Comparison of scattering diagrams of large non-spherical particles calculated by VCRM and MLFMA

Journal of Quantitative Spectroscopy & Radiative Transfer ] (]]]]) ]]]–]]]

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Contents lists available at ScienceDirect

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Journal of Quantitative Spectroscopy & Radiative Transfer

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journal homepage: www.elsevier.com/locate/jqsrt

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Comparison of scattering diagrams of large non-spherical particles calculated by VCRM and MLFMA

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Minglin Yang a,b, Yueqian Wu a, Xinqing Sheng a, Kuan Fang Ren b,n

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a b

Center for Electromagnetic Simulation, School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China UMR 6614/CORIA, CNRS - Université et INSA de Rouen, BP 12, 76801 Saint Etienne du Rouvray, France

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a r t i c l e i n f o

abstract

Article history: Received 17 December 2014 Received in revised form 25 January 2015 Accepted 30 January 2015

The vectorial complex ray model (VCRM) developed in recent years improves considerably the precision and the applicability of the ray model in the light scattering by large nonspherical particles. But its results for particles of complex shape remain to be validated. The surface integral equation (SIE) with parallel multilevel fast multipole algorithm (MLFMA) is a rigorous numerical method so it permits us to obtain accurate results of particles of any shape. With new improvement we have made, our parallelized code of MLFMA allows us to calculate the scattering of non-spherical dielectric particles of size much larger than the incident wavelength. This paper is devoted to the validation of the VCRM by comparison of the scattering diagrams with the MLFMA results. The scattering diagrams of large ellipsoidal particles with different size parameters and aspect ratios are computed by using the two methods. Good agreements between them prove the ability of both MLFMA and VCRM for large non-spherical particles. & 2015 Published by Elsevier Ltd.

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Keywords: Light scattering Vectorial complex ray model Multilevel fast multipole algorithm Non-spherical particle

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1. Introduction

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In various research fields, we need to solve light scattering by particles of complex shape, such as multiphase flow, atmospheric radiative transfer, bio-optics and climatology. Particles encountered in nature, in industry and in the laboratories (spray of atomization, ice crystals, dust aerosols, raindrops and cells in biophysical systems) are often irregularly shaped and have a wide range of sizes. Various theories and models have been developed for solving such problems which can be categorized into three groups: rigorous theories, approximate methods and numerical methods. Rigorous methods are based on the solution of Maxwell's equations usually by variable separation. The Lorenz–Mie theory (LMT) is one of the most famous

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n

Corresponding author. E-mail address: [email protected] (K. Fang Ren).

theories to describe the interaction between a homogeneous isotropic sphere and an electromagnetic plane wave [1]. In 1980s, it is generalized to take into account the shape of the incident wave. One of the most famous formalisms is the so-called Generalized Lorenz–Mie theory (GLMT) [2]. However, such kind of rigorous theories are limited to the particles of simple shapes (sphere, spheroid, infinite cylinder, etc.) [3–6], and the calculable size of particles is often limited to some tens of wavelengths except the sphere and the circular infinite cylinder. Approximate methods are precision limited, but it is flexible to particle shape and it may give physical insight on the mechanism of the interaction of the light with the particle. High frequency approximate methods, such as the geometrical optics (GO), have been therefore used for predicating light scattering by spheres [7], absorbing spheres [8], Platonic solids [9] or combined with electromagnetic wave methods for the scattering of ice crystals [10]. In principle, GO can be applied to deal with the interaction of light with objects of any shape. But in reality,

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http://dx.doi.org/10.1016/j.jqsrt.2015.01.024 0022-4073/& 2015 Published by Elsevier Ltd.

61 Please cite this article as: Yang M, et al. Comparison of scattering diagrams of large non-spherical particles calculated by VCRM and MLFMA. J Quant Spectrosc Radiat Transfer (2015), http://dx.doi.org/10.1016/j.jqsrt.2015.01.024i

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it is rarely used in quantitative study of the scattering diagram by non-spherical, non-circular cylindrical particles. The key problem is the divergence/convergence of a wave on the surface of the particle. Ren et al. have introduced a new wave property – wave front curvature and developed a new model: the vectorial complex ray model (VCRM) [11]. This model has already been applied for computing scattering diagrams of large ellipsoids and infinite long elliptical cylinders with the plane wave or shaped beam incidences [11,12]. However, the results of VCRM are to be validated because neither theoretical nor numerical methods exist to predict with precision the scattering properties of very large non-spherical particles. Numerical methods, such as the T-matrix technique, the finite-difference time-domain (FDTD) and the discrete dipole approximation (DDA), though time consuming and needing large amounts of CPU memory, are very flexible in the shape of the particle. The T-matrix method has been well developed in a series of papers for scattering problems [13–16]. This technique usually relies on central expansions of the electromagnetic field in the vector spherical wave functions (VSWF). Arbitrary shaped particles of low rotational symmetry will increase time required to evaluate the T-matrix. The FDTD method developed by Yee in 1966 [17] is a widely used and simple numerical method for computational electromagnetism [18–20]. The finite spatial discretization and approximate boundary conditions will affect the accuracy of the FDTD and make it poorly suitable for achieving high and controllable numerical accuracy. The DDA was proposed by Purcell and Pennypacker which divided the scatterer by a set of point dipoles [21]. Later, developments and improvements on DDA have been done by many authors [22–25]. Both DDA and FDTD are volume discretized methods. Their number of unknowns are therefore related to the volume of the particle(Oðx3 ) with x ¼ 2π a=λ). They are also flexible and robust for inhomogeneous, anisotropic particles but the computation resource need increases rapidly as a function of the particle size, so limited to the scattering of small particles. Besides, the cubical subvolume meshes also make them not flexible or efficient enough for those complex shaped particles with sharp corners or wedges. On the other hand, the method of moments (MoM), an implementation of surface integral equations (SIEs), is very powerful and has been widely used for computing electromagnetic scattering by homogeneous objects, because of its accuracy and efficiency [26–29]. It is more flexible and efficient for large homogeneous particles, especially for particles with large refractive index. Moreover, the discretization with triangles makes it easier to generate meshes for modeling irregular shape particles. It can also be efficiently accelerated with various acceleration methods, among which the multilevel fast multipole algorithm (MLFMA) has been well developed and can be used to solve challenging problems [30–33]. The SIEs with MLFMA have already been successfully applied for computing radiation pressure force and surface stress for large nonspherical particles of size parameter larger than 600 [34,35]. This paper is devoted therefore to validate the VCRM computations by comparison of scattering diagrams of large non-spherical particles with the MLFMA results.

The body of this paper is organized as follows: Sections 2 and 3 give respectively a brief introduction to each of the two models VCRM and MLFMA. After the general description of the principle of VCRM and MLFMA, the comparisons between the scattering diagrams calculated by MLFMA and VCRM for ellipsoidal particles will be given and discussed in Section 4. The last section is devoted to the conclusions.

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2. Vectorial complex ray model

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In VCRM, all waves are described by bundles of vectorial complex rays and each ray has five properties. Besides the direction, the amplitude, the polarization and the phase, the wavefront curvature of the wave is introduced. Furthermore, the wave vector k is used to describe the direction of a vectorial complex ray which simplifies considerably the calculation. According to the continuity of the tangent component of the wave vector on the interface of the particle, the Snell law can be written as

73

0

kτ ¼ kτ

ð1Þ 0

where kτ and kτ are respectively the tangent components 0 of the wave vectors k and k of the rays before and after interaction (reflection or refraction). Then the normal component of the emergent ray is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 02 2 ð2Þ kn ¼ k  kτ We suppose that the curvature of the particle surface in the vicinity of the incident point of a ray is described by matrix C; the curvatures of the incident and refracted/ reflected wave fronts are expressed respectively by the matrices Q and Q 0 (see Fig. 1). The projection matrix between the base unitary vectors of the coordinates systems on the planes tangent to the wavefront (t 1 ; t 2 ) and to the dioptric surface (s1 ; s2 ) is ! s1  t 1 s1  t 2 Θ¼ ð3Þ s2  t 1 s2  t 2 Then the relation of these three curvature matrices is given by the following wavefront matrix equation [11]:

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75 77 79 81 83 85 87 89 91 93 95 97 99 101 103

ð4Þ

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where n is the normal of dioptric surface, the superscript T represents the transposed matrix, and the letters with or without the prime represent respectively the values after or before interaction of the ray with the surface. In VCRM the divergence factor D is introduced to describe the divergence/convergence of the wave and it

107

0

0

0T

0

ðk  kÞ  nC ¼ k Θ Q 0 Θ kΘ Q Θ T

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Fig. 1. Schematic of the interaction of the wavefront and surface.

Please cite this article as: Yang M, et al. Comparison of scattering diagrams of large non-spherical particles calculated by VCRM and MLFMA. J Quant Spectrosc Radiat Transfer (2015), http://dx.doi.org/10.1016/j.jqsrt.2015.01.024i

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1

D¼

7 9 11 13

R01p R02p R011 R021 R012 R022  ⋯ R12 R22 R13 R23 ðr þR01p Þðr þR02p Þ

65 ð5Þ 67

where r is the distance between the origin of the coordinate system and the observation point. The index p is the order of the ray. R1j and R2j (j ¼ 1; 2; …; p) represent the two curvature radii of the wavefront of incident ray at jth interaction with the surface. R01j and R02j are the corresponding curvature radii of the refracted ray. To take into account the effect of interference, generally, the phase of each ray must be computed which is composed of four parts:

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Φ ¼ Φinc þ Φpth þ ΦFrs þ Φfcl

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where Φinc is the phase of the incident wave, Ψpth the phase due to the optical path, ΦFrs the phase in the Fresnel coefficients and Φfcl the phase due to the focal lines. The amplitude of a ray is calculated by the product of four terms: SG the amplitude of the incident wave, εX;p the pffiffiffiffiffiffi Fresnel coefficients, Dp the divergence factor and Ap the absorption factor. So the complex amplitude of a ray is given by pffiffiffiffiffiffi SX;p ¼ Dp SG εX;p Ap eiΦ ð7Þ

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ð6Þ

where X ¼1 or 2 corresponds respectively to the perpendicular or parallel polarization. It is worth to point out that the absorption factor Ap is evaluated by the attenuation factor as a function of the distance of the ray in the particle according to ! p X 0 ð8Þ Ap ¼ exp  mi kn;q dq q¼1

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is determined directly by the curvature radii of wavefronts as follows:

3 5

3

where mi is the imaginary part of the refractive index, dq 0 the distance that the ray crosses in the particle, and kn;q the component of the wave vector normal to the particle surface. The total amplitude of the scattering field at a given angle is just the summation of all rays. 3. Combined tangential formulation with MLFMA For a given homogeneous dielectric object, using either the equivalence principle or the vector Green's theorem, we can formulate a set of integral equations to calculate the electric and magnetic fields ðE; HÞ in terms of equivalent electric and magnetic currents ðJ; MÞ on the boundary surface S of the dielectric body (Fig. 2). Four basic integral equations are expressed in the homogeneous medium: the electric field integral equation outside the dielectric body (EFIE-O), the magnetic field integral equation outside the dielectric body (MFIE-O), the electric field integral equation inside the dielectric body (EFIE-I), and the magnetic field integral equation inside the dielectric body (MFIE-I) [27]:

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EFIE  O :

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MFIE  O :

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EFIE  I :

E1  Z 1 L1 ðJÞ þK1 ðMÞ ¼ Ei H1 Z 1 1 L1 ðMÞ  K1 ðJÞ ¼ Hi E2  Z 2 L2 ðJÞ þ K2 ðMÞ ¼ 0

ð9Þ ð10Þ ð11Þ

69 71 73 Fig. 2. Sketch of three dimensional homogeneous dielectric body.

H2  Z 2 1 L2 ðMÞ  K2 ðJÞ ¼ 0

MFIE  I :

ð12Þ

where Z l ¼ ðμl =εl Þ1=2 denote the wave impedances in medium l with l ¼1 and 2 respectively for medium outside and inside object. The operators Ll and Kl are defined as # Z " 1 Xðr0 Þ þ 2 ∇ð∇0  Xðr0 ÞÞ Gl ðr; r0 Þ dr0 ð13Þ Ll fXgðrÞ ¼ jkl S kl Z Kl fXgðrÞ ¼

S

Xðr0 Þ  ∇0 Gl ðr; r0 Þ dr0

ð14Þ

pffiffiffiffiffiffiffiffi where j ¼ 1, kl ¼ ωðμl =ϵl Þ1=2 , X is either the equivalent electric current J or the equivalent magnetic current M on the surface of the object, and Gðr; r0 Þ ¼

expð  jkl jr  r0 jÞ 4π jr  r0 j

ð15Þ

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93 Various forms of combined field integral equations can be constructed, such as the Poggio–Miller–Chang–Harring95 ton–Wu–Tsai (PMCHWT) equations [26], the combined tangential formulation (CTF), the combined normal for97 mulation (CNF) and the electric and magnetic current combined-field integral equations (JMCFIE) [36]. Among Q2 99 those forms of combined equations, the nature of CTF equations is close to a first-kind integral equation and its 101 accuracy is the best, especially when dealing with objects with sharp edges, corners or large refractive index. The 103 equation form of CTF can be expressed as follows: ( 1  1 Z 1 t^  ðEFIE  OÞ þ Z 2 t^  ðEFIE  IÞ 105 ð16Þ Z 1 t^  ðMFIE  OÞ þ Z 2 t^  ðMFIE  IÞ where t^ is the tangential vector at a point on the surface. Eq. (16) can be discretized by first expanding (J; M) with the Rao, Wilton and Glisson (RWG) vector basis functions gi as [37] Ns X

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ð17Þ

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where Ns denotes the total number of edges on S. By substituting Eq. (17) into Eq. (16) and applying gi as the trial functions for the tangential field equations, a complete matrix equation system can be obtained. This numerical solution process is the well known method of moment (MoM). The MoM is conventionally limited to dealing with electrically small dielectric objects due to the computation and storage complexities which are both in the order

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J¼

i¼1

gi J i

M¼

Ns X

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gi M i

i¼1

Please cite this article as: Yang M, et al. Comparison of scattering diagrams of large non-spherical particles calculated by VCRM and MLFMA. J Quant Spectrosc Radiat Transfer (2015), http://dx.doi.org/10.1016/j.jqsrt.2015.01.024i

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M. Yang et al. / Journal of Quantitative Spectroscopy & Radiative Transfer ] (]]]]) ]]]–]]]

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OðN2 Þ, with N being the number of unknowns. To circumvent this problem, matrix-vector multiplications are performed with various acceleration methods, such as the fast Fourier transform (FFT), the adaptive integral method (AIM) and the MLFMA [38–40]. Among these acceleration methods, the MLFMA has been well developed. By using the MLFMA, both the time and memory complexities can be reduced to OðNlogNÞ [41,27,38]. For electrical large problems with more than tens of millions unknowns to be solved, a sequential implementation of MLFMA is not sufficient. In order to solve such extremely large problems, MLFMA can be parallelized on distributed-memory architectures with different strategies to increase the particle size from tens of millions to hundreds of millions in the last decade [31–33,42]. In this paper, the flexible and efficient hybrid message passing interface (MPI) and open multi-processing (OpenMP) paralleled MLFMA is used [43].

19 4. Numerical results 21 23 25 27 29 31 33 35 37 39

In this section, a series of numerical experiments are performed. We consider a triaxial ellipsoid of the semiaxes a; b; c along the x, y and z directions respectively. The shape of the particle is characterized by a couple of aspect ratios, defined as κ 1 ¼ b=a; κ 2 ¼ c=a. Hence we have κ 1 ¼ κ 2 ¼ 1 for a sphere, κ 1 ¼ κ 2 for a spheroid and κ 1 a κ 2 for a triaxial ellipsoid. The size parameter is defined in terms of a volume-equivalent sphere for all the cases. The observation is fixed in xz plane. The step of the scattering angle is set to 0.11. In all the results computed by VCRM, we have checked that 12 orders of rays and 4000 total number of rays are sufficient for the cases under study. The diffraction effect, which is very important in the forward direction, is also taken into consideration by using the simple model given in [11]. Before the study of the effects of the size parameter and the aspect ratios on the scattering properties for large spheroids, we validate our code by comparison with the results of an independent code, i.e. the open source

software ADDA [25]. Fig. 3 shows the scattering diagrams calculated by MLFMA and DDA. The size parameter of the ellipsoid is 12.56 and its aspect ratios are κ 1 ¼ 1:25 and κ 2 ¼ 1:5. The incident angle is set to θ ¼301. It is found that the agreement between the two methods is excellent for the two polarizations. Then we will focus on the validation of VCRM by comparing the scattering diagrams of large ellipsoidal particles calculated with the results of MLFMA code we developed. The first example we give here is for the comparison of VCRM and MLFMA with LMT for a large sphere. A water droplet with a radius 30 μm is computed. The results are plotted in Fig. 4. It can be seen from this figure that both VCRM and MLFMA are in agreement with LMT especially in the forward and backward direction. But there are slight differences between VCRM and LMT around 801 and Alexander's dark region while MLFMA agrees perfectly with LMT. The difference around 801 is due to the surface wave effect which has not yet been taken into consideration in VCRM while the discrepancy near rainbow angles and Alexander region is due to the caustics which must be remedied by taking into account the diffraction effect near the rainbow angles. It is obvious that better agreement is achieved for the perpendicular polarization than the parallel polarization. Similar phenomenon has been observed in the scattering by infinite long cylinders [12]. A spherical water droplet can deform to prolate or oblate spheroids. Next we study the influence of aspect ratios and evaluate the precision of VCRM by using MLFMA results as the reference data. We suppose that all the spheroidal particles in the next several examples have the same volume equal to that of a sphere with a radius of 30 μm. We first set the aspect ratios of a prolate spheroid to κ 1 ¼ 1:0, κ 2 ¼ 1:1. In such a condition, we have a ¼ b ¼ 29:06188 μm, c ¼ 31:96807 μm. The computed results for parallel and perpendicular polarization as a function of scattering angles are plotted in Fig. 5. Again we can observed that better agreement is achieved for the perpendicular polarization than that for the parallel

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0 -60 60 Scattering Angle (degree)

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0 -60 60 Scattering Angle (degree)

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Fig. 3. Comparison of scattering matrix elements computed by MLFMA and DDA for a triaxial ellipsoid (x¼ 12.56, κ1 ¼ 1:25, κ 2 ¼ 1:5 and m¼ 1.33) illuminated by the plane wave of wavelength 0.785 μm. The incident angle is set to θ¼ 301 and the observation is in xz plane. (a) Perpendicular polarization. Q4 (b) Parallel polarization.

Please cite this article as: Yang M, et al. Comparison of scattering diagrams of large non-spherical particles calculated by VCRM and MLFMA. J Quant Spectrosc Radiat Transfer (2015), http://dx.doi.org/10.1016/j.jqsrt.2015.01.024i

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LMT VCRM MLFMA

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log10(I)

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Scattering Angle (degree)

Fig. 4. Comparison of scattering diagrams computed by LMT, VCRM and MLFMA for a sphere with a radius of 30 μm illuminated by the plane wave of wavelength 0.785 μm. The refractive index is 1.33. The incident plane wave propagates along z-axis with the observation in xz plane. The results of VCRM and MLFMA are shifted by 102 and 10  2 respectively for clarity. (a) Perpendicular polarization. (b) Parallel polarization.

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polarization. Compared with these for a sphere, there are greater discrepancies between the two curves in Alexander's dark region and near the caustics. The band of Alexander's dark region between primary and secondary rainbows is also wider than for the sphere. We can also observe slight differences between the results of VCRM and MLFMA in the strong oscillation region caused by the interference of the diffracted and the reflected waves near the forward direction. To quantify the discrepancy between the two methods, we can use the global root mean square (RMS) error defined as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u No u1 X     I cal ðθÞ I ref ðθÞ   RMS ¼ t ð18Þ e2 θi with e θ ¼ No maxI ref ðθÞ i¼1

with No being the number of scattering angles. Ical and Iref are respectively the intensity calculated by the VCRM and the intensity of reference (here the results of MLFMA). RMS in Fig. 5(a) and (b) are respectively 0.24% and 0.2%, so we can conclude that the global agreement between VCRM and MLFMA is very good. However, the global RMS is not suitable to evaluate the difference of the two methods in the range of scattering angles where they have visible discrepancy since the mean scattering intensity will be four to six orders smaller than that in the forward direction. For example, it is clear that discrepancy around 801 in Fig. 5(b) is more important than that in Fig. 5(a). Here we define an average mean relative error to give quantitatively evaluation of the precision of VCRM as     I cal ðθÞ I ref ðθÞ   ; e θ ¼ maxI ref ðθÞ

θ ¼ 01; 11; 21⋯

ð19Þ

where I cal and I ref are the average values of the scattering intensities in dB calculated for each degree by VCRM and MLFMA respectively. This choice is to avoid the high

oscillations in the scattering diagrams. They are calculated by 11   1 X I θ ¼ log10 I θ þ ði  1Þ  0:11 11 i ¼ 1

 

85 87

ð20Þ

The calculated relative error as a function of scattering angles for Fig. 5 is plotted in Fig. 6. We find that VCRM and MLFMA are in better agreement for the perpendicular polarization than the parallel polarization around 801. We extend further the long axes of the prolate such that the aspect ratios are κ 1 ¼ 1:0, κ 2 ¼ 1:2 (a ¼ b ¼ 28:23108 μm, c ¼ 33:87730 μm). The scattering diagrams are shown in Fig. 7. We can observe similar phenomenon as in Fig. 5(a), Alexander's dark region becomes even wider. The global agreement between VCRM and MLFMA is still very good with a RMS of 0.33%. Fig. 8 shows the scattering diagrams to the same particle but the incident wave is inclined 301 relative to the symmetric axis of the prolate. The scattering diagrams are no longer symmetric as they should be, so they are given in all directions (  1801 to 1801). The two Alexander dark regions in the two sides of the scattering diagram are no longer symmetric. It is very narrow and not very discernible in one side from 97.51 to  93.91, but very large and clearly visible on the other side from 161.81 to 186.21 (  173.81). This can be seen more clearly from the isolate order intensity given in Fig. 9. Also, the two curves by VCRM and MLFMA are again in very good agreement except in Alexander dark regions and near  1501. These effects can be seen directly from the ray tracing graph given in Fig. 10. The rainbow of the first order (blue) around  1001 is much important than in 1701. The peaks around 140–1501 are due to the rays corresponding to the rainbow of second order p ¼3. But the structures are very different from that of a sphere. If we increase further the incident angle to 601, as shown in Fig. 11, there is only one Alexander dark region in one side of the scattering diagram ranging from  178.31 to  160.61

Please cite this article as: Yang M, et al. Comparison of scattering diagrams of large non-spherical particles calculated by VCRM and MLFMA. J Quant Spectrosc Radiat Transfer (2015), http://dx.doi.org/10.1016/j.jqsrt.2015.01.024i

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Fig. 6. Relative error between VCRM and MLFMA in Fig. 5 for different polarizations.

and is clearly visible. The other Alexander dark region disappears because both the rays of orders p¼2 and p¼3 cover the region from  77.41 to 55.91, as shown in Fig. 12. Again we can see that the agreement between the two curves of VCRM and MLFMA are very good. When the incident angle is set to 901 (Fig. 13), the results of VCRM and MLFMA are in better agreement than in Fig. 11. This is because there is no dark Alexander region in such condition. For a water droplet (m ¼1.33), if the incident is along the symmetric axis of a prolate with aspect ratios κ 1 ¼ 1:0, κ 2 ¼ 1:5 (a ¼ b ¼ 26:20741 μm, c ¼ 39:31112 μm), there are neither visible rainbows nor Alexander's dark region, the scattered light of the orders p ¼2 and p¼3 are confined almost all in the backward and the forward direction respectively (Fig. 14). This can be seen clearly from the ray tracing graph given in Fig. 15. Now we examine the scattering diagram of oblate spheroids calculated by the two methods. We set κ 1 ¼ 0:9, κ 2 ¼ 1:0 always for a particle of volume equivalent to a sphere of radius 30 μm such that the three axes are a ¼ c ¼ 31:07232 μm, b ¼ 27:96509 μm. The scattering

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VCRM MLFMA

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0.1 0.05

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Scattering Angle (degree)

0.15

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0

log10(I)

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Relative Error

0.25

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0

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Fig. 5. Comparison of scattering diagrams computed by VCRM and MLFMA for a prolate spheroid (κ 1 ¼ 1:0, κ 2 ¼ 1:1 and m¼1.33) with its volume being equal to that of a sphere with a radius of 30 μm illuminated by the plane wave of wavelength 0.785 μm. The incident plane wave propagates along z-axis polarized in x- and y-axis. The observed plane is fixed in xz plane. The results of VCRM are shifted by 102 for clarity. (a) Perpendicular polarization. (b) Parallel polarization.

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Fig. 7. Same parameters as in Fig. 5(a) except the aspect ratio, which is set to κ1 ¼ 1:0, κ2 ¼ 1:2.

diagrams calculated by VCRM and MLFMA are shown in Fig. 16. Compared with that for a sphere in Fig. 4(a), the discrepancy between the two methods for observation angles from 701 to about 1201 is more visible. There is also a peak for the curve of VCRM at about 421 which differs greatly from the curve of MLFMA. Around these angles are the rainbow of third and fourth orders (p¼4 and p¼ 5) where the intensity tends to be very large and a singularity appears in VCRM. The case κ 1 ¼ 1:0, κ 2 ¼ 0:9 shown in Fig. 17 is equivalent to set the observation plane in yz plane in the case above. We found that the agreements between the two methods in Fig. 17 is better than that in Fig. 16. Our VCRM code can also calculate the scattering diagrams of large ellipsoids when the incident plane wave remains in a plane defined by any two axes of the ellipsoid. Figs. 18 and 19 show respectively the scattering diagrams of an ellipsoid with κ 1 ¼ 1:1, κ 2 ¼ 1:2 (a ¼ 27:34827 μm, b ¼ 30:08310 μm, c ¼ 32:81793 μm) and κ 1 ¼ 1:2, κ 2 ¼ 1:1 (a ¼ 27:34827 μm, b ¼ 32:81793 μm, c ¼ 30:08310 μm). The incident plane wave is perpendicularly polarized. Similarly, good agreement can be observed.

Please cite this article as: Yang M, et al. Comparison of scattering diagrams of large non-spherical particles calculated by VCRM and MLFMA. J Quant Spectrosc Radiat Transfer (2015), http://dx.doi.org/10.1016/j.jqsrt.2015.01.024i

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Fig. 8. Comparison of scattering diagrams computed by VCRM and MLFMA for a prolate spheroid (κ1 ¼ 1:0, κ 2 ¼ 1:2 and m¼1.33) with the incident angle being set to 301. Other parameters are the same as in Fig. 7.

75 77

Fig. 11. Comparison of scattering diagrams computed by VCRM and MLFMA for a prolate spheroid (κ1 ¼ 1:0, κ 2 ¼ 1:2 and m¼1.33) with the incident angle set to 601. Other parameters are the same as in Fig. 7.

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Scattering Angle (degree) Fig. 9. Scattered intensities of rays of orders p ¼ 2 and p¼ 3 for the prolate spheroid in Fig. 8.

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Fig. 12. Scattered intensities of rays of orders p ¼2 and p ¼ 3 for the prolate spheroid in Fig. 11.

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Fig. 10. Ray tracing for the prolate spheroid in Fig. 8. Incident rays: cyan, p ¼0: red, p ¼ 2: blue, p ¼ 3: green, p ¼4: yellow. (For interpretation of the references to color in this figure caption, the reader is referred to the web Q5 version of this paper.)

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Before studying the influence of size parameter on the scattering diagrams we would point out that though the theoretical solution exists for a spheroidal particle, the evaluation of spheroidal wave functions is beset with numerical instabilities for large particle. So the validation using the MLFMA is important for spheroids, as well as for

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Scattering Angle (degree) Fig. 13. Comparison of scattering diagrams computed by VCRM and MLFMA for a prolate spheroid (κ1 ¼ 1:0, κ 2 ¼ 1:2 and m¼1.33) with the incident angle set to 901. Other parameters are the same as in Fig. 7.

ellipsoids. The next example is for a prolate spheroid with its volume equal to a sphere of radius 50 μm (x¼ 400). The aspect ratios are κ 1 ¼ 1:0, κ 2 ¼ 1:2, which correspond to a ¼ b ¼ 47:05180 μm, c ¼ 56:46216 μm. The scattering diagrams computed by VCRM and MLFMA are shown in

Please cite this article as: Yang M, et al. Comparison of scattering diagrams of large non-spherical particles calculated by VCRM and MLFMA. J Quant Spectrosc Radiat Transfer (2015), http://dx.doi.org/10.1016/j.jqsrt.2015.01.024i

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Fig. 14. Same parameters as in Fig. 7 except the aspect ratio which is set to κ 1 ¼ 1:0, κ2 ¼ 1:5.

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Fig. 17. Comparison of scattering diagrams computed by VCRM and MLFMA for an oblate spheroid (κ1 ¼ 1:0, κ 2 ¼ 0:9). Other parameters are the same as in Fig. 16.

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Fig. 15. Ray tracing for rays of orders p ¼2 and p ¼3 for a prolate spheroid of equivalent radius 30 μm and aspect ratio κ 1 ¼ 1:0, κ2 ¼ 1:5. The incident plane wave comes from the left, the rays of order p ¼2 (blue in color version) are in the backward direction and the rays p ¼ 3 (green in color version) are in the forward direction. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

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Fig. 18. Comparison of scattering diagrams computed by VCRM and MLFMA for an ellipsoid (κ 1 ¼ 1:1, κ 2 ¼ 1:2) illuminated by a perpendicular polarized plane wave. Other parameters are the same as those in Fig. 5(a).

VCRM MLFMA

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Scattering Angle (degree) Fig. 16. Comparison of scattering diagrams computed by VCRM and MLFMA for an oblate spheroid (κ1 ¼ 0:9, κ 2 ¼ 1:0). Other parameters are the same as in Fig. 5(a).

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Fig. 20. Compared to the results in Fig. 7 for the smaller prolate spheroid of 30 μm, better agreements are observed in the scattering angles near 1201, as the mean relative error given in Fig. 21. This is because the bigger the scatter is, the better the result of the ray model. Alexander's dark region is still clearly visible. Then we show in Fig. 22 the scattering diagram of an ellipsoid. The two aspect ratios are κ 1 ¼ 1:1, κ 2 ¼ 1:2

Fig. 19. Comparison of scattering diagrams computed by VCRM and MLFMA for an ellipsoid (κ 1 ¼ 1:2, κ2 ¼ 1:1). Other parameters are the same as those in Fig. 18.

(a ¼ 45:58046 μm, b ¼ 50:13850 μm, c ¼ 54:69655 μm). Again better agreements can be observed between VCRM and MLFMA compared to the results in Fig. 18, especially for scattering angles range from 901 to 1201.

Please cite this article as: Yang M, et al. Comparison of scattering diagrams of large non-spherical particles calculated by VCRM and MLFMA. J Quant Spectrosc Radiat Transfer (2015), http://dx.doi.org/10.1016/j.jqsrt.2015.01.024i

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M. Yang et al. / Journal of Quantitative Spectroscopy & Radiative Transfer ] (]]]]) ]]]–]]]

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Scattering Angle (degree) Fig. 21. Relative error between scattering diagrams for a prolate spheroid (κ1 ¼ 1:0, κ 2 ¼ 1:2) with different size parameters by VCRM and MLFMA.

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Fig. 23. Comparison of scattering diagrams computed by VCRM and MLFMA for an ellipsoid (κ 1 ¼ 1:1, κ2 ¼ 1:2 and m¼ 1.33) with its volume being equal to a sphere with a radius of 50 μm illuminated by the plane wave of wavelength 0.785 μm. The incident plane wave with the incident angle 301 propagates along z-axis polarized in y-axis. The observed plane is fixed in xz plane. The results of VCRM are shifted by 102 for clarity.

Memory (GB)

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Nu

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Table 1 List of computation resources used by MLFMA for the large ellipsoid particles: Nu–number of unknowns and Ni–number of iterations.

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Fig. 20. Comparison of scattering diagrams computed by VCRM and MLFMA for a prolate spheroid (κ 1 ¼ 1:0, κ 2 ¼ 1:2 and m¼1.33) with its volume being equal to a sphere with a radius of 50 μm illuminated by the plane wave of wavelength 0.785 μm. The incident plane wave propagates along z-axis polarized in y-axis. The observed plane is fixed in xz plane. The results of VCRM are shifted by 102 for clarity.

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Fig. 22. Comparison of scattering diagrams computed by VCRM and MLFMA for an ellipsoid (κ1 ¼ 1:1, κ 2 ¼ 1:2 and m¼ 1.33) with its volume being equal to a sphere with a radius of 50 μm illuminated by the plane wave of wavelength 0.785 μm. The incident plane wave propagates along z-axis polarized in y-axis. The observed plane is fixed in xz plane. The results of VCRM are shifted by 102 for clarity.

In the last example, we show in Fig. 23 the scattering diagram of an ellipsoid (κ 1 ¼ 1:1, κ 2 ¼ 1:2 (a ¼ 45:58046 μm, b ¼ 50:13850 μm, c ¼ 54:69655 μm) with an incident angle of 301. Still good agreements can be observed between the two methods. From all the numerical experiments shown in Figs. 4–22 we can conclude that VCRM can predict with precision the scattering diagrams of a sphere, prolate or oblate spheroids, ellipsoids of size as small as some tens of wavelengths. The larger the size of the particle, the better the precision. However, in ray model, there are singularities near rainbow angles for particles of refractive index greater than that of the surrounding medium and near critical angles where the derivative of intensity is not continuous. These problems are under study by taking into account wave effect. Another important advantages of VCRM is its rapidity. For all the calculations presented in the papers, the computation time is less than one second on a personal computer. MLFMA however needs much more computer resources and CPU times. For the calculation in Fig. 23, there are 34:5  106 unknowns and the calculation has taken 200.8 GB memory and 18.9 h CPU time with 36 MPI processes and 2 threads for each process. For smaller particle the CPU time and the number of iterations are both reduced. To give an idea about the computation resource need, we compile in Table 1 the parameters used in the calculations for a large prolate spheroid and an large ellipsoid. These two calculations are performed with the same computer resources. We can found that, with the same size parameter and the same number of unknown,

Please cite this article as: Yang M, et al. Comparison of scattering diagrams of large non-spherical particles calculated by VCRM and MLFMA. J Quant Spectrosc Radiat Transfer (2015), http://dx.doi.org/10.1016/j.jqsrt.2015.01.024i

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the number of iterations and the CPU time increases with the complexity (dissymmetry here) of the problem.

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5. Conclusions

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In this paper, scattering diagrams calculated by VCRM for large ellipsoidal particles are compared with the results of MLFMA to evaluate its precision. In general, good agreements are observed between the two methods. Numerical results prove that VCRM can predict with precision the scattering diagrams of a spherical or nonspherical particle (spheroids and ellipsoids) of size as small as some tens of wavelengths. Larger the size of the particle, better the precision of VCRM. However, visible discrepancies are always observed near rainbow angles. We are working on the improvement of VCRM in that regions by taking into account the Airy-like wave effect.

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Acknowledgments

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This work is supported by the French National Research Agency under the Grant ANR-13-BS090008-01 (AMOCOPS). This work has also been partially supported by substantial computation facilities from CRIHAN (Centre de Ressources Informatiques de Haute-Normandie), the National Basic Research Program (973) of China under Grants 2012CB720702 and 61320601-1 and 111 Project of China under the Grant B14010.

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Please cite this article as: Yang M, et al. Comparison of scattering diagrams of large non-spherical particles calculated by VCRM and MLFMA. J Quant Spectrosc Radiat Transfer (2015), http://dx.doi.org/10.1016/j.jqsrt.2015.01.024i

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