The phenomenon of laser-driven heating wave propagation in a plasma with a density less than the critical density is considered for the case of a finite radius of the laser beam. Based on computational and theoretical studies, the effect of channeling the heating wave propagation in the central axial region of plasma due to the reflection of laser radiation on density gradient, formed by plasma motion, at the tuning point was found. Unlike the self-focusing phenomenon, where the laser flux is concentrated without ray intersection, in refractive channeling, this occurs due to ray intersection. This is similar to the creation of a high flux density during multi-beam irradiation of the laser target. It is shown that the longitudinal velocity of heating wave in the channeling region is significantly larger compared with that in the peripheral plasma regions located at a distance of about laser beam radius.

## I. INTRODUCTION

The problem of the interaction of plasma-producing laser radiation flux with a half-space of subcritical-density substance in the absence of motion has a self-similar solution for a plane heating wave.^{1} The properties of such a wave are determined by the dependence of inverse bremsstrahlung absorption coefficient for laser radiation $\kappa a$ on plasma density $\rho $ and temperature *T*: $\kappa a\u221d\rho 2/T3/2$. The heating wave velocity $Di$ and the temperature behind its front increase with laser intensity *I*, respectively, as $I3/5$ and $I2/5$.^{1} At the same time, the oppositely directed dependencies of absorption coefficient on temperature and density lead to oppositely directed dependencies of the wave velocity and temperature behind its front on time, as well as on plasma density and laser radiation wavelength. The wave velocity decreases with time as $t\u22122/5$, and the temperature, on the contrary, increases as $t2/5$.^{1} Finally, with increasing density and radiation wavelength, the wave velocity decreases as $\rho \u22127/5\lambda \u22124/5$, and the temperature behind its front increases as $\rho 2/5\lambda 4/5$.^{1–3}

Under the irradiation of a half-space by the laser beam of finite radius in the conditions of developing motion of the produced plasma, it should be expected that the nature of heating wave propagation will be significantly complicated due to a number of effects and, first of all, those associated with laser light propagation and absorption. Hydrodynamic expansion leads to the formation of the two-dimensional (2D) spatial distributions of density and temperature of heated region substance. The angle that makes up the direction of density gradient with respect to the beam optical axis varies from 0° on the beam axis to the values close to 90° at the boundary of the heated expanding region.

In this case, the conditions arise inside the heated region for laser light refraction directed to the beam axis. Moreover, the refraction effect increases with increasing distance of the rays from the beam axis. The effect is absent on the beam axis and reaches the maximum degree for the rays of the beam's peripheral part. As a consequence, such refraction leads to preferential heating of the central plasma region located near the beam axis that provides there the increased heating wave velocity compared with the velocity in the peripheral plasma region. For a central part of the laser beam, the latter circumstance leads to the formation of the axial channel of propagation with the length exceeding the longitudinal size of the surrounding area of the heated plasma. At the same time, the refracted rays from beam's peripheral part have the opportunity to fall directly into such a channel, heating it, and further increasing the heating wave velocity. With self-focusing^{4} of various types, the concentration of the laser radiation flux occurs as a result of the refraction of rays without their intersection due to the inhomogeneity of the refractive index of a low-density plasma. Refractive channeling is based on the phenomenon of reflection of rays at the turning point with subsequent intersection as they propagate in an enough dense plasma with a large density gradient caused by hydrodynamic motion.

The paper is devoted to computational and theoretical studies of a finite-radius laser beam propagation in plasma with a subcritical density as well as to substantiation and description of the phenomenon of the refractive channeling of the laser-driven heating wave. The phenomenon is of interest for studies of laser beam propagation in the earth's atmosphere as well as in extended plasma of a dense gas jet, gas-filled microbaloon, and fast-homogenized micro-size porous substance. The last case is associated with the practical use of low-density porous media in modern research on inertial confinement fusion (ICF) and studies of the equation of state of matter (EOS). Under terawatt laser pulse heating, homogenization of near-critical-density porous substance with submicron pore's size occurs in times of several tens of picoseconds.^{2} So, the produced plasma shows itself as a continuous media very fast in the case of heating laser pulse of nanosecond duration. In ICF, low-dense porous media are used as a material of the outer layer of ICF target intended to smooth the nonuniformities of target heating with a finite number of laser beams.^{5–7} In the field of EOS research, the use of low-density media is associated with the application of a well-known method of increasing the pressure of a shock wave during its transition to a substance with a higher density.^{8} With regard to laser irradiation, in the simplest scheme, this method is based on the use of target containing layers of substances with different densities, when the generation of the primary shock wave occurs in the substance with a lower density.^{9–11}

## II. THE PHENOMENON OF LASER-DRIVEN HEATING WAVE CHANNELING IN PLASMA WITH SUBCRITICAL DENSITY

Two-dimensional problem of heating of subcritical-density flat target by the laser beam of finite radius *R* is first considered without taking into account refraction. It is assumed that a laser beam with a circular cross section falls normally on the surface of a half-space of a homogeneous substance, whose density is less than the critical plasma density corresponding to the radiation wavelength (Fig. 1). The intensity distribution over beam cross section is assumed to be uniform, and the power of laser pulse is constant in time. Plasma formation can be approximately described using the scales of two rates of heated region expansion in the longitudinal direction along the direction of laser beam incidence and in the transverse (radial) direction (Fig. 1).

^{1}The velocity of energy transfer wave in the transverse direction is the velocity $Dsw$ of shock wave propagating from a cylindrical plasma region heated by the laser-driven heating wave. For estimation, it is considered the initial stage of plasma formation with duration $tm$, in the approximation of small energy losses on the motion of matter, when

^{1}Using inverse bremsstrahlung absorption coefficient of laser radiation,

^{2,3}

*T*is the electron temperature of the plasma measured in keV,

*Z*and

*A*are the charge and atomic number of plasma ions, $\rho 0$ is plasma density in g cm

^{−3}( $\rho 0<\rho c$), $\lambda \mu m$ is laser radiation wavelength measured in microns, and $\rho c\u22481.83\xd710\u22123A/(Z\lambda \mu m2)$ is the critical plasma density; in the approximations of one-temperature plasma, absence of substance motion and electron conductivity and space-time constant laser intensity gives that the velocity of plane heating wave $Di$ depends on time as

^{3}

^{−2}. The space-averaged temperature behind the front of such a wave in the approximation of its homogeneous spatial distribution is determined by the expression:

^{3}

*T*are obtained as

*T*into the formula for a strong shock wave

^{8}gives the scale of transverse shock wave velocity $Dsw$

*R*. At the large values of radius, the angle $\alpha $ decreases with time and with increasing intensity

*I*. As an estimate, the angle at $t=tm$ could be used

^{−3}. The densities $\rho 0=2$ mg cm

^{−3}and $\rho 0=4$ mg cm

^{−3}are considered. At an intensity of $1013$ W cm

^{−2}, the angle $\alpha m$ is 21° and 47°, respectively. At an intensity of $1014$ W cm

^{−2}, the angle $\alpha m$ is 7° and 22°, respectively.

Further, the conditions for appearance of strong refraction, leading to the heating wave channeling, are considered. The density gradient direction is characterized by its angle $\theta 0$ with *z*-axis or with the direction of ray incidence, since the incident rays are parallel to *z*-axis (Fig. 1). Replacing the plasma density distribution in the region of ray refraction by plane-distribution with gradient directed at an angle $\theta 0$ to *z*-axis, the law of refraction could be written as $n\u2009sin\u2009\theta =const$, where $n=\epsilon $ is the refractive index, $\epsilon =1\u2212\rho /\rho c$ is the plasma permittivity, $\rho $ is the plasma density, and $\theta $ is the current (along the ray propagation) angle between the ray direction and the density gradient line. At the turning point $\theta t=90\xb0$, $sin\u2009\theta t=1$. At the beginning of ray entry into the heated region (at $\theta =\theta 0$), plasma density is small, and it can be assumed that $n0=1$. Then from the law of refraction, it could be concluded that $sin\u2009\theta 0=nt=(1\u2212\rho t/\rho c)1/2$. Refracted ray has a direction that makes an angle of $\theta r$ with *z*-axis (Fig. 1). Since the angle of incidence relative to the density gradient line is equal to the reflection angle, then $\theta r=180\xb0\u22122\theta 0$.

The rays of a central (near-axis) part of the beam initially directed at the end surface of the heated cone propagate in plasma at $sin\u2009\theta 0\u22480$ ( $\rho t\u2248\rho c$) and refract weakly. For an axial ray, the density gradient at $r=0$ is directed along the *z*-axis, and this ray will not deviate from its original direction. On the contrary, the rays of a peripheral part of the beam with the initial direction of incidence on the lateral surface of the heated cone can have significant refraction, the degree of which depends on the cone angle $\alpha $. In the approximation of plane-layered plasma near a conical surface, $\theta 0=90\xb0\u2212\alpha $, since the density gradient is perpendicular to this surface. Then, the angles $\theta r$ and $\alpha $ are connected as $\theta r=2\alpha $. Using the above estimations for cone angle $\alpha m$ when the beam of $2\omega $ Nd–laser harmonic radiation beam with a radius $R=100$ $\mu $m and nanosecond pulse duration irradiates the low-density target, it can be seen that the refraction angles for peripheral rays can be very significant. At an intensity of 10^{13} W cm^{−2}, the angle $2\alpha m$ can reach 42° and 94° for the density of a substance, respectively, 2 and 4 mg cm^{−3}, and at an intensity of $1014$ W cm^{−2}, this angle can reach 14° and 44° for the density of a substance, respectively, 2 and 4 mg cm^{−3}. It should be noted that the value of the angle $\theta r$ depends on the ratio $\rho t/\rho c$, which increases with the increase in the initial density of the target. Therefore, with an increase in the initial density, the angle $\theta 0$ may be less than 45° and the angle $\theta r$ may be larger than 90°. In this case, the refracted rays can exit the plasma through the plasma–vacuum boundary.

## III. NUMERICAL SIMULATION RESULTS AND DISCUSSION

Numerical simulations were carried out using two codes for calculating the 2D hydrodynamics of laser–plasma interaction, namely, the code in Lagrangian coordinates ATLANT-HE^{12} and the code in Eulerian coordinates PM^{2} (Package for Multidimensional Plasma Modeling), built on the NUT platform.^{13,14} Both codes are based on the equations of single-fluid, two-temperature plasma hydrodynamics with electron and ion thermal conductivity and temperature relaxation due to electron–ion collisions and contain modules for calculating the laser light absorption due to inverse bremsstrahlung mechanism and light refraction along the ray trajectories in plasma. At the same time, the PM^{2} code provides a graphical visualization of ray trajectories. The calculations were carried out in an axisymmetric geometry using cylindrical coordinates $(r,z)$ under the assumption of axial symmetry of laser beam at its normal incidence on a flat layer of low-density CH-matter with subcritical density. The laser beam of $2\omega $ Nd–laser in the majority of calculations had a Gaussian intensity distribution over beam cross section with a characteristic radius $a=100$ $\mu $m: $I(r)=I0\u2009exp(\u2212r2/a2)$. The beam boundary was at $r/a=2$, and at $r/a>2$, the intensity was assumed to be zero. The rays in the beam were almost parallel to the *z*-axis (a very long-focus focusing system was assumed). The intensity $I0$ in the beam was independent of time. The calculations with a super-Gaussian intensity distribution over beam cross section were also carried out. The calculations are performed for the values of laser intensity $I0$ ranging from 10^{13} to 10^{14} W cm^{−2} and for the values of initial density $\rho 0$ ranging from 1 to 10 mg cm^{−3} for fully ionized substance, which is less than the critical density $\rho c=12.7$ mg cm^{−3}. Calculations using both the codes based on Lagrangian as well as Eulerian coordinates showed similar results, confirming the effect of refractive channeling of the laser-driven heating wave in subcritical plasma.

Below the regularities are presented of refractive channeling based on the results of numerical simulations performed for a Gaussian laser beam at an intensity $I0=1013$ W cm^{−2} and a density of CH-matter $\rho 0=2$ mg cm^{−3}. Figure 2 shows the spatial distributions of plasma density (g cm^{−3}) and electron temperature (keV) for three moments of time (0.5, 1, and 1.5 ns) obtained in simulations for the above conditions using the ATLANT-HE code. Figure 3 for the same formulation of the problem shows the distribution of plasma density, supplemented by the trajectories of laser rays, and the temperature distribution obtained in calculation using the PM^{2} code. From these results, it follows that in addition to a wide heated region with a transverse size scale corresponding to the radius of laser beam, there is a narrow heated region near the *z*-axis, in which heating wave velocity is significantly larger compared with a wide heated region. This is due to an increase in the concentration of laser energy deposition near the axis of symmetry caused by refraction of light toward the *z*-axis. This concentration leads to heating and pressure increase in the plasma, which leads to transverse expansion and channel formation. In addition, when the ray direction makes a certain angle with the direction of plasma density gradient (angle of incidence $\theta 0$), then there is a point of total reflection of the ray (turning point) at the density $\rho t$, depending on the angle of incidence, as indicated above, according to the ratio $sin\u2009\theta 0=nt=(1\u2212\rho t/\rho c)1/2$. At zero angle of incidence, this density is equal to the critical density. At an angle of incidence other than zero, this density is less than critical density and may lie in the range of densities corresponding to the formation of special distribution as a result of plasma motion with subcritical density. Table I shows the values of $\theta 0$, $\theta r$ and $\rho t$ corresponding to refraction of peripheral rays at the characteristic directions of density gradient near the lateral surface of a wide heated area determined from the data of Fig. 2 of the ATLANT-HE calculation for the time moments 0.5, 1, and 1.5 ns. The angle $\theta 0$ increases with time, while the density at which reflection occurs decreases. The refraction angle $\theta r$ decreases with time, remaining large enough for the refracted ray to reach the *z*-axis. Since there is an axial symmetry, the ray can be mirrored from the *z*-axis. After that, the ray is directed to the plasma with a larger density and smaller temperature at a small angle relative to the density gradient. Therefore, the mirrored ray will be effectively absorbed.

t (ns)
. | $\theta 0$ (°) . | $\theta r$ (°) . | $\rho t$ (mg cm^{−3})
. |
---|---|---|---|

0.5 | 68 | 44 | 1.78 |

1.0 | 75 | 30 | 0.85 |

1.5 | 79 | 22 | 0.46 |

t (ns)
. | $\theta 0$ (°) . | $\theta r$ (°) . | $\rho t$ (mg cm^{−3})
. |
---|---|---|---|

0.5 | 68 | 44 | 1.78 |

1.0 | 75 | 30 | 0.85 |

1.5 | 79 | 22 | 0.46 |

The plasma near the *z*-axis turns out to be largely heated by the refracted rays from the peripheral part of the beam. So, the rays of the near-axis part of the beam, which refract weakly, pass through this heated region with a small energy loss. As a result, these rays are absorbed deeper compared to the depth of a wide area. This nature of near *z*-axis ray propagation leads to the fact that a narrow and long heated channel is formed from the heated area bottom deep into the target. The trajectories of the refracted rays have the form of broken lines. In the near-axis region, the ray trajectories are close to straight lines. The described nature of ray refraction is clearly illustrated by the results of PM^{2} code calculations, shown in Figs. 3(b) and 3(c). The transverse size of a wide heated area, which is analogous to the heated cone discussed in Sec. II, has a scale corresponding to the scale of laser beam radius. A narrow channel near the *z*-axis is heated much deeper along the z-axis compared to a wide heated area. An additional confirmation of the above is the results of PM^{2} code calculations performed without taking into account refraction (Fig. 4). In the absence of refraction, the isolines of thermodynamic characteristics have a smooth character without fractures, which appear in the case of refractive channeling. The heated region has a regular conical shape, which transmits the temporal dynamics of the shock wave propagation in the transverse direction.

It should be noted that in Fig. 2(c), the thermal front near the boundary with the undisturbed substance has a slight filamentation. This is caused by the competition of two factors, on which the intensity of the laser radiation coming to the front depends. The first factor is the Gaussian dependence of the intensity incident on the target. The second factor is the different fractions of absorption along the rays due to the different opacities. This leads to a non-monotonic spatial dependence of the laser flux coming to the front. However, the size of the filaments at the front is comparable to the size of the numerical mesh intervals. Therefore, in this task, it seems, it is not worth paying special attention to these filaments.

The model of the heated cone described in Sec. II agrees well with the numerical simulation results. This, in particular, follows from the analysis of the data in Table I. Indeed, the angle $\theta 0$ at time $t=0.5$ ns for the density $\rho t=1.78$ mg cm^{−3} is equal to 68°. Since this density is close to the initial density of the substance $\rho 0=2$ mg cm^{−3}, density isoline $\rho t=1.78$ mg cm^{−3} is almost identical to the lateral surface of the heated cone. Hence, at time $t=0.5$ ns, the heated cone angle in the numerical calculation should be approximately close to the value 90°–68° = 22°. On the other hand, the estimation by formula (8) for time $t=0.5$ ns gives the heated cone angle of about 20.5°. In addition, the direct determination of the heated cone angle from the data in Fig. 2 shows that in the time range from 0.5 to 1.5 ns, this angle changes slightly (in the range of 21°–15°). Estimations according to formula (8) give an interval of changing that angle of 24°–20°.

The main feature of refractive channeling is a reflection of rays at the turning point with subsequent intersection as they propagate in enough dense plasma with a large density gradient caused by hydrodynamic motion. Such a regime of the concentration of laser radiation flux differs from the case of self-focusing, when the refraction of rays occurs without their intersection due to the inhomogeneity of the refractive index of low-density plasma. It is useful to estimate the condition for the occurrence of refractive channeling. According to the above-mentioned condition of ray reflection at the turning point, the ratio of depth and radius of the wide heated region, which corresponds to the fulfillment of this condition, is $Lch/Rch=tan\u2009\theta 0=(\rho c/\rho t\u22121)1/2$ (angle $\theta 0$ is shown in Fig. 1). The maximum plasma density in the heated region is achieved at its boundary and represents the density behind the shock front $\rho max=k\rho 0$ ( $\rho 0$ is the initial density of the target, $k=(\gamma +1)/(\gamma \u22121)$, for $\gamma =5/3k=4$). If $k\rho 0\u2264\rho c$ then at $\rho t=k\rho 0$, when the turning point has just appeared, the ratio $Lch/Rch$ is $Lch*/Rch*=(\rho c/(k\rho 0)\u22121)1/2$. Starting from this value of $Lch*/Rch*$, the self-focusing mode switches to the refractive channeling regime. If $k\rho 0>\rho c$, then $\rho t=\rho c$, since the density at the turning point cannot exceed critical density. In this case, $Lch*/Rch*=0$ and the refractive channeling regime immediately appears. For the critical density $\rho c=12.7$ mg cm^{−3}, $\rho 0=2$ mg cm^{−3}, and $k=4$, the ratio $Lch*/Rch*$ is 0.77. According to the data in Fig. 2, the ratio $Lch/Rch$ is 3, 4, and 4.6 in the moments 0.5, 1, and 1.5 ns, respectively. These values are significantly larger than the value $Lch*/Rch*=0.77$. So, at $\rho 0=2$ mg cm^{−3}, the refractive channeling regime is established at the very beginning of heated region formation. Under the considered irradiation conditions, the value of $Lch*/Rch*$ grows with decreasing density $\rho 0$ and it equals to 3.86 and 12.6, respectively, at $\rho 0=0.2$ mg cm^{−3} and $\rho 0=0.02$ mg cm^{−3}. It follows that the self-focusing regime can only manifest itself in a very low-density plasma with a density significantly lower than the critical one.

The channel formation features are confirmed by the comparison of spatial distributions of thermodynamic characteristics obtained in 2D and 1D calculations made under the same statement of the problem. The 1D calculation was performed using the RAPID code,^{15} the 2D calculation was performed using the ATLANT-HE code. There are $2\omega $ Nd–laser intensity of 10^{13} W cm^{−2} and CH-target initial density of 2 mg cm^{−3} in both simulations. The characteristic laser beam radius is equal to 100 $\mu $m in 2D simulation. Figure 5 shows these distributions at time $t=1$ ns: (a) for 1D calculation and (b) for 2D calculation (along the *z*-axis at $r=0$). In 1D calculation, the average density is close to the initial density $\rho 1D\u22482$ mg cm^{−3}, the average electron temperature $Te1D$ is about 0.66 keV. In 2D calculation, these values are $\rho 2D\u22480.8$ mg cm^{−3} and $Te2D\u22480.33$ keV. Thus, the ratios of thermodynamic plasma parameters in the interaction region of one-dimensional and two-dimensional calculations are $\rho 1D/\rho 2D=2.5$ and $Te1D/Te2D=2$. Keeping in mind that the length of inverse bremsstrahlung absorption $La=1/\kappa a\u223cTe3/2/\rho 2$ [see (2)], the ratio of absorption length scales in 2D and 1D calculations is $La2D/La1D=2.2$. According to the data in Fig. 5, during the time of 1 ns, an absorption wavefront passes a distance of $\Delta z1D=0.082$ cm (the initial target surface is located at $z=0.3$ cm) in 1D calculation and $\Delta z2D=0.18$ cm in 2D calculation.

The ratio of these distances is $\Delta z2D/\Delta z1D=2.2$, which coincides in amount with the ratio of characteristic absorption lengths $La2D/La1D$. The density and temperature of plasma in the 2D case are lower compared to the 1D case due to transverse plasma expansion. However, the absorption length turns out to be larger than in the 1D case. In the absence of refraction of rays near the *z*-axis, this leads to the formation of a narrow heated channel. Note that the formula (2) for the absorption length used for the analytical estimates in Sec. II agrees well with the data of numerical calculations. In the case of 1D calculation for $\rho 1D=2$ mg cm^{−3} and $Te1D=0.66$ keV, the absorption length estimation gives 0.095 cm, and in the case of a 2D calculation for $\rho 2D\u22480.8$ mg cm^{−3} and $Te2D\u22480.33$ keV, estimation gives 0.21 cm. Figure 5 also shows the spatial distributions of laser intensity $qL$ and energy deposition $Ed$. In the two-dimensional case, only the energy deposition is given, since the radiation comes to the *z* axis from different sides and at different angles. At the same time, the concept of intensity loses the meaning that it had in the one-dimensional case. A comparison of energy depositions shows that in the two-dimensional case on the *z*-axis (at $r=0$), it is significantly higher than in the one-dimensional case [the scales of the $Ed$ axes in Figs. 5(a) and 5(b) are different]. The ratio of the maximum values of $Ed$ in the range $0.22<z<0.3$ cm is about 7. This confirms the important role of refraction, which leads to a significant concentration of laser energy deposition near the *z-axis*.

Figure 6 shows the results of the calculation already mentioned, which was performed using the PM^{2} code at a radiation flux density of 10^{13} W cm^{−2} and an initial CH-target density of 2 mg cm^{−3} (see Fig. 3) related now to the time evolution of channel size and the dynamics of energy balance in the channel.

This figure shows the time dependencies of the depth $Lch$ and the radius of the base of the channel $Rch$ (hereinafter the channel radius). To analyze the energy balance features, it is useful to consider the heating of channel by two parts of the laser beam. These parts are the inner part with a radius equal to the channel radius $Rch(t)$, depending on time, and the outer surrounding part from the radius $Rch(t)$ to the beam radius $Rb=200$ $\mu $m. The energies of the inner and outer parts of laser beam, respectively, $E1(t)$ and $E2(t)$ depend on time. The PM^{2} code calculates the energies $E1ch(t)$ and $E2ch(t)$ absorbed in the channel from, respectively, the inner and outer parts of beam. The energy of the outer part of beam $E2(t)$ enters the channel due to refraction. Figure 6 shows the time dependencies of the energy fractions $e1=E1ch/E1$ and $e2=E2ch/E2$ released in the channel, respectively, from the inner and outer parts of beam. Also, it is shown the time dependence of the ratio $e3=E2ch/(E1ch+E2ch)$, which characterizes the energy contribution to the heating of channel from the outer part of beam compared to the energy released in the channel from the entire beam. The dependencies are given for the time interval from the beginning of a distinct channel formation at 0.6 up to 1.75 ns, when the channel radius is compared with beam radius. At the initial stage of channel formation from 0.6 to 0.64 ns, the radius and depth of channel are close in magnitude, and by the time of 0.64 ns are approximately equal to 50 $\mu $m. The subsequent channel evolution occurs at a significantly larger rate of increase in its depth, which is caused by heating wave propagation, compared with the rate of increase in its radius, which is caused by shock wave propagation (Fig. 6, curves $Lch$ and $Rch$). The formation of channel in the time interval from 0.6 to 1.1 ns occurs when the energy of the inner part of beam $E1(t)$ increases with the growth of the channel radius, but remains less than the energy of the outer part of beam $E2(t)$. By the time of 1.1 ns, the channel radius reaches a value of 120 $\mu $m, and its depth is 700 $\mu $m. The average values of the velocities of the increasing channel depth and radius are, respectively, $1.3\xd7108$ and $1.4\xd7107$ cm s^{−1}. The fraction of the energy of the inner part of beam $e1$ absorbed in the channel (Fig. 6, curve $e1$) is several times larger than the fraction of the energy of the outer part of beam $e2$ absorbed in the channel (Fig. 6, curve $e2$). Both values grow with time and reach the values of $e1=0.24$ and $e2=0.037$ at the time of 1.1 ns. However, since the energy entering the target from the outer part of beam significantly exceeds the energy coming from the inner part of beam, the radiation from the outer part of beam makes a significant contribution to the heating of channel (up to 30%, Fig. 6, curve $e3$). The time dependence of the energy fraction $e3$ of the outer part of beam released in the channel in relation to the total energy released in channel has a non-monotonic character with two maxima. The first maximum is reached at the initial stage of channel formation at $t=0.62$ ns. The decrease in the $e3$ energy fraction in the 0.62–0.64 ns time interval is due to an increase in the channel depth at a relatively small channel radius compared to the beam radius. At the same time, the penetration of refracted rays into the channel becomes more complicated. At the stage of subsequent channel evolution, the $e3$ energy fraction increases until a decrease in the energy of the outer part of beam caused by an increase in the radius of its inner part begins to affect. At $t>1.1$ ns, the absorption of radiation from the inner part of beam begins to play a dominant role in the formation of channel, which, with an increase in the radius of channel, is accompanied by refraction of radiation from this part of beam directly in the channel. As a result, the rate of increase in the channel depth increases markedly compared to the time interval of 0.6–1.1 ns. In the time interval from 1.1 to 1.32 ns, the average value of this rate is $2.5\xd7108$ cm s^{−1}. This is due to an increase in the energy of the inner part of beam with an increase in the channel radius, which reaches a value of 170 $\mu $m by the time of 1.32 ns. At this radius, the energy of the inner part of beam is more than 90% of the total energy of beam entering the target. The whole beam becomes composed almost of its inner part. The growth of the $e2$ energy fraction is saturated, and the $e3$ energy fraction is monotonically decreasing. At 1.75 ns, the channel radius becomes equal to the boundary Gaussian beam radius. From this moment, a new cycle of channel formation begins, accompanied by refraction of enter laser beam, directly, in the channel created during the first cycle. The average rate of channel depth increase in the time interval 1.32–1.75 ns is $1.4\xd7108$ cm s^{−1}, and this value is close to the rate at the initial stage of the first step of channel formation.

The dependencies of the channel characteristics on laser pulse intensity and target initial density were investigated by means numerical calculations performed for the ranges of intensity $I0$ from $1013$ to $1014$ W cm^{−2} and the initial density $\rho 0$ from 1 up to 10 mg cm^{−3}. Figure 7 shows the spatial distributions of plasma density and temperature at 1 and 1.5 ns, calculated using the PM^{2} code at $\rho 0=2$ mg cm^{−3} and $I0=1014$ W cm^{−2}.

With increasing intensity, all the processes of channel formation occur faster. The creation of a distinct channel configuration occurs at earlier moments in time. In the calculation at intensity of 10^{14} W cm^{−2}, the formation of a channel occurs at 0.2 ns, and this is much earlier than in the calculation discussed above at the same density of matter, but at intensity of $1013$ W cm^{−2} (0.6 ns). The completion of the first cycle of channel formation, when its radius reaches the radius of laser beam, occurs by about 1 ns. It is also much earlier than at an intensity of $1013$ W cm^{−2} (1.7 ns). The rates of increasing the channel depth and radius are, respectively, $3.5\xd7108$ and $3\xd7107$ cm s^{−1}. These values are $2.5\u22123$ times larger than calculated at intensity of $1013$ W cm^{−2}. The rate of increase in the channel depth increases with increasing intensity, approximately as $I1/2$, which is a slightly slower dependence compared to the dependence of the velocity of the plane heating wave [ $Di\u221dI3/5$, see (3)]. This is due to the fact that most of the incident radiation does not enter the channel, but is absorbed in the heated cone.

Figure 8 shows the spatial distributions of plasma density and temperature at 2 ns, calculated using the ATLANT-HE code at $\rho 0=4$ mg cm^{−3} and $I0=1013$ W cm^{−2}.

Calculations have shown that the 2D nature of the heating and movement of the target substance leads to a weaker dependence of the heating wave velocity in channel plasma on the initial density compared to solution (3) for a plane heating wave. The calculated dependence of the wave velocity along the *z*-axis on the density $\rho 0$ is close to an inversely proportional dependence, and the plane heating wave velocity depends on the density as $\rho 0\u22127/5$.

Calculations for the super-Gaussian intensity distribution over the beam cross section [ $I(s)=I0$ at $0\u2264s\u22641$, $I(s)=I0\u2009exp[\u2212(s3\u22121)2]$ at $1\u2264s\u22641.5$, $I(s)=0$ at $s>1.5$, where $s=r/a$, $a=100$ $\mu $m] showed results that practically do not differ from the case of a Gaussian beam. This once again confirms the decisive role of refraction in plasma heating, in which the intensity distribution in the incident beam at plasma boundary does not affect the spatial dependence of the energy release in the plasma.

## IV. CONCLUSION

Based on computational and theoretical studies of laser beam interaction with plasma of subcritical density, the effect of refractive channeling the heating wave in the central axial region of the plasma due to the reflection of rays on density gradient, formed during the hydrodynamic motion of the plasma, at the turning point was found. It is shown that refraction directed to the beam axis under the conditions of developed hydrodynamic motion leads to preferential heating of a central plasma region located near the beam axis. This provides an increased velocity of the longitudinal heating wave compared with the velocity of a peripheral part of the wave removed from the axis at a distance of the order of beam radius. This circumstance leads to the formation of a channel for propagating the laser beam near its axis. The channel length significantly exceeds the longitudinal size of the surrounding heated plasma. At the same time, part of the refracted radiation of the peripheral part of the beam has the ability to fall directly into such a channel, heat it, and further increase the propagation velocity of the heating wave.

It is shown that under the conditions of an experiment on the interaction of a laser beam of a nanosecond Nd–laser pulse with a plane target of subcritical density, the velocity of the longitudinal heating wave in the channel can be several times higher than the velocity of the peripheral part of the wave.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**S. Yu Gus'kov:** Formal analysis (equal); Supervision (lead); Writing – original draft (lead). **P. A. Kuchugov:** Investigation (equal); Software (equal); Validation (equal); Visualization (lead); Writing – review & editing (equal). **N. N. Demchenko:** Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal).

## DATA AVAILABILITY

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

## REFERENCES

*Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena*