PHYSICAL REVIEW E 78, 046609 共2008兲

Electromagnetic transparency by coated spheres with radial anisotropy L. Gao,1,2,* T. H. Fung,1 K. W. Yu,1,† and C. W. Qiu3,‡

1

Department of Physics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong, China Jiangsu Key Laboratory of Thin Films, Department of Physics, Soochow University, Suzhou 215006, China 3 Department of Electrical and Computer Engineering, National University of Singapore, Kent Ridge, 119260, Singapore 共Received 30 May 2008; published 22 October 2008兲 2

We establish an account of electromagnetic scattering by coated spheres with radial dielectric and magnetic anisotropy. Within full-wave scattering theory, we show that the total scattering cross section Qs is strongly dependent on both the dielectric anisotropy and magnetic anisotropy. As a consequence, by a suitable adjustment of the radius ratio, one may make the anisotropic coated particle nearly transparent or invisible. In the quasistatic case, we take one step forward to derive the effective permittivity and permeability for the coated particle, and the near-zero scattering radius ratio can be well described within effective medium theory. To one’s interest, the introduction of radial anisotropy is helpful to achieve better transparency quality such as a much smaller Qs and wider range of near-zero scattering ratio. Moreover, when the coated particle is anisotropic, the position of the near-zero scattering radius ratio can be tunable, resulting in a tunable electromagnetic cloaking. DOI: 10.1103/PhysRevE.78.046609

PACS number共s兲: 41.20.Jb, 42.25.Fx, 42.25.Bs

I. INTRODUCTION

The creation of an electromagnetic cloak of invisibility has received much attention in recent years because of its potential applications in nanotechnology and engineering. For instance, planes and weapons with cloaks may be invisible to radar, which is very important for military purposes. To achieve “invisibility” or “low observability” for an object in electromagnetic waves, various methods or schemes were put forward such as the coordinate transformation 关1–3兴, tunneling light transmittance 关4,5兴, partial resonance 关6,7兴, and zero-scattering mechanism in the dipolar limit 关8兴. Later, Cai et al. proposed the design of a nonmagnetic cylindrical cloak operating at optical frequencies based on a coordinate transformation 关9,10兴. The optical cloak is of great potential interest and brings us one step closer to the ultimate illusion of optical invisibility. In addition, based on Mie scattering theory 关11,12兴, the use of coating materials with metamaterials or plasmonic materials can drastically reduce the total scattering section of spherical or cylindrical objects, and hence make the objects “invisible” or “transparent” 关13兴. Since the realization of transparency relies on the nonresonant mechanism, it is almost invariant with the change of the shape, geometrical, and electromagnetic properties of the cloaked object 关14兴. Further investigation on cloaking and transparency was made for more realistic systems such as collections of particles with metamaterial and plasmonic covers 关15兴, multilayered spheres, coated spheroids, and two-phase random mixtures 关16兴. In the quasistatic case, the transparency condition, under which the total scattering section of the composite particles is zero, was derived based on “neutral inclusion” idea 关16兴. Moreover, in metal and dielectric microspheres, with

*Corresponding author: [email protected] †

[email protected] [email protected]

‡

1539-3755/2008/78共4兲/046609共11兲

the proper design of the metal and dielectric shell, the dispersion spectra of the system can be tailored to make the forward-scattering cross section suppressed, resulting in plasmon-assisted transparency 关17兴. More recently, achieving transparency and maximizing scattering with metamaterialcoated conducting cylinders has been given 关18兴. In this paper, in order to achieve better transparency or invisibility, we would like to consider coated spheres with radial anisotropy in physical properties including both permittivity and permeability tensors. To one’s interest, here the anisotropic tensors are assumed to be radially anisotropic; i.e., they are diagonal in spherical coordinates with values ⑀r 共␮r兲 in the radial direction and ⑀t 共␮t兲 in the tangential directions. Actually, such kind of anisotropy was indeed found in phospholipid vesicle systems 关19,20兴 and in cell membranes containing mobile charges 关21,22兴. Furthermore, the radial anisotropy can be easily established from a problem of graphitic multishells 关23兴, spherically stratified medium 关24兴. In the quasistatic limit, the third-order nonlinear optical susceptibility in graded mixtures 关25兴 and the second and thirdharmonic generations for a suspensions of coated particles 关26兴 were investigated, and it was found that the choice of radial anisotropy plays a role in determining the magnitude of nonlinearity enhancement and resonant frequencies 关25,26兴. Motivated by the recent progress in an analytical demonstration of perfect invisibility for Pendry’s cloak 关27兴, the interactions of electromagnetic waves with the coated sphere of radial anisotropies in both electric and magnetric parameters have been established based on full-wave electromagnetic scattering theory 关11兴. Incidentally, scattering by solid particles of radial anisotropy was investigated analytically and numerically for parametric studies by the concept anisotropy ratio 关28兴. More recently, peculiar light scattering and the role of anisotropy in plasmonic resonances were studied 关29兴, and dyadic Green’s functions for arbitrarily mulitlayered radially anisotropic spheres were established 关30兴. In addition to the analytical establishment of Debye potentials and the scattering coefficients, we aim at the effects

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©2008 The American Physical Society

PHYSICAL REVIEW E 78, 046609 共2008兲

GAO et al.

field vector normal to and tangent to the local optical axis for i = c , s. For a harmonic electromagnetic wave 共i.e., E ⬃ e−i␻t兲, with Maxwell equations, the time-independent parts of the local electric and magnetic fields are written as

Z

a

b


cr,
ct

x

cr, ct sr, st

incident wave

共3兲

of anisotropic parameters in the core and/or the shell on the reduction of the total scattering section, so as to make the objects nearly “transparent” or “invisible.” In the quasistatic limit, we present an effective medium theory that simultaneously determines the effective permittivity and permeability of coated particles with radial anisotropy. As a consequence, the approximate transparency conditions can be derived as a first step to design the reduction of total scattering section of coated particles whose dimensions are comparable with the wavelength of operation. We turn now to the body of the paper. We derive the wave equations for the coated sphere with dielectric and magnetic anisotropies in both the core and the shell in Sec. II. In Sec. III, from the boundary conditions, Mie scattering coefficients are determined and the far-field solution is given. An effective medium theory for radially anisotropic magnetodielectric coated spheres is proposed and the near-transparency condition is derived in the quasistatic limit in Sec. IV. In Sec. V, numerical results are shown. The paper ends with a discussion and conclusion in Sec. VI.

冊

冊

We consider electromagnetic scattering of a plane wave by a coated spherical particle with radial anisotropy when the polarized wave with unit amplitude Ei = exeik0z is incident upon it 共see Fig. 1兲, where k0 ⬅ ␻冑⑀0␮0 = ␻ / c and ⑀0 and ␮0 are the permittivity and permeability for a vacuum. The coated particle is composed of a core of radius a and permitJ c兲 and a shell of radius b tivity 共permeability兲 tensors J ⑀c 共␮ J s兲. Here we assume that the core and the and tensors J ⑀s 共␮ shell are a kind of rotationally uniaxial material characterized by radial anisotropy,

冣

␮ir 0 0 J i = 0 ␮it 0 , ␮ 0 0 ␮it

共1兲

where ⑀ir 共␮ir兲 and ⑀it 共␮it兲 stand for the permittivity 共permeability兲 elements corresponding to the electric- 共magnetic-兲

冊

1 ⳵共rH␪兲 ⳵Hr − = − i␻⑀itE␾ r ⳵r ⳵␪

and

冉 冉 冉

共4兲

冊

1 ⳵共rE␾ sin ␪兲 ⳵共rE␪兲 − = − i␻␮irHr , r sin ␪ ⳵␪ ⳵␾ 2

冊

1 ⳵Er ⳵共rE␾ sin ␪兲 − = − i␻␮itH␪ , r sin ␪ ⳵␾ ⳵r

冊

1 ⳵共rE␪兲 ⳵Er − = − i␻␮itH␾ . r ⳵r ⳵␪

共5兲

We shall solve Eqs. 共4兲 and 共5兲 together with boundary conditions including the continuities of E␪, E␾, H␪, and H␾. Actually, the solution of the above equations can be regarded as a superposition of two linearly independent fields such as 共ETM , HTM兲 and 共ETE , HTE兲, possessing the properties Er,TM = Er, Hr,TM = 0 for transverse magnetic 共TM兲 waves and Er,TE = 0, Hr,TE = Hr for transverse electric 共TE兲 ones 关32,34兴. Then the Debye potential for the TM case ⌽TM can be defined as 关31兴

II. EQUATIONS FOR DEBYE POTENTIALS AND FIELDS

冣 冢

冉 冉 冉

1 ⳵Hr ⳵共rH␾ sin ␪兲 − = − i␻⑀itE␪ , r sin ␪ ⳵␾ ⳵r

E

FIG. 1. Geometry of scattering of a plane wave by a coated sphere with permittivity and permeability tensors.

冢

J iH. ⵱ ⫻ E = i␻␮

⳵共rH␾ sin ␪兲 ⳵共rH␪兲 1 − = − i␻⑀irEr , r sin ␪ ⳵␪ ⳵␾ 2

k

⑀ir 0 0 J ⑀i = 0 ␧it 0 , 0 0 ⑀it

共2兲

In spherical polar coordinates, Eqs. 共1兲 and 共2兲 become 关31–33兴


sr,
st
0, 0

⵱ ⫻ H = − i␻J ⑀iE,

E␪,TM =

H␪,TM = −

1 ⳵2共r⌽TM兲 , r ⳵r⳵␪

E␾,TM =

i␻⑀it ⳵共r⌽TM兲 , r sin ␪ ⳵␾

1 ⳵2共r⌽TM兲 , r sin ␪ ⳵r⳵␾

H␾,TM =

i␻⑀it ⳵共r⌽TM兲 . 共6兲 ⳵␪ r

By means of Eqs. 共2兲–共6兲, it can be shown that ⌽TM satisfies

冉

冊

1 1 ⳵ ⳵⌽TM ⳵2⌽TM ⑀ir 1 ⳵2共r⌽TM兲 + 2 sin ␪ + 2 2 2 ⳵␪ ⑀it r ⳵r r sin ␪ ⳵␪ r sin ␪ ⳵␾2 + ␻2⑀ir␮it⌽TM = 0.

共7兲

The Debye potential for the TE case ⌽TM has a similar form as Eq. 共7兲, i.e.,

冉

冊

1 1 ⳵ ⳵⌽TE ⳵2⌽TE ␮ir 1 ⳵2共r⌽TE兲 + sin ␪ + ⳵␪ ␮it r ⳵r2 r2 sin ␪ ⳵␪ r2 sin2 ␪ ⳵␾2

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+ ␻2␮ir⑀it⌽TE = 0.

共8兲

ELECTROMAGNETIC TRANSPARENCY BY COATED …

PHYSICAL REVIEW E 78, 046609 共2008兲

By separation of variables, the solutions of Eqs. 共7兲 and 共8兲 are ⬁

r⌽TM = 兺

l

兺

l=0 m=−l

l 共k itr兲 关cTM l ␺vi1

+

共m兲 l 共k itr兲兴P dTM l ␹vi1 l 共cos

␪兲

TM TM cos共m␾兲 + bm sin共m␾兲兴, ⫻关am ⬁

r⌽TE = 兺

共9兲

III. MIE COEFFICIENTS AND SCATTERING CROSS SECTION

When the incident plane polarized wave, propagating along the positive z axis, has its electric vector of unit amplitude vibrating parallel to the x axis, it is characterized by 关31,35兴 Ein = exeik0r cos ␪,

l

兺 关cTE l ␺v

l=0 m=−l

l i2

共m兲 共kitr兲 + dTM l ␹vl 共kitr兲兴Pl 共cos ␪兲

in r⌽TM =

where a, b, c, and d are coefficients whose values are determined by the relevant boundary conditions, P共m兲 l 共cos ␪兲 are the associated Legendre polynomials, kit = ␻冑⑀it␮it, l = vi1

冑

l = vi2

冑

␮it 1 1 l共l + 1兲 + − , ␮ir 4 2 共10兲

and ␺vi and ␹vi are the Ricatti-Bessel functions defined by

␺v共x兲 =

冑

␲x Jv+1/2共x兲, 2

␹v共x兲 = −

冑

E␾ =

in r⌽TE =

共13兲

⳵2共r⌽TM兲 + ␻2⑀it␮itr⌽TM , ⳵r2

1 ⳵2共r⌽TM兲 i␻␮it ⳵共r⌽TE兲 + , r ⳵r⳵␪ r sin ␪ ⳵␾

⬁

1

2l + 1

兺 il−1 l共l + 1兲 ␺l共k0r兲P共1兲 l 共cos ␪兲cos ␾ , k2 l=1 ⬁

1

k20

i 冑␮0/⑀0 兺 l=1

l−1

2l + 1 ␺l共k0r兲P共1兲 l 共cos ␪兲sin ␾ , l共l + 1兲 共14兲

while for the scattering wave, they should be written as

␲x Nv+1/2共x兲, 2

where Jv+1/2共x兲 and Nv+1/2共x兲 are the Bessel functions and Neumann functions. Once the potentials for the TM and TE cases are given, the complete solution of the fields can be written in the form by adding the two fields,

E␪ =

⑀0 ik r cos ␪ e 0 . ␮0

0

sc =− r⌽TM

共11兲

Er =

冑

Correspondingly, the Debye potentials for incident fields can be expressed as

i2

TE TE cos共m␾兲 + bm sin共m␾兲兴, ⫻关am

⑀it 1 1 l共l + 1兲 + − , ⑀ir 4 2

Hin = ey

sc r⌽TE =−

⬁

1

⬁

1

k20

2l + 1

共1兲 兺 il−1 l共l + 1兲 ATM l ␨l共k0r兲Pl 共cos ␪兲cos ␾ ,

k20 l=1

冑␮0/⑀0

2l + 1

il−1 ATE␨l共k0r兲P共1兲 兺 l 共cos ␪兲sin ␾ , l共l + 1兲 l l=1 共15兲

1 共x兲, with the firstwhere ␨l共x兲 ⬅ ␺l共x兲 − i␹l共x兲 = 冑␲x / 2Hl+1/2 1 kind Hankel functions Hl+1/2共x兲. For the core and the shell, the Debye potentials are described by Eq. 共9兲. However, due to orthogonality between 共m兲 P共1兲 l 共x兲 and Pl 共x兲 for m ⫽ 1, only the term with m = 1 survives. As a result, for the shell, the Debye potentials are s =− r⌽TM

1 ⳵2共r⌽TM兲 i␻␮it ⳵共r⌽TE兲 − ⳵␪ r sin ␪ ⳵r⳵␾ r

1

⬁

2l + 1

兺 il−1 l共l + 1兲 关DTM l ␺v

kst2 l=1

l s1

共1兲 共kstr兲 + ETM l ␹vl 共kstr兲兴Pl s1

⫻共cos ␪兲cos ␾ , s =− r⌽TE

⳵ 共r⌽TE兲 Hr = + ␻2⑀it␮itr⌽TE , ⳵r2 2

⬁

1

kst2

冑␮0/⑀0

2l + 1

关DTE␺v il−1 兺 l共l + 1兲 l l=1

l s2

共kstr兲

共1兲 + ETE l ␹vl 共kstr兲兴Pl 共cos ␪兲sin ␾ ,

共16兲

s2

H␪ = −

H␾ =

i␻⑀it ⳵共r⌽TM兲 1 ⳵2共r⌽TE兲 + , r sin ␪ ⳵␾ r ⳵r⳵␪

i␻⑀it ⳵共r⌽TM兲 1 ⳵2共r⌽TE兲 + . ⳵␪ r r sin ␪ ⳵r⳵␾

while for the core, they are given by c r⌽TM

=−

1

⬁

2l + 1

兺 il−1 l共l + 1兲 FTM l ␺v

k2ct l=1

共12兲

To this point, we have derived the equations for Debye potentials and the electric and magnetic vectors in terms of Debye potentials. In what follows, we shall apply the formulas to the coated sphere with radial anisotropy in both the permittivity and permeability tensors.

c r⌽TE =−

1

k2ct

冑␮0/⑀0

⬁

l c1

共kctr兲P共1兲 l 共cos ␪兲cos ␾ ,

2l + 1

FTE il−1 兺 l ␺v l共l + 1兲 l=1

l c2

⫻共cos ␪兲sin ␾ .

共kctr兲P共1兲 l 共17兲

ATE l

ATM l ,

To derive the scattering coefficients and the boundary conditions 关which can be derived from Eq. 共12兲兴

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PHYSICAL REVIEW E 78, 046609 共2008兲

GAO et al.

s in sc ⑀str⌽TM = ⑀0r共⌽TM + ⌽TM 兲,

must be applied. For the present model, they are c s ⑀ctr⌽TM = ⑀str⌽TM , c s ⳵共r⌽TM 兲 ⳵共r⌽TM 兲 = , ⳵r ⳵r

c s ␮ctr⌽TE = ␮str⌽TE ,

c s ⳵共r⌽TE 兲 ⳵共r⌽TE 兲 = , ⳵r ⳵r

s in sc ␮str⌽TE = ␮0r共⌽TE + ␾TE 兲,

s in sc ⳵共r⌽TM 兲 ⳵共r⌽TM + r⌽TM 兲 = , ⳵r ⳵r

at r = a,

s in sc ⳵共r⌽TE 兲 ⳵共r⌽TE + r⌽TE 兲 = , ⳵r ⳵r

共18兲

at r = b.

共19兲

Substituting Eqs. 共14兲–共17兲 into Eqs. 共18兲 and 共19兲 yields

and

ATM l =

冨

l 共k stb兲 ␮0␹vl 共kstb兲 ␮st␺l共k0b兲 ␮0␺vs1 s1

kst␺l⬘共k0b兲

冨 冨 冨

s1

k0␹⬘vl 共kstb兲

0

s1

0

␮ct␺vl 共ksta兲 ␮ct␹vl 共ksta兲 − ␮st␺vl 共kcta兲 s1

s1

0

kct␺⬘vl 共ksta兲

kct␹⬘vl 共ksta兲

s1

c1

− kst␺v⬘l 共kcta兲

s1

c1

l 共k stb兲 ␮0␹vl 共kstb兲 ␮st␨l共k0b兲 ␮0␺vs1 s1

kst␨l⬘共k0b兲

ATE l =

k0␺⬘vl 共kstb兲

0

k0␺⬘vl 共kstb兲 s1

0

k0␹⬘vl 共kstb兲

0

s1

␮ct␺vl 共ksta兲 ␮ct␹vl 共ksta兲 − ␮st␺vl 共kcta兲

0

kct␺⬘vl 共ksta兲 kct␹⬘vl 共ksta兲 − kst␺v⬘l 共kcta兲

s1

s1

c1

s1

c1

l 共k stb兲 ⑀0␹vl 共kstb兲 ⑀st␺l共k0b兲 ⑀0␺vs2 s2

0

kst␺l⬘共k0b兲 k0␺⬘vl 共kstb兲 k0␹⬘vl 共kstb兲

0

s2

s2

0

⑀ct␺vl 共ksta兲 ⑀ct␹vl 共ksta兲 − ⑀st␺vl 共kcta兲

0

kct␺⬘vl 共ksta兲 kct␹⬘vl 共ksta兲 − kst␺v⬘l 共kcta兲

s2

s2

s2

c2

s2

c2

l 共k stb兲 ⑀0␹vl 共kstb兲 ⑀st␨l共k0b兲 ⑀0␺vs2 s2

0

kst␨l⬘共k0b兲 k0␺⬘vl 共kstb兲 k0␹⬘vl 共kstb兲

0

s2

s2

0

⑀ct␺vl 共ksta兲 ⑀ct␹vl 共ksta兲 − ⑀st␺vl 共kcta兲

0

kct␺⬘vl 共ksta兲 kct␹⬘vl 共ksta兲 − kst␺v⬘l 共kcta兲

s2

s2

s2

c2

s2

where the primes on ␺, ␹, and ␨ denote differentiations with respect to the arguments. The full-wave total scattering cross section of the coated particles is defined by

c2

⬁

S 1共 ␪ 兲 = 兺 l=1 ⬁

S 2共 ␪ 兲 = 兺 l=1

Qs =

2␲

冨 冨 冨

0

s1

冨

,

,

冋 冋

TE 2 2 兺 共2l + 1兲共兩ATM l 兩 + 兩Al 兩 兲.

册 册

共1兲 P共1兲 2l + 1 l 共cos ␪兲 TM dPl 共cos ␪兲 ATE + A . l l共l + 1兲 l sin ␪ d␪

共23兲

共22兲

In addition, the scattering 共the far field兲 of linearly polarized light by the coated particle can be well described by the two basic scattering amplitudes 关31,35兴

共21兲

P共1兲 dP共1兲 2l + 1 l 共cos ␪兲 l 共cos ␪兲 ATM + ATE , l l l共l + 1兲 sin ␪ d␪

⬁

k20 l=1

共20兲

IV. EFFECTIVE MEDIUM THEORY IN THE QUASISTATIC LIMIT

In this section, we aim at deriving the effective permittivity and permeability for the coated particles in long-

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moments proportional to 共k0b兲2l+1 are expected to be quite small, and we may keep only the dipole terms l = 1. In this sense, the dipole field coefficient for the TM case is

-wavelength and low-frequency limit. Note that the effective permittivity and permeability are isotropic for radial anisotropy 关36兴. In the long-wavelength limit k0b Ⰶ 1, higher-order

冏 冏

1 共k stb兲 ␮st␺1共k0b兲 ␮0␺vs1

ATM 1 =

k0␺⬘v1 共kstb兲

kst␺1⬘共k0b兲

s1

1 共k stb兲 ␮st␨1共k0b兲 ␮0␺vs1

k0␺⬘v1 共kstb兲

kst␨1⬘共k0b兲

s1

冏 冏 冏 冏 P1 −

P1 −

1 共k stb兲 ␮st␺1共k0b兲 ␮0␹vs1

kst␺1⬘共k0b兲

k0␹⬘v1 共kstb兲 s1

1 共k stb兲 ␮st␨1共k0b兲 ␮0␹vs1

kst␨1⬘共k0b兲

k0␹⬘v1 共kstb兲 s1

冏 冏

P2 共24兲

,

P2

with P1 = ␮stkct␺v1 共kcta兲␹⬘v1 共ksta兲 − ␮ctkst␹v1 共ksta兲␺⬘v1 共kcta兲, c1

s1

s1

c1

P2 = ␮stkct␺v1 共kcta兲␺⬘v1 共ksta兲 − ␮ctkst␺v1 共ksta兲␺⬘v1 共kcta兲. c1

s1

s1

共25兲

c1

Incidentally, the dipole coefficient for the TE case ATE 1 can be obtained from Eqs. 共24兲 and 共25兲 by replacements of ⑀ → ␮ and vi1 → vi2, respectively. In the long-wavelength 共k0b Ⰶ 1兲 and low-frequency 共kitb Ⰶ 1兲 limits, since the arguments for the functions ␺, ␹, and ␨ are small, the leading terms for these functions can only be retained, that is,

冑␲x/2

冉冊

x ⌫共n + 3/2兲 2

␺n共x兲 =

n+1/2

,

␹n共x兲 =

冑␲x/2⌫共n + 1/2兲 ␲

冉冊 2 x

n+1/2

as x → 0,

共26兲

where ⌫共¯兲 is the Euler gamma function. Substituting Eq. 共26兲 into Eqs. 共24兲 and 共25兲 and those for ATE 1 leads to a + b

+ ⑀sr共1 +

1 兲兴 vs1

a + b

1 ␮0兲关␮crvc2

+ ␮sr共1 +

1 兲兴 vs2

a + b

1 2␮0兲关␮crvc2

+ ␮sr共1 +

1 兲兴 vs2

a + b

1 ⑀0兲关⑀crvc1

1 2⑀0兲关⑀crvc1

−

ATM 1 =C 1 共⑀srvs1

1 共␮srvs2

+

−

ATE 1 =C 1 共␮srvs2

+

冉冊 冉冊

+ ⑀sr共1 +

1 兲兴 vs1

1 共⑀srvs1

冉冊 冉冊

1 共2vs1 +1兲

1 共2vs1 +1兲

1 共2vs2 +1兲

1 共2vs2 +1兲

1 1 1 关⑀0 + 共1 + vs1 兲⑀sr兴共⑀crvc1 − ⑀srvs1 兲

, 关2⑀0 − 共1 +

1 1 兲⑀sr兴共⑀srvs1 vs1

1 ⑀crvc1 兲

−

1 1 1 关␮0 + 共1 + vs2 兲␮sr兴共␮crvc2 − ␮srvs2 兲

, 关2␮0 − 共1 +

1 1 兲␮sr兴共␮srvs2 vs2

−

共27兲

1 ␮crvc2 兲

with C = 32 i共k0b兲3. To search for the effective responses for coated particles, one always assumes that the coated particles of radial anisotropy are embedded in an effective medium with isotropic effective permittivity ⑀ef f and permeability ␮ef f . In this sense, they can be TE determined by the condition that both ATM 1 and A1 vanish, if ⑀0 and ␮0 are replaced by ⑀ef f and ␮ef f 关37兴. As a result, we have 1

⑀ef f =

1 1 1 1 1 1 ⑀srvs1 关⑀crvc1 + 共1 + vs1 兲⑀sr兴 + 共a/b兲共2vs1+1兲共1 + vs1 兲⑀sr共⑀crvc1 − ⑀srvs1 兲 1

1 1 1 1 ⑀crvc1 + 共1 + vs1 兲⑀sr − 共a/b兲共2vs1+1兲共⑀crvc1 − ⑀srvs1 兲

共28兲

,

1

␮ef f =

1 1 1 1 1 1 ␮srvs2 关␮crvc2 + 共1 + vs2 兲␮sr兴 + 共a/b兲共2vs2+1兲共1 + vs2 兲␮sr共␮crvc2 − ␮srvs2 兲 1

1 1 1 1 ␮crvc2 + 共1 + vs2 兲␮sr − 共a/b兲共2vs2+1兲共␮crvc2 − ␮srvs2 兲

.

共29兲

Now, the coated sphere particle is regarded as a uniform sphere in the long-wavelength and low-frequency limit, and the normalized scattering cross section may be simplified as 关31兴 046609-5

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冉 冊 冉冏

Qs 64␲4 b = 3 ␭0 ␭20

6

⑀ef f − ⑀0 ⑀ef f + 2⑀0

冏 冏 2

+

␮ef f − ␮0 ␮ef f + 2␮0

冏冊

3

2

,

2 共30兲

1
eff/
0

where ␭0 = 2␲ / k0. It is evident that for ⑀ef f = ⑀0 and ␮ef f = ␮0, Qs is zero, making the coated particles invisible or transparent to an outside observer. However, for magnetodielectric coated particles, the conditions ⑀ef f = ⑀0 and ␮ef f = ␮0 cannot be satisfied simultaneously for a given a / b. Consequently, the coated particle may be nearly invisible if Qs achieves a minimum for certain a / b. The condition for near transparency is determined by the relation

冎

for nonmagnetic particles and a = b

再

1 1 1 共␮srvs2 − ␮0兲关␮crvc2 + ␮sr共vs2 + 1兲兴 1 1 1 共␮srvs2 − ␮crvc2 兲关␮0 + ␮sr共1 + vs2 兲兴

冎


ct=6
0



共31兲

-4

(b)

-6 -8 -10

共32兲

-12 0

1 1/共2vs2 +1兲



1 1 1 共⑀srvs1 − ⑀crvc1 兲关⑀0 + ⑀sr共1 + vs1 兲兴

1 1/共2vs1 +1兲


ct=4
0

-3 -2

log10
Qs/ 

再

1 1 1 共⑀srvs1 − ⑀0兲关⑀crvc1 + ⑀sr共vs1 + 1兲兴


ct=2
0

-2

For nonmagnetic 共or pure magnetic兲 coated particles, ␮ef f 共⑀ef f 兲 is equal to ␮0 共⑀0兲. In this connection, to make the coated particle 共nearly兲 transparent, from Eq. 共31兲 one yields ⑀ef f = ⑀0 共␮ef f = ␮0兲 corresponding to a = b

0 -1

log10
Qs/ 

共␮ef f − ␮0兲␮0 d␮ef f 共⑀ef f − ⑀0兲⑀0 d⑀ef f + = 0. 3 共⑀ef f + 2⑀0兲 d共a/b兲 共␮ef f + 2␮0兲3 d共a/b兲

(a)

共33兲

for magnetic materials. It is evident that for isotropic coated particles, Eq. 共32兲 is exactly the same as the transparency condition derived by Alu and Engheta 关13兴 and Zhou and Hu 关16兴, respectively.

-2

(c)

-4 1.2

V. NUMERICAL RESULTS

Qs/0

In what follows, we perform numerical calculations for the normalized scattering section Qs / ␭20 with Eq. 共22兲 共valid for general full-wave scattering兲, and Eq. 共30兲 共valid for the quasistatic limit including long-wavelength and lowfrequency limits兲. In Fig. 2, Qs is shown for the coated dielectric sphere with plasmonic shell and the core of radial dielectric anisotropy as a function of the the radius ratio a / b for various particles sizes. We find that for small particle sizes such as b = 0.01␭0 关see Fig. 2共b兲兴, Eq. 共30兲 yields the same results as the Mie full-wave scattering theory, Eq. 共22兲, as expected. The scattering cross section is almost zero, indicating the transparency or “invisibility” of the particles, when a / b takes some values, which can be exactly determined by the relation that ⑀ef f = ⑀0 关see Fig. 2共a兲兴. In addition, for small a / b, one can get the effective permittivity ⑀ef f of coated particles to be −2⑀0. As a result, strong resonant behavior takes place as predicted from Eq. 共30兲. For large particles in Figs. 2共c兲 and 2共d兲, the number of multipolar terms contributing to the scattering increases rapidly, and hence one cannot resort to effective medium theory. In this situation, we still find that Qs exhibits a significant reduction at small radius ratio, in

2

1.0

(d)

0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

a/b FIG. 2. 共Color online兲 共a兲 The effective permittivity ⑀ef f of the coated dielectric particle versus radius ratio a / b for ⑀ct = 2⑀0 共black solid line兲, 4⑀0 共red dashed line兲, and 6⑀0 共green dotted line兲, 共b兲 normalized scattering section Qs / ␭20 of the coated particle for the full-wave case with b = 0.01␭0 for ⑀ct = 2⑀0 共black solid line兲, 4⑀0 共red dashed line兲, and 6⑀0 共green dotted line兲 and for the effective medium for ⑀ct = 2⑀0 共black dash-dotted line兲, 4⑀0 共red short dashed line兲, and 6⑀0 共green short dotted line兲, and 共c兲 and 共d兲 Qs / ␭20 for the full-wave case with b = 0.1␭0 and b = 0.2␭0 for ⑀ct = 2⑀0 共black solid line兲, 4⑀0 共red dashed line兲, and 6⑀0 共green dotted line兲. Other parameters are ⑀cr = 4⑀0, ⑀sr = ⑀st = −3⑀0, and ␮cr = ␮ct = ␮sr = ␮st = ␮0.

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4

9
st=5
0
st=15
0

3 0

3


st=10
0
eff/
0


eff/
0

6

log10
Qs/ 

(b)





log10
Qs/ 

0 -8

-6 -8 (b)

-10 -12 -14

-12 0

(c)



log10
Qs/ 

-2



log10
Qs/ 


ct=6
0

2 1

-4

-2

-4


ct=4
0

(a)

-3 -2

-10


ct=2
0

(a)

-4

-6

(c)

0.06

1.2

(d) 2

Qs/0

0.6

0.04

Qs/0

2

0.9 (d)

0.02

0.3 0.0 0.5

0.6

0.7

0.8

0.9

0.00 0.5

1.0

0.6

0.7

0.8

0.9

1.0

a/b

a/b FIG. 3. 共Color online兲 Similar as Fig. 2, but for the radial dielectric anisotropy in the shell. Other parameters are ⑀cr = ⑀ct = −3⑀0, ⑀sr = 10⑀0, and ␮cr = ␮ct = ␮sr = ␮st = ␮0.

comparison with the quasistatic case. To one’s interest, when the dielectric anisotropy is taken into account, decreasing ⑀ct may result in a much lower scattering section 共near zero scattering section兲 and thereby better transparency at the cost of large size of plasmonic shell. For instance, for small dielectric anisotropy ⑀ct = 2, one achieves a smaller scattering section than the one for the isotropic case, ⑀ct = 4. One further notes that from Fig. 2共d兲, close to the “near-zero-scattering” ratio, a sharp peak exists, resulting from the resonant phenomenon of ATM 3 共not shown here兲. In Fig. 3, we apply the transparency phenomenon to a plasmonic particle with a coating shell of radial dielectric

FIG. 4. 共Color online兲 Same as Fig. 2, but for coated magnetic particles with radial magnetic anisotropy in the core. Other parameters are ␮cr = 4␮0, ␮sr = ␮st = 0.5␮0, and ⑀cr = ⑀ct = ⑀sr = ⑀st = ⑀0.

anisotropy. In the quasistatic limit, for an isotropic shell ⑀sr = ⑀st = 10, one would expect that the coated particle is transparent for a / b ⬇ 0.825, at which ⑀ef f = ⑀0, as shown in Fig. 3共a兲. Good agreement is again found between full-wave theory and effective medium theory. Incidentally, since the core is plasmonic, the resonance takes place in the thin shell limit a / b → 1. A more interesting phenomenon is that through the suitable adjustment of dielectric anisotropy in the shell, one can achieve a much small scattering section and tune the near-zero scattering radius ratio. In detail, the minimum of Qs for the anisotropic case with ⑀st = 5, which occurs at a / b ⬇ 0.69, is one order smaller than the one for the

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1.5

-2
st=0.2
0

log10
Qs/
 


st=0.5
0

(a)
eff/
0




st=0.8
0

1.0

(a)

-6 -8 -10 1.5



log10
Qs/ 

0.5 -8

-4

-10

(b) (b)

Qs/


2

-12

1.2

-14

0.9

-4

0.3 1.0

(c) TE

|A | or |A |



log10
Qs/ 

0.6

0.02

TE

A1

TM

A2

0.6

TE

A2

0.4

TM

A3

TE

(d)

A3

0.2

2

Qs/0

A1

(c)

TM

-6

TM

0.8

0.01

0.0 0.0

0.2

0.4

0.6

0.8

1.0

a/b

FIG. 5. 共Color online兲 Same as Fig. 4, but for coated magnetic particles with radial magnetic anisotropy in the shell. Parameters are ␮cr = ␮ct = 4␮0, ␮sr = 0.5␮0, and ⑀cr = ⑀ct = ⑀sr = ⑀st = ⑀0.

FIG. 6. Qs / ␭20 for 共a兲 b = 0.01␭0 and 共b兲 b = 0.2␭0, and 共c兲 contributions of the several scattering coefficients such as ATM 1 共solid TM TE line兲, ATE 共dashed line兲, A 共dotted line兲, A 共dash-dotted line兲, 1 2 2 TE ATM 共short dashed line兲, and A 共short dotted line兲 versus a / b for 3 3 the magnetodielectric coated particle with both radially dielectric and magnetic anisotropy. Parameters are ⑀cr = 4⑀0, ⑀ct = 2⑀0, ␮cr = ␮ct = ␮0, ⑀sr = ⑀st = −3⑀0, ␮sr = 0.2␮0, and ␮st = 0.5␮0.

isotropic case with ⑀st = 10 at a / b = 0.825, thus resulting in much better transparent behavior. Therefore, to get better transparency, we require the permittivity in the radial direction to be larger than the one in the transverse direction. This should be in contrast with that in Ref. 关3兴, in which the permittivity in the radial direction is smaller than the tangential one. Actually, the latter is based on the coordinate transformation technique, while our work is based on the dipolecanceling mechanism 关13兴. As for large coated particles, more resonant peaks or bands are predicted due to high-order resonant modes 关see Fig. 3共d兲兴. Here we emphasize that since the transparency phenomenon does not result from the reso-

nant effect, one would expect a relatively broad range for the radius ratio, around which the scattering section is almost zero. By decreasing the shell anisotropy ⑀st, the “near-zeroscattering” ratio band becomes much broader, accompanied with much less scattering. As a result, the adjustment of shell anisotropy may be helpful to improve the transparency quality. In Figs. 4 and 5, only permeability has radial anisotropy. It is observed that in the quasistatic limit 关see Figs. 4共b兲 and 5共b兲兴, the scattering section can be well described by effective medium theory and the transparency phenomenon takes place at the radius ratio, corresponding to the one at which

0.00 0.5

0.6

0.7

0.8

a/b

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-7

8

1.0x10

-8

8.0x10

shell outer region region

core region 6

-8

Eloc/E0

I1

6.0x10

-8

4.0x10

-8

2.0x10

0.0

4

2

-7

1.0x10 8.0x10 I2

(a)

a/b=0.80 a/b=0.82 a/b=0.84

-8

0 4

-8

6.0x10

-8

4.0x10

core region

-8

0.0

Eloc/E0

2.0x10

0

30

60

90

120

150

180

shell region

outer region

shell region

outer region

2




FIG. 7. 共Color online兲 Normalized differential cross sections I1 = 兩S1共␪兲兩2 / 共k20␲b2兲 and I2 = 兩S2共␪兲兩2 / 共k20␲b2兲 as a function of the scattering angle ␪ for b = 0.01␭0 and a / b = 0.80 共black solid line兲, 0.82 共red dashed line兲, and 0.84 共green dotted line兲. Other parameters are the same as in Fig. 6.

0 12 10

core region

8 Eloc/E0

the effective permeability is taken as ␮0 instead of ⑀0 关see Figs. 4共a兲 and 5共a兲兴. Due to the choice of the positive permeability for the core and the shell, no resonance can be excited. As a result, there are no sharp peaks for these cases. On the other hand, for large particle size 关see Figs. 4共c兲 and 5共c兲兴, a near-zero scattering section, making the particle invisible or transparent, is still found in a wide range of radius ratio a / b. In Fig. 6, we examine the case for the magnetodielecric coated particle in which the permittivity and permeability are radially anisotropic. In the quasistatic limit, both the fullwave expression, Eq. 共22兲, and effective medium theory, Eq. 共30兲, predict that there exist one enhancement peak and one near-transparent position, characterized by a nearly zero Qs. Since no magnetic resonance occurs, the resonant peak is due to the electric resonance. However, magnetic and electric spectra contribute to the electromagnetic transparency of the coated particle at a / b = 0.82, which should be determined by Eq. 共31兲, rather than a simple formula ⑀ef f = ⑀0 or ␮ef f = ␮0. For large coated particles, a high-scattering resonant peak appears due to the ATM 3 term as shown in Fig. 6共c兲. In addition, the minimum of Qs is not zero, but with an appreciable scattering. This mainly results from scattering terms such as TM ATE 1 and A2 . Therefore, one may expect to tune the anisotropic parameters to decrease the magnitudes of ATE 1 and ATM 2 , so as to realize the near-transparency condition in this region. Compared with the isotropic coated sphere 关13兴, the anisotropic coated sphere has introduced more physical param-

(b)

6 4 2 (c) 0 0

2

4

6

8

10

12

distance from particle center (nm) FIG. 8. Spatial dependence of the the electric-field ratio in and around coated particle at transparency condition for 共a兲 ⑀ct = 2⑀0, 共b兲 ⑀ct = 4⑀0, and 共c兲 ⑀ct = 6⑀0. The solid curve for the field parallel to the applied electric field polarization direction E0, and the dotted line for the one perpendicular to E0. Other parameters are the same as in Fig. 2.

eters for us to achieve transparency. For large particles, and ATM may tend to higher-order scattering coefficients ATM l l zero by our suitable adjustment of these anisotropic physical parameters. Next, we would like to aim at the normalized differential scattering sections I1 = 兩S1共␪兲兩2 / 共k20␲b2兲 共the scattering pattern

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in the yz plane兲 and I2 = 兩S2共␪兲兩2 / 共k20␲b2兲 共the scattering pattern in the xz plane兲 in Fig. 7. These parameters are very important because they can be used to calculate the various experimentally observable scattering variables 关19兴. In the quasistatic limit, as we have shown, the coated particles possess a near-zero scattering section for a / b ⬇ 0.82 and thus are transparent. However, as to the discussion of the differential scattering section, such a choice is not always perfect. For instance, from Fig. 7共a兲, we find that the scattered power for a / b = 0.82 共at which Qs ⬇ 0兲 is larger than the one for a / b = 0.80 in the backward direction and the one for a / b = 0.84 in the forward direction. A similar discussion was performed by Zhou and Hu for acoustic wave transparency 关38兴. However, our model is quite different from Pendry’s cloak, in which zero backscattering is always found even when a type of loss is introduced 关27兴. In the end, electric field distributions for the dielectric coated particle with radial anisotropy in the core is shown in Fig. 8. For simplicity, the quasistatic case is studied. For isotropic case 关see Fig. 8共b兲兴, it is found that the local field is uniform in the core. However, the introduction of the dielectric anisotropy leads to large fluctuations in the local field, which may be useful for the enhancement of optical nonlinearity. Here, the principal observation from these curves is that the field outside the particles is nothing but the applied field, which proves that the scattered fields are indeed canceled. Therefore, the cloaking mechanism here is distinguished from Pendry’s idea 关1兴, where the incident fields cannot penetrate into the core and the fields in the core are always zero. VI. CONCLUSION AND DISCUSSION

In this paper, we have established electromagnetic scattering theory by coated particles of radial electric and magnetic anisotropies. Effects of anisotropic physical parameters in both the core and the shell on the total scattering section are systematically investigated. Based on full-wave scattering theory, we show that by a suitable adjustment of the radius ratio, one may make the coated particle nearly transparent or invisible. In the quasistatic case, the effective medium concept is valid, and we derive effective permittivity and permeability for the coated particle. The near-zero scattering radius ratio can be well described within effective medium theory. It shows that the introduction of radial anisotropy may be helpful to achieve much better transparency such as much

关1兴 J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 共2006兲. 关2兴 U. Leonhardt, Science 312, 1777 共2006兲. 关3兴 D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, Science 314, 977 共2006兲. 关4兴 F. J. Garcia de Abajo, G. Gomez-Santos, L. A. Blanco, A. G. Borisov, and S. V. Shabanov, Phys. Rev. Lett. 95, 067403 共2005兲.

lower Qs and wider near-zero scattering ratio, and to adjust the position of the radius ratio, exhibiting a tunable electromagnetic transparency. Here we would like to add a few comments. The key to realize the electromagnetic transparency of the coated particles lies in nearly zero value of the numerator of the scattering coefficients. On the other hand, if one needs large scattering, the anomalous plasmonic resonance should be induced. In this connection, the strong electromagnetic resonance in a large collection of coated particles may create the negative permeability. As a consequence, one may realize double-negative metamaterials with coated nonmagnetic spheres of radially dielectric anisotropy 关39兴. Therefore, it is of interest to develop effective medium theory for coated particles of radial anisotropy beyond the quasistatic limit and to investigate the effect of anisotropic parameters on the resonant behavior of the effective permittivity and permeability. Due to the reduction of both backscattering and forward scattering, two-dimensional cylindrical cloaks were realized in experiment 关3兴. Accordingly, full-wave, finite-element numerical simulations for cylindrical invisibility cloaks were done 关40兴, and mirage effect whereby the source seems to radiate from a shifted location, were observed 关41兴. Theoretically, Zhang et al. 关42兴 and Ruan et al. 关43兴 investigated the electromagnetic response of cylindrical invisibility cloaks within the framework of electromagnetic wave scattering theory. In this regard, our work can be generalized to twodimensional cylindrical invisibility cloaks with radial anisotropy without any difficulty. In the quasistatic case, for coated cylinders with radially dielectric anisotropy, partial resonance conditions are derived as s⑀sr + ⑀0 = 0 and s⑀sr + ⑀c = 0 with s = 冑⑀st / ⑀sr and ⑀c the permittivity of the isotropic core 关44兴. When the partial resonance is satisfied, the cloaking may be proved for finite collections of polarizable line dipoles that lie within a specific distance from a coated cylinder with radial anisotropy 关7兴. ACKNOWLEDGMENTS

This work was supported by the Research Grants Council of Hong Kong SAR Government 共L.G. and K.W.Y.兲, the National Natural Science Foundation of China under Grant No. 10674098 共L.G.兲, the National Basic Research Program under Grant No. 2004CB719801 共L.G.兲, and the Natural Science of Jiangsu Province under Grant No. BK2007046 共L.G.兲.

关5兴 J. W. Lee, M. A. Seo, J. Y. Sohn, Y. H. Ahn, D. S. Kim, S. C. Jeoung, Ch. Lienau, and Q-Han Park, Opt. Express 13, 10681 共2005兲. 关6兴 G. W. Milton and N. A. P. Nicorovici, Proc. R. Soc. London, Ser. A 426, 3027 共2006兲. 关7兴 N. A. P. Nicorovici, G. W. Milton, R. C. Mcphedran, and L. C. Botten, Opt. Express 15, 6314 共2007兲. 关8兴 M. Kerker, J. Opt. Soc. Am. 65, 376 共1975兲.

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ELECTROMAGNETIC TRANSPARENCY BY COATED …

PHYSICAL REVIEW E 78, 046609 共2008兲

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