Active flux schemes on moving meshes with applications to geometric optics

An Energy Conservative hp-method for Liouville’s Equation of Geometrical Optics

“…Recently, numerical schemes for Liouville's equation were developed that incorporate the optical interfaces. In [33] van Lith et al derived a first-order upwind finite difference scheme and in [32] van Lith et al introduced a third-order active flux finite volume scheme on moving meshes. The third-order active flux scheme was proved to be faster and more accurate compared to classical ray tracing for obtaining an energy distribution on phase space.…”

“…This restriction invites us to use spherical coordinates to represent the momentum vector p as p = ( p, p z ) = n (sin θ cos ϕ, sin θ sin ϕ, cos θ ) , (7) where θ represents the polar angle, describing the angle between the z-axis and p measured from the z-axis, and ϕ the azimuthal angle for describing the direction in the q-plane. Therefore, at a given position z 0 along the optical axis, one can visualise the phase space coordinates on the screen that is perpendicular to the z-axis and intersects the z-axis at z 0 , where q is the position on the screen and p describes the projection of p on the screen [32]. The restriction of the momentum for physical rays p to Descartes' sphere also implies that the two-dimensional momentum vector is restricted by | p| ≤ n, describing a region known as Descartes' disc [35].…”

“…In the expression for ψ the plus sign should be taken for rays that propagate in the positive z-direction, while the minus sign should be taken for rays that propagate in the negative z-direction. Furthermore, the sign of n 2 , i.e., sgn(n 2 ), in the first case of (51a) can be used to accommodate embedded mirrors in a medium of refractive index n 1 ≥ 1 by taking n 2 = −n 1 , see [32]. Note that there is a so-called critical momentum p c when n 1 ≥ n 2 so that δ = 0.…”

“…To illustrate that the strategy outlined in Sect 4 for handling optical interfaces is energy conservative, we apply it to a test case. The test case 'bucket of water' introduced by van Lith et al [32,33] is a suitable choice. The refractive index for this problem is given by…”

An energy conservative hp-scheme for light propagation using Liouville’s equation for geometrical optics

“…Standard forward ray-tracing methods for geometrical optics, based on Monte Carlo methods with bin counting, are in general not energy conservative and exhibit a slow convergence. A recently new approach is based on a phasespace description of light, where each light ray propagating through an optical system evolves according to an optical Hamiltonian [1,2]. The 'time' parameter for Hamilton's equations can be the arc length of the ray, or in our case it will be the distance along a chosen optical axis, denoted by z.…”