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

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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) [CrossRef]  ]. 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 $\Delta D$ is defined that measures the disagreement between the two methods. A threshold value of 5% for $\Delta D$ 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 ${0.97} \le a/c \le {1.03}$. For the fourth-order rainbow, the application range of the Lock approximation is ${0.99} \le a/c \le {1.01}$ 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.

© 2022 Optical Society of America

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