A sub-grid 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 sub-wavelength 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 self-similar 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.

Low-density 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 conditions7 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 interaction11,12 is computationally expensive due to the necessity to spatially resolve the density differences in the foam micro-structure. Also, it cannot account for all micro-physics 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 foam13–15 are limited to a description of an idealized one-dimensional 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 sub-grid (two-scale) 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 sub-grid micro-scale calculation is applied to the non-homogenized cells, and it is stopped once the foam cell is fully homogenized. This way the sub-grid approach describes a continuous transformation of a cold foam into a hot plasma.

The first sub-grid model by Velechovský et al.16 is based on a numerical calculation of laser absorption and expansion of planar-shaped solid foam elements, following the theory proposed by Gus'kov et al.13 The model proposed in Refs. 14, 17, and 18 describes the laser-driven 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 30 50 μm, where the phase of the expansion of solid foam elements is not considered as important. However, its application to foams with micrometer-size pores is problematic. This model was implemented in the radiation hydrodynamic code MULTI-FM 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 self-similar isothermal solid element expansion and applied it to foams with small pores 1 μm and with small average density 2 mg/cm3. It is the first model to introduce the foam-specific 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 sub-wavelength-sized objects needs special attention. The wave-based description according to the Mie theory24 and detailed kinetic simulations25 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 wire-like cylindrical solid elements by including the processes of ionization, heating, expansion, and ablation in a self-consistent microscopic model and by using the theory of an electromagnetic wave interaction with sub-wavelength objects to describe laser absorption and scattering. In our micro-scale 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 self-similar 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 sub-grid module in the hydrodynamic codes PALE26 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 small-pore 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.

The main idea of a two-scale 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 macro-scale, 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 micro-scale. Both models describe the dynamics of the exact same volume and mass of the fluid.

The micro-scale contains an additional information about the subdivision of the mass and energy inside the pore, whereas the macro-scale only uses the values averaged over the pores. The micro-scale computes the dynamics of each cell separately, and, therefore, it relies on the macro-scale 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 micro-scale 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 macro-scale computation.

The hydrodynamic model on the macro-scale is given by the two-temperature Euler equations written in Lagrangian coordinates for the plasma density ϱ, flow velocity U, and internal energies of ions and electrons, ε i and ε e,
1 ϱ d ϱ d t = U ,
(1)
ϱ d U d t = ( p e + p i ) ,
(2)
ϱ d ε i d t = p i U + G e i ( T e T i ) + ( κ i T i ) ,
(3)
ϱ d ε e d t = p e U + G i e ( T i T e ) + ( κ e T e ) I las .
(4)
These equations are supplemented by the plasma-specific terms describing the ion and electron heat conductivity, laser energy deposition, and electron–ion temperature relaxation. The pure hydrodynamics is given by the first terms on the right hand side of equations (1)–(4). The electron-ion temperature relaxation is given by the second terms on the right hand side of the ion and electron internal energy Eqs. (3) and (4). The heat conductivity is given by the third terms and the laser source by the fourth term on the right hand side of the electron internal energy Eq. (4). The electron/ion pressures p e , i and temperatures T e , i are connected to primary conservative variables by the ideal gas equation of state.

The hydrodynamic Eqs. (1)–(4) are solved in two-dimensional (2D) cylindrical r, z geometry. The code PALE (Prague Arbitrary Lagrangian Eulerian code)26 solves them in the Lagrangian form, and the code FLASH27 solves them in their Eulerian form.

The macro-scale 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 micro-scale. Initially all cells are in the cold state—the solid elements are at room temperature, and they are surrounded by vacuum or a low-density gas. When the laser or heat wave reaches the cold state cell, its state turns into the intermediate state described by our new micro-scale model. In these cells, the micro-scale and macro-scale models run in parallel. Some aspects of the foam internal dynamics (including the laser deposition and electron-ion energy redistribution) are handled by the micro-model (when active) and are, thus, omitted from the macro-scale to avoid including the same terms twice. We assume that the macro-scale 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 macro-scale cell when the micro-scale model is active (see more details about the modifications at the macro-scale 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 micro-scale model is turned off, and the corresponding macro-scale 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 = K 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.

Each macro-scale computation cell has its corresponding micro-scale cell. As schematically shown in Fig. 1(a), the micro-scale cell has a form of a parallelepiped with a square l p × l p cross section and a length l c. The initial macro-scale cell dimensions in the radial and axial directions are chosen to be equal to the pore size l p, i.e., Δ r = Δ 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 micro-cell has the same mass and volume V pore = l p 2 l c at its parent macro-scale cell. In the 2D axially symmetric geometry, the macro-cell volume (given by the rotation of the cell Δ r × Δ 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 macro-scale cell. (Foam elements are longer further away from the symmetry axis.)

FIG. 1.

Cylinder and plasma region of the micro-scale cell: (a) three-dimensional view of the micro-scale cell; and (b) one-dimensional cut parallel to the parallelepiped face.

FIG. 1.

Cylinder and plasma region of the micro-scale cell: (a) three-dimensional view of the micro-scale cell; and (b) one-dimensional cut parallel to the parallelepiped face.

Close modal
The solid foam element—a cylinder with radius a and length l c—is placed in the center of pore and is surrounded by a low-density background plasma that fills the rest of the pore. (We will refer to it as the ambient plasma.) The cylinder contains initially a solid material with a density ϱ s. The initial radius of the cylinder is defined by the average foam density ϱ ¯ as
a 0 = l p ϱ ¯ / π ϱ s .
The total mass of the micro-scale cell m 0 is divided into the mass of the cylinder and mass of the plasma
m 0 = π a 0 2 ϱ s l c = m cyl + m pl M cell = V pore ϱ ¯ .
(5)
The initial cylinder mass is equal to 0.97 m 0. We put a small fraction 0.03 m 0 in the ambient plasma in order to avoid nonphysical divergences. The micro-scale cell volume V pore is divided into the cylinder volume V cyl and the plasma volume V pl,
V pore = l p 2 l c , V cyl = π a 2 l c , V pl = V pore V cyl = l p 2 l c π a 2 l c .
The cylinder radius a ( t ), mass m cyl ( t ), density ϱ cyl ( t ) = m cyl / ( π a 2 l c ), and temperature T cyl ( t ) are constant in space, see Fig. 1(b). The fluid velocity v cyl inside the cylinder is a linear function of radius and approaches the value of expansion velocity at the cylinder edge r = a.

Evolution in time of the cylinder parameters is described by ordinary differential equations. The homogenization is modeled at the micro-scale as mass and energy exchange between the cylinder and the ambient plasma. The latter is modeled with density ϱ 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 ambient plasma mass increases with time due to ablation and leads to an increase in plasma density,
ϱ pl ( t ) = ( m 0 m cyl ) / V pl .
(6)

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 two-phase model, this three-step 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.

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 sub-wavelength 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 particle-in-cell code. In the micro-scale 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 0.5 1.5 and Q sca 1 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 2.3, overestimating its actual value.

The laser power absorbed and scattered on the cylinder is defined as
P abs = Q abs 2 a l c I las and P sca = Q sca 2 a l c I las .
(7)
Laser rays entering the pore have a chance to be scattered on the cylinder, deflecting their direction by angle θ. This probability of scattering is given by a ratio of the scattered power to the laser power entering in the cell through the surface l p × l c,
p sca ( θ ) = s ( θ ) P sca l p l c I las s ( θ ) [ 1 exp ( Q sca 2 a l p ) ] .
(8)
Such a functional form is chosen in order to limit the probability to the maximum value of 1. The angular distribution of scattered rays s ( θ ) in the plane perpendicular to the cylinder axis obtained from the Mie theory is shown in Fig. 2(a) for a representative case of a homogeneous cylinder of a radius a = 0.32 λ, density ϱ cyl / ϱ cr = 36, and collision frequency ν e / ω = 1, where λ and ω are the laser wavelength and frequency, ϱ cr is the critical density, and ν e is the electron collision frequency. It is peaked in a forward direction in a cone of 25 ° with an approximately isotropic distribution outside the cone and weakly depends on laser polarization. The cumulative scattering function calculated as an arithmetic average over the laser polarizations, s c ( θ ) = 0 θ s ( θ ) d θ , is shown in Fig. 2(b). It is approximated in the code by a two-slope profile (dashed line), which corresponds to a small-angle and a large-angle scattering with approximately the same probability.
FIG. 2.

(a) Angular distribution s ( θ ) of scattered rays for S and P polarization (blue and red, respectively). (b) Cumulative distribution function s c ( θ ) for the average polarization and its two-step approximation.

FIG. 2.

(a) Angular distribution s ( θ ) of scattered rays for S and P polarization (blue and red, respectively). (b) Cumulative distribution function s c ( θ ) for the average polarization and its two-step approximation.

Close modal
Laser absorption in the plasma region is due to the inverse bremsstrahlung. The absorption coefficient is weighted by the ratio of volumes V pl / V pore to account for the fact that the laser encounters the opaque cylinder before it can penetrate the full distance of the pore size, effectively reducing the absorption length in the plasma compared to the macro-scale ray length. This correction results in the following laser absorption rate,
K pl = K ib ( ϱ pl , T pe ) ( 1 π a 2 l c V pore ) ,
(9)
where K ib is computed from the plasma permittivity ϵ as follows:
K ib ( ϱ pl , T pe ) = 2 ω c Im ϵ , ϵ = 1 ω pe 2 ω ( ω + i ν e ) .
(10)
Here, ω pe is the electron plasma frequency and the electron collision frequency ν e given by a standard expression for the electron–ion collisions in the plasma
ν ei = Z pl 2 e 4 ϱ pl ln Λ 3 ( 2 π ) 3 / 2 ϵ 0 2 m e 1 / 2 m i ( k B T pe ) 3 / 2 ,
(11)
where k B is the Boltzmann constant, Z pl is the ion charge in plasma, m e and m i are the electron and ion masses, and ϵ 0 is the vacuum permittivity. This expression corrected by two factors accounting for non-ideal effects. First, the Coulomb logarithm ln Λ is limited by a minimum value of 2; second, the value of ν ei is limited from above by a value estimated from the ab initio calculations of the static electric conductivity in plastic CH2 at electron temperatures 1 10 eV.31 The calculated value of 2 × 10 5 Ω 1 m 1 corresponds to the electron collision frequency of 10 16 s−1, assuming the ion charge Z cyl = 1 and density ϱ s 1 g/cm3. This value of collision frequency, linearly extrapolated to the actual cylinder density, is considered as maximum value of the electron collision frequency
ν e = min { 10 16 ϱ cyl / ϱ s , ν ei } ,
(12)
where collision frequency is in s−1.
The laser absorption process in the cylinder depends on its density. When the cylinder is overcritical, the absorption in the cylinder is described by the effective absorption rate, which is the ratio of the effective absorption cross section to the pore volume,
ϱ cyl > ϱ cr : K cyl = Q abs 2 a l c V pore .
(13)
Once the cylinder becomes sub-critical, the total absorption is calculated as the inverse bremsstrahlung absorption weighted by the volume ratio V pl / V pore,
ϱ cyl < ϱ cr : K cyl = K ib ( ϱ cyl , T cyl ) π a 2 l c V pore .
(14)
The volume ratio determines the average fraction of the laser ray trajectory in the cylinder compared to the pore size. In a sub-critical cylinder, the laser deposition is no longer confined to the vicinity of the critical surface and the absorbed energy is homogeneously distributed over the cylinder volume. The transition from surface to volume absorption is done by setting ζ las = 1 in the definition of energy fluxes in Eq. (19). This also turns off the laser contribution to the ablation flux.
The micro-scale model, thus, provides the total absorption rate per unit length, which is a sum of the contributions from the cylinder and the plasma
K tot = K cyl + K pl .
(15)
The ray-tracing routine evaluates the absorbed laser power P las as a sum of powers deposited by all rays crossing the cell, P las = P ray dep. The power deposited by a ray P ray dep is calculated by using the total absorption coefficient K tot as
P ray dep = P ray ( 1 e K tot L ray ) ,
where P ray is the ray power entering the macro-scale cell and L ray is the length of the ray trace inside the cell. In the micro-scale model, the total absorbed power is divided into two parts: the power deposited to the cylinder P abs = η cyl P las and the power absorbed in plasma P ib = η ib P las, with the partition coefficients defined as
η cyl = K cyl K tot , η ib = K pl K tot , η cyl + η ib = 1.
(16)
The energy flux entering the surface of the cylinder
q cyl = P abs 2 π a l c + q th .
(17)
is a sum of the laser power P abs absorbed on the surface of the cylinder and the local electron heat flux q th from the ambient plasma, calculated as a fraction of the free-streaming limit
q th = f lim Z max ϱ pl m i k B v te ( T p e T cyl ) ,
(18)
where v te = ( k B T pe / m e ) 1 / 2 is the electron thermal velocity in plasma. The energy flux q cyl is divided into a part q heat, which heats bulk of the cylinder and the rest, which is used for the surface ablation of the cylinder. We introduce two separate parameters ζ las and ζ th for splitting of the laser and thermal heat fluxes, and we define the energy flux heating the cylinder
q heat = ζ las P abs 2 π a l c + ζ th q th ,
(19)
and the energy flux causing the ablation
q abl = q abl , las + q abl , th = ( 1 ζ las ) P abs 2 π a l c + ( 1 ζ th ) q th .
(20)
The factor ζ las is set to 1, if the cylinder density is smaller than the critical density, and it is set to 0, if the density of ambient plasma is larger than the critical density.
The cylinder radius evolves in time due to ablation and expansion. Ablation corresponds to a mass removal from the surface with velocity v abl. Expansion corresponds to increase in the cylinder radius with velocity v exp without changing mass. Thus, the cylinder mass loss is defined by the ablation velocity
d m cyl d t = 2 π a l c v abl ϱ cyl = 2 v abl a m cyl , m pl = m 0 m cyl ,
(21)
and the evolution of cylinder radius is due to a combined effect of expansion and ablation
d a d t = v exp v abl .
(22)
We assume a linear profile of the velocity inside the cylinder, which allows us to use the self-similar solution of isothermal expansion,15,32
d v exp d t = 4 3 ε cyl a Θ ( T cyl T exp ) .
(23)
Here, ε cyl is the cylinder specific internal energy and Θ is Heaviside step function that suppresses expansion at low temperatures when the cylinder is not sufficiently ionized. We have chosen T exp = 5 eV as a minimum temperature for plasma expansion, which corresponds to the ionization level Z cyl 1 of solid material, see Sec. IV E.

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 counter-pressure are relatively small in the parameter domain of interest, and these are omitted in the present version of the micro-scale model.

Ablation is considered as a stationary process, and simple models are employed to describe the mass flow normal to the cylinder surface. It is assumed that the inward energy flux q abl (20) is balanced by the outward energy flux of the ablated plasma, including the enthalpy flux, ion kinetic energy, and electron heat flux. In the self-similar model of laser ablation,33,34 it is assumed that laser absorption takes place at the critical density ϱ cr, where the plasma flow velocity v i ( ϱ cr ) is equal to the local sound velocity c 0. Then, the energy balance reads q abl , las = 4 ϱ cr c 0 3, and the mass flow conservation ϱ cr v i ( ϱ cr ) = ϱ cr c 0 = ϱ cyl v abl provides expression for the mass ablation rate m ̇ = ϱ cr v i ( ϱ cr ) and for the velocity of laser ablation,
v abl , las = ξ a ( 2 q abl , las ρ cr 2 ) 1 / 3 2 ϱ cyl Θ ( ϱ cyl ϱ cr ) ,
(24)
where we have introduced a free parameter ξ a 1 accounting for a possible deviation of the cylindrical non-stationary ablation from the ideal planar ablation model. As mentioned above, the laser ablation is stopped when the cylinder density becomes under-critical.
A self-similar model of electron heat-driven ablation has been developed by Betti et al.35 It is supposed that the electron heat flux penetrates the cylinder, and the vapors are ejected with the internal energy ε vap equal to the internal energy of the ambient plasma ε pl. Then, the condition of the energy balance q abl , th = 2 3 ϱ cyl v abl , th ε vap is complemented with the condition ε vap = ε pe + ε pi. The mass ablation rate is defined as m ̇ ϱ cyl v abl , th, and expression for the velocity of thermal ablation reads
v abl , th = ξ a 3 q abl , th 2 ϱ cyl ( ε pe + ε pi ) .
(25)
Numerically, we impose positiveness of the thermal ablation velocity, v abl , th 0. The contributions from the laser (24) and electron heat (25) ablation are summed up and the maximum ablation velocity is limited by the condition v lim = ( q abl / 4 ϱ cyl ) 1 / 3 accounting for the fact that the density of ablated vapors has to be smaller than the cylinder density,
v abl = min { v abl , las + v abl , th , ( q abl / 4 ϱ cyl ) 1 / 3 } .
(26)
With our assumptions of a constant density and linear velocity inside the expanding cylinder (with value v exp at the cylinder surface), we obtain the kinetic energy of the cylinder 2 π 0 a ( ϱ cyl v exp r 2 / a ) d r = 1 4 m cyl v exp 2 by integrating over the volume. The total energy of the cylinder is E cyl = m cyl ( ε cyl + 1 4 v exp 2 ). The cylinder energy E cyl increases due to the laser and heat fluxes (represented by q heat), and it decreases by the energy carried with the ablated mass. The energy balance equation can be written as
d E cyl d t = 2 π a l c q heat + d m cyl d t ( ε cyl + 1 2 v exp 2 ) .
By using Eqs. (23) and (21) for the expansion velocity and the mass ablation rate, we obtain the following ordinary differential equation for the cylinder specific internal energy
d ε cyl d t = 1 2 a v abl v exp 2 2 3 v exp a ε cyl Θ ( T cyl T exp ) + 2 π a l c m cyl q heat .
(27)
Variation in time of the total energy of ambient plasma E pl = m pl ( ε pe + ε pi ) is due to the laser collisional absorption P ib, the heat flux from the adjacent pores W e , i macro (calculated by the macroscopic model), the local heat electron flux q th, and the energy outflow from the cylinder carried with the ablated mass,
d m cyl d t ( ε cyl + 1 2 v exp 2 ) + 2 π a l c q abl .
This term is split between electrons and ions of plasma as 1 ζ i and ζ i. The value of this parameter ζ i 0.6 is chosen according to the kinetic simulations.25 Consequently, equation for the electron internal plasma energy can be expressed as
d ε pe d t = 2 v abl a m cyl m pl ( 1 ζ i ) ( ε cyl + 1 2 v exp 2 ) 2 v abl a m cyl m pl ε pe + ( 1 ζ i ) 2 π a l c m pl q abl ,
(28)
2 π a l c m pl q th + P ib m pl W ei + W e macro .
In a similar way, the plasma ion internal energy equation reads
d ε pi d t = 2 v abl a m cyl m pl ε pi + 2 v abl a m cyl m pl ζ i ( ε cyl + 1 2 v exp 2 ) +  ζ i 2 π a l c m pl q abl + W ei + W i macro .
(29)
The first term on the right hand side of these equations represents the decrease in the internal energy due to inflow of ablated mass. The second and third terms represent the energy flux carried with the ablated mass. The fourth and fifth terms in the electron equation represent the electron energy loss due to the cylinder heating and laser collisional absorption. Finally, the term W e i accounts for the electron-ion energy exchange,
W ei = 3 ν ei ( m e / m i 2 ) Z max k B ( T pe T pi ) ,
(30)
with the collision frequency defined by Eq. (11). All terms involved in the micro-scale energy balance are schematically represented in Fig. 3.
FIG. 3.

Mass and energy balance in the micro-scale cell.

FIG. 3.

Mass and energy balance in the micro-scale cell.

Close modal

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 2-photon 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 0.1 proceeds on a subpicosecond timescale when the laser intensity exceeds 10 12 10 13 W/cm2, 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 2 T cyl / χ H for Z cyl Z max, where χ H 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 time-dependent foam ionization20,36 can be added to the model.

The temperature dependence of ionization is accounted for in the relation between the cylinder temperature T cyl and the internal energy ε cyl,
T cyl = max { 2 3 m i ε cyl ( Z max + 1 ) k B , ( m i χ H 3 k B ε cyl ) 2 / 3 } .
The ambient plasma is assumed to be fully ionized, and the temperature is defined by the equation of state of an ideal gas with the polytrope index 5 / 3,
T pe = 2 3 m i Z max k B ε pe , T pi = 2 3 m i k B ε pi .
(31)
Connection between the macro- and micro-scales is based on the mass (5) and energy conservation
M cell ( ε e + ε i ) = m cyl ( ε cyl + 1 4 v exp 2 ) + m pl ( ε pe + ε pi ) .
(32)
The full set of hydrodynamic Eqs. (1)–(4) is solved in the part of computational domain where the foam cells are homogenized. In non-homogenized cells, all source terms are already covered by the micro-scale model and we suppress the movement of the foam cells by setting the hydrodynamic velocity to zero. These assumptions reduce the full macroscopic equations to two equations of the macro-scale heat transport
ϱ d ε e , i d t = ( κ T e , i ) .
(33)
Their solution determines the macro-scale heat flux sources in the micro-scale equations
W e , i macro = V pore m pl ( κ T e , i ) .
(34)
By the nature of the problem, Eq. (33) is solved in the whole computational domain. As the majority of the energy source terms and interaction terms were moved to be calculated on the micro-scale, all information about the absorbed laser energy and its partition between species is present exclusively in the micro-scale variables. The macro-scale internal energies ε e, ε i in Eq. (32), therefore, need to be properly updated whenever we compute the heat transport to accurately reflect the energy contained inside the ambient plasma. The internal and kinetic energy of the cylinder are temporarily omitted as we assume that only ambient plasma interacts with adjacent cells. The cylinder energy is, thus, effectively not included in the macro-scale energy balance, and it is added at the end of homogenization stage. Assuming the same electron-ion partition on macro- scale and micro-scale and using the relation between the micro- and macro-scale energies M cell ε e = m pl ε p e, M cell ε i = m pl ε p i, the update of macro-scale internal energy of electrons is given by the following equation:
ϱ d ε e d t = m pl V pore ( d ε pe d t + 2 v abl a m cyl m pl ε pe ) ,
(35)
where d ε pe / d t is taken from the micro-scale electron energy Eq. (28). The same approach is applied to macro-scale ion internal energy
ϱ d ε i d t = m pl V pore ( d ε pi d t + 2 v abl a m cyl m pl ε pi ) .
(36)
During the homogenization stage, the cylinder density is greater than the micro-scale plasma density ϱ cyl > ϱ pl. The homogenization is finished when the cylinder density is equal to the plasma density, that is, ϱ cyl ϱ pl ϱ ¯. From that moment, the micro-scale part of the model is no longer computed, and the cell is turned into the plasma state. At this moment, the cylinder energy, hidden from the macro-scale during the homogenization stage, is added to the macro-scale energies according to Eq. (32)
M cell ( Δ ε e + Δ ε i ) = m cyl ( ε cyl + 1 4 v exp 2 ) .
(37)
The cylinder internal energy is redistributed between electrons and ions. The cylinder kinetic energy is only deposited to ions since the ion–ion collisions are the dominant process of slowing down the expanding plasma flow when it encounters a counterstreaming plasma from the neighboring pores. For the domain of parameters of interest, the characteristic ion–ion collision time is in the range of 10 100 ps,14 which is compatible with the cell homogenization time. After this redistribution, the one-time increments to the macro-scale energy have the following form:
Δ ε e = m cyl M cell Z cyl Z cyl + 1 ε cyl ,
(38)
Δ ε i = m cyl M cell ( 1 Z cyl + 1 ε cyl + 1 4 v exp 2 ) .
(39)
Inclusion of the missing cylinder energy ensures a global energy conservation on the macro-scale.

One macro-scale time step of the full computation procedure for the hybrid model can be summarized as follows:

  1. Compute the macro-scale heat conductivity (33) in all macro-scale cells.

  2. Compute the laser absorption coefficient K tot (15) from micro-scale variables.

  3. Compute the laser power deposition (16) in every macro-scale cell by the ray tracing routine.

  4. Compute the micro-scale 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 fourth-order Runge–Kutta method with a reduced time step, so that we use multiple micro-scale time steps for each macro-scale time step.

  5. Compute the macro-scale 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 macro-scale electron and ion energies (35) and (36).

  6. 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 micro-scale 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 single-pore kinetic simulations25 or guided from experimental data as discussed in Sec. VI. The model is tested first in single-cell simulations as discussed in Sec. V, and then integrated simulations are compared with the data available from several experiments.

TABLE I.

Empirical factors used in the micro-scale model and their recommended values.

Q abs = 0.5 1.5  Laser absorption efficiency factor of the cylinder (7) 
Q sca = 1 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) 
ζ las [ 0 , 1 ]  Fraction of the absorbed laser energy used for bulk cylinder heating (19) 
ζ th [ 0 , 1 ]  Fraction of the heat flux used for the bulk cylinder heating (19) 
ξ a = 0.1 0.5  Coefficient controlling the laser-driven ablation velocity (26) 
ζ i = 0.6  Fraction of energy of ablated plasma deposited to ions (29) 
Q abs = 0.5 1.5  Laser absorption efficiency factor of the cylinder (7) 
Q sca = 1 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) 
ζ las [ 0 , 1 ]  Fraction of the absorbed laser energy used for bulk cylinder heating (19) 
ζ th [ 0 , 1 ]  Fraction of the heat flux used for the bulk cylinder heating (19) 
ξ a = 0.1 0.5  Coefficient controlling the laser-driven ablation velocity (26) 
ζ i = 0.6  Fraction of energy of ablated plasma deposited to ions (29) 

The micro-scale model is tested for an isolated foam pore with a well-defined continual energy input given by the laser of intensity of 10 14 W/cm2. The micro-scale 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 ϱ ¯ = 10 mg/cm3, pore size l p = 2 μm, solid density ϱ s = 1 g/cm3, material is plastic with average charge Z = 3.85 and mass number A = 7.2, and laser wavelength λ = 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 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 ion-electron energy partition parameter ζ 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 ξ a = 0.5 is chosen by comparing the ablation velocity and the incident laser fluxes.

Parameters ζ las and ζ 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.

FIG. 4.

Simulation results of the micro-scale model for a single pore in three regimes: expansion (green), combined (red), and ablation (blue). The upper row presents the time evolution of the cylinder radius a, the expansion velocity v exp, and the ablation velocity v abl. The middle row shows the time evolution of the mass of the cylinder m cyl, cylinder density ϱ cyl (solid line), plasma density ϱ pl (dashed line), and integrated absorption efficiency. The bottom row presents the time evolution of the cylinder temperature T cyl, plasma electron temperature T pe (solid line), and ion temperature T pi (dotted line).

FIG. 4.

Simulation results of the micro-scale model for a single pore in three regimes: expansion (green), combined (red), and ablation (blue). The upper row presents the time evolution of the cylinder radius a, the expansion velocity v exp, and the ablation velocity v abl. The middle row shows the time evolution of the mass of the cylinder m cyl, cylinder density ϱ cyl (solid line), plasma density ϱ pl (dashed line), and integrated absorption efficiency. The bottom row presents the time evolution of the cylinder temperature T cyl, plasma electron temperature T pe (solid line), and ion temperature T pi (dotted line).

Close modal

The expansion regime is achieved by setting ζ 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 sub-grid models.15,16,20 Qualitatively, this regime is comparable to the kinetic simulations25 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 macro-scale model. The expansion regime is relevant for very low-density foams with thin solid elements, where the assumption of volumetric absorption is justified.

The combined ablation–expansion regime is attained by setting ζ las = 0, i.e., by switching the laser absorption to surface deposition and cylinder ablation, and by using ζ th = 0.1 1.0. We will discuss the results for ζ 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 simulations25 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 simulations25 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 high-temperature 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 ζ 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.

The ablation regime is achieved by further suppression of bulk cylinder heating by setting ζ las = 0 and ζ th < 0.1. Here, we discuss the results for ζ th = 10 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 three-step 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.

The hybrid ablation–expansion micro-scale model has been implemented in the ALE code PALE26 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.

The first simulation setup was chosen to match the experiments with foam targets at the Shenguang III Prototype (SGIII-P) laser facility.6 The targets were made of a 800 μm thick TMPTA foam of average density of 10 mg/cm3. They were symmetrically irradiated from two sides by laser beams at wavelength λ = 351 nm and total energy 3.2 kJ. We consider here a half of the original target of thickness 400 μm irradiated from one side. The density of solid elements is ϱ s = 1 g/cm3, the average pore size is l p = 2 μm, and the ion charge to mass ratio is Z / A = 3.85 / 7.2. The multi-beam setup is approximated by a single laser beam of equivalent parameters, as described in Ref. 6. This laser beam has a fourth order super-Gaussian spatial profile with a radius of 350 μm 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 × 10 14 W/cm2. The measured velocity of propagation of the ionization front inside the foam is about 0.33 μm/ps.

Our hybrid ablation–expansion model has several empirical parameters, which are summarized in Table I. Two parameters are set according to the single-pore simulations: ζ 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 ζ las , ζ th , ξ 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 μm 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 η tr = 100 % η abs η 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.

TABLE II.

Results of the integrated simulations for the conditions of experiment at SGIII-P6—the ionization front velocity v ioniz, the averaged ion and electron temperatures T i , T e, the total absorption efficiency η abs, and the fraction of reflected laser energy η refl. Variation of four control parameters: ζ th (a), ξ a (b), Q abs (c), and Q sca (d). For the empirical parameters, which are not displayed in the table, we use the values f lim = 0.01 and ζ i = 0.6.

Regime ζ las ζ th    ξ a Q abs Q sca v ioniz ( μm/ps) T i (keV) T e (keV) η abs (%) η refl (%)
(a)  Homogeneous  ⋯  ⋯  ⋯  ⋯  ⋯  1.08  0.57  0.85  93.7  ⋯ 
  Expansion  0.0  0.0  0.75  3.0  0.87  0.77  1.02  97.1  0.7 
  Combined  0.5  0.1  0.75  3.0  0.58  2.02  1.13  94.4  3.8 
  Combined  0.1  0.1  0.75  3.0  0.51  2.63  1.04  95.5  4.5 
Ablation  0.01  0.1  0.75  3.0  0.39  3.13  1.06  93.4  6.6 
  Ablation  0.005  0.1  0.75  3.0  0.35  3.59  1.01  91.8  8.1 
  Ablation  0.001  0.1  0.75  3.0  0.28  4.73  0.96  88.7  11.1 
(b)  Ablation  0.01  0.3  0.75  3.0  0.66  1.59  1.15  89.7  3.8 
  Ablation  0.01  0.2  0.75  3.0  0.53  2.28  1.12  91.7  5.3 
Ablation  0.01  0.1  0.75  3.0  0.39  3.13  1.06  93.4  6.6 
(c)  Ablation  0.01  0.1  0.5  3.0  0.42  2.92  1.07  90.5  7.8 
Ablation  0.01  0.1  0.75  3.0  0.39  3.13  1.06  93.4  6.6 
  Ablation  0.01  0.1  1.0  3.0  0.37  3.35  1.04  94.1  5.7 
  Ablation  0.01  0.1  1.5  3.0  0.33  3.70  1.02  95.4  4.6 
(d)  Ablation  0.01  0.1  0.75  0.60  2.15  1.02  87.7 
  Ablation  0.01  0.1  0.75  2.0  0.43  2.93  1.07  92.1  5.2 
Ablation  0.01  0.1  0.75  3.0  0.39  3.13  1.06  93.4  6.6 
  Ablation  0.01  0.1  0.75  4.0  0.36  3.30  1.04  92.3  7.6 
Regime ζ las ζ th    ξ a Q abs Q sca v ioniz ( μm/ps) T i (keV) T e (keV) η abs (%) η refl (%)
(a)  Homogeneous  ⋯  ⋯  ⋯  ⋯  ⋯  1.08  0.57  0.85  93.7  ⋯ 
  Expansion  0.0  0.0  0.75  3.0  0.87  0.77  1.02  97.1  0.7 
  Combined  0.5  0.1  0.75  3.0  0.58  2.02  1.13  94.4  3.8 
  Combined  0.1  0.1  0.75  3.0  0.51  2.63  1.04  95.5  4.5 
Ablation  0.01  0.1  0.75  3.0  0.39  3.13  1.06  93.4  6.6 
  Ablation  0.005  0.1  0.75  3.0  0.35  3.59  1.01  91.8  8.1 
  Ablation  0.001  0.1  0.75  3.0  0.28  4.73  0.96  88.7  11.1 
(b)  Ablation  0.01  0.3  0.75  3.0  0.66  1.59  1.15  89.7  3.8 
  Ablation  0.01  0.2  0.75  3.0  0.53  2.28  1.12  91.7  5.3 
Ablation  0.01  0.1  0.75  3.0  0.39  3.13  1.06  93.4  6.6 
(c)  Ablation  0.01  0.1  0.5  3.0  0.42  2.92  1.07  90.5  7.8 
Ablation  0.01  0.1  0.75  3.0  0.39  3.13  1.06  93.4  6.6 
  Ablation  0.01  0.1  1.0  3.0  0.37  3.35  1.04  94.1  5.7 
  Ablation  0.01  0.1  1.5  3.0  0.33  3.70  1.02  95.4  4.6 
(d)  Ablation  0.01  0.1  0.75  0.60  2.15  1.02  87.7 
  Ablation  0.01  0.1  0.75  2.0  0.43  2.93  1.07  92.1  5.2 
Ablation  0.01  0.1  0.75  3.0  0.39  3.13  1.06  93.4  6.6 
  Ablation  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 ζ th is the key factor for the slowing down the ionization front and increasing laser energy deposition. Decreasing ζ 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 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 ξ a, see Table II(b), which is controlling the laser-driven component of the ablation velocity. The value ξ 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 ζ th = 0.001. Results compatible with the experimentally measured ionization front velocity are obtained by reducing ξ 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 non-polarized laser). The kinetic simulations and one-pore tests were performed for the laser intensity of 10 14 W/cm2 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 non-polarized. These factors explain the necessity of using smaller values of ablation velocity, ξ a 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 follow-up simulations, we use Q sca = 3 and Q abs = 0.75, which correspond to theoretical values averaged over the laser polarization.

Inclusion of the foam-induced 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.

FIG. 5.

Absorbed laser power (in units of W/cm3) at time of 300 ps for foam target modeled without laser scattering on foam elements (a) and foam target modeled with the scattering enabled (b). The laser is incident from the right.

FIG. 5.

Absorbed laser power (in units of W/cm3) at time of 300 ps for foam target modeled without laser scattering on foam elements (a) and foam target modeled with the scattering enabled (b). The laser is incident from the right.

Close modal

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 ζ th = 0.005 provides the best result in terms of the propagation speed. However, the case of ζ th = 0.01 provides a better agreement with the estimated value of average ion temperature, T i 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.

FIG. 6.

Simulated ionization wave propagation in the under-dense foam target for the experiment at SGIII-P.6 The laser comes from the top, the front edge of the target is at z = 0, and the target thickness is 400 μm. The temporal shape of the laser pulse is shown as a yellow contour.

FIG. 6.

Simulated ionization wave propagation in the under-dense foam target for the experiment at SGIII-P.6 The laser comes from the top, the front edge of the target is at z = 0, and the target thickness is 400 μm. The temporal shape of the laser pulse is shown as a yellow contour.

Close modal

The spatial profiles of the electron and ion macro-scale 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 expansion-related 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.

FIG. 7.

Electron temperature (left column), ion temperature (middle column), and absorbed laser power (right column) computed with the PALE code at the time of 300 ps for a foam target modeled as homogeneous medium (first row) and for a foam target modeled by the hybrid model in the expansion regime (second row), combined regime (third row), and ablation regime (last row).

FIG. 7.

Electron temperature (left column), ion temperature (middle column), and absorbed laser power (right column) computed with the PALE code at the time of 300 ps for a foam target modeled as homogeneous medium (first row) and for a foam target modeled by the hybrid model in the expansion regime (second row), combined regime (third row), and ablation regime (last row).

Close modal
FIG. 8.

Lineouts at r = 0 from the results shown in Fig.7 for homogeneous simulation (black), expansion regime (green), combined regime (red), and ablation regime (blue). Electron temperature (a), ion temperature (b), and absorbed laser power (c) are plotted at the time of 300 ps.

FIG. 8.

Lineouts at r = 0 from the results shown in Fig.7 for homogeneous simulation (black), expansion regime (green), combined regime (red), and ablation regime (blue). Electron temperature (a), ion temperature (b), and absorbed laser power (c) are plotted at the time of 300 ps.

Close modal

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 C24H16O6 with ion charge to mass ratio Z / A = 4.54 / 8.73, pore size 2 μm, and solid density of 1 g/cm3. Two target average densities are considered: 9.1 mg/cm3 of thickness 400 μm and 4.5 mg/cm3 and thickness 380 μm. 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 μm 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 × 10 15 W/cm2.

The plasma self-emission from the ionization front is measured with an x-ray streak camera. The ionization front velocity for the 4.5 mg/cm3 foam is 1.3 μm/ps, which corresponds to the front break-out time of 500 ps at the rear side of the target. In the case of the 9.1 mg/cm3 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 1.4 ns, resulting in average velocity of 0.33 0.37 μm/ps of the laser-driven heat front.

The original data from the 2007 experiment30 have been recently re-processed using the approach described in Ref. 38 to determine the ion temperature from the Doppler broadening of the intercombination He- α y line of chlorine. In the case of the 9.1 mg/cm3 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 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/cm3 foam, the optimal reference regime is the ablation one as for the case of SGIII-P experiment in Sec. VI A, while for 4.5 mg/cm3 foam, the combined regime is the optimal one. This is due to a thinner solid elements in the 4.5 mg/cm3 foam, resulting in earlier transition from laser ablation to expansion. The reference regimes are marked by the asterisk in the tables.

FIG. 9.

Simulated ionization wave propagation in a foam target with a density of 9.1 mg/cm3 (a) and 4.5 mg/cm3 (b) for the conditions of PALS experiments.37 The laser comes from the top, the front edge of the target is at z = 0, and the target thickness is 400 μm for 9.1 mg/cm3 (a) and 380 μm for 4.5 mg/cm3 (b). The temporal shape of the laser pulse is shown with a yellow contour.

FIG. 9.

Simulated ionization wave propagation in a foam target with a density of 9.1 mg/cm3 (a) and 4.5 mg/cm3 (b) for the conditions of PALS experiments.37 The laser comes from the top, the front edge of the target is at z = 0, and the target thickness is 400 μm for 9.1 mg/cm3 (a) and 380 μm for 4.5 mg/cm3 (b). The temporal shape of the laser pulse is shown with a yellow contour.

Close modal
TABLE III.

Results of integrated simulations for the experiment on PALS laser facility37 with a foam of average density of 9.1 mg/cm3—the average ionization front velocity v ioniz, the averaged ion and electron temperatures T i and T e, the total absorption efficiency η abs, and the fraction of reflected laser energy η refl.

Regime ζ las ζ th    ξ a Q abs Q sca v ioniz ( μm/ps) T i (keV) T e (keV) η abs (%) η refl (%)
Homogeneous  ⋯  ⋯  ⋯  ⋯  ⋯  1.47  0.35  0.46  85.7  0.17 
Expansion  0.75  3.0  1.21  0.77  1.21  89.6  1.0 
Combined  0.1  0.1  0.75  3.0  0.75  4.27  1.20  95.3  4.4 
* Ablation  0.01  0.1  0.75  3.0  0.36  4.57  0.88  93.2  6.8 
Regime ζ las ζ th    ξ a Q abs Q sca v ioniz ( μm/ps) T i (keV) T e (keV) η abs (%) η refl (%)
Homogeneous  ⋯  ⋯  ⋯  ⋯  ⋯  1.47  0.35  0.46  85.7  0.17 
Expansion  0.75  3.0  1.21  0.77  1.21  89.6  1.0 
Combined  0.1  0.1  0.75  3.0  0.75  4.27  1.20  95.3  4.4 
* Ablation  0.01  0.1  0.75  3.0  0.36  4.57  0.88  93.2  6.8 
TABLE IV.

Results of integrated simulations for the experiment on PALS laser facility37 with a foam of average density of 4.5 mg/cm3—the average ionization front velocity v ioniz, the averaged ion and electron temperatures T i and T e, the total absorption efficiency η abs, and the fraction of reflected laser energy η refl.

Regime ζ las ζ th    ξ a Q abs Q sca v ioniz ( μm/ps) T i (keV) T e (keV) η abs (%) η refl (%)
Homogeneous  ⋯  ⋯  ⋯  ⋯  ⋯  2.70  0.30  0.41  43.2  0.7 
Expansion  0.75  3.0  1.92  0.76  1.0  55.1  10.4 
* Combined  0.1  0.1  0.75  3.0  1.31  4.6  1.1  63.1  11.4 
Ablation  0.01  0.1  0.75  3.0  1.02  7.1  1.1  76.6  12.8 
Regime ζ las ζ th    ξ a Q abs Q sca v ioniz ( μm/ps) T i (keV) T e (keV) η abs (%) η refl (%)
Homogeneous  ⋯  ⋯  ⋯  ⋯  ⋯  2.70  0.30  0.41  43.2  0.7 
Expansion  0.75  3.0  1.92  0.76  1.0  55.1  10.4 
* Combined  0.1  0.1  0.75  3.0  1.31  4.6  1.1  63.1  11.4 
Ablation  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 μm/ps for the 9.1 mg/cm3 target and 1.31 μm/ps for the 4.5 mg/cm3 target and correspond reasonably well to the values from experiment, 0.33 0.37 and 1.3 μm/ps for the 9.1 and 4.5 mg/cm3 targets, respectively. The ratio of ion to electron temperature in the simulation is around 5 for 9.1 mg/cm3 target, which is comparable to the ratio T i / T e 3 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 4.2 for the combined regime of the 4.5 mg/cm3 foam.

We present a novel approach to the numerical modeling of the laser interaction with under-critical foams. It combines a macroscopic hydrodynamic description of the homogenized cells with microscopic description of the cells undergoing the homogenization process. The non-homogenized cells contain two phases: solid structural elements represented by homogeneous cylinders and ambient low-density 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 sub-wavelength size are calculated from the Mie theory.

The micro-scale model combines self-similar 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 micro-scale model is controlled by several empirical parameters that are chosen by comparison with kinetic single-pore simulations and with experiments. Three regimes of operation are identified in single-pore 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 low-density foams with very thin structural elements. A variation of two parameters ζ las, ζ 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 two-scale model is evaluated by comparing the simulation results with the experiments at the SGIII-P 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 SGIII-P 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.

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 AWP21-ENR-01-CEA-02 “Advancing shock ignition for direct-drive 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.

The authors have no conflicts to disclose.

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).

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

1.
D.
Batani
,
A.
Balducci
,
W.
Nazarov
,
T.
Löwer
,
T.
Hall
,
M.
Koenig
,
B.
Faral
,
A.
Benuzzi
, and
M.
Temporal
,
Phys. Rev. E
63
,
046410
(
2001
).
2.
H. F.
Robey
,
L. B.
Hopkins
,
J. L.
Milovich
, and
N. B.
Meezan
,
Phys. Plasmas
25
,
012711
(
2018
).
3.
M.
Olazabal-Loumé
,
P.
Nicolaï
,
G.
Riazuelo
,
M.
Grech
,
J.
Breil
,
S.
Fujioka
,
A.
Sunahara
,
N.
Borisenko
, and
V.
Tikhonchuk
,
New J. Phys.
15
,
085033
(
2013
).
4.
B.
Delorme
,
M.
Olazabal-Loumé
,
A.
Casner
,
P.
Nicolaï
,
D. T.
Michel
,
G.
Riazuelo
,
N.
Borisenko
,
J.
Breil
,
S.
Fujioka
,
M.
Grech
et al,
Phys. Plasmas
23
,
042701
(
2016
).
5.
V.
Tikhonchuk
,
Y. J.
Gu
,
O.
Klimo
,
J.
Limpouch
, and
S.
Weber
,
Matter Radiat. Extremes
4
,
045402
(
2019
).
6.
V.
Tikhonchuk
,
T.
Gong
,
N.
Jourdain
,
O.
Renner
,
K.
Pan
,
W.
Nazarov
,
L.
Hudec
,
J.
Limpouch
,
R.
Liska
,
M.
Krus
et al,
Matter Radiat. Extremes
6
,
025902
(
2021
).
7.
R. E.
Olson
and
R. J.
Leeper
,
Phys. Plasmas
20
,
092705
(
2013
).
8.
F.
Perez
,
J. D.
Colvin
,
M. J.
May
,
S.
Charnvanichborikarn
,
S. O.
Kucheyev
,
T. E.
Felter
, and
K. B.
Fournier
,
Phys. Plasmas
22
,
113112
(
2015
).
9.
O. N.
Rosmej
,
N. E.
Andreev
,
S.
Zaehter
,
N.
Zahn
,
P.
Christ
,
B.
Borm
,
T.
Radon
,
A.
Sokolov
,
L. P.
Pugachev
,
D.
Khaghani
et al,
New J. Phys.
21
,
043044
(
2019
).
10.
O. N.
Rosmej
,
M.
Gyrdymov
,
M. M.
Günther
,
N. E.
Andreev
,
P.
Tavana
,
P.
Neumayer
,
S.
Zähter
,
N.
Zahn
,
V. S.
Popov
,
N. G.
Borisenko
et al,
Plasma Phys. Controlled Fusion
62
,
115024
(
2020
).
11.
R. J.
Mason
,
R. A.
Kopp
,
H. X.
Vu
,
D. C.
Wilson
,
S. R.
Goldman
,
R. G.
Watt
,
M.
Dunne
, and
O.
Willi
,
Phys. Plasmas
5
,
211
(
1998
).
12.
T.
Kapin
,
M.
Kucharik
,
J.
Limpouch
, and
R.
Liska
,
Czech. J. Phys.
56
,
B493
(
2006
).
13.
S. Y.
Gus'kov
,
J.
Limpouch
,
P.
Nicolaï
, and
V. T.
Tikhonchuk
,
Phys. Plasmas
18
,
103114
(
2011
).
14.
S. Y.
Gus'kov
,
M.
Cipriani
,
R.
De Angelis
,
F.
Consoli
,
A. A.
Rupasov
,
P.
Andreoli
,
G.
Cristofari
, and
G.
Di Giorgio
,
Plasma Phys. Controlled Fusion
57
,
125004
(
2015
).
15.
M. A.
Belyaev
,
R. L.
Berger
,
O. S.
Jones
,
S. H.
Langer
, and
D. A.
Mariscal
,
Phys. Plasmas
25
,
123109
(
2018
).
16.
J.
Velechovsky
,
J.
Limpouch
,
R.
Liska
, and
V. T.
Tikhonchuk
,
Plasma Phys. Controlled Fusion
58
,
095004
(
2016
).
17.
M.
Cipriani
,
S. Y.
Gus'kov
,
R.
De Angelis
,
F.
Consoli
,
A. A.
Rupasov
,
P.
Andreoli
,
G.
Cristofari
, and
G.
Di Giorgio
,
Phys. Plasmas
25
,
092704
(
2018a
).
18.
M.
Cipriani
,
S. Y.
Gus'kov
,
R.
De Angelis
,
F.
Consoli
,
A. A.
Rupasov
,
P.
Andreoli
,
G.
Cristofari
,
G.
Di Giorgio
, and
F.
Ingenito
,
Laser Part. Beams
36
,
121
(
2018
).
19.
R. D.
Angelis
,
F.
Consoli
,
S. Y.
Gus'kov
,
A. A.
Rupasov
,
P.
Andreoli
,
G.
Cristofari
, and
G. D.
Giorgio
,
Phys. Plasmas
22
,
072701
(
2015
).
20.
M. A.
Belyaev
,
R. L.
Berger
,
O. S.
Jones
,
S. H.
Langer
,
D. A.
Mariscal
,
J.
Milovich
, and
B.
Winjum
,
Phys. Plasmas
27
,
112710
(
2020
).
21.
A. M.
Khalenkov
,
N. G.
Borisenko
,
V. N.
Kondrashov
,
Y. A.
Merkuliev
,
J.
Limpouch
, and
V. G.
Pimenov
,
Laser Part. Beams
24
,
283
(
2006
).
22.
N. G.
Borisenko
,
I. V.
Akimova
,
A. I.
Gromov
,
A. M.
Khalenkov
,
Y. A.
Merkuliev
,
V. N.
Kondrashov
,
J.
Limpouch
,
J.
Kuba
,
E.
Krouský
,
K.
Mašek
et al,
Fusion Sci. Technol.
49
,
676
(
2006
).
23.
P.
Nicolaï
,
M.
Olazabal-Loumé
,
S.
Fujioka
,
A.
Sunahara
,
N.
Borisenko
,
S.
Gus'kov
,
A.
Orekhov
,
M.
Grech
,
G.
Riazuelo
,
C.
Labaune
et al,
Phys. Plasmas
19
,
113105
(
2012
).
24.
H. C.
van der Hulst
,
Light Scattering by Small Particles
(
Dover Publications
,
1957
).
25.
S.
Shekhanov
,
A.
Gintrand
,
L.
Hudec
,
R.
Liska
,
J.
Limpouch
,
S.
Weber
, and
V.
Tikhonchuk
,
Phys. Plasmas
30
,
012708
(
2023
).
26.
J.
Fort
,
J.
Kuchařik
,
J.
Halama
,
R.
Herbin
, and
F.
Hubert
, in
Finite Volumes for Complex Applications VI Problems & Perspectives FVCA 6, International Symposium
,
2011
.
27.
B.
Fryxell
,
K.
Olson
,
P.
Ricker
,
F. X.
Timmes
,
M.
Zingale
,
D. Q.
Lamb
,
P.
MacNeice
,
R.
Rosner
,
J. W.
Truran
, and
H.
Tufo
,
Astrophys. J., Suppl. Ser.
131
,
273
(
2000
).
28.
A.
Dubey
,
K.
Antypas
,
M. K.
Ganapathy
,
L. B.
Reid
,
K.
Riley
,
D.
Sheeler
,
A.
Siegel
, and
K.
Weide
,
Parallel Comput.
35
,
512
(
2009
).
29.
P.
Tzeferacos
,
M.
Fatenejad
,
N.
Flocke
,
G.
Gregori
,
D.
Lamb
,
D.
Lee
,
J.
Meinecke
,
A.
Scopatz
, and
K.
Weide
,
High Energy Density Phys.
8
,
322
(
2012
).
30.
J.
Limpouch
,
O.
Renner
,
N.
Borisenko
,
D.
Klir
,
V.
Kmetik
,
E.
Krousky
,
R.
Liska
,
K.
Masek
,
W.
Nazarov
, and
J.
Ullschmied
,
J. Phys.: Conf. Ser.
112
,
042056
(
2008
).
31.
D. V.
Knyazev
and
P. R.
Levashov
,
Phys. Plasmas
22
,
053303
(
2015
).
32.
A. V.
Farnsworth
,
Phys. Fluids
23
,
1496
(
1980
).
33.
W.
Manheimer
,
D.
Colombant
, and
J. H.
Gardner
,
Phys. Fluids
25
,
1644
(
1982
).
34.
R.
Fabbro
,
C.
Max
, and
E.
Fabre
,
Phys. Fluids
28
,
1463
(
1985
).
35.
R.
Betti
,
M.
Umansky
,
V.
Lobatchev
,
V.
Goncharov
, and
R.
McCrory
,
Phys. Plasmas
8
,
5257
(
2001
).
36.
G.
Duchateau
,
S. X.
Hu
,
A.
Pineau
,
A.
Kar
,
B.
Chimier
,
A.
Casner
,
V.
Tikhonchuk
,
V. N.
Goncharov
,
P. B.
Radha
, and
E. M.
Campbell
,
Phys. Rev. E
100
,
033201
(
2019
).
37.
J.
Limpouch
,
O.
Renner
,
E.
Krousky
,
W.
Nazarov
,
N.
Borisenko
,
N.
Demchenko
,
S.
Gus'kov
,
D.
Klir
,
V.
Kmetik
,
R.
Liska
et al,
34th EPS Conference 31F, O
(
2006
).
38.
V. V.
Gavrilov
,
A. Y.
Gol'tsov
,
N. G.
Koval'skii
,
S. N.
Koptyaev
,
A. I.
Magunov
,
T. A.
Pikuz
,
I. Y.
Skobelev
, and
A. Y.
Faenov
,
Quantum Electron.
31
,
1071
(
2001
).
Published open access through an agreement with Ceske Vysoke Uceni Technicke v Praze Fakulta jaderna a fyzikalne inzenyrska