Fig. 1. Optical paths of second-order refracted light p=2 and direct reflected light p=0 around geometrical rainbow angle when a droplet is illuminated by plane wave and their rainbow signals. (a) Optical paths; (b) typical standard rainbow signal of single droplet

Fig. 2. Division of 5 scattering angle regions for second-order refracted light p=2

Fig. 3. Rainbow signals and FFT spectrum of the standard rainbow simulated for a droplet with n=1.33 and D=100 μm. (a) Signals of standard rainbow, Airy rainbow (p=2) and low-pass filtered Airy rainbow; (b) spectrum of the standard rainbow in Fig. 3(a) after FFT processing

Fig. 4. Inversion flowchart of rainbow signal

Fig. 5. Inversion results and error analysis of standard rainbow signals of droplets with different refractive indices (n=1.33, 1.34, 1.35 and 1.36) and droplet sizes (D=30‒150 μm). (a) Absolute error of refractive index; (b) absolute and relative error of droplet size

Fig. 6. Distribution preset inversion results of global rainbow signals of dispersed droplets with different size distributions. (a) Normal distribution; (b) log-normal distribution

Fig. 7. Non-distribution preset inversion results of global rainbow signals of dispersed droplets with different log-normal distributions. （a） μ=60 μm，σ=0.1 μm and μ=100 μm，σ=0.1 μm； （b） μ=80 μm，σ=0.1 μm and μ=120 μm，σ=0.1 μm； （c） μ=40 μm，σ=0.2 μm and μ=80 μm，σ=0.2 μm；（d）μ=60 μm，σ=0.2 μm and μ=100 μm，σ=0.2 μm

Fig. 8. Non-distribution preset inversion results of global rainbow signals of dispersed droplets with different normal distributions. （a） μ=40/60/80/100/120/140/160 μm，σ=2 μm； （b） μ=40/60/80/100/120/140/160 μm，σ=5 μm； （c） μ=60/100/140 μm， σ=10 μm

Fig. 9. Non-distribution preset inversion results of global rainbow signals of dispersed droplets with multi-distributions. (a) Trimodal distribution; (b) binormal distribution

Fig. 10. Experimental results for monodisperse droplets.(a) Standard rainbow image; (b) comparison between experimental standard rainbow signal and fitted signal

Fig. 11. Experimental results for polydisperse droplets. (a) Global rainbow image; (b) comparison between experimental global rainbow signal and fitted signal; (c) non-distribution preset inversion result