Particle-in-cell simulations of density peak formation and ion heating from short pulse laser-driven ponderomotive steepening

Ultraintense laser interactions with dense targets represent an interesting regime, from both a fundamental and an applied perspective, which has not yet been exhaustively explored. One less explored phenomenon in this regime is the formation of electron and ion density peaks due to a laser pulse that strongly reflects from a dense target. There are papers that discuss this process—sometimes called ponderomotive steepening—going back to Estabrook et al. (1975).¹ Figure 1 provides a qualitative sketch of the physics involved in this laser-plasma interaction. First, a normally incident, linearly polarized laser makes a strong reflection from a dense plasma. The interference between the incident and reflected pulses produces a standing wave pattern [Fig. 1(a)]. The ponderomotive force associated with this standing wave has a strong effect on the electron distribution [Fig. 1(b)], and over time, peaks form in the density of both the electrons and ions [Fig. 1(c)]. Readers who are familiar with Kruer's 1988 textbook² will recall the discussion of this phenomenon there. Ponderomotive steepening also draws many parallels to theoretical and computational work that considers the standing electromagnetic (EM) wave formed by crossing two laser pulses to generate plasma optics such as plasma gratings^3,4 and the so-called transient plasma photonic crystals^5–7 which are phenomena that may have useful applications in the future (see discussions in Refs. 4 and 6). Drawing parallels to this work, we consider the simplified case of a laser propagating through a constant density preplasma shelf and reflecting off of a highly reflective overdense target as illustrated in Figs. 1(d)–1(f). From an experimental point of view, ponderomotive steepening only requires one laser pulse, and the high densities near the critical surface allow for larger transverse electric fields than with counterpropagating lasers in low density media.

We are motivated to return to this topic with fresh eyes in part due to the maturation of technologies to produce intense laser pulses at mid-infrared (IR) wavelengths (⁠$2 μm ≲ λ ≲ 10 μm$⁠).⁸ This presents an opportunity to examine the wavelength dependence of intense laser-matter interactions to see if theoretical models developed from studying laser interactions at shorter wavelengths remain valid at longer wavelengths (e.g., Ref. 9 and ongoing research efforts¹⁰). As discussed later, the density peaks that form with ponderomotive steepening are separated by approximately half the laser wavelength. It is therefore challenging to detect and resolve these density peaks in near-IR or shorter-wavelength laser interactions. There have been many experiments that confirm that the ponderomotive force does steepen the plasma profile near the target as expected (e.g., Refs. 11–15), and researchers have found evidence in experiments with counterpropagating near-IR laser pulses that the interference shapes the plasma distribution in a low density medium (e.g., Ref. 16). However, multiply peaked ponderomotive steepening,^1,2,17–20 where multiple electron and ion density peaks are formed in the preplasma, has not yet been directly observed with interferometry or by other means. We aim to provide useful analytic insights for experimentalists working to demonstrate this effect.

A challenge for connecting theory to observation is that multiply peaked ponderomotive steepening is the simplest to model and has larger longitudinal electric field strengths when the laser interactions are at normal incidence, whereas at the highest intensities, normal incidence experiments are rare because of the potential damage that the reflected pulse could do to optical elements. There are, however, methods to protect optics from the reflected pulse. Normal incidence experiments were conducted, for example, at $≈1018$ W cm⁻² peak intensities with $≈3$ mJ pulses at a kHz repetition rate in Refs. 21 and 22. Although the present paper is not tied to modeling interactions from a particular laser system, it is important to note that normal incidence experiments can be performed.

As will be discussed, the initial dynamics of the ions in the preplasma for multiply peaked ponderomotive steepening are not accounted for in steady state models,^1,2,17–20 but some ions do reach significant energies due to the charge separation caused by the ponderomotive force, and experiments could investigate this regime. According to estimates that agree with our 2D(3v) particle-in-cell (PIC) simulations, under the right plasma conditions and laser parameters, these interactions have the potential to accelerate ions to energies exceeding 100 keV. Experiments of this kind would also be interesting as a new type of code validation experiment for high intensity laser-plasma interactions. Both during and after the laser interaction, ions move and the electron and ion density profiles change over time which can be investigated with interferometry^23–25 and measurements of escaping ion energies (e.g., Ref. 26). The simplicity and symmetry of normal incidence interactions would be helpful for comparing experiment to simulation and theory in a straightforward way.

In Sec. II, we provide a brief review of the physics of ponderomotive steepening and identify the relevant timescales for ion motion using simple analytic models. In Sec. III, we describe 2D(3v) PIC simulations that exhibit multiply peaked ponderomotive steepening. In Sec. IV, the simulation results are presented and compared to the analytic models discussed in Sec. II. Finally, we address implications of our results in the concluding sections.

The traditional analytic approach for ponderomotive steepening considers a steady state solution to the fluid equations, to which a term for the ponderomotive force is added. The electric field is then assumed to take a particular form based on the geometry of the problem and to allow for numerical solutions or approximate solutions.^1,2,17–20 These approximations limit the validity of the conclusions, and the steady state solution provides little insight into the dynamics of the phenomenon. We investigate these dynamics by developing a simple model to estimate the longitudinal electric fields experienced by the ions and comparing the predictions to PIC simulations.

Our simple model is similar in many ways to a more sophisticated analytic model described in a recent paper by Ref. 7 that considers the dynamics of the electron motion for the case of lower intensity counterpropagating laser beams in a low density medium with 1D Vlasov simulations. Our paper is complementary to theirs because we consider standing waves that form from the normal incidence reflection of intense laser pulses from an overdense target, rather than counterpropagating laser pulses. While we probe the same physics, we consider much shorter pulses, higher densities,²⁷ and higher intensity lasers, as discussed in Sec. III A. Also, we focus on the dynamics of the ions, which move much more quickly because of these parameters.

Our 2D(3v) PIC simulations include the focusing of the laser and transverse extent of the density perturbations, which are not captured in 1D Vlasov simulations. Where appropriate, we provide comments for those wishing to compare our work to Ref. 7. The timescales and intensity thresholds we develop are very similar to their models. We also acknowledge that these electron density perturbation models could be modified to account for different pulse shapes following the work by Refs. 28 and 29.

As sketched in Fig. 1, the laser creates a charge imbalance due to the ponderomotive force on the electrons, which in turn creates a longitudinal electric field to accelerate the ions toward the electron peaks. We develop a simple model that balances the ponderomotive force with the Coulomb force associated with the charge separation.

A charged particle in an inhomogeneous EM field experiences the ponderomotive force, which is a cycle-averaged force that models the motion of these particles on timescales larger than the laser period. For a particle of mass m with charge e and an electric field with frequency ω and amplitude E, the ponderomotive force is given by

where the electric field is cycle-averaged. While this effect is experienced by both electrons and ions, for the laser intensities we are concerned with here, the much more massive ions are hardly affected by the ponderomotive force.

We consider a linearly polarized plane electromagnetic wave propagating in the +x direction and reflecting off of a semi-infinite overdense plasma at x > 0. Similar to Refs. 30 and 31, we assume that the plasma is a perfect conductor and reflects 100% of the light, resulting in a standing EM wave (x <0) with electric and magnetic fields described by

where E₀ is the electric field strength of the laser if there had been no reflection and B₀ = E₀∕c. In real experiments, we do not expect 100% reflectivity. Yet, the reflectivity can be high for laser interactions near, but not significantly above, the threshold for relativistic effects since the high temperatures produce a nearly collisionless plasma, but relativistic absorption is not yet pronounced.^32,33 Inserting Eqs. (2) into (1) yields the longitudinal ponderomotive force associated with the standing wave

which will be compared to the Coulomb force associated with the charge separation.

We assume that before reaching the overdense plasma at x > 0, the laser travels through a constant, subcritical density shelf. We estimate the strength and spatial dependence of the electrostatic force in this shelf by choosing a distribution to perfectly balance the ponderomotive force when integrated with the one-dimensional (1D) Poisson equation. This produces a sinusoidal electron density modulation of the form

where n₀ is the average electron density in the plasma (i.e., the electron density at that location in the plasma before the laser pulse arrives) and n_e describes the amplitude of the density modulation. Equation (5) is useful for gaining qualitative insights into the ponderomotive steepening process. We remind the reader that the ponderomotive force is time averaged, so this simple model does not fully capture the physics involved. For a more accurate description of the electron density distribution, we refer the reader to the methods discussed in Refs. 28 and 29, which can account for the pulse shape and temporal profile.

Note that because the local electron density must always be greater than or equal to zero, n_e in Eq. (5) must not exceed n₀ as you cannot remove more electrons than are available in the plasma. Since the laser only travels in the subcritical-density region of the plasma, n₀ must also be less than the critical density, $ncrit=4π2εomec2/λ2e2,$ and the maximum electron density is limited.³⁴ 

Integrating Eq. (5) with the one-dimensional Poisson equation (for experimental and simulated beam profiles, we assume that the laser spot size is much larger than $λ/2$⁠) results in a quasistatic electric field in the longitudinal (x) direction of the form

According to Eq. (6) the peak longitudinal electric field is expressed by

which is equivalent to

Note that n_e depends on the intensity of the laser. If we equate the peak ponderomotive force [Eq. (4)] to the electrostatic force from [Eq. (6)], one finds that for this model, the laser is limited to displacing electron densities up to

where the laser intensity $I=cε0|E0|2/2$ is used to simplify the expression and n_e,max is less than the initial electron density in the plasma. For this model, the density modulation is saturated with a critical intensity of

For this critical intensity, the normalized vector potential a₀ for the laser (⁠$a0=eE0/meωc$⁠) is

or in terms of the electron plasma frequency, $ωpe=nee2/meε0$ (using n_e = n₀), $a0,crit=1/2 ωpe/ω$⁠. Since $a0,crit ≲ 0.7$⁠, it is clear that the applicability of this model does not extend to the strongly relativistic regime.

Intensities somewhat above this limit are considered in Sec. II A 2. We note the similarity of this estimate with the wave-breaking limit in laser wakefield acceleration,³⁵ and in Ref. 7, this intensity threshold relates to the transition between what they call the “collective electron” regime to the “single electron bouncing” regime. For high electron temperatures, this type of model could be extended by considering the Bohm-Gross frequency³⁶ like in Ref. 7. Laser driven instabilities would also play a role in certain regimes.^2,37

At laser intensities significantly above the critical estimate derived in Sec. II A 1, the electrons are more strongly peaked than predicted by Eq. (5), and our sinusoidal model breaks down, as demonstrated by simulation results that will be presented later. Although the sinusoidal model breaks down, there is a simple way to determine the maximum longitudinal electric fields in this limiting case. If all of the available electrons are evacuated to the peaks, the maximum electric field in the maximum depletion regime is a factor of π greater than the sinusoidal model. This comes from integrating the charge density in the depletion region (⁠$en0×λ/4)$⁠, giving the maximum longitudinal electric field to be

This result is notable simply in that it implies that the longitudinal electric field is enhanced [relative to Eq. (8)] at intensities slightly exceeding I_crit from Eq. (10), rather than being suppressed.

This subsection determines a time scale for ion motion (for ions to reach an electron peak), which will be useful for comparison to the duration of the laser pulse. If the laser pulse duration is shorter than the time scale for ion motion, then we characterize this as the “short pulse” regime. If instead, the laser pulse duration is significantly longer than this time scale, we label this as the “long pulse” regime.

We assume the plasma to be an initially neutral mixture of electrons and ions with charge +Ze, where Z is the average ionization. Now, we consider the electrostatic force on an ion of mass m_i between two of the electron peaks. Following the sinusoidal model, we focus on the ions at a distance of $λ/8$ or less away from an electron peak as they will reach the electron peak more quickly (the farthest away ions are considered in  Appendix B), and we approximate the electric field as linear in this region [matching the slope of Eq. (6) near its root], or

This results in simple harmonic motion with an angular frequency of

We use this equation to compute the oscillation period of the ion motion. Since we are primarily interested in the dynamics of the ion density peak growth, we are concerned with the time scale for an ion to move from its initial location to the electron density peak. This time scale is equivalent one-quarter of the ion oscillation period ( Appendix A), which is

where we note that this formula is only valid for $I≤Icrit$⁠, where $Icrit$ is given by Eq. (10). For higher intensities, as discussed in Sec. II A 1, the maximum electron density that the laser displaces is limited by the number of available electrons and critical density. This results in a minimum time scale of

where in the approximation, we have for convenience assumed Z = 1 and $mi≈2mp$⁠, where m_p is the mass of a proton.

Equations (15) and (16) are represented in Fig. 2 which illustrates the division between the short pulse regime and the long pulse regime as a function of laser intensity and wavelength. As examined earlier, we do not expect the time scale to be significantly decreased for $I≫Icrit$ due to the depletion of electrons. This limit is represented with the horizontal lines. We also note that relativistic effects should be considered in this region for $a0≳1$⁠. The laser wavelength does not appear in Eq. (15), which is why at low intensities in Fig. 2, the time scale does not depend on the wavelength. At higher intensities, our minimum time scale [Eq. (16)] depends on the initial plasma density, where $ncrit$ does depend on the wavelength. Above this critical intensity, according to the sinusoidal model, the maximum electron density that the laser could displace exceeds the available number of electrons in the plasma near the electron peak. It should be noted that Eq. (15) has the same scaling with parameters as the time estimate in Ref. 7 for an ion “grating” to develop in the standing wave.

We assumed simple harmonic motion to obtain $τion$ in Eq. (15). This approach also provides a characteristic ion energy which can be compared to our simulations. Assuming simple harmonic motion with an amplitude of $λ/8$ and an available electron density of n_e, we have an ion velocity that increases with time as

where $t≤tSW$ is the time elapsed since the standing wave fields began. Ions are assumed to move at a constant velocity after the standing wave dissipates (⁠$t>tSW$⁠). Although Eq. (17) does not explicitly depend on the laser wavelength, as mentioned earlier, our expression for $τion$ is only valid at laser intensities below the critical intensity [Eq. (10)] which does depend on the wavelength. Since shorter wavelength lasers have a higher critical intensity, one can reach much smaller values of $τion$ as illustrated in Fig. 2, which would allow the ion velocity [Eq. (17)] to grow more quickly. However, this growth is limited by the duration of the laser pulse if we are considering the short pulse regime (⁠$tSW≪τion$⁠).

For a sufficiently long duration laser pulse, the standing wave fields will last long enough that t_SW approaches $τion$⁠. From Eq. (17), it is straightforward to show that this implies a maximum kinetic energy exceeding 100 keV

Interestingly, this expression is independent of wavelength except for the wavelength dependence of n_crit.

Multiply peaked ponderomotive steepening is examined numerically with implicit 2D(3v) PIC simulations performed with the LSP PIC code.³⁸ The initial conditions are such that we are in the short pulse regime of our model and we have exceeded the critical intensity for our model (Fig. 2). For these simulations, an x − z Cartesian geometry is used, where the laser propagates in the + x direction and the polarization is in the z direction. The simulations have a spatial resolution of 25 nm × 25 nm (⁠$λ/32×λ/32$⁠) and were run for 400 fs with a 0.1 fs time step.

To isolate the dynamics of the ion peak formation process, we consider an idealized geometry of a rectangular target with an extended preplasma shelf. The plasma is assumed to be singly ionized with fixed ionization. This choice is made to prevent the critical surface from moving significantly due to ionization caused by the laser pulse. Ponderomotive steepening still occurs in simulations when the critical surface moves forward from ionization (e.g., Refs. 9 and 33), but we ignore this effect in order to focus on the electron and ion dynamics. In the laser propagation direction, the target consists of a $7 μm$ long constant subcritical density plasma shelf (⁠$n=8.594×1019$ cm⁻³ $≈ncrit/20$⁠) with a sharp interface in a $15 μm$ overdense target (n = 10²³ cm⁻³ $≈60ncrit$⁠) as illustrated in Fig. 3. In the polarization direction, the target is $30 μm$ wide. Ion-ion collisions are not considered for reasons discussed in  Appendix C.

We describe three different simulations with targets composed of fully ionized hydrogen, deuterium, or tritium ions in order to investigate different charge-to-mass ratios. All these simulations keep the laser intensity and initial target electron and ion number densities constant. The overdense region is given a number density similar to our group's previous work.³³ The simulations were initialized with 81 particles per cell for the electrons and 49 particles per cell for the ions with an initial thermal energy of 1 eV.

We consider an 800 nm wavelength, normally incident laser pulse propagating in the +x direction which would reach a peak intensity of $1018Wcm−2$ if no target were present. The pulse duration is 42 fs full width at half maximum (FWHM) with a sine-squared envelope and a Gaussian spot size of $1.5 μm$ (FWHM). The laser has a Gaussian beam radius of $w0=1.27 μm$ and a Rayleigh length (⁠$zr=πw02/λ)$ of $6.3 μm$⁠. These parameters are similar to those of the Ti:Sapphire kHz repetition rate laser system described in Refs. 21, 22, and 33. We use these parameters to explore the short pulse regime of our model with $tSW<τion.$

The laser focus is set at the back of the target, as shown in Fig. 3, in order to create a larger transverse region over which ponderomotive steepening can occur. For these 2D(3v) simulations, the laser comes to focus as $I2D=I0/1+x2/zr2$⁠, which is less sharply than 3D Gaussian beams [$I3D=I0/(1+x2/zr2)$], as illustrated in Ref. 9. This results in an intensity of $≈2.7×1017$ W cm⁻² at the beginning of the underdense shelf (⁠$x=−23 μm$⁠) and $≈3.8×1017$ W cm⁻² where the laser reflects (⁠$x=−15.5 μm$⁠).

In our simulations, the intensity near the reflection point is measured to be roughly $≈2.6×1017W cm−2$⁠. According to Eq. (10), for our wavelength and plasma density, $Icrit≈5×1017W cm−2$⁠. Our simulations therefore explore the regime where the intensity is about five times larger than this threshold. Regarding the time scale of ion motion, for these simulations, $τion=129$ fs $×mion/mp$⁠. This time scale in all three simulations is longer than the 42 fs FWHM laser pulse (and even the full simulated 84 fs pulse with a sine-squared envelope), making these interactions well within the short pulse regime as illustrated in Fig. 1. We are able to explore the interesting nonlinear regime identified in Ref. 7, but with a much denser plasma (⁠$≈1/20th$ of critical density compared to $10−4ncrit$⁠), much higher intensity laser pulse (⁠$a0≈0.35$ compared to their a₀ = 0.03), and a much shorter laser pulse duration (42 fs vs 3200 fs).

Figure 4 (Multimedia view) shows the snapshots of the electron density and ion density from the deuterium simulation at three different times. The standing EM wave causes the electrons to form peaks which, over time, produce peaks in the ion density separated by approximately $λ/2$ throughout the underdense region. The hydrogen and tritium simulations show similar behavior, with the growth of the ion peaks happening sooner for lighter ions and later for more massive ions. In all three simulations, we observe more than 10 ion density peaks in the $7 μm$ long underdense region.

As discussed in Sec. IV B, an examination of the ion trajectories confirms that ions accelerated from both sides of the peak are streaming past each other. Figure 5 shows this happening in all three simulations, albeit on different timescales. For each simulation, the peak ion density increases to $≈2.5×1020 cm−3$ (approximately three times the initial density), which lasts for tens to hundreds of femtoseconds and then begins to decrease. Multiply peaked ponderomotive steepening in this short pulse regime is therefore a highly transient effect.

To better understand the dynamics of the peak formation process, we consider the motion of the ion macroparticles in the simulation. In particular, if we consider the ion trajectories (Fig. 6), we see that the ions are accelerated toward the electron peaks, while the standing wave is present. Later, as the standing wave dissipates, the inertia of the ions allows them to continue to travel with a roughly constant velocity. We see from Fig. 6 that many of the ions travel through the peak before the end of the simulation, which produces the broadening observed in Fig. 4. We note that the transverse movement of the ions is negligible compared to the spot size of the laser.

The energy distribution of the ions is represented in Fig. 7 which highlights results from the deuterium simulation and overlays the average ion energies from the hydrogen and tritium simulations. In all three simulations, the ions are accelerated, while electron density peaks from the standing wave are present, reaching kilo-electron-volt energies. The average ion energy decreases slightly as the standing wave dissipates. The transverse motion of the ions is considered in Sec. IV E. The conversion efficiencies from laser energy (>100 eV) to ion energy were approximately 0.10%, 0.06%, and 0.04% for the three simulations at 200 fs, respectively. We did not include ion-ion collisions in these simulations, which could potentially change the behavior of the ions and potentially lengthen the duration of the peak. In  Appendix C, we determine that the mean free path of ion-ion collisions for our conditions is larger than the scale of the peak for the higher energy ions in the shelf region.

Figure 8 shows a line out of the longitudinal electric field along the laser axis from the deuterium simulation compared to various models for context. As mentioned, the intensity of this standing wave exceeds I_crit by about a factor of 5, which means that we do not expect the sinusoidal model to be accurate in this case. As seen in Fig. 8, the peak sustained longitudinal electric fields along the laser axis in the simulation are close to $2×1011$ V m⁻¹ which is larger than one would expect in this case from the sinusoidal model (10¹¹ V m⁻¹) by about a factor of 2. This is still somewhat below the peak electric field of the “maximum depletion” model shown in Fig. 2 which is near $3.1×1011$ V m⁻¹. This model is described in Sec. II A 2, and it concludes that the peak electric fields are up to a factor of π larger than the sinusoidal model as a limiting case. The results from the simulation lie between these two bounds. On the critical surface, where there are more available electrons, larger fields are present as shown in Fig. 8, although there are oscillations in the field. When moving further away from the laser axis, there are oscillations in the longitudinal electric field.

Equation (18) estimates the maximum ion energies from the interaction that we compare to the PIC simulations; however, this estimate requires some assumption for how long the standing wave is in place (t_SW). This is a difficult number to uniquely establish because the intensity envelope of the laser pulse is sine-squared, and there is no abrupt turn on and turn off of the standing wave. From considering the results of Fig. 7, using the 42 fs FWHM of the laser pulse as the duration of the standing wave is too short because the ion energies continue to grow even 42 fs after it begins to rise. Using the 84 fs full pulse duration of the laser pulse as the duration of the standing wave is too long, both empirically from Fig. 7 and from the reality that the standing waves are created by the overlap of the forward and reflected laser pulse. One could also do a more careful calculation following Refs. 28 and 29. In Table I, we therefore consider both of these timescales for the standing wave in our model in order to bracket the possible ion energies. We empirically find that choosing t_SW to be 76 fs yields particularly accurate estimates for the max ion energies in all three simulations.

One-dimensional models of ponderomotive steepening and transient photonic crystal growth provide useful insights to explain simulation results and plan experiments, but there are higher dimensional effects indicated by our simulations that must also be considered. The focusing of the laser limits the longitudinal extent and uniformity of density perturbations as shown in Fig. 4, which is especially important for high intensity lasers. We do observe density peaks throughout the underdense region with fairly significant transverse widths as shown in Fig. 4.³⁹ 

One-dimensional models do not account for the transverse acceleration of ions, which is nonzero as shown in Fig. 9, where we see a similarly shaped distribution for all three simulations. We observe that the maximum transverse velocity is roughly one-fifth of the maximum longitudinal velocity, although comparing these results to models and determining how this generalizes to different systems is the subject of future work. For the time scale of our simulations, there is minimal transverse movement of the ions compared to the transverse width of the density peaks as mentioned previously. These effects may be important for plasma grating research considering much longer pulses.

A multiply peaked density modulation is observed in our simulations throughout the underdense shelf region for these initial conditions. The short pulse regime for ponderomotive steepening identified in this theoretical work shows large longitudinal electric fields (potentially up to $≈1012$ V m⁻¹ for 800 nm light near the critical density) that accelerate ions to tens to hundreds of kilo-electron-volt in energy when the above conditions are satisfied. The consequences of these conditions seem to be overlooked in the literature. From a peak ion energy standpoint, this mechanism is not as appealing as conventional laser-based acceleration schemes such as Target Normal Sheath Acceleration (TNSA)^40,41 and Radiation Pressure Acceleration (RPA),^42,43 but because the energies are still sufficient to produce fusion, experiments of this kind may be useful, for example, for producing neutrons with a very small source size.

Largely because the spacing between density peaks is close to $λ/2$⁠, features like these have not yet been observed in optical interferometry. By using intense mid-IR laser systems to produce these modulations this may be possible, so long as one is careful to consider that the peaks are a highly transient effect and that in a freely expanding plasma, the density will drop off significantly if the interaction region is extended to tens to hundreds of micrometer. In our simulations with 800 nm wavelengths, 42 fs FWHM pulse durations, and peak intensities near 10¹⁸ W cm⁻², the features persist for less than a picosecond. While there are interferometric systems that can operate at this short time scale (e.g., Ref. 23), experiments at longer wavelengths, lower intensities, and involving ions with lower charge-to-mass ratios can be designed to make the ion acceleration happen over a longer time scale in order to study the evolution of these peaks. The interferometric data would be useful as a novel validation test of kinetic plasma codes, especially if the experiment can be performed at normal incidence.

There are papers in the literature that study the presence of periodicity in the density distribution from the overlap of two crossed laser pulses (e.g., Refs. 4 and 16) because this produces a kind of transient “plasma grating” that can be detected with probe light. The growth of this plasma grating is similar in many ways to the peaks that form via ponderomotive steepening, and we outline a number of parallels in the present paper to recent work by Ref. 7 who consider overlapping laser pulses through a low density medium. This phenomenon has interesting potential applications as discussed in Refs. 4 and 6.

Compared to approaches with counterpropagating laser pulses, there are some advantages to producing these density modulations through the reflection of laser light from an overdense target. Specifically, less total laser energy is required because the reflected laser pulse interferes with itself, and there is no need to carefully time the overlap of the pulses since the laser naturally reflects from an overdense surface. The other advantage of overdense targets, as we have explored in this paper, is simply that the density of the shelf or medium the laser travels through can be significantly larger than counterpropagating laser experiments would allow. Larger densities allow for significantly larger longitudinal electric fields for accelerating ions. The density of the medium in experiments with overlapping laser pulses is typically a few orders of magnitude below critical density because of the need to avoid the intensity dependent index of refraction effects. Experiments with overdense targets are not as constrained by this because irradiating an overdense target with an appropriate “prepulse” produces a few-to-many-micrometer subcritical density plasma in front of it. Besides increasing the peak ion energies, the other advantage of producing density modulations in a higher density medium is that the difference between the peak and minimum density will be larger, which should produce more easily detectable fringe shifts in efforts to perform interferometric imaging.

We have emphasized the novelty of performing experiments of this kind in the mid-IR (⁠$2 μm ≲ λ ≲ 10μm$⁠). Our results also imply that it would be interesting to investigate ponderomotive steepening with shorter wavelengths as well. Shorter wavelength lasers are able to propagate into denser regions, and as previously discussed, denser plasmas produce larger peak electric fields which are advantageous for accelerating ions. This detail is important for the possibility of using experiments of this kind to create a neutron source with a very small source size because, as is well known, neutron yields increase significantly with ion energy.⁴⁴ In a suitably designed experiment, one could try to produce neutrons from the collision of counterstreaming ions in the density peaks. However, as considered in  Appendix C, the mean free path for these collisions is large compared to $λ/2$⁠. Neutron-producing fusion reactions are more likely to come from ions that stream toward the first peak near the target and continue into the overdense region. This would be a “pitcher-catcher” type configuration where the pitcher and catcher are separated by only $≈λ/2$⁠. We note that the expected yield is likely less than conventional schemes as shown in  Appendix D but could be detected and is still interesting from a fundamental perspective.

We note that our model is limited as it neglects the motion of ions that are initially further than $λ/8$ from an electron peak, which require a longer pulse for maximum energy. We also approximate the field as linear. Alternatively, one could find the maximum energy from the work done by the electric field, and this is included in  Appendix B which predicts slightly larger velocities. When considering more realistic laser pulses, there are additional considerations. For example, our model ignores the transverse pulse shape (e.g., see Ref. 29) and does not account for the temporal pulse shape of the incoming beam. In practice, our model can be extended by replacing the time scale with an effective time scale (as in this work), or similarly, the ponderomotive force could be averaged over the pulse to get an effective intensity to be used in our model. A more accurate model could expand the work by Refs. 28, 29, and 7 to better capture the dynamics of the electron density distribution.

Our 1D model assumes that the spot size is much larger than $λ/2$ and also is the most applicable near the reflection point of the laser and along the laser axis where we have the full effect of the standing wave. Hence, our model should only serve as a rough estimate (or upper bound) for potential ion velocities, particularly for experiments, which have additional considerations. A crucial assumption of this work is that the plasma remains highly reflective. This is certainly true of our simulations, but it is well known that the intensity and wavelength of the laser are important factors for the reflectivity. To make more reliable extrapolations to shorter and longer wavelengths and smaller and larger intensities than we consider in the simulations we present here, one would need to carefully consider the scaling of the reflectivity with various parameters (e.g., Ref. 32). While it is outside the scope of the present work, this remains an important priority for future investigations.

The formation of multiply peaked density modulations associated with ponderomotive steepening is of fundamental interest as a basic plasma process and of practical interest as a means to modify the density profile of a target and to accelerate ions. Our PIC simulations indicate that these peaks are especially transient, lasting less than a picosecond after the end of a short-pulse laser interaction. This is important to factor into the design of future experiments to detect this phenomenon. We also find that the large longitudinal electric fields that are produced in these laser interactions accelerate ions to few kilo-electron-volt energies in short pulse laser interactions and potentially up to hundreds of kilo-electron-volt energies in longer duration interactions. In our simulations, these fields reach $2×1011$ V m⁻¹.

We outline a simple model to estimate the time scale of ion motion and peak energies of ions in these interactions. This model matches the peak ion energies in our simulations reasonably well. We also comment on extensions to this model which provide some insight even when the laser intensity exceeds a critical value. The model indicates that higher field strengths are achieved with shorter wavelength interactions due to the increased critical density. Ion acceleration should be much less pronounced in longer wavelength interactions, but this may still be an interesting regime to perform interferometric imaging as a novel validation test of plasma codes if the experiments are performed at normal incidence.

Multiply peaked ponderomotive steepening has many parallels to studies of counterpropagating laser pulses which is a phenomenon with interesting potential applications for the field.^4,16 A key difference is that interference from reflection occurs at a comparatively higher density. As a result, the longitudinal electric field strengths are much larger, as just mentioned, and there are important subtleties to analytically modeling this phenomenon and challenges in experimentally probing it that we have outlined.

This research was supported by the Air Force Office of Scientific Research under LRIR Project NO. 17RQCOR504 under the management of Dr. Riq Parra. This project also benefited from a grant of time at the Onyx supercomputer (ERDC) and the Ohio Supercomputer Center.⁴⁵ Support was also provided by the DOD HPCMP Internship Program and the AFOSR summer faculty program.

The time scale for ion motion found in Sec. II B makes a linear approximation for the force on ions a distance $λ/8$ or closer to the peak. Alternatively, to represent the time scale of the ion motion, we could consider the ion plasma oscillation frequency

We replace Zn_i with n_e and then use Eq. (9) to write this as a function of laser intensity

providing a time scale of (one-quarter of the ion oscillation period)

which agrees with the time scale found in Sec. II B.

To calculate the maximum ion energy, one may also calculate the work done by the electric field on an ion traveling from a valley to an electron peak. For the sinusoidal model, this results in a maximum energy of

which, as expected, is slightly higher than predicted with the energy predicted from the linear approximation in Eq. (13). If $ne≈n0,$ then the field would shrink at later times, due to ion movement, reducing this maximum energy.

Similarly, if we calculated the energy from the maximum depletion model, we find

where we see that the ion acceleration associated with ponderomotive steepening appears to be primarily a sub-millielectron volt acceleration mechanism.

Following the ion-ion mean free path estimate from Refs. 46 and 47, the ion-ion mean free path for colliding flows with mass number A and velocity $v$ before the collision is approximated as

assuming that the Coulomb logarithm is ten⁴⁸ and that the temperature of the counterstreaming flows is much smaller than the energy of the ions due to the bulk flow velocity. For example, if we look at the deuterium simulation (A_z = 2, Z = 1) and consider the maximum peak density to be $≈2.5×1020$ cm⁻³ and average velocity to be $≈3×105$ m/s, then we find $λmfp≈65 μm$⁠, which is orders of magnitude larger than the width of the peaks. This mean free path is shorter for the low energy ions, although these are of less interest for this work. For higher densities, such as in the bulk of the target, or with a higher density preplasma, collisions would be more significant.

The accelerated ions from this phenomenon could be used to create neutrons with a pitcher-catcher type configuration. One could make an experiment where there is a preplasma “pitcher” in front of the solid density target which would act as the catcher. For this case, the pitcher and catcher are only separated by $≈λ/2$ (for the first peak). For an order of magnitude estimate of potential neutron yield, we consider a deuterated polyethylene (CD₂) target as discussed in Ref. 44.

From Eq. (18), we expect the maximum energy deuterons from this phenomenon to be $≈100$ keV, which would have a yield of $≈4×10−8$ neutrons per deuteron.⁴⁴ From these simulations, we make a very generous estimate of a conversion efficiency of 10⁻⁴ for laser energy to 100 keV ions. This results in $O(102$⁠) neutrons per Joule of laser energy. This is a few orders of magnitude lower than typical schemes producing $∼105−107$ neutrons per Joule of laser energy.⁴⁴ This suggests that this phenomenon would not outperform conventional schemes in terms of neutron yield, but it may be useful for certain applications.