Scattering by a spheroidal particle illuminated with a couple of on-axis Gaussian beams

Optics & Laser Technology 44 (2012) 1290–1293

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Scattering by a spheroidal particle illuminated with a couple of on-axis Gaussian beams Haihua Wang a, Xianming Sun a,n, Huayong Zhang b a b

School of Electrical and Electronic Engineering, Shandong University of Technology, ZiBo, Shandong 255049, China School of Electronics and Information Engineering, AnHui University, HeFei, AnHui 230039, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 10 May 2011 Received in revised form 19 December 2011 Accepted 28 December 2011 Available online 20 January 2012

An on-axis Gaussian beam for any angle of incidence is expanded in terms of the spheroidal vector wave functions. With such an expansion, the problem of interaction between a couple of on-axis Gaussian beams and a spheroidal particle is studied in the framework of the generalized Lorenz–Mie theory (GLMT). The scattering characteristics are described in detail, and numerical results of the normalized differential scattering cross section are presented. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Scattering Gaussian beam Spheroidal particle

1. Introduction

2. Formulation

The generalized Lorenz–Mie theory (GLMT) framework [1] published by Gouesbet is effective for describing the interaction between an arbitrarily shaped beam and a class of regular scatterers including spheres [2,3], multilayered spheres [4], infinite cylinders [5,6] and spheroids [7–10]. Various applications of the GLMT have also been discussed [11], such as optimizing the rate at which morphology-dependent resonances (MDRs) are excited [12], laser trapping and particle manipulation [13,14] and optical particle sizing with Phase-Doppler Anemometry (PDA) [15–19]. In a PDA system, two coherent focused laser beams described by Gaussian beam models intersect to form a measurement volume (MV). When a particle passes through the MV, both the particle velocity and particle size can be determined by measuring the phase difference of the scattered light by the particle. Obviously, it is essential that an exact description of the light scattering properties of the particle be given. This is the aim of this paper to provide an analytical solution to the scattering of a couple of on-axis Gaussian beams by a spheroidal particle. The paper is organized as follows. In Section 2, the theoretical procedure is presented for the determination of the scattered fields of a couple of on-axis Gaussian beams by a spheroid. In Section 3, numerical results of the scattering properties are presented. Section 4 presents the conclusion.

2.1. Expansion of on-axis Gaussian beam in terms of spheroidal vector wave functions Fig. 1 shows the geometry of the scatterer. The center of the spheroid is located at origin O of the Cartesian coordinate system Oxyz, and the major axis is along the z axis and the semifocal distance; semimajor and semiminor axes are denoted by f, a and b. The system Ox0 y0 s0 is obtained by rotating Oxyz through Euler angles a ¼ j, b ¼ y, g ¼0 [20]. A Gaussian beam propagates in free space and from the negative z0 to the positive z0 axis of Ox0 y0 s0 , with the middle of its beam waist O0 located on the z0 axis, i.e., at (0,0,z0) in Ox0 y0 s0 (on-axis case). In this paper, we assume that the time-dependent part of the electromagnetic fields is exp(  iot). We have obtained an expansion in [9] of the electromagnetic fields of an incident Gaussian beam in terms of the spheroidal vector wave functions Mrð1Þ ðc, z, Z, fÞ and Nrð1Þ ðc, z, Z, fÞ with e e o

Corresponding author. E-mail address: [email protected] (X. Sun).

0030-3992/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2011.12.048

o

mn

respect to the system Oxyz, in the following forms: Ei ¼ E0

1 1 X X

0m rð1Þ rð1Þ in ½Gm n,TE Memn ðc, z, Z, fÞGn,TE Momn ðc, z, Z, fÞ

m¼0n¼m m rð1Þ rð1Þ þiG0m n,TM Nemn ðc, z, Z, fÞ þ iGn,TM Nomn ðc, z, Z, fÞ

Hi ¼ E0 n

mn

1 1 X k X

om m ¼ 0 n ¼ m

ð1Þ

m rð1Þ rð1Þ in ½G0m n,TM Memn ðc, z, Z, fÞ þGn,TM Momn ðc, z, Z, fÞ

0m rð1Þ rð1Þ iGm n,TE Nemn ðc, z, Z, fÞ þiGn,TE Nomn ðc, z, Z, fÞ

ð2Þ

H. Wang et al. / Optics & Laser Technology 44 (2012) 1290–1293

1291

Within the GLMT framework for a spheroidal particle, the scattered fields as well as the fields within the spheroid can be expanded in terms of appropriate spheroidal vector wave functions as follows [9,10]: 1 1 X X

Es ¼ E0

rð3Þ in ½bmn Memn ðc, z, Z, fÞbmn Mrð3Þ omn ðc, z, Z, fÞ 0

m¼0n¼m þi 0mn Nrð3Þ emn ðc, z,

a

rð3Þ Z, fÞ þ iamn Nomn ðc, z, Z, fÞ

1 1 X k X

Hs ¼ E0

om m ¼ 0 n ¼ m

ð5Þ

rð3Þ in ½a0mn Memn ðc, z, Z, fÞ þ amn Mrð3Þ omn ðc, z, Z, fÞ

rð3Þ ibmn Nrð3Þ emn ðc, z, Z, fÞ þibmn Nomn ðc, z, Z, fÞ 0

1 1 X X

Ew ¼ E0

rð1Þ rð1Þ in ½dmn Memn ðc1 , z, Z, fÞdmn Momn ðc1 , z, Z, fÞ 0

m¼0n¼m þi 0mn Nrð1Þ emn ðc1 , z,

g

Hw ¼ E0 Fig. 1. Major axis of a spheroid is along the z axis of the Cartesian coordinate system Oxyz, and the system Ox0 y0 z0 is obtained by a rigid-body rotation of Oxyz through Euler angles a ¼ j, b ¼ y, g ¼ 0. A Gaussian beam propagates from the negative z0 to the positive z0 axis of Ox0 y0 z0 , and the Cartesian coordinates of the middle of its beam waist O0 are (0,0,z0) in Ox0 y0 z0 .

By considering the fact that the system Oxyz can at the same time be obtained by rotating Ox0 y0 z0 through Euler angles a ¼0, b ¼  y, g ¼  j, the expansion coefficients or beam shape coeffi0m 0m m cients Gm n,TE , Gn,TE , Gn,TM and Gn,TM , for the TE mode as an example, can be expressed explicitly as follows: 1 dP m ðcos yÞ ð2dm0 Þ r þ mdy cos mj B C B 0m C B ð2d Þ dPmrþ m ðcos yÞ sinmj C 1 mn B Gn,TE C B C m0 2 X 0 dr ðcÞ dy B C B C g B 0m, C ¼ C Pm r þ m ðcos yÞ B Gn,TM C N mn r ¼ 0,1 ðr þ mÞðr þ mþ 1Þ r þ m B 2m sin m j B C @ A sin y @ A m m Gn,TM P r þ m ðcos yÞ 2m sin y cos mj 0

Gm n,TE

0

1

ð3Þ When the Davis–Barton model of the Gaussian beam is used [21], the gr þ m coefficients in Eq. (3) can be computed using the localized approximation as [3] " # 1 s2 ðr þm þ 1=2Þ2 gr þ m ¼ expðikz0 Þexp ð4Þ 12isz0 =w0 12isz0 =w0 where s¼1/kw0, and w0 is the beam waist radius. It is interesting to note that in Eq. (4) gr þ m ¼ 1, as w0-N and z0 ¼0, then Eqs. (1)–(3) (gr þ m ¼1) become the expansions of the TE mode description for plane wave incidence and the corresponding expansion coefficients. 2.2. Scattering by spheroidal particle illuminated with a couple of on-axis Gaussian beams Let us assume that the spheroid in Fig. 1 is illuminated by a couple of on-axis Gaussian beams. Gaussian beam I, without any loss of generality, can be described in the Cartesian coordinate system Ox1y1z1, which is obtained by rotating Oxyz through Euler angles a ¼0, b ¼ y1, g ¼ 0, and Gaussian beam II in the system Ox2y2z2 by a ¼ j2, b ¼ y2, g ¼ 0. The electromagnetic fields of Gaussian beam I, Ei1 and Hi1 have an expansion of the form as in Eqs. (1) and (2), and so do the electromagnetic fields Ei2 and Hi2 for Gaussian beam II. We denote the expansion coefficients of 0mðIÞ 0mðIÞ mðIÞ Gaussian beam I by GmðIÞ n,TE , Gn,TE , Gn,TM and Gn,TM , and those of

GmðIIÞ n,TE ,

G0mðIIÞ n,TE ,

0mðIIÞ Gn,TM

and Gaussian beam II by obtained by following Eqs. (1) and (2).

mðIIÞ Gn,TM ,

all easy to be

ð6Þ

rð1Þ Z, fÞ þ igmn Nomn ðc1 , z, Z, fÞ

1 1 X k1 X

om m ¼ 0 n ¼ m

ð7Þ

n

rð1Þ i ½g0mn Mrð1Þ emn ðc1 , z, Z, fÞ þ gmn Momn ðc 1 , z, Z, fÞ

rð1Þ idmn Nrð1Þ emn ðc1 , z, Z, fÞ þ idmn Nomn ðc 1 , z, Z, fÞ 0

ð8Þ

~ c1 ¼fk1 and n~ is the refractive index of the material where k1 ¼ kn, of the spheroidal particle relative to that of free space. 0 The unknown coefficients bm(z), bm ðzÞ, a0m ðzÞ, am(z), dm(z), 0 0 dm ðzÞ, gm ðzÞ and gm(z) in Eqs. (5)–(8) can be determined by applying the boundary conditions of continuity of the tangential electromagnetic fields over the surface z ¼ z0, with z0 as the radial coordinate of the boundary surface of the spheroid. The boundary conditions at z ¼ z0 are described by 9 Ei1f þ Ei1f þEsf ¼ Ew Ei1Z þ Ei2Z þEsZ ¼ Ew Z, f = ð9Þ i i s w ; at z ¼ z0 Hi1Z þ Hi2Z þ HsZ ¼ Hw Z , H1f þ H2f þ Hf ¼ Hf By virtue of the field expansions in Eqs. (1) and (2) and in Eqs. (5)–(8), the above boundary conditions can be written as 2 ð1Þ,t 3 2 3 U mn ðcÞ amn 6 ð1Þ,t 7 6 7 1 1 6 X n 6 V mn ðcÞ 7 X n 6 bmn 7 7 7 ð10Þ i ½G6 i ½G6 ð1Þ,t 7 6 dmn 7 ¼ 6 7 X ðcÞ 4 5 n¼m n¼m 4 mn 5 gmn Y ð1Þ,t mn ðcÞ 2

a0mn

2

3

U ð1Þ,t mn ðcÞ

3

6 ð1Þ,t 7 6 0 7 1 6V X ðcÞ 7 6 bmn 7 7 n 0 6 mn 7¼ in ½G6 i ½G  6 ð1Þ,t 7 0 6d 7 6 7 X ðcÞ 4 mn 5 n ¼ m n¼m mn 4 5 ð1Þ,t g0mn Y mn ðcÞ 1 X

ð11Þ

where the matrixes [G] and [G] are given by 2

mðIIÞ GmðIÞ n,TE þ Gn,TE

6 6 GmðIÞ þ GmðIIÞ 6 n,TM n,TM ½G ¼ ð1Þ6 6 0 6 4 0

mðIIÞ GmðIÞ n,TM þ Gn,TM

0

mðIIÞ GmðIÞ n,TE þ Gn,TE

0

0

mðIIÞ GmðIÞ n,TE þGn,TE

0

mðIIÞ GmðIÞ n,TM þ Gn,TM

0

3

7 7 7 7 mðIÞ mðIIÞ 7 Gn,TM þ Gn,TM 7 5 mðIIÞ GmðIÞ n,TE þ Gn,TE 0

ð12Þ 2

ð3Þ,t V mn ðcÞ 6 ð3Þ,t 6 U mn ðcÞ 6 ½G ¼ 6 ð3Þ,t 6 Y mn ðcÞ 4 ð3Þ,t ðcÞ X mn

U ð3Þ,t mn ðcÞ V ð3Þ,t mn ðcÞ X ð3Þ,t mn ðcÞ Y ð3Þ,t mn ðcÞ

U ð1Þ,t mn ðc1 Þ  kk1

V ð1Þ,t mn ðc 1 Þ X ð1Þ,t mn ðc 1 Þ k1 ð1Þ,t  k Y mn ðc1 Þ

ð1Þ,t V mn ðc1 Þ

3

7 7  kk1 U ð1Þ,t mn ðc 1 Þ 7 7 7 Y ð1Þ,t ðc Þ 1 mn 5 k1 ð1Þ,t  k X mn ðc1 Þ

ð13Þ

mðIIÞ The matrix [G0 ] can be obtained by replacing GmðIÞ n,TE , Gn,TE with mðIÞ mðIIÞ 0mðIÞ 0mðIIÞ G0mðIIÞ n,TE , and Gn,TM , Gn,TM with Gn,TM , Gn,TM in Eq. (12). ðjÞ,t ðjÞ,t ðjÞ,t ðjÞ,t The parameters U mn , V mn , X mn , Y mn (j ¼1 or 3 according to the ðjÞ radial function Rmn ðc, zÞ of the first or third kind) are given by

G0mðIÞ n,TE ,

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H. Wang et al. / Optics & Laser Technology 44 (2012) 1290–1293

Asano and Yamamoto in [22], and Eqs. (10) and (11), given the value of m, are valid for each of t Z0. By taking t to be sufficiently large, an adequate number of relations between the unknown coefficients is generated, then, the unknown expansion coefficients of the scattered and internal electromagnetic fields can be determined.

Of practical interest in the PDA is the behavior of the scattered wave at relatively large distances from the scatterer (far field), which can be deduced by taking the asymptotic form of Es, as

102

mn

o

10-1

mn

order higher than 1/r can be neglected, then, from Eq. (5) the asymptotic forms of the scattered electric field Es are obtained as follows:   1 X 1  il 2pr X dS ðc,cos yÞ EsZ ¼ E0 exp i amn mn dy 2pr l m¼0n¼m  Smn ðc,cos yÞ sin mf þ mbmn sin y   dSmn ðc,cos yÞ Smn ðc,cos yÞ 0 þ a0mn þmbmn cos mf ð14Þ dy sin y   1 X 1  il 2pr X Smn ðc,cos yÞ exp i mamn sin y 2pr l m¼0n¼m  dSmn ðc,cos yÞ cos mf þ bmn dy    dSmn ðc,cos yÞ Smn ðc,cos yÞ 0  bmn þma0mn sin mf dy sin y

10-2

1 1  X X

amn

m¼0n¼m

60 80 100 120 140 Scattering angle θ (degree)

160

180

104

πσ (θ,φ)/λ2

ð15Þ

φ=0 φ=π/2

101 100 10-1

ð16Þ

 dSmn ðc,cos yÞ Smn ðc,cos yÞ þ mbmn sin mf dy sin y ð17Þ

ð18Þ 1 1  X X m¼0n¼m

40

102

 1 1  X X Smn ðc,cos yÞ dSmn ðc,cos yÞ mamn þ bmn cos mf T 2 ðy, fÞ ¼ sin y dy m¼0n¼m

T 3 ðy, fÞ ¼

20

103

where T 1 ðy, fÞ ¼

0

Fig. 2. Normalized differential scattering cross sections ps(y,0)/l2 and ps(y,(p/ 2))/l2 for a spheroid (ka¼ 10, n~ ¼ 1:33, a/b ¼ 2) illuminated by a couple of Gaussian beams (w0 ¼2l, y1 ¼ p/6, y2 ¼ p/2, j1 ¼ j2 ¼0) (TE mode).

Esf ¼ E0

The differential scattering cross section is defined by  s 2 E  sðy, fÞ ¼ 4pr2   E0 l2 n 2 2 2 2 ¼ 9T 1 ðy, fÞ9 þ 9T 3 ðy, fÞ9 þ 9T 2 ðy, fÞ9 þ 9T 4 ðy, fÞ9 p þ2Re½T 1 ðy, fÞT 3 ðy, fÞn T 2 ðy, fÞT 4 ðy, fÞn 

101 100

ðc, z, Z, fÞand Nrð3Þ ðc, z, Z, fÞ, as cz-N, terms of cz-N. In Mrð3Þ e e o

φ=0 φ=π/2

103

πσ (θ,φ)/λ2

3. Numerical results

104

 dS ðc,cos yÞ Smn ðc,cos yÞ 0 a0mn mn þ mbmn cos mf dy sin y

10-2

0

20

40

60 80 100 120 140 Scattering angle θ (degree)

160

180

Fig. 3. Normalized differential scattering cross sections ps(y,0)/l2 and ps(y,(p/ 2))/l2 for a spheroid (ka¼ 15.7, n~ ¼ 1:33, a/b¼ 2) illuminated by a couple of Gaussian beams (w0 ¼ 2l, y1 ¼ 0, j1 ¼ 0, y2 ¼ p/2, j2 ¼ p/2) (TE mode).

The normalized differential scattering cross section ps(y,f)/l2 is shown in Fig. 3 for parallel (y1 ¼0) and normal (y2 ¼ p/2) incidence of a couple of Gaussian beams (TE mode, w0 ¼2l) on a spheroidal particle. The spheroid has the maximum scattering in the angle regions around y ¼0 and y ¼ p/2.

ð19Þ T 4 ðy, fÞ ¼

 1 1  X X Smn ðc,cos yÞ dSmn ðc,cos yÞ 0 ma0mn þ bmn sin mf sin y dy m¼0n¼m ð20Þ

The normalized differential scattering cross section ps(y,f)/l2 is thereafter evaluated in the coordinate system attached to the spheroidal particle. In the following calculations, the incident Gaussian beam is TE polarized and z0 ¼0. Fig. 2 shows the normalized differential scattering cross section ps(y,f)/l2 for a spheroidal particle illuminated by a couple of Gaussian beams (TE mode, w0 ¼2l).

4. Conclusion In the GLMT framework, an approach to compute the scattering by a spheroidal particle illuminated with a couple of on-axis Gaussian beams is presented. The expansion coefficients for the scattered wave are determined by a solution to a properly truncated system of equations derived from boundary conditions. The normalized differential scattering cross section is plotted. As a result, this study is suggestive and useful for interpretation of electromagnetic scattering by non-spherical particles as well as analysis of the operation of a PDA system.

H. Wang et al. / Optics & Laser Technology 44 (2012) 1290–1293

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