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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 2 — Jan. 16, 2012
  • pp: 718–731
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Scattering of Gaussian beam by arbitrarily shaped particles with multiple internal inclusions

Yiping Han, Zhiwei Cui, and Wenjuan Zhao  »View Author Affiliations


Optics Express, Vol. 20, Issue 2, pp. 718-731 (2012)
http://dx.doi.org/10.1364/OE.20.000718


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Abstract

In this paper, we introduce an efficient numerical method based on surface integral equations to characterize the scattering of an arbitrarily incident Gaussian beam by arbitrarily shaped particles with multiple internal inclusions. The incident Gaussian beam is described by the Davis–Barton fifth-order approximation in combination with rotation Euler angles. For numerical purposes, the surfaces of the host particle and the inclusions are modeled using small triangular patches and the established surface integral equations are discretized with the method of moments. The resultant matrix equation is solved by using a parallel implementation of conjugate gradient method on distributed-memory architectures. Some numerical results are included to illustrate the validity and capability of the developed method. These results are also expected to provide useful insights into the scattering of Gaussian beam by composite particles.

© 2012 OSA

1. Introduction

The scattering of light by various particles is an active and important subject of research with myriad practical applications in fields ranging from atmospheric optics to optical particle sizing to aerosol detection and Raman scattering diagnostics. With regard to the internal structure of the particles, the problem of light scattering from composite particles consisting of one or multiple inclusions within a host particle, for incidence of a plane wave or a focused Gaussian beam, has attracted much interest in recent years [1

1. F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, “Optical properties of spheres containing a spherical eccentric inclusion,” J. Opt. Soc. Am. A 9(8), 1327–1335 (1992). [CrossRef]

24

24. B. Yan, H. Y. Zhang, and C. H. Liu, “Scattering of Gaussian beam by a spheroidal particle with a spherical inclusion at the center,” Opt. Commun. 284(16–17), 3811–3815 (2011). [CrossRef]

]. This is partially due to the fact that such particles are very common in our daily life as well as in the research for industry or environment concerns. Specific examples include raindrops, dust grains, ice crystals, daily cosmetics, medicinal sprays, biological cells and aerosols in the atmosphere.

During the years the scattering of plane wave by various composite particles has been investigated extensively by many researchers, see [1

1. F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, “Optical properties of spheres containing a spherical eccentric inclusion,” J. Opt. Soc. Am. A 9(8), 1327–1335 (1992). [CrossRef]

15

15. S. Xian-Ming, W. Hai-Hua, L. Wan-Qiang, and S. Jin, “Light scattering by a spherical particle with multiple densely packed inclusions,” Chin. Phys. B 18(3), 1040–1044 (2009). [CrossRef]

] to quote a few of them. For the case of an incident focused Gaussian beam, an early study was carried out by Khaled et al. [16

16. E. E. Khaled, S. C. Hill, and P. W. Barber, “Light scattering by a coated sphere illuminated with a Gaussian beam,” Appl. Opt. 33(15), 3308–3314 (1994). [CrossRef] [PubMed]

]. In that paper, they applied the T-matrix method to examine the scattering of an off-axis focused Gaussian beam by a spherical particle with a concentric spherical inclusion. Later, within the framework of the generalized Lorenz-Mie theory (GLMT) [17

17. G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories (Springer, Berlin, 2011).

], Gouesbet and Gréhan [18

18. G. Gouesbet and G. Gréhan, “Generalized Lorenz-Mie theory for a sphere with an eccentrically located spherical inclusion,” J. Mod. Opt. 47(5), 821–837 (2000).

], Han et al. [19

19. G. X. Han, Y. P. Han, J. Y. Liu, and Y. Zhang, “Scattering of an eccentric sphere arbitrarily located in a shaped beam,” J. Opt. Soc. Am. B 25(12), 2064–2072 (2008). [CrossRef]

], Yan et al. [20

20. B. Yan, X. E. Han, and K. F. Ren, “Scattering of a shaped beam by a spherical particle with an eccentric spherical inclusion,” J. Opt. A 11(1), 015705 (2009). [CrossRef]

], and Wang et al. [21

21. J. J. Wang, G. Gouesbet, Y. P. Han, and G. Gréhan, “Study of scattering from a sphere with an eccentrically located spherical inclusion by generalized Lorenz-Mie theory: internal and external field distribution,” J. Opt. Soc. Am. A 28(1), 24–39 (2011). [CrossRef] [PubMed]

, 22

22. J. J. Wang, G. Gouesbet, G. Gréhan, Y. P. Han, and S. Saengkaew, “Morphology-dependent resonances in an eccentrically layered sphere illuminated by a tightly focused off-axis Gaussian beam: parallel and perpendicular beam incidence,” J. Opt. Soc. Am. A 28(9), 1849–1859 (2011). [CrossRef]

] investigate the light scattering by a spherical particle with an eccentrically located spherical inclusion illuminated by a Gaussian beam with arbitrary incidence. Subsequently, Zhang and Liao [23

23. H. Y. Zhang and T. Q. Liao, “Scattering of Gaussian beam by a spherical particle with a spheroidal inclusion,” J. Quant. Spectrosc. Radiat. Transf. 112(9), 1486–1491 (2011). [CrossRef]

] employed the GLMT to study the scattering of Gaussian beam by a spherical particle with a spheroidal inclusion. Yan et al. [24

24. B. Yan, H. Y. Zhang, and C. H. Liu, “Scattering of Gaussian beam by a spheroidal particle with a spherical inclusion at the center,” Opt. Commun. 284(16–17), 3811–3815 (2011). [CrossRef]

] also studied the case of a spheroidal particle with a spherical inclusion. Despite some studies, as reviewed herein, have been carried out on the Gaussian beam scattering by several kinds of composite particles, these studies mainly focused on the cases of a sphere or a spheroid with only one internal inclusion. To the best of our knowledge, the scattering of a focused Gaussian beam by arbitrarily shaped particles with multiple internal inclusions of arbitrary shape has not been reported.

In this paper, we present an efficient numerical method for the treatment of Gaussian beam scattering by arbitrarily shaped host particles containing multiple arbitrarily shaped inclusions. Specifically, the arbitrarily incident Gaussian beams are described by using the Davis-Barton fifth-order approximate expressions [25

25. J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66(7), 2800–2802 (1989). [CrossRef]

] in combination with rotation Euler angles [26

26. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, 1957).

, 27

27. G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz-Mie theories through rotations of coordinate systemsIII. Special Euler angles,” Opt. Commun. 283(17), 3235–3243 (2010). [CrossRef]

] and the scattering problems involving arbitrarily shaped particles with multiple inclusions of arbitrary shape are formulated by utilizing the surface integral equations [28

28. S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. 30(3), 409–418 (1982). [CrossRef]

32

32. Z. W. Cui, Y. P. Han, and M. L. Li, “Solution of CFIE-JMCFIE using parallel MOM for scattering by dielectrically coated conducting bodies,” J. Electromagn. Waves Appl. 25(2), 211–222 (2011). [CrossRef]

]. The solution procedure uses the well-known method of moments (MOM) [33

33. R. F. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1968).

] with Galerkin’s type testing. In implementation, the surfaces of the host particle and each inclusion are modeled by using small triangular patches, and the unknown equivalent electromagnetic currents are expanded in Rao-Wilton-Glisson (RWG) basis functions [28

28. S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. 30(3), 409–418 (1982). [CrossRef]

]. The resulting matrix equations are solved iteratively by employing a parallel implementation of conjugate gradient method (CGM) [32

32. Z. W. Cui, Y. P. Han, and M. L. Li, “Solution of CFIE-JMCFIE using parallel MOM for scattering by dielectrically coated conducting bodies,” J. Electromagn. Waves Appl. 25(2), 211–222 (2011). [CrossRef]

] on distributed-memory architectures.

This paper is organized as follows. In Section 2, a detailed description of the arbitrarily incident Gaussian beam is given. Section 3 is devoted to describe an extension of the surface integral equation method for the simulation of light scattering by arbitrarily shaped particles with multiple internal inclusions. Section 4 presents some numerical results to demonstrate the validity and capability of the present method. Finally, Section 5 concludes the paper.

2. Description of the incident Gaussian beam

For the purpose of the present numerical study, a detailed mathematical description of the electromagnetic field components of the incident Gaussian beam which accurately satisfies Maxwell’s equations is required. When the beam waist radius is much greater than the wavelength, the Davis first-order Gaussian beam description [34

34. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19(3), 1177–1179 (1979). [CrossRef]

] has been found to give good results. However, for tightly focused beams, the Davis first-order Gaussian beam description satisfies Maxwell’s equations less accurately and the numerical calculations using the method described below likewise become less accurate. To accurately describe the incident Gaussian beam, particularly for tightly focused conditions, one should consider the use of higher-order approximate expressions. It is also possible to use the localized beam model developed by Gouesbet et al [17

17. G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories (Springer, Berlin, 2011).

]. In this study, we adopt the Davis-Barton fifth-order approximation [25

25. J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66(7), 2800–2802 (1989). [CrossRef]

] in combination with rotation Euler angles [26

26. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, 1957).

, 27

27. G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz-Mie theories through rotations of coordinate systemsIII. Special Euler angles,” Opt. Commun. 283(17), 3235–3243 (2010). [CrossRef]

] to describe the arbitrarily incident focused Gaussian beam. Mathematically, details of the description are the following. Referring to Fig. 1
Fig. 1 Geometry of Cartesian coordinates of the beam and particle.
, a focused Gaussian beam with its electric field polarized in theu direction at the waist is assumed to propagate along thewaxis in its own Cartesian coordinate systemOGuvw. The beam waist centerOGis located at (x0,y0,z0) in the Cartesian coordinate system of the particleOxyz.The frame systemOxyzcan be obtained from the beam systemOGuvwby rotations through Euler angles(α,β,γ) followed by a translation of (x0,y0,z0). To be specific, we write the rotation relationship between these two coordinate systems as
[xx0yy0zz0]=A[uvw]
(1)
where the transformation matrixAis introduced as follows:

A=[cosαsinα0sinαcosα0001][cosβ0sinβ010sinβ0cosβ][cosγsinγ0sinγcosγ0001]
(2)

On the bases of such a transformation, the beam description of Einc(Eu,Ev,Ew) and Hinc(Hu,Hv,Hw)in the beam coordinate systemOGuvwcan be transformed to their counterpartsEinc(Ex,Ey,Ez)andHinc(Hx,Hy,Hz)in the particle coordinate system Oxyzvia the following formula:

[ExEyEz]=A[EuEvEw],[HxHyHz]=A[HuHvHw]
(3)

For a fifth-order Davis-Barton beam [25

25. J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66(7), 2800–2802 (1989). [CrossRef]

], the electromagnetic field components in the OGuvwsystem read as
Eu=E0{1+s2(ρ2Q2+iρ4Q32Q2ξ2)+s4[2ρ4Q4i3ρ6Q50.5ρ8Q6+(8ρ2Q4i2ρ4Q5)ξ2]}ψ0eikw
(4)
Ev=E0{s2(2Q2ξη)+s4[(8ρ2Q4i2ρ4Q5)ξη]}ψ0eikw
(5)
Ew=E0{s(2Qξ)+s3[(6ρ2Q3i2ρ4Q4)ξ]+s5[(20ρ4Q5+i10ρ6Q6+ρ8Q7)ξ]ψ0eikw
(6)
Hu=H0{s2(2Q2ξη)+s4[(8ρ2Q4i2ρ4Q5)ξη]}ψ0eikw
(7)
Hv=H0{1+s2(ρ2Q2+iρ4Q32Q2η2)+s4[2ρ4Q4i3ρ6Q50.5ρ8Q6+(8ρ2Q4i2ρ4Q5)η2]}ψ0eikw
(8)
Hw=H0{s(2Qη)+s3[(6ρ2Q3i2ρ4Q4)η]+s5[(20ρ4Q5+i10ρ6Q6+ρ8Q7)η]ψ0eikw
(9)
whereE0andH0are the amplitude of the electric field and magnetic field strength, respectively, and
s=1kω0
(10)
Q=1i+2ζ
(11)
ρ=ξ2+η2
(12)
ψ0=iQexp(iρ2Q)
(13)
ξ=uω0,η=vω0,ζ=wkω02
(14)
in whichω0is the beam waist radius andk=2π/λis the wave number, withλbeing the wavelength of the incident beam.

3. Surface integral equation method

Now, let us consider the problem of Gaussian beam scattering by an arbitrarily shaped particle with multiple internal inclusions, as illustrated in Fig. 2
Fig. 2 Configuration of an arbitrarily shaped particle with multiple internal inclusions.
. The host particle is characterized by permittivity and permeability(εp,μp), all the internal inclusions are assumed to be formed by homogeneous dielectric medium and theith(i=1,2,,m) inclusion is characterized by permittivity and permeability(εi,μi), withmbeing the total number of the inclusions. For simplicity, the surrounding medium is considered to be free space with parametersε0andμ0. LetSpdenote the surface of the host particle and Sidenote the surface of theithinclusion. Also, the free space region is denoted as Ω0, the region occupied by the ith internal inclusion is denoted asΩi, and the region occupied by the host particle except those occupied by all the internal inclusions is denoted asΩp. To solve this problem by means of the surface integral equation method, we introduce equivalent electromagnetic currents(Jp,Μp)onSp and(Ji,Μi)onSi(i=1,2,,m). Through the Stratton-Chu representation formulas, the scattered fields in each region can be expressed in terms of the equivalent electromagnetic currents. Specifically, the scattered fields in regionΩ0, due to the equivalent electric and magnetic currents Jp and ΜponSp, can be expressed as
E0sca=Z0L0Sp(Jp)K0Sp(Mp)
(15)
H0sca=K0Sp(Jp)+1Z0L0Sp(Mp)
(16)
whereZ0=μ0/ε0, and the integral operatorsL0SpandK0Spare defined as
L0Sp(X)=ik0Sp[X(r')+1k02('X(r'))]G0(r,r')dS'
(17)
K0Sp(X)=SpX(r')×G0(r,r')dS'
(18)
in whichk0=ωε0μ0, withωbeing the angular frequency, the vectorsrandr'are the field and source points, respectively, and
G0(r,r')=e-ik0|r-r'|4π|r-r'|
(19)
is the homogeneous space Green’s function. The scattered fields in regionΩpare produced by the equivalent currents(Jp,Μp)onSpand(Ji,Μi)onSi(i=1,2,,m), and can be expressed as
Epsca=[ZpLpSp(Jp)KpSp(Mp)]+i=1m[ZpLpSi(Ji)KpSi(Mi)]
(20)
Hpsca=[KpSp(Jp)+1ZpLpSp(Mp)]+i=1m[KpSi(Ji)+1ZpLpSi(Mi)]
(21)
whereZp=μp/εpand the operators LpSp(i)and KpSp(i)are defined similarly toL0Spand K0Sp, provided that all the subscripts are changed from “0” to “p”. Accordingly, the scattered fields in regionΩi(i=1,2,,m)can be expressed in terms of the equivalent electric and magnetic currentsJiandΜias
Eisca=ZiLiSi(Ji)KiSi(Mi)
(22)
Hisca=KiSi(Ji)+1ZiLiSi(Mi)
(23)
whereZi=μi/εiand the operatorsLiSiandKiSiare also defined similarly toL0Spand K0Sp, provided that all the subscripts are changed from “0” to “i”.

By enforcing boundary condition on the surface of the host particle, namely that the total tangential fields across the interface are continuous, we can obtain two boundary integral equations as follows:
(E0sca+Einc)|tan(Sp)=Epsca|tan(Sp)
(24)
(H0sca+Hinc)|tan(Sp)=Hpsca|tan(Sp)
(25)
where the subscript “tan(Sp)” stands for tangential components of the fields onSp, EincandHincrepresent the fields of the incident focused Gaussian beam, and the field components are given by (3). Similarly, enforcing boundary condition on the surface of each internal inclusion yields the following two boundary integral equations:
Epsca|tan(Si)=Eisca|tan(Si)
(26)
Hpsca|tan(Si)=Hisca|tan(Si)
(27)
where the subscript “tan(Si)” stands for tangential components of the fields onSi.

{Jt}={J1Jm},{Mt}={M1Mm}
(32)

Therefore, Eq. (31) can take the form

[ZJpJpZJpMpZJpJtZJpMtZMpJpZMpMpZMpJtZMpMtZJtJpZJtMpZJtJtZJtMtZMtJpZMtMpZMtJtZMtMt]{JpMpJtMt}={bEbH00}
(33)

4. Numerical results and discussion

In this section, we present some of our preliminary numerical results for the scattering of Gaussian beam by arbitrarily shaped particles with multiple internal inclusions. The results to be presented are in terms of the differential scattering cross section (DSCS) in the particle systemOxyz, the E-plane corresponds to thexOz-plane and the H-plane corresponds to the yOz-plane. The DSCS is defined as

σ=limr4πr2|Efarsca|2|E0|2
(36)

Without loss of generality, the amplitude of the electric field for the incident beam is assumed to be unity. In what follows, all the computations are performed on a HP Workstation. It has 16 nodes and each node is Intel Xeon MP 3.0 GHz, 4 GB memory. Iterations are carried out until the residual error is reduced to below 0.001.

In order to demonstrate the validity of the present method for the case of Gaussian beam illumination, we first consider a spherical particle with a concentric spherical inclusion. The radii of the host sphere and the inclusion arer=1.0λand r1=0.5λwith refractive indices of m=1.33andm1=1.55, respectively, withλbeing the wavelength of the incident beam. The particle is illuminated by an obliquely incident Gaussian beam with ω0=1.0λ. The beam waist center is located at the origin of the particle system with x0=y0=z0=0 and the rotation Euler angles are specified as α=0o,β=45oandγ=0o. For numerical solution, the surfaces of the host particle and the inclusion are respectively discretized into 3788 and 1024 triangular patches using the mesh density of 12 parts per wavelength. As a result, a total of 14436 unknowns are generated. Performing the parallel CGM on the HP Workstation with 40 processors, 112 s is needed to solve the resultant matrix equation. Figure 4
Fig. 4 Comparison of the DSCSs for a spherical particle with a concentric spherical inclusion obtained from the present method and that from the GLMT.
presents the computed DSCSs as a function of the scattering angle in both the E-plane and the H-plane. The results obtained from the GLMT are also presented. As is evident from the figure, good agreement is obtained between the results obtained from the two methods.

Now, we consider the problem of Gaussian beam scattering by a spheroidal particle with two identical spherical inclusions, as illustrated in Fig. 5
Fig. 5 Geometry of a spheroidal particle with two spherical inclusions.
. The semimajor axis and the semiminor axis of the host spheroid area=2.0λandb=1.0λ, respectively, and the refractive index ism=1.414. Both inclusions are located on thez-axis in the particle system withz1=1.0λandz2=1.0λ, respectively. Each spherical inclusion has a radius ofri=0.5λand the refractive index ofmi=2.0. The beam waist is centered at (x0,y0,z0)=(0.0,0.0,0.0), and the angle set of the beam isα=β=γ=0o. The computed DSCSs for different values of the beam waist radius are given in Fig. 6
Fig. 6 DSCSs for a spheroidal particle with two spherical inclusions illuminated by a plane wave and a Gaussian beam: (a) E-plane, (b) H-plane.
. For comparison purpose, the results for the case of plane wave illumination obtained with the commercial soft FEKO are given in the same figure. It is obvious that the DSCS for Gaussian beams is smaller than that for a plane wave because of the influence of the beam shape coefficients. As expected, the results in the case of Gaussian beam incidence with a relatively large waist radius ofω0=20.0λare in excellent agreement with the results in the case of plane wave incidence. This further demonstrates the validity the present method.

Next, we present three examples to illustrate the capabilities of the present method. The first one is a spherical particle with four inclusions of different shape, as depicted in Fig. 7
Fig. 7 Geometry of a spherical particle with four inclusions of different shape.
. The radius of the host sphere isr=2.0λand the refractive index ism=1.5. These four internal inclusions are assumed to be formed by identical homogeneous medium characterized by refractive indexmi=1.8. The first inclusion is a sphere with radiusr1=0.5λ, the second one is a cylinder with radiusr2=0.25λand heighth2=1.0λ, the third one is an oblate spheroid with semi-axesa3=0.5λandb3=0.4λ, and the fourth inclusion is a cone with radiusr4=0.5λand heighth4=1.0λ. The positions of the internal inclusions with respect to the global coordinate system of the host particle is specified by the coordinates (x1,y1,z1)=(1.0,0.0,0.0)λ,(x2,y2,z2)=(0.0,0.0,1.0)λ, (x3,y3,z3)=(1.0,0.0,0.0)λand (x4,y4,z4)=(0.0,0.0,0.0)λ. The particle is illuminated by a focused Gaussian beam at incidence anglesα=β=γ=0owith beam waistω0=1.5λ. The location of the beam waist center is a varied parameter. Figure 8
Fig. 8 DSCSs for a spherical particle containing four different inclusions with the location of the beam waist center as a varied parameter: (a) E-plane, (b) H-plane.
shows the effects of the beam waist center positioning on the DSCS distributions for both the E and H-planes. It is found that the position offset of the beam waist center makes a small difference to the DSCS by reducing it for most but not all angles. This occurs because the beam waist center deviate the center of the particle leading to the change of the distribution of the scattering intensity.

The second example considered is the scattering of a focused Gaussian beam by a cubic particle containing 27 randomly distributed spherical inclusions, as shown in Fig. 9
Fig. 9 Illustration of a cubic particle containing 27 randomly distributed spherical inclusions.
. All the inclusions are assumed to be uniform and the positions are generated from the Monte Carlo method [39

39. L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).

] with fractional volumef=6.0%. The side length of the host cube is l=3.0λ and the refractive index ism=1.2i0.2. Each spherical inclusion has a radius ri=0.25λ and the refractive indexm=1.5i0.1. Both the centers of the host cube and the beam waist are located at the origin of the particle system withx0=y0=z0=0. The incident focused Gaussian beam has a beam waist of ω0=2.0λand the incident angle is taken in the E-plane, that isα=γ=0o. Thus, the E-plane scattering property is what we are most interested in. Figure 10
Fig. 10 Angular distributions of the DSCS for a cubic particle containing 27 randomly distributed spherical inclusions in the E-plane with Euler anglesα=γ=0oandβas a parameter.
presents angular distributions of the DSCS in the E-plane for the Gaussian beam with different values of incident angleβ. It can be observed that the scattering angle corresponding to the extreme value of the DSCS is just coincident with the incident direction of the beam, which indicates that most of the scattering energy concentrates forward. This is identical with the general idea of light scattering theory.

In the last example, we apply the present method to simulate the scattering of an off-axis obliquely incident Gaussian beam by a hexagonal prism containing a fractal aggregate that consists of 100 identical spherical particles, as illustrated in Fig. 11
Fig. 11 Illustration of a hexagonal prism with a fractal aggregate of spherical inclusions: (a) host hexagonal prism, (b) internal fractal aggregate.
. Internal aggregate positions were generated by using the cluster-cluster aggregation (CCA) algorithm described by Mackowski [40

40. D. W. Mackowski, “A simplified model to predict the effects of aggregation on the absorption properties of soot particles,” J. Quant. Spectrosc. Radiat. Transf. 100(1–3), 237–249 (2006). [CrossRef]

], with prefactor constantkf=1.19and fractal dimensionDf=1.82. The radius and the height of the host hexagonal prism arer=2.0λandh=4.0λ, respectively, and the refractive index ism=1.313. Each primary particle of the internal fractal aggregate has radiusri=0.2λand refractive indexm=1.50i0.65. The beam waist is centered at(x0,y0,z0)=(1.0,1.0,1.0)λ with a beam waist radius of ω0=1.5λ, and the rotation Euler angles are specified as α=0o,β=45oandγ=0o. Figure 12
Fig. 12 Differential scattering cross sections for a hexagonal prism with a fractal aggregate of spherical inclusions.
presents the simulated DSCSs as a function of the scattering angle in both the E-plane and the H-plane.

5. Conclusion

In this paper, we present an efficient numerical method for the simulation of light scattering by arbitrarily shaped particles with multiple internal inclusions illuminated by a focused Gaussian beam. Specifically, the Davis-Barton fifth-order approximate expressions in combination with rotation Euler angles is employed to describe the arbitrarily incident Gaussian beam. The surface integral equations are applied to formulate the scattering problems involving arbitrarily shaped particles with multiple internal inclusions and are numerically discretized by the MoM with the RWG basis functions defined on triangular patches. To reduce computational burden, the resultant matrix equation is solved by a parallel implementation of the CGM on distributed-memory architectures. The present method is validated and its capability illustrated in several characteristic examples. This work is helpful for further research on the scattering of an arbitrarily incident focused Gaussian beam by composite particles.

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A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a sphere with an eccentric spherical inclusion,” J. Opt. Soc. Am. A 24(6), 1695–1703 (2007). [CrossRef] [PubMed]

13.

D. K. Wu and Y. P. Zhou, “Forward scattering light of droplets containing different size inclusions,” Appl. Opt. 48(15), 2957–2965 (2009). [CrossRef] [PubMed]

14.

M. Mikrenska and P. Koulev, “Simulation of light scattering by large particles with randomly distributed spherical or cubic inclusions,” J. Quant. Spectrosc. Radiat. Transf. 110(14–16), 1411–1417 (2009). [CrossRef]

15.

S. Xian-Ming, W. Hai-Hua, L. Wan-Qiang, and S. Jin, “Light scattering by a spherical particle with multiple densely packed inclusions,” Chin. Phys. B 18(3), 1040–1044 (2009). [CrossRef]

16.

E. E. Khaled, S. C. Hill, and P. W. Barber, “Light scattering by a coated sphere illuminated with a Gaussian beam,” Appl. Opt. 33(15), 3308–3314 (1994). [CrossRef] [PubMed]

17.

G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories (Springer, Berlin, 2011).

18.

G. Gouesbet and G. Gréhan, “Generalized Lorenz-Mie theory for a sphere with an eccentrically located spherical inclusion,” J. Mod. Opt. 47(5), 821–837 (2000).

19.

G. X. Han, Y. P. Han, J. Y. Liu, and Y. Zhang, “Scattering of an eccentric sphere arbitrarily located in a shaped beam,” J. Opt. Soc. Am. B 25(12), 2064–2072 (2008). [CrossRef]

20.

B. Yan, X. E. Han, and K. F. Ren, “Scattering of a shaped beam by a spherical particle with an eccentric spherical inclusion,” J. Opt. A 11(1), 015705 (2009). [CrossRef]

21.

J. J. Wang, G. Gouesbet, Y. P. Han, and G. Gréhan, “Study of scattering from a sphere with an eccentrically located spherical inclusion by generalized Lorenz-Mie theory: internal and external field distribution,” J. Opt. Soc. Am. A 28(1), 24–39 (2011). [CrossRef] [PubMed]

22.

J. J. Wang, G. Gouesbet, G. Gréhan, Y. P. Han, and S. Saengkaew, “Morphology-dependent resonances in an eccentrically layered sphere illuminated by a tightly focused off-axis Gaussian beam: parallel and perpendicular beam incidence,” J. Opt. Soc. Am. A 28(9), 1849–1859 (2011). [CrossRef]

23.

H. Y. Zhang and T. Q. Liao, “Scattering of Gaussian beam by a spherical particle with a spheroidal inclusion,” J. Quant. Spectrosc. Radiat. Transf. 112(9), 1486–1491 (2011). [CrossRef]

24.

B. Yan, H. Y. Zhang, and C. H. Liu, “Scattering of Gaussian beam by a spheroidal particle with a spherical inclusion at the center,” Opt. Commun. 284(16–17), 3811–3815 (2011). [CrossRef]

25.

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66(7), 2800–2802 (1989). [CrossRef]

26.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, 1957).

27.

G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz-Mie theories through rotations of coordinate systemsIII. Special Euler angles,” Opt. Commun. 283(17), 3235–3243 (2010). [CrossRef]

28.

S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. 30(3), 409–418 (1982). [CrossRef]

29.

S. M. Rao, C. C. Cha, R. L. Cravey, and D. Wilkes, “Electromagnetic scattering from arbitrary shaped conducting bodies coated with lossy materials of arbitrary thickness,” IEEE Trans. Antenn. Propag. 39(5), 627–631 (1991). [CrossRef]

30.

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general composite metallic and dielectric structures with junctions,” Prog. Electromagn. Res. 52, 81–108 (2005). [CrossRef]

31.

J. Rivero, J. M. Taboada, L. Landesa, F. Obelleiro, and I. García-Tuñón, “Surface integral equation formulation for the analysis of left-handed metamaterials,” Opt. Express 18(15), 15876–15886 (2010). [CrossRef] [PubMed]

32.

Z. W. Cui, Y. P. Han, and M. L. Li, “Solution of CFIE-JMCFIE using parallel MOM for scattering by dielectrically coated conducting bodies,” J. Electromagn. Waves Appl. 25(2), 211–222 (2011). [CrossRef]

33.

R. F. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1968).

34.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19(3), 1177–1179 (1979). [CrossRef]

35.

Z. W. Cui, Y. P. Han, and H. Y. Zhang, “Scattering of an arbitrarily incident focused Gaussian beam by arbitrarily shaped dielectric particles,” J. Opt. Soc. Am. B 28(11), 2625–2632 (2011). [CrossRef]

36.

Z. W. Cui, Y. P. Han, and Q. Xu, “Numerical simulation of multiple scattering by random discrete particles illuminated by Gaussian beams,” J. Opt. Soc. Am. A 28(11), 2200–2208 (2011). [CrossRef] [PubMed]

37.

D. A. Dunavant, “High degree efficient symmetrical Gaussian quadrature rules for the triangle,” Int. J. Numer. Methods Eng. 21(6), 1129–1148 (1985). [CrossRef]

38.

R. D. Graglia, “On the numerical integration of the linear shape functions times the 3-D green’s function or its gradient on a plane triangle,” IEEE Trans. Antenn. Propag. 41(10), 1448–1455 (1993). [CrossRef]

39.

L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).

40.

D. W. Mackowski, “A simplified model to predict the effects of aggregation on the absorption properties of soot particles,” J. Quant. Spectrosc. Radiat. Transf. 100(1–3), 237–249 (2006). [CrossRef]

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(290.5850) Scattering : Scattering, particles
(140.3295) Lasers and laser optics : Laser beam characterization

ToC Category:
Scattering

History
Original Manuscript: November 3, 2011
Revised Manuscript: December 8, 2011
Manuscript Accepted: December 18, 2011
Published: January 3, 2012

Citation
Yiping Han, Zhiwei Cui, and Wenjuan Zhao, "Scattering of Gaussian beam by arbitrarily shaped particles with multiple internal inclusions," Opt. Express 20, 718-731 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-718


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References

  1. F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, “Optical properties of spheres containing a spherical eccentric inclusion,” J. Opt. Soc. Am. A9(8), 1327–1335 (1992). [CrossRef]
  2. N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Indirect mode-matching solution to scattering from a dielectric sphere with an eccentric inclusion,” J. Opt. Soc. Am. A11(6), 1859–1866 (1994). [CrossRef]
  3. F. Borghese, P. Denti, and R. Saija, “Optical properties of spheres containing several spherical inclusions,” Appl. Opt.33(3), 484–493 (1994). [CrossRef] [PubMed]
  4. G. Videen, D. Ngo, P. Chylek, and R. G. Pinnick, “Light scattering from a sphere with an irregular inclusion,” J. Opt. Soc. Am. A12(5), 922–928 (1995). [CrossRef]
  5. D. Ngo, G. Videen, and P. Chýlek, “A FORTRAN code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun.99(1), 94–112 (1996). [CrossRef]
  6. A. Macke, M. I. Mishchenko, and B. Cairns, “The influence of inclusions on light scattering by large ice particles,” J. Geophys. Res.101(D18), 23311–23316 (1996). [CrossRef]
  7. M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles:Ttheory, Measurements, and Applications (Academic, San Diego, 2000).
  8. D. R. Prabhu, M. Davies, and G. Videen, “Light scattering calculations from oleic-acid droplets with water inclusions,” Opt. Express8(6), 308–313 (2001). [CrossRef] [PubMed]
  9. M. P. Ioannidou and D. P. Chrissoulidis, “Electromagnetic-wave scattering by a sphere with multiple spherical inclusions,” J. Opt. Soc. Am. A19(3), 505–512 (2002). [CrossRef] [PubMed]
  10. T. Weigel, J. Schulte, and G. Schweiger, “Inelastic scattering on particles with inclusions,” J. Opt. Soc. Am. A22(6), 1048–1052 (2005). [CrossRef] [PubMed]
  11. A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, Berlin, 2006).
  12. A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a sphere with an eccentric spherical inclusion,” J. Opt. Soc. Am. A24(6), 1695–1703 (2007). [CrossRef] [PubMed]
  13. D. K. Wu and Y. P. Zhou, “Forward scattering light of droplets containing different size inclusions,” Appl. Opt.48(15), 2957–2965 (2009). [CrossRef] [PubMed]
  14. M. Mikrenska and P. Koulev, “Simulation of light scattering by large particles with randomly distributed spherical or cubic inclusions,” J. Quant. Spectrosc. Radiat. Transf.110(14–16), 1411–1417 (2009). [CrossRef]
  15. S. Xian-Ming, W. Hai-Hua, L. Wan-Qiang, and S. Jin, “Light scattering by a spherical particle with multiple densely packed inclusions,” Chin. Phys. B18(3), 1040–1044 (2009). [CrossRef]
  16. E. E. Khaled, S. C. Hill, and P. W. Barber, “Light scattering by a coated sphere illuminated with a Gaussian beam,” Appl. Opt.33(15), 3308–3314 (1994). [CrossRef] [PubMed]
  17. G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories (Springer, Berlin, 2011).
  18. G. Gouesbet and G. Gréhan, “Generalized Lorenz-Mie theory for a sphere with an eccentrically located spherical inclusion,” J. Mod. Opt.47(5), 821–837 (2000).
  19. G. X. Han, Y. P. Han, J. Y. Liu, and Y. Zhang, “Scattering of an eccentric sphere arbitrarily located in a shaped beam,” J. Opt. Soc. Am. B25(12), 2064–2072 (2008). [CrossRef]
  20. B. Yan, X. E. Han, and K. F. Ren, “Scattering of a shaped beam by a spherical particle with an eccentric spherical inclusion,” J. Opt. A11(1), 015705 (2009). [CrossRef]
  21. J. J. Wang, G. Gouesbet, Y. P. Han, and G. Gréhan, “Study of scattering from a sphere with an eccentrically located spherical inclusion by generalized Lorenz-Mie theory: internal and external field distribution,” J. Opt. Soc. Am. A28(1), 24–39 (2011). [CrossRef] [PubMed]
  22. J. J. Wang, G. Gouesbet, G. Gréhan, Y. P. Han, and S. Saengkaew, “Morphology-dependent resonances in an eccentrically layered sphere illuminated by a tightly focused off-axis Gaussian beam: parallel and perpendicular beam incidence,” J. Opt. Soc. Am. A28(9), 1849–1859 (2011). [CrossRef]
  23. H. Y. Zhang and T. Q. Liao, “Scattering of Gaussian beam by a spherical particle with a spheroidal inclusion,” J. Quant. Spectrosc. Radiat. Transf.112(9), 1486–1491 (2011). [CrossRef]
  24. B. Yan, H. Y. Zhang, and C. H. Liu, “Scattering of Gaussian beam by a spheroidal particle with a spherical inclusion at the center,” Opt. Commun.284(16–17), 3811–3815 (2011). [CrossRef]
  25. J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys.66(7), 2800–2802 (1989). [CrossRef]
  26. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, 1957).
  27. G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz-Mie theories through rotations of coordinate systemsIII. Special Euler angles,” Opt. Commun.283(17), 3235–3243 (2010). [CrossRef]
  28. S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag.30(3), 409–418 (1982). [CrossRef]
  29. S. M. Rao, C. C. Cha, R. L. Cravey, and D. Wilkes, “Electromagnetic scattering from arbitrary shaped conducting bodies coated with lossy materials of arbitrary thickness,” IEEE Trans. Antenn. Propag.39(5), 627–631 (1991). [CrossRef]
  30. P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general composite metallic and dielectric structures with junctions,” Prog. Electromagn. Res.52, 81–108 (2005). [CrossRef]
  31. J. Rivero, J. M. Taboada, L. Landesa, F. Obelleiro, and I. García-Tuñón, “Surface integral equation formulation for the analysis of left-handed metamaterials,” Opt. Express18(15), 15876–15886 (2010). [CrossRef] [PubMed]
  32. Z. W. Cui, Y. P. Han, and M. L. Li, “Solution of CFIE-JMCFIE using parallel MOM for scattering by dielectrically coated conducting bodies,” J. Electromagn. Waves Appl.25(2), 211–222 (2011). [CrossRef]
  33. R. F. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1968).
  34. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A19(3), 1177–1179 (1979). [CrossRef]
  35. Z. W. Cui, Y. P. Han, and H. Y. Zhang, “Scattering of an arbitrarily incident focused Gaussian beam by arbitrarily shaped dielectric particles,” J. Opt. Soc. Am. B28(11), 2625–2632 (2011). [CrossRef]
  36. Z. W. Cui, Y. P. Han, and Q. Xu, “Numerical simulation of multiple scattering by random discrete particles illuminated by Gaussian beams,” J. Opt. Soc. Am. A28(11), 2200–2208 (2011). [CrossRef] [PubMed]
  37. D. A. Dunavant, “High degree efficient symmetrical Gaussian quadrature rules for the triangle,” Int. J. Numer. Methods Eng.21(6), 1129–1148 (1985). [CrossRef]
  38. R. D. Graglia, “On the numerical integration of the linear shape functions times the 3-D green’s function or its gradient on a plane triangle,” IEEE Trans. Antenn. Propag.41(10), 1448–1455 (1993). [CrossRef]
  39. L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
  40. D. W. Mackowski, “A simplified model to predict the effects of aggregation on the absorption properties of soot particles,” J. Quant. Spectrosc. Radiat. Transf.100(1–3), 237–249 (2006). [CrossRef]

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