Möbius shifts associated with the third-order and the fourth-order rainbows of a spheroidal droplet computation

Abstract

Vector-ray tracing (VRT) is employed to calculate Möbius shifts of the third-order and the fourth-order rainbows for a spheroidal droplet. When the aspect ratio of a spheroidal droplet is small, approximation expressions for calculating the Möbius shift (i.e., deviation of the geometrical rainbow angle for a spheroidal droplet and that for a spherical droplet) were given by Lock and Können [Appl. Opt. 56, G88 (2017)APOPAI0003-693510.1364/AO.56.000G88]. The assessment of applicability ranges of the Lock approximation is obtained by comparing with a VRT simulation for a water droplet with the refractive index m = 1.333 . For this, a parameter Δ D is defined that measures the disagreement between the two methods. A threshold value of 5% for Δ D is chosen below which the agreement is considered to be good. For the third-order rainbow, it is shown that this is the case for the Lock approximation for water droplets ( m = 1.333 ) with aspect ratios in the range of 0.97 ≤ a / c ≤ 1.03 . For the fourth-order rainbow, the application range of the Lock approximation is 0.99 ≤ a / c ≤ 1.01 for water droplets. For the first-order and second-order rainbows, the application ranges are briefly revisited with the current method. The influence of the droplet refractive index on the Möbius shift is also investigated.