Fig. 1: Evolution of scattering networks.
a, A material composed of n identical point particles in the spatial domain Ω. b–d, Three different states of impulse scattering responses depending on the spatial ordering of materials: forward (|k′| < \(\sqrt 2\)ko for 0 ≤ θ ≤ π/2 or 3π/2 ≤ θ ≤ 2π) (b), backward (\(\sqrt 2\)ko ≤ |k′| < 2ko for π/2 ≤ θ ≤ 3π/2) (c) and zero (|k′| ≥ 2ko) (d) scattering states. Yellow points denote the impulse Sn(k) = δ(k – k′). Red arrows represent the allowed incident and scattering wavevectors connected through the impulse scattering response ±k′. Red, white and black solid circles have the radii of ko, \(\sqrt 2\)ko and 2ko, respectively, where the red one represents the light cone. Red dashed circles are the shifted light cones due to the scattering events described by Sn(k) = δ(k – k′). e, An (n + 1)-particle material evolved from the material in a, by adding the (n + 1)th point particle (blue sphere). f, Network modeling of wave scattering from a material with scatterer nodes and k-dependent links. Orange and green arrows denote incident and scattering waves, respectively. Red and blue solid lines represent the positive and negative signs of existing link weights defined by equation (2), respectively. Red and blue arrows also represent the positive and negative signs of newly included link weights after adding the (n + 1)th particle, respectively. Only the links with large values of |wp,qK| are assumed to be plotted because a scattering network is fully connected. The black arrow describes the k-impulse component cos[k ∙ (rp – rq)] of the link weight between the pth and qth particles. The transparency of the solid lines and arrows denotes the magnitude of the weights.
