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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Franco Gori
  • Vol. 31, Iss. 1 — Jan. 1, 2014
  • pp: 89–100
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Scattering of an electromagnetic plane wave by a homogeneous sphere made of an orthorhombic dielectric–magnetic material

A. D. Ulfat Jafri and Akhlesh Lakhtakia  »View Author Affiliations


JOSA A, Vol. 31, Issue 1, pp. 89-100 (2014)
http://dx.doi.org/10.1364/JOSAA.31.000089


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Abstract

An exact transition matrix was formulated for electromagnetic scattering by an orthorhombic dielectric–magnetic sphere whose permeability dyadic is a scalar multiple of its permittivity dyadic. Calculations were made for plane waves incident on the sphere. As the size parameter increases, the role of anisotropy evolves; multiple lobes appear in the plots of the differential scattering efficiency in any scattering plane; the total scattering, extinction, and forward-scattering efficiencies exhibit a prominent maximum each; and the absorption efficiency generally increases with weak undulations. Certain orientations of the sphere with respect to the directions of propagation and the electric field of the incident plane wave make it highly susceptible to detection in a monostatic configuration, whereas other orientations make it much less vulnerable to detection. Impedance match to the ambient free space decreases backscattering efficiency significantly, although anisotropy prevents null backscattering.

© 2013 Optical Society of America

1. INTRODUCTION

The homogeneous sphere is ubiquitous in electromagnetic-scattering literature, whether the sphere is electrically small [1

1. O. F. Mossotti, “Recherches théoriques sur l’induction électro-statique, envisagée d’après idées de Faraday,” Supp. Biblio. Univer. Genève Arch. Sci. Phys. Natur. 16, 193–198 (1847).

3

3. M. Faraday, “Experimental relations of gold (and other materials) to light,” Philos. Trans. R. Soc. London 147, 145–181 (1857). [CrossRef]

] or electrically huge [4

4. A. Walther, “Optical applications of solid glass spheres,” Ph.D. thesis (Delft University of Technology, 1959).

,5

5. “Spherical glass solar energy generator by andre rawlemon,” August25, 2012, http://www.designboom.com/technology/spherical-glass-solar-energy-generator-by-rawlemon/ (accessed on June 3, 2013).

] or anywhere in between in size [6

6. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

,7

7. M. Kerker, ed., Selected Papers on Light Scattering, Part 1 (SPIE, 1988).

]. Exact analytical solutions are known for scattering by isotropic dielectric [8

8. L. V. Lorenz, “Lysvevægelsen i og uden for en af plane lysbølger belyst kugle,” K. Dan. Vidensk. Selsk. Forh. 6, 1–62 (1890).

10

10. G. Mie, “Contributions on the optics of turbid media, particularly colloidal metal solutions—translation,” Sandia Laboratories, Albuquerque, New Mexico, 1978, SAND78-6018. National Translation Center, Chicago, Illinois, Translation 79–21946.

], isotropic dielectric–magnetic [11

11. J. A. Stratton, Electromagnetic Theory (McGraw–Hill, 1941), Chap. 7.

,12

12. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), Chap. 4.

], and bi-isotropic spheres [13

13. C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974). [CrossRef]

], because vector spherical wavefunctions are available in closed form for these materials. However, similarly analytical solutions for scattering by anisotropic spheres remain unknown, because the relevant vector spherical wavefunctions in closed form have not been found.

Recently, however, closed-form vector spherical wavefunctions were found for orthorhombic dielectric–magnetic material with gyrotropic-like magnetoelectric properties [14

14. A. Lakhtakia and T. G. Mackay, “Vector spherical wavefunctions for orthorhombic dielectric-magnetic material with gyrotropic-like magnetoelectric properties,” J. Opt. 41, 201–213 (2012). [CrossRef]

], thereby extending our options in solving scattering problems analytically. For the simplest application of the newly found wavefunctions, we decided to calculate scattering by a homogeneous sphere of an orthorhombic dielectric–magnetic material described by the frequency-domain constitutive relations
D(r,ω)=ϵ0ϵr(ω)(ω)·(ω)·E(r,ω)B(r,ω)=μ0μr(ω)(ω)·(ω)·H(r,ω)},
(1)
where ϵ0 and μ0 are the permittivity and the permeability of free space, respectively; the scalars ϵr(ω) and μr(ω) are complex-valued scalar functions of the angular frequency ω; the diagonal dyadic,
(ω)=αx1(ω)x^x^+αy1(ω)y^y^+z^z^,
(2)
contains the real-valued scalars αx(ω)>0 and αy(ω)>0 that are also frequency dependent; and the Cartesian unit vectors are written as x^, y^, and z^.

Belonging to the magnetic point group D2h or D2h [15

15. T. G. Mackay and A. Lakhtakia, Electromagnetic Anisotropy and Bianisotropy: A Field Guide (World Scientific, 2010).

, p. 68], the chosen material is anisotropic but pathologically unirefringent because its permeability dyadic is a scalar multiple of its permittivity dyadic [16

16. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Plane waves and canonical sources in a gyroelectromagnetic uniaxial medium,” Int. J. Electron. 71, 853–861 (1991). [CrossRef]

]. Suppose that a plane wave propagates in an arbitrary direction in this material with wavenumber kc. Then, kc is bounded by either (i) k and kαx or (ii) k and kαy, where k=ωϵ0μ0ϵrμr(αxαy)1. In naturally occurring materials, both |αx21| and |αy21| are expected be very small (0.1), based on available data for solid dielectric crystals [17

17. C. D. Gribble and A. J. Hall, Optical Mineralogy, Principles and Practice (University College London Press, 1992).

]. However, both of these quantities could be larger for manufactured composite materials. An example is a composite material made by dispersing a mixture of electrically small springs that are aligned along any one of the three Cartesian axes, provided that every spring has a coaligned enantiomorph present in close proximity. Although simple techniques for making composite materials containing aligned springs [18

18. J. C. Bose, “On the rotation of plane of polarisation of electric waves by a twisted structure,” Proc. R. Soc. London, Ser. A 63, 146–152 (1898). [CrossRef]

] have been around for at least a century, nano/microscale additive manufacturing techniques [19

19. O. S. Ivanova, C. B. Williams, and T. A. Campbell, “Additive manufacturing (AM) and nanotechnology: promises and challenges,” Rapid Prototyping 19, 353–364 (2013). [CrossRef]

] should play a major role in the very near future toward engineering complex materials. Furthermore, suspensions of ferromagnetic grains in liquid crystals [20

20. F. Brochard and P. G. de Gennes, “Theory of magnetic suspensions in liquid crystals,” J. Phys. 31, 691–708 (1970). [CrossRef]

] are also candidate materials.

2. THEORY

Let a homogeneous sphere of radius a be located with its center at the origin of the spherical coordinate system (r,θ,ϕ). The sphere is made of a material that obeys the constitutive relations stated via Eqs. (1) and (2). The region r>a is vacuous.

A. Incident Electromagnetic Field

With the assumption that the source of the incident field lies in the region r>b>a, the incident electric field can be expressed as
Einc(r)=s{e,o}n=1m=0n{Dmn[Asmn(1)Msmn(1)(k0r)+Bsmn(1)Nsmn(1)(k0r)]},r<b,
(3)
and the incident magnetic field as
Hinc(r)=k0iωμ0s{e,o}n=1m=0n{Dmn[Asmn(1)Nsmn(1)(k0r)+Bsmn(1)Msmn(1)(k0r)]},r<b.
(4)
The vector spherical wavefunctions for free space used in these two equations are defined as [11

11. J. A. Stratton, Electromagnetic Theory (McGraw–Hill, 1941), Chap. 7.

,12

12. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), Chap. 4.

]
Memn(1)(k0r)=×[rjn(k0r)Pnm(cosθ)cos(mϕ)]Momn(1)(k0r)=×[rjn(k0r)Pnm(cosθ)sin(mϕ)]Nsmn(1)(k0r)=k01×Msmn(1)(k0r),s{e,o}},m{0,1,2,,n},n{1,2,3,}
(5)
with jn(·) denoting the spherical Bessel function of order n, and Pnm(·) the associated Legendre function of order n and degree m. The normalization factor
Dmn=(2δm0)(2n+1)(nm)!4n(n+1)(n+m)!
(6)
employs the Kronecker delta δpq.

The coefficients Asmn(1) and Bsmn(1) are presumed to be known. Suppose that the incident electromagnetic field is that of a plane wave with a propagation vector
kinc=k0(x^sinθinccosϕinc+y^sinθincsinϕinc+z^cosθinc)
(7)
with θinc[0,π] and ϕinc[0,2π). If the incident electric field is written as
Einc(r)=eincexp(ikinc·r)
(8)
with einc·kinc=0, then the incident magnetic field is
Hinc(r)=kinc×eincωμ0exp(ikinc·r).
(9)
The identity [22

22. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. II (McGraw-Hill, 1953).

, p. 1866]
eincexp(ikinc·r)=4s{e,o}n=1m=0n{inDnmn(n+1)[Csmn(θinc,ϕinc)Msmn(1)(k0r)iBsmn(θinc,ϕinc)Nsmn(1)(k0r)]},
(10)
where [22

22. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. II (McGraw-Hill, 1953).

, p. 1899]
Bemn(θ,ϕ)=1n(n+1)[dPnm(cosθ)dθcos(mϕ)θ^mPnm(cosθ)sinθsin(mϕ)ϕ^]Bomn(θ,ϕ)=1n(n+1)[dPnm(cosθ)dθsin(mϕ)θ^+mPnm(cosθ)sinθcos(mϕ)ϕ^]Cemn(θ,ϕ)=1n(n+1)[mPnm(cosθ)sinθsin(mϕ)θ^+dPnm(cosθ)dθcos(mϕ)ϕ^]Comn(θ,ϕ)=1n(n+1)[mPnm(cosθ)sinθcos(mϕ)θ^dPnm(cosθ)dθsin(mϕ)ϕ^]},
(11)
then yields
Asmn(1)=4inn(n+1)einc·Csmn(θinc,ϕinc)Bsmn(1)=4in1n(n+1)einc·Bsmn(θinc,ϕinc)}.
(12)

B. Scattered Electromagnetic Field

With the assumption that the field scattered by the sphere does not interact with (and modify) the source of the incident field, we can state the scattered electric and magnetic fields as follows:
Esca(r)=s{e,o}n=1m=0n{Dmn[Asmn(3)Msmn(3)(k0r)+Bsmn(3)Nsmn(3)(k0r)]},r>a,
(13)
Hsca(r)=k0iωμ0s{e,o}n=1m=0n{Dmn[Asmn(3)Nsmn(3)(k0r)+Bsmn(3)Msmn(3)(k0r)]},r>a,
(14)
In these expressions [11

11. J. A. Stratton, Electromagnetic Theory (McGraw–Hill, 1941), Chap. 7.

,12

12. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), Chap. 4.

],
Memn(3)(k0r)=×[rhn(1)(k0r)Pnm(cosθ)cos(mϕ)]Momn(3)(k0r)=×[rhn(1)(k0r)Pnm(cosθ)sin(mϕ)]Nsmn(3)(k0r)=k01×Msmn(3)(k0r),s{e,o}},m{0,1,2,,n},n{1,2,3,},
(15)
where hn(1)(·) is the spherical Hankel function of the first kind and order n, but the coefficients Asmn(3) and Bsmn(3) have to be determined by the solution of a boundary-value problem [14

14. A. Lakhtakia and T. G. Mackay, “Vector spherical wavefunctions for orthorhombic dielectric-magnetic material with gyrotropic-like magnetoelectric properties,” J. Opt. 41, 201–213 (2012). [CrossRef]

].

In the far zone, the relation
limk0r[k0rexp(ik0r)hn(1)(k0r)]=(i)n+1
(16)
leads to
limk0r[k0rexp(ik0r)Msmn(3)(k0r)]=(i)n+1n(n+1)Csmn(θ,ϕ)limk0r[k0rexp(ik0r)Nsmn(3)(k0r)]=(i)nn(n+1)Bsmn(θ,ϕ)}.
(17)
Accordingly, in the far zone the scattered electric field may be approximated as
Esca(r)Fsca(θ,ϕ)exp(ik0r)r
(18)
and the scattered magnetic field as
Hsca(r)η01r^×Fsca(θ,ϕ)exp(ik0r)r,
(19)
where r^=r/r and the vector far-field scattering amplitude
Fsca(θ,ϕ)=k01s{e,o}n=1m=0n{(i)nDmnn(n+1)[iAsmn(3)Csmn(θ,ϕ)+Bsmn(3)Bsmn(θ,ϕ)]}.
(20)

The foregoing expressions can be used to verify the satisfaction of the Sommerfeld radiation conditions [23

23. S. H. Schot, “Eighty years of Sommerfeld’s radiation condition,” Hist. Math. 19, 385–401 (1992).

]
limk0r|rEsca(r)ik0Esca(r)|O(r2)limk0r|rHsca(r)ik0Hsca(r)|O(r2)}
(21)
as well as of the Silver–Müller radiation conditions [24

24. A. Rubinowicz, “A weaker formulation of the electromagnetic radiation conditions,” Rep. Math. Phys. 2, 63–77 (1971). [CrossRef]

]
limk0r|Esca(r)+η0r^×Hsca(r)|O(r2)limk0r|η0Hsca(r)r^×Esca(r)|O(r2)}
(22)
by Eqs. (13) and (14).

C. Internal Electromagnetic Field

The electric and magnetic fields excited inside the sphere are represented by [14

14. A. Lakhtakia and T. G. Mackay, “Vector spherical wavefunctions for orthorhombic dielectric-magnetic material with gyrotropic-like magnetoelectric properties,” J. Opt. 41, 201–213 (2012). [CrossRef]

]
Eexc(r)=s{e,o}n=1m=0n[bsmnMsmn(r)+csmnNsmn(r)],r<a,
(23)
Hexc(r)=k0iωμ0ϵrμrs{e,o}n=1m=0n[bsmnNsmn(r)+csmnMsmn(r)],r<a,
(24)
where the coefficients bsmn and csmn are not known.

The vector spherical wavefunctions appearing in these expressions are given as
Msmn(r)=Jn(kr)f1(ϕ)1·{r^[f4(ϕ)f12(ϕ)f2(θ,ϕ)sinθcosθQsmn(θ,ϕ)(αxαy)sinθsinϕcosϕRsmn(θ,ϕ)]+θ^[f4(ϕ)cos2θ+f12(ϕ)sin2θf2(θ,ϕ)Qsmn(θ,ϕ)(αxαy)cosθsinϕcosϕRsmn(θ,ϕ)]+ϕ^[αxαyf2(θ,ϕ)cosθsinϕcosϕQsmn(θ,ϕ)f4(ϕ)Rsmn(θ,ϕ)]}
(25)
and
Nsmn(r)=1·(r^{Jn(kr)kr[cos2θ+f4(ϕ)sin2θf22(θ,ϕ)]Psmn(θ,ϕ)+Kn(kr)f1(ϕ)[f4(ϕ)f12(ϕ)f2(θ,ϕ)sinθcosθRsmn(θ,ϕ)+(αxαy)sinθsinϕcosϕQsmn(θ,ϕ)]}+θ^{Jn(kr)kr[f4(ϕ)1f22(θ,ϕ)sinθcosθ]Psmn(θ,ϕ)+Kn(kr)f1(ϕ)[f4(ϕ)cos2θ+f12(ϕ)sin2θf2(θ,ϕ)Rsmn(θ,ϕ)+(αxαy)cosθsinϕcosϕQsmn(θ,ϕ)]}+ϕ^{Jn(kr)kr[αxαyf22(θ,ϕ)sinθsinϕcosϕ]Psmn(θ,ϕ)+Kn(kr)f1(ϕ)[αxαyf2(θ,ϕ)cosθsinϕcosϕRsmn(θ,ϕ)+f4(ϕ)Qsmn(θ,ϕ)]}),
(26)
where
Jn(kr)=jn[krf2(θ,ϕ)],
(27)
Kn(kr)=n+1krf2(θ,ϕ)Jn(kr)Jn+1(kr),
(28)
Psmn(θ,ϕ)=n(n+1)Pnm[cosθf2(θ,ϕ)]Vsm(ϕ),
(29)
Qsmn(θ,ϕ)=mPnm[cosθf2(θ,ϕ)]f2(θ,ϕ)f1(ϕ)sinθUsm(ϕ),
(30)
Rsmn(θ,ϕ)=1f1(ϕ)sinθ{(nm+1)f2(θ,ϕ)Pn+1m[cosθf2(θ,ϕ)](n+1)cosθPnm[cosθf2(θ,ϕ)]}Vsm(ϕ),
(31)
Usm(ϕ)={sin[mf3(ϕ)]cos[mf3(ϕ)]},s={eo,
(32)
Vsm(ϕ)={cos[mf3(ϕ)]sin[mf3(ϕ)]},s={eo,
(33)
f1(ϕ)=+(αx2cos2ϕ+αy2sin2ϕ)1/2,
(34)
f2(θ,ϕ)=+[f12(ϕ)sin2θ+cos2θ]1/2,
(35)
f3(ϕ)=tan1(αyαxtanϕ),
(36)
f4(ϕ)=αxcos2ϕ+αysin2ϕ.
(37)
The angle f3(ϕ) must lie in the same quadrant as its argument. Let us also point out that Msmn(r) and Nsmn(r) are provided here in closed form and do not require the solution of any eigenvalue problem [25

25. G. W. Ford and S. A. Werner, “Scattering and absorption of electromagnetic waves by a gyrotropic sphere,” Phys. Rev. B 18, 6752–6769 (1978).

,26

26. J. L.-W. Li, W.-L. Ong, and K. H. R. Zheng, “Anisotropic scattering effects of a gyrotropic sphere characterized using the T-matrix method,” Phys. Rev. E 85, 036601 (2012). [CrossRef]

].

D. Solution of Boundary–Value Problem

Application of the standard boundary conditions,
θ^·Eexc(r)=θ^·[Einc(r)+Esca(r)]ϕ^·Eexc(r)=ϕ^·[Einc(r)+Esca(r)]θ^·Hexc(r)=θ^·[Hinc(r)+Hsca(r)]ϕ^·Hexc(r)=ϕ^·[Hinc(r)+Hsca(r)]},r=a,
(38)
across the closed surface r=a as well as the orthogonalities of the functions Bsmn(θ,ϕ) and Csmn(θ,ϕ) on the unit sphere leads to the following simultaneous algebraic equations for every combination of j{1,3}, s{e,o}, n{1,2,3,}, and m{0,1,2,,n} [14

14. A. Lakhtakia and T. G. Mackay, “Vector spherical wavefunctions for orthorhombic dielectric-magnetic material with gyrotropic-like magnetoelectric properties,” J. Opt. 41, 201–213 (2012). [CrossRef]

]:
Asmn(1)=s{e,o}n=1m=0n[Ismn,smn(1)bsmn+Jsmn,smn(1)csmn],
(39)
Bsmn(1)=s{e,o}n=1m=0n[Ksmn,smn(1)bsmn+Lsmn,smn(1)csmn],
(40)
Asmn(3)=s{e,o}n=1m=0n[Ismn,smn(3)bsmn+Jsmn,smn(3)csmn],
(41)
Bsmn(3)=s{e,o}n=1m=0n[Ksmn,smn(3)bsmn+Lsmn,smn(3)csmn].
(42)
In these equations, the quantities Ismn,smn(1), etc., are the integrals
Ismn,smn(j)=i(k0a)2π02πdϕ0πdθsinθ{Nsmn()(k0ar^)·[r^×Msmn(ar^)]+ϵrμrMsmn()(k0ar^)·[r^×Nsmn(ar^)]},
(43)
Jsmn,smn(j)=i(k0a)2π02πdϕ0πdθsinθ{Nsmn()(k0ar^)·[r^×Nsmn(ar^)]+ϵrμrMsmn()(k0ar^)·[r^×Msmn(ar^)]},
(44)
Ksmn,smn(j)=i(k0a)2π02πdϕ0πdθsinθ{Msmn()(k0ar^)·[r^×Msmn(ar^)]+ϵrμrNsmn()(k0ar^)·[r^×Nsmn(ar^)]},
(45)
Lsmn,smn(j)=i(k0a)2π02πdϕ0πdθsinθ{Msmn()(k0ar^)·[r^×Nsmn(ar^)]+ϵrμrNsmn()(k0ar^)·[r^×Msmn(ar^)]},
(46)
where j{1,3}, =j+2(mod4){3,1}, and r^=(x^cosϕ+y^sinϕ)sinθ+z^cosθ is the unit radial vector. Let us note here that the radial components of Msmn(ar^) and Nsmn(ar^) do not show up in the foregoing integrals.

After truncating the summations over n{1,2,3,} to n{1,2,3,,N} and restricting n{1,2,3,} to n{1,2,3,,N} similarly, Eqs. (39)–(42) can be written symbolically as the matrix equations
[A(1)B(1)]=[Y(1)][bc],[A(3)B(3)]=[Y(3)][bc],
(47)
whence
[A(3)B(3)]=[Y(3)][Y(1)]1[A(1)B(1)]=[T][A(1)B(1)].
(48)
In this equation, [T]=[Y(3)][Y(1)]1 is the T matrix of the chosen sphere embedded in free space. It has to be evaluated for increasing values of N until the various cross sections converge to preset tolerances.

E. Uniaxial Dielectric–Magnetic Sphere

When the sphere is made of a uniaxial dielectric–magnetic material, i.e., αx=αy, then the integrals (43)–(46) show symmetries with respect to the indices s and m as follows:
Ismn,smn(j)δss(δmm+δm,m1+δm,m+1),
(49)
Jsmn,smn(j)(1δss)(δmm+δm,m1+δm,m+1),
(50)
Ksmn,smn(j)(1δss)(δmm+δm,m1+δm,m+1),
(51)
Lsmn,smn(j)δss(δmm+δm,m1+δm,m+1).
(52)
Thus, all four of these integrals vanish if m and m differ by more than unity. Furthermore, Iemn,omn(j)=Iomn,emn(j)=Lemn,omn(j)=Lomn,emn(j)=0 and Jemn,emn(j)=Jomn,omn(j)=Kemn,emn(j)=Komn,omn(j)=0. These symmetries simplify the storage of and reduce the time to calculate the matrices [Y(j)] defined in Eqs. (47).

F. Isotropic Dielectric–Magnetic Sphere

When αx=αy=1, the symmetries (49)–(52) expand further to
Ismn,smn(j)δssδmmδnn,
(53)
Jsmn,smn(j)(1δss)δmmδnn,
(54)
Ksmn,smn(j)(1δss)δmmδnn,
(55)
Lsmn,smn(j)δssδmmδnn,
(56)
and these integrals can then be obtained analytically because the simplifications Msmn(r)=Msmn(1)(kr) and Nsmn(r)=Nsmn(1)(kr) emerge. We have verified that the usual Lorenz–Mie solution [12

12. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), Chap. 4.

] for an isotropic dielectric–magnetic sphere embedded in free space is recovered.

G. Scattering Cross Sections and Efficiencies

The plane wave scattering response is usually quantified through the differential scattering cross section [27

27. J. Van Bladel, Electromagnetic Fields (Hemisphere, 1985).

, Section 8.8]
σD(θ,ϕ)=4πFsca(θ,ϕ)·Fsca*(θ,ϕ)einc·einc*
(57)
and the total scattering cross section
σsca=1einc·einc*ϕ=02πθ=0π[Fsca(θ,ϕ)·Fsca*(θ,ϕ)]sinθdθdϕ,
(58)
=1einc·einc*πk02s{e,o}n=1m=0n[Dmn(|Asmn(3)|2+|Bsmn(3)|2)].
(59)
The differential scattering efficiency is defined as QD=σD/πa2, and the total scattering efficiency as Qsca=σsca/πa2.

H. Extinction and Absorption Cross Sections and Efficiencies

In order to quantify the removal of energy from the incident plane wave through both absorption and scattering, the extinction cross section
σext=4πk0Im[Fsca(θinc,ϕinc)·einc*einc·einc*]
(60)
is calculated. The extinction efficiency is defined as Qext=σext/πa2.

The excess of σext over σsca is the absorption cross section σabs, which is null valued if the sphere is made of a nondissipative material (i.e., both ϵr and μr are both positive and real). When σabs is normalized by πa2, the result is the absorption efficiency Qabs.

3. NUMERICAL RESULTS AND DISCUSSION

In order to compute the T matrix as well as the differential, total scattering, and extinction cross sections, we set up a Mathematica program. For that purpose, we made the replacement
s{e,o}n=1m=0ns{e,o}n=1Nm=0n
(61)
for the representation of all fields. The value of N would be incremented in steps of unity as the T matrix and the backscattering efficiency
Qb=σD(π+θinc,π+ϕinc)/πa2
(62)
were calculated for fixed {k0a,ϵr,μr,αx,αy,θinc,ϕinc}. The procedure would be terminated once the backscattering cross section σb=Qbπa2 had converged within a preset tolerance of 0.1%. In general, the larger the deviation of from and/or the larger the deviation of |ϵrμr| from unity, the greater the adequate value of N for converged results. For all results presented here, N11. The program was implemented on a laptop computer.

For partial validation of the Mathematica program, αx=αy=1 was set therein so that the scattering sphere was taken to be made of an isotropic dielectric–magnetic material. First, the forward-scattering efficiency,
Qf=σD(θinc,ϕinc)/πa2,
(63)
the backscattering efficiency, and the total scattering efficiency were checked for an isotropic dielectric sphere (μr=1) against data computed using the Lorenz–Mie solution and published in standard works [12

12. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), Chap. 4.

,28

28. V. V. Varadan, A. Lakhtakia, and V. K. Varadan, eds., Field Representations and Introduction to Scattering (North-Holland, 1991).

]. Second, a Mathematica program was written to implement the Lorenz–Mie solution for an isotropic dielectric–magnetic sphere (ϵr1, μr1) and was used to check the results yielded by the program written for the anisotropic dielectric–magnetic sphere. The latter program satisfied both tests.

Additional tests for validation involved symmetry. For example, for a uniaxial dielectric–magnetic sphere (αx=αy1), we found that the total scattering cross section does not change if (i) Einc=x^exp(ik0z) is replaced by Einc=y^exp(ik0z), (ii) Einc=z^exp(ik0x) is replaced by Einc=z^exp(ik0y), or (iii) Einc=x^exp(ik0y) is replaced by Einc=y^exp(ik0x). Similarly, we verified that σD(θ,0) for Einc=x^exp(ik0z) remains the same as σD(θ,π/2) for Einc=y^exp(ik0z), provided that the values of αx and αy are interchanged.

Finally, we also verified that the absorption cross section σabs is null valued if the sphere is made of a nondissipative material.

A. Rayleigh Scattering

Scattering by an electrically small object [6

6. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

, Section 6.4] is called Rayleigh scattering. Not only is it studied to explain the electromagnetic response of particulate materials [29

29. C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987). [CrossRef]

,30

30. C. F. Bohren, “Understanding colors in nature,” Pigment Cell Res. 1, 214–222 (1988).

], but it is also used to design composite materials [31

31. H. Frohlich, Theory of Dielectrics (Oxford University Press, 1958).

,32

32. L. Ward, The Optical Constants of Bulk Materials and Films (Adam Hilger, 1988).

].

A long-wavelength approximation is very useful to obtain closed-form analytical results for scattering by homogeneous and electrically small objects [33

33. A. Lakhtakia, “Rayleigh scattering by a bianisotropic ellipsoid in a biisotropic medium,” Int. J. Electron. 71, 1057–1062 (1991). [CrossRef]

]. Accordingly, scattering by an electrically small sphere described by the constitutive relations (1) is equivalent to radiation jointly by an electric dipole moment
peqvt=4πa3ϵ0(ϵr·)·(ϵr·+2)1·einc
(64)
and a magnetic dipole moment
meqvt=4πa3μ0(μr·)·(μr·+2)1·(kinc×eincωμ0),
(65)
both located at the centroid of the sphere, with denoting the identity dyadic. Therefore, the Rayleigh estimate of the vector far-field scattering amplitude is
FscaRayleigh(r^)=ω2μ04π[r^×(r^×peqvt)+ϵ0μ0r^×meqvt]=k02a3(r^×)·[(r^×)·(ϵr·)·(ϵr·+2)1+(μr·)·(μr·+2)1·(k^inc×)]·einc,
(66)
where k^inc=kinc/k0. The cross sections σDRayleigh(θ,ϕ), σscaRayleigh, and σextRayleigh can be calculated thereafter by using Eq. (66) in Eqs. (57), (58), and (60), respectively. These expressions can be expected to hold for [6

6. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

, Section 10.6.1]
k0aπ/5k0aϵrμrmax{αx1,αy1,1}π/5},
(67)
i.e., when the radius a is equal to or less than a tenth of the smallest wavelength in the ambient medium (vacuum) and the scattering material.

We computed the differential scattering, total scattering, and extinction efficiencies, exactly using Eq. (20) and approximately using Eq. (66), of an electrically small uniaxial dielectric–magnetic sphere (ϵr=4, μr=1.1, αx=αy=1.2) illuminated by a plane wave with Einc=x^exp(ik0z)Vm1. No distinction can be seen in the curves of Qsca and QscaRayleigh versus k0a(0,0.5] presented in Fig. 1(a). The same curves also hold for Einc=y^exp(ik0z)Vm1. Although the long-wavelength approximation is expected to be valid for k0a0.3 according to the inequalities (67), the Rayleigh expressions hold well in Fig. 1(a) for k0a as high as 0.5.

Fig. 1. Total scattering efficiency—computed exactly using Eq. (20) and approximately using Eq. (66)—as a function of k0a, when ϵr=4 and μr=1.1. (a) Einc=x^exp(ik0z)Vm1 and αx=αy=1.2. (b) Einc=x^exp(ik0z)Vm1, αx=1.1, and αy=1.2.

For Fig. 1(a), the sphere is transversely isotropic with respect to the direction of the incident plane wave, because αx=αy and k^incz^. By setting αx and αy unequal without any other changes, the condition of transverse isotropy can be changed. Figure 1(b) presents Qsca and QscaRayleigh versus k0a(0,0.5] for an electrically small biaxial dielectric–magnetic sphere (ϵr=4, μr=1.1, αx=1.1, αy=1.2) illuminated by a plane wave with Einc=x^exp(ik0z)Vm1. Clearly now, the long-wavelength approximation is highly satisfactory for k0a0.3, as predicted by the inequalities (67), but the discrepancies between the exact and the approximate results begin to grow as k0a increases further.

The biaxiality of the electrically small dielectric–magnetic sphere in Fig. 1(b) also enhances the roles of the directions of both einc and kinc. Comparing the plots of Qsca versus k0a(0,0.5] presented in
  • Figure 1(b) for Einc=x^exp(ik0z)Vm1,
  • Figure 2(a) for Einc=x^exp(ik0y)Vm1, and
  • Figure 2(b) for Einc=z^exp(ik0y)Vm1
for the same sphere (ϵr=4, μr=1.1, αx=1.1, αy=1.2), we see that the total scattering efficiency is significantly affected by the directions of both einc and kinc for k0a as low as the upper limit predicted by the inequalities (67).

Although we presented data in Subsection 3.A on only the total scattering efficiencies of electrically small spheres here, the conclusions drawn on the adequacy of the Rayleigh expression (66) therefrom are also applicable to the differential scattering and extinction efficiencies.

Fig. 2. Total scattering efficiency—computed exactly using Eq. (20) and approximately using Eq. (66)—as a function of k0a, when ϵr=4, μr=1.1, αx=1.1, and αy=1.2. (a) Einc=x^exp(ik0y)Vm1, (b) Einc=z^exp(ik0y)Vm1.

B. Differential Scattering Efficiency

In order to present representative results for differential scattering efficiency, we fixed k^incz^, ϵr=4, μr=1.1, αx=1.1, and αy=1.2. Plots of QD(θ,0°) and QD(θ,90°) are presented in Figs. 3 and 4 for k0a{0.5,2.5,4.5} and either e^incx^ or e^incy^, where e^inc=einc/|einc|. A very notable feature of these plots is the appearance of multiple lobes as the size parameter k0a increases. This happens in any scattering plane, as exemplified by the plots of QD versus θ in the ϕ=0° and ϕ=90° planes. However, when the electrical size is small, lobes may not appear in certain scattering planes—e.g., in the ϕ=90° plane in Fig. 3(b) and the ϕ=0° plane in Fig. 4(a). Even though the sphere is made of an anisotropic material lacking transverse isotropy with respect to the direction of propagation of the incident plane wave, the sphere appears to be isotropic just because it is electrically small. As the size parameter increases, anisotropy asserts its presence.

Fig. 3. Differential scattering efficiency as a function of θ, when Einc=x^exp(ik0z)Vm1, ϵr=4, μr=1.1, αx=1.1, and αy=1.2. The dashed red line is for k0a=0.5, the solid blue line is for k0a=2.5, and the dashed-and-dotted black line is for k0a=4.5. (a) ϕ=0°, (b) ϕ=90°.
Fig. 4. Same as Fig. 3, except that Einc=y^exp(ik0z)Vm1.

With the size parameter fixed at k0a=4.5, we also compared the effects of uniaxiality and biaxiality. Figures 5 and 6 show plots of QD(θ,0°) and QD(θ,90°), when k^incz^, either e^incx^ or e^incy^, ϵr=4, and μr=1.1. The sphere is transversely isotropic with respect to the direction of propagation of the incident plane wave if the material is uniaxial (αx=αy=1.1), but not if it is biaxial (αx=1.1, αy=1.2). The curve of QD(θ,0°) versus θ for e^incx^ in Fig. 5(a) is identical to that of QD(θ,90°) versus θ for e^incy^ in Fig. 6(b), and the curve of QD(θ,90°) versus θ for e^incx^ in Fig. 5(b) is identical to that of QD(θ,0°) versus θ for e^incy^ in Fig. 6(a), for the uniaxial sphere. This symmetry is not evident for the biaxial sphere.

Fig. 5. Differential scattering efficiency as a function of θ, when Einc=x^exp(ik0z)Vm1, k0a=4.5, ϵr=4, and μr=1.1. The red dashed line is for αx=αy=1.1, and the solid blue line is for αx=1.1 and αy=1.2. (a) ϕ=0°, (b) ϕ=90°.
Fig. 6. Same as Fig. 5, except that Einc=y^exp(ik0z)Vm1.

C. Total Scattering and Extinction Efficiencies

The total scattering and extinction efficiencies calculated using Eqs. (58) and (60), respectively, are shown in Figs. 79 as functions of k0a for six combinations of the directions of kinc and einc. The sphere was taken to be dissipative and biaxial: ϵr=4(1+i0.1), μr=1.1, αx=1.1, and αy=1.2.

Fig. 7. Extinction, total scattering, absorption, backscattering, and forward-scattering efficiencies as functions of k0a, when k^incz^, ϵr=4(1+i0.1), μr=1.1, αx=1.1, and αy=1.2. (a) e^incx^, (b) e^incy^.
Fig. 8. Same as Fig. 7, except that k^incx^. (a) e^incy^, (b) e^incz^.
Fig. 9. Same as Fig. 7, except that k^incy^. (a) e^incx^, (b) e^incz^.

Because of dissipation, Qext>Qsca for all k0a>0 in these figures. Furthermore, Qext tracks Qsca so well that the absorption efficiency Qabs rises steadily with k0a[0,2] and becomes a weak function of k0a[2,4.5], regardless of the directions of kinc and einc.

All six panels in Figs. 79 clearly indicate that Qext and Qsca are maximum at some k0a[2.2,3.2]. The plots suggest that both Qext and Qsca would settle down to show a weak dependence on k0a as that parameter increases further—just like an isotropic sphere [12

12. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), Chap. 4.

, Section 4.4]—but calculations could not be carried out on a laptop computer for k0a>4.5 in a reasonable period of time. This limitation arises from the functions Msmn(r) and Nsmn(r) not displaying orthogonality properties on the surface of any sphere centered at the origin when [14

14. A. Lakhtakia and T. G. Mackay, “Vector spherical wavefunctions for orthorhombic dielectric-magnetic material with gyrotropic-like magnetoelectric properties,” J. Opt. 41, 201–213 (2012). [CrossRef]

]. We conjectured from Figs. 79 that the highest peaks of Qext and Qsca versus k0a are obtained when the direction of propagation of the incident plane wave is such that the scalar k^inc··k^inc has a minimum value. This conjecture was verified by several calculations for different values of αx(1,1.5] and αy(1,1.5].

In contrast to the clear maximums of both Qext and Qsca versus k0a in Figs. 79 the absorption efficiency Qabs—despite some undulations—shows a generally increasing trend with k0a for six combinations of the directions of kinc and einc. In all six panels in these figures, Qsca first increases slower than Qext and then decreases faster than Qext, as k0a increases.

D. Backscattering and Forward-Scattering Efficiencies

The backscattering and forward-scattering efficiencies computed using Eqs. (62) and (63), respectively, are also shown in Figs. 79 as functions of k0a for six combinations of the directions of kinc and einc, when ϵr=4(1+i0.1), μr=1.1, αx=1.1, and αy=1.2.

We observe that Qf keeps on generally increasing as the size parameter k0a increases up to a limit and then exhibits undulations with further increases of k0a. The undulations are most pronounced for the smallest value of k^inc··k^inc.

In contrast to Qf, the plots of Qb versus k0a are oscillatory regardless of the directions of kinc and einc. Monostatic detection of the sphere would be easier at frequencies where Qb peaks, but it will be difficult at frequencies for which Qb reduces to a minimum, which can be substantially smaller than the adjacent peak values. In this characteristic, a sphere made of the chosen material responds similarly to a sphere made of an isotropic achiral material [28

28. V. V. Varadan, A. Lakhtakia, and V. K. Varadan, eds., Field Representations and Introduction to Scattering (North-Holland, 1991).

, p. 246]. But the anisotropy of the chosen material shows its effect too. Higher Qb peaks are favored by the highest value of e^inc··e^inc, as can be concluded from Fig. 10. Even more importantly, the backscattering efficiency of a sphere of a specific size parameter can be halved from its maximum value simply by reorienting the sphere with respect to the directions of the wave vector and the electric field of the incident plane wave. Thus, certain orientations of the sphere make it highly susceptible to detection in a monostatic configuration, whereas other orientations make it much less vulnerable to detection.

Fig. 10. Backscattering efficiency as a function of k0a, when ϵr=4(1+i0.1), μr=1.1, αx=1.1, and αy=1.2, for six different combinations of e^inc and k^inc.

E. Impedance Match

When ϵr=μr, the chosen material can be said to be impedance-matched to the ambient medium (i.e., free space). If = additionally, the sphere would be made of an isotropic dielectric–magnetic material, and then Qb0 by virtue of a theorem of Weston [34

34. V. H. Weston, “Theory of absorbers in scattering,” IEEE Trans. Antennas Propag. 11, 578–584 (1963). [CrossRef]

]. Figure 11 shows plots of Qb versus k0a for six different combinations of k^inc and e^inc, when ϵr=μr=2(1+i0.1), αx=1.1, and αy=1.2. Whereas Qb is independent of e^inc in Fig. 11, it does depend on k^inc. Comparing with Fig. 10, we conclude that the independence of Qb from e^inc is due to the impedance match with the ambient free space.

Fig. 11. Same as Fig. 10, except that ϵr=μr=2(1+i0.1), αx=1.1, and αy=1.2.

Even though the values of k/k0 are quite similar in Figs. 10 and 11, a significant consequence of impedance match to the ambient free space is a tenfold decrease in the backscattering efficiency. That Qb did not actually drop to zero is because anisotropy makes the chosen sphere noninvariant under a rotation by π/2 about an axis parallel to the direction of incidence, a condition required by Weston’s theorem [34

34. V. H. Weston, “Theory of absorbers in scattering,” IEEE Trans. Antennas Propag. 11, 578–584 (1963). [CrossRef]

] for null backscattering.

4. CONCLUDING REMARKS

Electromagnetic scattering by an orthorhombic dielectric–magnetic sphere whose permeability dyadic is a scalar multiple of its permittivity dyadic was formulated exactly in terms of a transition matrix, simplifying a recent development wherein closed-form vector spherical wavefunctions were formulated to capture anisotropy. Calculating the total scattering, extinction, differential scattering, backscattering, and forward-scattering efficiencies, we examined the evolving role of anisotropy as the size parameter increases.

Multiple lobes appear in the plots of the differential scattering efficiency in any scattering plane, as the size parameter increases. However, lobes may not appear in certain scattering planes when the sphere is electrically small in size. As the size parameter increases, the total scattering, extinction, and forward-scattering efficiencies exhibit a prominent maximum each, but the absorption efficiency generally increases with weak undulations. Certain orientations of the sphere with respect to the directions of propagation and the electric field of the incident plane wave make it highly susceptible to detection in a monostatic configuration, whereas other orientations make it much less vulnerable to detection. Finally, the backscattering efficiency can be reduced significantly, if the sphere is impedance-matched to the ambient free space.

ACKNOWLEDGMENTS

A. D. U. J. acknowledges the support of the Higher Education Commission of Pakistan for sponsoring a six-month visit to the Pennsylvania State University, where this work was done. A. L. is grateful to the Charles Godfrey Binder Endowment at the Pennsylvania State University for ongoing support of his research.

REFERENCES

1.

O. F. Mossotti, “Recherches théoriques sur l’induction électro-statique, envisagée d’après idées de Faraday,” Supp. Biblio. Univer. Genève Arch. Sci. Phys. Natur. 16, 193–198 (1847).

2.

O. F. Mossotti, “Discussione analitica sull influenza che l’azione di un mezzo dielettrico ha sulla distribuzione dell’elettricita alla superficie di più corpi elettrici disseminati in esso,” Mem. Mat. Fis. Modena 24, 49–74 (1850).

3.

M. Faraday, “Experimental relations of gold (and other materials) to light,” Philos. Trans. R. Soc. London 147, 145–181 (1857). [CrossRef]

4.

A. Walther, “Optical applications of solid glass spheres,” Ph.D. thesis (Delft University of Technology, 1959).

5.

“Spherical glass solar energy generator by andre rawlemon,” August25, 2012, http://www.designboom.com/technology/spherical-glass-solar-energy-generator-by-rawlemon/ (accessed on June 3, 2013).

6.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

7.

M. Kerker, ed., Selected Papers on Light Scattering, Part 1 (SPIE, 1988).

8.

L. V. Lorenz, “Lysvevægelsen i og uden for en af plane lysbølger belyst kugle,” K. Dan. Vidensk. Selsk. Forh. 6, 1–62 (1890).

9.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. Lpz. 25, 377–445 (1908).

10.

G. Mie, “Contributions on the optics of turbid media, particularly colloidal metal solutions—translation,” Sandia Laboratories, Albuquerque, New Mexico, 1978, SAND78-6018. National Translation Center, Chicago, Illinois, Translation 79–21946.

11.

J. A. Stratton, Electromagnetic Theory (McGraw–Hill, 1941), Chap. 7.

12.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), Chap. 4.

13.

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974). [CrossRef]

14.

A. Lakhtakia and T. G. Mackay, “Vector spherical wavefunctions for orthorhombic dielectric-magnetic material with gyrotropic-like magnetoelectric properties,” J. Opt. 41, 201–213 (2012). [CrossRef]

15.

T. G. Mackay and A. Lakhtakia, Electromagnetic Anisotropy and Bianisotropy: A Field Guide (World Scientific, 2010).

16.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Plane waves and canonical sources in a gyroelectromagnetic uniaxial medium,” Int. J. Electron. 71, 853–861 (1991). [CrossRef]

17.

C. D. Gribble and A. J. Hall, Optical Mineralogy, Principles and Practice (University College London Press, 1992).

18.

J. C. Bose, “On the rotation of plane of polarisation of electric waves by a twisted structure,” Proc. R. Soc. London, Ser. A 63, 146–152 (1898). [CrossRef]

19.

O. S. Ivanova, C. B. Williams, and T. A. Campbell, “Additive manufacturing (AM) and nanotechnology: promises and challenges,” Rapid Prototyping 19, 353–364 (2013). [CrossRef]

20.

F. Brochard and P. G. de Gennes, “Theory of magnetic suspensions in liquid crystals,” J. Phys. 31, 691–708 (1970). [CrossRef]

21.

K. Aydin and A. Hizal, “On the completeness of the spherical vector wave functions,” J. Math. Anal. Appl. 117, 428–440 (1986). [CrossRef]

22.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. II (McGraw-Hill, 1953).

23.

S. H. Schot, “Eighty years of Sommerfeld’s radiation condition,” Hist. Math. 19, 385–401 (1992).

24.

A. Rubinowicz, “A weaker formulation of the electromagnetic radiation conditions,” Rep. Math. Phys. 2, 63–77 (1971). [CrossRef]

25.

G. W. Ford and S. A. Werner, “Scattering and absorption of electromagnetic waves by a gyrotropic sphere,” Phys. Rev. B 18, 6752–6769 (1978).

26.

J. L.-W. Li, W.-L. Ong, and K. H. R. Zheng, “Anisotropic scattering effects of a gyrotropic sphere characterized using the T-matrix method,” Phys. Rev. E 85, 036601 (2012). [CrossRef]

27.

J. Van Bladel, Electromagnetic Fields (Hemisphere, 1985).

28.

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, eds., Field Representations and Introduction to Scattering (North-Holland, 1991).

29.

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987). [CrossRef]

30.

C. F. Bohren, “Understanding colors in nature,” Pigment Cell Res. 1, 214–222 (1988).

31.

H. Frohlich, Theory of Dielectrics (Oxford University Press, 1958).

32.

L. Ward, The Optical Constants of Bulk Materials and Films (Adam Hilger, 1988).

33.

A. Lakhtakia, “Rayleigh scattering by a bianisotropic ellipsoid in a biisotropic medium,” Int. J. Electron. 71, 1057–1062 (1991). [CrossRef]

34.

V. H. Weston, “Theory of absorbers in scattering,” IEEE Trans. Antennas Propag. 11, 578–584 (1963). [CrossRef]

OCIS Codes
(160.1190) Materials : Anisotropic optical materials
(260.2110) Physical optics : Electromagnetic optics
(290.5825) Scattering : Scattering theory

ToC Category:
Scattering

History
Original Manuscript: October 8, 2013
Revised Manuscript: November 8, 2013
Manuscript Accepted: November 8, 2013
Published: December 10, 2013

Citation
A. D. Ulfat Jafri and Akhlesh Lakhtakia, "Scattering of an electromagnetic plane wave by a homogeneous sphere made of an orthorhombic dielectric–magnetic material," J. Opt. Soc. Am. A 31, 89-100 (2014)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-31-1-89


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References

  1. O. F. Mossotti, “Recherches théoriques sur l’induction électro-statique, envisagée d’après idées de Faraday,” Supp. Biblio. Univer. Genève Arch. Sci. Phys. Natur. 16, 193–198 (1847).
  2. O. F. Mossotti, “Discussione analitica sull influenza che l’azione di un mezzo dielettrico ha sulla distribuzione dell’elettricita alla superficie di più corpi elettrici disseminati in esso,” Mem. Mat. Fis. Modena 24, 49–74 (1850).
  3. M. Faraday, “Experimental relations of gold (and other materials) to light,” Philos. Trans. R. Soc. London 147, 145–181 (1857). [CrossRef]
  4. A. Walther, “Optical applications of solid glass spheres,” Ph.D. thesis (Delft University of Technology, 1959).
  5. “Spherical glass solar energy generator by andre rawlemon,” August25, 2012, http://www.designboom.com/technology/spherical-glass-solar-energy-generator-by-rawlemon/ (accessed on June 3, 2013).
  6. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).
  7. M. Kerker, ed., Selected Papers on Light Scattering, Part 1 (SPIE, 1988).
  8. L. V. Lorenz, “Lysvevægelsen i og uden for en af plane lysbølger belyst kugle,” K. Dan. Vidensk. Selsk. Forh. 6, 1–62 (1890).
  9. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. Lpz. 25, 377–445 (1908).
  10. G. Mie, “Contributions on the optics of turbid media, particularly colloidal metal solutions—translation,” Sandia Laboratories, Albuquerque, New Mexico, 1978, SAND78-6018. National Translation Center, Chicago, Illinois, Translation 79–21946.
  11. J. A. Stratton, Electromagnetic Theory (McGraw–Hill, 1941), Chap. 7.
  12. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), Chap. 4.
  13. C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974). [CrossRef]
  14. A. Lakhtakia and T. G. Mackay, “Vector spherical wavefunctions for orthorhombic dielectric-magnetic material with gyrotropic-like magnetoelectric properties,” J. Opt. 41, 201–213 (2012). [CrossRef]
  15. T. G. Mackay and A. Lakhtakia, Electromagnetic Anisotropy and Bianisotropy: A Field Guide (World Scientific, 2010).
  16. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Plane waves and canonical sources in a gyroelectromagnetic uniaxial medium,” Int. J. Electron. 71, 853–861 (1991). [CrossRef]
  17. C. D. Gribble and A. J. Hall, Optical Mineralogy, Principles and Practice (University College London Press, 1992).
  18. J. C. Bose, “On the rotation of plane of polarisation of electric waves by a twisted structure,” Proc. R. Soc. London, Ser. A 63, 146–152 (1898). [CrossRef]
  19. O. S. Ivanova, C. B. Williams, and T. A. Campbell, “Additive manufacturing (AM) and nanotechnology: promises and challenges,” Rapid Prototyping 19, 353–364 (2013). [CrossRef]
  20. F. Brochard and P. G. de Gennes, “Theory of magnetic suspensions in liquid crystals,” J. Phys. 31, 691–708 (1970). [CrossRef]
  21. K. Aydin and A. Hizal, “On the completeness of the spherical vector wave functions,” J. Math. Anal. Appl. 117, 428–440 (1986). [CrossRef]
  22. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. II (McGraw-Hill, 1953).
  23. S. H. Schot, “Eighty years of Sommerfeld’s radiation condition,” Hist. Math. 19, 385–401 (1992).
  24. A. Rubinowicz, “A weaker formulation of the electromagnetic radiation conditions,” Rep. Math. Phys. 2, 63–77 (1971). [CrossRef]
  25. G. W. Ford and S. A. Werner, “Scattering and absorption of electromagnetic waves by a gyrotropic sphere,” Phys. Rev. B 18, 6752–6769 (1978).
  26. J. L.-W. Li, W.-L. Ong, and K. H. R. Zheng, “Anisotropic scattering effects of a gyrotropic sphere characterized using the T-matrix method,” Phys. Rev. E 85, 036601 (2012). [CrossRef]
  27. J. Van Bladel, Electromagnetic Fields (Hemisphere, 1985).
  28. V. V. Varadan, A. Lakhtakia, and V. K. Varadan, eds., Field Representations and Introduction to Scattering (North-Holland, 1991).
  29. C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987). [CrossRef]
  30. C. F. Bohren, “Understanding colors in nature,” Pigment Cell Res. 1, 214–222 (1988).
  31. H. Frohlich, Theory of Dielectrics (Oxford University Press, 1958).
  32. L. Ward, The Optical Constants of Bulk Materials and Films (Adam Hilger, 1988).
  33. A. Lakhtakia, “Rayleigh scattering by a bianisotropic ellipsoid in a biisotropic medium,” Int. J. Electron. 71, 1057–1062 (1991). [CrossRef]
  34. V. H. Weston, “Theory of absorbers in scattering,” IEEE Trans. Antennas Propag. 11, 578–584 (1963). [CrossRef]

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