Supernumerary spacing of rainbows produced by an elliptical-cross-section cylinder I Theory

Supernumerary spacing of rainbows produced by an elliptical-cross-section cylinder II Experiment

“…Similarly, in Ref. 6 we found that the supernumerary spacing parameter of the p ϭ 2 rainbow is approximated accurately in the geometrical-optics regime for small ⑀ and 1.25 Յ n Յ 1.7 by…”

“…6 we theoretically examined the dependence of the supernumerary spacing parameter h of the first-order rainbow of an elliptical cross-sectional cylinder on the cylinder's rotation angle . We numerically traced a collection of closely spaced rays in the vicinity of the rainbow ray through the cylinder and computed the optical path length of the rays with respect to that of the rainbow ray.…”

“…The principal result of Ref. 6 was that h is a remarkably delicate feature of the rainbow caustic, and its accurate determination requires that the relative position of a number of supernumerary maxima be measured with great accuracy. Similarly, our principal result here is that, although small local inhomogeneities in the rod's refractive index affect the rainbow angle only negligibly, the supernumerary spacing parameter appears to be surprisingly sensitive to them.…”

Debye series for light scattering by a spheroid

“…In GO, the stationary deflection of the emergent rays with respect to the incidence angle marks the position of the Descartes rainbow ⌰ D , where the two-ray-zero-ray transition causes the discontinuity of the intensity curve and the intensity infinity due to the infinitely dense emergent rays. In wave theory the intensity in the vicinity of the primary rainbow peak is assumed to be proportional to the square of the Airy function Ai͑−u͒ [34]:…”

“…Same as in Fig. 8, but the spheroid radius is 15.0 m. expression for h D is difficult to obtain, a numerical solution is available via calculating the optical path length of the incident rays from the entrance plane to the exit plane, multiplying it by the wave number to get the phase of the rays around the rainbow ray, and determining h D as the coefficient multiplying the cubic term in series expansion of the phase [34]. As compared in Fig.…”