## 6.2 Perspectives

In this thesis, I have shown good agreement between VCRM and LMT for large spherical particles. However, compared with rigorous theories, discrepancies can still be observed: (i) in the rainbow angle where the intensity tends to infinite, (ii) at the border of the particle where the variation of the amplitude of the incident wave is abrupt, (iii) in the critical angle where the total reflection occurs, so the amplitude of the reflected wave is not continuous. In these regions, supplementary theories or models are needed to correct the prediction of VCRM. The fundamental idea is to take into account the wave effect near the singularity point or caustics by analytical or semi-analytical expression. To profit the advantage of the VCRM to deal with the scattering of shaped beams by particles of arbitrary shape, wave effect near the singularity point need to be added.

In addition, for better understanding of the influences of particle's non-sphericity on rainbow patterns, further investigation of the relations between supernumerary bows (positions and intensities) and spheroidal particle parameters is needed. The property, like that the intensity ratio of different orders of rainbows is very sensible to the particle ellipticity, deserves more study and can be applied to characterize the deformation of particles.

Finally, experimental measurement and data processing procedure in this thesis need to be improved. The scattering patterns of different liquids and sizes will be examined. Most of all, our current numerical implementation of the VCRM only addresses rays propagating within the equatorial plane of an ellipsoidal particle. Due to this, our experimental study is also limited to the symmetric plane of pendant drops. In experiment, we observe many interesting phenomena near the rainbow angle and in forward direction. For example when pendant drop becomes bigger and is elongated in the vertical direction, correspondingly, the scattering patterns changes. The second order rainbow is deformed with increase of the particle size. However by the limitation of the numerical simulation with 3D VCRM, our study is limited to the equatorial plane of the drops. For further studies of the influence of particle non-sphericity on the rainbow pattern, we need to extend our code of VCRM to 3D scattering of complex objects.
# Appendices
<figure><img src="image_45.png" /></figure>
# Appendix A

## Interpolation of scattered field

In the calculation of the total field of scattering light, the interpolation of the data obtained by ray tracing is necessary (see section 3.2.3). In my code of VCRM in C#, this subroutine of interpolation is translated from the code of Professor REN in language Pascal and Lagrange polynomials are used for this interpolation.

For a given set of $(x_j, y_j)$, the Lagrange polynomial is the polynomial of the lowest degree that assumes at each $x_j$ the corresponding value $y_j$ (i.e. the functions coincide at each point). Given a set of $n + 1$ data points, $(x_0, y_0), \dots, (x_j, y_j), \dots, (x_n, y_n)$, where all values of $x_j$ are different. The interpolation polynomial in the Lagrange form is a linear combination given by

$$L(x) = \sum_{j=0}^{k} y_j l_j(x) \qquad (A.1)$$

$l_j(x)$ is the Lagrange basis polynomials

$$l_j(x) = \prod_{m=0, m \neq j}^{n} \frac{x - x_m}{x_j - x_m} = \frac{x - x_0}{x_j - x_0} \dots \frac{x - x_{j-1}}{x_j - x_{j-1}} \cdot \frac{x - x_{j+1}}{x_j - x_{j+1}} \dots \frac{x - x_n}{x_j - x_n} \qquad (A.2)$$

Only when $x = x_j$, we have

$$l_j(x_j) = \prod_{m=0, m \neq j}^{n} \frac{x_j - x_m}{x_j - x_m} = 1 \qquad (A.3)$$

Otherwise, when $x = x_i$ and $i \neq j$,

$$l_j(x_i) = \frac{x_i - x_0}{x_j - x_0} \dots \frac{x_i - x_i}{x_j - x_i} \dots \frac{x_i - x_n}{x_j - x_n} = 0 \qquad (A.4)$$

Therefore

$$L(x_j) = y_0 l_0(x_j) + \dots + y_j l_j(x_j) + \dots + y_n l_n(x_j) = y_j l_j(x_j) = y_j \qquad (A.5)$$
proving that $L(x)$ interpolates the function exactly.

In VCRM, after ray tracing, we get several sets of data points of amplitude and phase, as functions of emergent rays, i.e., deviation angle $\theta_{p,i}$ of the emergent ray of order $p$ for an incident ray $i$, with their corresponding amplitude $A_{p,i}$, and the phase $\Phi_{p,i}$. Referred to Fig. 3.2 in section 3.2.3, we know that, to calculate the summation of the amplitude of all the rays in a given angle, a careful interpolation of the amplitudes and the phase shifts of each order emergent ray is necessary. Besides, in the implementation of interpolation algorithm to the amplitude and phase of each order emergent ray, we should noted that, normally the interpolation algorithm applies to the monotone interval. However from Fig. 3.2 we can find that two rays of the same order may arrive at the same angle, that is because the deviation angle $\theta_{p,i}$ increases as function of emergent rays and then decreases after the rainbow angle. Therefore we must split the data obtained by ray tracing into monotone groups according to rainbow angle and interpolate the data respectively.

An interpolation of the second degree Lagrange polynomials is applied for points between the first and last two points. And an interpolation of the three degree Lagrange polynomials is applied for points between two intermediate points. For example, if $\theta$ is a point located between the first two points $\theta_0$ and $\theta_1$, then its amplitude is calculated by

$$A_{\theta} = A_{\theta_0} \cdot \frac{(\theta - \theta_1)(\theta - \theta_2)}{(\theta_0 - \theta_1)(\theta_0 - \theta_2)} + A_{\theta_1} \cdot \frac{(\theta - \theta_0)(\theta - \theta_2)}{(\theta_1 - \theta_0)(\theta_1 - \theta_2)} + A_{\theta_2} \cdot \frac{(\theta - \theta_0)(\theta - \theta_1)}{(\theta_2 - \theta_0)(\theta_2 - \theta_1)} \quad (A.6)$$

If $\theta$ is a point located between two intermediate points $\theta_j$ and $\theta_{j+1}$, we have

$$A_{\theta} = A_{\theta_{j-1}} \cdot \frac{(\theta - \theta_j)(\theta - \theta_{j+1})(\theta - \theta_{j+2})}{(\theta_{j-1} - \theta_j)(\theta_{j-1} - \theta_{j+1})(\theta_{j-1} - \theta_{j+2})} + A_{\theta_j} \cdot \frac{(\theta - \theta_{j-1})(\theta - \theta_{j+1})(\theta - \theta_{j+2})}{(\theta_j - \theta_{j-1})(\theta_j - \theta_{j+1})(\theta_j - \theta_{j+2})} + A_{\theta_{j+1}} \cdot \frac{(\theta - \theta_{j-1})(\theta - \theta_j)(\theta - \theta_{j+2})}{(\theta_{j+1} - \theta_{j-1})(\theta_{j+1} - \theta_j)(\theta_{j+1} - \theta_{j+2})} + A_{\theta_{j+2}} \cdot \frac{(\theta - \theta_{j-1})(\theta - \theta_j)(\theta - \theta_{j+1})}{(\theta_{j+2} - \theta_{j-1})(\theta_{j+2} - \theta_j)(\theta_{j+2} - \theta_{j+1})} \quad (A.7)$$

In this method, we can calculate the amplitudes and the phases of all the rays arriving at the same angle by interpolation. And their summation is the complex amplitude of the total field.
# Appendix B

## Macros of ImageJ for image processing

This thesis is devoted to investigate the influences of particle ellipticity on the intensity distribution of scattered light. Scattering patterns near the rainbow angles, and images of drop, are recorded by two synchronized cameras. For further comparisons between the theoretical scattering diagrams and the experimental results, two principle curvature radii of drop in the equatorial plane of the drops are essential. Besides, it is needed to get intensity distribution in the equatorial plane from the images of scattering pattern. Finally the noise from the environment can not be ignored. Noise reduction is therefore necessary.

To meet these demands, three plugins of ImageJ are developed in Macro language. The ImageJ Macro language is a script built into ImageJ. It allows to automate a series of ImageJ commands. Programs written in macros, Macros for short in this thesis, can be used to perform sequences according to our design. The first Macro is designed to find the equatorial plane. The curvature radii of drop in this plane are then calculated. The second one is programed for batch background noise reduction of scattering pattern. And the third one is to find the equatorial plane in the images of scattering pattern, to get its intensity distribution, and to calculate the intensity ratio between the first and the second order rainbows for comparison. In the following sections, I will give a detailed description of these three Macros.

### B.1 Macro for Batch background noise reduction

In our experiment, the light scattered by the drop is recorded by a camera mounted on a rail fixed to the rotation platform. Between the camera and the drop, a white screen is installed in order to assure a sufficient large field of view. In such configuration the light reflected by the tube can not be negligible. Therefore it is needed to subtract
the noise from the images of scattering pattern.

The principle of this macro is that we define the image without drop as a back-
ground, and then subtract it from the other images. The existing ImageJ command
can realize the calculation between two images, but it is limited to the number of
images and can't meet our demand of batch processing (more than 100 images). So a
macro of images batch processing is developed, to automate the subtraction of noise
from our images.

Image B.1(a) is an original image of scattering pattern near rainbow angles. Noise from the environment, particularly the reflected light from the tube in the middle of the image, can be observed obviously. After the application of our background noise reduction macro, most of noise is deleted in the image B.1(b). Further comparison is made for their scattering diagram measured in the middle of images. Images B.2(a) and B.2(b) are respectively scattering diagrams of image B.1(a) and B.1(b). We can find that, with the help of our macro, the main noise is eliminated and a typical Airy structure of the first and the second order rainbows is much better.

<figure><img src="image_46.png" /><figcaption>Figure B.1: Scattering pattern near the rainbow angle before and after treatment.</figcaption></figure>

<figure><img src="image_47.png" /><figcaption>Figure B.2: Scattering diagram in the equatorial plane before and after treatment.</figcaption></figure>
## B.2 Macro for the treatment of images of drops

This macro is developed to get the profiles of drops from the images recorded in experiment and find their curvature radii in the equatorial plane.

**1. Make binary** A binary image is a digital image that has only two possible values for each pixel. The procedure of making binary is to convert the current gray scale image to a binary image. The output image replaces all pixels in the input image with luminance greater than threshold level with the value 255 and replaces all other pixels with the value 0. This threshold level can be determined by analyzing the histogram of a selected region or set by the user. Making binary can simplify the data processing. Image B.3(a) is an image of drop recorded in experiment. After application of binary image processing, it is converted to Image B.3(b). In this way, the object can be separated from the background easily, for further extraction of drop's contour.

<figure><img src="image_48.png" /><figcaption>Figure B.3: The image of drop before and after making binary</figcaption></figure>

**2. Extract the contour of a drop near equatorial plane by Moving Average Method** After previous procedure, a image recorded in experiment is converted to a binary image. In this step, we want to determine the contour of drop from its image. To calculate the scattering intensity of a sphere in the equatorial plane in VCRM, only local curvature radii of the particle surface in this plane is needed. Therefore, just contour near equatorial plane is necessary. To reduce the noise outside of the drop, the Moving Average method is applied. It uses a moving window to search values and averages all data points within the window. Here it is used to average intensities of *n* successive pixels in a row. The window is moved according to the scanning direction and step value. When the average value is greater than or equal to
<figure><img src="image_49.png" /><figcaption>Figure B.4: Partial enlarged detail of a drop after applying Moving Average method. Black part in the right is part of drop. The other black pixels out of it are noise introduced in experiment. And the red pixels selected are the contour extracted.</figcaption></figure>

a indicated value (for example 127.5 here, half intensity of a pixel within the object), the frontier between the points of gray scale 0 and 256 is considered as the contour of drop. This method is very efficient to suppress the noise from the environment. Fig. B.4 is the partial enlarged detail of a drop after applying Moving Average method. The black part in the right is a part of the drop. Outside of the drop, the back pixels are the noise introduced in the experiment. After apply the Moving Average method, the red points are pixels selected for the contour of drop. We can find that even if the noise exist, the result is hardly affected.

Fig. B.5 is the contour near the equatorial plane extracted from a pendant drop image. Here we are interested only in the scattering intensity distribution in the equatorial plane of pendant drop. In this circumference, only the geometry of the drop in this plane plays its role. We need only the local curvature radii in the horizontal and vertical directions. Because a pendant drop is axial symmetric and has a circular cross section in the horizontal plane, the radius of drop in the equatorial plane is then horizontal curvature radius. The curvature radius of contour at a point of the equator is the vertical curvature radius. To get curvature radii of contour, an approximating function is needed.

**3. Curve fitting** The contour of the drop obtained in the above section is not smooth. To calculate the curvatures, the curve fitting is necessary. We should firstly find a mathematical function to fit the contour data, then the curvature radii in the equatorial plane can be calculated.
<figure><img src="image_50.png" /><figcaption>Figure B.5: Contour near the equatorial plane extracted from a pendant drop image.</figcaption></figure>

<figure><img src="image_51.png" /><figcaption>Figure B.6: Left contour (in black) extracted from the pendant drop image and its fitted curve (in red). The abscissa and ordinate of the contour is exchanged to ease the explanation.</figcaption></figure>

We find that our contour is a quadratic like curve, therefore a second degree polynomial function may be suitable. Suppose that $P(x, y)$ is a pixel on one of the contour in Fig. B.5. We aim to find a suitable approximation function $y = ax^2+bx+c$ which has the best fit to the contour. For ease of explanation, we exchange the horizontal and vertical axes of the Fig. B.5 and take only the left cure of the two branches, as shown by the black curve in Fig. B.6. Here we use directly the curve fitting tool CurveFitter provided by ImageJ. The red curve in Fig. B.6 is the fitted curve. From the comparison, we can find that the curve obtained from our method fits very well with the contour extracted from the image of drop, proving that our quadratic function is suitable to describe the contour of drop near the equatorial plane.

**4. Calculations of curvature radii** The symmetrical axis of the drop is vertical in Fig. B.5 and horizontal in Fig. B.6. So the equatorial plane corresponds to the point on the fitted curve where the tangent is zero, $y' = 2ax + b = 0$, i.e., $x = -\frac{b}{2a}$,
$y = \frac{4ac-b^2}{4a}$. Its abscissa $x$ shows the position of equatorial plane in Fig.B.5. And the curvature radius of this curve, is the vertical curvature radius of drop in the equatorial plane. We know that if a curve is given in Cartesian coordinates as $y(x)$, then the radius of curvature $R_c$ at any point $x$ is

$$R_c = \left| \frac{(1 + y')^2}{y''} \right| \qquad (B.1)$$

where $y' = \frac{dy}{dx}$, $y'' = \frac{d^2y}{dx^2}$. For our quadratic equation $y = ax^2 + bx + c$, its curvature radius $R_{qua}$ is then

$$R_q = \left| \frac{[1 + (2ax + b)^2]^{\frac{3}{2}}}{2a} \right| \qquad (B.2)$$

At the equatorial plane $x = -\frac{b}{2a}$, the curvature radius of curve is therefore

$$R = \left| \frac{1}{2a} \right| \qquad (B.3)$$

The same method can be applied to the right part of Fig. B.5 and we should obtain the same result. The good agreement between them can show the symmetric of the pendant drop. Difference of the ordinates of the two points in the equatorial plane is the diameter of the circular cross section of drop in this plane. Besides, it should be noted that, the curvature radii got from the images of drops are in pixels. They should be converted to real size with help of a reference in the experiment.

## B.3 Macro for scattering pattern

This macro is designed to get the intensity distribution of the scattered light in the equatorial plane, more specifically, near the rainbow angle. However for pendant drops of different shapes, the positions of their equatorial planes are different. Fig. B.7 illustrates four images of a series of scattering diagrams. Image B.7(a) is a typical image of scattering diagrams taken near the rainbow angle when the drop is almost spherical. The rainbows are similar to what we observe in the nature. Two visible bows correspond to the primary (right) and the secondary (left) order rainbows. Each of them has a fold structure, well known as Airy structure. It consists of a main peak and its subsidiary supernumerary bows. With growing of drop, the rainbows in image B.7(b), notably the second one on the left, are more curved. Finally the rainbow structures in images B.7(c) and B.7(d) are completely twisted due to further deformation of the particle. Therefore in this macro, to get intensity distribution of scattered light in the equatorial plane, we have to at first locate its corresponding region in the images of scattering pattern, and then export its scattering diagram. Finally the intensity ratio of the first to the second order rainbow is calculated.
<figure><img src="image_52.png" /><figcaption>Figure B.7: Scattering pattern for suspended drops of different shapes. The scattering patterns of main peak in the region of interest (yellow rectangle) are extracted (red curve) and curve fitted (green curve).</figcaption></figure>
**1. Extract the scattering pattern of the first order rainbow by Moving Average method** Through our analysis, we find that if we fit the scattering pattern of main peak of first order rainbow near the equatorial plane with a quadratic function, its vertex is on the equator. In this way, the equatorial plane can be located. Hence the first step is to extract the scattering pattern of main peak by Moving Average method. Firstly a region of interest (ROI) is selected manually in Fig. B.7. By adjusting the range of ROI, the interference from the second order rainbow can be eliminated. Then our former mentioned Moving Average method is applied in the selected region. It averages intensities of a indicated number of successive pixels in each row. When the average intensity of a window is largest in this row, the window center is selected as the main peak. Finally we can get the scattering pattern of the main peak near the equatorial plane (red curve in Fig. B.7).

**2. Curve fitting and calculation of equatorial plane** The scattering pattern of the first order rainbow near the equatorial plane is fitted by a quadratic function. The principle is the same as our aforementioned curve fitting method in section B.2. From Fig. B.7, we can find that our equation (green curve) fits the scattering pattern very well. The extreme point on the fitted curve indicates the position of the equatorial plane.

**3. Export of intensity distribution diagram and calculation of intensity ratio between first and second order rainbow** As soon as the equatorial plane is determined, the intensity on this plane is extracted as described in section B.1 and then we can calculate the intensity ratio of the two orders of rainbow.
Bibliography

[1] Fabrice RA Onofri, Kuan Fang Ren, Matthias Sentis, Quentin Gaubert, and Chantal Pelcé. Experimental validation of the vectorial complex ray model on the inter-caustics scattering of oblate droplets. *Optics express*, 23(12):15768–15773, 2015.

[2] Cameron Tropea. Optical particle characterization in flows. *Annual Review of Fluid Mechanics*, 43:399–426, 2011.

[3] H-E Albrecht, Nils Damaschke, Michael Borys, and Cameron Tropea. *Laser Doppler and phase Doppler measurement techniques*. Springer Science & Business Media, 2013.

[4] Franz Durst and Mohammed Zaré. Laser Doppler measurements in two-phase flows. In *The accuracy of flow measurements by laser Doppler methods*, volume 1, pages 403–429, 1976.

[5] Amir Naqwi, Franz Durst, and Xiaozhi Liu. Two optical methods for simultaneous measurement of particle size, velocity, and refractive index. *Applied optics*, 30(33):4949–4959, 1991.

[6] Cameron Tropea, Tian-Hua Xu, Fabrice Onofri, Gérard Géhan, Per Haugen, and Manfred Stieglmeier. Dual-Mode Phase-Doppler Anemometer. *Particle & particle systems characterization*, 13(2):165–170, 1996.

[7] Fabrice Onofri, Thierry Girasole, Gérard Gréhan, Gérard Gouesbet, Günter Brenn, Joachim Domnick, Tian-Hua Xu, and Cameron Tropea. Phase-Doppler Anemometry with the Dual Burst Technique for measurement of refractive index and absorption coefficient simultaneously with size and velocity. *Particle & particle systems characterization*, 13(2):112–124, 1996.

[8] Norbert Roth, Klaus Anders, and Arnold Frohn. Refractive-index measurements for the correction of particle sizing methods. *Applied optics*, 30(33):4960–4965, 1991.

[9] Subramanian V Sankar, Dale H Buermann, and William D Bachalo. Application of rainbow thermometry to the study of fuel droplet heat-up and evaporation
characteristics. *Journal of Engineering for gas turbines and power*, 119(3):573-584, 1997.

[10] JPAJ Van Beeck, D Giannoulis, Laurent Zimmer, and ML Riethmuller. Global rainbow thermometry for droplet-temperature measurement. *Optics letters*, 24(23):1696-1698, 1999.

[11] JPAJ Van Beeck and ML Riethmuller. Nonintrusive measurements of temperature and size of single falling raindrops. *Applied optics*, 34(10):1633-1639, 1995.

[12] F Onofri and S Barbosa. Chapter II: Optical particle characterization. *Laser Metrology in Fluid Mechanics*, *Boutier, A* (ed.), ISBN: 9781848213982, *Wiley-ISTE, London*, 2012.

[13] Gustav Mie. Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen. *Annalen der physik*, 330(3):377-445, 1908.

[14] L Lorenz. Det Kongelige Danske Videnskabernes Selskabs Skrifter 6, 1890.

[15] Shoji Asano and Giichi Yamamoto. Light scattering by a spheroidal particle. *Applied optics*, 14(1):29-49, 1975.

[16] James R Wait. Scattering of a plane wave from a circular dielectric cylinder at oblique incidence. *Canadian Journal of Physics*, 33(5):189-195, 1955.

[17] Craig F Bohren and Donald R Huffman. Absorption and scattering by a sphere. *Absorption and Scattering of Light by Small Particles*, pages 82-129, 1983.

[18] Peter W Barber and Steven C Hill. *Light scattering by particles: computational methods*, volume 2. World scientific, 1990.

[19] C Yeh. Backscattering cross section of a dielectric elliptical cylinder. *Journal of the Optical Society of America*, 55(3):309-314, 1965.

[20] A Sebak and L Shafai. Generalized solutions for electromagnetic scattering by elliptical structures. *Computer physics communications*, 68(1):315-330, 1991.

[21] Charles L Adler, James A Lock, Justin K Nash, and Kirk W Saunders. Experimental observation of rainbow scattering by a coated cylinder: twin primary rainbows and thin-film interference. *Applied optics*, 40(9):1548-1558, 2001.

[22] Renxian Li, Xiang'e Han, Huifen Jiang, and Kuan Fang Ren. Debye series of normally incident plane-wave scattering by an infinite multilayered cylinder. *Applied optics*, 45(24):6255-6262, 2006.

[23] Gérard Gouesbet, Bruno Maheu, and Gérard Gréhan. Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation. *Journal of the Optical Society of America A*, 5(9):1427-1443, 1988.
[24] F Onofri, G Gréhan, and G Gouesbet. Electromagnetic scattering from a multi-layered sphere located in an arbitrary beam. *Applied Optics*, 34(30):7113-7124, 1995.

[25] Yiping Han and Zhensen Wu. Scattering of a spheroidal particle illuminated by a Gaussian beam. *Applied optics*, 40(15):2501-2509, 2001.

[26] Feng Xu, Kuanfang Ren, Gerard Gouesbet, Xiaoshu Cai, and Gerard Grehan. Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam. *Physical Review E*, 75(2):026613, 2007.

[27] KF Ren, G Gréhan, and G Gouesbet. Scattering of a Gaussian beam by an infinite cylinder in the framework of Generalized Lorenz-Mie theory: formulation and numerical results. *Journal of the Optical Society of America A*, 14(11):3014-3025, 1997.

[28] Gérard Gouesbet, Loïc Mees, Gérard Gréhan, and Kuan-Fang Ren. Localized approximation for Gaussian beams in elliptical cylinder coordinates. *Applied optics*, 39(6):1008-1025, 2000.

[29] Loïc Mees, Gerard Gouesbet, and Gerard Gréhan. Scattering of laser pulses (plane wave and focused Gaussian beam) by spheres. *Applied optics*, 40(15):2546-2550, 2001.

[30] G Gouesbet and G Grehan. Generalized Lorenz-Mie theory for assemblies of spheres and aggregates. *Journal of Optics A: Pure and Applied Optics*, 1(6):706, 1999.

[31] Peter Debye. Der lichtdruck auf kugeln von beliebigem material. *Annalen der physik*, 335(11):57-136, 1909.

[32] Herch Moyses Nussenzveig. High-Frequency scattering by a transparent sphere. I. direct reflection and transmission. *Journal of Mathematical Physics*, 10(1):82-124, 1969.

[33] James A Lock, J Michael Jamison, and Chih-Yang Lin. Rainbow scattering by a coated sphere. *Applied optics*, 33(21):4677-4690, 1994.

[34] James A Lock. Debye series analysis of scattering of a plane wave by a spherical Bragg grating. *Applied optics*, 44(26):5594-5603, 2005.

[35] Renxian Li, Xiang Han, and Kuan Fang Ren. Generalized Debye series expansion of electromagnetic plane wave scattering by an infinite multilayered cylinder at oblique incidence. *Physical Review E*, 79(3):036602, 2009.

[36] Feng Xu, James A Lock, and Cameron Tropea. Debye series for light scattering by a spheroid. *Journal of the Optical Society of America A*, 27(4):671-686, 2010.
[37] Feng Xu, James A Lock, and Gérard Gouesbet. Debye series for light scattering by a nonspherical particle. *Physical Review A*, 81(4):043824, 2010.

[38] AR Jones. Calculation of the ratios of complex Riccati-Bessel functions for Mie scattering. *Journal of Physics D: Applied Physics*, 16(3):L49, 1983.

[39] PC Waterman. Matrix formulation of electromagnetic scattering. *Proceedings of the IEEE*, 53(8):805-812, 1965.

[40] Peter C Waterman. Symmetry, unitarity, and geometry in electromagnetic scattering. *Physical review D*, 3(4):825, 1971.

[41] PW Barber and Cavour Yeh. Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies. *Applied Optics*, 14(12):2864-2872, 1975.

[42] Michael I Mishchenko, Larry D Travis, and Andrew A Lacis. *Scattering, absorption, and emission of light by small particles*. Cambridge university press, 2002.

[43] Adrian Doicu, Thomas Wriedt, and Yuri A Eremin. *Light scattering by systems of particles: null-field method with discrete sources: theory and programs*, volume 124. Springer, 2006.

[44] MI Mishchenko. Light scattering by randomly oriented axially symmetric particles. *Journal of the Optical Society of America A*, 8(6):871-882, 1991.

[45] Lei Bi, Ping Yang, George W Kattawar, and Michael I Mishchenko. Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles. *Journal of Quantitative Spectroscopy and Radiative Transfer*, 116:169-183, 2013.

[46] Michael Morgan and K Mei. Finite-element computation of scattering by inhomogeneous penetrable bodies of revolution. *IEEE Transactions on Antennas and Propagation*, 27(2):202-214, 1979.

[47] Kane S Yee et al. Numerical solution of initial boundary value problems involving Maxwells equations in isotropic media. *IEEE Trans. Antennas Propag*, 14(3):302-307, 1966.

[48] Roger F Harrington and Jan L Harrington. *Field computation by moment methods*. Oxford University Press, 1996.

[49] Edward M Purcell and Carlton R Pennypacker. Scattering and absorption of light by nonspherical dielectric grains. *The Astrophysical Journal*, 186:705-714, 1973.

[50] Jiming Song, Cai-Cheng Lu, and Weng Cho Chew. Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects. *IEEE Transactions on Antennas and Propagation*, 45(10):1488-1493, 1997.
[51] Xiao-Min Pan and Xin-Qing Sh. A sophisticated parallel MLFMA for scattering by extremely large targets [EM Programmer's Notebook]. *IEEE Antennas and Propagation Magazine*, 50(3):129-138, 2008.

[52] Minglin Yang, Kuan Fang Ren, Mingjiang Gou, and Xinqing Sheng. Computation of radiation pressure force on arbitrary shaped homogenous particles by multilevel fast multipole algorithm. *Optics letters*, 38(11):1784-1786, 2013.

[53] Minglin Yang, Kuan Fang Ren, Yueqian Wu, and Xinqing Sheng. Computation of stress on the surface of a soft homogeneous arbitrarily shaped particle. *Physical Review E*, 89(4):043310, 2014.

[54] Lord Rayleigh. V. On the incidence of aerial and electric waves upon small obstacles in the form of ellipsoids or elliptic cylinders, and on the passage of electric waves through a circular aperture in a conducting screen. *The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science*, 44(266):28-52, 1897.

[55] George E Davis. Scattering of light by an air bubble in water. *Journal of the Optical Society of America*, 45(7):572-581, 1955.

[56] van de Hulst HC. *Light scattering by small particles*. Dover Publications, New York, 1957.

[57] Werner J Glantschnig and Sow-Hsin Chen. Light scattering from water droplets in the geometrical optics approximation. *Applied Optics*, 20(14):2499-2509, 1981.

[58] A Ungut, G Grehan, and G Gouesbet. Comparisons between geometrical optics and Lorenz-Mie theory. *Applied optics*, 20(17):2911-2918, 1981.

[59] Edward A Hovenac. Calculation of far-field scattering from nonspherical particles using a geometrical optics approach. *Applied optics*, 30(33):4739-4746, 1991.

[60] Feng Xu, Xiaoshu Cai, and Kuanfang Ren. Geometrical-optics approximation of forward scattering by coated particles. *Applied optics*, 43(9):1870-1879, 2004.

[61] Arthur L Aden and Milton Kerker. Scattering of electromagnetic waves from two concentric spheres. *Journal of Applied Physics*, 22(10):1242-1246, 1951.

[62] Xiangzhen Li, Xiang'e Han, Renxian Li, and Huifen Jiang. Geometrical-optics approximation of forward scattering by gradient-index spheres. *Applied optics*, 46(22):5241-5247, 2007.

[63] Haitao Yu, Jianqi Shen, and Yuehuan Wei. Geometrical optics approximation of light scattering by large air bubbles. *Particuology*, 6(5):340-346, 2008.
[64] Ye Grynko and Yu Shkuratov. Scattering matrix calculated in geometric optics approximation for semitransparent particles faceted with various shapes. *Journal of Quantitative Spectroscopy and Radiative Transfer*, 78(3):319-340, 2003.

[65] Ping Yang and KN Liou. Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals. *Applied Optics*, 35(33):6568-6584, 1996.

[66] Ping Yang, Heli Wei, Hung-Lung Huang, Bryan A Baum, Yong X Hu, George W Kattawar, Michael I Mishchenko, and Qiang Fu. Scattering and absorption property database for nonspherical ice particles in the near-through far-infrared spectral region. *Applied optics*, 44(26):5512-5523, 2005.

[67] Kuan Fang Ren, Fabrice Onofri, Claude Rozé, and Thierry Girasole. Vectorial complex ray model and application to two-dimensional scattering of plane wave by a spheroidal particle. *Optics letters*, 36(3):370-372, 2011.

[68] Aydin M Sayili. The Aristotelian explanation of the rainbow. *Isis*, 30(1):65-83, 1939.

[69] David C Lindberg. Roger Bacon's Theory of the Rainbow: Progress or Regress? *Isis*, 57(2):235-248, 1966.

[70] H Moyses Nussenzveig. The theory of the rainbow. *Scientific American*, 236(4):116-127, 1977.

[71] René Descartes and Et Gilson. *Discours de la méthode*. Vrin, 1987.

[72] Jearl D Walker. Multiple rainbows from single drops of water and other liquids. *American Journal of Physics*, 44(5):421, 1976.

[73] Philip Laven. Supernumerary arcs of rainbows: Youngs theory of interference. *Applied Optics*, 56(19):G104-G112, 2017.

[74] George Biddell Airy et al. On the intensity of light in the neighbourhood of a caustic. *Transactions of the Cambridge Philosophical Society*, 6:379, 1838.

[75] Ru T Wang and HC Van de Hulst. Rainbows: Mie computations and the Airy approximation. *Applied Optics*, 30(1):106-117, 1991.

[76] N Roth, K Anders, and A Frohn. Simultaneous measurement of temperature and size of droplets in the micrometer range. *Journal of Laser Applications*, 2(1):37-42, 1990.

[77] F Corbin, A Garo, G Gouesbet, and G Grehan. Réfractométrie darc-en-ciel: application au diagnostic des gouttes avec gradient dindice. *Recueil des actes du 5e congrès Francophone de Vélocimétrie Laser*, Rouen, France E, 1, 1996.
[78] JPAJ Van Beeck and ML Riethmuller. Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity. *Applied optics*, 35(13):2259-2266, 1996.

[79] Michael I Mishchenko, Joachim W Hovenier, and Larry D Travis. *Light scattering by nonspherical particles: theory, measurements, and applications*. Academic press, 1999.

[80] Philip L Marston and Robert E Apfel. Quadrupole resonance of drops driven by modulated acoustic radiation pressureExperimental properties. *The Journal of the Acoustical Society of America*, 67(1):27-37, 1980.

[81] Philip L Marston. Rainbow phenomena and the detection of nonsphericity in drops. *Applied optics*, 19(5):680-685, 1980.

[82] Philip L Marston and Eugene H Trinh. Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops. *Nature*, 312(5994):529-531, 1984.

[83] JF Nye. Rainbow scattering from spheroidal dropsan explanation of the hyperbolic umbilic foci. *Nature*, 312(5994):531-532, 1984.

[84] Philip L Marston. Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering. *Optics letters*, 10(12):588-590, 1985.

[85] Harry J Simpson and Philip L Marston. Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes. *Applied optics*, 30(24):3468-3473, 1991.

[86] Philip L Marston. Transverse cusp diffraction catastrophes: Some pertinent wave fronts and a Pearcey approximation to the wave field. *The Journal of the Acoustical Society of America*, 81(2):226-232, 1987.

[87] Gregory Kaduchak, Philip L Marston, and Harry J Simpson. E6 diffraction catastrophe of the primary rainbow of oblate water drops: observations with white-light and laser illumination. *Applied optics*, 33(21):4691-4696, 1994.

[88] Cleon E Dean and Philip L Marston. Opening rate of the transverse cusp diffraction catastrophe in light scattered by oblate spheroidal drops. *Applied optics*, 30(24):3443-3451, 1991.

[89] JF Nye. Rainbows from ellipsoidal water drops. In *Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences*, volume 438, pages 397-417. The Royal Society, 1992.

[90] James A Lock and Feng Xu. Optical caustics observed in light scattered by an oblate spheroid. *Applied optics*, 49(8):1288-1304, 2010.
[91] Haitao Yu, Feng Xu, and Cameron Tropea. Optical caustics associated with the primary rainbow of oblate droplets: simulation and application in non-sphericity measurement. *Optics express*, 21(22):25761-25771, 2013.

[92] Kenneth Sassen. Angular scattering and rainbow formation in pendant drops. *Journal of the Optical Society of America*, 69(8):1083-1089, 1979.

[93] Dean S Langley and Philip L Marston. Generalized tertiary rainbow of slightly oblate drops: observations with laser illumination. *Applied optics*, 37(9):1520-1526, 1998.

[94] James A Lock and Judith R Woodruff. Non-Debye enhancements in the Mie scattering of light from a single water droplet. *Applied optics*, 28(3):523-529, 1989.

[95] CW Chan and WK Lee. Measurement of a liquid refractive index by using high-order rainbows. *Journal of the Optical Society of America B*, 13(3):532-535, 1996.

[96] PH Ng, MY Tse, and WK Lee. Observation of high-order rainbows formed by a pendant drop. *Journal of the Optical Society of America B*, 15(11):2782-2787, 1998.

[97] Pak-hong Ng, Pui-yiu So, Chiu-wah Chan, and Wing-kee Lee. Interference of the eleventh-and higher-order rainbows formed by a pendant water drop. *Journal of the Optical Society of America B*, 20(11):2395-2399, 2003.

[98] Jeronimus Petrus Antonius Johannes van Beeck. Rainbow phenomena: development of a laser-based, non-intrusive technique for measuring droplet size, temperature and velocity. 1997.

[99] Willy Möbius. Zur Theorie des Regenbogens und ihrer experimentellen Prüfung. *Annalen der Physik*, 338(16):1493-1558, 1910.

[100] Craig F Bohren and Donald R Huffman. *Absorption and scattering of light by small particles*. John Wiley & Sons, 2008.

[101] Wolfram Hergert and Thomas Wriedt. *The Mie Theory: basics and applications*, volume 169. Springer, 2012.

[102] Edward A Hovenac and James A Lock. Assessing the contributions of surface waves and complex rays to far-field Mie scattering by use of the Debye series. *Journal of the Optical Society of America A*, 9(5):781-795, 1992.

[103] Renxian Li, Xiang'e Han, Huifen Jiang, and Kuan Fang Ren. Debye series for light scattering by a multilayered sphere. *Applied optics*, 45(6):1260-1270, 2006.
[104] Kuan Fang Ren. ABSphere, software licence: IDDN. FR. 001. 130022. 000. R. P. 2014. 000. 30000.

[105] René Descartes. *Discourse on method, optics, geometry, and meteorology*. Hackett Publishing, 2001.

[106] Kuan Fang Ren. VCRMell2D, software licence: IDDN. FR. 001. 130023. 000. R. P. 2014. 000. 30000.

[107] X Han, KF Ren, L Mees, G Gouesbet, and G Gréhan. Surface waves/geometrical rays interferences: numerical and experimental behaviour at rainbow angles. *Optics communications*, 195(1):49-54, 2001.

[108] Philip L Marston. Critical angle scattering by a bubble: physical-optics approximation and observations. *Journal of the Optical Society of America*, 69(9):1205-1211, 1979.

[109] Fabrice RA Onofri, Mariusz Krzysiek, Janusz Mroczka, Kuan-Fang Ren, Stefan Radev, and Jean-Philippe Bonnet. Optical characterization of bubbly flows with a near-critical-angle scattering technique. *Experiments in fluids*, 47(4-5):721, 2009.

[110] Feng Xu, Kuan Fang Ren, and Xiaoshu Cai. Extension of geometrical-optics approximation to on-axis Gaussian beam scattering. I. By a spherical particle. *Applied optics*, 45(20):4990-4999, 2006.

[111] Feng Xu, Kuan Fang Ren, Xiaoshu Cai, and Jianqi Shen. Extension of geometrical-optics approximation to on-axis Gaussian beam scattering. II. By a spheroidal particle with end-on incidence. *Applied optics*, 45(20):5000-5009, 2006.

[112] James A Lock. Ray scattering by an arbitrarily oriented spheroid. I. Diffraction and specular reflection. *Applied optics*, 35(3):500-514, 1996.

[113] James A Lock. Ray scattering by an arbitrarily oriented spheroid. II. Transmission and cross-polarization effects. *Applied optics*, 35(3):515-531, 1996.

[114] Keli JIANG. *Theoretical study of light scattering by an elliptical cylinder*. PhD thesis, Texas A&M University, USA, 2013.
# Résumé

Cette thèse est dédiée à l'extension du Tracé de Rayons Vectoriels Complexes (TRVC) développé au laboratoire et son application à la caractérisation des particules non-sphériques.

Dans divers domaines de recherche, tels que la mécanique des fluides et la combustion, nous devons mesurer les caractéristiques des particules. Parmi les différents types de techniques de mesure, la métrologie optique est largement utilisée grâce à sa précision et sa fiabilité. Cependant, la forme des particules est souvent considérée comme sphérique principalement à cause du manque de moyens pour prédire avec précision l'interaction de la lumière avec de grandes particules non-sphériques. TRVC a été développ pour répondre à ces besoins. Il est basé sur le modèle de rayons mais une amélioration radicale a été apporté dans ce nouvel modèle par l'introduction d'une nouvelle propriété dans la notion de rayons lumineux - la courbure de front d'onde.

Dans cette thèse, le module est appliqué réexaminer la théorie d'Airy. Il a été montré que TRVC prédit mieux les intensités et les positions des lobes secondaires dans les arcs-en-ciel d'une particule sphérique que la théorie d'Airy. Ensuite, TRVC est appliqué à l'étude des propriétés des arcs-en-ciel formés par les ellipsoïdes allongé et aplati. Il est montré que TRVC peut prédire analytiquement les positions et les intensités de lobes secondaires dans les arcs-en-ciel d'une particule sphérodales lorsquelle est éclairée par une onde plane dans le plan symétrique. Les pics dus à l'effet focal d'une particule sphéroïdale aplati ont galement été évalué analytiquement en utilisant le facteur de divergence de TRVC.

Un système de mesure est aussi mis en place dans le laboratoire pour la diffusion de la lumière par des gouttes pendantes. Il a été montré que le rapport des intensités des deux premiers arcs-en-ciel est sensible à l'ellipticité d'un sphéroïde équivalent. Ceci ouvre un champ d'application potentiel pour caractériser la déformation d'une goutte pendante.

**Mots-clés:** diffusion de lumière, Tracé de Rayons Vectoriels Complexes, particule non-sphérique, théorie d'Airy, goutte pendante, métrologie optique
Abstract

This thesis is dedicated to the extension of Vectorial Complex Ray Model (VCRM) developed in the laboratory and its application in the characterization of large non-spherical particles.

In various research domains, such as the fluid mechanics and the combustion field, we need to measure the characteristics of the particles. Among different kinds of measurement techniques, the optical metrology is largely employed due to its advantage of being accurate, reliable and non-intrusive. However, in many optical techniques, the shape of the particles is assumed to be spherical. The main reason of this limit is due to the lack of theoretical model to describe with precision the interaction of light with large non-spherical particles. The Vectorial Complex Ray Model has been developed to reply this demand. It is based on the ray model but a radical improvement has been done by introducing a new property: the wave front curvatures in the ray model.

In this thesis, the model is firstly applied to reexamine the Airy theory. It is shown that even in the case of spherical particle VCRM predicts better the intensity and positions of supernumerary bows of rainbow than the Airy theory. Then VCRM is applied to investigate the properties of the rainbows formed by a spheroidal (prolate or oblate) particle. It is shown that VCRM can predict analytically the positions and the intensity of supernumerary bows and the peaks due to the focal effect when it is illuminated by a plane wave in the symmetrical plane.

The theoretical research results have been also applied to the experimental characterization of a pendant drop. The intensity ratio of the two first orders of rainbow is shown sensible to the aspect ratio of the equivalent spheroid and it opens a potential to develop a measurement technique to characterize the deformation of pendant drop.

**Keywords:** light scattering, Vectorial Complex Ray Model, non-spherical particle, Airy theory, pendant drop, optical metrology