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Frontiers in Physics ·

Volume 14 - 2026 | https://doi.org/10.3389/fphy.2026.1849759

Hot electron model for ion emission (Hermione): an effective model for target normal sheath acceleration ion spectra from femtosecond to picosecond lasers

  • C. L. C. Lacoste1,2*corresponding author
  • N. Iwata3
  • M. Bardon1
  • E. D’Humières1
  • Y. Sentoku3
  • P. Antici2
  1. 1CELIA, University of Bordeaux-CNRS-CEA, Talence, France
  2. 2Institut National de la Recherche Scientifique, INRS EMT, Quebec, QC, Canada
  3. 3Institute of Laser Engineering, The University of Osaka, Osaka, Japan
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Abstract

Recent advances in high-power lasers for secondary particle generation highlight the need for reliable and affordable particle sources. In laser-driven ion acceleration, the most routinely obtainable acceleration mechanism is the so-called Target Normal Sheath Acceleration (TNSA) mechanism. Predicting the characteristics of particle sources produced through this mechanism is essential for modeling experiments at existing and future laser facilities. In this work, we present a versatile and fast predictive model, Hermione, capable of reproducing proton spectra across several laser facilities, including the laser facilities PETAL, Apollon, LFEX, and ALLS. Our results show very good agreement with experimental data for laser pulses longer than 100 fs, while a slight overestimation of the proton yield is observed for shorter pulses (<100 fs). We also demonstrate the model’s ability to reproduce wavelength-dependent effects for lasers operating between 0.8 μ m and 2 μ m. Hermione delivers results on very short timescales (less than 1 minute on a standard laptop), making it an efficient complementary tool to more complex and computationally demanding Particle-In-Cell (PIC) simulations. This enables rapid optimization of the TNSA acceleration regime by varying laser energy, pulse duration, and focal spot size. The code is open access, allowing the community to refine and adapt it to specific applications.

Keywords

1 Introduction

The interaction of ultra-intense, short-duration laser pulses with solid targets has paved new ways for compact and tunable ion acceleration sources [1, 2]. Among the various acceleration mechanisms, the Target Normal Sheath Acceleration (TNSA) process [3, 4] stands out for its robustness, reproducibility, and well-understood physics. In this scheme, laser-accelerated hot electrons propagate through the target and form a quasi-static sheath field at the rear surface, reaching electric field amplitudes on the order of several TV/m. This sheath field accelerates ions up to a hundred of MeV energies within a few picoseconds [5, 6].

TNSA is a promising acceleration scheme because of its compactness and simplicity [7–9], a high number of particles per bunch ( > 1 0 11 ), high ion energies (up to 100 MeV [6]), low emittance [10], and short duration [5]. Applications of laser-accelerated ions include isochoric heating [11, 12], isotope [13–15] or neutron production [16–18], plasma radiography [19, 20], nuclear fusion in a fast ignition scheme [21, 22], material science [23–25] and cultural heritage [26–29].

A global challenge in the TNSA acceleration regime is to optimize the laser parameters (intensity, pulse duration and focal spot size) to enhance the maximum ion energy and conversion efficiency. Analytical and numerical studies have established several scaling laws linking the proton cut-off energy to these parameters. Pioneering works have proposed both, empirical [30] and analytical [31–34] scalings, to link the laser parameters to the number and cut-off energy of the protons depending on the plasmas parameters.

The commonly used scaling laws proposed by Wilks et al. [4] and Mora et al. [31], later refined by Fuchs [30] et al. and by Robson et al. [34] provide a convenient framework to estimate the maximum proton energy in TNSA. However, for picosecond and kilojoule-class laser systems operating at intensities of 1 0 19 W/c m 2 or below, those models significantly underestimate the proton cut-off energy, typically predicting values below 10 MeV, in clear contradiction with numerous experimental observations [35, 36]. Conversely, when extrapolated to femtosecond, multi-tens-of-joule laser systems at ultra-relativistic intensities exceeding 1 0 21 W/c m 2, these scaling laws tend to overestimate the achievable cut-off energies, predicting proton energies on the order of 100 M e V or higher. Such predictions have also been shown to be inconsistent with experimental results [17], highlighting the limitations of these models outside their original domain of applicability. These discrepancies emphasize the need for more refined descriptions that account for the complex interplay between laser temporal structure, energy coupling efficiency, and the multiple electron populations involved in sheath formation.

In the picosecond regime, experimental works [35, 37] have shown that multi-picosecond laser pulses can significantly modify hot-electron recirculation, thereby enhancing ion acceleration efficiency beyond the classical ponderomotive scaling. These studies revealed that the temporal extension of the laser pulse allows for successive bursts of electron injection into the sheath, maintaining the accelerating field over longer timescales.

Furthermore, recent numerical studies [38, 39] demonstrated that, in the picosecond regime, the electron density near the target rear surface increases depending on the spot radius and the electron temperature and decreases with the thickness of the TNSA target. This study revealed that the picosecond and kJ laser pulse increases the electron density creating a Weibel filamentation of the laser beam -like instability. This instability generates a strong magnetic field which in turn confines the plasma. They proposed an analytical model incorporating this density enhancement, leading to a modified expression for the electron density.

In the present work, it is this modified expression of the electron density which permits to investigate how proton acceleration evolves depending on the laser parameters.

In the femtosecond regime, and particularly for tightly focused laser pulses with small spot radii, it has been shown that the effective hot-electron temperature can fall below the classical ponderomotive scaling [40]. This deviation arises from the reduced interaction volume and the limited efficiency of electron acceleration under extreme focusing conditions. Experimental investigations of femtosecond TNSA have been reported [41–44], demonstrating the feasibility of ion acceleration in this regime.

However, for many of the applications mentioned before, a further optimization of the proton spectrum, both in terms of cut-off energy and particle number, is required. To address this challenge, several target-based strategies have been proposed, including the use of helical coil targets [45–47], which have demonstrated significant enhancements in proton energy. Other approaches rely on target shaping, such as hemispherical or curved rear-surface targets [11, 48], that propose to increase the proton beam flux by focusing. Additional concepts, including nanostructured surfaces [49–51] or foam [52, 53], have also been explored to enhance laser absorption. Nevertheless, an alternative and complementary strategy consists in first optimizing the TNSA mechanism itself by understanding and controlling its dependence on laser parameters. This calls for the development of a model that can predict TNSA parameters given different laser input parameters.

In this study, we couple the main theoretical scaling models for TNSA [30, 31] to understand the proton spectrum and cut-off energy dependencies with the laser parameters, the preplasma length studies [40, 54] and the lateral confinement of the sheath [38]. All these previous works permit to build our theoretical model presented in Section 2. The model is implemented in a new Python code named Hermione [55]. The results of the model are described in Section 3. In Section 4, we study the proton spectra variations with laser wavelength variations or even as function of the pulse duration and the spot radius to get an optimized proton spectrum.

2 Theoretical and empirical scaling laws

For relativistic intensities ( a 0 ≳ 1 ), the ponderomotive scaling [56] gives an electron temperature in the sheath of the TNSA of:where a 0 = e E L m e c ω L = e m e c ω 0 2 I c ε 0 is the normalized vector potential, I = E L τ L π R 2is the laser intensity, ω L = 2 π c / λ L, E L and τ L are respectively the laser frequency, the laser energy and the pulse duration, R the focal spot radius and λ L the laser wavelength. Equation 1, can be corrected at lower a 0 by the empirical form [57]:yielding a smooth transition between the non-relativistic and relativistic regimes.

T pond = m e c 2 1 + a 0 2 2 − 1 , (1)

T smooth = 0.47 m e c 2 a 0 2 2 / 3 , (2)

In addition, experimental studies have shown that the preplasma scale length L μ m strongly modifies the hot-electron temperature and absorption fraction. The term ”preplasma” refers to the front-side plasma formed by the laser prepulse. Furthermore, this length corresponds to the effective preplasma where the electrons are accelerated by the laser and which contributes only to the sheath (and not with escaped electrons [58, 59]). Since we consider only the part of the preplasma contributing to the sheath, our effective preplasma scale length is one order of magnitude shorter than in the literature [60]. To account for this, we use in this study the generalized empirical scaling [40, 54]:

T h = 0.34 I λ L 2 1 0 18 W c m − 2 μ m 2 0.55 τ L p s 0.27 L μ m 0.32 M e V , (3)

This expression recovers the ponderomotive scaling and introduces an enhancement for longer L μ m and a reduction for shorter L μ m. In Hermione it is possible to choose between Equation 1, Equation 2 or Equation 3 for the electron temperature. However, in this study, we focus on the Equation 3 which take into account the effective preplasma scale length. In Particle in-Cell (PIC) simulations, the preplasma density profile is often fitted using several exponential components, reflecting the presence of different electron populations and therefore multiple characteristic scale lengths. In the present work, we adopt a simplified description by considering a single effective preplasma scale length associated with the electron population responsible for the sheath formation. In practice, we retain only the shortest scale length, which governs the hot electron temperature relevant for the sheath field. Furthermore, Antici et al. [61] and Passoni et al. [32] demonstrated that laser-plasma interactions can generate multiple electron populations characterized by different temperatures. In particular, electron populations with significantly higher temperatures are associated with much larger effective preplasma scale lengths and are instead preferentially ejected electron population. Therefore these electrons do not play a significant role in the formation or sustainment of the accelerating sheath field. Consequently, in our model, only the electron temperature corresponding to the sheath-forming population is retained to define a single, effective preplasma scale length.

The duration over which ions are accelerated depends on the relation between the laser pulse duration and the characteristic ion response time (the typical time required for ions to begin responding dynamically). For laser pulses longer than the ion plasma period, the acceleration time can be described by the empirical scaling proposed by Fuchs et al. [62], which accounts for the extended lifetime of the sheath field. In contrast, for ultrashort pulses where the laser duration is shorter than the ion plasma period, a refined description of the early-time sheath dynamics must be used, as proposed by Simpson et al. [63].

Accordingly, the acceleration time is written as:where,is the ion plasma frequency and u bal = v t h ⁡ sin θ ̄ is the ballistic component of the electron velocity in the preplasma which denotes the transverse electron expansion velocity with v t h = c 1 − 1 T h m e c 2 + 1 2 the thermal electron velocity and θ ̄ the average scattering angle (see Figure 1).

τ acc n e = τ L 2 + 6 ω p i 2 + 2 R u bal 2 , 63 for τ L < 2 π / ω p i , F acc τ L + 60 f s , 62 otherwise , (4)

ω p i n e = e 2 n e / 2 m p ε 0 , (5)

Electron emission scheme during the TNSA process. In light blue is represented the electron emission with a dispersion angle of θ. u bal is represented in dark blue and represents the projection on the transverse side of the electron emission with a mean velocity v t h.
Figure 1. Electron emission scheme during the TNSA process. In light blue is represented the electron emission with a dispersion angle of θ. u bal is represented in dark blue and represents the projection on the transverse side of the electron emission with a mean velocity v t h.

In the long-pulse regime, the multiplicative factoris used to correct the model for the acceleration process. While F acc slightly decreases with increasing laser intensity, the actual sheath lifetime grows with the pulse duration. This factor therefore accounts for the influence of the sheath field in long-pulse interactions.

F acc I = max − 6 × 1 0 − 20 I W / c m 2 + 3.12 , 1.3 , (6)

The computation of the electron density is a key element in the TNSA model. In this work, we consider different temporal regimes of the laser pulse duration τ L.

Next, for a given pulse duration τ L, the electron density grows linearly from zero at t = 0 to its saturation value n sat at the characteristic time t s h = 2 π / ω p i ( n sat ), where n sat = γ n c, γ = 1 + T h / ( m e c 2 ), and n c = ε 0 m e ω 2 / e 2.

Finally, for τ L > t s h, the electron density is considered as a combination of two contributions: a linearly increasing density from 0 at t = 0 to n sat at t s h, followed by a constant density n sat from t s h up to the total acceleration time τ acc ( n sat ).

Furthermore, a more refined model [38] of electron density is implemented in the code which includes a confinement time t conf obtained from the balance between ballistic and diffusive transport due to the filamentation of the laser beam for high energy and long pulse laser facilities such as PETAL or LFEX.where Θ = 1 / sin θ ̄ tan θ ̄ = 0.76, and where the average scattering angle is then given bywhich is about 57°, L tot is the effective plasma length, determined iteratively from

n e conf = γ n c Θ R 2 L tot 2 1 + 2 u dif u bal , (7)

θ ̄ = ∫ 0 π / 2 θ ⁡ sin ⁡ θ d θ ∫ 0 π / 2 ⁡ sin ⁡ θ d θ = 1 r a d , (8)

t conf = R 3 u dif 1 + 3 u dif u bal , L tot = L + 2 c s t conf . (9)

Here, c s = T h / ( 2 m p ) is the ion-acoustic speed, and u dif = L 2 R tan θ ̄ u bal is the diffusive component of the electron velocity in the preplasma and L is the target thickness.

In the domain of validity of Equation 7 presented by N. Iwata et al. [38] (for R / L tot > tan ( θ ) and t conf < L / ( 2 c s ), τ L > t conf), as it represents the maximal value of the electron density in the sheath and as our model does not take into account the time dynamics, we take the average value in time of Equation 7, n e = n e conf τ acc ( n sat ) τ acc ( n sat ) − t conf / 2.

To summarize, we will distinguish 4 different regimes for the electron density:

n e τ L = n sat τ L t s h , if 0 < τ L < t s h , n sat τ acc n sat τ acc n sat − t s h 2 − 1 e x if τ L ≥ t s h , n e , c o n f τ acc n sat × τ acc n sat − t conf 2 if R L tot > tan θ , t conf < L 2 c s , and τ L > t conf . (10)

Finally, the proton spectrum is determined by the plasma expansion model [30, 31] which gives, for the proton cut-off energy:where T h is defined in Equation 3, τ acc in Equation 4, ω p i in Equation 5 and n e in Equation 10. And at last, a proton spectrum expressed as:where the sheath area is S sheath = π ( R + L ⁡ tan ⁡ θ ) 2, with θ = 6.45 ⁡ ln ( I ( W / c m 2 ) ) − 257.42 [63, 64], which reproduces the quasi-exponential shape observed in experimental TNSA spectra and provides a direct link between the hot-electron population and the ion beam characteristics.

E p max = 2 T h ln t p + t p 2 + 1 2 , t p = τ acc ω p i 2 e , (11)

d N p d E = n e c s τ acc S sheath 2 E T h exp − 2 E T h , (12)

3 Model results

All the previous equations have been implemented in a python code named Hermione [55] (for Hot ElectRon Model for ION Emission). In this section, we present the results for ALLS, Apollon, PETAL and LFEX facilities, four facilities with different laser characteristics, from 22 fs to 6 ps, from 3.2 J to 600 J on target and from 2.5 μm of spot radius to 52 μm. The electron density equations used during these computations, which depend directly of the laser pulse duration as described in Equation 10. For LFEX the Equation 10, for PETAL the Equation 10, and for ALLS and Apollon Equation 10, were used.

In this study, the effective preplasma scale length is not self-consistently calculated in the present model but rather treated as a free parameter. In practice, it is manually adjusted to reproduce the experimental observations, in particular to obtain an electron temperature in the sheath and a resulting proton spectrum consistent with measurements. Accordingly, a natural extension of this work would be to establish scaling laws for the characteristic preplasma length, enabling this parameter to be directly inferred from the laser properties rather than a free parameter.

The model is designed to be applicable over a laser intensity range spanning from 1 0 18 to 1 0 22 W / c m 2. Beyond 1 0 22 W / c m 2, additional physical processes such as radiation reaction and electron-positron pair production become significant and are not included in the present framework, thereby limiting its validity in this regime. Within its domain of applicability, the model provides reasonably robust predictions: the proton yield is typically accurate within about 50 %, while the cut-off energy is reproduced with a better agreement, on the order of 20 %.

3.1 ALLS facility

Figure 2 shows the TNSA spectrum computed for the ALLS laser, producing 3.2 J of energy on target, a pulse duration of 22 fs, a wavelength of 800 nm, a focal spot diameter of 5 μm [44], and a pre-plasma scale length of 0.3 μm (this parameter was manually adjusted in order to obtain an electron temperature in the sheath and a proton spectrum consistent with the experimental measurements [44, 49, 65, 66]) and compares it to additional experimental results obtained from previous experimental studies [65] for a 2 μm thickness target. Under these conditions, our model predicts an electron temperature of T e ≈ 1.14 MeV which corresponds to the expected values obtained in previously published simulations [49, 66].

Energy distribution of the accelerated protons, the blue curve shows the result obtained with the Hermione code with E L = 3.2 J, τ L = 22 fs, λ L = 800 nm, R = 5 μm and L μ m = 0.3 μm, while the red curve corresponds to a experimental measurement obtained during a previous experimental campaign [65] on ALLS with the same parameters as described for Hermione.
Figure 2. Energy distribution of the accelerated protons, the blue curve shows the result obtained with the Hermione code with E L = 3.2 J, τ L = 22 fs, λ L = 800 nm, R = 5 μm and L μ m = 0.3 μm, while the red curve corresponds to a experimental measurement obtained during a previous experimental campaign [65] on ALLS with the same parameters as described for Hermione.

The resulting proton spectrum exhibits the typical exponential-like decrease of TNSA, with a cut-off located at 6.71 MeV which corresponds to the experimental measurement included in the figure and as obtained in other experiments [44]. However, the proton flux lies in the range of 1 0 11 particles/MeV/sr at low energy, consistent with expectations for few-joule, tightly focused laser systems but 6 times higher than the experimental measurement. The total number of accelerated protons is of 1.5 × 1 0 11 which is comparable with previous experimental results [44, 65].

3.2 Apollon facility

Figure 3 shows the proton energy spectrum obtained for the Apollon laser using the following parameters: an incident energy of 40 J, a pulse duration of 23.7 fs, a wavelength of 850 nm, a focal spot size of 2.5 μm [17], and an effective preplasma scale length of 0.15 μm (this parameter was manually adjusted in order to obtain an electron temperature in the sheath and a proton spectrum consistent with the experimental measurements [17]) for a 5 μm thickness target. Under these conditions, our model predicts an electron temperature of T e ≈ 8.23 MeV. The resulting TNSA spectrum exhibits the quasi-exponential decay characteristic of this acceleration mechanism, with a proton flux reaching 2.5 × 1 0 12 particles/MeV/sr at low energy.

Energy distribution of the accelerated protons, the blue curve represents the result obtained with the Hermione code using E L = 40 J, τ L = 23.7 fs, R = 2.5 μm and L μ m = 0.15 μm, whereas the red dots corresponds to previously reported experimental measurements [17] on Apollon with the same parameters as described for Hermione.
Figure 3. Energy distribution of the accelerated protons, the blue curve represents the result obtained with the Hermione code using E L = 40 J, τ L = 23.7 fs, R = 2.5 μm and L μ m = 0.15 μm, whereas the red dots corresponds to previously reported experimental measurements [17] on Apollon with the same parameters as described for Hermione.

The proton curve displays a cut-off in the range of 49.2 MeV, consistent with the proton spectrum measurement achieved on Apollon in this configuration. The total number of accelerated protons (integrated over the full spectrum) is on the order of 5.6 × 1 0 12 which is at least 10 times higher than the experimental measurement.

3.3 PETAL facility

Figure 4 presents the TNSA spectrum obtained with the PETAL laser [67] using an energy of 409 J, a pulse duration of 660 fs, a wavelength of 1.053 μm, a focal spot diameter of 52 μm [36], and a pre-plasma scale length of 1.5 μm (this parameter was manually adjusted in order to obtain an electron temperature in the sheath and a proton spectrum consistent with the experimental measurements [36]) for a 10 μm thickness target. In this picosecond regime, our model yields a moderate electron temperature of T e ≈ 1.1 MeV.

Energy distribution of the accelerated protons, the blue curve represents the result obtained with the Hermione code with E L = 409 J, τ L = 660 fs, λ L = 1.053 μm, R = 52 μm and L μ m = 1.5 μm, whereas the red curve corresponds to previously reported experimental measurements [36] on PETAL with the same parameters as described for Hermione.
Figure 4. Energy distribution of the accelerated protons, the blue curve represents the result obtained with the Hermione code with E L = 409 J, τ L = 660 fs, λ L = 1.053 μm, R = 52 μm and L μ m = 1.5 μm, whereas the red curve corresponds to previously reported experimental measurements [36] on PETAL with the same parameters as described for Hermione.

The proton spectrum shows a cut-off around 43.15 MeV, reflecting a combination of a high electron temperature (for this characteristic of laser) due to a long preplasma length and a high number of accelerated protons. The total number of produced protons is about 1.8 × 1 0 14, which is well fitting with experimental measurement.

3.4 LFEX facility

Figure 5 shows the proton energy spectrum obtained for the LFEX laser using the following parameters: for Figure 5a), an incident energy of 150 J on target, a pulse duration of 1.5 ps, a focal spot size of 45 μm [35], for Figure 5b), an incident energy of 300 J on target, a pulse duration of 3 ps, a focal spot size of 45 μm [35], for Figure 5c), an incident energy of 600 J on target, a pulse duration of 6 ps, a focal spot size of 45 μm [35]. These three computations were performed using a pre-plasma scale length of 0.15 μm for 1.5 ps pulse duration and 0.4 μm for 3 and 6 ps pulse durations and for a 5 μm thickness target and a wavelength of 1.053 μm,. The pre-plasma scale length parameters was manually adjusted in order to obtain an electron temperature in the sheath and a proton spectrum consistent with the experimental measurements [35]. Under these conditions, our model predicts an electron temperature of respectively 0.28, 0.46 and 0.56 MeV. The resulting TNSA spectrum exhibits values similar to previous published experimental results [35] in energy and number.

Proton energy spectrum for λ L = 1.053 μm, **(a)** E L = 150 J, τ L = 1.5 ps,R = 45 μm, L μ m = 0.15 μm **(b)** E L = 300 J, τ L = 3 ps,R = 45 μm, L μ m = 0.4 μm and **(c)** E L = 600 J, τ L = 6 ps,R = 45 μm and L μ m = 0.4 μm. The blue curve shows the result obtained with the Hermione code using the parameters described above, while the red curve shows previously reported experimental results [35] on LFEX with the same parameters describe above.
Figure 5. Proton energy spectrum for λ L = 1.053 μm, (a) E L = 150 J, τ L = 1.5 ps,R = 45 μm, L μ m = 0.15 μm (b) E L = 300 J, τ L = 3 ps,R = 45 μm, L μ m = 0.4 μm and (c) E L = 600 J, τ L = 6 ps,R = 45 μm and L μ m = 0.4 μm. The blue curve shows the result obtained with the Hermione code using the parameters described above, while the red curve shows previously reported experimental results [35] on LFEX with the same parameters describe above.

All the results presented in this section are summarized in the Table 1. Laser-ion acceleration experiments exhibit significant shot-to-shot fluctuations. Therefore, the comparison between experiments and simulations will be further improved in future studies including parametric studies and implementing new scalling laws. The objective of this model is to provide a framework that can progressively be enriched by the community. It is intended to evolve and improve over time through additional experimental data, refined scaling laws, and broader theoretical and numerical developments.

TABLE 1. Comparison between Hermione predictions and experimental measurements for each facility: proton cut-off energy, total number of accelerated protons, and electron temperature. In the experiments, the electron temperature is measured using an electron spectrometer which detects the emitted electrons. The temperature given by Hermione is representative of the electron population in the sheath, which is intrinsically colder. Consequently, the electron temperature predicted by Hermione is expected to be lower than the experimentally measured values.

Facility (E L, τ L, R)E cut Herm. (MeV)E cut exp (MeV)N p Herm.N p expT e Herm. (MeV)T e biblio (MeV)
ALLS (3.2 J, 22 fs, 5 μm)6.716.71.5 × 1 0 11∼ 5 − 12 × 1 0 101.14∼ 1.4
Apollon (40 J, 23.7 fs, 2.5 μm)49.2495.6 × 1 0 12∼ 1 × 1 0 118.235.3 − 20
PETAL (409 J, 660 fs, 52 μm)43.15431.8 × 1 0 14∼ 1 × 1 0 131.103.29
LFEX (150 J, 1.5 ps, 45 μm)14133 × 1 0 11 > 5 M e V∼ 3 × 1 0 11 > 5 M e V0.280.41
LFEX (300 J, 3 ps, 45 μm)31293 × 1 0 12 > 5 M e V∼ 5 × 1 0 12 > 5 M e V0.461.13
LFEX (600 J, 6 ps, 45 μm)47331 × 1 0 13 > 5 M e V∼ 8 × 1 0 12 > 5 M e V0.560.97

4 Discussion

The present analytical model highlights the critical role played by the effective preplasma scale length in determining the main characteristics of the TNSA process. By adjusting the preplasma length, the model allows to consistently estimate the hot-electron temperature in the sheath as well as the resulting proton cut-off energy. For the range of laser parameters considered in this work, both quantities are found to be in good agreement with experimental observations reported in the literature.

In particular, the dependence of the hot-electron temperature on the laser intensity, pulse duration, and effective preplasma scale length leads to electron temperatures in the MeV range, consistent with experimental diagnostics and particle-in-cell simulations. The corresponding proton cut-off energies, derived from standard analytical TNSA models, follow the expected scaling laws and reproduce the behaviour and tendency observed in experiments, thereby validating the use of an effective acceleration time.

In contrast, the total number of accelerated protons predicted by the model for the ALLS and Appolon cases is overestimated by approximately one order of magnitude when compared to experimental measurements. This difference can be attributed to the simplifying assumptions inherent to a purely static 0-D approach. In particular, the model assumes a uniform electron density and temperature and an exponential shape preplasma (∝ e z − z 0 L μ m, where z is the laser propagation axis perpendicular to the target and z 0 is the target position).

Nevertheless, the ability of the model to correctly predict the electron temperature and proton cut-off energy, indicates that the essential physics of the acceleration process is well reproduced.

Furthermore, we note that the sheath field is primarily governed by the electron dynamics. On the timescales relevant for the acceleration of the highest energy protons, the lighter ion species respond much more rapidly than the heavier ones. Indeed, the electron plasma frequency typically satisfies ω p e / ω p i , p ∼ 40 for protons (where ω p e and ω p i , p denote the electron and proton plasma frequencies, respectively), and ω p e / ω p i , C ∼ 100 for carbon ions. As a consequence, the sheath field evolves on electron timescales and is only weakly influenced by the slower response of heavier ion species. We therefore expect the multi–species effect on the sheath formation and on proton acceleration to be negligible within the present model.

This model permits also to predict future experimental results. For example, the studies on the proton spectrum variations with the laser wavelength. However, in the femtoseconde cases (<100 fs), the cut-off energy remains well captured, but the proton yield is not yet accurately reproduced and requires further improvement. This highlights the need for dedicated refinements of the model for ultrashort pulses.

4.1 Wavelength study

The laser wavelength is a fundamental parameter governing the dynamics of high-intensity laser–solid interactions and plays a central role in determining the dominant particle acceleration mechanisms. While most experimental and theoretical studies of TNSA have historically focused on near-infrared laser systems, recent advances in high-power laser technology [68] have renewed interest in exploring wavelength-dependent effects on electron heating, sheath formation, and ion acceleration [69].

In this context, a recent work [69] performed a comprehensive study of the wavelength dependence of TNSA and highlights an increased cut-off energy with 2 microns wavelength laser compared to 1 microns.

Therefore, in Figure 6, assuming an effective preplasma scale length of 0.4 µm in both cases and at same intensity (same energy, same duration, same spot size), a similar effect than the one described by Wilks et al. [69] is observed. The electron temperature increases from 0.46 MeV with a 1 μm laser to 0.97 MeV with a 2 μm laser. Furthermore, the proton cut-off energy rises from 30.3 MeV to 54.5 MeV for the 2 μm laser. This represents a very significant enhancement in the TNSA performance.

Energy distribution of the accelerated protons and electrons for E L = 300 J, τ L = 3 ps, R = 45 μm using a laser wavelength of 2 microns (red curve) or one microns (blue curve).
Figure 6. Energy distribution of the accelerated protons and electrons for E L = 300 J, τ L = 3 ps, R = 45 μm using a laser wavelength of 2 microns (red curve) or one microns (blue curve).

4.2 Optimized parameters

Hermione also allows for a systematic investigation of the impact of laser parameters on TNSA performance by varying the laser energy, pulse duration, and focal spot radius. These studies can be performed either assuming a fixed effective preplasma scale length or considering a preplasma length that scales with the laser pulse duration as L μ m = c s τ L. This flexibility enables rapid exploration of a wide parameter space and helps identify optimal conditions for electron heating and proton acceleration.

Figure 7 presents maps of the hot-electron temperature T h (MeV) (a), the normalized hot-electron temperature T h / T pond (MeV) (b), the accelerated proton yield N p (c), and the proton cut-off energy E p max (MeV) (d) as functions of both, the focal spot radius and pulse duration for E L = 400 J and λ L = 1.053 μm (corresponding to the PETAL laser). A constant effective preplasma scale length of 1.5 μm (choosen to fit with the PETAL laser facility) is assumed across the entire parameter range.

Hot-electron temperature T h (MeV) **(a)**, normalized hot-electron temperature T h / T pond **(b)**, accelerated proton yield N p **(c)** and proton cut-off energy E p max (MeV) **(d)** as a function of focal spot radius and pulse duration for E L = 400 J, λ L = 1.053 μm, a target thickness of 5 μm and a fixed preplasma length of L μ m = 1.5 μm. The white solid lines correspond to the iso-values of the laser intensities. The upper right area represents intensity lower than 1 0 18 W/c m 2. However, our study only focuses on regimes with intensity above this threshold.
Figure 7. Hot-electron temperature T h (MeV) (a), normalized hot-electron temperature T h / T pond (b), accelerated proton yield N p (c) and proton cut-off energy E p max (MeV) (d) as a function of focal spot radius and pulse duration for E L = 400 J, λ L = 1.053 μm, a target thickness of 5 μm and a fixed preplasma length of L μ m = 1.5 μm. The white solid lines correspond to the iso-values of the laser intensities. The upper right area represents intensity lower than 1 0 18 W/c m 2. However, our study only focuses on regimes with intensity above this threshold.

The Hermione results are presented in Figure 7. As shown in Figure 7a, tighter focusing and shorter pulses lead to higher electron temperatures. This result is supported by Figure 7b, the normalized temperature T h / T pond: considering the scaling law presented in Equation 3 a decreased temperature for short pulse duration and an increased temperature for longer pulse duration compared to the ponderomotrice temperature presented in Equation 1 are obtained. This result is perfectly fitting with the meta-analysis of Rusby et al. [40].

Furthermore, Figure 7c shows that the proton number is optimal both for long pulse duration (>1 ps) and small spot radii (optimum between 10 and 30 μm). The increased number of protons is a consequence of the filamentation of the laser beam. Indeed, this filamentation generates a high focusing magnetic field and thus laterally confines the electrons. This phenomena leads to an increased electron density in the sheath as described in Equation 7. Therefore, as the cut-off energy depends both on electron temperature and electron density, an optimized cut-off energy region is shown on Figure 7d for small spot radius and higher pulse duration, a compromise between high electron temperature and high electron density.

Similar results were also obtained at lower laser energy, as illustrated in Figure 8. In this case, simulations were performed using laser parameters representative of the Apollon laser facility France, namely, a pulse energy of 40 J, a central wavelength of 850 nm, and a pulse duration of 23.7 fs, interacting with a 5 µm-thick target and a constant effective preplasma scale length of 0.3 μm. Despite the reduced energy, the main features of the acceleration process remain consistent, highlighting the robustness of the observed trends across different laser conditions. However, the number of protons for pulse duration lower than 100 fs remains overestimated.

Hot-electron temperature T h (MeV) **(a)**, accelerated proton yield N p **(b)** and proton cut-off energy E p max (MeV) **(c)** as a function of focal spot radius and pulse duration for E L = 40 J, λ L = 850 nm, a target thickness of 5 μm and a fixed preplasma length of L μ m = 0.2 μm. The white solid lines correspond to the iso-values of the laser intensities. The upper right area represents intensity lower than 1 0 18 W/c m 2. However, our study only focuses on regimes with intensity above this threshold.
Figure 8. Hot-electron temperature T h (MeV) (a), accelerated proton yield N p (b) and proton cut-off energy E p max (MeV) (c) as a function of focal spot radius and pulse duration for E L = 40 J, λ L = 850 nm, a target thickness of 5 μm and a fixed preplasma length of L μ m = 0.2 μm. The white solid lines correspond to the iso-values of the laser intensities. The upper right area represents intensity lower than 1 0 18 W/c m 2. However, our study only focuses on regimes with intensity above this threshold.

The sharp transitions observed in these parametric maps arise from the change of analytical model used to describe the electron density and acceleration time as a function of the pulse duration. Since each temporal regime relies on a distinct formulation, the boundaries between them produce discontinuities that would not appear in an experimental scan. Future developments of the model will include smooth interpolations between these regimes to ensure continuous and physically consistent transitions across the parameter space.

4.3 Parametric studies

The Hermione code can also be used to support parametric studies previously reported in the literature. In particular, it provides a useful framework for investigating the dependence of the proton cut-off energy on the incident laser energy. As an example, the code may be employed to support and interpret the experimental trends observed in the work of N. Dover et al. [43], where the evolution of the proton cut-off energy with an increasing laser energy was investigated. As shown in Figure 9, a good agreement is found between experimental data [43] and the Hermione simulations using an efficient preplasma scale length of 0.1 μm.

Proton cut-off energy as function of the incident laser energy considering τ L = 40 fs, R = 1.5 μm using a laser wavelength of 0.8 microns obtained experimentally (magenta dots) or with Hermione using a fixed preplasma length of L μ m = 0.1 μm (red curve) for the J-KAREN-P laser.
Figure 9. Proton cut-off energy as function of the incident laser energy considering τ L = 40 fs, R = 1.5 μm using a laser wavelength of 0.8 microns obtained experimentally (magenta dots) or with Hermione using a fixed preplasma length of L μ m = 0.1 μm (red curve) for the J-KAREN-P laser.

In another example of experimental parametric study, in the recent work by M. Ehret et al [70], the same overall trends are observed and can be reproduced by Hermione as shown in Figure 10. In particular, both the cut-off energy (see Figure 10a) and the proton charge (see Figure 10b) increase with increasing laser pulse duration, in agreement with the tendencies observed in the experimental parametric study [70]. However these tendencies are a qualitative tendency agreement. 25% increased values in cut-off energy are obtained with Hermione compared to the experimental data [70].

Parametric study of proton acceleration as a function of laser parameters. **(a)** Proton cut-off energy as a function of the laser pulse duration for a laser energy of 6.9 J and a focal spot radius of 6.5 μm. **(b)** Number of proton as a function of the laser pulse duration for a laser energy of 6.9 J and a focal spot radius of 6.5 μm.
Figure 10. Parametric study of proton acceleration as a function of laser parameters. (a) Proton cut-off energy as a function of the laser pulse duration for a laser energy of 6.9 J and a focal spot radius of 6.5 μm. (b) Number of proton as a function of the laser pulse duration for a laser energy of 6.9 J and a focal spot radius of 6.5 μm.

5 Conclusion

In summary, we have developed and validated Hermione, a fast and flexible phenomenological model intended to predict general trends for proton acceleration via the TNSA mechanism across multiple laser facilities. The model reproduces key experimental trends, including wavelength dependence with results consistent with PIC simulations but obtained in a fraction of the computational time (about 15 s). In this study, we compare our results to several laser facilities, i.e., PETAL, ALLS, Apollon and LFEX, which correspond to four different laser characteristics.

While very good agreement is observed for pulses longer than 100 fs, refinement of the model for ultrashort pulses (< 100 fs) is needed in future work. The open-access nature of Hermione [55] encourages the community to further develop and optimize it, providing a valuable tool for both theoretical studies and experimental planning in laser-driven proton acceleration.

Furthermore, an important extension of this work would be to develop scaling laws for the characteristic preplasma length and directly relate this quantity to the laser parameters. Ultimately, this would enhance the predictive power of the model, making it fully constrained by the laser characteristics alone, while improving its relevance for the interpretation and planning of experiments. In a future work, a generalization of the present model to heavier ion species could also be the subject of further investigation, provided that appropriate species-dependent scaling laws are established and validated.

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Footnotes

Edited by: Massimiliano Sciscio', Energy and Sustainable Economic Development (ENEA), Italy

Reviewed by: Francesco Mirani, Politecnico di Milano, Italy

Michael Ehret, ELI Beamlines, Czechia

Statements

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The author ED’H declared that they were an editorial board member of Frontiers at the time of submission. This had no impact on the peer review process and the final decision.

Funding

The author(s) declared that financial support was received for this work and/or its publication. This work is supported by the CEA/DAM laser plasma experiments validation project and the CEA/DAM basic technical and scientific studies project. This work is also supported by the National Sciences and Engineering Research Council of Canada (NSERC) (Grant—RGPIN-2023-05459, ALLRP 556340 – 20, Projet R27-46-012-PME), PRIMA Québec, the Digital Research Alliance of Canada (Job: pve-323-ac), the Canada Foundation for Innovation (CFI), the Ministère de l’Économie, de l’Innovation et de l’Énergie (MEIE) and the Fonds de recherche du Québec - secteur Nature et technologies (FRQNT), coopération franco-québécoise - programme Samuel-De Champlain. We acknowledge the financial support of the IdEx University of Bordeaux/Grand Research Program ”GPR LIGHT”, and of the Graduate Program on Light Sciences and Technologies of the University of Bordeaux. This work is partly supported by JSPS KAKENHI Grant Nos. JP23K03354, JP24H00204, and JP23K20038.

Data availability

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

CL: Investigation, Software, Conceptualization, Writing – review and editing, Writing – original draft, Formal Analysis, Data curation. NI: Writing – original draft, Conceptualization, Formal Analysis, Supervision, Validation, Writing – review and editing, Methodology. MB: Funding acquisition, Supervision, Validation, Writing – original draft. ED’H: Methodology, Validation, Conceptualization, Supervision, Writing – original draft, Funding acquisition, Writing – review and editing, Investigation. YS: Supervision, Writing – original draft, Writing – review and editing, Investigation, Methodology, Conceptualization, Validation. PA: Writing – original draft, Validation, Writing – review and editing, Supervision, Funding acquisition.

Acknowledgements

The authors are grateful to F. Légaré, D. Raffestin, Y. Takagi, P. Nicolaï, M. Huault, H. Larreur and K. Aliane for their help and useful discussions.

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Citation

Lacoste, C. L. C., et al. (2026). Hot electron model for ion emission (Hermione): an effective model for target normal sheath acceleration ion spectra from femtosecond to picosecond lasers. Frontiers in Physics, Volume 14. https://doi.org/
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14 - 2026

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Copyright © 2026 Lacoste, Iwata, Bardon, D’Humières, Sentoku and Antici.

License: https://creativecommons.org/licenses/by/4.0/

*Correspondence: C. L. C. Lacoste

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