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  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 8, Iss. 8 — Sep. 4, 2013
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Internal and near-surface electromagnetic fields for a uniaxial anisotropic cylinder illuminated with a Gaussian beam

Huayong Zhang, Zhixiang Huang, and Yuan Shi  »View Author Affiliations


Optics Express, Vol. 21, Issue 13, pp. 15645-15653 (2013)
http://dx.doi.org/10.1364/OE.21.015645


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Abstract

Within the generalized Lorenz-Mie theory (GLMT) framework, an analytical solution to the scattering by a uniaxial anisotropic cylinder, for oblique incidence of an on-axis Gaussian beam, is constructed by expanding the incident Gaussian beam, scattered fields as well as internal fields in terms of appropriate cylindrical vector wave functions (CVWFs). The unknown expansion coefficients are determined by virtue of the boundary conditions. For a localized beam model, numerical results are provided for the normalized internal and near-surface field intensity distributions, and the scattering characteristics are discussed concisely.

© 2013 OSA

1. Introduction

The paper is organized as follows. Section 2 provides the theoretical procedure for the determination of the scattered and internal fields of a uniaxial anisotropic cylinder illuminated by a Gaussian beam. In Section 3, the normalized internal and near-surface field intensity distributions are displayed. Section 4 is the conclusion.

2. Formulation

2.1 Expansions of Gaussian beam, scattered fields and internal fields in cylindrical coordinates

As shown in Fig. 1
Fig. 1 Uniaxial anisotropic cylinder illuminated by an on-axis incident Gaussian beam.
, an infinite uniaxial anisotropic cylinder of radius r0 is attached to the Cartesian coordinate system Oxyz. An incident Gaussian beam propagates in free space and along the positive z axis in the xOz plane, with the middle of its beam waist located at origin O on the z axis. Origin O has a coordinate z0 on the axis Oz (on-axis case), and the angle made by the axis Oz with the axis Oz is β. In this paper, the time-dependent part of the electromagnetic fields is assumed to be exp(iωt).

We have obtained an expansion in [17

17. H. Y. Zhang, Y. P. Han, and G. X. Han, “Expansion of the electromagnetic fields of a shaped beam in terms of cylindrical vector wave functions,” J. Opt. Soc. Am. B 24(6), 1383–1391 (2007). [CrossRef]

] of the electromagnetic fields of an incident Gaussian beam (a focused TEM00 mode laser beam) in terms of the CVWFs with respect to the system Oxyz, in the following form
Ei=E0m=0π[Im,TE(ζ)mmλ(1)(h)+Im,TM(ζ)nmλ(1)(h)]eihzdζ,
(1)
Hi=iE0k0ωμ0m=0π[Im,TE(ζ)nmλ(1)(h)+Im,TM(ζ)mmλ(1)(h)]eihzdζ,
(2)
where λ=k0sinζ, h=k0cosζ, k0 and μ0 are the free-space wave number and permeability, respectively, and In,TEm, In,TMm are the beam shape coefficients.

For a TE polarized Gaussian beam, the coefficients In,TEm, In,TMm are
Im,TE=(i)m+1k0n=|m|(nm)!(n+m)!2n+12n(n+1)×gn[m2Pnm(cosβ)sinβPnm(cosζ)sinζ+dPnm(cosβ)dβdPnm(cosζ)dζ],
(3)
Im,TM=(i)m+1k0mn=|m|(nm)!(n+m)!2n+12n(n+1)×gn[Pnm(cosβ)sinβdPnm(cosζ)dζ+dPnm(cosβ)dβPnm(cosζ)sinζ],
(4)
where gn are the Gaussian beam shape coefficients in spherical coordinates, and for the Davis-Barton model of the Gaussian beam [19

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

], can be described by the localized approximation as [20

20. G. Gouesbet, “Validity of the localized approximation for arbitrary shaped beam in the generalized Lorenz-Mie theory for spheres,” J. Opt. Soc. Am. A 16(7), 1641–1650 (1999). [CrossRef]

, 21

21. G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” JQSRT 112(1), 1–27 (2011). [CrossRef]

]
gn=11+2isz0/w0exp(ikz0)exp[s2(n+1/2)21+2isz0/w0],
(5)
where s=1/(k0w0), and w0 is the beam waist radius.

For a TM polarized Gaussian beam, the corresponding expansions can be obtained only by replacing In,TEm in Eqs. (1) and (2) by iIn,TMm, and In,TMm by iIn,TEm.

It is worth mentioning that the coefficients In,TEm, In,TMm in Eqs. (3) and (4) are expressed in terms of beam shape coefficients in spherical coordinates, obviously an extrinsic method. Gouesbet et al. introduced a cylindrical localized approximation to evaluate the coefficients, in which an intrinsic method is used [22

22. G. Gouesbet, G. Gréhan, and K. F. Ren, “Rigorous justification of the cylindrical localized approximation to speed up computations in the generalized Lorenz–Mie theory for cylinders,” J. Opt. Soc. Am. A 15(2), 511–523 (1998). [CrossRef]

].

By following Eqs. (1) and (2), an appropriate expansion of the scattered fields in terms of the CVWFs can be written as
Es=E0m=0π[αm(ζ)mmλ(3)+βm(ζ)nmλ(3)]eihzdζ,
(6)
Hs=iE0k0ωμ0m=0π[αm(ζ)nmλ(3)+βm(ζ)mmλ(3)]eihzdζ,
(7)
where αm(ζ), βm(ζ) are the unknown expansion coefficients to be determined.

Let’s consider that a uniaxial anisotropic medium is characterized by a permittivity tensor ε¯=x^x^εt+y^y^εt+z^z^εz in the system Oxyz and a scalar permeability μ=μ0.

From Appendix, the electromagnetic fields within the uniaxial anisotropic cylinder can be expanded in terms of the CVWFs (refer to Appendix), as follows:
Ew=E0q=12m=0πFmq(ζ)[αqe(ζ)mmλq(1)+βqe(ζ)nmλq(1)+γqe(ζ)lmλq(1)]eihqzdζ,
(8)
Hw=iE0q=12m=0πkqωμ0Fmq(ζ)[βqe(ζ)(ζ)mmλq(1)+αqe(ζ)nmλq(1)]eihqzdζ,
(9)
where

h1=h2=h=k0cosζ,
(10)
a12=ω2εtμ0,  a22=ω2εzμ0,
(11)
k1=a1,k2=1a1a12a22+(a12a22)k02cos2ζ,
(12)
λ1=a12k02cos2ζ,  λ2=k2sinθk=a2a1a12k02cos2ζ,
(13)
α1e(ζ)=1, β1e(ζ)=γ1e(ζ)=α2e(ζ)=0,
(14)
β2e(ζ)=ia12a2(a12k02cos2ζ)[a12a22+(a12a22)k02cos2ζ],
(15)
γ2e(ζ)=a12a22a12a1a2k0cosζa12k02cos2ζa12a22+(a12a22)k02cos2ζ,
(16)

2.2 On-axis Gaussian beam scattering by a uniaxial anisotropic cylinder

The unknown expansion coefficients αm(ζ), βm(ζ) in Eqs. (6) and (7) as well as Fm1(ζ), Fm2(ζ) in Eqs. (8) and (9) can be determined by using the following boundary conditions
Eϕi+Eϕs=Eϕw,Ezi+Ezs=EzwHϕi+Hϕs=Hϕw,Hzi+Hzs=Hzw}atr=r0,
(17)
By virtue of the field expansions, the above boundary conditions can be written as
ξddξJm(ξ)Im,TE+hmk0Jm(ξ)Im,TM+ξddξHm(1)(ξ)αm(ζ)+hmk0Hm(1)(ξ)βm(ζ)=Fm1(ζ)ξ1ddξ1Jm(ξ1)+Fm2(ζ)β2e(ζ)hmk2Jm(ξ2)Fm2(ζ)γ2e(ζ)imJm(ξ2),
(18)
ξ2[Jm(ξ)Im,TM+Hm(1)(ξ)βm(ζ)]=Fm2(ζ)k0k2ξ22Jm(ξ2)[β2e(ζ)+γ2e(ζ)ihk2λ22],
(19)
hmk0Jm(ξ)Im,TE+ξddξJm(ξ)Im,TM+hmk0Hm(1)(ξ)αm(ζ)+ξddξHm(1)(ξ)βm(ζ)=hmk0Fm1(ζ)Jm(ξ1)+k2k0Fm2(ζ)β2e(ζ)ξ2ddξ2Jm(ξ2),
(20)
ξ2[Jm(ξ)Im,TE+Hm(1)(ξ)αm(ζ)]=Fm1(ζ)ξ12Jm(ξ1),
(21)
where ξ=λr0, ξ1=λ1r0, and ξ2=λ2r0

From the system consisting of Eqs. (18)-(21), the expansion coefficients of the scattered and internal fields can be determined. By substituting them into Eqs. (6)-(9), the solution to the scattering of an on-axis Gaussian beam by a uniaxial anisotropic cylinder is obtained.

3. Numerical results

In this paper, we will focus our attention on the normalized internal and near-surface field intensity distributions, which are described, respectively, by
|Ew/E0|2=(|Erw|2+|Eϕw|2+|Ezw|2)/|E0|2,
(22)
and

|(Ei+Es)/E0|2=(|Eri+Ers|2+|Eϕi+Eϕs|2+|Ezi+Ezs|2)/|E0|2,
(23)

The subscripts r, ϕ and zin Eqs. (22) and (23) denote the r, ϕ and z components of the electric fields. By taking Eq. (22) as an example, the r, ϕ and z components of the internal electric field Ew are expressed as

Erw=E0m=eimϕ0π{Fm1(ζ)imrJm(λ1r)+Fm2(ζ)rJm(λ2r)[β2e(ζ)ihk2+γ2e(ζ)]}eihzdζ
(24)
Eφw=E0m=eimϕ0π{Fm1(ζ)rJm(λ1r)+Fm2(ζ)mrJm(λ2r)[β2e(ζ)hk2+iγ2e(ζ)]}eihzdζ
(25)
Ezw=E0m=eimϕ0πFm2(ζ)Jm(λ2r)[β2e(ζ)λ22k2+ihγ2e(ζ)]eihzdζ
(26)

In a similar way, from Eqs. (1) and (6) the r, ϕ and z components of the electric fields Ei and Es can also be obtained.

Figures (2)
Fig. 2 |Ew/E0|2 and |(Ei+Es)/E0|2for a uniaxial anisotropic cylinder illuminated by a TM polarized Gaussian beam.
and (3)
Fig. 3 Same model as in Fig. 2 but illuminated by a TE polarized Gaussian beam.
, sharing the same colorbar, show the normalized internal and near-surface field intensity distributions |Ew/E0|2 and |(Ei+Es)/E0|2 in the xOz plane for a uniaxial anisotropic cylinder, illuminated by a TM and TE polarized Gaussian beam, respectively. The radius of the cylinder r0 and the beam waist radius w0 are assumed to be five and two times the wavelength of the incident Gaussian beam, and a1=3k0, a2=2k0, β=π/4, z0=0.

From Figs. 2 and 3 we can see that a TM polarized Gaussian beam is reflected less than a TE polarized one, so it is reasonable to conclude that in the same case the transmitted power is larger for the former. Another noticeable difference lies in the fact that the middle of the transmitted beam for a TM polarized Gaussian beam is lower than that for a TE polarized one, demonstrating that the former is refracted more strongly, which dose not appear for a dielectric isotropic cylinder.

4. Conclusion

An approach to compute the on-axis Gaussian beam scattering by a uniaxial anisotropic cylinder is presented. Numerical results of the normalized internal and near-surface field intensity distributions show that, compared with a TE polarized Gaussian beam, a TM polarized one is reflected less and refracted more strongly. As a result, this study provides an exact analytical model for interpretation of Gaussian beam scattering phenomena for a uniaxial anisotropic cylinder.

Appendix

In [7

7. Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 056609 (2004). [CrossRef] [PubMed]

], the eigen plane wave spectrum representation of the electric field in uniaxial anisotropic media is given by the Fourier transform, as follows:
Ew=E0q=12kq2sinθkdθk02πFqe(θk,ϕk)fq(θk,ϕk)eikqrdϕk,
(27)
where the wave numbers kq and electric field eigenvectors Fqe(θk,ϕk)eikqr(q=1,2) are

k1=a1,    k2=a1a21a12sin2θk+a22cos2θk,
(28)
Fqe(θk,ϕk)eikqr=(Fqxex^+Fqyey^+Fqzez^)eikqr,
(29)
Fqxe={sinϕk,q=1a22a12cosθksinθkcosϕk,q=2
(30)
Fqye={cosϕk,q=1a22a12cosθksinθksinϕk,q=2
(31)
Fqze={0,q=11,q=2
(32)

The unknown angular spectrum amplitude fq(θk,ϕk) in Eq. (27) is periodic with respect to ϕk, so we use a Fourier expansion for the fq(θk,ϕk) as

fq(θk,ϕk)=n=Gnq(θk)einϕk,
(33)

A substitution of Eq. (33) into Eq. (27) leads to

Ew=E0q=12n=Gnq(θk)kq2sinθkdθk02πFqe(θk,ϕk)eikqreinϕkdϕk,
(34)

An arbitrary plane electromagnetic wave of unit amplitude can be represented in a convergent series of the CVWFs, as follows [5

5. W. Ren, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 47(1), 664–673 (1993). [CrossRef] [PubMed]

, 6

6. X. B. Wu and K. Yasumoto, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: An analytical solution,” J. Appl. Phys. 82(5), 1996–2003 (1997). [CrossRef]

]:
x^eikr=m=(amxmmλ(1)+bmxnmλ(1)+cmxlmλ(1))eihz,
(35)
y^eikr=m=(amymmλ(1)+bmynmλ(1)+cmylmλ(1))eihz,
(36)
z^eikr=m=(amzmmλ(1)+bmznmλ(1)+cmzlmλ(1))eihz,
(37)
where the expansion coefficients are determined by

[amxbmxcmx]=im1eimϕkk[1sinθksinϕkicosθksinθkcosϕksinθkcosϕk],
(38)
[amybmycmy]=im1eimϕkk[1sinθkcosϕkicosθksinθksinϕksinθksinϕk],
(39)
[amzbmzcmz]=im1eimϕkk[0icosθk].
(40)

Substituting Eqs. (35)-(37) into Eq. (34) and considering that 02πei(nm)ϕkdϕk equals 2π when m=n and 0 when mn, we end up with
Ew=E0q=12m=Gmq(θk)[Aqe(θk)mmλq(1)+Bqe(θk)nmλq(1)+Cqe(θk)lmλq(1)]eihqzdθk,
(41)
where Gmq(θk)=2πim+1Gmq(θk)kq, λq=kqsinθk, hq=kqcosθk, and

A1e(θk)=1,  B1e(θk)=C1e(θk)=A2e(θk)=0,
(42)
B2e(θk)=ia12sin2θk+a22cos2θka12sinθk,
(43)
C2e(θk)=a12a22a12sinθkcosθk,
(44)

Due to the phase continuity over the cylinder surface when using the boundary conditions, we have hq=kqcosθk=h=k0cosζ (q=1,2). Then, Eq. (41) is transformed into Eq. (8), where A1e(θk)=α1e(ζ), B1e(θk)=β1e(ζ), C1e(θk)=γ1e(ζ), A2e(θk)=α2e(ζ), B2e(θk)=β2e(ζ), C2e(θk)=γ2e(ζ), and Gmq(θk)dθk=Fmq(ζ)dζ.

Acknowledgments

The authors would acknowledge the support by the NSFC (Nos.60931002, 61101064, 61201122), Distinguished Natural Science Foundation (No.1108085J01), Universities Natural Science Foundation of Anhui Province (Nos.KJ2011A002, KJ2011A242, KJ2012A013), DFMEC (No.20123401110009) and NCET (No.NCET-12-0596) of China.

References and links

1.

R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77(5), 750–760 (1989). [CrossRef]

2.

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antenn. Propag. 37(6), 800–802 (1989). [CrossRef]

3.

S. N. Papadakis, N. K. Uzunoglu, and C. N. Capsalis, “Scattering of a plane wave by a general anisotropic dielectric ellipsoid,” J. Opt. Soc. Am. A 7(6), 991–997 (1990). [CrossRef]

4.

C. M. Rappaport and B. J. McCartin, “FDFD analysis of electromagnetic scattering in anisotropic media using unconstrained triangular meshes,” IEEE Trans. Antenn. Propag. 39(3), 345–349 (1991). [CrossRef]

5.

W. Ren, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 47(1), 664–673 (1993). [CrossRef] [PubMed]

6.

X. B. Wu and K. Yasumoto, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: An analytical solution,” J. Appl. Phys. 82(5), 1996–2003 (1997). [CrossRef]

7.

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 056609 (2004). [CrossRef] [PubMed]

8.

Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antenn. Propag. 57(2), 572–576 (2009). [CrossRef]

9.

Y. L. Geng and C. W. Qiu, “Extended Mie theory for a gyrotropic-coated conducting sphere: An analytical approach,” IEEE Trans. Antenn. Propag. 59(11), 4364–4368 (2011). [CrossRef]

10.

Z. J. Li, Z. S. Wu, and H. Y. Li, “Analysis of electromagnetic scattering by uniaxial anisotropic bispheres,” J. Opt. Soc. Am. A 28(2), 118–125 (2011). [CrossRef] [PubMed]

11.

Z. J. Li, Z. S. Wu, Y. Shi, L. Bai, and H.-Y. Li, “Multiple scattering of electromagnetic waves by an aggregate of uniaxial anisotropic spheres,” J. Opt. Soc. Am. A 29(1), 22–31 (2012). [CrossRef]

12.

Z. S. Wu, Q. K. Yuan, Y. Peng, and Z. J. Li, “Internal and external electromagnetic fields for on-axis Gaussian beam scattering from a uniaxial anisotropic sphere,” J. Opt. Soc. Am. A 26(8), 1778–1787 (2009). [CrossRef] [PubMed]

13.

Q. K. Yuan, Z. S. Wu, and Z. J. Li, “Electromagnetic scattering for a uniaxial anisotropic sphere in an off-axis obliquely incident Gaussian beam,” J. Opt. Soc. Am. A 27(6), 1457–1465 (2010). [CrossRef] [PubMed]

14.

G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” J. Optics (Paris) 26(5), 225–239 (1995). [CrossRef]

15.

K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz-Mie theory: formulation and numerical results,” J. Opt. Soc. Am. A 14(11), 3014–3025 (1997). [CrossRef]

16.

J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder,” J. Opt. Soc. Am. A 14(3), 640–652 (1997). [CrossRef]

17.

H. Y. Zhang, Y. P. Han, and G. X. Han, “Expansion of the electromagnetic fields of a shaped beam in terms of cylindrical vector wave functions,” J. Opt. Soc. Am. B 24(6), 1383–1391 (2007). [CrossRef]

18.

H. Y. Zhang and Y. P. Han, “Scattering of shaped beam by an infinite cylinder of arbitrary orientation,” J. Opt. Soc. Am. B 25(2), 131–135 (2008). [CrossRef]

19.

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

20.

G. Gouesbet, “Validity of the localized approximation for arbitrary shaped beam in the generalized Lorenz-Mie theory for spheres,” J. Opt. Soc. Am. A 16(7), 1641–1650 (1999). [CrossRef]

21.

G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” JQSRT 112(1), 1–27 (2011). [CrossRef]

22.

G. Gouesbet, G. Gréhan, and K. F. Ren, “Rigorous justification of the cylindrical localized approximation to speed up computations in the generalized Lorenz–Mie theory for cylinders,” J. Opt. Soc. Am. A 15(2), 511–523 (1998). [CrossRef]

OCIS Codes
(140.3430) Lasers and laser optics : Laser theory
(260.2110) Physical optics : Electromagnetic optics
(290.4020) Scattering : Mie theory

ToC Category:
Physical Optics

History
Original Manuscript: May 13, 2013
Revised Manuscript: June 12, 2013
Manuscript Accepted: June 14, 2013
Published: June 21, 2013

Virtual Issues
Vol. 8, Iss. 8 Virtual Journal for Biomedical Optics

Citation
Huayong Zhang, Zhixiang Huang, and Yuan Shi, "Internal and near-surface electromagnetic fields for a uniaxial anisotropic cylinder illuminated with a Gaussian beam," Opt. Express 21, 15645-15653 (2013)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-13-15645


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References

  1. R. D.  Graglia, P. L. E.  Uslenghi, R. S.  Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77(5), 750–760 (1989). [CrossRef]
  2. V. V.  Varadan, A.  Lakhtakia, V. K.  Varadan, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antenn. Propag. 37(6), 800–802 (1989). [CrossRef]
  3. S. N.  Papadakis, N. K.  Uzunoglu, C. N.  Capsalis, “Scattering of a plane wave by a general anisotropic dielectric ellipsoid,” J. Opt. Soc. Am. A 7(6), 991–997 (1990). [CrossRef]
  4. C. M.  Rappaport, B. J.  McCartin, “FDFD analysis of electromagnetic scattering in anisotropic media using unconstrained triangular meshes,” IEEE Trans. Antenn. Propag. 39(3), 345–349 (1991). [CrossRef]
  5. W.  Ren, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 47(1), 664–673 (1993). [CrossRef] [PubMed]
  6. X. B.  Wu, K.  Yasumoto, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: An analytical solution,” J. Appl. Phys. 82(5), 1996–2003 (1997). [CrossRef]
  7. Y. L.  Geng, X. B.  Wu, L. W.  Li, B. R.  Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 056609 (2004). [CrossRef] [PubMed]
  8. Y. L.  Geng, C. W.  Qiu, N.  Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antenn. Propag. 57(2), 572–576 (2009). [CrossRef]
  9. Y. L.  Geng, C. W.  Qiu, “Extended Mie theory for a gyrotropic-coated conducting sphere: An analytical approach,” IEEE Trans. Antenn. Propag. 59(11), 4364–4368 (2011). [CrossRef]
  10. Z. J.  Li, Z. S.  Wu, H. Y.  Li, “Analysis of electromagnetic scattering by uniaxial anisotropic bispheres,” J. Opt. Soc. Am. A 28(2), 118–125 (2011). [CrossRef] [PubMed]
  11. Z. J.  Li, Z. S.  Wu, Y.  Shi, L.  Bai, H.-Y.  Li, “Multiple scattering of electromagnetic waves by an aggregate of uniaxial anisotropic spheres,” J. Opt. Soc. Am. A 29(1), 22–31 (2012). [CrossRef]
  12. Z. S.  Wu, Q. K.  Yuan, Y.  Peng, Z. J.  Li, “Internal and external electromagnetic fields for on-axis Gaussian beam scattering from a uniaxial anisotropic sphere,” J. Opt. Soc. Am. A 26(8), 1778–1787 (2009). [CrossRef] [PubMed]
  13. Q. K.  Yuan, Z. S.  Wu, Z. J.  Li, “Electromagnetic scattering for a uniaxial anisotropic sphere in an off-axis obliquely incident Gaussian beam,” J. Opt. Soc. Am. A 27(6), 1457–1465 (2010). [CrossRef] [PubMed]
  14. G.  Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” J. Optics (Paris) 26(5), 225–239 (1995). [CrossRef]
  15. K. F.  Ren, G.  Gréhan, G.  Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz-Mie theory: formulation and numerical results,” J. Opt. Soc. Am. A 14(11), 3014–3025 (1997). [CrossRef]
  16. J. A.  Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder,” J. Opt. Soc. Am. A 14(3), 640–652 (1997). [CrossRef]
  17. H. Y.  Zhang, Y. P.  Han, G. X.  Han, “Expansion of the electromagnetic fields of a shaped beam in terms of cylindrical vector wave functions,” J. Opt. Soc. Am. B 24(6), 1383–1391 (2007). [CrossRef]
  18. H. Y.  Zhang, Y. P.  Han, “Scattering of shaped beam by an infinite cylinder of arbitrary orientation,” J. Opt. Soc. Am. B 25(2), 131–135 (2008). [CrossRef]
  19. L. W.  Davis, “Theory of electromagnetic beam,” Phys. Rev. A 19(3), 1177–1179 (1979). [CrossRef]
  20. G.  Gouesbet, “Validity of the localized approximation for arbitrary shaped beam in the generalized Lorenz-Mie theory for spheres,” J. Opt. Soc. Am. A 16(7), 1641–1650 (1999). [CrossRef]
  21. G.  Gouesbet, J. A.  Lock, G.  Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” JQSRT 112(1), 1–27 (2011). [CrossRef]
  22. G.  Gouesbet, G.  Gréhan, K. F.  Ren, “Rigorous justification of the cylindrical localized approximation to speed up computations in the generalized Lorenz–Mie theory for cylinders,” J. Opt. Soc. Am. A 15(2), 511–523 (1998). [CrossRef]

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