Development of a light scattering solver applicable toparticles of arbitrary shape on the basis of thesurface-integral equations method of Müllertype. I. Methodology, accuracy of calculation,
and electromagnetic current on the particle surface
Takashi Y. Nakajima,1,2,* Teruyuki Nakajima,3 Kyu Yoshimori,4 Sumit K. Mishra,5
and Sachchida N. Tripathi5
1Research and Information Center, Tokai University, 2-28-4 Tomigaya, Shibuya-ku, Tokyo 151-0063, Japan2Department of Atmospheric Sciences, Colorado State University, 1371 Campus Delivery, Fort Collins,
Colorado 80523-1371, USA3Center for Climate System Research, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8568, Japan4Department of Computer and Information Science, Iwate University, 4-3-5 Ueda, Morioka, Iwate 020-8551, Japan
5Department of Civil Engineering, Indian Institute of Technology Kanpur, India, 208016, India
Light scattering by small particles is an importantresearch subject in many scientific and engineeringfields, such as astronomy and geophysics. For
example, in Earth observation and remote sensingwith spaceborne satellites, it is necessary to calcu-late the radiative transfer, which includes scatter-ing of electromagnetic waves by particles in theatmosphere, such as clouds and aerosols, in addi-tion to scattering from molecules and interactionsat the ground surface. In fact, many radiativetransfer solvers have been used for simulating
satellite-measured signals with respect to such com-plex scattering processes. In such calculations, it isnecessary to consider that, generally, the shape ofmost scattering particles in nature is nonspherical,with the exception of particular types of particles,such as warmwater cloud droplets. Thus, an efficientand accurate method for calculating the light-scat-tering properties of nonspherical particles has beenpursued for many years. In the context of this paper,the term “scattering properties” denotes the effi-ciency factors of the scattering and the phasefunction.There are two possible approaches to obtaining the
light-scattering properties for nonspherical particles.One approach is based on an approximation method,while the other involves an exact solution. A typicalmethod used in the former approach is an algorithmbased on the geometrical optics approximation(GOA), which is applicable to particles with verylarge size parameters (α ¼ 2πr=λ, where r is theequivalent-volume-sphere radius of the nonsphericalparticle and λ is the wavelength of the incident light).Liou and Takano [1] applied GOA to homogeneoushexagonal columns, hollow columns, bullet rosettes,dendrites, and capped columns. The method has alsobeen applied to more complex shapes, such as poly-hedral ice crystals [2] and randomly shaped roughparticles [3,4]. Although the GOA technique has along history, it is still valuable for estimating the ap-pearance of visible optical phenomena, such as halosand/or arcs, in the results of exact methods [5,6], aswell as for calculating the scattering properties forlarge particles [7]. The most precise theories and nu-merical algorithms corresponding to the latter ap-proach are based on solving Maxwell’s equations.Analytical solutions have been examined for only afew simple cases. Mie [8] derived a solution for homo-geneous isotropic spheres, while Wait [9] obtained afull solution for infinite homogenous isotropic circu-lar cylinders. Furthermore, Asano and Yamamoto[10] successfully derived a general solution for homo-geneous isotropic spheroids. In addition to exactanalytical solutions, many numerical algorithmshave also been developed for solving this problem.The T-matrix method [11,12] and the finite-difference time-domain (FDTD) method [13] are twoefficient methods for performing such calculations,and a comprehensive scattering database has beendeveloped by using a combination of FDTD, T-matrix,and GOA [7]. Among the numerical algorithms,methods based on volume or surface integrals areconsidered to be efficient for such calculations. Oneof the popular algorithms for methods based onvolume-integral equations, which are applicable toparticles of arbitrary shapes, is discrete dipole ap-proximation (DDA) [14,15]. This method involvespartitioning a particle into N dipoles. However, sinceN is proportional to α3, in principle, the computingtime drastically increases following the increase ofα. On the contrary, a method based on surface inte-gral equations requires less computational resources
than that based on volume integral equations sincethe number of matching nodal points (NMNP) is pro-portional to α2. Mano [16] developed an algorithm fora method based on surface-integral equations andobtained the scattering properties for a homogeneoushexagonal column with a size parameter of up to 40.FDTD can be applied to both the volume- andsurface-integral methods [17]. Results obtained fromseveral methods have been used for the data analysisof atmospheric observations. For example, Dubovik[18] applied nonspherical models to the Aerosol Ro-botic Network (AERONET) sky-radiometer data andshowed that the mixture of spheroids allows accuratefitting of measured spectral and angular dependen-cies of observed intensity and polarization.
We have focused on developing an algorithm, thesurface-integral equation method for Müller type(SIEM/M), for a method based on surface-integralequations. The target particles are shaped as spheresor hexagonal columns in this paper. There are threeobjectives in this paper. The first is the elucidation ofthe relationship between the accuracy of the ob-tained scattering properties and NMNP. Regardingthis objective, Mano [16] has shown that six nodesper wavelength are necessary in order to obtainthe scattering properties with very high accuracy.He used the symmetry characteristic of hexagonalparticles in order to drastically reduce the requiredcomputational resources (both the calculation timeand the amount of required memory). Since one ofthe aims of this paper is the elucidation of the appli-cability of the method based on surface-integralequations to particles with arbitrary shapes withlimited computer resources, we do not use the sym-metry of the particles. Thus, the number of nodes perwavelength is set to less than six in some calcula-tions. Although this might degrade the calculationaccuracy or restrict the calculations to smaller valuesof the size parameter as compared with those inMano [16], the results provide significant indicationswhen the method based on surface-integral equa-tions is adopted to particles of arbitrary shape, whichentail higher requirements for computer memory.The second objective is the derivation of the electro-magnetic currents on the surface of scattering parti-cles and the scattering phase functions for allscattering directions. The results reveal the impor-tance of the smoothness of the electromagnetic cur-rents on the surface of the particle with respect toobtaining the scattering parameters with high accu-racy. Finally, the phase functions of randomly ori-ented hexagonal column particles are presentedand discussed.
2. Formulations and Numerical Calculation Method
A. Combined Surface-Integral Equations of the Müllertype
In this section, we introduce an integral equationmethod based on obtaining numerical solutions ofMaxwell’s equations. The important advantage of
surface-integral equations is that the dimensionalityof the problem is reduced by 1 as compared to thecase of volume-integral equations, and the numberof unknowns is proportional to the square ratherthan to the cube of the particle size. We choseFredholm equations of the second kind, which wereobtained by Müller [19]. These equations are basedon the following formulations:
iðrÞ × EincðrÞ ¼ −12ð ~m2 þ 1ÞKðrÞ − iðrÞ
×Zsfjk20Jðr0Þð ~m2G1 −G0Þ þ k0Kðr0Þ
×∇0ð ~m2G1 −G0Þ þ jðJðr0Þ·∇0Þ∇0ðG1 −G0Þgds0; ð1Þ
iðrÞ ×HincðrÞ ¼ JðrÞ − iðrÞ
×Zsfjk20Kðr0Þð ~m2G1 −G0Þ − k0Jðr0Þ
×∇0ðG1 −G0Þþ jðKðr0Þ ·∇0Þ∇0ðG1 −G0Þgds0;
ð2Þwhere r and r0 are the positions of matching and in-tegration points, respectively. Einc and Hinc denotethe incident plane electromagnetic wave. J and Kare unknown surface electric and magnetic currentsat r, j is the imaginary unit, ~m is the complex refrac-tive index of the scattering particle, i is an outwardunit vector normal to the surface of the scatteringparticle at r, and k0 is the wavenumber of the inci-dent electromagnetic wave. G1 and G0 representthe Green’s function of the three-dimensional Helm-holtz equation for the incident wavenumber k0 forthe inside (subindex of 1) and the outside (subindexof 0) of the scattering object:
G1ðr; r0Þ ¼e−j ~mk0jr−r0 j
4πk0jr − r0j ; ð3Þ
G0ðr; r0Þ ¼e−jk0jr−r0j
4πk0jr − r0j : ð4Þ
B. Methodology of Numerical Calculation
For the numerical calculation, J and K are discre-tized as
JðrÞ ¼XMm¼1
amfmðrÞ; ð5Þ
KðrÞ ¼XMm¼1
bmfmðrÞ; ð6Þ
where f mðrÞ is the local-domain basis function withnumber of M ¼ 100 (10 × 10 ¼ 100 around the inte-gral point). We adopted the three-dimensionalB-spline function for f. Equations (1) and (2) canbe discretized and written by using residual equa-tions. In our calculations, a point matching methodis adopted.
The coordinates of every nodal point are expressedas rðx; y; zÞ ¼ rðs; tÞ. Here, r is parametrically repre-sented by rðs; tÞ ¼ ðxðs; tÞ; yðs; tÞ; zðs; tÞÞ, where s andt are scalar parameters along the latitudinal andlongitudinal curves on the surface of particles.Figure 1 schematically illustrates the “latitude–longitude” type of particle definition with s and tas parameters and presents a parametrical illustra-tion of the defined particle surface for a sphere and ahexagonal column. In this example, the number ofpoints defined along the s and t curves is 19 and24 for the sphere and 25 and 24 for the hexagonalcolumn and, thus, the surface currents J and K areset at 19 × 24 ¼ 456 and 25 × 24 ¼ 600 NMNP forthe sphere and the hexagonal column in these cases.The small surface facet used for integrating Eqs. (1)and (2) can be described by means of a Jacobianand dsdt:
Δs0 ¼����∂r
0ðs; tÞ∂s
×∂r0ðs; tÞ
∂t
����dsdt: ð7Þ
The Gaussian quadrature integration was applied toavoid the singular point that appeared when jr − r0j ¼0 in the Green’s function. In fact, dense and coarsenumerical integrations had been performed aroundand far from the target matching nodal point r.The number of Gauss points along both the s and tparameters were 20 at the four meshes that sur-round the target matching nodal point r, and 4 atthe other meshes.
The definition of particles (generally, “mesh gen-eration”) can be accomplished with ease in our com-putation program. First, the user is required to setthe position of the nodes by using an ðx; y; zÞ coordi-nate system. At this time, only the minimal numberof nodes sufficient for representing the framework ofthe particle (referred to as “fundamental nodes”) isnecessary. Users can add more nodes (“additional
Fig. 1. Models of a sphere and a hexagonal column with normalvectors at all matching nodal points. A parametric spline is usedfor defining the particle size. k0 denotes the incident wavenumber.
nodes”) between every fundamental node by simplydefining the number of divisions between fundamen-tal nodes. When the user wishes to distort the shapeof the particle, it is necessary to alter the coordinatesof one or more fundamental nodes. The three-dimensional parametric spline function connectsall fundamental and additional nodes smoothly, re-gardless of whether the user has implemented anydistortions. This simple method of particle definitionallows us to generate particles with arbitrary shapesin a few simple steps.Equations (1) and (2) can be expressed as algebraic
equations for solving am and bm in Eqs. (5) and (6):
y ¼ Zx: ð8ÞEquation (8) can be solved by performing LU decom-position, after which the results of the LU-decomposed matrix Z can be used for solving theequation for an arbitrary propagation direction ofthe incident electromagnetic waves. This is usefulfor obtaining the optical parameters of randomly or-iented particles.Once J and K are obtained by solving Eq. (8) by
using Eqs. (5) and (6), the vector scattering ampli-tude F, the scattering cross sectionCs, and the extinc-tion cross section Ce are given by the optical theorem[20]:
FðrÞ ¼ jk204π
�ir × ξir ×
Zs
J expðjk0r0 · irÞds0 þ ir
×Zs
K expðjk0r0 · irÞds0�; ð9Þ
Cs ¼1
k20A20
ZjFj2dΩ; ð10Þ
Ce ¼ −4π
k20A20
ImfA0 · Fðr0Þg; ð11Þ
where ir is a unit vector of the scattering direction rand A0 is the polarized component of the incidentelectric field. Here, ξ ¼ ffiffiffiffiffiffiffiffiffiffiffiffi
μ0=ε0p
, where μ0 ¼ 1:26×10−6 ½H=m� and ε0 ¼ 8:85 × 10−12 ½F=m� are thepermeability and the permittivity of vacuum,respectively.
C. Efficiency Factors and Size Parameters
The efficiency factors of extinction Qext and scatter-ing Qsca can be expressed in a simple manner asfollows:
Qext ¼ Ce=ðπr2Þ; ð12Þ
Qsca ¼ Cs=ðπr2Þ: ð13Þ
We defined r in Eqs. (12) and (13) as the radius of thesphere in the case of a spherical particle or that of anequivalent-volume sphere in the case of a nonspheri-cal particle. The scattering properties are character-ized by the ratio of the particle size and thewavelength λ of the electromagnetic wave. Thus,we defined the size parameter α as follows:
α≡ k0r ¼2πrλ : ð14Þ
For example, α ¼ 1, 10, and 100 for a wavelengthλ∼ 3:7 μm of shortwave infrared waves correspond toa particle radius of about 0.6, 6, and 60 μm, respec-tively. One of the shapes of particles targeted in thispaper is that of a hexagonal column. The defined di-mensions are illustrated in Fig. 2. In our definition, ahexagonal column is described by the lengths L andD, while the size parameters are defined asin Eq. (14).
In order to confirm validity of our SIEM/M algo-rithmwe compared our result with the result appear-ing in Fig. 6(a) of Mano, under the same optical andparticle conditions. Figure 3 shows the phase func-tion of the hexagonal column obtained by theSIEM/M algorithm under the conditions of ~m ¼ð1:3;−0:0Þ, L=D ¼ 0:866 (corresponding to L=D ¼1:0 by the Mano definition), and πL=λ ¼ 3:4. The re-sult is quite similar to Mano, so that our calculationis consistent with that study.
3. Results
A. Accuracy with Respect to Qext versus NMNP
We assumed a wavelength of 3:7 μm for the calcula-tions since this wavelength is important for retriev-ing cloud particle size in the application of cloudremote sensing with visible-to-infrared imaging sen-sors. The extinction efficiency factor Qext of a homo-geneous spherical particle was calculated with a
Fig. 2. Definition of the hexagonal column dimension. Aspect ra-tio is described by L=D.
complex index of refraction ~m ¼ ð1:395;−6:99 × 10−3Þby assuming an ice-phase particle at a wavelengthof 3:7 μm of the electromagnetic waves, forvalues of the parameter α in the range between 1and 30. Figure 4 shows Qext as a function of the sizeparameter α [Figs. 4(a) and 4(b)], as well as the rela-tive error of Qext as obtained with SIEM/M againstthe exact solution obtained from the Mie theory.The results are shown as a function of NMNP[Figs. 4(c) and 4(d)] for electromagnetic waves withpolar incidence (PI) [Figs. 4(a) and 4(c)], and equatorincidence (EI) [Figs. 4(b) and 4(d)]; Fig. 1 contains adefinition of PI and EI. The relationships betweenthe total NMNP and the number of nodes alongthe s and t curves are summarized in Table 1.As shown in Fig. 4, Qext as obtained with SIEM/M
is similar to an exact Qext when the size parameter isless than about 10, in the case of both PI and EI. Thedifferences increase as the size parameter increases,and drastically decrease following the increase ofNMNP. The errors decrease faster for PI than EIin most cases. Since these differences depend onthe distribution of nodal points on the particle sur-face with respect to the direction of propagation ofthe incident light as well as total NMNP, the errorsdo not always linearly decrease with NMNP. How-ever, the general trend is that Qext errors for PI ap-pear to be smaller than those for EI. The oscillationsof Qext along α are smoother for EI than for PI. Thisfact is due to the different distribution of nodal pointswith respect to the direction of propagation of the in-cident light. This feature is discussed in Subsec-tion 3.B. by showing the electric currents J asobtained for the particle surface. Figures 4(c) and
4(d) show that the relative percent error of Qext de-creases as NMNP increases. When NMNP ¼ 3136for PI and EI, the errors become 1% or less for valuesof size parameter α lower than about 20 and 17, re-spectively. On the basis of this evaluation, our SIEM/M is considered to be applicable for values of the sizeparameter lower than about 20, with an error of a fewpercent in the case of spherical particles. In this case,the node density was about four nodes per wave-length, as shown in Table 1. We also performed cal-culations without absorption ~m ¼ ð1:395;−0:0Þ inorder to estimate the convergence level of the calcu-lations. The relative difference value of jQext −
Qscaj=Qext will be nearly zero if the calculation is pre-cise. Here, the values were 6:7 × 10−4, 2:3 × 10−3,6:6 × 10−3, 1:0 × 10−2, 2:0 × 10−2, and 2:7 × 10−2 for va-lues of the size parameter α ¼ 1, 5, 9, 13, 17, and 21when total NMNP ¼ 2408, and were 4:9 × 10−4, 2:1×10−3, 6:3 × 10−3, 9:5 × 10−3, 1:4 × 10−2, and 2:0 × 10−2
when total NMNP ¼ 3136.
B. Electric Current on the Surface of a Particle andScattering Phase Function
The electric and magnetic currents J and K are ob-tained by solving Eq. (8) with the aid of Eqs. (5)and (6). Figure 5 illustrates the real part of the ob-tained J on the surface of a spherical particle for(a) PI and (b) EI, with a complex index of refraction~m ¼ ð1:395;−6:99 × 10−3Þ. The color on the surfacedenotes the magnitude of
����ReðJÞ���� normalized by its
maximum value. The particle was rotated in sucha way that the incident light illuminated the targetfrom the front-right region of each viewgraph, as in-dicated with thick blue arrows in the figure. We canidentify concentric ripples on the particle surface,which appeared when α was larger than 5, wherethe number of ripples increased as the size para-meter α increased. The difference of theQext accuracybetween PI and EI as noted in Subsection 3.A. can beexplained by considering the different distribution ofnodal points with respect to the direction of propaga-tion of the incident light. For PI, the density of nodesis higher at the “north pole,” so that the incident lightwill be dealt with more efficiently. Moreover, theripples were smoothly distributed on the surface ofthe particle since each wavefront and t curve sharedthe same central point. On the contrary, the distribu-tion of nodes is coarse at the surface that faces theincident light, and the wavefronts have a roughstructure for EI. Thus, the better accuracy of Qextfor PI was obtained as a result of the density of nodesat the specific area and the smooth appearance of theelectric and magnetic currents on the surface of theparticle. Despite the better accuracy of Qext for PI ascompared with that for EI, the oscillation phases ofQext as a function of α are smoother for EI than for PI,as seen in Fig. 4. This is due to the regular intervalsbetween the nodal points along the s direction, whichis in the direction of propagation in the PI case. Assize parameter varies, this spacing is not able to ade-quately represent the propagating wave. It is
Fig. 3. Phase function of a hexagonal column obtained by SIEM/M. The obtained curve corresponds to that in Fig. 6(a) in Mano[16]. The conditions of the particle are ~m ¼ ð1:3;−0:0Þ, L=D ¼0:866 (corresponding to L=D ¼ 1:0 by the Mano definition), andπL=λ ¼ 3:4.
remarkable that the EI case, for which the nodalpoints are more irregularly spaced in the directionof propagation, gives a smoother representation ofthe oscillation of Qext as a function of α. In general,however, the PI alignment is considered superiorto the EI alignment for spherical particles. Our resultindicates that the accuracy fluctuates in accordance
with the distribution of nodal points on the particlesurface with respect to the direction of propagation ofthe incident light.
Figure 6 shows the phase functions for a sphere[Fig. 6(a)] and a hexagonal column with an aspect ra-tio L=D of 1.0 [Fig. 6(b)] for all scattering directions.The center and the circumference of each panel
Fig. 4. (a), (b)Qext as a function of the size parameter α and (c), (d) relative error ofQext as obtained by SIEM/M against the exact solutionobtained with the Mie theory as a function of the number of matching nodal point for electromagnetic waves with (a), (c) polar incidenceand (b), (d) equator incidence.
Table 1. Average Number of Nodes per Wavelength Along s Parameter of the Spline Function at the Number of Matching Nodal Points
denote forward and backward scattering, respec-tively. The values of the size parameter α are thesame as those in Fig. 5. The total NMNP is 2408(¼43 × 56) for the sphere and 2352 (¼49 × 48) forthe hexagonal column, and the average number ofnodes per wavelength along the s curve is 105.4,21.1, 11.7, 8.1, 6.2, and 5.0 for values of size para-meter α ¼ 1, 5, 9, 13, 17, and 21 in the case of a hex-agonal column. Since the number of nodes perwavelength is similar to that of a sphere with2408 nodes, the calculation accuracy for a hexagonalcolumn is considered to be similar to that for asphere. In the case of a hexagonal column, the inci-
dent light propagates toward the particles along thez axis by adopting a fixed geometrical alignment (seeFig. 7). Rayleigh-like scattering (dipole scattering)occurs both in the case of a sphere and the case ofa hexagonal column when the particle size is com-paratively small (for example, when α ¼ 1). Thephase function for the sphere becomes gradually iso-tropic as α increases, whereas the phase function fora hexagonal column for larger values of α has a morecomplex and distorted structure. The nonsphericityof the phase function appears around α ≥ 3 and gra-dually increases as α increases. The other differencebetween the characteristics of the phase function for
Fig. 5. Real part of the obtained electric currents J for the spherical particle in the case of (a) polar incidence (PI) and (b) equator incidence(EI), with complex index of refraction ~m ¼ ð1:395;−6:99 × 10−3Þ. The particle was rotated in such a way that the incident light propagatedtoward the target particle from the front-right part in each viewgraph. The total number of matching nodal points is 3136.
a sphere and a hexagonal column is the number andthe locations of the “cold spots,” where the phasefunction takes remarkably small values. Whenα ¼ 17, these spots appear at azimuth angles of45°, 135°, 225°, and 350° with elevation angles of90° for a hexagonal column, and at 0°, 90°, 180°,and 270° with elevation angles of 120° for a sphere.
4. Discussion of the Results for Randomly OrientedParticles
The scattering properties of a hexagonal column fordifferent orientations of the particle with respect tothe incident electromagnetic waves can be solved bycontrolling the terms Einc and Hinc on the left side ofEqs. (1) and (2) since the particle orientation is
Fig. 6. Phase functions for (a) a sphere and (b) a hexagonal column with an aspect ratio of L=D ¼ 1:0 for all scattering directions. Thecenter and the circumference of each panel denote forward and backward scattering, respectively. The size parameters α are 1, 5, 9, 13, 17,and 21, with a refractive index ~m ¼ ð1:395;−6:99 × 10−3Þ. The total number of matching nodal points was 2408 (¼43 × 56) for the sphereand 2352 (¼49 × 48) for the hexagonal column.
equivalent to that of the incident electromagneticwaves. We specified the orientation of the particlewith respect to the coordinate system by using tworotational angles. As a brief outline, a rotationaround the z axis at an angle θz ∈ ½0; 2π� and aroundthe y axis at an angle θy ∈ ½0; π� defines an arbitraryrotation of the target particle. In order to obtain thecharacteristic scattering features of a particle shapedas a hexagonal column, we obtained a solution withrandomly oriented incident electromagnetic wavesand averaged scattering properties. In order to esti-mate the degree of convergence, we defined a conver-gence parameter ε≡ j1� �Qextðnþ 1Þ=�QextðnÞj, where�QextðnÞ represents Qext averaged over the nth rota-tion. When ε ≤ 1:0 × 10−4 consecutively five times,the calculation was terminated. A few hundred tothousand rotations were needed for α < 21. Figure 8illustrates the phase function P for values of the sizeparameter of (a) α ¼ 1, (b) 5, (c) 9, and (d) 21 of thesphere (Mie) and the hexagonal column (SIEM/M),where the complex index of refraction is ~m ¼ð1:313;−0:0Þ, assuming an ice-phase particle and awavelength of 0:5 μm for the electromagnetic wave.For comparison of Mie and SIEM/M, the phase func-tions P were normalized by factor c ¼ 0:01 and c ¼1:0 for Mie and SIEM/M, respectively, using theformulation
Pnormalized ¼ c ×PR
P sin θdθ ; ð15Þ
where θ is the scattering angle.Fast oscillations in the phase function for each
fixed orientation (see the results obtained by Mie)at relatively larger values of αweaken upon the aver-aging of the phase functions obtained in each ran-domly oriented in the case of a hexagonal column,with the exception of a noteworthy region around20° for the scattering angle. The perturbations ap-pearing in this angle interval are a well-known opti-
cal phenomenon referred to as “halo,” which is aresult of the hexagonal structure of the particle.Mishchenko et al. [21] investigated the phase func-tion of circular cylinders by using T matrices andGOA and concluded that well-defined halo opticalphenomena will appear when the value of the sizeparameter of a nonspherical particle is 100 andmore.In this regard, a faint halo appeared slowly butsteadily in the case of a hexagonal after a sufficientlyhigh number of random reorientations. Similar re-sults have been obtained with the FDTD methodby Yang and Liou [13] for hexagonal columns witha size parameter of 10 and an aspect ratio of 2.The SIEM/Mmethod also confirmed that a randomlyoriented hexagonal particle generates a faint butnoticeable signal for the halo phenomenon despitethe relatively small particle size.
5. Summary and Concluding Remarks
A numerical calculation program based on a com-bined method involving SIEM/M was developed. Inorder to evaluate the accuracy of calculations withSIEM/M, we compared the scattering properties ofspheres with respect to SIEM/M-derived and exactMie solutions. The results showed that the accuracydepends on the NMNP and the error was lower thana few percent for NMNP of four or more per wave-length. It was shown that PI alignments yield morefavorable solutions than EI alignments in most casesfor spherical particles with the “latitude–longitude”type of node definition. This can be explained by thefact that the density of nodes is higher at the “northpole” so that the incident light will be dealt withmore efficiently and the isotropic rippled structureof the electric and magnetic current on the surfaceof the particle was smooth in the case of PI and roughin the case of EI. It is shown that the calculation ac-curacy depends on the distribution of the nodalpoints on the surface with respect to the directionof propagation. Thus optimization of the node den-sity with respect to the shape of the particle willbe one of the tunable aspects in our algorithm.
A comparison of the respective phase functions fora sphere and a hexagonal column with a size para-meter between 1 and 21 showed a Rayleigh-like scat-tering with comparatively small α∼ 1 for both thesphere and the hexagonal column. Nonsphericityappeared in the phase function in the region of α ≥
3 and gradually increased with the increase of α.Another difference in the characteristics of the phasefunction between a sphere and a hexagonal columnwas the number and the locations of “cold spots,”at which the phase function takes remarkably smallvalues.
An averaged phase function for a hexagonal col-umn was obtained by solving SIEM/M. There ap-peared a remarkable optical phenomenon knownas “halo” for scattering angles around 20° and rela-tively large values for the size parameter, such as 20.Our results suggested that the values of the sizeparameter at which halos appear are comparatively
Fig. 7. Geometrical alignment of the incident electromagneticwaves and the hexagonal column used in Fig. 6.
smaller than previously considered. Until recently,the upper limit for the size parameter of SIEM/Mwas around 20 when using computers with ∼8 giga-bytes of memory, and this limit will increase follow-ing the increased availability of computer resources.Using the symmetric structure of the particles is ex-pected to be an excellent way of increasing the sizeparameter. However, this study has been carriedout from the perspective of applying this methodto particles with arbitrary shape in the future withthe aim that the computation holds for the full sizeof the scattering matrix [Z in Eq. (8)]. It is clear thatthis is a problem involving a trade-off between com-puter resources and degrees of asymmetry of the tar-get particles. In part II of our paper, we will present
some results of SIEM/M for particles with more com-plex shapes, with small to moderate size parameters.
The authors are grateful to Yuzo Mano from theMeteorological Research Institute (MRI), Tsukuba,Japan, for his support in developing our SIEM/Mcode. The authors also thank Graeme L. Stephensfrom Colorado State University for his encourage-ment toward the advancement of our research. Thisresearch is supported by the Advanced Earth Obser-ving Satellite-II (ADEOS-II) Science Project (2006,2007), the Earth Cloud, Aerosol and RadiationExplorer (EarthCARE) Science Project (2007, 2008)of the Japan Aerospace Exploration Agency (JAXA),Tsukuba, Japan, the Greenhouse Gases ObservingSatellite (GOSAT) Science Project of the National
Fig. 8. Phase function of Mie (lower curves) and hexagonal column particles with L=D ¼ 1:0 (upper curves) with a size parameter(a) α ¼ 1, (b) 5, (c 9, and (d) 21. Phase functions were normalized by 0.01 (Mie) and 1.0 (SIEM/M). The complex index of refraction is~m ¼ ð1:313;−0:0Þ. Phase functions parallel (thin dashed curves) and perpendicular (thin solid curves) to Einc are shown. Azimuthallyaveraged results of the phase functions are also shown by thick solid curves.
Institute of Environmental Studies, Tsukuba, Japan(2006, 2007, 2008), and the Indian Space ResearchOrganization (ISRO) Megha-Tropiques (MT) project.
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