Wave-based laser absorption method for high-order transport–hydrodynamic codes

Abstract

Models of the laser propagation and absorption are a crucial part of the laser–plasma interaction models. Hydrodynamic codes are afflicted by usage of the simplified, not self-consistent, models of the geometrical optics, limiting their physical accuracy. A robust and efficient method is presented for computing the stationary wave solution, not restricted to this field of application exclusively. The method combines the semi-analytic and high-order differential approaches to benefit from both. Flexibility of the discretization is maintained, including the discontinuous methods. Performance of the model is evaluated for the problem of a transition layer by comparison with the analytic solution. Reliable results on coarse computational meshes and high convergence rates on fine meshes are obtained. The relevance to the current fusion research and non-local energy transport is pointed out.

Appendix: Analytic solution of the absorption in the transition layer

The absorption in the transition layer with the dielectric constant profile defined by (28) has an analytic solution as described in [7, 14]. Specifically, the form used in this text represents the antisymmetric case, where ε monotonously changes from ε^l to ε^r. The solution is not derived directly for the Maxwell equations (2–3), but the Helmholtz equation (1) for E arising from them, which is being equivalent provided the solution is sufficiently smooth. To briefly outline the inference of the solution, the exchange of the primary variable for γ = exp(ζ) yields with $E(x)=\bar {E}(\gamma )$:

$$ \bar{E}^{\prime\prime} + \frac{1}{\gamma}\bar{E}^{\prime} + \frac{{\Delta} \xi^{2}}{\gamma}\left( \frac{\varepsilon_{l}}{\gamma} + \frac{\varepsilon_{r}-\varepsilon_{l}}{\gamma+ 1} \right)\bar{E}= 0 ~. $$

(1)

The substitution $\bar {E}(\gamma )=\gamma ^{a}f(\gamma )$ gives the hypergeometric equation:

$$ \gamma(\gamma+ 1)f^{\prime\prime}+(2a + 1)(\gamma+ 1)f^{\prime}+(a+b)(a-b)f = 0 ~, $$

(2)

where $a=i{\Delta } \xi \sqrt {\varepsilon _{l}}$ and $b=i{\Delta } \xi \sqrt {\varepsilon _{r}}$. The solution can be found in terms of the hypergeometric functions then. The structure of the solution is the following:

$$ \bar{E} = \left\{\begin{array}{ll} C_{E} (P_{1} + C_{PP} P_{2}) & \gamma < 1~(x < x_{0})\\ C_{E} C_{PQ} Q_{2} & \gamma \geq 1~(x \geq x_{0}) \end{array}\right. ~, $$

(3)

where P₁ is the incoming wave (from the left-hand side), P₂ reflected, and Q₂ refracted (₂F₁ is the Gaussian hypergeometric function):

$$\begin{array}{@{}rcl@{}} P_{1} &=& \gamma^{a} {{~}_{2}F_{1}}\left( \left.\begin{array}{cc} a+b, a-b\\ 2a + 1& \end{array}\right|- \gamma \right) \end{array} $$

(4)

$$\begin{array}{@{}rcl@{}} P_{2} &=& \gamma^{-a} {{~}_{2}F_{1}}\left( \left.\begin{array}{cc} -a+b, -a-b\\ -2a + 1 & \end{array}\right|- \gamma \right) \end{array} $$

(5)

$$\begin{array}{@{}rcl@{}} Q_{2} &=& \gamma^{b} {{~}_{2}F_{1}}\left( \left.\begin{array}{cc} a-b, -a-b\\ -2b + 1 & \end{array}= \right|-\gamma^{-1} \right) \end{array} $$

(6)

The constants C_PP and C_PQ denote respectively the total reflection coefficient and the scaling constant to match the limits of the two branches of the solution at x₀. The latter must be evaluated directly by comparison of the two branches at x₀ and the former is given by the analytic formula [14]:

$$ C_{PP}=\frac{{\Gamma}(2a){\Gamma}(-a-b){\Gamma}(-a-b + 1)}{{\Gamma}(-2a){\Gamma}(a-b){\Gamma}(a-b + 1)} ~. $$

(7)

Fig. 4

The profiles of (a) the electric field $\tilde {E}$, (b) the magnetic field $\tilde {H}$, (c) the incoming wave function $\tilde {P}$, and (d) the reflection coefficient $\tilde {V}$ for the transition layer problem from Section 3. The real parts are plotted by full lines and the imaginary ones are dashed. See the accompanying text for more details

The constant C_E serves for scaling the solution to match it with the electric field of the incoming wave in the limit x →−∞, i.e., $C_{E}=\displaystyle \lim _{x\rightarrow -\infty }P(x)/P_{1}(x)$. However, P cannot be identified with P₁ in general, but only in the limit of the homogeneous medium, where the two functions have the limit behavior $\sim \exp (ik_{0}\sqrt {\varepsilon _{l}} x)$. The proper transforming relations must be used otherwise following (3) and (4):

$$\begin{array}{@{}rcl@{}} P &=& \frac{1}{2}(E-H/\beta) = \frac{1}{2}\left( E-\frac{i}{\beta k_{0}}E^{\prime}\right) ~, \end{array} $$

(8)

$$\begin{array}{@{}rcl@{}} R &=& \frac{1}{2}(E+H/\beta) = \frac{1}{2}\left( E+\frac{i}{\beta k_{0}}E^{\prime}\right) ~. \end{array} $$

(9)

These relations allow determination of C_E at x_min instead of the limit and satisfying the boundary condition directly.

The analytic profiles of the electric and magnetic fields $\tilde {E},\tilde {H}$ and the functions $\tilde {P},\tilde {V}$ are plotted in Fig. fig˙app˙EHPV for the parameters given in Section 3. The profiles are normalized to give $\tilde {P}(\xi _{\min })= 1$. Although not being of the main interest, the dependency of the reflection coefficient $\tilde {V}$ on the width of the transition layer σ can be clearly identified. This behavior is in agreement with the limit $\displaystyle \lim _{x\rightarrow -\infty } V(x)=C_{PP}$ governed by (7).