A subgrid foam model is developed to describe numerically a sequence of processes transforming solid elements in the foam into a hot plasma under the energy deposition by lasers. We account for three distinct processes: accurate laser energy absorption and scattering on the subwavelength cylindrical solid elements, expansion of the foam element due to the energy deposition in its volume, and ablation of the solid element due to the energy deposition on its surface. The foam element dynamics is modeled via a selfsimilar isothermal expansion combined with a stationary ablation model, and it is described by a set of ordinary differential equations. The microscopic single pore model is incorporated in the macroscopic hydrodynamic codes, and numerical simulations show a good agreement with available experimental data.
I. INTRODUCTION
Lowdensity foams have interesting properties that make them attractive for fundamental studies of the laser–plasma interaction and for various applications, such as inertial confinement fusion and bright sources of x rays. Foams are used for launching strong shock waves in high energy density physics experiments,^{1} lining hohlraums in the indirect drive fusion experiments to prevent undesired fast expansion of the walls,^{2} covering spherical shells in the direct drive experiments to reduce the laser imprint,^{3,4} and studying the laser–plasma interaction processes.^{5,6} Foams allow us to improve the fusion ignition conditions^{7} or increase the efficiency of conversion laser energy into hot electrons and x rays.^{8–10}
The process of transformation of a cold foam into a hot plasma is complicated. It is known from experiments and numerical simulations that the ionization of foam by laser proceeds slower than ionization of a homogeneous material of the same average density, thus allowing us to store more laser energy in the same plasma volume. This behavior can be explained by the homogenization process—plasma from overcritical solid elements needs to fill the empty voids inside pores before the laser can penetrate further into the target.
The direct numerical modeling of the laser–foam interaction^{11,12} is computationally expensive due to the necessity to spatially resolve the density differences in the foam microstructure. Also, it cannot account for all microphysics processes due to the intrinsic limitations of hydrodynamic models that are not designed to operate at kinetic spatial and temporal scales.
Analytic macroscopic models for the laser interaction with foam^{13–15} are limited to a description of an idealized onedimensional laser interaction with a static target and without effects like heat conduction. An extension of these models to a full numerical simulation consists in a subgrid (twoscale) approach, where a simplified (analytic or numerical) model for the single pore dynamics is embedded into a radiation hydrodynamic code that computes the macroscopic interaction between the pores or numerical cells. The subgrid microscale calculation is applied to the nonhomogenized cells, and it is stopped once the foam cell is fully homogenized. This way the subgrid approach describes a continuous transformation of a cold foam into a hot plasma.
The first subgrid model by Velechovský et al.^{16} is based on a numerical calculation of laser absorption and expansion of planarshaped solid foam elements, following the theory proposed by Gus'kov et al.^{13} The model proposed in Refs. 14, 17, and 18 describes the laserdriven foam homogenization as an adiabatic mixing of two plasma flows coming from neighboring pores mediated by the ion viscosity. It was successfully applied to foams with a large pore size $ \u2273 30 \u2212 50 \u2009 \mu $m, where the phase of the expansion of solid foam elements is not considered as important. However, its application to foams with micrometersize pores is problematic. This model was implemented in the radiation hydrodynamic code MULTIFM in one spatial dimension and showed good agreement with experiments with overcritical foams.^{19}
Belyaev et al.^{15,20} have proposed another foam homogenization model based on a selfsimilar isothermal solid element expansion and applied it to foams with small pores $ \u2264 1 \u2009 \mu $m and with small average density $ \u223c 2$ mg/cm^{3}. It is the first model to introduce the foamspecific ion heating due to dissipation of kinetic energy of foam element expansion. It is implemented in the electromagnetic paraxial code pF3D in three spatial dimensions and demonstrated a good agreement with the experiment.^{20} However, this model overestimates the ionization front velocity when applied to other experiments with foams of larger density.^{6,21–23}
In all the models mentioned above, the cross section for the laser absorption in the solid elements is determined by the geometrical size of the overcritical region without considering the exact physics of the laser interaction with the foam element. However, the width of solid elements is typically smaller or comparable to the laser wavelength. The laser interaction with subwavelengthsized objects needs special attention. The wavebased description according to the Mie theory^{24} and detailed kinetic simulations^{25} shows that the absorption process strongly depends on laser polarization, shape, and internal structure of the solid elements. The resonant laser deposition near the critical surface and the suppression of the heat flux inside the solid element play an important role in the overall dynamics, resulting in an ablation of solid elements and in a longer pore homogenization time compared to a volumetric absorption and isothermal expansion.
In this paper, we revise the model of foam homogenization for wirelike cylindrical solid elements by including the processes of ionization, heating, expansion, and ablation in a selfconsistent microscopic model and by using the theory of an electromagnetic wave interaction with subwavelength objects to describe laser absorption and scattering. In our microscale model, each foam pore is divided into two regions with separate masses, densities, and temperatures—the central cylinder represents the expanding solid element, and the outer plasma region acts as the ambient plasma in the pore. The motion of the boundary between these two environments is controlled by the selfsimilar expansion, while the mass transfer between regions is given by a stationary ablation model. Pores are considered to be homogenized when the cylinder and the outer plasma reach the same density.
This new microscopic foam homogenization model is included as a subgrid module in the hydrodynamic codes PALE^{26} and FLASH.^{27–29} The results of numerical simulations performed with the code PALE are compared in Sec. VI with the available experimental data on the laser interaction with smallpore subcritical foams.^{6,16,30} The comparison shows that the model correctly captures the physical processes responsible for foam homogenization and calculates the foam ionization velocity in agreement with experimental data.
II. THE TWOSCALE METHOD
The main idea of a twoscale model is to describe the dynamics of the foam simultaneously on two spatial and temporal scales. The first scale, which we refer to as the macroscale, considers the foam as homogeneous medium of the average properties and does not intent to resolve the phenomena related to internal foam dynamics. It is computed by the standard methods for plasma hydrodynamics described by partial differential equations, and it is meant to provide connection between the foam pores when they have been fully homogenized. The second scale describes the internal cell dynamics: the laser interaction with the individual solid elements and their transformation into a plasma state. We refer to it as the microscale. Both models describe the dynamics of the exact same volume and mass of the fluid.
The microscale contains an additional information about the subdivision of the mass and energy inside the pore, whereas the macroscale only uses the values averaged over the pores. The microscale computes the dynamics of each cell separately, and, therefore, it relies on the macroscale to provide information about the state of adjacent pores, which is necessary to correctly estimate the incident laser intensity and energy transferred by heat conductivity. The microscale model is formulated in terms of ordinary differential equations. This allows us to solve it efficiently and with better temporal resolution, all without affecting the performance of the hydrodynamic code used for the macroscale computation.
III. MACROSCALE HYDRODYNAMIC MODEL
The hydrodynamic Eqs. (1)–(4) are solved in twodimensional (2D) cylindrical r, z geometry. The code PALE (Prague Arbitrary Lagrangian Eulerian code)^{26} solves them in the Lagrangian form, and the code FLASH^{27} solves them in their Eulerian form.
The macroscale computational cells can be in three different states: cold, intermediate, and fully homogenized plasma state. These are based on the homogenization state of the solid foam elements inside the cells, which are modeled by the microscale. Initially all cells are in the cold state—the solid elements are at room temperature, and they are surrounded by vacuum or a lowdensity gas. When the laser or heat wave reaches the cold state cell, its state turns into the intermediate state described by our new microscale model. In these cells, the microscale and macroscale models run in parallel. Some aspects of the foam internal dynamics (including the laser deposition and electronion energy redistribution) are handled by the micromodel (when active) and are, thus, omitted from the macroscale to avoid including the same terms twice. We assume that the macroscale cells are not moving during the homogenization state, that is, their mass remains constant and the flow velocity is zero. With all these assumptions and simplifications, only the equations for the laser and heat transport remain to be solved in a macroscale cell when the microscale model is active (see more details about the modifications at the macroscale in Sec. IV F). When the foam element density drops below the average foam density or when it equalizes with the density of the ambient plasma, the homogenization is terminated, the microscale model is turned off, and the corresponding macroscale cell turns into the plasma state. We solve the complete hydrodynamic system (1)–(4) in plasma state cells.
The laser energy transport and absorption are modeled by the ray tracing method. The laser beam is split into a set of independent rays, and their trajectories are governed by the ray equation of geometric optics. Attenuation of the ray power $ P ray$ along its path is described by $ d P ray / d s = \u2212 K \u2009 P ray$, where the absorption rate K depends on the cell state—the standard inverse bremsstrahlung absorption coefficient is used for cells in the plasma state, while a microscopic absorption coefficient, combining absorption in the solid foam element (cylindrical wire) and the ambient plasma, is employed for cells in the intermediate state. See Sec. IV A for more details.
IV. MICROSCALE HYBRID ABLATION–EXPANSION MODEL
Each macroscale computation cell has its corresponding microscale cell. As schematically shown in Fig. 1(a), the microscale cell has a form of a parallelepiped with a square $ l p \xd7 l p$ cross section and a length $ l c$. The initial macroscale cell dimensions in the radial and axial directions are chosen to be equal to the pore size $ l p$, i.e., $ \Delta r = \Delta z = l p$, to guarantee that each macroscopic cell will have exactly one solid element and, therefore, correspond to one foam pore. By definition, the microcell has the same mass and volume $ V pore = l p 2 l c$ at its parent macroscale cell. In the 2D axially symmetric geometry, the macrocell volume (given by the rotation of the cell $ \Delta r \xd7 \Delta z$ around the z axis by 1 radian) depends on its distance from the z axis, making the length $ l c$ equal to the radius of the center of the macroscale cell. (Foam elements are longer further away from the symmetry axis.)
Evolution in time of the cylinder parameters is described by ordinary differential equations. The homogenization is modeled at the microscale as mass and energy exchange between the cylinder and the ambient plasma. The latter is modeled with density $ \u03f1 pl$, electron $ T pe$, and ion $ T pi$ temperatures and zero flow velocity, see Fig. 1(b). The assumption of zero velocity of the ambient plasma is a major simplification of the model as in reality the ablated plasma escapes the cylinder with a large velocity and may collide with flows arriving from neighboring cells. The flows are eventually damped and thermalized. We assume that this relaxation process due to ion–ion collisions or turbulence is faster than the time of foam homogenization. This hypothesis has to be revised when considering foams with large pores.
The motion of the cylinder boundary is controlled by the radial expansion and mass ablation as described in Sec. IV B. A fraction of the laser power absorbed at the surface of the cylinder and a fraction of the thermal heat flux from the ambient plasma drive the ablation process. This surface energy deposition is responsible for the acceleration of ablated mass, thus effectively converting the absorbed energy into a kinetic energy of accelerated particles (electrons and ions). Due to collisions with the background plasma, the kinetic energy of the ablated vapors is eventually converted into a plasma thermal energy. Within our twophase model, this threestep process is instantaneous and corresponds to a continuous conversion of the ablated mass and energy flux into a homogeneous ambient plasma with increasing mass and ion and electron internal energy. This assumption of continuous energy and mass exchange between two phases is physically justified since the ambient plasma is hot and the ablation flow is subsonic.
A. Laser scattering and absorption
When the size of foam solid elements is comparable to the laser wavelength, the absorption efficiency depends strongly on the shape of the structure and its orientation with respect to the laser polarization. In the Mie theory,^{24} the interaction of a plane electromagnetic wave with a subwavelength solid obstacle is characterized by the efficiency factors $ Q abs$ and $ Q sca$, which are a ratio of the absorption and scattering cross section to the geometrical cross section. A detailed study of the laser interaction with an inhomogeneous expanding cylinder under the conditions relevant to the laser interaction with foams is presented in the companion paper,^{25} where the efficiency factors and the rate of cylinder ablation/expansion were analyzed by an analytical approach and by numerical simulations performed with a particleincell code. In the microscale model, we use a simplified version for laser absorption and scattering by taking constant values for the efficiency factors averaged over the angle of incidence and laser polarization: $ Q abs \u2243 0.5 \u2212 1.5$ and $ Q sca \u2243 1 \u2212 4$. If the efficiency factors were calculated from the cross section of the overcritical part of the foam element, i.e., according to the geometrical approximation used in the existing foam models, the absorption factor would be equal to $ Q abs = 2.0 \u2212 2.3$, overestimating its actual value.
B. Energy fluxes
C. Cylinder ablation and expansion
A more accurate form of Eq. (23) can be obtained from the momentum conservation of the expanding cylinder. On the right hand side, there will be two more terms accounting for the pressure of ablated vapors and the pressure of ambient plasma. Both of them slow down the expansion. We found, however, that these effects of counterpressure are relatively small in the parameter domain of interest, and these are omitted in the present version of the microscale model.
D. Energy equations
E. Equation of state for the microscopic model
The cold dielectric cylinders forming the foam are scattering the laser radiation. Absorption is forbidden as a laser photon energy is smaller than the energy gap. As laser intensity increases, the 2photon absorption is activated. This is a very fast process, and the cylinder becomes opaque when the free electron density becomes greater than the critical density. The ionization to $ Z \u223c 0.1$ proceeds on a subpicosecond timescale when the laser intensity exceeds $ 10 12 \u2212 10 13$ W/cm^{2}, and it provides a population of free electrons large enough to make the collisional ionization the dominant effect. The thermal (collisional) ionization of the foam solid elements can be approximated by expression $ Z cyl \u2243 2 T cyl / \chi H$ for $ Z cyl \u2264 Z max$, where $ \chi H \u2243 13.6$ eV is the ionization potential of hydrogen.^{13} Such an approximation is sufficient for modeling the laser–foam interaction of laser pulses with duration exceeding 100 ps. For laser pulses of shorter duration or lower intensity, a dynamic equation describing timedependent foam ionization^{20,36} can be added to the model.
F. Microscale and macroscale connection
G. Summary of the algorithm
One macroscale time step of the full computation procedure for the hybrid model can be summarized as follows:

Compute the macroscale heat conductivity (33) in all macroscale cells.

Compute the laser absorption coefficient $ K tot$ (15) from microscale variables.

Compute the laser power deposition (16) in every macroscale cell by the ray tracing routine.

Compute the microscale model in the cells that are in the intermediate homogenization state. This consists of solving 6 ordinary differential equations (21), (22), (23), (27), (28), and (29) by using a fourthorder Runge–Kutta method with a reduced time step, so that we use multiple microscale time steps for each macroscale time step.

Compute the macroscale model without the heat conduction. For the fully homogenized cells, this model consists of the hyperbolic part of Eqs. (1)–(4) without the heat conduction in the electron energy Eq. (4). For the cells in the intermediate state, the model consists of two equations to update the macroscale electron and ion energies (35) and (36).

For the cells, which have been fully homogenized at the given time step, increase the macroscopic internal electron and ion energies by the increments (38) and (39).
The microscale model contains seven empirical parameters those are summarized in Table I along with their ranges of the recommended values. They control the thermal and laser energy deposition in solid element in the pore and its partition during the homogenization process. The values for these parameters are either chosen from detailed singlepore kinetic simulations^{25} or guided from experimental data as discussed in Sec. VI. The model is tested first in singlecell simulations as discussed in Sec. V, and then integrated simulations are compared with the data available from several experiments.
$ Q abs = 0.5 \u2212 1.5$  Laser absorption efficiency factor of the cylinder (7) 
$ Q sca = 1 \u2212 4$  Laser scattering efficiency factor of the cylinder (7) 
$ f lim = 0.01$  Limiting factor in the expression for the local electron heat flux (18) 
$ \zeta las \u2208 [ 0 , 1 ]$  Fraction of the absorbed laser energy used for bulk cylinder heating (19) 
$ \zeta th \u2208 [ 0 , 1 ]$  Fraction of the heat flux used for the bulk cylinder heating (19) 
$ \xi a = 0.1 \u2212 0.5$  Coefficient controlling the laserdriven ablation velocity (26) 
$ \zeta i = 0.6$  Fraction of energy of ablated plasma deposited to ions (29) 
$ Q abs = 0.5 \u2212 1.5$  Laser absorption efficiency factor of the cylinder (7) 
$ Q sca = 1 \u2212 4$  Laser scattering efficiency factor of the cylinder (7) 
$ f lim = 0.01$  Limiting factor in the expression for the local electron heat flux (18) 
$ \zeta las \u2208 [ 0 , 1 ]$  Fraction of the absorbed laser energy used for bulk cylinder heating (19) 
$ \zeta th \u2208 [ 0 , 1 ]$  Fraction of the heat flux used for the bulk cylinder heating (19) 
$ \xi a = 0.1 \u2212 0.5$  Coefficient controlling the laserdriven ablation velocity (26) 
$ \zeta i = 0.6$  Fraction of energy of ablated plasma deposited to ions (29) 
V. SINGLE PORE SIMULATIONS
The microscale model is tested for an isolated foam pore with a welldefined continual energy input given by the laser of intensity of $ 10 14$ W/cm^{2}. The microscale Eqs. (21), (22), (23), (27), (28), and (29) are solved until the full pore homogenization. The interaction with neighboring cells is neglected, and the role of free parameters listed in Table I is investigated. The foam and laser parameters are chosen according to the experiment:^{5} the average density $ \u03f1 \xaf = 10$ mg/cm^{3}, pore size $ l p = 2 \u2009 \mu $m, solid density $ \u03f1 s = 1$ g/cm^{3}, material is plastic with average charge $ Z = 3.85$ and mass number $ A = 7.2$, and laser wavelength $ \lambda = 351$ nm.
The free parameters of the model determine the regime of operation. Some of the parameters were set to the appropriate value obtained from results of PIC simulations for a single foam pore.^{25} The appropriate value of absorption factor $ Q abs$ averaged over S and P polarizations is in the range $ 0.5 \u2212 1.5$ according to the Mie theory. Kinetic simulations narrow down the value to $ Q abs = 0.75$, which is what we use for the simulations shown in this section. Increase of $ Q abs$ results in slowing down the ionization front velocity but decreases the homogenization time of an individual pore. The ionelectron energy partition parameter $ \zeta i$ is set to 0.6 according to the kinetic simulations. The thermal flux limiting factor $ f lim$ (18) is set to 0.01 following the kinetic simulations. Such a low value is explained by suppression of the heat flux due to a high electron collision frequency in the cold cylinder. The value for the parameter $ \xi a = 0.5$ is chosen by comparing the ablation velocity and the incident laser fluxes.
Parameters $ \zeta las$ and $ \zeta th$ control the regime of pore homogenization—the homogenization time and the amount of mass transferred from cylinder to the ambient plasma. Three characteristic regimes can be defined: expansion dominated regime, combined expansion–ablation regime, and ablation dominated regime. The results of single pore simulations are shown in Fig. 4.
A. Expansion dominated regime
The expansion regime is achieved by setting $ \zeta las = 1$. The results of simulations are shown in Fig. 4 with green lines. This regime corresponds to an efficient laser absorption and bulk cylinder heating, similarly to the other subgrid models.^{15,16,20} Qualitatively, this regime is comparable to the kinetic simulations^{25} for P polarization or to the model by Belyaev et al.^{20} applied to cylindrical foam elements, as they all correspond to isothermal expansion. However, they differ in the homogenization times due to vastly different models for the laser absorption in the cylinder.
This regime has a very short homogenization time of $ t h = 7$ ps, mainly because of the fast cylinder heating and the steep increase in expansion velocity. As expected for the case of bulk heating, ablation is suppressed, the cylinder mass remains unchanged, and the density decreases with time. The conversion of the cylinder kinetic energy into the ion thermal energy takes place at the end of the homogenization stage, but it is not shown in these results because it is part of the macroscale model. The expansion regime is relevant for very lowdensity foams with thin solid elements, where the assumption of volumetric absorption is justified.
B. Combined regime
The combined ablation–expansion regime is attained by setting $ \zeta las = 0$, i.e., by switching the laser absorption to surface deposition and cylinder ablation, and by using $ \zeta th = 0.1 \u2212 1.0$. We will discuss the results for $ \zeta th = 0.5$, the equal distribution of the local heat flux between cylinder bulk heating and ablation. As shown in Fig. 4 with red lines, this regime is characterized by a smaller expansion velocity and a longer homogenization time $ t h = 20$ ps. It is in good agreement with the kinetic simulations^{25} for the case of S polarization in terms of the fraction of ablated mass (up to 75%) and the laser absorption efficiency (up to 10%). The longer time of pore homogenization is result of the fact that the characteristic ablation velocity is significantly smaller than the expansion velocity.
The most important features of the original simulations^{25} are also well replicated by our model: the large predicted discrepancy in the electron and ion temperatures of the plasma and the cylinder core remaining relatively cold despite being directly exposed to the hightemperature plasma. The cylinder temperature is initially below the threshold for phase transition, which effectively delays the start of expansion by approximately 10 ps. It increases slowly due to the heat flux from the ambient plasma but remains in the range of a few tens of electronvolts. The ablation velocity increases at later times when the cylinder density decreases due to expansion. A jump of ablation velocity around 18 ps is caused by the cylinder reaching the critical density. At that moment, the laser energy deposition is switched from ablation to volumetric heating by changing $ \zeta las$ from 0 to 1, and ablation is maintained only by the contribution of the electron heat flux. The volumetric cylinder heating is then responsible for an abrupt increase in cylinder temperature and expansion velocity, accelerating the equilibration of cylinder and plasma densities.
It is important to note the high ion temperature of the ambient plasma in the combined and ablation dominated regimes, which is significantly higher than the electron temperature. This ion overheating is a consequence of the direct laser energy transfer to ions in the ablation process. It is similar to the mechanism of the ion heating due to the kinetic energy dissipation in the expansion process;^{20} however, it is more efficient as less energy is spent on the cylinder heating. The ion heating is the major specificity in the laser interaction with structured materials, which results in a more effective laser energy deposition compared to a homogeneous media.
C. Ablation dominated regime
The ablation regime is achieved by further suppression of bulk cylinder heating by setting $ \zeta las = 0$ and $ \zeta th < 0.1$. Here, we discuss the results for $ \zeta th = 10 \u2212 3$. As shown in Fig. 4 with blue lines, this choice of parameters results in freezing of cylinder expansion, relatively small laser absorption, slow ablation resulting in a consistent decrease in the cylinder radius, and a very long homogenization time. The process ends when all cylinder mass is ablated. The threestep conversion of laser energy into the flow of ablated mass contributes to strong electron and ion heating of the ambient plasma. All these properties lead to the slowest propagation of the ionization front in full scale simulations, as is shown in Sec. VI.
This regime could be appropriate for description of dense foams with thick solid elements where the complete inhibition of cylinder heating can be justified. It also shows the flexibility of our model and the possibility of adopting it to different materials and foam structures.
VI. INTEGRATED SIMULATIONS OF LASERDRIVEN FOAM IONIZATION
The hybrid ablation–expansion microscale model has been implemented in the ALE code PALE^{26} and in the Eulerian code FLASH.^{27–29} In this section, we show the performance of the hybrid model when applied to realistic conditions of the laser–foam interaction. We chose two different experimental setups to cover a variety of different foam properties (average densities) and laser parameters (laser intensities and pulse temporal shapes). The simulations presented in this paper were performed with PALE in the 2D cylindrical configuration. Results obtained from the hybrid model in the FLASH code are similar and not shown in this paper.
A. Shenguang III laser experiment
The first simulation setup was chosen to match the experiments with foam targets at the Shenguang III Prototype (SGIIIP) laser facility.^{6} The targets were made of a $ 800 \u2009 \mu $m thick TMPTA foam of average density of 10 mg/cm^{3}. They were symmetrically irradiated from two sides by laser beams at wavelength $ \lambda = 351$ nm and total energy 3.2 kJ. We consider here a half of the original target of thickness $ 400 \u2009 \mu $m irradiated from one side. The density of solid elements is $ \u03f1 s = 1$ g/cm^{3}, the average pore size is $ l p = 2 \u2009 \mu $m, and the ion charge to mass ratio is $ Z / A = 3.85 / 7.2$. The multibeam setup is approximated by a single laser beam of equivalent parameters, as described in Ref. 6. This laser beam has a fourth order superGaussian spatial profile with a radius of $ 350 \u2009 \mu $m and a trapezoidal temporal shape with 150 ps rise time, 1 ns constant, and 150 ps fall as shown with a yellow shadow in Fig. 6. The maximum intensity is $ 2.1 \xd7 10 14$ W/cm^{2}. The measured velocity of propagation of the ionization front inside the foam is about $ 0.33 \u2009 \mu $m/ps.
Our hybrid ablation–expansion model has several empirical parameters, which are summarized in Table I. Two parameters are set according to the singlepore simulations: $ \zeta i = 0.6$ and $ f lim = 0.01$. Variation of other five control parameters allows us to cover a large range of interaction regimes. The following characteristics of the laser–foam interaction are considered: velocity of propagation of the ionization front in the foam, the ratio of the ion to electron temperature, and the efficiency of laser absorption and reflection. Table II summarizes how variation of parameters $ \zeta las , \zeta th , \xi a , Q abs$, and $ Q sca$ influences the characteristics of the laser–foam interaction. The ionization front velocity is computed from the burnthrough time, that is, the time when the 400 $\mu $m thick foam is fully ionized. The ion and electron temperatures are averaged in space over a region with electron temperature higher than 100 eV and over the whole simulation time. We show data only for absorption and reflection in Table II, the fraction of laser light transmitted through the target can be calculated as $ \eta tr = 100 % \u2212 \eta abs \u2212 \eta refl$. Colors of regimes (black for homogeneous, green for expansion, red for combined, and blue for ablation regimes) correspond to the colors of lines showing these regimes in Fig. 6. The reference (best performing) interaction regime is marked by asterisk and is repeated in all parts (a)–(d) of the table for easier comparison.
.  Regime .  $ \zeta las$ .  $ \zeta th$ .  $ \xi a$ .  $ Q abs$ .  $ Q sca$ .  $ v ioniz$ ( $\mu $m/ps) .  $ T i$ (keV) .  $ T e$ (keV) .  $ \eta abs$ (%) .  $ \eta refl$ (%) . 

(a)  Homogeneous  ⋯  ⋯  ⋯  ⋯  ⋯  1.08  0.57  0.85  93.7  ⋯ 
Expansion  1  0.0  0.0  0.75  3.0  0.87  0.77  1.02  97.1  0.7  
Combined  0  0.5  0.1  0.75  3.0  0.58  2.02  1.13  94.4  3.8  
Combined  0  0.1  0.1  0.75  3.0  0.51  2.63  1.04  95.5  4.5  
*  Ablation  0  0.01  0.1  0.75  3.0  0.39  3.13  1.06  93.4  6.6 
Ablation  0  0.005  0.1  0.75  3.0  0.35  3.59  1.01  91.8  8.1  
Ablation  0  0.001  0.1  0.75  3.0  0.28  4.73  0.96  88.7  11.1  
(b)  Ablation  0  0.01  0.3  0.75  3.0  0.66  1.59  1.15  89.7  3.8 
Ablation  0  0.01  0.2  0.75  3.0  0.53  2.28  1.12  91.7  5.3  
*  Ablation  0  0.01  0.1  0.75  3.0  0.39  3.13  1.06  93.4  6.6 
(c)  Ablation  0  0.01  0.1  0.5  3.0  0.42  2.92  1.07  90.5  7.8 
*  Ablation  0  0.01  0.1  0.75  3.0  0.39  3.13  1.06  93.4  6.6 
Ablation  0  0.01  0.1  1.0  3.0  0.37  3.35  1.04  94.1  5.7  
Ablation  0  0.01  0.1  1.5  3.0  0.33  3.70  1.02  95.4  4.6  
(d)  Ablation  0  0.01  0.1  0.75    0.60  2.15  1.02  87.7  0 
Ablation  0  0.01  0.1  0.75  2.0  0.43  2.93  1.07  92.1  5.2  
*  Ablation  0  0.01  0.1  0.75  3.0  0.39  3.13  1.06  93.4  6.6 
Ablation  0  0.01  0.1  0.75  4.0  0.36  3.30  1.04  92.3  7.6 
.  Regime .  $ \zeta las$ .  $ \zeta th$ .  $ \xi a$ .  $ Q abs$ .  $ Q sca$ .  $ v ioniz$ ( $\mu $m/ps) .  $ T i$ (keV) .  $ T e$ (keV) .  $ \eta abs$ (%) .  $ \eta refl$ (%) . 

(a)  Homogeneous  ⋯  ⋯  ⋯  ⋯  ⋯  1.08  0.57  0.85  93.7  ⋯ 
Expansion  1  0.0  0.0  0.75  3.0  0.87  0.77  1.02  97.1  0.7  
Combined  0  0.5  0.1  0.75  3.0  0.58  2.02  1.13  94.4  3.8  
Combined  0  0.1  0.1  0.75  3.0  0.51  2.63  1.04  95.5  4.5  
*  Ablation  0  0.01  0.1  0.75  3.0  0.39  3.13  1.06  93.4  6.6 
Ablation  0  0.005  0.1  0.75  3.0  0.35  3.59  1.01  91.8  8.1  
Ablation  0  0.001  0.1  0.75  3.0  0.28  4.73  0.96  88.7  11.1  
(b)  Ablation  0  0.01  0.3  0.75  3.0  0.66  1.59  1.15  89.7  3.8 
Ablation  0  0.01  0.2  0.75  3.0  0.53  2.28  1.12  91.7  5.3  
*  Ablation  0  0.01  0.1  0.75  3.0  0.39  3.13  1.06  93.4  6.6 
(c)  Ablation  0  0.01  0.1  0.5  3.0  0.42  2.92  1.07  90.5  7.8 
*  Ablation  0  0.01  0.1  0.75  3.0  0.39  3.13  1.06  93.4  6.6 
Ablation  0  0.01  0.1  1.0  3.0  0.37  3.35  1.04  94.1  5.7  
Ablation  0  0.01  0.1  1.5  3.0  0.33  3.70  1.02  95.4  4.6  
(d)  Ablation  0  0.01  0.1  0.75    0.60  2.15  1.02  87.7  0 
Ablation  0  0.01  0.1  0.75  2.0  0.43  2.93  1.07  92.1  5.2  
*  Ablation  0  0.01  0.1  0.75  3.0  0.39  3.13  1.06  93.4  6.6 
Ablation  0  0.01  0.1  0.75  4.0  0.36  3.30  1.04  92.3  7.6 
Analysis of the table shows that model of a homogeneous material with an equivalent average density strongly overestimates the ionization front velocity and underestimates the ion to electron temperature ratio. Suppression of the bulk cylinder heating via decreasing $ \zeta th$ is the key factor for the slowing down the ionization front and increasing laser energy deposition. Decreasing $ \zeta th$ from 0.5 to 0.001, see Table II(a), results in a decrease in the ionization front velocity by a factor of 2 and in an increase in the ion to electron temperature ratio by a factor of 5. This is related to the increase in the pore homogenization time as observed in single pore simulations. One can also see an effect on the laser reflectivity and absorption, which is relatively weak. The amount of backscattered light is proportional to the time needed to destroy the foam structure. For this reason, a higher reflectivity $ \u223c 10 %$ is observed for the regimes with a slower ionization front propagation. The absorption efficiency is greater than $ 90 %$ for most cases.
The same trend is observed while varying the factor $ \xi a$, see Table II(b), which is controlling the laserdriven component of the ablation velocity. The value $ \xi a = 0.5$ suggested in kinetic simulations in the case of S polarization results in a too fast ionization front propagation even with a low value of the heat flux parameter $ \zeta th = 0.001$. Results compatible with the experimentally measured ionization front velocity are obtained by reducing $ \xi a$ to 0.1. The ablation velocity depends on laser polarization and angle of incidence on the cylinder, and the case of S polarization corresponds to the most efficient ablation. (It is slower for a nonpolarized laser). The kinetic simulations and onepore tests were performed for the laser intensity of $ 10 14$ W/cm^{2} and for the absorption coefficient corresponding to S polarization. In the integrated simulation, the homogenization process starts at much lower laser intensity and laser is nonpolarized. These factors explain the necessity of using smaller values of ablation velocity, $ \xi a \u2243 0.1$, in the integrated simulations.
One can also observe the effect of laser absorption and scattering in Table II, parts (c) and (d). An increase in either $ Q sca$ or $ Q abs$ leads to a decrease in ionization front velocity and an increase in the ion to electron temperature ratio. Both factors contribute to the increase in laser extinction, affecting the width of the absorption region. In the followup simulations, we use $ Q sca = 3$ and $ Q abs = 0.75$, which correspond to theoretical values averaged over the laser polarization.
Inclusion of the foaminduced laser scattering is crucial for the foam simulations as it reduces the laser penetration and decreases the width of the homogenization zone. It can reduce the front velocity by almost a factor of 2, while constituting less than a 7% energy loss due to backscattering/reflection. Its effect on the laser penetration is shown in Fig. 5, where the laser deposition in target is shown for two cases: without laser scattering, $ Q sca = 0$ (a) and for $ Q sca = 4$ (b), which correspond to the first and fourth line in part (d) of Table II. The laser scattering near the homogenization front also decreases the depth of laser preheat ahead of the ionization front.
We have chosen four representative cases for each of the four regimes marked in color in Table II. These are a homogeneous regime (black), an expansion regime (green), a combined regime (red), and an ablation regime (blue). Figure 6 presents the position of the ionization front as a function of time for each of the four regimes. It is defined in the simulation as 200 eV isoline of the electron temperature. Three color curves show the results for the regimes of the hybrid model. They can be compared with the case of a homogeneous media of the same average density shown in black. Variation of control parameters allows us to reduce the ionization front velocity by a factor of 3. The ablation regimes provide the best agreement with this experiment. The case of $ \zeta th = 0.005$ provides the best result in terms of the propagation speed. However, the case of $ \zeta th = 0.01$ provides a better agreement with the estimated value of average ion temperature, $ T i \u223c Z T e$ (the ion temperature was not measured in this experiment), and seems to be also applicable to the conditions of other experiments with similar foams, as shown in Sec. VI B.
The spatial profiles of the electron and ion macroscale temperatures and of the absorbed laser power at an early time of 300 ps are presented in Fig. 7. Different regimes of the interaction can be compared with the corresponding line cuts along the laser beam axis plotted in Fig. 8. Note that the absorbed laser power is plotted in the logarithmic scale in Fig. 7 and in the linear scale in Fig. 8. The early time has been chosen so that one can see the results for all interaction regimes before the foam is burnthrough. For the homogeneous and expansion regimes, the maximal electron temperature is around 1.4 keV, and the maximal ion temperature is 0.9 and 1.3 keV for the homogeneous and expansion cases, respectively. Ion heating is weak in the homogeneous case as it is related to the collisional energy exchange with electrons, which are heated by the laser. It is slightly stronger for the expansion regime due to inclusion of the expansionrelated ion heating, but it is not as effective as the regimes with ablation because of the shorter pore homogenization times. For the combined and ablation regimes, the maximal electron temperature stays around 1.4 keV, while the maximal ion temperature reaches 9 keV for the combined regime and 14 keV for the ablation regime. The ion temperature considerably decreases at later times for both regimes when foam homogenization finishes; see the averaged temperatures in Table II.
B. PALS experiments
In this section, we present the results of simulations for the laser–foam interaction experiments conducted at the PALS facility.^{21,22,30,37} The targets were made out of a TAC foam C_{24}H_{16}O_{6} with ion charge to mass ratio $ Z / A = 4.54 / 8.73$, pore size $ 2 \u2009 \mu $m, and solid density of 1 g/cm^{3}. Two target average densities are considered: 9.1 mg/cm^{3} of thickness $ 400 \u2009 \mu $m and 4.5 mg/cm^{3} and thickness $ 380 \u2009 \mu $m. The laser pulse of energy 170 J at the wavelength 438 nm (third harmonic of iodine laser) was focused to a focal spot of radius $ 150 \u2009 \mu $m with a Gaussian temporal profile of the duration of 320 ps FWHM. The simulation starts 500 ps before the laser maximum. These parameters correspond to the maximum intensity of $ 1.1 \xd7 10 15$ W/cm^{2}.
The plasma selfemission from the ionization front is measured with an xray streak camera. The ionization front velocity for the 4.5 mg/cm^{3} foam is $ 1.3 \u2009 \mu $m/ps, which corresponds to the front breakout time of 500 ps at the rear side of the target. In the case of the 9.1 mg/cm^{3} target, the determination of the ionization front velocity is more complicated because the target was not burnthrough by the laser. The heat wave, however, reached the rear target side at $ 1.3 \u2212 1.4$ ns, resulting in average velocity of $ 0.33 \u2212 0.37 \u2009 \mu $m/ps of the laserdriven heat front.
The original data from the 2007 experiment^{30} have been recently reprocessed using the approach described in Ref. 38 to determine the ion temperature from the Doppler broadening of the intercombination He $ \alpha y$ line of chlorine. In the case of the 9.1 mg/cm^{3} TAC foam targets, the calculated value of the time and spatially averaged ion temperature is approximately 3 keV, corresponding to the ratio of ion to electron temperature of $ 3 \u2212 4$.
Figure 9 and Tables III and IV show the results of integrated simulations and variation of control parameters for the experiments conducted at PALS laser facility.^{37} They confirm the choice of parameters suggested in Sec. VI A and their validity for other experiments. For 9.1 mg/cm^{3} foam, the optimal reference regime is the ablation one as for the case of SGIIIP experiment in Sec. VI A, while for 4.5 mg/cm^{3} foam, the combined regime is the optimal one. This is due to a thinner solid elements in the 4.5 mg/cm^{3} foam, resulting in earlier transition from laser ablation to expansion. The reference regimes are marked by the asterisk in the tables.
Regime .  $ \zeta las$ .  $ \zeta th$ .  $ \xi a$ .  $ Q abs$ .  $ Q sca$ .  $ v ioniz$ ( $\mu $m/ps) .  $ T i$ (keV) .  $ T e$ (keV) .  $ \eta abs$ (%) .  $ \eta refl$ (%) . 

Homogeneous  ⋯  ⋯  ⋯  ⋯  ⋯  1.47  0.35  0.46  85.7  0.17 
Expansion  1  0  0  0.75  3.0  1.21  0.77  1.21  89.6  1.0 
Combined  0  0.1  0.1  0.75  3.0  0.75  4.27  1.20  95.3  4.4 
* Ablation  0  0.01  0.1  0.75  3.0  0.36  4.57  0.88  93.2  6.8 
Regime .  $ \zeta las$ .  $ \zeta th$ .  $ \xi a$ .  $ Q abs$ .  $ Q sca$ .  $ v ioniz$ ( $\mu $m/ps) .  $ T i$ (keV) .  $ T e$ (keV) .  $ \eta abs$ (%) .  $ \eta refl$ (%) . 

Homogeneous  ⋯  ⋯  ⋯  ⋯  ⋯  1.47  0.35  0.46  85.7  0.17 
Expansion  1  0  0  0.75  3.0  1.21  0.77  1.21  89.6  1.0 
Combined  0  0.1  0.1  0.75  3.0  0.75  4.27  1.20  95.3  4.4 
* Ablation  0  0.01  0.1  0.75  3.0  0.36  4.57  0.88  93.2  6.8 
Regime .  $ \zeta las$ .  $ \zeta th$ .  $ \xi a$ .  $ Q abs$ .  $ Q sca$ .  $ v ioniz$ ( $\mu $m/ps) .  $ T i$ (keV) .  $ T e$ (keV) .  $ \eta abs$ (%) .  $ \eta refl$ (%) . 

Homogeneous  ⋯  ⋯  ⋯  ⋯  ⋯  2.70  0.30  0.41  43.2  0.7 
Expansion  1  0  0  0.75  3.0  1.92  0.76  1.0  55.1  10.4 
* Combined  0  0.1  0.1  0.75  3.0  1.31  4.6  1.1  63.1  11.4 
Ablation  0  0.01  0.1  0.75  3.0  1.02  7.1  1.1  76.6  12.8 
Regime .  $ \zeta las$ .  $ \zeta th$ .  $ \xi a$ .  $ Q abs$ .  $ Q sca$ .  $ v ioniz$ ( $\mu $m/ps) .  $ T i$ (keV) .  $ T e$ (keV) .  $ \eta abs$ (%) .  $ \eta refl$ (%) . 

Homogeneous  ⋯  ⋯  ⋯  ⋯  ⋯  2.70  0.30  0.41  43.2  0.7 
Expansion  1  0  0  0.75  3.0  1.92  0.76  1.0  55.1  10.4 
* Combined  0  0.1  0.1  0.75  3.0  1.31  4.6  1.1  63.1  11.4 
Ablation  0  0.01  0.1  0.75  3.0  1.02  7.1  1.1  76.6  12.8 
The velocity of ionization front propagation is influenced by the laser intensity profile. The ionization front moves slowly in the beginning of the laser pulse, see Fig. 9. We compute the average ionization front velocity from the front position at $ t = 200$ ps and from the final burnthrough time. The velocities from simulations are $ 0.36 \u2009 \mu $m/ps for the 9.1 mg/cm^{3} target and $ 1.31 \u2009 \mu $m/ps for the 4.5 mg/cm^{3} target and correspond reasonably well to the values from experiment, $ 0.33 \u2212 0.37$ and $ 1.3 \u2009 \mu $m/ps for the 9.1 and 4.5 mg/cm^{3} targets, respectively. The ratio of ion to electron temperature in the simulation is around 5 for 9.1 mg/cm^{3} target, which is comparable to the ratio $ T i / T e \u223c 3 \u2212 4$ measured in the experiment. The difference can be explained by a longer time of the experimental measurement (several ns) during which the plasma cools down, and the temperature difference is reduced. The calculated temperature ratio is $ T i / T e \u2243 4.2$ for the combined regime of the 4.5 mg/cm^{3} foam.
VII. CONCLUSIONS
We present a novel approach to the numerical modeling of the laser interaction with undercritical foams. It combines a macroscopic hydrodynamic description of the homogenized cells with microscopic description of the cells undergoing the homogenization process. The nonhomogenized cells contain two phases: solid structural elements represented by homogeneous cylinders and ambient lowdensity plasmas. The mass exchange between these phases is due to the ablation of solid elements induced by laser absorption and heat flux from the ablated plasma. Laser absorption in the homogenized cells is due to the inverse bremsstrahlung, while absorption and scattering on structural elements of subwavelength size are calculated from the Mie theory.
The microscale model combines selfsimilar expansion of structural foam elements with their surface ablation. Competition between these two processes defines energy partition between electrons and ions in the ambient plasma, the ionization front velocity, and the rate of laser energy deposition. Dominance of ablation leads to slowing down of the ionization front and to an increase in the internal energy of the downstream plasma.
The microscale model is controlled by several empirical parameters that are chosen by comparison with kinetic singlepore simulations and with experiments. Three regimes of operation are identified in singlepore tests: expansion dominated regime, combined expansion–ablation regime, and ablation dominated regime. The latter applies to foams with a size of structural elements larger than the laser skin layer, while the combined regime is more suitable for very lowdensity foams with very thin structural elements. A variation of two parameters $ \zeta las$, $ \zeta th$ allows for a continuous transition between the three characteristic regimes. Taking into account the laser scattering on foam structural elements is crucial part of the model as it reduces the laser penetration and decreases the width of the homogenization zone.
Performance of the hybrid twoscale model is evaluated by comparing the simulation results with the experiments at the SGIIIP and PALS laser facilities. Two major characteristics are the ionization front velocity and the ion to electron temperature ratio in the downstream plasma. Optimal set of model parameters is chosen by the comparison of these characteristics to the SGIIIP experiment, and it is successfully used with minor modifications for modeling other experiments. This is a demonstration that the model correctly describes the physics of laser interaction with foams, and it can be used for predictive simulations.
ACKNOWLEDGMENTS
This research was supported by the projects ADONIS (Advanced research using high intensity laser produced photons and particles, CZ.02.1.01/0.0/0.0/16_019/0000789), by High Field Initiative (HiFI, CZ. $ 02.1.01 / 0.0 / 0.0 / 15 _ 003 / 0000449$), and by CAAS (Centre of Advanced Applied Sciences, CZ. $ 02.1.01 / 0.0 / 0.0 / 16 _ 019 / 0000778$), all from European Regional Development Fund. This research was also supported in part by the Czech Technical University in Prague project SGS22/184/OHK4/3T/14. We also acknowledge partial funding via EUROfusion Enabling research project AWP21ENR01CEA02 “Advancing shock ignition for directdrive inertial fusion,” within the framework of the EUROfusion Consortium, funded the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200—EUROfusion). Views and opinions expressed are, however, those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Additional support from MEYS CR (Project No. 9D22001) is acknowledged.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Lubomir Hudec: Conceptualization (supporting); Data curation (equal); Investigation (equal); Software (lead); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Antoine Gintrand: Conceptualization (supporting); Investigation (equal); Software (supporting); Validation (equal). Jiri Limpouch: Funding acquisition (equal); Investigation (equal); Supervision (supporting); Validation (equal). Richard Liska: Investigation (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (supporting). Sviatoslav Shekhanov: Conceptualization (supporting); Investigation (equal); Software (supporting); Validation (equal). Vladimir T. Tikhonchuk: Conceptualization (lead); Formal analysis (equal); Investigation (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Stefan Weber: Funding acquisition (equal); Investigation (equal); Supervision (supporting); Validation (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.