1. INTRODUCTION
The rapid development of ultrahigh-intensity (I > 1018 W/cm2) short-pulse laser, whose intensity greatly exceeding the strength required to induce atomic-scale relativistic behavior, has greatly motivated the discovery of laser–plasma interaction. Many researches, including fast electron production and transport during the interaction (Cohen et al., Reference Cohen, Kemp and Divol2010; Sakagami et al., Reference Sakagami, Okada, Kaseda, Taguchi and Johzaki2012; Fox et al., Reference Fox, Robinson and Pasley2013), ion acceleration driven by intense laser pulse (Lefebvre et al., Reference Lefebvre, Gremillet, Lévy, Nuter, Antici, Carrié, Ceccotti, Drouin, Fuchs, Malka and Neely2010; d'Humières et al., Reference d'Humières, Brantov, Bychenkov and Tikhonchuk2013; Macchi et al., Reference Macchi, Borghesi and Passoni2013), have been extensively studied as the potential applications in fast ignition of inertial confinement fusion (JinQing et al., Reference Jinqing, Weimin, Lihua, Zongqing, Leifeng, Lianqiang, Dongxiao, Xiaolin, Bin and Yuqiu2012; Robinson et al., Reference Robinson, Strozzi, Davies, Gremillet, Honrubia, Johzaki, Kingham, Sherlock and Solodov2014) and other related fields (Daido, Reference Daido2012; Jinqing et al., Reference Jinqing, Weimin, Lihua, Zongqing, Leifeng, Lianqiang, Dongxiao, Xiaolin, Bin and Yuqiu2012). In laser–plasma interaction, ionization plays a critical role on plasma formation. And the formed plasma will further affect the laser–plasma interaction (Lichtenberg, Reference Lichtenberg2005). Hence, understanding the whole interaction process including ionization is critical for the researches on laser–plasma interaction and particle acceleration.
In modeling the laser–plasma interaction with particle-in-cell (PIC) code, the target is usually assumed to be pre-ionized plasma. This is because the ionization process generally has finished before the main laser pulse reaching the target (Esarey et al., Reference Esarey, Schroeder and Leemans2009). However, ionization will affect the propagation of the main laser pulse in the plasma and the subsequent processes, including electron impact ionization, Coulomb collision, hot electron production, scattering, and so on. Thus, assuming pre-ionized plasma will ignore many interesting physics phenomena. It is necessary to study the ionization process in laser–plasma interaction. Effects of field ionization on the acceleration process (McGuffey et al., Reference McGuffey, Thomas, Schumaker, Matsuoka, Chvykov, Dollar, Kalintchenko, Yanovsky, Maksimchuk, Krushelnick, Glazyrin and Karpeev2010; Psikal et al., Reference Psikal, Klimo and Limpouch2011, Reference Psikal, Klimo and Limpouch2012) and effects of laser pulse on field ionization process (Petrov et al., Reference Petrov, Davis and Petrova2009) have been considered in the past. However, to the best of our knowledge, the effect of target density, which is also important to understand the whole interaction process, has not been analyzed before.
In this paper, we studied the effects of target density and laser pulse intensity on field ionization with PIC/Monte Carlo collisional (MCC) simulations using a relativistic and fully electromagnetic 1D3V PIC/MCC code Bumblebee (Xiaolin et al., Reference Xiaolin, Tao, Wenlong, Jianqing, Hualiang, Bin and Zhonghai2015). The Ammosov-Delone-Krainov (ADK) rate formula (Ammosov & Krainov, Reference Ammosov, Delone and Krainov1986) is applied to model the field ionization process. The helium (He) material is used in the simulation to discuss the physical issues. The field ionization model is illustrated in Section 2. To illustrate the effect of target density, we will present the ion density and its change rate for different target densities at a given laser intensity in Section 3. The results of different laser intensities are compared in Section 4.
2. DYNAMICS MODEL OF FIELD IONIZATION
The field ionization model has been implemented in Bumblebee. Ionization rate is given by the ADK formula (Kemp et al., Reference Kemp, Pfund and Meyer-ter-Vehn2004)
$$\eqalign{{\rm \omega} _{\rm i} = 6.6 \times 10^{16}\displaystyle{{Z^2} \over {n_{{\rm eff}}^{4.5}}} \left[ {10.87\displaystyle{{Z^3} \over {n_{{\rm eff}}^4}} \displaystyle{{E_{\rm h}} \over {\left \vert {\mathop {E} \limits ^{\rightharpoonup}} (t) \right \vert}}} \right]^{2n_{{\rm eff}} - 1.5} &\cr \times \exp \left[ { - \displaystyle{2 \over 3}\displaystyle{{Z^3} \over {n_{{\rm eff}}^3}} \displaystyle{{E_{\rm h}} \over {\left \vert {\mathop {E} \limits^{\rightharpoonup}(t)} \right \vert}}} \right], &}$$
where
$E_{\rm h} \equiv m_{\rm e}^2 e^5\hbar ^{ - 4} = 5.14 \times 10^9\,{\rm V}/{\rm cm}$
is the atomic field strength, e and m
e are the charge and mass of electron, respectively,
$\hbar $
is Planck constant,
$n_{{\rm eff}} = Z/\sqrt {W_{\rm i}/I_{\rm H}} $
is the effective main quantum number of the new produced electron by ionization (called “ionized electron” below), Z is the ion charge after ionization, W
i is the ionization energy for a given process, I
H is the ionization potential of hydrogen and
${\mathop {E} \limits^{\rightharpoonup} (t)}$
is the instantaneous electric field at the position of the traced particle (ion or atom).
In order to fulfill energy conservation, energy loss of field ionization ΔW
ion is assumed to be equal to the electromagnetic field energy change ΔW
field in each cell and each time step, that is, ΔW
field = ΔW
ion. An artificial ionization current density
${\mathop {J} \limits^{\rightharpoonup}}_{\rm ion}$
, which is directed along the electric field
${\mathop {E} \limits^{\rightharpoonup} (t)}$
, is introduced to calculate the field energy change during ionizations (Kemp et al., Reference Kemp, Pfund and Meyer-ter-Vehn2004). The ionization current density is determined by
$$\int {dt} \int {dV{{\mathop {J} \limits^{\rightharpoonup}}}_{{\rm ion}}(t) \cdot {\mathop {E} \limits^{\rightharpoonup}}(t)} = \Delta W_{{\rm field}},$$
where the integrals are taken over one time step and one cell, respectively. The energy loss of field ionization is given by
$$\Delta W_{{\rm ion}}{\rm =} \sum\limits_j {w_j\left( {W_{i,j} + {\rm \varepsilon} _j} \right)}.$$
Here
$\sum\nolimits_j {} $
is the sum over all ionization processes in one cell and one time step, w
j
and W
i,j
are the weight of the ionized macro-particle (called “particle” below) and the ionization energy for the jth ionization event, respectively, ε
j
is the initial kinetic energy of the ionized electron. We assume that the velocity direction of the ionized electron is parallel to that of ion and the velocity magnitude is equal to that of ion. Then the value of ε
j
can be obtained. According to energy conservation, the relation between ionization current density and electric field should satisfy
$${\mathop {J} \limits^{\rightharpoonup}}_{{\rm ion}}\left( t \right) \cdot {\mathop {E} \limits^{\rightharpoonup}} \left( t \right)\Delta t\Delta V = \sum\limits_j {w_j\left( {W_{i,j} + {\rm \varepsilon} _j} \right)},$$
where Δt and ΔV are the time step and the cell size in PIC simulation, respectively. Equation (4) can be deduced to
$${\mathop {J} \limits^{\rightharpoonup}}_{\rm ion}\left( t \right) = \displaystyle{{\sum\nolimits_j {w_j(W_{i,j} + {\rm \varepsilon} _j)} } \over {\Delta t\Delta V{\left\vert {\mathop {E} \limits^{\rightharpoonup} (t)} \right\vert}^2}}{\mathop {E} \limits ^{\rightharpoonup}} \left( t \right).$$
By adding the ionization current density to the deposited particle current density, the physical current in each cell and each time step can be obtained.
3. EFFECT OF TARGET DENSITY
3.1. Simulation model
A laser pulse of 20T is used in the simulations, where T is the laser period. Figure 1 gives the configurations of the laser and target. The laser wavelength is λ = 0.8 µm. The laser pulse envelope rises to peak in the first 5T and maintains in the next 10T, then drops to zero in the last 5T. The peak laser pulse intensity is given by Iλ2 = a 21.37 × 1018 W µm2/cm2, here a is the dimensionless maximum field amplitude. The laser is normally incident on a 30λ thick neutral target, corresponding to the blue region in Figure 1. The neutral atom density n is normalized by the critical density n c, which is defined as n c = ε 0 m eω2/e 2, ε0 is the permittivity of free space, and ω is the laser frequency.
Fig. 1. Laser and target configuration.
The ionization electric field for He2+ and He+ are 2.57 × 1011 and 1.05 × 1011 V/m, respectively, which can be estimated using the formula E i = (W i/2I h)2 E h/4i (Petrov et al., Reference Petrov, Davis and Petrova2009). And i represents the ion charge after ionization. The incident laser electric field amplitude (E l = am eωc/e ≈ 2.2 × 1012 V/m) is almost 22 and 8.6 times the ionization electric field for He+ and He2+, respectively. Therefore, the helium atom can be fully ionized if the laser interacts with the neutral target completely.
3.2. Results and discussion
The ionization rate is independent on the target density as seen in Eq. (1). However, target density affects the transport of laser pulse and further affects the field ionization process. Based on this consideration, we investigated the influences of different initial target densities on field ionization. We firstly analyzed the evolution of ion density to understand the ionization process. After that, to illustrate the ionization speed, the space-averaged ion density and the ion density change rate of He gas target, particularly near the target front surface (represented by a 1λ region, see the shadowed region of Fig. 1), were discussed.
We considered different He gas target densities (n = 0.1, n = 1 and n = 10) with a constant laser intensity a = 0.5. The neutral atom density n is normalized by the critical density n c. The spatio-temporal evolutions of He+ and He2+ density are shown in Figure 2. Two factors affect the ionizations of He atoms, one is the laser field, and the other is the electrostatic field caused by the transport of energetic electrons.
Fig. 2. The spatio-temporal evolutions of He+ for (a) n = 0.1; (b) n = 1; (c) n = 10. The spatio-temporal evolutions of He2+ (bottom) for (d) n = 0.1; (e) n = 1; (f) n = 10. The color bar represents the particle number density.
Firstly, we will discuss the effect of laser field. He+ appearance and disappearance processes are clearly shown in Figure 2a. At the beginning, no ionization happens, as the laser field is not strong enough to ionize the target. Shortly after the laser intensity reaches the field-ionization intensity threshold (Posthumus et al., Reference Posthumus, Frasinski, Giles and Codling1995) (t ~ 5.2T, see the inset of the figure), He+ ions begin to appear. It should be noted that the laser intensity does not reach the threshold when the laser just arrives at the target (t = 5T), because there is vacuum region of 5λ on the left of the simulation region and the laser pulse envelope is not a flat-top at the first 5T (see Fig. 1). He+ is ionized to He2+ as the laser transmitting into the target after 5.2T. As a result, He+ ions decrease with time, as seen in Figure 2a. And He2+ ions are observed at x ≈ 5.2λ with the density continuously increasing from 5.6 to 5.7T, then the density of He2+ maintains at 0.1n c throughout the simulation, see Figure 2d. When atom density is approximately equal to n c (n = 1), He+ almost cannot be ionized to He2+ behind x = 5.7λ (Fig. 2e), so most He+ ions remain unchanged (Fig. 2b) as only a small part of laser energy could transport into the target. For the case of n = 10, the laser could only propagate into the target for a short distance (nearly the plasma skin length). As a result, the laser pulse is stopped at such high density. Thus, only a very thin target layer can be ionized at n = 10 (right column of Fig. 2). In conclusion, the thickness of the ionized region is determined by laser parameters and target density.
After the laser irradiation is over, the ions are affected by the electrostatic field caused by the transport of energetic electrons. As can be seen in the right two plots of Figure 2, He+ ions increase with time till the atoms being fully ionized to He+ at density ~10, even no incident laser after t = 25T, which is different from the case of He2+. Ionized electrons are accelerated forward by laser pulse after the generation of He+, resulting in the formation of space-charge field inside the target. The space-charge field further ionizes He atoms to He+ after the laser irradiation is over, while the field amplitude has not reached the requirement for second ionization. Thus, the region of He+ is much larger than that of He2+. Moreover, we can find a small region with high density of He2+ (n ~ 28) in Figure 2f. To find out the reason of such high density, we plot the temporal evolutions of the total particle numbers in the simulation region in Figure 3. We can clearly see that the total number of He2+ nearly sustains constant for n = 10. Therefore, the high density of He2+ is resulted from ions bunching, which is caused by electrostatic field push.
Fig. 3. The temporal evolutions of He+ and He2+ numbers.
We note that the target front surface (see the shadowed region of Fig. 1) is the initial location of the ionization process and the most intense part of the laser–plasma interaction. Additionally, the tendency of ion density versus time shares similar characteristic at different positions for low target density (see Fig. 2a, 2d). Hence, the space-averaged ion densities at four typical positions near the target front surface, which are calculated over an interval of 0.25λ and presented in Figure 4, have been obtained to analyze the ionization process. For clarity, the averaged ion density is plotted only before 10T for n = 0.1 because of its unchanged tendency after 10T.
Fig. 4. The temporal evolutions of He+ average density for (a) n = 0.1; (b) n = 1; (c) n = 10. The temporal evolutions of He2+ average density (bottom) for (d) n = 0.1; (e) n = 1; (f) n = 10.
As we can see, the averaged ion densities for various target densities in Figure 4 are significantly different. For He+ (Fig. 4a–c), the averaged density becomes difficult to decrease with increasing initial target density after it reaches the peak value, and even increase with time for n = 10. The reason is the same as the explanation of Figure 2, that is, the higher the target density is, the less the laser energy could transport into the target. Hence, He+ is difficult to be ionized to He2+. For He2+ in Figure 4f), due to the effect of skin depth, He+ ions can be still ionized to He2+ ions within the skin length for n = 10. Therefore, we can obviously observe a black line whose peak value is higher than other lines in Figure 4f. In summary, the results in Figure 4 are generally consistent with the analysis of Figure 2.
In addition, one can see from Figure 4f that the black line increases as a stair-step shape, and the oscillation frequency is twice of the laser frequency. This is probably caused by the injected laser with a sin-shaped ramp amplitude at the laser pulse front. The incident laser amplitude profile with time is shown as Figure 5a, and the space-averaged total field energy (E 2 + B 2, E and B are the total electric field and magnetic field, and normalized by m eωc/e and m eω/e, respectively) versus time in the plasma region is shown in Figure 5b. The black lines (x = 5.125λ) in Figures 5b and 4f display the same stair-step shape, which means the laser field stimulates He2+ density distribution of x = 5.125λ in Figure 4f. Compared with the result of x = 5.125λ, the space-averaged field energy for the other three curves in Figure 5b is almost equal to zero. For the cases of n = 1 and n = 10 (corresponding target density n ≥ n c ), atoms at the target front are ionized quickly. The ion density consequently increases with time and becomes higher than critical density n c , as a result, the laser can hardly propagate into the target. As the laser field has been stopped by high-density plasma, no effective ionization events take place inside the target except the target front surface.
Fig. 5. The temporal evolutions of (a) the laser pulse amplitude and (b) the field amplitude with the density n = 10.
Figure 6 shows the temporal evolution of the ion density change rate for He+ and He2+ with lower initial target density n < n c near the target front surface (x < 0.1λ). The ion density change rate is calculated by (n k − n k−1)/Δt, where n k is the averaged density for the kth time step. In summary, the ion density change rate changes much faster with increasing target density for n < n c.
Fig. 6. The ion density change rates of (a) He+ and (b) He2+ for different initial target densities as x < 0.1λ.
4. EFFECT OF LASER PULSE INTENSITY
To observe the effect of laser pulse intensity on ionization, the target density is fixed to n = 10 with laser intensity varying from a = 2 to a = 10 in this section. Figure 7 shows the spatio-temporal density evolutions of He+ and He2+ at different laser intensities. In laser–plasma interaction, the electron ponderomotive temperature T e is proportional to the square root of the laser intensity I. Therefore, higher laser intensity results in higher temperature of forward electron and consequently larger space-charge field, which ionizes He+ to He2+. Hence, larger ionized region is formed, which can be found from He+ and He2+ density distributions for different laser intensities in Figures 2c, 2f and 7. Also, we can find that high-density region of He2+ for a = 2 and a = 10 (Fig. 7c, 7d) is larger than the case of a = 0.5 (Fig. 2f). The formation of such high density has been discussed above.
Fig. 7. The spatio-temporal evolutions of He+ for (a) a = 2; (b) a = 10. The spatio-temporal evolutions of He2+ (bottom) for (c) a = 2; (d) a = 10. The color bar represents the number particle density.
The space-averaged ion density averaged over an interval of 0.25λ at different positions is shown in Figure 8. The results in Figure 8 are consistent with the analysis of Figure 7. Furthermore, the comparison of averaged density of He+ in Figures 4c, 8a, and 8b indicates that ionization process is much faster for higher laser intensity.
Fig. 8. The temporal evolutions of He+ average density for (a) a = 2; (b) a = 10. The temporal evolutions of He2+ average density (bottom) for (c) a = 2; (d) a = 10.
5. CONCLUSIONS
Plasma formation due to field ionization has been studied with 1D3V PIC/MCC code Bumblebee in this paper. The effects of target density and laser intensity on ionization have been discussed in detail. The simulation results indicate that the laser intensity will affect field ionization as expected. Besides, the target density will also affect field ionization. For the case of n ≥ n c, an ionization region is formed near the target front surface. While for the case of n < n c, the ionization speed increases with the initial target density. In addition, the greater the laser intensity is, the faster the ionization speed and the larger the ionization region will be. This work may be benefit for the understanding of plasma formation with intense laser pulse and the setup of initial plasma distribution before numerical modeling of the laser–plasma interaction.
ACKNOWLEDGMENT
This work was supported by National Natural Science Foundation of China (Grant Nos 61201003 and 61301054).
SUPPLEMENTARY MATERIAL
The supplementary material for this article can be found at http://dx.doi.org/10.1017/S0263034616000392.
