Periodicity of interference structure of the Mie extinction cross section

Abstract

Slow oscillation of the normalized extinction cross section of a dielectric spherical particle (extinction efficiency) is usually referred to as an interference structure. Sharp resonance peaks superimposed on this smoothly varying curve form the so-called ripple structure of the extinction curve. The van de Hulst anomalous diffraction approximation predicts correctly the interference structure and its periodicity Δx = π / (m − 1 ), where x is the size parameter and m is the refractive index. The fact that the above expression, derived under the assumption m − 1 ≪ 1 (anomalous diffraction approximation), provides results in agreement with numerical calculations for refractive indices up to about m = 2.5 suggests that it can be derived under less restrictive assumptions on the refractive index m. Using asymptotic expansions of the Riccati-Bessel functions valid for size parameter x ≫ n, we show that the whole expression Re(an + bn) becomes independent of the summation index n. For refractive indices 0.5 ⪯ m ⪯ 2.5 the Re(an + bn) contains only one term periodic in the size parameter x. Periodicity of this term is Δx = π /(m − 1). Consequently, periodicity of the interference structure of the extinction curve is Δx = π/(m − 1) for all refractive indices 0.5 ⪯ m ⪯ 2.5.