with helium [13–15]. Ionization injection has a lower intensity threshold with $a_o \sim 2$. A few papers did produce and characterize synchrotron X-ray beams generated by focusing the laser pulse into a pure nitrogen gas target [16–18].

In this paper, we discuss the physical processes affecting the generation of intense and stable X-ray beam during the propagation phase of the high intensity ultrashort pulse in the gas jet target. We demonstrate stable propagation in gas jets of a relativistic laser pulse through self-guiding on length well larger than the dephasing and depletion lengths, which allows producing very intense beams of hard X-rays. The experimental scaling law obtained for the photon yield in the 10-20 keV range are presented and the level of X-ray emission at the 1 PW laser peak power level is estimated.

2. Experimental conditions and parameters

The experimental campaigns have been performed at the Advanced Laser Light Source (ALLS) facility at INRS-EMT, using a high power Ti:sapphire laser. This laser system can deliver on target as much as 500 TW (9 J, 18 fs, and 2.5 Hz). In these experimental series, in order to explore the laser-based synchrotron X-ray emission experimental scaling, the on-target laser parameters have been limited to instantaneous power P between 80 and 200 TW, energy between 2 and 4 J, and pulse duration $\tau _L$ between 20 and 35 fs. The pulse duration is changed by shaping the width of the laser spectrum. All the possible combination are not necessarily present in the data points: for example the 80 TW instantaneous power is obtained with 2.75 J combined with 35 fs pulse duration, and the 170 TW instantaneous power is obtained with 3.6 J combined with 21 fs pulse duration.

The laser pulse rise time is an important parameter in our experiments since the laser pedestal ionizes the gas and the main pulse is thus propagating in a plasma. The laser pulse intensity time profile is measured by a third order auto-correlator. The laser pulse contrast ratio is $10^{-6}$ at $\sim$ 8-10 ps before the laser pulse peak intensity.

A deformable mirror is used to monitor and control the laser beam phase front and the laser focusing. The far field distribution is imaged at nominal energy using an imaging system. In the focal plane, the full width half maximum (FWHM) spot size is 15 $\mu \textrm {m}$ with 80% of the total energy contained within an area limited by the $1/e^2$ radius. We also observe that the far field distribution presents some low energy rings outside the main spot. The energy outside the most intense spot may contribute to produce a large volume of ionization, which could affect the propagation of the main pulse.

The experimental geometry has already been presented in detail in another publication [18]. The laser pulse is in linear polarization and is focused by an off-axis parabola (f/12 with 1.5 m focal length) onto a supersonic gas jet inside a vacuum vessel. The supersonic gas jet is produced by a 7-mm diameter nozzle, which results in a density profile well defined by a 7.5-mm-long electron density plateau. On-target laser intensity $I_L$ has been varied between $2 \times 10^{19}$ and $6\times 10^{19}$ W/cm$^2$. The corresponding normalized vector potential amplitude $a_0$ is between $3$ and 5. In practical units, it is given by $a_0 = 8.5 \times 10^{-10} \lambda _L [\mu m] I_L^{1/2} [W/cm^2]$, where $\lambda _L$ is the 800 nm laser fundamental central wavelength. The gas jet electron density is adjusted between $2\times 10^{18}$ and $7\times 10^{18}$ cm$^{-3}$ in order to explore a range of ratio $P/P_c$ between 6 and $\sim$ 40. $P_c$ is the critical power for the onset of relativistic self-focusing, and $P_c = 16.2 \frac {n_c}{n}$ in GW ($n$ is the electron density, and $n_c$ is the plasma critical density) [19]. The focus has been optimized for maximum X-ray yield. This position remains the same for all data point. It is close to the entrance of the gas jet: 1 mm from the beginning of the gas jet density plateau. All the experiments presented here have been done with $N_2$ gas target.

Various diagnostics allow to measure the electrons and X-ray beams parameters. An electron spectrometer is used to measure the electron spectra, divergence, and charge. An X-ray CCD is used to determine the X-ray spatial distribution, spectra, and yield. An auxiliary beam with a pulse duration similar to the main laser pulse monitors the laser propagation inside the gas jet. It is used to realize transverse shadowgraphs of the plasma at different delays compared to the main laser peak intensity from ps up to ns range. An imaging system filtered by an interferometric filter centred at the laser fundamental wavelength is also used to measure the integrated Thomson scattering emitted by the plasma. The spatial resolution for these two imaging systems are respectively 15 and 11 $\mu$m.

3. Intense laser beam propagation in a gas jet

Laser pulses in current TW and PW systems are spatially and temporally far from an ideal Gaussian beam. However, numerical calculations are usually done with spatially Gaussian intensity distribution, which results in ideal far field spot, spatial beam quality and beam phase front. As an example, 3D PIC simulations by Ferri et al. indicate that the performances on the electron acceleration and the laser-based synchrotron X-ray emission are strongly degraded by the non-Gaussian features of the laser beam [20]. Simulations using the experimental laser spot and phase show a decrease on the X-ray photon number by almost one order of magnitude compared to the Gaussian case. This reduction is related to the less efficient and delayed self-focusing that occurs with the realistic laser in comparison with the Gaussian beam. The injection starts later and less charge is accelerated to high energy.

In high peak power laser systems, the pulse intensity increases by several order of magnitude in a few picoseconds before the laser pulse peak intensity because of the coherent contrast ratio which is inherent to the compression process. During the laser pulse rise time, the intensity increases, with some radial distribution, above the ionization threshold. Any small intensity or electron density fluctuation across the beam can start to increase through a filamentation instability (thermal or ponderomotive) [21]. This instability is non-linearly coupling the spatial intensity and the electron density fluctuations. It usually develops when the plasma length $L$ is sufficiently large: $\frac {L}{\lambda _L} > 2.2 \times 10^2 (I_{14} \lambda _L^2)^{-1/3}$, where $\lambda _L$ is the laser fundamental wavelength normalized in $\mu$m, and $I_{14}$ is the laser intensity normalized to $10^{14}$ W/cm$^2$ [22]. For $N_2$ molecules, we can consider that the ionization intensity threshold is around $10^{13}$ W/cm$^2$ at 800 nm wavelength [23]. The corresponding threshold plasma length is around 500 $\mu$m, which is easily reached in the experimental conditions used in LWFA experiments with a gas jet target.

If the intensity and power conditions are appropriate, the radial intensity distribution outside the main focal spot can produce in the focal plane a large diameter ionized channel which can extend up to a few hundreds of $\mu$m. The filamentation instability could develop inside this channel generating some profiling of the plasma density before the arrival of the main pulse. The filamentation instability optimized radial fluctuation wave number $k_r$, or spatial wavelength $\lambda _r$, is given by $\frac {k_r^2}{k_L^2} = \frac {1}{4}\left (\frac {v_{os}}{v_{th}} \right )^2\frac {n}{n_c}$, where $k_L$ is the laser wave number, $v_{os}$ is the quiver velocity of an electron in the laser field, and $v_{th}$ is the electron thermal velocity. We obtain a radial fluctuation with a typical wavelength around 70 $\mu$m for an intensity of $10^{13}$ W/cm$^2$, 800 nm wavelength, electron density of $6 \times 10^{18}$ cm$^{-3}$, and electron temperature $T=20$ eV. The longitudinal spatial growth rate along the laser propagation axis is $k_i= \frac {k_L}{8} \left (\frac {v_{os}}{v_{th}}\right )^2\frac {n}{n_c}$, which for a 70 $\mu$m mode is around $0.5$ mm$^{-1}$.

When a short laser beam is incident at relativistic intensity on a gas target, the electrons are efficiently accelerated by laser wakefield and the so-called bubble, or blowout, regime is reached when the laser field amplitude $a_0$ is greater than 2 [24]. The laser beam propagates into a plasma, created by the pulse pedestal and rising intensity edge. Then it experiences multi self-focusing and defocusing effects as can be observed in PIC code simulation [20,25]. This produces very strong field amplitude variations along the propagation axis. The laser self-focus is due partly to the radial phase velocity gradient inducing a curvature of the wave front. The defocus is primary due to the plasma created by the forward part of the pulse, which is defocusing the trailing part of the pulse. This effect is important and related to the final X-ray yield.

We show on Fig. 1 transverse shadowgraphs measured at several delays (17, 167, and 360 ps) after the passage of the laser pulse through the gas jet. They illustrate the complex propagation of the laser pulse which is experiencing filamentation and self-focusing. The radial distribution of the ionized region shows strong radial perturbations of the density whose imprint appears on the beam [26].

 

Fig. 1. Shadowgraphs measured with the auxiliary laser beam at several delays. a)- 17 ps. b)- 167 ps. c)- 360 ps. d)- Integrated image recording the plasma emission to observe Thomson scattering. All the images a), b), c), and d) are at the same scale and are positioned at the same initial position. The laser beam is incident from the left side of each image. The white line length in a) is $7.5$ mm long.

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The radial perturbation distribution can be measured on the shadowgraphs at the different probing times. With a 17 ps delay, the beam has travelled only through 5 mm of the gas jet. The observed radial fluctuations are mainly related to the laser pulse rising edge, which should extend over $\sim 3$ mm. The radial distribution at a later time of 167 ps shows the same filamentation structure generated during the laser rise time. Figure 2 presents the variation of the spatial wavelength for the radial fluctuation as a function of the electron density n. It is obtained with 20 fs pulse duration and peak intensity around $4.5\times 10^{19}$ W/cm$^2$. The spatial wavelength is measured at the same longitudinal position in the middle of the gas jet. On Fig. 2, we plotted the fitted curve (thin dashed line) corresponding to a power function and obtained $n^{a}$ with $a=-0.49 \pm 0.07.$ The radial perturbation wavelength changes as $n^{-1/2}$ as expected from the scaling laws for convective filamentation instability. We also plotted the calculated curve (thick continuous curve) corresponding to the expected filamentation radial frequency. It is calculated for $T_e = 20$ eV and an intensity of $10^{13}$ W/cm$^2$, which is reached between 8-10 ps before the laser pulse peak intensity. The filamentation instability, by modulating the radial electron density profile of the gas jet before the arrival of the main pulse during the laser rising edge, could play an important role at higher peak power and laser intensities by assisting and increasing the self-guiding of the peak of the pulse.

 

Fig. 2. Radial wavelength due to the perturbation distribution versus electron density. The thin dashed line is the data fit to a power law. The thick continuous line is the calculated curve with a $n^{-1/2}$ dependency.

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In addition, with a 167 ps delay (images 1(b) and 1(c)), we start to observe dark spots, not present at earlier time, along the laser propagation axis and starting where the laser is incident onto the gas jet. We interpret the spherical structures we observe along the laser axis as series of micro-plasma related to the laser field value evolution during its propagation in the plasma with a succession of focusing and defocusing effects as shown schematically on Fig. 1 from Fourmaux et al. proceeding [27]. These spherical structures are not related to the filamentation instability as they occur after the laser pule propagation. We claim that this series of micro plasma is simply the signature of the laser intensity variation along the axis during propagation in the plasma, which we call guiding. The line out along the propagation axis shows an axial high frequency modulation that changes as a function of the distance along the gas jet and also with the gas jet parameters [26]. The integrated imaging system is used to image plasma emission corresponding to Thomson scattering as can been seen in Fig. 1(d). It shows qualitatively similar macroscopic fluctuations with strong focusing and defocusing effects along the propagation axis with high frequency fluctuations. On longer time scales, we also measure a larger ionized zone and each micro-plasma is expanding spherically. After some time, the micro-plasma expansions merge in a cylindrical channel expanding radially as a Taylor-Sedov blast wave [27,28].

We note that instabilities such as electron vortex and electromagnetic solitons, or the hosing instability are not pertinent for our experimental conditions [29,30]. The first ones appear to be similar in shape with the observed spherical structures when measured with a shadowgraphic diagnostic but they are observed with larger electron densities and for a short duration ($\sim$ 6 ps) with no evolution of their size. This is not the case of what we observe: the micro-plasma appears $\sim$ 80 ps after the laser pulse propagation and their size evolution scale like the square root of the time delay [27]. The hosing instability is usually observed at higher density compared to the one we use here. This instability could explain the small position shift of the micro-plasma along the normal axis compared to the laser axis but this can’t explain the presence of the micro-plasma.

4. Experimental scaling of LWFA based X-ray emission

The parameters governing the non-linear propagation of an ultrafast laser beam with a gas medium include $a_0$, $P/P_c$, $\tau _L$, and $\lambda _L$. In the LWFA process, the electrons are accelerated over the dephasing length $L_d$ at large energies up to hundreds of MeV. The laser energy is efficiently transferred to the electrons via the LWFA process over the depletion length $L_p$. After the laser depletion length, plasma wakefield acceleration can occur: the electron charge and energy is lost due to the creation of a wakefield driven by the electron field. Both processes: laser wakefield and plasma wakefield can contribute to the X-ray generation as shown in several publications [20,25]. In the blowout regime (where $a_0 \geq 2$), we have $L_d \sim a_0^{1/2} \times n^{-3/2}$ and $L_p \sim a_0 \times n^{-1}$ [31]. Various guiding techniques of the laser pulse through the gas jet or gas cell using low electron density have been proposed to mitigate the laser pulse diffraction and avoid defocusing before the end of the plasma. One technique has been proposed by Debus et al. to mitigate the dephasing and depletion lengths limits [32]. Along the propagation axis, the LWFA process left behind a train of micro-plasmas. In images 1(b) and 1(c), the distance on which the focusing-defocusing modulations effect, or self-guiding, extend is named the focusing-defocusing length $L_{sf}$. This length starts from the point one can observe them close to the beginning of the gas jet plateau until the last clearly observable black dot structure. This can occurs at the end of the gas jet plateau. This length is around 4.2 mm, for $a_0 = 3.2$, $P/P_c \sim 20$, and a density of $5 \times 10^{18}$ cm$^{-3}$. This is well beyond the dephasing and depletion lengths, which are respectively around 2.3 and 3.4 mm in the present case. $L_{sf}$ is decreasing when the gas density increases [26]. With a higher field amplitude: $a_0 = 5$, the self-guiding effect can be generated over a length $L_{sf}$ limited by the nozzle length.

The wiggling of the electrons accelerated in the plasma field behind the laser pulse in the LWFA bubble regime produces synchrotron radiation in the X-ray energy range. The X-ray theoretical scaling is related to the dephasing length. The number of X-ray photons $N_X$ is given by $N_X = N_bKN_e$, where $N_b$ is the number of Betatron oscillations ($N_b = \frac {L_d}{\lambda _b}$, $\lambda _b$ being the wiggler period) and $N_e$ is the number of accelerated electrons. The strength parameter $K$ defines emission properties, here the wiggler regime. For the photon number, the theoretical scaling law is found to be $N_X \sim a_0^{5/2} \times n^{-3/2}$ according to Corde et al. [31] and $N_X \sim a_0^{1/2} \times n^{-7/4}$ according to Kneip et al. [33].

We are looking for the relationship between the focusing-defocusing modulation length $L_{sf}$ and the parameters $a_0$ and $P/P_c$ (the later being related to $n$). For this purpose, we plot on Fig. 3 the parameter $L_{sf}$ as a function of $a_0^{1/2}\times n^{-1}$. The error bars on the graph correspond to the standard deviation as each data corresponds to 5 to 10 laser shots in order to improve the statistical error. As can be seen from the graph, for $a_0 \geq 2$ and $P/P_c \geq 10$, our experimental data indicate that the self-guiding length is $L_{sf} \sim a_0^{1/2}\times n^{-1}$. The curve corresponding to the power fit for which we obtained $\left (a_0^{1/2}\times P/P_c\right )^{a}$ with $a=1.06 \pm 0.1$ is plotted on the graph. This relationship can also be expressed as $L_{sf} \sim \lambda _L^{1/2} \times \tau _L^{-1/4} \times n^{-1}$.

 

Fig. 3. Relationship between the length over which focusing-defocusing modulations extends $L_{sf}$ and the experimental parameters laser intensity ($a_0$) and electron density ($P_c$). We show the curve corresponding to the power law best fit with $L_{sf} \sim a_0^{1/2}\times P_c$.

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In our experimental conditions, we typically used on-target power of 160 TW power, $4.6 \times 10^{19}$ W/cm$^2$ intensity, and $P/P_c$ around 25. We measured an electron mean energy of $\sim$ 340 MeV, X-ray critical energy $E_c$ around 15 keV, a photon number at 10-20 keV $N_X \sim 2\times 10^9$ photons/(sr$\times 0.1\%BW\times$ shot), and a divergence of the X-ray beam of $58 \times 54$ mrad [18]. The X-ray beam pointing stabilities are below one percent rms. The X-ray source has been operated at a repetition rate of 2.5 Hz, equal to the laser repetition rate.

We found that the photons number $N_X$ in the 10-20 keV band is directly governed by the focusing-defocusing self-guiding length $L_{sf}$. This energy band corresponds to the detector best detection efficiency (no filtering). On Fig. 4, we show the X-ray yield for the 10-20 keV band versus $L_{sf}$. The depletion length for these data is below 4 mm, whereas $L_{sf}$ can be much higher than this value. Thus, a large part of the X-ray emission can be due to the plasma wakefield process [20,25]. $L_{sf}$ is larger than the classical LWFA depletion and dephasing lengths but it is not an indication that the wakefield occurs over this same distance. This is just a value that is correlated to the X-ray flux. The curve corresponding to the power fit $N_X \sim \left ( L_{sf}\right )^a$ with $a=1.98 \pm 0.23$ is plotted on the graph. It follows that $N_X \sim a_0\times n^{-2}$, which can also be expressed as $N_X \sim \lambda _L^{-1/2}\times n^{-2}$. This empiric scaling, although different from the theoretical ones, has a similar trend as a function of the laser field amplitude and the gas density. The synchrotron emission is a direct result of the wakefield performance (both laser wakefield and plasma wakefield) and is therefore indirectly linked to the laser propagation. What we observe is that a larger $L_{sf}$ is correlated with a better X-ray emission. We argue that a larger $L_{sf}$ is the signature of a better laser propagation because enough laser energy is still there to produce the succession of micro-plasma.

 

Fig. 4. X-ray yield versus $L_{sf}$. We show the curve corresponding to the power law best fit with $\left ( L_{sf}\right )^a$.

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The laser peak potential evolves significantly during the laser propagation in the plasma, as shown macroscopically in some previous 3D simulations [20,25]. Here the normalized vector potential amplitude $a_0$ used in the experimental scaling law is just the one already defined in section 2 and based on the vacuum focus. One could develop a more sophisticated model like Bloom et al. where the peak potential evolution is taken into account, but this is out of the scope of the present article [34]. The available published theoretical scaling laws for Betatron radiation are developed in term of $a_0$ and $n$ ($P_c$ being equivalent to $1/n$). This is easier to compare our scaling using a similar formalism and this give a practical way to relate the laser intensity to the final X-ray yield.

To check this scaling, using the experimental series we realized, we determined the number of photons in the 10-20 keV energy band and in the 30-40 keV energy band. This last measurement is achieved with a Ross filter (Nd/Sn). In the 10-20 keV energy band, we obtain $N_X = b \times a_0\times n^{-2}$ with $b = 1.7 \times 10^{46}$ and $\sigma /b=0.3$ where $\sigma$ is the standard deviation calculated from all the available data points. We also check the validity of this scaling in the 30-40 keV energy band and obtain $b = 5.7 \times 10^{45}$ and $\sigma /b=0.4$. To illustrate these results we plot on Fig. 5 the number of X-ray photons as a function of $a_0\times n^{-2}$. We plot the scaling law for each energy band (continuous line) and the experimental data (plain circle marks). The agreement is reasonable. The obtained scaling law $N_X \sim a_0\times n^{-2}$ can also be expressed as $N_X \sim E_L^{0.5}\times \lambda \times d^{-1}\times \tau _L^{-0.5}\times n^{-2}$, where $E_L$ is the laser energy, and $d$ is the focal spot diameter. Note that the critical energy $E_c$ for these data extend from 9 to 20 keV. We anticipate that for higher energy band, around 80 keV for example, the number of photons will vary strongly depending of the critical energy and the scaling we found might not apply anymore.

 

Fig. 5. Validation of the scaling law $N_X \sim a_0\times n^{-2}$ using the experimental data series both in the 10-20 keV and 30-40 keV energy bands. The continuous lines corresponds to the average factor to fit the scaling law $N_X = b \times a_0\times n^{-2}$. The data points (plain circle marks) correspond to the data measured in all the available experimental series.

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In this work, a systematic parametric study has been achieved only for a 7.5 mm-long gas jet. We expect that using a shorter nozzle length won’t change the observed results as long as the gas jet density entrance profile is similar. Behaviour similar to the one observed on Figs. 3 and 4 should be obtained with a shorter nozzle. In this case $L_{sf}$ will be limited by the nozzle length. For a larger nozzle length at one point the electron charge and energy will be completely lost in the wakefield during the plasma wakefield process and the X-ray emission will stop. Concerning the scaling law validity in term of laser pulse duration, 20-35 fs is a range that covers most of the available pulse duration of high power laser system facilities working on LWFA. We are not aware of any high power laser system (100 TW range) suitable for laser wakefield operating in a range much shorter than this, for example, 10 fs. Thus there is a practical applicability of our experimental scaling law for most of the high peak power laser systems. One may obtain a completely different regime by working with shorter pulse duration (10 fs range), but this is something which remains to be explored and our results in the 20-35 fs range of pulse duration gives a strong reference to scale any experiment and estimate the X-ray flux.

To visualize the empiric scaling obtained with our experimental parameters, we established a chart correlating the number of photons $N_X$ to the laser power on target $P$ and to the ratio $P/P_c$. Writing our scaling for the photon number as a function of the ratio $P/P_c$ gives $N_X \sim a_0^5\times (P/P_c)^{-2}$. We found that $P/(P_c)^2$ value for our experimental data are mainly close to $3.5$ ($P$ and $P_c$ being expressed in TW). Thus, an interesting scaling of the X-ray photon number $N_X$ and of the energy $E_X$ contained in the X-ray beam in the 10-20 keV band as a function of the laser power can be obtained from this chart by considering a trajectory for which $P/(P_c)^2$ is constant. Assuming constant focusing parameters, one obtains in those conditions $N_X \sim \alpha ^{-5/2}\times (P/P_c)^3$ or $E_X \sim \alpha ^{-5/2}\times (P/P_c)^3$ where $\alpha = P/(P_c)^2$. The Fig. 6 presents the scaling of the energy $E_X$ as a function of $P/P_c$ for various values of the $\alpha$ parameter and laser power. We also indicated on this figure the experimental data with the corresponding errors. The error bars on the graph along the vertical energy axis is the standard deviation obtained over 5 to 10 laser shots. The error bars along the horizontal $P/P_c$ is the measurement error for each data point.

 

Fig. 6. Scaling chart for $E_X$ the energy contained in the X-ray beam in the 10-20 keV band versus $P/P_c$. The continuous lines (in black) corresponds to the scaling law $N_X \sim \alpha ^{-5/2}\times (P/P_c)^3$ or $E_X \sim \alpha ^{-5/2}\times (P/P_c)^3$ with alpha respectively equal to 1, 2.5, 5, and 10. We also show the line of constant instantaneous power (in red) ranging from 80 TW up to 1.5 PW. We added all the experimental points used to validate the experimental scaling law. The open circle marks are data obtained with P ranging from 135 to 195 TW. The triangle and the square marks corresponds to respectively P=170 and 80 TW.

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With such a scaling trajectory where $\alpha$ is constant, a 1.5 PW power (30 J in 20 fs) on target with $\alpha = 2.5$ ($P/P_c = 61$, $a_0 = 10$, $I_L = 2 \times 10^{20}$ W/cm$^2$) gives an energy of $0.7$ mJ per shot in the 10-20 keV band. It should be noted that lower $\alpha$ value requires higher $P_c$, which can be obtained by lowering the electron density $n$ or by increasing the critical density and thus decreasing the laser wavelength.

5. Conclusion

The recent laser technology developments indicate that 10 Hz repetition rate will be soon available at the PW level with the Ti:Sapphire laser technology, and thus, it would be possible to reach, with PW LWFA-based synchrotron radiation, a $1-10$ mW average power in the 10-20 keV energy band. The scaling can also be used to measure in situ the laser intensity in a long focal length configuration, as the one used here, from the knowledge of the gas jet density ($P_c$) and from the measurement of $L_{sf}$. Indeed, the scaling law gives us $I\lambda _L^2 = 648 (L_{sf}/P_c)^4$, where $I$ is in $10^{18}$ W/cm$^2$, $\lambda _L$ is in $\mu$m, $L_{sf}$ in mm and $P_c$ is in TW. The knowledge of the gas density and the measurement of the self-focusing length $L_{sf}$ gives an in-situ assessment of the laser intensity. It is also clear that at higher intensities, the laser pulse rise time could affect the interaction and the X-ray generation process. Frequency doubling followed by post pulse compression to minimize the energy loss is a possible interesting avenue, which could give an increase of the X-ray yield, in those conditions, while keeping a higher density and a strong self-focusing regime and minimizing the detrimental effects of the pulse rise time at higher laser intensities.