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Research Article

Mie-Type Calculation of the Generalized Electromagnetic

Nonlocal Conductivity Tensor for a Sphere and Its Equivalence

to the T-Matrix Operator

E. Gutiérrez-Reyes ,1Rubén G. Barrera,2and A. Garc-a-Valenzuela3

1CONACYT-Centro de Investigaci´

on Cient´

ıca y de Educaci´

on Superior de Ensenada, Baja California, Unidad La Paz,

Miraores No. 334 e/Muleg´

e y La Paz, C.P. 23050, La Paz, BCS, Mexico

2Instituto de F´

ısica, Universidad Nacional Aut´

onoma de M´

exico, Apartado Postal 20-364, 01000 Ciudad de M´

exico, Mexico

3Instituto de Ciencias Aplicadas y Tecnolog´

ıa, Universidad Nacional Aut´

onoma de M´

exico, Apartado Postal 70-168,

04510 Ciudad de M´

exico, Mexico

Correspondence should be addressed to E. Guti´

errez-Reyes; edahigure@gmail.com

Received 13 September 2018; Accepted 23 December 2018; Published 6 February 2019

Academic Editor: Francesca Vipiana

Copyright © E. Guti´

errez-Reyes et al. is is an open access article distributed under the Creative Commons Attribution

License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

cited.

e purpose of this article is to calculate the generalized nonlocal conductivity tensor of a spherical particle made of isotropic

and linear materials. e generalized conductivity tensor is a crucial element in the formulation of the mean-eld theories of the

electromagnetic response of random particulate systems. is is equivalent to what is called the T-Matrix in multiple scattering

theories. Here, a new method is proposed for nding explicit expressions for this tensor directly from its denition including the

magnetic response of the spheres. Its relation with the -Matrix in the theory of single scattering is stated as a generalization. Several

approximations and limit cases of possible interest in specic systems are analyzed and the results of some calculations are presented

as a numerical example.

1. Introduction

In a previous paper [], a reference was made to the gen-

eralized nonlocal conductivity tensor, as the main physical

concept in the development of a formalism for the cal-

culation of the eective bulk electromagnetic response of

randomly located discrete scatterers such as turbid colloids.

is approach was later extended to the calculation of the

reection and transmission amplitudes of the average electric

eld of colloidal suspensions conned in a half-space []. e

generalized conductivity tensor continued to play a major

role in the framework of the theoretical description [].

Colloidal systems are dened as two-phase systems in which a

sparse phase is immersed within a continuous one []. While

thecontinuousphaseisusuallycalledthematrix,thesparse

phaseissaidtobecomposedofcolloidalparticles.When

an external electromagnetic eld is incident upon a colloidal

system, it induces currents within the randomly located

colloidal particles and these currents radiate electromagnetic

elds which act back upon the particles themselves. e total

electromagnetic eld conformed in this way can be split

into two components, an average component with a smooth

spatial variation and traveling in a denite direction, called

the coherent beam, and a uctuating component with abrupt

spatial variations and traveling in all dierent directions,

called the diuse eld.

e work cited above [], related to the reection and

transmission amplitudes from colloidal suspensions, dealt

with the reection and transmission amplitudes of the coher-

ent beam for a system of randomly located identical spheres,

within a homogeneous matrix with a at interface occupying

a half-space. e procedure devised for this calculation

involved the solution of an integral equation posed in terms

of the generalized nonlocal conductivity tensor of an isolated

sphere and valid for a low concentration of spheres. Also,

it has been pointed out that this integral equation for the

Hindawi

Mathematical Problems in Engineering

Volume 2019, Article ID 1530821, 20 pages

https://doi.org/10.1155/2019/1530821

Mathematical Problems in Engineering

average electric eld is analogous to the one used in multiple-

scattering-theory (MST), to solve similar problems. e

integral equation in MST is usually written in terms of the so-

called T-matrix operator, whose calculation for the case of an

isolated sphere has already been performed [] by solving the

integral equation obeyed by this operator. Since a wide variety

of systems are usually modeled as a collection of spheres, the

calculation of the T-matrix for an isolated sphere becomes a

necessary ingredient in this type of calculations.

Here, we rst recall that the T-matrix operator and

the generalized nonlocal conductivity tensor are actually

proportional to each other [] obeying, essentially, the same

integral equation. en, we propose a quite straightforward

procedure to calculate the generalized nonlocal conductivity

tensor of an isolated sphere, making use of its physical

interpretation. Instead of solving the integral equation, our

procedure is based on a Mie-type scattering boundary-value

problem in the presence of external currents, yielding closed-

form expressions for all the components of the generalized

nonlocal conductivity tensor, or, equivalently, the T-matrix

operator. We consider the sphere to be characterized by

both, a local electric permittivity and a local magnetic

permeability. erefore, the method described here poses

an alternative to the calculation of the T-matrix operator,

by simply nding the induced current inside a sphere given

an arbitrary external electric eld, yielding attractive closed-

form expressions for its components. Finally, we want to point

outthatalthoughthecomponentsoftheT-matrixoperator

have been already calculated for the case of a sphere with no

intrinsic magnetic response [], here we provide new closed-

form expressions for these components for the more general

case of a sphere with an intrinsic magnetic local permeability

dierent from the one of its surroundings, yielding, besides

abulkcontribution,alsoasurfacecontributioncomingfrom

induced surface currents. It is also important to notice that

all this is possible due, essentially, to the simplicity of the

calculation procedure proposed here.

2. Basic Concepts

e nonlocal conductivity tensor of an isolated sphere is

dened as the linear response of the internal induced current

to an external electric eld; in its most general form it can be

written as

→

𝐼→

;=𝑉𝑠←→

𝑁𝐿 →

,→

;⋅→

𝑒𝑥𝑡 →

;3()

where the dependence on the frequency on a given quantity

denotes the -component of the corresponding time Fourier

transform. Here, →

𝐼is the total internal induced current,

←→

𝑁𝐿 is the generalized nonlocal conductivity tensor, and

→

𝑒𝑥𝑡 is the external electric eld; it could also be called the

exciting eld. e term generalized means that in →

𝐼all

internal induced currents are included, even those that give

rise to a magnetic response. In this sense Eq. () can be

regarded as a generalized nonlocal Ohm’s law.

etotalinternalinducedcurrent→

𝐼can be calculated,

also, by using the local relationship

→

𝐼→

;=𝑙𝑜𝑐

𝑠→

;→

𝐼→

; ()

where

𝑙𝑜𝑐

𝑠→

;=

𝑠()<

0>, ()

where is the radius of the sphere, 𝑠() is its bulk local

conductivity, and 𝐼is the internal electric eld, that is, the

electric eld within the sphere. But the electric eld at any

point →

is given by

→

→

;

=→

𝑒𝑥𝑡 →

;

+0𝑉𝑠←→

0→

,→

;⋅→

𝐼→

;3

()

Here 𝑠is the volume of the sphere and also

←→

0→

,→

;=←→

1+1

2

0∇∇→

,→

; ()

isthefreeGreen’sfunctiondyadic,0=/,and

→

,→

;=exp 0→

−→

4→

−→

.()

erefore, when one uses Eq. () to iterate Eq. (), one can

nd the equation for ←→

𝑁𝐿, as dened by Eq. (). First, since

equation Eq. () also holds inside the sphere |→

|<,then,if

we multiply Eq. () by 𝑠,weobtain

→

𝐼→

;

=𝑉𝑠3→

−→

𝑙𝑜𝑐

𝑠→

;→

𝑒𝑥𝑡 →

;

+0𝑙𝑜𝑐

𝑠→

;

⋅𝑉𝑠←→

0→

,→

;⋅→

𝐼→

;3.

()

Substituting now Eq. () into Eq. (), we have that

𝑉𝑠←→

𝑁𝐿 →

,→

;⋅→

𝑒𝑥𝑡 →

;3

=𝑉𝑠3𝑙𝑜𝑐

𝑠→

;→

−→

←→

⋅→

𝑒𝑥𝑡 →

;

+0𝑠→

𝑉𝑠←→

0→

,→

;

⋅𝑉𝑠←→

𝑁𝐿 →

,→

;

⋅→

𝑒𝑥t→

;33.

()

Mathematical Problems in Engineering

By interchanging the order of integration and comparing the

terms within the 3integral, we nd

←→

𝑁𝐿 →

,→

;=𝑙𝑜𝑐

𝑠→

;→

−→

←→

+00𝑉𝑠←→

0→

,→

;

⋅←→

𝑁𝐿 →

,→

;3.

()

Now, we recall the denition of the T-matrix through the

following integral relation [, Eq. ..]:

→

𝑆→

;=𝑉𝑠3𝑉𝑠3←→

0→

,→

;

⋅←→

→

,→

;⋅→

𝑒𝑥𝑡 →

;. ()

Here →

𝑒𝑥𝑡 corresponds to the external eld within the sphere.

→

𝑆is the electric eld scattered by the sphere for |→

|>.

Since one can identify the scattered eld →

𝑆with the second

term in the right-hand side of Eq. (), by replacing →

𝐼in the

right-hand side of Eq. () with the expression given in Eq. ()

it follows immediately that Eq. () yields

0←→

𝑁𝐿 →

,→

;=←→

→

,→

;. ()

isrelationisvalidfor|→

| ≤ and |→

|≤. ere-

fore, the calculation of all the components of ←→

𝑁𝐿 can

be performed by solving the integral equation given in

Eq. (). is procedure has been already followed for the

corresponding integral equation for T-matrix operator [].

ItispertinenttopointoutthatthedenitionoftheT-

matrix operator is given, sometimes, as the linear integral

relationship between the scattered eld and the external eld

[]. In this case the Green’s function dyadic is incorporated in

the denition of the T-matrix operator; thus its relationship

with the generalized nonlocal conductivity has to be modied

correspondingly.

3. Calculation Procedure

Here we propose an alternative and simple way to evaluate

the components of the nonlocal generalized conductivity

tensor ←→

𝑁𝐿 for a sphere, by taking advantage of its physical

interpretation. is means that one has only to calculate the

current density induced within the sphere by an arbitrary

external electric eld. In order to see how this is done, let

us start by assuming that all the elds are in the frequency

domain. Now we consider that an external electromagnetic

planewaveisincidentuponthesphere,andwewriteits

electric eld as →

𝑒𝑥𝑡 =

𝛽𝑖→

𝑝⋅→

𝑟()

where →

isthewavevector,

𝛽is a unit vector along

the direction of polarization, and we are assuming a unit

amplitude. en, according to Eq. (), the current density →

𝐼

induced within the sphere by this external plane wave is given

by

→

𝐼→

,→

;

𝛽=𝑉𝑠←→

𝑁𝐿 →

,→

⋅

𝛽𝑖→

𝑝⋅→

𝑟3.()

where

𝛽as an argument in →

𝐼recalls the polarization of the

external electric eld. Now, we dene the Fourier transform

pair of ←→

𝑁𝐿 as

←→

𝑁𝐿 →

,→

= 1

(2)6

⋅R3R3←→

𝑁𝐿 →

,→

𝑖→

𝑝⋅→

𝑟−𝑖→

𝑝⋅→

𝑟33. ()

←→

𝑁𝐿 →

,→

=R3R3←→

𝑁𝐿 →

,→

−𝑖→

𝑝⋅→

𝑟𝑖→

𝑝⋅→

𝑟33. ()

And by taking the Fourier transform in Eq. (), one can write

→

𝐼→

,→

;

𝛽=←→

𝑁𝐿 →

,→

⋅

𝛽.()

Finally, the component of the above equation becomes

→

𝐼𝛼 →

,→

;

𝛽≡

𝛼⋅→

𝐼→

,→

;

𝛽

=

𝛼⋅←→

𝑁𝐿 →

,→

⋅

𝛽

≡𝑁𝐿

𝛼𝛽 →

,→

()

is means that the component of the nonlocal general-

ized conductivity tensor ←→

𝑁𝐿(→

,→

)has a straightforward

physical interpretation: it is equal to the component of the

→

Fourier transform of the current density induced within

the sphere by an electromagnetic plane wave with wavevector

→

and polarization

𝛽.

But the induced current →

𝐼can be written in terms of the

polarization and magnetization material elds, →

and →

,as

→

𝐼→

=−→

→

+∇×→

→

()

where →

→

=𝑠−0(1−(−))→

𝐼→

()

and →

→

=(1−(−))1

0−1

𝑠→

𝐼→

()

Here 𝑠and 𝑠refer to the local electric permittivity and the

local magnetic permeability of the sphere which are assumed

torespondlocallyandlinearlytotheelectricandmagnetic

Mathematical Problems in Engineering

elds (→

𝐼and →

𝐼)withinthesphere.Herewehavelocatedthe

centerofthesphereattheoriginofthecoordinatesystem,and

while 𝑠and 𝑠are the local electric permittivity and the local

magnetic permeability of the sphere, 0and 0denote the

corresponding response functions of the medium outside the

sphere;herewehaveasharpdiscontinuityofthesefunctions.

erefore, by plugging Eqs. () and () into Eq. (), the

induced current density can be written as

→

𝐼→

=−𝑠−0→

𝐼→

+1

0−1

𝑠∇

×→

𝐼→

−1

0−1

𝑠(−)

𝑟×→

𝐼.()

where →

𝐼and →

𝐼are the electric and magnetic elds

inside the sphere and the last term, on the right-hand side,

corresponds to an induced surface current. erefore, the

induced current within the sphere is completely determined

by the values of the internal electric and magnetic elds, and

the calculation of →

𝐼is reduced to the calculation of →

𝐼and

→

𝐼.

is means that the calculation of all the components

𝑁𝐿

𝛼𝛽 (→

,→

) can be set as a scattering problem, that is,

as the problem of calculating the scattered and internal

electromagnetic elds induced by an electromagnetic plane

wave incident upon a sphere, for three dierent polarizations

𝛽of the incident plane wave. is problem is very similar

to the classical problem of the scattering of a plane wave

by a sphere, solved rst by Gustav Mie in []. e

main dierence between this problem and the classical

Miesolutionisthatherethewavevector→

and the wave

frequency of the incident plane wave are independent of

each other, while, in the problem solved by Mie, the incident

planewaveisaself-propagatingwavewhosewavevector

and frequency are related through the dispersion relation of

the medium surrounding the sphere. is also means that

here the incident plane wave has to be generated by external

currents capable of xing its wavevector →

to any given value.

For example, if

𝛽are chosen as the cartesian unit vectors

and the sphere is immersed in a medium with permittivity

0and permeability 0, then the external electric eld →

𝑒𝑥𝑡

must satisfy the Maxwell’s equations

∇×→

𝑒𝑥𝑡 =0→

𝑒𝑥𝑡 ()

∇×→

𝑒𝑥𝑡 =→

𝑒𝑥𝑡 −0→

𝑒𝑥𝑡 ()

where one includes as a source the external current →

𝑒𝑥𝑡.

us, the external electric eld satises the vectorial equation

∇×∇×→

𝑒𝑥𝑡 +200→

𝑒𝑥𝑡 =0→

𝑒𝑥𝑡 ()

and the source →

𝑒𝑥𝑡 is chosen in order to obtain the desired

external electric eld, say, a plane wave

𝛽𝑖𝑝𝑧 propagating

along the axis, with polarization

𝛽taking each one of the

polarizations {

𝑥,

𝑦,

𝑧}.Itcanbeshownthatinthiscase→

𝑒𝑥𝑡

becomes →

𝑒𝑥𝑡 →

,→

;

𝑥= 1

02−2

0𝑖𝑝𝑧

𝑥()

→

𝑒𝑥𝑡 →

,→

;

𝑦= 1

02−2

0𝑖𝑝𝑧

𝑦()

→

𝑒𝑥𝑡 →

,→

;

𝑧=− 1

02

0𝑖𝑝𝑧

𝑧,()

as it could be veried by direct substitution. Note that in

order to solve this problem in its most general way one

has to accept the presence of longitudinal currents that will

give rise to an external eld with a longitudinal component.

Nevertheless, the calculation procedure of the scattered and

internal elds follows the standard mathematical process by

rst expanding the electromagnetic elds on a spherical basis

and then setting boundary conditions both at innity and on

the surface of the sphere.

4. Spherical Basis and Boundary Conditions

epurposeofthissectionistosolveMaxwell’sequations

for a sphere of radius located at the origin, and with the

external currents →

𝑒𝑥𝑡 given by Eqs. ()-(). We start by

writing Maxwell’s equations as

∇×∇×→

+2𝑠→

;𝑠→

;→

=0→

𝑒𝑥𝑡 ()

where we have assumed, as above, that all elds have an

oscillatory time dependence −𝑖𝜔𝑡,and

𝑠→

;=

𝑠()<

0()>

and 𝑠→

;=

𝑠()<

0()>.

()

us we start by expanding the external plane wave in a vector

spherical harmonic basis; that is, we write 𝑖𝑝𝑧

𝑥,𝑖𝑝𝑧

𝑦,𝑖𝑝𝑧

𝑧

in the functional basis [, p. ]

→

𝑒𝑛𝑚 =∇×→

𝑒𝑛𝑚,

→

𝑜𝑛𝑚 =∇×→

𝑜𝑛𝑚, ()

→

𝑒𝑛𝑚 =1

∇×→

𝑒𝑛𝑚,

→

𝑜𝑛𝑚 =1

∇×→

𝑜𝑛𝑚,()

→

𝑒𝑛𝑚 =1

∇𝑒𝑛𝑚,

→

𝑜𝑛𝑚 =1

∇𝑜𝑛𝑚,()

Mathematical Problems in Engineering

where

𝑒𝑛𝑚 =cos 𝑚

𝑛(cos )𝑛,()

𝑜𝑛𝑚 =sin (cos )𝑛,()

where isthewavenumberassociatedtothewaveequation

that satises the vector spherical harmonics

∇×∇×→

−2→

=0, ()

for →

𝑒𝑛𝑚,→

𝑜𝑛𝑚,→

𝑒𝑛𝑚,→

𝑜𝑛𝑚,and

∇∇⋅→

+2→

=0, ()

for →

𝑒𝑛𝑚,→

𝑜𝑛𝑚.Here𝑚

𝑛are the associated Legendre poly-

nomials as dened in [, p. ] and 𝑛denote either one of

sphericalBesselfunctions:

𝑛=

2𝑛+1/2 ()

𝑛=

2𝑛+1/2 . ()

It can be shown by using the orthogonality property of the

vector spherical harmonics that the expansion of a plane

wave for the three dierent polarizations {

𝑥,

𝑦,

𝑧}in a

vector spherical harmonic basis is given by [][pg. , Eq.

(36)-(38)].

→

𝑒𝑥𝑡 →

;

𝑥=𝑖𝑝𝑧

𝑥

=∞

𝑛=1 −2+1

(+1)𝑛→

𝑜𝑛1 ,→

−→

𝑒𝑛1 ,→

()

→

𝑒𝑥𝑡 →

;

𝑦=𝑖𝑝𝑧

𝑦

=∞

𝑛=1 (2+1)

(+1)𝑛→

𝑒𝑛1 ,→

+→

𝑜𝑛1 ,→

()

→

𝑒𝑥𝑡 →

;

𝑧=𝑖𝑝𝑧

𝑧=∞

𝑛=0 (2+1)𝑛−1→

𝑒𝑛0 ,→

()

wherethesecondvariableintheargumentof→

𝑒𝑥𝑡 denotes the

polarization.emagneticeldsarecalculatedwithFaraday’s

law →

𝑒𝑥𝑡 =1

0∇×→

𝑒𝑥𝑡.()

We include the result in Appendix A, in Eqs. (A.), (A.),

and(A.).Now,weproceedtosolvethescatteringproblem

by assuming that the electric eld outside the sphere is the

incident external electric eld →

𝑒𝑥𝑡 plus a scattered electric

eld →

𝑠, while inside the sphere is →

𝐼, and they must obey

the electromagnetic boundary conditions at the surface of the

sphere. Inside the sphere, the electric eld →

𝐼must fulll the

inhomogeneous vectorial wave equation given by Eq. (),

whose solution is the sum of the homogeneous equation,

expressed as an expansion of vector spherical harmonics, plus

aparticularsolution

→

𝑝𝑎𝑟𝑡 →

;

𝑥=𝑠

02−2

0

2−2

𝑠→

𝑒𝑥𝑡 →

;

𝑥

=𝑠

02−2

0

2−2

𝑠𝑖𝑝𝑧

𝑥,()

→

𝑝𝑎𝑟𝑡 →

;

𝑦=𝑠

02−2

0

2−2

𝑠→

𝑒𝑥𝑡 →

;

𝑦

=𝑠

02−2

0

2−2

𝑠𝑖𝑝𝑧

𝑦,()

→

𝑝𝑎𝑟𝑡 →

;

𝑧=0

𝑠→

𝑒𝑥𝑡 →

;

𝑧=0

𝑠𝑖𝑝𝑧

𝑧,()

which can be written in terms of the external eld →

𝑒𝑥𝑡.Here

2

0=

200while 2

𝑠=

2𝑠𝑠.Here we are using the

notation →

𝑝𝑎𝑟𝑡(→

;

𝑥)meaning that the particular solution

of the electric eld depends on the polarization

𝑥of the

exciting eld. e magnetic eld associated to this electric

eldisgivenbyEqs.(A.),(A.),and(A.)inAppendixA.

erefore, the electric eld induced inside a sphere with

radius can be written as

→

𝐼→

;

𝑥=∞

𝑛=1 −2+1

(+1)𝑛𝑛→

𝑜𝑛1 𝑠,→

−𝑛→

𝑒𝑛1 𝑠,→

+𝑠

02−2

0

2−2

𝑠𝑖𝑝𝑧

𝑥

()

→

𝐼→

;

𝑦=∞

𝑛=1 (2+1)

(+1)𝑛𝑛→

𝑒𝑛1 𝑠,→

+𝑛→

𝑜𝑛1 𝑠,→

+𝑠

02−2

0

2−2

𝑠𝑖𝑝𝑧

𝑦

()

→

𝐼→

;

𝑧=∞

𝑛=0 (2+1)𝑛−1𝐿

𝑛→

𝑒𝑛0 𝑠,→

+0

𝑠

⋅𝑖𝑝𝑧

𝑧

()

where the coecients 𝑛,𝑛,𝐿

𝑛are determined by the bound-

ary conditions. e corresponding magnetic eld is given by

Eqs. (A.), (A.), and (A.), where the magnetic eld →

𝑒𝑥𝑡

is given by (A.)-(A.), for the three dierent polarizations

of the external electric eld →

𝑒𝑥𝑡. We now assume that the

scatteredeldcanbewrittenas

→

𝑠→

;

𝑥=∞

𝑛=1 −2+1

(+1)𝑛𝑛→

(3)

𝑒𝑛1 0,→

−𝑛→

(3)

𝑜𝑛1 0,→

, ()

Mathematical Problems in Engineering

→

𝑠→

;

𝑦=∞

𝑛=1 −(2+1)

(+1)𝑛𝑛→

(3)

𝑜𝑛1 0,→

+𝑛→

(3)

𝑒𝑛1 0,→

, ()

→

𝑠→

;

𝑧=∞

𝑛=1 (2+1)𝑛−1𝐿

𝑛→

(3)

𝑒𝑛0 0,→

, ()

where the index (3)indicates that the functional dependence

of the vector spherical harmonics is through the spherical

Hankel function 𝑛(0). e scattered magnetic eld is a

consequence of Faraday’s law and is given by Eqs. (A.),

(A.), and (A.) in Appendix A.

Now we apply the boundary conditions for the electro-

magnetic eld that consist in the continuity of the parallel

component, of both the electric and magnetic elds, at the

surface of the sphere =.Inasphericalbasisthese

boundary conditions can be written as

→

𝑒𝑥𝑡𝜃+→

𝑠𝜃=→

𝐼𝜃

and →

𝑒𝑥𝑡𝜃+→

𝑠𝜃=→

𝐼𝜃,()

and

→

𝑒𝑥𝑡𝜙+→

𝑠𝜙=→

𝐼𝜙

and →

𝑒𝑥𝑡𝜙+→

𝑠𝜙=→

𝐼𝜙.()

e rst set of boundary conditions given in Eqs. () yields

the following expressions for the expansion coecients:

𝑛,0,𝑠= (1−)∗

𝑛𝑠/𝑠𝑛𝑠−1−∗/0𝑛∗

𝑛𝑠

∗

𝑛0𝑠/𝑠𝑛𝑠−0/0𝑛0∗

𝑛𝑠 ,()

𝑛,0,𝑠= (1−)𝑛𝑠/𝑠∗

𝑛𝑠−1−∗/0∗

𝑛𝑛𝑠

𝑛0𝑠/𝑠∗

𝑛𝑠−0/0∗

𝑛0𝑛𝑠 ,()

𝑛,0,𝑠= 𝑛01−∗/0∗

𝑛−0/0∗

𝑛0(1−)𝑛

𝑛0𝑠/𝑠∗

𝑛𝑠−0/0∗

𝑛0𝑛𝑠 ,()

𝑛,0,𝑠= ∗

𝑛01−∗/0𝑛−0/0𝑛0(1−)∗

𝑛

∗

𝑛0𝑠/𝑠𝑛𝑠−0/0𝑛0∗

𝑛𝑠 ,()

where

∗

𝑛= 1

𝑛,

∗

𝑛= 1

𝑛, ()

=𝑠

02−2

0

2−2

𝑠,

∗=2−2

0

2−2

𝑠.()

e second set of boundary conditions given in Eqs. ()

yields, for an external electric eld of the form

𝑧𝑖𝑝𝑧,the

following expressions for the expansion coecients of the

scattered eld in Eq. () and the internal electric eld in Eq.

():

𝐿

𝑛,0,𝑠= 0

𝑠𝑠

0𝑛𝑠

𝑛0𝐿

𝑛, ()

𝐿

𝑛,0,𝑠

=1−0/𝑠𝑛/𝑛0

∗

𝑛𝑠𝑛0−0/𝑠𝑠/0𝑛𝑠∗

𝑛0.()

Once the internal electric eld has been calculated, we are

able to calculate the current density →

𝐼inducedwithinthe

sphere with the aid of Eq. () that we rewrite as

→

𝐼→

=−𝑠−0→

𝐼→

+𝑠

0−1∇

×→

𝐼→

+1−𝑠

0(−)

𝑟

×→

𝐼→

,

()

where →

𝐼is given in Eqs. (A.)-(A.) for each polarization of

the external electric eld.

5. The Induced Current

Now one substitutes the internal electric eld for each polar-

ization, as given in Eqs. (), (), and (), into equation

(), to arrive at the following result.

e induced current inside the sphere for each polariza-

tion {

𝑥,

𝑦,

𝑧}of the incident eld with arbitrary wave vector

→

is

→

𝐼→

;

𝑥

=𝑘

∞

𝑛=1 −2+1

(+1)𝑛𝑛→

𝑜𝑛1 𝑠,→

Mathematical Problems in Engineering

−𝑛→

𝑒𝑛1 𝑠,→

+𝑝𝑠

02−2

0

2−2

𝑠𝑖𝑝𝑧

𝑥

+1−𝑠

0(−)

𝑟×→

𝐼→

;

𝑥,

()

→

𝐼→

;

𝑦=𝑘

∞

𝑛=1 (2+1)

(+1)𝑛𝑛→

𝑒𝑛1 𝑠,→

+𝑛→

𝑜𝑛1 𝑠,→

+𝑝𝑠

02−2

0

2−2

𝑠𝑖𝑝𝑧

𝑦

+1−𝑠

0(−)

𝑟×→

𝐼→

;

𝑦,

()

→

𝐼→

;

𝑧=𝑘

∞

𝑛=0 (2+1)𝑛−1𝐿

𝑛→

𝑒𝑛0 𝑠,→

+𝑝𝑧 0

𝑠𝑖𝑝𝑧

𝑧+1−𝑠

0(−)

𝑟×→

𝐼→

;

𝑧,

()

where 𝑘,𝑝,𝑝𝑧 are given by

𝑘=1

02

𝑠−2

0, ()

𝑝=1

020𝑠−0+1−0

𝑠2, ()

𝑝𝑧 =1

020𝑠−0. ()

is result allows us to express the conductivity tensor in a

mixed Fourier representation as

←→

𝑁𝐿 →

,→

⋅

𝑥=→

𝐼→

;

𝑥, ()

←→

𝑁𝐿 →

,→

⋅

𝑦=→

𝐼→

;

𝑦, ()

←→

𝑁𝐿 →

,→

⋅

𝑧=→

𝐼→

;

𝑧, ()

and one must remember that the amplitude of the incident

planewavehasbeentakenasunity.

5.1. Fourier Transform of the Induced Current. Once we have

an expression for the internal induced current →

𝐼,wecan

take its Fourier transform and, with the aid of Eq. (), we

obtain an expression for the nonlocal conductivity tensor

←→

𝑁𝐿(→

,→

) in Fourier space. Since →

𝐼is given as an

expansion in vector spherical harmonics, the calculation of

its Fourier transform requires the calculation of the Fourier

transform of each spherical harmonic, and this is what we do

next.

First, following [, p. ], we dene the vectors

→

𝑛𝑚 ,=𝑚

𝑛(cos )𝑖𝑚𝜙

𝑟,()

→

𝑛𝑚 ,

=

𝑚

𝑛(cos )

𝜃+𝑚

𝑛(cos )

sin →

𝜙𝑖𝑚𝜙,()

→

𝑛𝑚 ,

=𝑚

𝑛(cos )

sin

𝜃−

𝑚

𝑛(cos )→

𝜙𝑖𝑚𝜙.()

en, the vector spherical harmonics can be written as

→

𝑛𝑚 ,,=𝑛→

𝑛𝑚 ,, ()

→

𝑛𝑚 ,,=(+1)𝑛

→

𝑛𝑚 ,

+1

𝑛→

𝑛𝑚 ,, ()

→

𝑛𝑚 ,,=

𝑛→

𝑛𝑚 ,

+𝑛

→

𝑛𝑚 ,, ()

andwerecallthatinournotation

→

𝑛𝑚 ,,=→

𝑒𝑛𝑚 ,,+→

𝑜𝑛𝑚 ,,,()

→

𝑛𝑚 ,,=→

𝑒𝑛𝑚 ,,+→

𝑜𝑛𝑚 ,,,()

→

𝑛𝑚 ,,=→

𝑒𝑛𝑚 ,,+→

𝑜𝑛𝑚 ,,. ()

Next, it can be shown [, p. ] that the functions (), (),

and()satisfythefollowingintegralrelations:

→

𝑛𝑚 ,,

=𝑛−1

4 2𝜋

0𝜋

0−𝑖𝑘𝑟̂

𝑟⋅̂

𝑟→

𝑛𝑚 ,sin ,()

→

𝑛𝑚 ,,

=𝑛

42𝜋

0𝜋

0−𝑖𝑘𝑟̂

𝑟⋅̂

𝑟→

𝑛𝑚 ,sin ,()

→

𝑛𝑚 ,,

=𝑛−1

4 2𝜋

0𝜋

0−𝑖𝑘𝑟̂

𝑟⋅̂

𝑟→

𝑛𝑚 ,sin ,()

where

⋅

=sin sin cos −+cos cos ,()

Mathematical Problems in Engineering

and this integral relation will be used to evaluate the angular

integration of the Fourier transform of (), (), and ().

us, by using Eq. () and taking its Fourier transform we

obtain

𝑉𝑎→

𝑛𝑚 𝑠−𝑖→

𝑝⋅→

𝑟3=2𝜋

0𝜋

0𝑎

0𝑛𝑠

⋅→

𝑛𝑚 ,−𝑖𝑝𝑟(

̂

𝑝⋅̂

𝑟) sin 2

=𝑎

0𝑛𝑠2𝜋

0𝜋

0→

𝑛𝑚 ,−𝑖𝑝𝑟(

̂

𝑝⋅̂

𝑟)

⋅sin 2.

()

Here we have written →

⋅→

=

(

⋅

) separating

the directional dependence from the magnitude. Also here

𝑎indicates that the integration volume is the volume of

a sphere of radius ,andwehaveusedthesphericalbasis

{,

𝑝,

𝑝}for evaluating the integrals. In this basis, →

=

sin

𝑝cos

𝑝

𝑥+

sin

𝑝sin

𝑝

𝑦+

cos

𝑝

𝑧,where

{

𝑥,

𝑦,

𝑧}is the vector basis in which the spherical harmonics

were dened, and

𝑧is chosen along the wave vector →

of the

incident plane wave.

NowweuseEqs.()and()toget

𝑉𝑎→

𝑛𝑚 𝑠−𝑖→

𝑝⋅→

𝑟3

=𝑎

0𝑛𝑠→

𝑛𝑚 ,

𝑝,

𝑝2

=4

𝑛𝑎

0𝑛𝑠𝑛2→

𝑛𝑚

𝑝,

𝑝.

()

us, one can nally write

→

𝑛𝑚 ,

𝑝,

𝑝

=43

𝑛𝑛,𝑛 𝑠,,2→

𝑛𝑚

𝑝,

𝑝, ()

where the argument of the function indicates the corre-

sponding Fourier space, ,

𝑝,

𝑝, which are the spherical

coordinates in the Fourier space. Also we have dened

𝑛,𝑛1,2,≡1

0𝑛1𝑛2𝑁. ()

erefore, using this result, the Fourier transform of →

𝑒𝑛𝑚

and →

𝑜𝑛𝑚 can be written as

→

𝑒𝑛𝑚 ,

𝑝,

𝑝

=43

𝑛𝑛,𝑛 𝑠,,2Re →

𝑛𝑚

𝑝,

𝑝, ()

→

𝑜𝑛𝑚 ,

𝑝,

𝑝

=43

𝑛𝑛,𝑛 𝑠,,2Im →

𝑛𝑚

𝑝,

𝑝. ()

Furthermore,itcanalsobeshown[]thattheintegral

𝑛,𝑛(𝑠,,2)inEq.()canbereadilyevaluatedtoyield

𝑛,𝑛 1,2,2= 1

2

1−2

22𝑛1𝑛−1 2

−1𝑛−1 1𝑛2. ()

OnecanalsocalculatetheFouriertransformof→

𝑛𝑚 using Eq.

()andthesameprocedureasabove.Aersomealgebraic

simplications, one can write

→

𝑒𝑛𝑚 ,

𝑝,

𝑝=43

𝑛−1 (+1)1,𝑠,

⋅Re →

𝑛𝑚

𝑝,

𝑝+2,𝑠,

⋅Re →

𝑛𝑚

𝑝,

𝑝,

()

→

𝑜𝑛𝑚 ,

𝑝,

𝑝=43

𝑛−1 (+1)1,𝑠,

⋅Im →

𝑛𝑚

𝑝,

𝑝+2,𝑠,Im →

𝑛𝑚

⋅

𝑝,

𝑝,

()

where we have dened

1,𝑠,= 1

3𝑎

0𝑛𝑠

𝑠

𝑛𝜌=𝑝𝑟

+1

𝑛𝜌=𝑘𝑠𝑟𝑛

𝜌=𝑝𝑟2

=𝑛𝑠

𝑠𝑛

,

()

and

2,𝑠,≡ 1

3𝑎

0(+1)𝑛𝑠

𝑠

⋅𝑛

+1

𝑛𝜌=𝑘𝑠𝑟

⋅1

𝑛𝜌=𝑝𝑟2.

()

In fact it is possible to evaluate 2,denedinEq.(),inclosed

form,asitwillbeseenlateron.Itturnsouttobegivenby

2,𝑠,

=

𝑠2−2

𝑛,𝑠

𝑛,𝑠,()

Mathematical Problems in Engineering

where

𝑛(,𝑠),

𝑛(,𝑠) are the coecients in the

result (), () calculated without magnetic response, i.e.,

𝑠=0.

Finally we need the Fourier transform of 𝑖𝑝𝑧,whichin

our coordinate system, where the axis coincides with the

propagation direction →

,couldbewrittenas𝑖→

𝑝⋅→

𝑟,anditis

given by

𝑖→

𝑝⋅→

𝑟=431→

−→

→

−→

.()

Here

indicates the Fourier transform operator. As partic-

ular case, if |→

| = |→

|=

0and is the angle between

them, this implies that |→

−→

|=20sin(/2)and the former

expression can be written as

1→

−→

→

−→

=120sin (/2)

20sin (/2).()

6. Results

With these expressions it is now possible to calculate the

Fourier transform of each vector spherical harmonic and

eachplanewavethatappearintheexpressionsfortheinternal

induced current given in Eqs. (), (), and (). We split the

nonlocal conductivity tensor as

←→

𝑁𝐿 →

,→

=←→

𝑉→

,→

+←→

Σ→

,→

, ()

where ←→

𝑉corresponds to the conductivity associated to the

induced currents in Eqs. (), (), and () located in the

bulk of the sphere while ←→

Σcorresponds to the conductivity

associated to the induced currents located at the surface of

the sphere, that is, the term with (−).

6.1. Nonlocal Conductivity Tensor. Finally, using Eq. () we

arrive at the main results of this work.

Bulk Contribution

←→

𝑉→

,→

⋅

𝑥=43𝑘

∞

𝑛=1 −2+1

(+1)𝑛

⋅𝑛,𝑛 𝑠,,2Im →

𝑛1

𝑝,

𝑝+𝑛

⋅(+1)1,𝑠,Re →

𝑛1

𝑝,

𝑝

+2,𝑠,Re →

𝑛1

𝑝,

𝑝+43𝑝𝑠

0

⋅2−2

0

2−2

𝑠

1→

−→

→

−→

𝑥,

()

er=

p

e

e

ey

ex

ez=

p

F : Figure of the coordinate systems employed.

←→

𝑉→

,→

⋅

𝑦=43𝑘

∞

𝑛=1 2+1

(+1)𝑛

⋅𝑛,𝑛 𝑠,,2Re →

𝑛1

𝑝,

𝑝−𝑛(

+1)1,𝑠,Im →

𝑛1

𝑝,

𝑝

+2,𝑠,Im →

𝑛1

𝑝,

𝑝+43𝑝𝑠

0

⋅2−2

0

2−2

𝑠

1→

−→

→

−→

𝑦,

()

←→

𝑉→

,→

⋅

𝑧=43𝑘

∞

𝑛=1 (2+1)𝐿

𝑛(

+1)1,𝑠,Re →

𝑛0

𝑝,

𝑝()

+2,𝑠,Re →

𝑛0

𝑝,

𝑝+43𝑝𝑧 0

𝑠

⋅1→

−→

→

−→

𝑧,()

where 𝑘,𝑝,𝑝𝑧 are given by Eqs. (), (), and (). We

must remember that the basis {

𝑥,

𝑦,

𝑧}is such that

𝑧

coincide with the →

direction. e angles

𝑝,

𝑝are the

spherical coordinates of →

with respect to →

;seeFigure.

Now we include the surface magnetization currents

induced upon the sphere; for this we must include the

contribution of the surface term () and take its Fourier

transform.

Because the surface term () includes a Dirac delta for

=, then the Fourier transform with variable →

for the

surface term is

→

Σ→

,→

=21− 𝑠

02𝜋

0𝜋

0

𝑟×→

𝐼−𝑖𝑝𝑎̂

𝑝⋅̂

𝑟sin . ()

Mathematical Problems in Engineering

e eld →

𝐼is given by (A.)-(A.), and by the fact that the

vector spherical harmonics can be expressed in terms of the

functions →

𝑚𝑛,→

𝑚𝑛,→

𝑚𝑛 by the expression

→

𝑛𝑚 ,,=𝑛→

𝑛𝑚 ,, ()

→

𝑛𝑚 ,,=(+1)𝑛

→

𝑛𝑚 ,

+1

𝑛→

𝑛𝑚 ,. ()

Also we have that

𝑟×→

𝑚𝑛 ,=0, ()

𝑟×→

𝑚𝑛 ,=−→

𝑚𝑛 ,, ()

𝑟×→

𝑚𝑛 ,=→

𝑚𝑛 ,, ()

and by (), () we get that

2𝜋

0𝜋

0

𝑟×→

𝑛𝑚𝑖𝑝𝑎̂

𝑝⋅̂

𝑟sin

=−4

𝑛→

𝑛𝑚 ,

𝑝,

𝑝, ()

2𝜋

0𝜋

0

𝑟×→

𝑛𝑚𝑖𝑝𝑎̂

𝑝⋅̂

𝑟sin

=4

𝑛−1 →

𝑛𝑚 ,

𝑝,

𝑝. ()

With all this we have the next result.

e surface term given by () associated to the surface

current and by the application of the integration formulas

()-() we nd that its Fourier transform, when we consider

that the external electric eld has a unitary amplitude, gives

the part of the nonlocal conductivity tensor associated to

these surface currents; consider the following.

Surface Contribution

←→

Σ→

,→

⋅

𝑥=421− 𝑠

0∞

𝑛=1 2+1

(+1)

⋅ 𝑠

𝑠𝑛∗

𝑛𝑠+∗

0∗

𝑛

⋅→

𝑜𝑛1 ,

𝑝,

𝑝

− 𝑠

𝑠𝑛𝑛𝑠+∗

0𝑛

⋅→

𝑒𝑛1 ,

𝑝,

𝑝

()

←→

Σ→

,→

⋅

𝑦=−421− 𝑠

0∞

𝑛=1 2+1

(+1)

⋅ 𝑠

𝑠𝑛∗

𝑛𝑠+∗

0∗

𝑛

⋅→

𝑒𝑛1 ,

𝑝,

𝑝

+ 𝑠

𝑠𝑛𝑛𝑠+∗

0𝑛

⋅→

𝑜𝑛1 ,

𝑝,

𝑝

()

←→

Σ→

,→

⋅

𝑧=420

𝑠−1 𝑠

0

∞

𝑛=0 (2+1)

⋅𝐿

𝑛𝑛𝑠→

𝑒𝑛0 ,

𝑝,

𝑝()

where

∗=2−2

0

2−2

𝑠.()

erefore, we have arrived to the main result of our work, the

complete nonlocal generalized conductivity tensor written

as the addition of the bulk conductivity and the surface

conductivity

←→

𝑁𝐿 →

,→

=←→

𝑉→

,→

+←→

Σ→

,→

.()

We must remember thatthe base {

𝑥,

𝑦,

𝑧}is such that vector