Towards direct spatial and intensity characterization of ultra-high intensity laser pulses using ponderomotive scattering of free electrons

High-intensity laser-plasma interactions have been a focus of research for several decades driven by their wide range of applications in fields such as fusion energy Zylstra et al. (2022), high-energy density physics Hoffmann et al. (2005), and table-top particle acceleration Esarey, Sprangle, and Krall (1995); Sarri et al. (2013); Tajima, Yan, and Ebisuzaki (2020); Robson et al. (2007). In all laser-plasma interactions, the intensity of the laser is a primary parameter that determines the outcome Danson et al. (2019); Yoon et al. (2021). Despite its fundamental importance, it remains an open problem to directly measure both the laser intensity, and the spatial mode on shot at full power.

Direct measurement (independent of other laser parameters) of the intensity of a high-power laser is not trivial as it can surpass values of $10^{18}$ Wcm^-2, far above the breakdown threshold of any known material. Any solid, liquid, or gaseous material placed within the vicinity of the laser focus is ionized and destroyed. For this reason, the peak intensity of almost every high-intensity laser is inferred from indirect measurements, that is, the intensity is determined from measurements of the pulse duration, pulse energy, and a measured spatial mode at the laser focus. These measurements are often made at reduced energies, or in equivalent planes, and can ignore nonlinear effects in the laser system when increasing the laser energy such as thermal distortion of the gratings and thermal lensing in the amplifiers. Any nonlinear changes in the near-field amplitude or phase induced when running at full power or high repetition rate can lead to significant changes to the far-field focal spot shape, size, pulse duration, and consequently, the peak intensity Tiwari et al. (2019).

Several techniques have been suggested to measure the laser intensity directly including nonlinear Thomson scattering Chen, Maksimchuk, and Umstadter (1998); He et al. (2019); Gao (2006); Har-Shemesh and Piazza (2012); Tarbox et al. (2015); Harvey (2018); Mackenroth and Holkundkar (2019), ionization state measurements Perry, Landen, and Szöke (1989); Chowdhury, Barty, and Walker (2001); Yamakawa et al. (2003); Ciappina et al. (2019); Ciappina, Peganov, and Popruzhenko (2020); Ouatu et al. (2022); Link et al. (2006); Smeenk et al. (2011); Alnaser et al. (2004), pair-production Aleksandrov and Andreev (2021), as well as electron, ion, and proton scattering Galkin et al. (2010); Kalashnikov et al. (2015); Vais et al. (2017); Ivanov et al. (2018); Vais and Bychenkov (2018); Krämer et al. (2018); Mackenroth, Holkundkar, and Schlenvoigt (2019); Vais et al. (2020); Vais and Bychenkov (2020); Bukharskii et al. (2022). These techniques have previously demonstrated successful direct measurements over a range of relativistic laser intensities. However, their implementation can require complicated diagnostics, extensive analyses, and an accurate, simple, and robust experimental demonstration of directly measuring laser intensities above $10^{18}$ Wcm^-2 has proven difficult. In addition to the peak intensity of the laser pulse, the spatio-temporal mode of the laser is also of great interest and has been shown to play a significant role in laser-plasma interactions Jeandet et al. (2022); Grace et al. (2022); Wilhelm and Durfee (2019). Several techniques have been deployed to measure the spatio-temporal properties of the pulse but are often conducted either in the laser near-field using pick-off mirrors before final focusing, or collecting light after the focus through re-imaging optics that can also aberrate the image and phase of the focal spot Grace et al. (2021); Pariente et al. (2016). The development of a diagnostic that can directly measure spatial, temporal, and intensity properties of a beam operating at full power is therefore of great interest.

In this work, we demonstrate through theory, simulations, and preliminary experiments how the distribution of scattered electrons ionized from ultra-low density gases $(\approx 10-100$nbar$)$ can be used to directly characterize the laser intensity, and possibly spatial profiles at full power. We derive back-of-the-envelope scaling laws that characterize the peak scattered electron energies, their corresponding forward scattering angles, and compare them to single particle simulations in paraxial laser fields. We demonstrate through simulations and experiments that for gases with sufficient ionization potentials, it is possible to make direct measurements of the laser’s peak intensity independent of laser energy, beam waist, and pulse duration. The method utilizes a single image plate detector as demonstrated in our complementary paper Ravichandran et al. (2023), and is shown to potentially reduce uncertainties in the measurements from current indirect methods. We extend our analysis to spatially aberrated beams using compositions of higher order laser modes and qualitatively demonstrate spatially modified scattering distributions indicating that it may be possible to directly measure the laser spatial mode at full power. We present preliminary experimental results obtained using two high power laser systems giving a clear demonstration of the models, methods, and a path towards future work.

As electrons interact with a laser pulse, they gain energy and momentum proportional to the field amplitude. If the electrons are born infinitely far from the laser pulse and there are no spatial gradients in the field amplitude, then according to the Lawson-Woodward theorem, no net energy can be imparted to the electrons after their interaction Esarey, Sprangle, and Krall (1995); Gibbon (2005); Macchi (2013). To transfer a net energy to the electrons the interaction must violate the Lawson-Woodward theorem either through limiting the interaction volume, utilizing the ponderomotive force of the laser, or using background electric or magnetic fields.

In the case of a high intensity laser pulse propagating in ultra-low density gas ($\approx 10-100$nbar), electrons can be ionized near the region of the peak intensity and then scattered away via the ponderomotive force with an angle relative to the laser $\hat{k}$ vector and proportional to the final energy Meyerhofer et al. (1996); Ivanov et al. (2018); Singh (2006); Lin et al. (2007); Mackenroth, Holkundkar, and Schlenvoigt (2019); Wang et al. (2002); He et al. (2003); Hartemann et al. (1995); Quesnel and Mora (1998); Yang, Craxton, and Haines (2011); Kibble (1966); Bauer, Mulser, and Steeb (1995); Popov et al. (2008); Hegelich, Labun, and Labun (2023). To calculate this scattering angle, we begin by approximating the laser as an infinite plane wave. The relativistic motion of a free electron in an infinite plane wave has been well studied, and its evolution as a function of time and space has known analytic solutions. We refer the reader to the many works on this subject and highlight some of the key results here Sarachik and Schappert (1970); Macchi (2013); Gibbon (2005); Hartemann et al. (1995); Gonoskov et al. (2022).

The motion of an electron in a general electromagnetic field can be given by the normalized Lorentz force,

where $\vec{p}=\vec{p}/m_{e}c$ is the normalized momentum, $\vec{a}=e\vec{E}/m_{e}c\omega$ is the normalized vector potential, $\vec{b}=e\vec{B}/m_{e}\omega$ is the normalized magnetic field, and the space and time variables are normalized such that $\vec{x}=k\vec{x}$, and $t=\omega t$. $\vec{E}$ and $\vec{B}$ are the oscillatory electric and magnetic fields of the wave, and $\gamma=\sqrt{1+|\vec{p}|^{2}}$, $m_{e}$, $e$, $c$, $k$, $\omega$ are the Lorentz factor, electron mass, electron charge, speed of light, and the laser wave-number and angular frequency respectively. Throughout this work we will maintain these normalization’s unless otherwise stated. We also neglect radiation reaction as we are working with intensities and initial electron energies well below the point at which these effects become important $(a_{0}\approx 100)$ Gonoskov et al. (2022).

The electric and magnetic fields of a pulsed infinite plane wave propagating along $\hat{z}$ and polarized along $\hat{x}$ can be written as the real parts of the function Jackson (1999),

where $f(t)$ is the temporal envelope of the laser, typically a Gaussian, $a_{0}\approx 0.855\lambda_{\mu m}\sqrt{I_{0}[10^{18}Wcm^{-2}]}$ is the peak normalized vector potential, $I_{0}$ the laser intensity, and $\lambda_{\mu m}$ is the laser wavelength in microns. Analyzing the motion of the electron in a linearly polarized continuous plane wave according to the Lorentz force shows that the electron will oscillate along the polarization direction due to the real part of the transverse electric field such that $p_{x}=a_{0}\cos\left(z(t)-t\right)$, where the longitudinal position of the electron is given by Macchi (2013); Gibbon (2005); Hartemann et al. (1995),

If $a_{0}\gtrsim 1$, the electron motion becomes relativistic and the $\vec{p}\times\vec{b}$ force becomes comparable to the electric force causing a significant forward drift along the $\hat{z}$ axis. If the initial electron momentum is much less than the oscillation momentum in the plane wave $(p_{0}\ll a_{0})$, one can derive the relationship Hartemann et al. (1995),

Taking the ratio of the electrons forward momentum to its transverse momentum, and the fact that in an infinite plane wave the forward momentum of the laser is conserved into the forward motion of the electron $(p_{z}=\gamma-1)$, we yield the well known result Hartemann et al. (1995); Moore, Knauer, and Meyerhofer (1995); Meyerhofer et al. (1996),

Given that the electron is constrained to the $\hat{x}-\hat{z}$ plane in a linearly polarized plane wave, we can write the Lorentz factor as,

As the Lorentz factor changes its value periodically, we look to find its average value over one laser cycle to calculate the average scattering energy. Combining (4) and (6), we can write the Lorentz factor as,

The motion of an electron in the laboratory frame calculated by numerically integrating (1) in plane waves of amplitude $a_{0}=1,2$ and $10$ are plotted in Fig.1 with a blue solid line, red dashed line, and black dotted line respectively. We see in Fig.1(a) that as the laser intensity increases, the momentum deviates from the sinusoidal trajectory into a more complicated curve given its transcendental dependence on $z(t)$. Similarly when we examine the value of $\gamma$ as a function of time in Fig.1(b), we find that its functional form changes from a quasi-sinusoid at non-relativistic intensity, into a non-elementary functional form at relativistic intensities.

To obtain an average energy of the scattered electron in these complicated trajectories, we derive 3 models. The first model assumes that a scattered electron maintains its peak oscillation energy in (7),

This average value of $\gamma$ appears to be what is used in previous works for calculating scattering angles for the electrons Kalashnikov et al. (2015); Ivanov et al. (2018). A second model calculates the average of (7) in the co-moving frame, $\xi=z(t)-t$, yielding the result,

Our third model calculates the time average of the electron momentum in the lab frame. This is difficult given the transcendental dependence on the electron position where we have limited analytic tools to perform the integration. We instead rely on numerical techniques to find an approximate formula for the time averaged $\gamma$ in an infinite plane wave,

For completeness, we also include the expression for $\gamma$ frequently used in relativistic mass expressions Kibble (1966); Meyerhofer et al. (1996), and plasma ponderomotive temperature scaling Macchi (2013); Gibbon (2005); Wilks et al. (1992),

These expressions can then be combined with (5) to estimate the scattering angle of the electrons relative to the laser $\hat{k}$. With these four plane wave models, we can compare the peak ejected values of $\gamma$ and the corresponding scattering angles to more realistic models including the effects of finite spot sizes, finite pulse durations, and longitudinal field components.

To model the realistic scattering of the electrons from tightly focused Gaussian laser pulses, we need to include the effects of the spatial profile and the additional field components introduced when focusing a laser pulse. In this work we will restrict ourselves to the paraxial regime $(kw_{0}\gg 1)$, where $w_{0}$ is the laser beam waist at best focus. There is an ongoing debate as to when and where the paraxial approximation breaks down after which the use of more sophisticated models of the electric an magnetic fields will be required Peatross et al. (2017); Popov et al. (2008). In this work, we treat beams with $5<f_{\#}<10$ as weakly paraxial and do not consider focal geometries with f-numbers smaller than 5. Under the paraxial approximation, we can write our electric and and magnetic fields in the form Peatross et al. (2017); Erikson and Singh (1994); Longman and Fedosejevs (2022),

where $\psi_{p}^{\ell}$ is the Laguerre-Gaussian basis set describing the laser spatial envelope given by,

Here we have introduced the beam waist at a plane $z$ as $w(z)=w_{0}\sqrt{1+z^{2}/z_{0}^{2}}$, the Rayleigh range $z_{0}=w_{0}^{2}/2$, the azimuthal and radial mode numbers $\ell\in\mathbb{Z}$ and $p\in|\mathbb{Z}|$ respectively with $\mathbb{Z}$ being the set of integers, the Gouy phase $\Psi=(1+|\ell|+2p)\tan^{-1}(z/z_{0})$, and the associated Laguerre polynomials $L_{p}^{\ell}(X)$ given by,

The Laguerre-Gaussian functions are an orthogonal set of solutions to the paraxial Helmholtz equation in cylindrical coordinates. Typically, we are concerned with the fundamental $(\ell=0,p=0)$ Gaussian mode as this closely resembles a typical focused laser pulse. As we will see in the following sections, it becomes necessary to include higher order laser modes to model non-Gaussian near-fields, Airy rings, aberrations, and other higher order effects typically found in realistic high-power laser pulses. Full derivations of the field equations from (12) for the Laguerre-Gaussian basis set is given in another work Longman and Fedosejevs (2022). Additional corrections to (12) can be introduced to correct for ultra-short pulse durations when $\tau\leq 2w_{0}$ Quesnel and Mora (1998). In this work we primarily work with laser pulses of duration $\tau=30fs$ and beam waists $w_{0}>4\mu$m which do not satisfy the inequality and so these corrections are neglected.

If the electron density is low enough such that the Debye screening length $\lambda_{D}$ is much larger than the interaction volume, we can treat the electrons as non-interacting single particles and directly integrate the Lorentz force without including collective effects. Assuming an electron temperature of 10 keV, a fully ionized nitrogen gas, a 100 $\mu$m screening length equates to an electron density of roughly $1\times 10^{14}cm^{-3}$. Using the ideal gas law at room temperature, we find that we need to operate at pressures roughly below $3\times 10^{-4}$mbar. This is easily achievable in current vacuum systems which routinely operate at and below $10^{-6}$mbar. Realistically, the interaction of a tightly focused ultra-high intensity laser and ionized electrons can probably be treated as single particles even in the $10^{-3}$mbar range.

Direct integration of (1) using the full 3D field equations given by (12) is performed using a high-order Runge-Kutta algorithm. On the order of 1 million electrons is found to be sufficient to effectively sample the maximum ejected energies and the corresponding ejection angle of the electrons over a large volume in space.

The simulation volume is specified in space centered about the focal point, typically $3w_{0}\times 3w_{0}\times 6z_{0}$ in $x,y,z$ respectively in which gaseous atoms are randomly distributed. A routine is run that identifies when electrons will be ionized based on their ionization potential and the position of each atom, after which it calculates the effective number of electrons scattered per unit pressure of the gas. While this volume is not sufficient to capture all electrons ionized by the laser, it is sufficient to sample the electrons ionized near the peak laser intensity and hence the electrons scattered with the highest energies and smallest angles relative to $\hat{k}$. Electrons ionized in either the temporal foot of the laser, far from the laser axis, or in any pre-pulse are normally ponderomotively swept away before they can interact with the peak laser intensity and for now, are not of interest.

Given the complexity of the ionization mechanism of the electrons, we opt to use a simple modification of the over-the-barrier ionization model such that the intensity at the ionization will occur is given by Gibbon (2005),

where $U_{eV}$ is the ionization potential of the $Z^{th}$ electron shell in electron volts, and $R$ is a random number selected between 1 and 10 for each individual electron. This random number loosely emulates the nature of the measured ionization populations as a function of intensity found in several works Augst et al. (1989, 1991). Future work may introduce a more sophisticated model of ionization such as the ADK model but as we will show, the ionization rate is used to determine the intensity range of the gas, rather than the actual intensity measurement, and as such is not the focus of the current work Tong, Zhao, and Lin (2002). Example values of electron ionization potentials, and the corresponding ionization intensities for $R=1$ are calculated and given in Table 1 using the NIST atomic database Kramida et al. (2022).

Using the laser wavelength and our best estimates of the peak vacuum intensity, we can use Table 1 to select a gas that has an ionization intensity below the estimated peak intensity, but as we will show, should be relatively close to the estimated peak intensity of the laser. Additional gases outside of Table 1, for example xenon, neon, and oxygen can also be used but are not listed for brevity.

An illustration of the simulation setup is given in Fig.2. A plane detector is placed a distance $d$ from the focal plane of the laser that registers the position and momentum of each electron passing it. Simulations were run for various Gaussian laser pulses to identify a suitable scaling model for the maximum electron energy and the corresponding minimum scattering angle. For the scope of this work, we are interested in values of $a_{0}$ in the range of 1-30 which roughly corresponds to intensities of $10^{18}$Wcm^-2 to $10^{21}$Wcm^-2 at 800nm. Intensities below this range are weakly relativistic and we find that any forward scattering of the electrons becomes negligible, while intensities above this range are possible to measure primarily via the electron energy alone as scattering angles become too small.

Fig.3 shows the scattering of 2 million electrons ionized from nitrogen gas from a laser pulse with $a_{0}=10$, $w_{0}=10\mu$m, $\tau=30$fs, and $\lambda=800$nm, where $\tau$ is the temporal intensity full-width half-maximum (FWHM). This corresponds to a peak intensity of $~{}2.1\times 10^{20}$Wcm^-2. The normalized flux of electrons striking the detector is given in Fig.3(a) showing the inner electron cut-off angle and some spatial structure in the electron distribution, notably a higher number of electrons deposited along the polarization axis. We note that unlike the prediction of the plane wave that restricts the electrons to the $\hat{E}-\hat{k}$ plane, the electrons scatter into a 3D ring in agreement with previous work Quesnel and Mora (1998).

Electron energies below 1MeV in Fig.3(a) are removed to emulate the electron stopping power of an aluminum light blocking shield used in experiments. This gives a lower electron energy cutoff and hence an outer ring as shown in Fig.3(a). An average radial lineout of Fig.3(a) is plotted in Fig.3(d) where the red dashed line indicates the measured cutoff radius of the electrons. For this set of laser parameters, we find the inner electron cutoff angle to be $\tan(\theta)=r/d\approx 0.26$.

Fig.3(b) and (c) give scatter plots of the scattered electron Lorentz factor as a function of the initial radius and longitudinal positions respectively, while the color map corresponds to the ionization shell highlighting that inner shell electrons shown in red are accelerated to the highest energies. We find that electrons born before the focal plane at $z_{inject}\approx-z_{0}/2$ are accelerated to the highest energies. Fig.3(e) gives the measured electron ejection angle $\theta_{eject}$ as a function of the ejected energy $\gamma_{eject}$. The red dashed line overlaid on the data points gives the scattered energy-angle relationship derived in (5). Although there is an excellent agreement between (5) and the data in Fig.3(e), we find that there is a broadening of the data points about the red dashed line. We believe this to be a result of the angular spread in the transverse laser $k_{\perp}$ vector components from the shorter f-number. Indeed, as the beam becomes non-paraxial and $w_{0}\to\lambda$, the spread of electrons about the red dashed line can become significant and even detrimental to our measurement of the peak intensity. Fig.3(f) gives the electron energy distribution where the red dashed line is a fitted Maxwell-Boltzmann temperature estimated to be 1.93MeV. This temperature further validates the use of the single particle approximation.

We have run simulations for various peak intensities between $a_{0}=2:30$ for 3 different beam waists, $4\mu$m, $8\mu$m, and $16\mu$m and for 3 different pulse durations, 30fs, 60fs, and 120fs. Simulations were run using nitrogen and krypton gas. Fig.4 summarizes the results of these simulations and plots the average scattered energy of the 100 highest energy electrons ejected from each of the interactions. Fig.4(a) gives the average final scattering energy for electrons ejected from a 30fs FWHM pulse with $4\mu$m, $8\mu$m, and $16\mu$m beam waists represented by the blue circles, red crosses, and yellow squares respectively. The 4 analytic models described in the previous section are overlaid using the 4 black lines. The $\langle\gamma\rangle_{peak}$ model (8) is displayed using a solid black line, $\langle\gamma\rangle_{\xi}$ (9) is displayed using a dashed black line, $\langle\gamma\rangle_{t}$ (10) is displayed using a dotted black line, and the $\gamma_{pond}$ model (11) is displayed using a dash-dot black line. We find that for laser intensities of $a_{0}\leq 10$, we have excellent agreement for all three beam waists with the $\langle\gamma\rangle_{\xi}$ and $\langle\gamma\rangle_{t}$ models. For the 30fs, $w_{0}=16\mu$m case, we find that the $\langle\gamma\rangle_{\xi}$ model agrees well all the way up to $a_{0}=30$. For smaller beam waists we find that the electrons tend to scatter with less energy when the laser amplitude is greater than $a_{0}=10$.

This is made apparent in the 60fs and 120fs cases with nitrogen in Fig.4(b) and (c) respectively. For example in the $4\mu$m 60fs case, the peak energy of the electrons essentially clamps at about $\gamma_{eject}\approx 30$. The same is true for both the $4\mu$m and $8\mu$m beams when scattered by a 120fs pulse in (c). In these situations, we believe the electrons are not sampling the peak intensity of the laser. Given that the ponderomotive force scales as $F_{p}\propto w_{0}^{-2}$, the smaller the beam waist, the faster the electron is expelled from the laser pulse. The inverse appears to be true considering the pulse duration, that is, the longer the pulse duration the less likely the electron will interact with the peak intensity before it is ponderomotively swept away. If the laser pulse is shorter, it requires less time for the electron to slip back to the peak intensity region and therefore is more likely to sample higher intensities before it is swept away.

To address this issue, we must select a gas in which the electrons are ionized close to the peak intensity as to increase the interaction probability. Switching to krypton gas in our simulations is shown in Fig.4 (d)-(f). In all pulse durations, we find that the scattered electron energy follows our $\langle\gamma\rangle_{\xi}$ and $\langle\gamma\rangle_{t}$ scalings much closer for all three beam waists, however, we find that between $a_{0}=8$ and $a_{0}=14$ there is a region where the electrons do not follow the scaling laws. This can be explained by the large gap in the ionization potentials between Kr²⁶⁺ and Kr²⁷⁺ as shown in Table 1. It is therefore more suitable to use a gas that does not have a large potential gap close to the peak intensity. While not shown, we find that xenon is a suitable candidate for measuring the peak intensity over the whole range of $a_{0}=1$ to 30 due to its large number of electrons with close ionization spacing in this regime.

This effect highlights the need for future models to introduce more advanced and realistic ionization algorithms as to effectively model the range over which a gas can be used to measure the intensity. Although our model does not capture this effect accurately, we believe that it has emphasized the necessity of an advanced ionization algorithm. We note that while the ionization rate can determine the intensity range over which a particular gas can be used, it does not appear to affect the scattering dynamics of the electrons and therefore does not affect the intensity measurement.

In Fig.5 we plot the average minimum scattering angle of the 100 highest energy electrons, and as in Fig.4, as a function of the pulse durations, beam waists and gas species. The scaling laws are computed using (5) and our 4 models of the various ejected values of $\langle\gamma\rangle$. These plots use the same marker and line styles as in Fig.4. In Fig.5(a) we plot the results of the electron scattering from a 30fs laser pulse in nitrogen. We find that all three beam waists follow both the $\langle\gamma\rangle_{\xi}$ and $\langle\gamma\rangle_{t}$ scaling laws well up to $a_{0}=20$. At $a_{0}\geq 20$ using a $4\mu$m beam waist we see a significant deviation from the scaling laws. This is due to the electrons being accelerated directly forward in this parameters, the exact physics of which is unclear at this point. As the pulse duration is increased to 60fs and 120fs in Fig.5(b) and (c) respectively, we see the smaller beam waists begin to deviate from the scaling laws similar to in Fig.4. This is due in part to the clamping of the electron energy, and also due to the increased spectrum of $k_{\perp}$ values in smaller f-number beams. Again, switching to krypton gas in Fig.5(d) to (f) shows the electrons following both the $\langle\gamma\rangle_{\xi}$ and the $\langle\gamma\rangle_{t}$ scalings closely for all three beam waists and pulse durations.

To summarize, we have found that the scalings given by $\langle\gamma\rangle_{\xi}$ and $\langle\gamma\rangle_{t}$ give good upper and lower approximations to the estimated peak scattering energy and scattering angle of the electrons provided they are ionized reasonably close to the peak laser intensity. A suitable choice of gas to ionize can be chosen based on the the over-the-barrier ionization intensities given by (15), however, more advanced ionization rate algorithms should be used to determine the exact intensity range of any given gas. For amplitudes of $a_{0}<10$, nitrogen, krypton, and argon look to be suitable gases while for $a_{0}>10$, krypton and xenon are more suitable.

While we have demonstrated numerically that the scattered electrons can make a suitable diagnostic for intensities in the range of $a_{0}=2:30$, and for f-numbers greater than 5, we have not demonstrated for intensities above this range ($I_{0}>10^{21}$Wcm^-2) or for non-paraxial beams. We note that while the scattering angle tends to zero at high intensities making precise measurements difficult, the electron energy increases linearly with the intensity. In principle it would be possible to use an electron spectrometer or a stack of image plates separated by various metal plates of various stopping powers to measure the peak intensity, however, as the intensity surpasses $10^{22}$Wcm^-2, radiation reaction and quantum electrodynamic (QED) effects can begin to play a role and our simple scalings (9), and (10) may break down. In addition, the flux of the highest energy electrons will likely reduce as the intensity increases making the measurement more challenging. More advanced models including non-paraxial beams, radiation reaction, and advanced ionization routines are therefore needed to explore the electron scattering beyond the scope of intensities, f-numbers, and pulse durations considered in this work.

Two experiments have been carried out to measure the scattering distributions of electrons from low pressure gases. In both experiments, image plates were used to record the spatial distributions of the scattered electrons averaged over many shots. The image plates measure the electron dose over a given area, that is, the convolved energy and flux of electrons. Despite only measuring dose, they offer several advantages over other methods given their high signal to noise ratio, dynamic range, aperture size, and overall cost per unit detector area. This allows us to image a large area of the electron distribution and is suitable for both multi-shot and possibly single-shot data acquisition. Fujifilm BAS MS and SR type Boutoux et al. (2015); Jackson (1999) image plates were used given their high sensitivity to electrons in the keV to MeV ranges, as well as long signal decay times Boutoux et al. (2015). The general method of using and characterizing image plates to capture electrons has been explored in our complementary paper Ravichandran et al. (2023).

Given that the image plates are sensitive to the laser light, low energy electrons and laser photons are filtered out using an aluminum foil mounted flush to the image plate. For these experiments, the image plates were mounted approximately $d\approx 40$mm from the laser focus and as such, the laser intensity was still sufficient to ablate the aluminum foil. As a result, a hole was cut in the center of the image plate and aluminum foil to allow the laser to pass through. This configuration restricts us to interactions where the electrons can only be scattered outside of the laser cone, however, one could place the image plate further away such that the aluminum foil is below the ablation threshold to avoid cutting a hole at a cost of signal strength.

As the electrons pass through the aluminum foil, they can lose energy, produce bremsstrahlung, and scatter within the foil adding both background and an uncertainty to our measurement. It is therefore desirable to keep the aluminum foil thin to reduce these effects. We can estimate the electron stopping power based on the aluminum thickness and using the NIST stopping power database Berger et al. (2005). Given that we expect electrons with energies well above 100keV, we can use foils on the order of 10-100$\mu$m thick to stop low energy electrons and sufficiently block laser photons. At this thickness, we expect the additional uncertainty introduced into the electron scattering angle to be minimal relative to the other experimental uncertainties. The basic experimental arrangement has been illustrated in Fig. 6.

To estimate the laser intensity using the standard indirect method, images were acquired of the focal spot in a reduced power mode using a long-working distance objective and a charge coupled device (CCD) camera. The magnification of the imaging system is determined using a known calibration allowing us to determine the effective focal spot area and distribution. We normalize the image of the focal spot to acquire an amplitude magnitude function for the laser focal spot profile,

where $\Delta x$ and $\Delta y$ are the magnified CCD pixel width and height respectively, and $I_{CCD}$ is the raw far-field image values.

We note that while we can retrieve the magnitude of the focal spot amplitude, we cannot retrieve its phase with a far-field image alone and require a second measurement to retrieve its phase such as a wavefront sensor, or a near-field amplitude measurement and then using a phase retrieval algorithm such as the Gerchberg-Saxton algorithm Fienup (1982).

Normalizing the relative intensity map then allows us to infer the peak laser intensity using the relation He et al. (2019),

where $U_{meas}$ is the measured laser energy at full power, and $\tau_{meas}$ is the measured full-width-full-maximum of the temporal intensity envelope, either at full power of a reduced power. The laser energy can be measured at full power upstream from the laser focus using leakage from a mirror and a pre-calibrated ratio of the energy in focus relative to the measurement position. The pulse duration is usually measured using an auto-correlator or a higher order measurement such as a frequency resolved optical gating (FROG) Trebino (2000). These measurements often use a small pick-off mirror placed in the beam so as not to disperse the pulse through transmissive optics. Estimating the pulse duration using this method is not trivial as the algorithms for pulse retrieval can have large uncertainties, as well as the pulse duration can change spatially within the beam if spatio-temporally aberrated.

We emphasize here that knowledge of the pulse energy, pulse duration, and the focal spot profile is needed to infer the peak intensity of the laser focus. These three measurements each have an uncertainty that can increase nonlinearly when operating at full power.

As we have seen experimentally, a laser rarely has a perfect Gaussian spatial profile unless heavily corrected or apodized and will likely contain aberrations from the laser near-field profile, misalignment’s in the optical system, and any thermal distortions from the optics when operating at full power. We have found numerically and experimentally that these aberrations play a fundamental role in the scattering of electrons. The role of aberrations in various laser-plasma experiments is relatively unexplored. However, there are certain works that claim enhanced effects due to aberrations in acceleration schemes such as wakefield acceleration Cummings and Thomas (2011), proton acceleration Grace et al. (2022), and vacuum ponderomotive acceleration Wilhelm and Durfee (2019).

Spatial aberrations are often described using the Zernike polynomials as they form a complete orthogonal basis set on the unit circle that can be applied to the laser near-field aperture Born and Wolf (1999). In this work, we omit the contributions of temporal, and spatio-temporal aberrations for brevity but acknowledge that they can be equally important in the electron scattering dynamics and will investigate them in future work Palastro et al. (2020).

To model spatial aberrations, we write the near-field complex amplitude as,

where $U_{1}(\rho,\phi)$ is the near-field spatial amplitude (often a Gaussian, super-Gaussian, or a circle), $\alpha_{m,n}$ is the aberration strength in units of $\lambda_{0}$, and $Z_{m}^{n}(\rho,\phi)$ is the Zernike polynomial given in Table 3. We note that the unit circle in the near-field can be arbitrarily specified relative to the size of the near-field profile if it is a Gaussian or super-Gaussian. Therefore, we normalize the Zernike unit circle radius to the near-field beam waist $R_{0}$ such that $U_{1}(\rho,\phi)\propto\exp\left(-\rho^{2}/R_{0}^{2}\right)^{N}$, where $N$ is the super-Gaussian parameter.

Using the angular spectrum method Goodman (2005), we can calculate the three dimensional far-field complex amplitude in the paraxial regime as,

where the subscript $\delta$ denotes the aberrated far-field complex amplitude, and $\mathfrak{F}^{-1}$ is the inverse Fourier transform. Several far-field profiles are plotted in Fig.11 and 12 for the case of a horizontal coma of strength $\alpha_{1,3}=0.05\lambda$, and an oblique astigmatism of strength $\alpha_{-2,2}=0.05\lambda$ respectively assuming a purely Gaussian near-field $(N=1)$, and an $f/10$ focusing optic. In Fig.11 and 12, we plot the normalized intensity in the focal plane in (a), and the the corresponding phase in (b). (c) and (d) give longitudinal slices of the normalized intensity profile in the $\hat{x}-\hat{z}$ plane and the $\hat{y}-\hat{z}$ plane respectively. In the case of coma in Fig.11, we find that the beam essentially bends as it propagates through space. As a result, the ponderomotive force that acts radially outwards is not only asymmetric in the focal plane, but changes as a function of longitudinal position.

In the case of astigmatism in Fig.12, the beam propagates in a straight line, but in this case the focal spot becomes elliptical outside of the focal plane as well as rotating 90 degrees before and after focus. It is not clear at this stage how these aberrations will affect the scattering of electrons but given that the ponderomotive force becomes asymmetric in both cases, we can predict that the aberrations may be imprinted into the scattered electron distribution.

To integrate the Lorentz force in (1) efficiently, we need an analytic description of the field equations in and around the laser focus. The result of (22) only gives the field amplitude and phase, not the electric and magnetic field equations. We could use (12) to numerically calculate the fields from our aberrated near-field and interpolate the fields at each time step within the Runge-Kutta, however we have found this to add a large numerical overhead.

An alternative approach would be to decompose the aberrated amplitude and phase into the previously given Laguerre-Gaussian (LG) functions (13). We use a modal decomposition into the LG basis set given by Longman and Fedosejevs (2017),

This requires the aberrated wavefunction is normalized such that $\braket{\psi_{\delta}}{\psi_{\delta}}=1$. The aberrated focal spot can then be reconstructed via summing over all LG modes weighted by the $\eta_{\ell,p}$ parameter. For completeness, this is written as,

The electric and magnetic fields are then computed as a linear superposition of the electric and magnetic fields of each mode. Using this technique, we find that the number of modes required for the decomposition can become large if the aberration is strong, so we restrict ourselves to weakly aberrated beams where $\alpha_{m,n}\leq 0.1\lambda_{0}$.

Simulations using aberrated focal spots were run for various strengths of coma and astigmatism, the two most common types of beam aberration found in the laboratory. The scattered distributions of 500000 electrons scattered into the screen placed at $z=d$ is shown in Fig.13. The figure is broken into 9 simulations where oblique astigmatism is increased horizontally as you move left to right, and horizontal coma is increased as you move vertically from top to bottom. All images utilize the same color scale. The laser parameters used in all simulations is $\lambda=800nm$, $P=500$TW, $f_{\#}\approx 20$, $\tau=30$fs, nitrogen gas, and cutting off electrons below 500keV, where $P\approx U/\tau$ is the laser power. For the aberrated beam simulations we opt to keep the laser power constant as the amplitude of each mode in the composition varies from aberration to aberration.

The plot in Fig.13 shows two unique characteristics of the scattering distributions of the electrons with aberrated beams. As the astigmatism is increased in Fig.13(a)-(c), we find the scattering distribution develops lobes of brighter flux in the horizontal and vertical planes. This contrasts the scattering from a laser pulse with coma as shown in Fig.13(d) and (g) in which we see an asymmetry form in the scattered distribution. On the left side of the distribution we see a bright lobe, whereas on the right side we see a shape that resembles a half moon, similar to what was observed experimentally in Fig.9(b). Combining the two aberrations produces distinct spatial distributions of the electrons as shown in Fig.13(e), (f), (h) and (i).

We note that the outside cutoff radius of the low energy electrons remains concentric with the laser $\hat{k}$, independent of the aberration in the beam, and we utilize this observation as a justification for centering the electron ring from VEGA3 using the outside cut-off rather than the highly aberrated inside cut-off angle. Referring to Fig.9(b), we notice that there is a high level of structure within the electron ring. To simulate this will likely require many laser modes and the introduction of higher order aberrations outside of just the coma and astigmatism as well as spatio-temporal aberrations. To achieve this we are developing new models that can solve for non-paraxial fields with temporal and spatial aberrations similar to other works Bukharskii et al. (2022).

We note in Fig.13 that as the aberration strength in the beam increases, the apparent inside cut-off angle of the two horizontal lobes increases. This is due to the aberration increasing the effective focal spot size area, and as a result decreases the focal spot intensity. Overlaid in the images are white dotted circles which indicate the theoretical electron cutoff from (9) and the known peak intensity of the aberrated beam. We note that despite the presence of coma or astigmatism, the ideal position for the cut-off measurement remains in the horizontal polarization plane.

At this stage we are still some distance away from solving the inverse problem of measuring the aberration given the scattered distribution of electrons. The scattered distribution will likely contain a wealth of information about the focal spot including the intensity, beam waist, spatial aberrations, and possibly temporal information about the laser pulse. For a complete reconstruction, it may be necessary to retrieve the full distribution function of the electrons, that is, their complete position-momentum phase space information Bukharskii et al. (2022). This is in principle possible using spatially resolved spectrometers Meyerhofer et al. (1996), but to build a device that can measure the spatial and spectral distributions with comparable resolution to the image plate will be challenging.

We believe that it may be possible to make quantitative estimates on the spatial characteristics of the beam based on the scattered distribution of the electron dose alone. By running a large number of simulations for various laser intensities, beam waists, and aberrations one could construct a suitable training set for a convolutional neural network (CNN). Once suitably trained, the CNN could solve the inverse problem and give quantitative information about the laser spatial profile at full power given the electron distribution on an image plate obtained experimentally.

In summary, we have qualitatively demonstrated that beam aberrations such as coma and astigmatism imprint distinct angular distributions into the scattered electrons and are likely the cause of the aberrated ring structure found in the VEGA3 data. The mapping between the aberrated focal spot and the scattered electron distribution will likely be highly nonlinear making analytic relations between the two unlikely. We propose the use of machine learning tools to solve the inverse problem through training a convolutional neural network on both simulation and experimental data sets. With a suitably trained network, it could be in practice possible to make quantitative estimates on both the spatial, and temporal aberrations in the beam when operating at full power. Experiments with user specified phase aberrations using spatial light modulators are currently being planned and should provide a verification of our current aberration models, and as training sets for neural networks.

With the increasing number of high-power, high-intensity laser facilities worldwide there is an increasing need to develop new methods for accurately and precisely characterizing the laser spatially, spectrally, and temporally both in the near-field, and the far-field. In this work, we have demonstrated the use of electrons ionized and scattered from high-intensity laser pulses propagating through ultra-low density gases as a way to directly measure the laser intensity independently from the laser pulse duration, energy and spatial mode, as well as possibly measuring the laser spatial mode at full-power.

Extracting information about the laser pulse intensity from the scattered distribution was achieved through an elementary analysis of single electrons scattering from plane waves relating the maximum ejected scattering energy and corresponding scattering angle to the peak intensity of the laser. Three models were developed and verified using numerical modeling of single electrons scattering from paraxial laser pulses highlighting two of the models as candidates to relate the laser intensity to the scattered electron energy and corresponding angle. The models were shown to work well for intensities in the range of $a_{0}=1:30$, pulse durations $\tau>30$fs at 800nm, and for f-numbers greater than 5. For shorter f-numbers, higher intensities, or shorter pulse durations, higher order corrections and more sophisticated models are likely necessary to measure the intensity.

The ionization rate was shown to play an important role in the scattered energy of electrons indicating that if the electrons are ionized at intensities much less than the peak laser intensity, they are ponderomotively scattered without effectively sampling the peak intensity. It is therefore necessary to select a gas with ionization levels reasonably close to the peak intensity. We showed through a simplified model that gases such as argon and nitrogen are suitable for measuring intensities of $a_{0}<10$, while krypton or xenon are more suitable for intensities above this range. More sophisticated models of the ionization rates and injection points are necessary to obtain more precise intensity bounds for a specific gas.

Two experimental campaigns utilized image plates to capture the convolved electron flux and energy (dose) in a plane approximately 40mm from the focus giving a two dimensional image of the scattered electron distribution. The analytic back-of-the-envelope models and simulations were then applied to the experimental results, the first of which demonstrated a good agreement between the inferred experimental intensity made from indirect measurements to our measured intensity using the direct method. The second experiment exhibited an aberrated focal spot which produced an asymmetric scattered electron distribution. This experiment yielded a measured intensity approximately $50\%$ of that inferred through indirect measurements. We believe this discrepancy is related to the fact that aberrations present in the focal spot observed at low power measurements may have worsened at high power leading to a reduction in the on shot peak intensity.

The measured doses on the image plates integrated from 70-100 shots indicate that single shot measurements for some laser systems may be possible if background pressures are on the order of $10^{-4}$mbar. It was shown through scaling arguments that the signal scales proportionally to the gaseous pressure and the $f_{\#}^{2}$ indicating that tightly focused beams may require more shots to get suitable signals for measurement.

Motivated by the experimental results of aberrated beams, we demonstrated qualitative numerical relationships between the focal spot aberrations and the scattered electron distributions in approximate agreement with the experimental results. The mapping from the aberration in the focal plane to the electron distribution at the measurement plane appears to be highly nonlinear and an analytic transform between the two may not exist. We propose the use of machine learning as a possible tool to solve the inverse mapping of the electron distribution to the focal plane. A convolutional neural network trained on both simulation and experimental data sets could be suitable way to make quantitative estimates on the spatial beam aberrations, and hence the laser phase on shot at full power.

We believe the use of image plates, coupled with our back-of-the-envelope scaling law, is a simple, cost-effective, and robust method to make direct measurements of the focal spot intensity at full power. Many high-power laser facilities are already equipped with image plate scanners allowing for the immediate implementation of the technique. Future work will push towards non-paraxial corrections to our scaling law, radiation reaction effects for intensities approaching and surpassing $a_{0}=100$, advanced ionization algorithms, and corrections for temporal effects. We believe that as we push towards single-shot, ultra-short (few cycle) laser pulses, additional effects such as the carrier-envelope phase may also begin to play a role, but for now they have been omitted for brevity.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.