A vector ray-tracing model (VRT) has been developed to compute the optical caustics associated with the primary rainbow for an oblate spheroidal water drop illuminated by a Gaussian beam. By comparing the optical caustic structures (in terms of limiting rainbow and hyperbolic umbilic fringes) for a water drop with a Gaussian beam (GB) illumination with that for the same drop, but with parallel beam (PB) illumination, the influence of the Gaussian beam on the optical caustics is investigated. For a water drop with GB illumination and different drop/beam ratios (i.e., the ratio between the drop equatorial radius and the Gaussian beam waist), the location of cusp points and the curvature of the limiting rainbow fringe are also studied. We anticipate that these results not only confirm the approach to compute optical caustics for oblate spheroidal drops illuminated by a shaped beam, but may also lead to a new method for measuring the aspect ratio of spheroidal drops.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Optical particle/drop diagnostic techniques are widely applied to atmospheric contamination monitoring, fuel sprays, spray drying of food and pharmaceuticals [1,2], etc. Understanding light scattering properties of drops is therefore an important basis for developing techniques to characterize drops in terms of refractive index, temperature, size, and shape. The shape of non-spherical liquid drops is often well approximated by a spheroid . For instance, using optical tweezers, biological cells take a spheroidal shape as they are stretched by the laser .
In 1984, Marston  described the hyperbolic umbilic and cusp diffraction catastrophes in the primary rainbow region of oblate spheroidal drops. Subsequently, many further optical caustics and their characteristics were investigated [6–11], which included the location of cusp point, opening rate of the cusp diffraction, lips event and V-shaped caustics etc. Nye  studied the critical aspect ratio of a spheroidal drop for generating the hyperbolic umbilic focus and predicted the occurrence of distinctive far-field caustic phenomena. Also, optical caustics have been observed from light scattering by oblate drops with white light illumination [12,13]. The rainbow caustics in the secondary and higher order rainbow regions [14–16] and the internal caustic structures of liquid drops  have been studied in detail. Moreover, the hyperbolic umbilic caustics in the primary rainbow region for drops with a vertically tilted laser beam illumination were studied by Jobe .
Based on the rainbow scattering from a spherical drop, drop information like size, refractive index and temperature can be determined, an approach known as the standard rainbow technique . For the measurement of clouds of drops in sprays, the global rainbow technique has been proposed [20,21]. Both of these techniques assume spherical drops and furthermore, assume that any non-spherical drops result in a uniform background scattering intensity. Several further enhancements to this technique have appeared. A phase rainbow refractometer was developed by Wu  to accurately measure the drop refractive index and minute size variations, allowing drop evaporation to be monitored. They also investigated the measurements of volume mixing ratios of drops using the global rainbow technique . A portable global rainbow instrument was designed for in-situ measurement . For a spheroidal drop, the refractive index and equatorial diameter were measured by the use of the generalized rainbow pattern . By using the rainbow pattern and the curvature of the rainbow fringes, the shape of the drop could also be measured by comparing to results obtained using a vector ray-tracing approach . Recently, a model for computing the optical caustics for the primary rainbow from a tilted spheroidal drop were proposed, and optical caustics were compared with those from a spheroidal drop with light beam propagating within the equatorial plane .
However, when a Gaussian beam is used as an illumination light source, the question arises whether there is any influence on the optical caustics and if so, could these changes be better used to quantify the non-sphericity? This is the question addressed in the present study. We anticipate that the results presented here will also be the first step to realizing instruments capable of performing such measurements.
This article is organized as follows: The vector ray-tracing model for an oblate spheroidal drop with Gaussian beam illumination is described in Section 2. In Section 3, the optical caustics for a drop illuminated by the Gaussian beam with different drop/beam ratios are obtained and compared with results for a drop illuminated by a parallel beam. The reason for the discrepancies between the two cases is discussed. The location of cusp points and the curvature of limiting rainbow fringe are also investigated. Section 4 provides a conclusion.
2. Vector ray-tracing model for a drop illuminated with a Gaussian beam
Suppose the center of a spheroidal drop to be the origin of the Cartesian coordinate system, which is noted as OP. We consider a TEM00 Gaussian beam with waist radius ω0, wavelength λ, polarization in the x direction, and propagation along the y direction. Its center OG is located at the center in the drop coordinate system OP-xyz (see Fig. 1). The spheroidal drop has radii a, b, and c and refractive index m. For an oblate spheroidal drop, a = b.
In the first-order approximation, the phase of the incident Gaussian beam φi at point A(x,y,z) is described by 
Within the framework of geometric optics, the Gaussian beam can be regarded as a combination of light ray bundles. When a light ray of the Gaussian beam encounters the spheroidal drop, its direction vector is normal to the local wavefront surface of the incident beam, which can be expressed by
The definition of the elevation angle and off-axis angle are the same as those given in Ref. . In the VRT model, the optical caustics (in terms of limiting rainbow and hyperbolic umbilic (HU) fringes) can be identified from an infinitively large number of the outgoing rays. Note that the limiting fringes only indicate the structure of the rainbow pattern without capturing the interference fringes of the rainbow. Two skew rays and two equatorial rays focusing in the same direction, give rise to the cusp point. For a drop with parallel beam illumination, an analytical solution to predict the location of cusp point was given based on Herzberger’s formalism .
3. Numerical results
Based on the VRT model, the optical caustics are computed for an oblate spheroidal drop with relative refractive index m=1.333 and equatorial radius a=100 μm. Here, the aspect ratio of the water drop is set to be 1.22 (i.e., a/c=1.22). The wavelength of the incident Gaussian beam (GB) is 0.6328 μm. About 108 incident light rays are computed and their corresponding outgoing rays are traced respectively. First, optical caustics are computed for a water drop illuminated by a GB with different drop/beam ratios, i.e. γ=a/ω0=10, 5, 2 and 1. These results are then compared with those obtained for a drop illuminated by a parallel beam (PB). Only part of optical caustics in the elevation angle range -10°≤ϕ≤10° will be presented, as these are the portions most likely to be significant for later realization for drop diagnostic instruments.
The optical caustics for a drop illuminated by either a PB or GB with a drop/beam ratio of γ=10 is shown in Fig. 2(a). There is an obvious difference between the limiting rainbow fringes. In the equatorial plane, the angle difference between the limiting rainbow fringes is about 0.97°. However, the two limiting HU fringes overlap almost exactly. The angle difference is about 0.25° for the cusp point. For drop/beam ratio γ=5, the limiting rainbow fringe for GB overlaps nearly completely with that for PB (see Fig. 2(b)). As shown in the inset of Fig. 2(b), the angle difference in the equatorial plane between the rainbow fringes is 0.063°. The angle difference between the two cusp points is approximately 0.026°. It can be seen from Fig. 2(c) that for γ=2, the limiting rainbow fringes for GB and PB illumination also overlap perfectly. Even in the small scattering range, the difference between the two results is not obvious (see the inset in Fig. 2(c)). The angle difference between the limiting rainbow fringes is only 0.0016°. As shown by Fig. 2(c), the limiting HU fringes still overlap completely. The angle difference between the two cusp points decreases to 0.0128°. Finally, for γ=1, it can be seen from the inset of Fig. 2(d) that there is no difference between the two results. Although it is not shown here, computations for other drops (i.e. different aspect ratios) illuminated by Gaussian beams with different drop/beam ratios also show that when the drop/beam ratio decreases, the difference between fringes from GB illumination and fringes from PB illumination becomes smaller. The reason for this is now discussed.
Here an angle α is defined to describe the deviation of an incident ray for the GB from that of the PB (see Fig. 1). The deviation angle is defined as
As known, the rainbow arises mainly from the rays in close proximity to the Descartes ray. For example, in a water drop with a=100 μm and m=1.333, the location of Descartes ray is z = 86.08 μm, y=-50.89 μm in the plane of x=0. From Eqs. (2) and (8), it can be seen that the deviation angle is affected by several parameters. For given ω0 and y0 but different x0, the deviation angles are plotted against z (see Fig. 3(a)). Here ω0=90 μm and y0=-50.89 μm. First, all the curves for deviation angles are symmetric about z=0, which is due to the symmetry of the GB. As x0 increases from 0 to 90 μm, the deviation angles increase, which means that the deviation of a light ray from a GB to the PB becomes larger and larger. For given ω0 and x0 but different y0, the deviation angles are shown in Fig. 3(b). As y0 increases, the deviation angles increase gradually. Moreover, all curves of deviation angles exhibit a slow increase and are symmetric about z=0. The influence of the drop/beam ratio on the deviation angles is presented in Fig. 3(c). It can be seen that, for drop/beam ratio γ=10, the deviation angles are around 2°. However, for γ=5 and 1 the deviation angles are close to zero. As expected, the deviation angle is related to the incident position (i.e., x0, y0 and z0) and the drop/beam ratio. For a GB with drop/beam ratio small enough, the light beam incident on the drop can be considered as a PB. Although it is not shown here, computations for drops with different radii reveal that when the drop/beam ratio is constant (i.e., γ=10, 5 or 1), the deviation angle will become smaller with increasing the drop radius.
Based on the simulation of optical caustics by VRT, the locations of cusp points for different drops are computed and shown in Fig. 4. For a water drop illuminated by a GB with drop/beam ratio γ=10, the cusp point first appears at the scattering angle θ=165.55° for a/c=1.069694. For γ=5 and 1, the cusp points appear with θ=165.54° for a/c=1.069693. However, for a drop with PB illumination, the cusp point first appears at θ=165.57° for a/c=1.069389 [7,26]. As the drop aspect ratio increases, the cusp point moves to smaller scattering angles (i.e. the forward direction). For the three cases (γ=10, 5, and 1) the difference between the cusp location can no longer be distinguished. When the aspect ratio further increases, the cusp location for the drop illuminated by a GB with drop/beam ratio γ=5 still agrees with that for the drop illuminated by a GB with γ=1. This is because the GB for these two cases can be considered as a PB. However, for γ=10, the difference is obvious for a/c=1.26. Once the aspect ratio is larger than 1.30, the cusp increases to larger scattering angles. However, when the aspect ratio reaches 1.392138, the cusp disappears for the drop illuminated by a GB with γ=10. As for γ=5 and 1, the cusps disappear only when the aspect ratio increases to 1.413391 and 1.414739, respectively. The cusp point reappears at θ=165.38° for drops with a/c=1.513154 and γ=10. For drop/beam ratio γ=5 and 1, the cusps reappear at θ=165.39° for a/c=1.513138 and 1.513137 respectively. Then, the scattering angles decrease with an increase of the aspect ratio. It is obvious that the cusp locations for GB illuminations with different drop/beam ratios are in good agreement. Furthermore, for γ=10, 5 and 1 respectively, the discrepancies between the cusp locations will become smaller as the drop equatorial radius increases. These results suggest therefore, that the relationship between the cusp location and the drop shape can be used for measuring drop aspect ratio, assuming a spheroidal shape.
The curvatures of the limiting rainbow fringe are shown in Fig. 5. As the aspect ratio gradually increases from 1.06 to 1.50, the curvature increases and then decreases. As the left inset in Fig. 5 shows, the curvature of the limiting rainbow fringe for a drop illuminated by a GB with drop/beam ratio γ=1 is in good agreement with that for the drop illuminated by a GB with γ=5. For the case of γ=10, a small difference can be observed. As the aspect ratio of drop increases, this difference becomes more pronounced. For a drop with the aspect ratio around 1.30, the limiting rainbow fringe becomes clearly distorted (see Fig. 5(f) in Ref. ). Therefore, the curvature is much larger than that for drops with other aspect ratios, which is not shown in Fig. 5. Further increasing the aspect ratio of drops, the curvature decreases quickly. For drops illuminated by a GB with γ=5 and 1 respectively, the curvatures of the limiting rainbow fringe agree very well with each other. However, there are differences between curvatures for the drop illuminated by a GB with γ=10 and that for the other two cases (see the right inset in Fig. 5). For a drop aspect ratio between 1.54 and 1.60, the curvature of the limiting rainbow fringe increases again and agreement for the conditions of γ=5 and 1 is also observed. The difference for the conditions of γ=10 and 1 also exists. With the increase of the drop radius but keeping drop/beam ratio as shown above, the curvatures of the limiting rainbow fringe for drop with GB illumination will gradually approach that for drop with PB illumination. Here, the effect of the drop/beam ratio on the limiting rainbow fringe curvature is demonstrated, an effect which can also potentially be utilized for drop shape measurement. For such a measurement, it is however evident, that the choice of beam waist must be made with consideration of the expected drop size.
Based on a VRT model, the optical caustic structure in the primary rainbow region is computed for oblate spheroidal drops illuminated by a Gaussian beam. Examining the caustics for different drop/beam ratios (i.e., γ=10, 5, 2 and 1), the influence of illumination with a Gaussian beam rather than a parallel beam is investigated for drops with an equatorial radius a=100 μm and aspect ratio a/c=1.22. As expected, the optical caustics for drops with Gaussian beam illumination agree well with those with parallel beam illumination when the drop/beam ratio decreases. By introducing a deviation angle for Gaussian beam illumination, the characteristic of incident beam forming the optical caustics is analyzed. Furthermore, the location of cusp point and curvature of the limiting rainbow fringe are also studied for drops with illumination of Gaussian beams of γ=10, 5 and 1. The relation between the location of cusp point and drop, as well as that between the curvature and drop can be used for measurement of spheroidal drop. The influence of the relative position of the drop and Gaussian beam on the optical caustic will be discussed in a future work. The model presented in this paper could also be applied to study the optical caustics for higher order rainbows and for drops illuminated by other shaped beams.
National Natural Science Foundation of China (51506129, 51476104); Natural Science Foundation of Shanghai (20ZR1455200).
The authors would like to thank Prof. Feng Xu from School of Meteorology, University of Oklahoma, for his useful discussions and support.
The authors declare that there are no conflicts of interest related to this article.
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