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The droplet size, size distribution, refractive index, and temperature can be measured simultaneously by the rainbow technique. In the present work, the rainbow scattering diagram for a spherical droplet in the secondary rainbow region is simulated by the use of the generalized Lorenz-Mie theory. For achieving high spatial resolution in denser droplet sprays, a focused Gaussian beam is used. For droplet characterization, different inversion algorithms are investigated, which includes trough-trough (
*θ*
_{min1} and
*θ*
_{min2}) method and inflection-inflection (
*θ*
_{inf1} and
*θ*
_{inf2}) method. For the trough-trough algorithm, the absolute error of the refractive index is between −6.4 × 10
^{−4} and 1.7 × 10
^{−4}, and the error of the droplet radius is only between −0.55% and 1.77%. For the inflection-inflection algorithm, the maximum absolute error of the inverted refractive index is less than −1.1 × 10
^{−3}. The error of the droplet radius is between −0.75% and 5.67%.

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The structure of this paper is as follows: In Section 2, inversion schemes for the characterization of droplets are given. In Section 3, based on the secondary rainbow, two inversion schemes are used to compute the refractive index and size of the spherical droplet. Section 4 is devoted to the conclusions.

Based on the geometrical optics, the deflection angle θ_{p} for a ray emerging from a spherical droplet is given as [

θ p ( θ i , m , p ) = ( p − 1 ) π + 2 θ i − 2 p arcsin ( sin θ i / m ) , p = 0 , 1 , 2 , ⋯ (1)

here θ_{i} is the incident angle, and m is the relative refractive index of the droplet. The physical meaning of p [

θ p = 2 π + 2 θ i − 6 arcsin ( sin θ i / m ) (2)

In the above formula, there is an incident angle that minimizes the deflection angle. The minimum deflection angle is called the geometrical optics rainbow angle, and the geometrical optics rainbow angle is given:

θ r g = 2 π + 2 sin − 1 ( 9 2 − m 2 8 ) − 6 sin − 1 [ 9 2 − m 2 8 m 2 ] (3)

In Airy theory, the light intensity distribution in the rainbow angle region is expressed by Airy integral. Airy integral F(z) [

F ( z ) = ∫ 0 ∞ cos [ π ( z t − t 3 ) / 2 ] d t (4)

where z is a dimensionless parameter, defined as:

z = ( − q ) [ 12 / ( h π 2 ) ] 1 / 2 α 2 / 3 ( θ − θ r g ) (5)

where,

α = 2 a π / λ , h = ( p 2 − 1 ) 2 ( p 2 − m 2 ) 1 / 2 / [ p 2 ( m 2 − 1 ) 3 / 2 ] (6)

Among them, the value of q is −1 or +1, depending on the angle of the emitted light relative to the incident light [_{0} = 100 μm, and the wavelength is 0.6328 μm. The eigenvalues of the light intensity distribution in the rainbow area (such as θ_{max1}, θ_{max2}) have a correspondence with the dimensionless parameter z.

Performing algebraic operations on the above formula, the geometrical optics rainbow angle can be obtained from the first two troughs (θ_{min1 }and θ_{min2}):

θ r g = θ min 1 − C 1 θ min 2 1 − C 1 (7)

The refractive index can be obtained from formula (3) and (7), and the droplet radius can be calculated:

a = 2 λ 3 3 [ ( 9 − m 2 ) 1 / 2 ( m 2 − 1 ) 3 / 2 ] 1 / 2 ( z min 2 − z min 1 | θ min 1 − θ min 2 | ) 3 / 2 (8)

here,

z min 1 = 2.4956 , z min 2 = 4.3632 , C 1 = 0.5719 (9)

For the another algorithm, the angles of the first two inflection points (i.e. θ_{inf1} and θ_{inf2}) of the intensity distribution are employed. The geometrical optics rainbow angle and droplet size are given by:

θ r g = θ inf 1 − C 2 θ inf 2 1 − C 2 (10)

and,

a = 2 λ 3 3 [ ( 9 − m 2 ) 1 / 2 ( m 2 − 1 ) 3 / 2 ] 1 / 2 ( z inf 2 − z inf 1 | θ inf 1 − θ inf 2 | ) 3 / 2 (11)

here,

z inf 1 = 0.3276 , z inf 2 = 1.8724 , C 2 = 0.3013 (12)

Firstly, according to the Debye series expansion, the scattered light intensity distribution for the secondary rainbow of the spherical droplet is calculated, and the influence of different Gaussian beam waist radius on the peak of the secondary rainbow light intensity distribution is explored. Take the droplet with refractive index of 1.333 and radius of 50 μm as an example, as shown in

The rainbow intensity distribution based on the Debye series expansion simulation only considers the contribution of specific rays. In actual measurement, the total light intensity distribution is produced by the interference of all light rays. These extra interference phenomena cause high-frequency oscillations superimposed on the rainbow pattern, that is, ripple structure. According to the generalized Lorenz-Mie theory (GLMT), the total light intensity distribution in the rainbow region (also known as the rainbow diagram) is calculated. The wavelength of the Gaussian beam is 0.6328 μm, and the beam waist radius ω_{0} = 100 μm. The size of the droplet is Between 50 μm and 200 μm. The rainbow diagram of the droplet with a radius of 50 μm is shown in

The condition of central incidence is studied, where the center of the Gaussian beam is located on the Déscartes ray [

For the trough-trough (θ_{min1} and θ_{min2}) algorithm, the absolute error of the refractive index is between −6.4 × 10^{−4} and 1.7 × 10^{−4} (see _{inf1} and θ_{inf2}) algorithm, the inverted refractive index is less than −1.1 × 10^{−3} (see

The generalized Lorenz-Mie theory is used to simulate the rainbow scattering pattern of the droplets in the secondary rainbow region. The low-pass filter passes the simulated rainbow pattern to obtain the characteristic angles. Two different inversion algorithms are studied to invert the droplet parameters. The results show that different inversion algorithms are also feasible to measure droplet information. In particular, for the first algorithm, the error of the droplet radius is even less than 1.77%. However, in this study, the selection of the cut-off frequency of the digital low-pass filter requires artificial evaluation of the angular position of the rainbow diagram. Therefore, we need to develop a better method to select the cutoff frequency or find a better filter, which we need to study further.

The authors declare no conflicts of interest regarding the publication of this paper.

Wang, W.T., Wang, J.Y. and Zhang, Y.D. (2021) Droplet Characterization Based on the Simulated Secondary Rainbows. Optics and Photonics Journal, 11, 133-139. https://doi.org/10.4236/opj.2021.116011