Buffon's Needle on Caustics and Torus Quantization

Abstract

The probability of n + 1 intersections of a long needle in the Buffon problem is the eikonal of a Hankel function which is the principal term in the uniform asymptotic expansion in powers of the small distance between the parallel lines. Evaluating this probability using the torus quantization conditions shows that in the physically meaningful region, where a closed convergence of rays covers the caustic circle, the probability is greater than unity. In addition, the method of steepest descent shows that the caustic and reflection indices are more general than the ones given by torus quantization. The distance from the light source to the center of caustic circle corresponds to the length of the needle, and n times the distance between the parallel lines is the radius of the caustic. Unlike diffraction problems, the solution cannot be extended to the shadow zone since the angles become imaginary. In the continuum limit where the distance between the parallels tends to zero, the number of intersections is governed by an arc sine law in which maximum number of intersections, or the maximum chord length in a circle of a given radius, is most probable.