Steepest-descent algorithm for simulating plasma-wave caustics via metaplectic geometrical optics

The design and optimization of radiofrequency-wave systems for fusion applications is often performed using ray-tracing codes, which rely on the geometrical-optics (GO) approximation. However, GO fails at wave cutoffs and caustics. To accurately model the wave behavior in these regions, more advanced and computationally expensive “full-wave” simulations are typically used, but this is not strictly necessary. A new generalized formulation called metaplectic geometrical optics (MGO) has been proposed that reinstates GO near caustics. The MGO framework yields an integral representation of the wavefield that must be evaluated numerically in general. We present an algorithm for computing these integrals using Gauss-Freud quadrature along the steepest-descent contours. Benchmarking is performed on the standard Airy problem, for which the exact solution is known analytically. The numerical MGO solution provided by the new algorithm agrees remarkably well with the exact solution and significantly improves on previously derived analytical approximations of the MGO integral.

Figure 1

Top row: The distribution of the wave intensity in the phase space $(x,k)$ for a wave near a fold-type Airy caustic (see Sec. 4b): (a) The exact Wigner function $W_{\psi }\propto \mathrm{Ai}\left[\mathcal{D}(x,k)\right]$; (b) GO solution $W_{\psi }\sim \delta \left[\mathcal{D}(x,k)\right]$; (c) GO solution in the phase space rotated by $\pi /2$. Bottom row: The corresponding wave fields: (d) the exact field ${\left|\psi \left(x\right)\right|}^2$; (e) the field intensity ${\left|\varphi \left(x\right)\right|}^2\doteq {\langle \left|\psi \left(x\right)\right|}^2\rangle$ in the GO approximation (where the angular brackets denote averaging over the local wavelength); (f) the field intensity in the spectral representation, $|\tilde{\varphi }{\left(k\right)|}^2\doteq \langle |\tilde{\psi }\left(k\right){|}^2\rangle$. The dispersion manifold $\mathcal{D}(x,k)=0$ is shown as the black dashed line. Clearly, ${\left|\varphi \left(x\right)\right|}^2$ diverges at $x\to 0$, but $|\tilde{\varphi }{\left(k\right)|}^2$ is well behaved everywhere.

Figure 2

Steepest-descent contours (orange) for the integrand phase function $f\left(\kappa \right)\doteq {\kappa }^a$ of Eq. (49) for various values of the parameter $a$, which characterizes the saddlepoint degeneracy. The background color represents the magnitude of the integrand $-\text{Im}\left(f\right)$, with green corresponding to larger values and blue corresponding to smaller values. The order $n=5$ quadrature nodes are shown as white dots, while the points $\kappa \left(l_{\pm }\right)$ [Eq. (42)] that are used to determine the rotation angles ${\sigma }_{\pm }$ [Eq. (44)] are shown as black dots. Unused steepest-descent contours are shown as dashed orange lines. As can be seen, the steepest-descent contour has a kink when $a$ is odd, which requires ${\sigma }_{+}$ and ${\sigma }_{-}$ to be calculated separately.

Figure 3

Comparison of the error in computing Eq. (49) using the quadrature rule of Eq. (47) for various values of $a$ and $b$. The error metric used is the relative error when $I(a,b)$ is nonzero and the absolute error otherwise. The shaded gray region marks the range of error over the entire range $b\in [0,2n-1]$ for which our quadrature rule is expected to be exact, while the dashed black lines bound the region obtained when only even values of $b$ are considered. The precision of the quadrature nodes and weights used is ${10}^{-15}$.

Figure 4

Same as Fig. 2 for the phase function $f(\epsilon ,p)$ [Eq. (60)] at various values of $p$. The white dots correspond here to the $n=10$ quadrature nodes. The steepest-descent contours evolve smoothly with $p$ and ultimately coalesce into a fold-type $A_2$ caustic at $p=0$.

Figure 5

(a) Comparison of the numerical MGO solution (orange) with the analytically approximated MGO solution (62) [31] (dashed pink), the standard GO solution (61) (dashed gray), and the exact solution (57) (black) for Airy's equation (55). The numerical MGO solution was obtained by applying the quadrature rule of Eq. (47) with order $n=10$ to Eqs. (58) and (59). The numerical MGO solution displays remarkable agreement with the exact solution compared with the analytical approximations, even near the fold-type caustic at $q=0$. (b) Error of the numerical MGO solution with respect to a scan over the quadrature order $n$. Note that the “pseudo error” is defined as the relative error with respect to the $n=10$ solution used in (a).