Abstract

Within the generalized Lorenz–Mie theory framework, an analytic solution to Gaussian beam scattering by a rotationally uniaxial anisotropic sphere is presented. The scattered fields as well as the fields within the anisotropic sphere are expanded in terms of infinite series with spherical vector wave functions by using an appropriate expansion of the incident Gaussian beam. The unknown expansion coefficients are determined from a system of linear equations derived from the boundary conditions. Numerical results of the normalized differential scattering cross section are shown, and the scattering characteristics are discussed concisely.

© 2012 Optical Society of America

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References

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  1. Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E. 70, 056609 (2004).
    [CrossRef]
  2. Y. L. Geng and C. W. Qiu, “Extended Mie theory for a gyrotropic-coated conducting sphere: an analytical approach,” IEEE Trans. Antennas Propag. 59, 4364–4368 (2011).
    [CrossRef]
  3. V. Schmidt and T. Wriedt, “T-matrix method for biaxial anisotropic particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 1392–1397 (2009).
    [CrossRef]
  4. H. S. Chen, B.-I. Wu, B. L. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
    [CrossRef]
  5. X. X. Cheng, H. S. Chen, and X. M. Zhang, “Cloaking a perfectly conducting sphere with rotationally uniaxial nihility media in monostatic radar system,” Progress Electromagn. Res. 100, 285–298 (2010).
    [CrossRef]
  6. G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A. 5, 1427–1443 (1988).
    [CrossRef]
  7. B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. 19, 59–67 (1988).
    [CrossRef]
  8. 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,” J. Quant. Spectrosc. Radiat. Transfer 112, 1–27 (2011).
    [CrossRef]
  9. G. Gouesbet, “T-matrix formulation and generalized Lorenz–Mie theories in spherical coordinates,” Opt. Commun. 283, 517–521 (2010).
    [CrossRef]
  10. L. W. Davis, “Theory of electromagnetic beam,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  11. A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. 36, 2971–2978 (1997).
    [CrossRef]
  12. H. Y. Zhang and Y. P. Han, “Addition theorem for the spherical vector wave functions and its application to the beam shape coefficients,” J. Opt. Soc. Am. B 25, 255–260 (2008).
    [CrossRef]
  13. K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized Lorenz–Mie theory,” Appl. Opt. 37, 4218–4225 (1998).
    [CrossRef]
  14. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

2011 (2)

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

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,” J. Quant. Spectrosc. Radiat. Transfer 112, 1–27 (2011).
[CrossRef]

2010 (2)

G. Gouesbet, “T-matrix formulation and generalized Lorenz–Mie theories in spherical coordinates,” Opt. Commun. 283, 517–521 (2010).
[CrossRef]

X. X. Cheng, H. S. Chen, and X. M. Zhang, “Cloaking a perfectly conducting sphere with rotationally uniaxial nihility media in monostatic radar system,” Progress Electromagn. Res. 100, 285–298 (2010).
[CrossRef]

2009 (1)

V. Schmidt and T. Wriedt, “T-matrix method for biaxial anisotropic particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 1392–1397 (2009).
[CrossRef]

2008 (1)

2007 (1)

H. S. Chen, B.-I. Wu, B. L. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef]

2004 (1)

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

1998 (1)

1997 (1)

1988 (2)

G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A. 5, 1427–1443 (1988).
[CrossRef]

B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. 19, 59–67 (1988).
[CrossRef]

1979 (1)

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

Chen, H. S.

X. X. Cheng, H. S. Chen, and X. M. Zhang, “Cloaking a perfectly conducting sphere with rotationally uniaxial nihility media in monostatic radar system,” Progress Electromagn. Res. 100, 285–298 (2010).
[CrossRef]

H. S. Chen, B.-I. Wu, B. L. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef]

Cheng, X. X.

X. X. Cheng, H. S. Chen, and X. M. Zhang, “Cloaking a perfectly conducting sphere with rotationally uniaxial nihility media in monostatic radar system,” Progress Electromagn. Res. 100, 285–298 (2010).
[CrossRef]

Davis, L. W.

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

Doicu, A.

Geng, Y. L.

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

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

Gouesbet, G.

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,” J. Quant. Spectrosc. Radiat. Transfer 112, 1–27 (2011).
[CrossRef]

G. Gouesbet, “T-matrix formulation and generalized Lorenz–Mie theories in spherical coordinates,” Opt. Commun. 283, 517–521 (2010).
[CrossRef]

K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized Lorenz–Mie theory,” Appl. Opt. 37, 4218–4225 (1998).
[CrossRef]

B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A. 5, 1427–1443 (1988).
[CrossRef]

Gréhan, G.

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,” J. Quant. Spectrosc. Radiat. Transfer 112, 1–27 (2011).
[CrossRef]

K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized Lorenz–Mie theory,” Appl. Opt. 37, 4218–4225 (1998).
[CrossRef]

B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A. 5, 1427–1443 (1988).
[CrossRef]

Guan, B. R.

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

Han, Y. P.

Kong, J. A.

H. S. Chen, B.-I. Wu, B. L. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef]

Li, L. W.

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

Lock, J. A.

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,” J. Quant. Spectrosc. Radiat. Transfer 112, 1–27 (2011).
[CrossRef]

Maheu, B.

B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A. 5, 1427–1443 (1988).
[CrossRef]

Qiu, C. W.

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

Ren, K. F.

Schmidt, V.

V. Schmidt and T. Wriedt, “T-matrix method for biaxial anisotropic particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 1392–1397 (2009).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Wriedt, T.

Wu, B.-I.

H. S. Chen, B.-I. Wu, B. L. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef]

Wu, X. B.

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

Zhang, B. L.

H. S. Chen, B.-I. Wu, B. L. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef]

Zhang, H. Y.

Zhang, X. M.

X. X. Cheng, H. S. Chen, and X. M. Zhang, “Cloaking a perfectly conducting sphere with rotationally uniaxial nihility media in monostatic radar system,” Progress Electromagn. Res. 100, 285–298 (2010).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (1)

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

J. Opt. (1)

B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. 19, 59–67 (1988).
[CrossRef]

J. Opt. Soc. Am. A. (1)

G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A. 5, 1427–1443 (1988).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Quant. Spectrosc. Radiat. Transfer (2)

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,” J. Quant. Spectrosc. Radiat. Transfer 112, 1–27 (2011).
[CrossRef]

V. Schmidt and T. Wriedt, “T-matrix method for biaxial anisotropic particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 1392–1397 (2009).
[CrossRef]

Opt. Commun. (1)

G. Gouesbet, “T-matrix formulation and generalized Lorenz–Mie theories in spherical coordinates,” Opt. Commun. 283, 517–521 (2010).
[CrossRef]

Phys. Rev. A (1)

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

Phys. Rev. E. (1)

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

Phys. Rev. Lett. (1)

H. S. Chen, B.-I. Wu, B. L. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef]

Progress Electromagn. Res. (1)

X. X. Cheng, H. S. Chen, and X. M. Zhang, “Cloaking a perfectly conducting sphere with rotationally uniaxial nihility media in monostatic radar system,” Progress Electromagn. Res. 100, 285–298 (2010).
[CrossRef]

Other (1)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

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Figures (4)

Fig. 1.
Fig. 1.

Cartesian coordinate system Oxyz is parallel to the Gaussian beam coordinate system Oxyz, and the Cartesian coordinates of O in Oxyz are (x0,y0,z0). An anisotropic sphere is natural to Oxyz.

Fig. 2.
Fig. 2.

Comparison between the normalized differential scattering cross sections πσ(θ,0)/λ2 and πσ(θ,π/2)/λ2 for an isotropic sphere (μt=μr=μ0, εt=εr=1.77ε0, r1=1.5λ) and those for an anisotropic sphere (μt=μr=μ0, εt=1.77ε0, εr=1.47ε0, r1=1.5λ), both illuminated by a Gaussian beam (w0=5λ, x0=y0=1.5λ, z0=2λ).

Fig. 3.
Fig. 3.

Normalized differential backscattering cross sections πσ(π,ϕ)/λ2 versus RP values for different anisotropic spheres with (μt=μr=μ0, εt=1.77ε0, RP=RTM, r1=2λ) (solid line), (εt=εr=1.77ε0, μt=μ0, RP=RTE, r1=2λ) (dotted line), and (μt=μ0, εt=1.77ε0, RP=RTE=RTM, r1=2λ) (dashed line), all illuminated by a Gaussian beam (w0=5λ, x0=y0=0, z0=5λ).

Fig. 4.
Fig. 4.

Normalized differential backscattering cross sections πσ(π,ϕ)/λ2 versus RP values for different anisotropic spheres with (μt=μr=μ0, εt=2ε0, RTM=RP, r1=2λ) (solid line) and (μt=μr=μ0, εt=2ε0, RTM=RP, r1=2.5λ) (dotted line), both illuminated by a Gaussian beam (w0=5λ, x0=y0=0, z0=5λ).

Equations (40)

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Ei=E0n=1m=nnFnm[ign,TEmmmnr(1)(kr)+gn,TMmnmnr(1)(kr)],
Hi=E0η0n=1m=nnFnm[gn,TEmnmnr(1)(kr)ign,TMmmmnr(1)(kr)],
Fnm={Fnm0(1)|m|(n+|m|)!(n|m|)!Fnm<0,
Fn=in12n+1n(n+1).
Es=E0n=1m=nn[iamnmmnr(3)(kr)+bmnnmnr(3)(kr)],
Hs=E0η0n=1m=nn[amnnmnr(3)(kr)ibmnmmnr(3)(kr)],
×E=iωB,
×H=iωD,
D=ϵ¯¯·E,
B=μ¯¯·H,
ϵ¯¯=ϵrr^r^+ϵtθ^θ^+ϵtϕ^ϕ^,
μ¯¯=μrr^r^+μtθ^θ^+μtϕ^ϕ^.
D=×(r^ΦTE),
B=×(r^ΦTM).
ω2ΦTE×r^=×{μ¯¯1·×[ϵ¯¯1·(ΦTE×r^)]}.
1RTE2ΦTEr2+1r2sinθθ(sinθΦTEθ)+1r2sin2θ2ΦTEϕ2+1RTEkt2ΦTE=0,
ΦTE=rjvTE(n)(ktr)Pnm(cosθ)eimϕ,
[mmvTM(n)r(1)(ktr)mmvTE(n)r(1)(ktr)]=×[r^ΦTMr^ΦTE],
[nmvTM(n)r(1)(ktr)nmvTE(n)r(1)(ktr)]=1kt×[mmvTM(n)r(1)(ktr)mmvTE(n)r(1)(ktr)].
Ew=E0n=1m=nn[idmnmmvTE(n)(ktr)+cmnktωϵ¯¯1·nmvTM(n)(ktr)],
Hw=E0n=1m=nn[dmnktωμ¯¯1·nmvTE(n)(ktr)icmnmmvTM(n)(ktr)].
Eθi+Eθs=Eθw,Eϕi+Eϕs=EϕwHθi+Hθs=Hθw,Hϕi+Hϕs=Hϕw}atr=r1,
Fnmgn,TEmjn(kr1)+amnhn(1)(kr1)=dmnjvTE(n)(ktr1),
Fnmgn,TMm1kr1dd(kr1)[kr1jn(kr1)]+bmn1kr1dd(kr1)[kr1hn(1)(kr1)]=cmnηt1ktr1dd(ktr1)[ktr1jvTM(n)(ktr1)],
Fnmgn,TEm1kr1dd(kr1)[kr1jn(kr1)]+amn1kr1dd(kr1)[kr1hn(1)(kr1)]=dmnη0ηt1ktr1dd(ktr1)[ktr1jvTE(n)(ktr1)],
Fnmgn,TMmjn(kr1)+bmnhn(1)(kr1)=η0cmnjvTM(n)(ktr1),
amn=Fnmgn,TEm×jn(kr1)η0ηtkktdd(ktr1)[ktr1jvTE(n)(ktr1)]jvTE(n)(ktr1)dd(kr1)[kr1jn(kr1)]dd(kr1)[kr1hn(1)(kr1)]jvTE(n)(ktr1)hn(1)(kr1)η0ηtkktdd(ktr1)[ktr1jvTE(n)(ktr1)],
bmn=Fnmgn,TMm×ηtη0kktdd(ktr1)[ktr1jvTM(n)(ktr1)]jn(kr1)dd(kr1)[kr1jn(kr1)]jvTM(n)(ktr1)dd(kr1)[kr1hn(1)(kr1)]jvTM(n)(ktr1)ηtη0kkthn(1)(kr1)dd(ktr1)[ktr1jvTM(n)(ktr1)].
σ(θ,ϕ)=4πr2|EsE0|2=λ2π(|T1(θ,ϕ)|2+|T2(θ,ϕ)|2),
T1(θ,ϕ)=n=1m=nn(i)n[iamnmsinθPnm(cosθ)+bmndPnm(cosθ)dθ]eimϕ,
T2(θ,ϕ)=n=1m=nn(i)n1[iamndPnm(cosθ)dθ+bmnmsinθPnm(cosθ)]eimϕ.
[gn,TMm,locign,TEm,loc]=12(i)m1exp(ik0z0)Knmψ¯00{Jm1(2Q¯R0ρnw02)exp[i(m1)ϕ0]Jm+1(2Q¯R0ρnw02)exp[i(m+1)ϕ0]},
ψ¯00=iQ¯exp(iQ¯R02/w02)exp[iQ¯(n+0.5)2/(k02w02)],
Knm={(i)|m|i/(n+0.5)|m|1m0in(n+1)/(n+0.5)m=0,
R0=x02+y02,
tanϕ0=y0x0,
ρn=(n+0.5)/k0,
Q¯=1i2z0/(k0w02).
mmv(n)=imsinθjv(n)(ktr)Pnm(cosθ)eimϕθ^jv(n)(ktr)dPnm(cosθ)dθeimϕϕ^,
nmv(n)=jv(n)(ktr)ktrn(n+1)Pnm(cosθ)eimϕr^+1ktrdd(ktr)[ktrjv(n)(ktr)]dPnm(cosθ)dθeimϕθ^+1ktrdd(ktr)[ktrjv(n)(ktr)]imsinθPnm(cosθ)eimϕϕ^,

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