Figure 4
Frame (a) shows how rays sent straight downwards toward the left end cap refract when hitting the scatterer. Frame (b) indicates the length of the each ray, l, inside the scatterer as a function of start position \(x_0\) with a logarithmic y-axis. Frame (c) also indicates the length of each ray as a function of start position. A selected interval within the \(x_0\)-interval in frame (b) has been magnified, a new \(x_0\)-interval is selected and magnified and so on. In frame (c), the \(x_0\)-interval has been magnified 7 times. The fractal structure is preserved. (See Supplementary Materials Sec. E for further magnifications revealing the fractal structure.) Frame (d) shows the determination of the fractal dimension of the set of long-lived rays based on the box-counting method. The blue and red lines show \(\ln (N)\) as a function of \(-\ln (\delta )\) where N is the number of intervals that contain at least one long-lived trajectory and \(\delta\) is the width of the interval. The slope of the blue and red lines are indicating the fractal dimension. The yellow line indicates that the slope is 0.65. \(N_{\mathrm{{10M}}}\), the blue line, indicates N for the case where 10 million rays were started in the interval and \(N_{\mathrm{{100M}}}\), the red line, where 100 million rays were started in the same interval. Frame (e) shows the average of \(\ln (D)\), where D is the distance between the two rays, as a function of path length, l, for 10 pairs of rays. The red, dashed line indicates the slope, i.e. the Lyapunov exponent, which in this case is 0.36. For all the frames, the refractive index of the scatterer is 1.8. Classical ray tracing investigations are wavelength independent and the length of the straight part of the stadium is 5 times the radius of the end caps. This corresponds to a system where \(d=50 \, {\upmu \rm m}\) and \(r=10\,\upmu \rm m\).
