Uniform line fillings

Deterministic fabrication of random metamaterials requires filling of a space with randomly oriented and randomly positioned chords with an on-average homogenous density and orientation, which is a nontrivial task. We describe a method to generate fillings with such chords, lines that run from edge to edge of the space, in any dimension. We prove that the method leads to random but on-average homogeneous and rotationally invariant fillings of circles, balls, and arbitrary-dimensional hyperballs from which other shapes such as rectangles and cuboids can be cut. We briefly sketch the historic context of Bertrand's paradox and Jaynes's solution by the principle of maximum ignorance. We analyze the statistical properties of the produced fillings, mapping out the density profile and the line-length distribution and comparing them to analytic expressions. We study the characteristic dimensions of the space between the chords by determining the largest enclosed circles and balls in this pore space, finding a lognormal distribution of the pore sizes. We apply the algorithm to the direct-laser-writing fabrication design of optical multiple-scattering samples as three-dimensional cubes of random but homogeneously positioned and oriented chords.

Figure 1

Bertrand's paradox: The top row illustrates three ways of filling a circle with random lines: (1) connect two random points on the circumference of the circle, (2) choose a random radius and angle from the center and then draw the line perpendicular to that, and (3) choose a random point inside the circle and find the chord of which the point is the midpoint. The bottom row shows a single realization with 500 lines according to the method above it. The resulting probability $p$ of finding a chord longer than the edge length of the enclosed equilateral triangle is different for the three methods. Jaynes argued that only the “radius method” (middle column) results in ensemble-averaged random line filling that is translationally invariant, i.e., uniformly dense.

Figure 2

Three-dimensional extension of the three methods of Bertrand. In (1) a line is drawn between two randomly chosen points at the surface of the sphere. In (2, 2*) a randomly oriented radial line is chosen and at a random radial distance $r$ a perpendicular line in arbitrary direction is selected. For method 2, $r$ is chosen from a uniform distribution $[0,1]$, while for method 2*, $r^2$ is chosen uniformly from $[0,1]$. In (3) a random point in the ball is chosen and then a random line with that point as midpoint is chosen. In 3D, both methods 1 and 2* give an on-average uniform distribution of lines.

Figure 3

Standard deviation over the mean of the line density for (a) 2D and (b) 3D. For 2D, Bertrands method 2 shows convergence, while both methods 1 and 2* show convergence for 3D.

Figure 4

(a) Probability density function of the angles of ${10}^7$ chords in a square produced with our Monte Carlo code following the “random radius” recipe (method 2), counting the contribution of each line once, together with the function $\frac{1}{4}\left(\right|sin\theta |+|cos\theta |)$ (red line). (b) The angular distribution weighing the lines with a factor proportional to their length. The result is a completely flat probability density function proving the angular homogeneity of the generated line filling.

Figure 5

Length distribution of a homogeneous distribution of random lines in a unit square. The symbols are the result of our Monte Carlo method, and the solid red line is the analytic result derived in the Appendix. The longest possible line has a length of $\sqrt{2}$ but has a vanishing probability. The integrable singularity at $l=1$ is caused by the many possible lines that run almost parallel to one of the sides and therefore have a length of slightly more, but never less than, 1. Inset: A set of randomly chosen lines in a unit square.

Figure 6

Length distribution of a homogeneous-on-average set of random lines in a unit cube. The symbols are the result of our Monte Carlo method, and the solid red line is the analytic result given in the Appendix. Inset: A set of randomly chosen lines in a unit cube.

Figure 7

Characterization of the void space by drawing the largest circles in the voids enclosed by the chords. In this sample distribution of 100 chords in a circle with unit radius, 1859 voids are identified. The line colors have no meaning and only serve to increase visibility. Inscribed circles cutting the large outer circle are discarded to avoid edge effects.

Figure 8

Probability density function of void radii in a 3D sphere of unit radius filled with a line density of $\rho =95$, 318, and 796 unit lengths per unit volume (symbols). The distributions are fitted with lognormal distributions (solid lines).

Figure 9

The most frequent radius $r_{\mathrm{MF}}$ in the probability density function of void sizes in a 3D sphere filled with a line density $\rho$ in unit length per unit volume in combination with the predicted scaling behavior $1/\sqrt{\rho }$.

Figure 10

Line coordinates acquired with method 2* and fabrication design for DLW. The line coordinates of this methods can be used to create a deterministic scattering medium composed with polymer lines, ensuring uniformity and rotational translation invariant despite its finite volume ${\left(20\mu \mathrm{m}\right)}^3$.

Figure 11

Construction to derive the length $L$ of a line segment (red) extending from edge to edge of a unit square (thick black) for a given angle $\varphi (0<\varphi <\pi /4)$ with the vertical and an intersection with the diagonal (dotted, blue) parametrized by $a$.

Figure 12

(a) The line length $l$ in a unit square as a function of the angle and parametrized entrance position, $a$. (b) The cumulative density function, $F_{L,\mathrm{\Phi }}(l,\varphi )$, for the probabilities to find a line up to length $l$, still as a function of the angle $\varphi$. Integration over $\varphi$, taking into account the a priori probability to find a line with angle $\varphi$, will yield the final $F_L\left(l\right)$.