# Mie-Type Calculation of the Generalized Electromagnetic Nonlocal Conductivity Tensor for a Sphere and Its Equivalence to the T-Matrix Operator

Authors: E. Gutiérrez-Reyes; Rubén G. Barrera; A. García-Valenzuela
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The purpose of this article is to calculate the generalized nonlocal conductivity tensor of a spherical particle made of isotropic and linear materials. The generalized conductivity tensor is a crucial element in the formulation of the mean-field theories of the electromagnetic response of random particulate systems. This is equivalent to what is called the T-Matrix in multiple scattering theories. Here, a new method is proposed for finding explicit expressions for this tensor directly from its definition including the magnetic response of the spheres. Its relation with theS-Matrix in the theory of single scattering is stated as a generalization. Several approximations and limit cases of possible interest in specific systems are analyzed and the results of some calculations are presented as a numerical example.


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## Body

## 1. Introduction

In a previous paper [1], a reference was made to the generalized nonlocal conductivity tensor, as the main physical concept in the development of a formalism for the calculation of the effective bulk electromagnetic response of randomly located discrete scatterers such as turbid colloids. This approach was later extended to the calculation of the reflection and transmission amplitudes of the average electric field of colloidal suspensions confined in a half-space [2]. The generalized conductivity tensor continued to play a major role in the framework of the theoretical description [3]. Colloidal systems are defined as two-phase systems in which a sparse phase is immersed within a continuous one [4]. While the continuous phase is usually called the matrix, the sparse phase is said to be composed of colloidal particles. When an external electromagnetic field is incident upon a colloidal system, it induces currents within the randomly located colloidal particles and these currents radiate electromagnetic fields which act back upon the particles themselves. The total electromagnetic field conformed in this way can be split into two components, an average component with a smooth spatial variation and traveling in a definite direction, called the coherent beam, and a fluctuating component with abrupt spatial variations and traveling in all different directions, called the diffuse field.The work cited above [2], related to the reflection and transmission amplitudes from colloidal suspensions, dealt with the reflection and transmission amplitudes of the coherent beam for a system of randomly located identical spheres, within a homogeneous matrix with a flat interface occupying a half-space. The 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 average electric field is analogous to the one used in multiple-scattering-theory (MST), to solve similar problems. The 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 [5] 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 first recall that the T-matrix operator and the generalized nonlocal conductivity tensor are actually proportional to each other [1] obeying, essentially, the same integral equation. Then, 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. Therefore, the method described here poses an alternative to the calculation of the T-matrix operator, by simply finding the induced current inside a sphere given an arbitrary external electric field, yielding attractive closed-form expressions for its components. Finally, we want to point out that although the components of the T-matrix operator have been already calculated for the case of a sphere with no intrinsic magnetic response [5], here we provide new closed-form expressions for these components for the more general case of a sphere with an intrinsic magnetic local permeability different from the one of its surroundings, yielding, besides a bulk contribution, also a surface contribution coming from 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

The nonlocal conductivity tensor of an isolated sphere is defined as the linear response of the internal induced current to an external electric field; in its most general form it can be written as(1)J→Ir→;ω=∫Vsσ⃡NLr→,r→′;ω·E→extr→′;ωd3r′where the dependence on the frequency ω on a given quantity denotes the ω-component of the corresponding time Fourier transform. Here, J→I is the total internal induced current, σ⃡NL is the generalized nonlocal conductivity tensor, and E→ext is the external electric field; it could also be called the exciting field. The term generalized means that in J→I all internal induced currents are included, even those that give rise to a magnetic response. In this sense Eq. (1) can be regarded as a generalized nonlocal Ohm’s law.The total internal induced currentJ→I can be calculated, also, by using the local relationship(2)J→Ir→;ω=σslocr→;ωE→Ir→;ωwhere(3)σslocr→;ω=σsωr<a0r>a,where a is the radius of the sphere, σs(ω) is its bulk local conductivity, and EI is the internal electric field, that is, the electric field within the sphere. But the electric field at any point r→ is given by(4)E→r→;ω=E→extr→;ω+iωμ0∫VsG⃡0r→,r→′;ω·J→Ir→′;ωd3r′Here Vs is the volume of the sphere and also(5)G⃡0r→,r→′;ω=1⃡+1k02∇∇gr→,r→′;ωis the free Green’s function dyadic, k0=ω/c, and(6)gr→,r→′;ω=exp⁡ik0r→-r→′4πr→-r→′.Therefore, when one uses Eq. (2) to iterate Eq. (4), one can find the equation for σ⃡NL, as defined by Eq. (1). First, since equation Eq. (4) also holds inside the sphere r→<a, then, if we multiply Eq. (4) by σs, we obtain(7)J→Ir→;ω=∫Vsd3r′δr→-r→′σslocr→′;ωE→extr→′;ω+iωμ0σslocr→;ω∫VsG⃡0r→,r→′;ω·J→Ir→′;ωd3r′.Substituting now Eq. (1) into Eq. (7), we have that(8)∫Vsσ⃡NLr→,r→′;ω·E→extr→′;ωd3r′=∫Vsd3r′σslocr→′;ωδr→-r→′I⃡·E→extr→′;ω+iωμ0σsr→∫VsG⃡0r→,r→′′;ω·∫Vsσ⃡NLr→′′,r→′;ω·E→extr→′;ωd3r′d3r′′.By interchanging the order of integration and comparing the terms within the d3r′ integral, we find(9)σ⃡NLr→,r→′;ω=σslocr→;ωδr→-r→′I⃡+iω0μ0∫VsG⃡0r→,r→′′;ω·σ⃡NLr→′′,r→′;ωd3r′′.Now, we recall the definition of the T-matrix through the following integral relation [6, Eq. 10.4.2]: (10)E→Sr→;ω=∫Vsd3r′∫Vsd3r′′G⃡0r→,r→′;ω·T⃡r→′,r→′′;ω·E→extr→′′;ω.Here E→ext corresponds to the external field within the sphere. E→S is the electric field scattered by the sphere for r→>a. Since one can identify the scattered field E→S with the second term in the right-hand side of Eq. (4), by replacing J→I in the right-hand side of Eq. (4) with the expression given in Eq. (1) it follows immediately that Eq. (10) yields(11)iωμ0σ⃡NLr→,r→′;ω=T⃡r→,r→′;ω.This relation is valid for r→≤a and r→′≤a. Therefore, the calculation of all the components of σ⃡NL can be performed by solving the integral equation given in Eq. (9). This procedure has been already followed for the corresponding integral equation for T-matrix operator [5]. It is pertinent to point out that the definition of the T-matrix operator is given, sometimes, as the linear integral relationship between the scattered field and the external field [5]. In this case the Green’s function dyadic is incorporated in the definition of the T-matrix operator; thus its relationship with the generalized nonlocal conductivity has to be modified correspondingly.


## 3. Calculation Procedure

Here we propose an alternative and simple way to evaluate the components of the nonlocal generalized conductivity tensorσ⃡NL for a sphere, by taking advantage of its physical interpretation. This means that one has only to calculate the current density induced within the sphere by an arbitrary external electric field. In order to see how this is done, let us start by assuming that all the fields are in the frequency domain. Now we consider that an external electromagnetic plane wave is incident upon the sphere, and we write its electric field as(12)E→ext=e^βeip→·r→where p→ is the wave vector, e^β is a unit vector along the direction of polarization, and we are assuming a unit amplitude. Then, according to Eq. (1), the current density J→I induced within the sphere by this external plane wave is given by(13)J→Ir→,p→;e^β=∫Vsσ⃡NLr→,r→′·e^βeip→·r→′d3r′.where e^β as an argument in J→I recalls the polarization of the external electric field. Now, we define the Fourier transform pair of σ⃡NL as(14)σ⃡NLr→,r→′=12π6∫R3∫R3σ⃡NLp→,p→′eip→·r→e-ip→′·r→′d3p′d3p.(15)σ⃡NLp→,p→′=∫R3∫R3σ⃡NLr→,r→′e-ip→·r→eip→′·r→′d3r′d3r.And by taking the Fourier transform in Eq. (13), one can write(16)J→Ip→′,p→;e^β=σ⃡NLp→′,p→·e^β.Finally, the α component of the above equation becomes(17)J→Iαp→′,p→;e^β≡e^α·J→Ip→′,p→;e^β=e^α·σ⃡NLp→′,p→·e^β≡σαβNLp→′,p→This means that the αβ component of the nonlocal generalized conductivity tensor σ⃡NL(p→′,p→) has a straightforward physical interpretation: it is equal to the α component of the p→′ Fourier transform of the current density induced within the sphere by an electromagnetic plane wave with wavevector p→ and polarization e^β.But the induced currentJ→I can be written in terms of the polarization and magnetization material fields, P→ and M→, as(18)J→Ir→=-iωP→r→+∇×M→r→where(19)P→r→=ϵs-ϵ01-θr-aE→Ir→and(20)M→r→=1-θr-a1μ0-1μsB→Ir→Here ϵs and μs refer to the local electric permittivity and the local magnetic permeability of the sphere which are assumed to respond locally and linearly to the electric and magnetic fields (E→I and B→I) within the sphere. Here we have located the center of the sphere at the origin of the coordinate system, and while ϵs and μs are the local electric permittivity and the local magnetic permeability of the sphere, ϵ0 and μ0 denote the corresponding response functions of the medium outside the sphere; here we have a sharp discontinuity of these functions. Therefore, by plugging Eqs. (19) and (20) into Eq. (18), the induced current density can be written as (21)J→Ir→=-iωϵs-ϵ0E→Ir→+1μ0-1μs∇×B→Ir→-1μ0-1μsδr-ae^r×B→I.where E→I and B→I are the electric and magnetic fields inside the sphere and the last term, on the right-hand side, corresponds to an induced surface current. Therefore, the induced current within the sphere is completely determined by the values of the internal electric and magnetic fields, and the calculation of J→I is reduced to the calculation of E→I and B→I.This means that the calculation of all the componentsσαβNL(p→′,p→) can be set as a scattering problem, that is, as the problem of calculating the scattered and internal electromagnetic fields induced by an electromagnetic plane wave incident upon a sphere, for three different polarizations e^β of the incident plane wave. This problem is very similar to the classical problem of the scattering of a plane wave by a sphere, solved first by Gustav Mie in 1908 [7]. The main difference between this problem and the classical Mie solution is that here the wave vector p→ and the wave frequency ω of the incident plane wave are independent of each other, while, in the problem solved by Mie, the incident plane wave is a self-propagating wave whose wave vector and frequency are related through the dispersion relation of the medium surrounding the sphere. This also means that here the incident plane wave has to be generated by external currents capable of fixing its wavevector p→ to any given value. For example, if e^β are chosen as the cartesian unit vectors and the sphere is immersed in a medium with permittivity ϵ0 and permeability μ0, then the external electric field E→ext must satisfy the Maxwell’s equations(22)∇×E→ext=iωμ0H→ext(23)∇×H→ext=J→ext-iωϵ0E→extwhere one includes as a source the external current J→ext. Thus, the external electric field satisfies the vectorial equation(24)∇×∇×E→ext+ω2ϵ0μ0E→ext=iωμ0J→extand the source J→ext is chosen in order to obtain the desired external electric field, say, a plane wave e^βeipz propagating along the z axis, with polarization e^β taking each one of the polarizations {e^x,e^y,e^z}. It can be shown that in this case J→ext becomes(25)J→extr→,p→;e^x=1iωμ0p2-k02eipze^x(26)J→extr→,p→;e^y=1iωμ0p2-k02eipze^y(27)J→extr→,p→;e^z=-1iωμ0k02eipze^z,as it could be verified 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 field with a longitudinal component. Nevertheless, the calculation procedure of the scattered and internal fields follows the standard mathematical process by first expanding the electromagnetic fields on a spherical basis and then setting boundary conditions both at infinity and on the surface of the sphere.


## 4. Spherical Basis and Boundary Conditions

The purpose of this section is to solve Maxwell’s equations for a sphere of radiusa located at the origin, and with the external currents J→ext given by Eqs. (25)-(27). We start by writing Maxwell’s equations as(28)∇×∇×E→+ω2ϵsr→;ωμsr→;ωE→=iωμ0J→extwhere we have assumed, as above, that all fields have an oscillatory time dependence e-iωt, and(29)ϵsr→;ω=ϵsωr<aϵ0ωr>aandμsr→;ω=μsωr<aμ0ωr>a.Thus we start by expanding the external plane wave in a vector spherical harmonic basis; that is, we write eipze^x, eipze^y, eipze^z in the functional basis [8, p. 23](30)M→enm=∇×r→ψenm,M→onm=∇×r→ψonm,(31)N→enm=1p∇×M→enm,N→onm=1p∇×M→onm,(32)L→enm=1p∇ψenm,L→onm=1p∇ψonm,where(33)ψenm=cos⁡mϕPnmcos⁡θznpr,(34)ψonm=sin⁡mϕcos⁡θznpr,where p is the wave number associated to the wave equation that satisfies the vector spherical harmonics(35)∇×∇×F→-p2F→=0,for M→enm, M→onm, N→enm, N→onm, and(36)∇∇·L→+p2L→=0,for L→enm,L→onm. Here Pnm are the associated Legendre polynomials as defined in [9, p. 437] and zn denote either one of spherical Bessel functions:(37)jnρ=π2ρJn+1/2ρ(38)hnρ=π2ρHn+1/2ρ.It can be shown by using the orthogonality property of the vector spherical harmonics that the expansion of a plane wave for the three different polarizations {e^x,e^y,e^z} in a vector spherical harmonic basis is given by [10][pg. 420, Eq. (36)-(38)].(39)E→extr→;e^x=eipze^x=∑n=1∞-2n+1nn+1inM→on1p,r→-iN→en1p,r→(40)E→extr→;e^y=eipze^y=∑n=1∞2n+1nn+1inM→en1p,r→+iN→on1p,r→(41)E→extr→;e^z=eipze^z=∑n=0∞2n+1in-1L→en0p,r→where the second variable in the argument of E→ext denotes the polarization. The magnetic fields are calculated with Faraday’s law(42)H→ext=1iωμ0∇×E→ext.We include the result in Appendix A, in Eqs. (A.1), (A.2), and (A.3). Now, we proceed to solve the scattering problem by assuming that the electric field outside the sphere is the incident external electric field E→ext plus a scattered electric field E→s, while inside the sphere is E→I, and they must obey the electromagnetic boundary conditions at the surface of the sphere. Inside the sphere, the electric field E→I must fulfill the inhomogeneous vectorial wave equation given by Eq. (28), whose solution is the sum of the homogeneous equation, expressed as an expansion of vector spherical harmonics, plus a particular solution(43)E→partr→;e^x=μsμ0p2-k02p2-ks2E→extr→;e^x=μsμ0p2-k02p2-ks2eipze^x,(44)E→partr→;e^y=μsμ0p2-k02p2-ks2E→extr→;e^y=μsμ0p2-k02p2-ks2eipze^y,(45)E→partr→;e^z=ϵ0ϵsE→extr→;e^z=ϵ0ϵseipze^z,which can be written in terms of the external field E→ext. Here k02=ω2ϵ0μ0 while ks2=ω2ϵsμs. Here we are using the notation E→part(r→;e^x) meaning that the particular solution of the electric field depends on the polarization e^x of the exciting field. The magnetic field associated to this electric field is given by Eqs. (A.4), (A.5), and (A.6) in Appendix A. Therefore, the electric field induced inside a sphere with radius a can be written as(46)E→Ir→;e^x=∑n=1∞-2n+1nn+1incnM→on1ks,r→-idnN→en1ks,r→+μsμ0p2-k02p2-ks2eipze^x(47)E→Ir→;e^y=∑n=1∞2n+1nn+1incnM→en1ks,r→+idnN→on1ks,r→+μsμ0p2-k02p2-ks2eipze^y(48)E→Ir→;e^z=∑n=0∞2n+1in-1dnLN→en0ks,r→+ϵ0ϵseipze^zwhere the coefficients cn,dn,dnL are determined by the boundary conditions. The corresponding magnetic field is given by Eqs. (A.7), (A.8), and (A.9), where the magnetic field H→ext is given by (A.1)-(A.2), for the three different polarizations of the external electric field E→ext. We now assume that the scattered field can be written as(49)E→sr→;e^x=∑n=1∞-2n+1nn+1inianN→en13k0,r→-bnM→on13k0,r→,(50)E→sr→;e^y=∑n=1∞-2n+1nn+1inianN→on13k0,r→+bnM→en13k0,r→,(51)E→sr→;e^z=∑n=1∞2n+1in-1anLN→en03k0,r→,where the index (3) indicates that the functional dependence of the vector spherical harmonics is through the spherical Hankel function hn(k0r). The scattered magnetic field is a consequence of Faraday’s law and is given by Eqs. (A.10), (A.11), and (A.12) in Appendix A.Now we apply the boundary conditions for the electromagnetic field that consist in the continuity of the parallel component, of both the electric and magnetic fields, at the surface of the spherer=a. In a spherical basis these boundary conditions can be written as(52)E→extθ+E→sθ=E→IθandH→extθ+H→sθ=H→Iθ,and(53)E→extϕ+E→sϕ=E→IϕandH→extϕ+H→sϕ=H→Iϕ.The first set of boundary conditions given in Eqs. (52) yields the following expressions for the expansion coefficients:(54)anpa,k0a,ksa=1-αJn∗paks/μsjnksa-1-α∗p/μ0jnpaJn∗ksaHn∗k0aks/μsjnksa-k0/μ0hnk0aJn∗ksa,(55)bnpa,k0a,ksa=1-αjnpaks/μsJn∗ksa-1-α∗p/μ0Jn∗pajnksahnk0aks/μsJn∗ksa-k0/μ0Hn∗k0ajnksa,(56)cnpa,k0a,ksa=hnk0a1-α∗p/μ0Jn∗pa-k0/μ0Hn∗k0a1-αjnpahnk0aks/μsJn∗ksa-k0/μ0Hn∗k0ajnksa,(57)dnpa,k0a,ksa=Hn∗k0a1-α∗p/μ0jnpa-k0/μ0hnk0a1-αJn∗paHn∗k0aks/μsjnksa-k0/μ0hnk0aJn∗ksa,where(58)Jn∗ρ=1ρddρρjnρ,Hn∗ρ=1ρddρρhnρ,(59)α=μsμ0p2-k02p2-ks2,α∗=p2-k02p2-ks2.The second set of boundary conditions given in Eqs. (53) yields, for an external electric field of the form e^zeipz, the following expressions for the expansion coefficients of the scattered field in Eq. (51) and the internal electric field in Eq. (48):(60)anLpa,k0a,ksa=μ0μsksk0jnksahnk0adnLpa,(61)dnLpa,k0a,ksa=1-ϵ0/ϵsjnpa/pahnk0aJn∗ksahnk0a-μ0/μsks/k0jnksaHn∗k0a.Once the internal electric field has been calculated, we are able to calculate the current density J→I induced within the sphere with the aid of Eq. (21) that we rewrite as(62)J→Ir→=-iωϵs-ϵ0E→Ir→+μsμ0-1∇×H→Ir→+1-μsμ0δr-ae^r×H→Ir→,where H→I is given in Eqs. (A.7)-(A.9) for each polarization of the external electric field.


## 5. The Induced Current

Now one substitutes the internal electric field for each polarization, as given in Eqs. (46), (47), and (48), into equation (62), to arrive at the following result.The induced current inside the sphere for each polarization{e^x,e^y,e^z} of the incident field with arbitrary wave vector p→ is(63)J→Ir→;e^x=γk∑n=1∞-2n+1nn+1incnpaM→on1ks,r→-idnpaN→en1ks,r→+γpμsμ0p2-k02p2-ks2eipze^x+1-μsμ0δr-ae^r×H→Ir→;e^x,(64)J→Ir→;e^y=γk∑n=1∞2n+1nn+1incnpaM→en1ks,r→+idnpaN→on1ks,r→+γpμsμ0p2-k02p2-ks2eipze^y+1-μsμ0δr-ae^r×H→Ir→;e^y,(65)J→Ir→;e^z=γk∑n=0∞2n+1in-1dnLpaN→en0ks,r→+γpzϵ0ϵseipze^z+1-μsμ0δr-ae^r×H→Ir→;e^z,where γk,γp,γpz are given by(66)γk=1iωμ0ks2-k02,(67)γp=1iωμ0ω2μ0ϵs-ϵ0+1-μ0μsp2,(68)γpz=1iωμ0ω2μ0ϵs-ϵ0.This result allows us to express the conductivity tensor in a mixed Fourier representation as(69)σ⃡NLr→,p→·e^x=J→Ir→;e^x,(70)σ⃡NLr→,p→·e^y=J→Ir→;e^y,(71)σ⃡NLr→,p→·e^z=J→Ir→;e^z,and one must remember that the amplitude of the incident plane wave has been taken as unity.


### 5.1. Fourier Transform of the Induced Current

Once we have an expression for the internal induced currentJ→I, we can take its Fourier transform and, with the aid of Eq. (16), we obtain an expression for the nonlocal conductivity tensor σ⃡NL(p→′,p→) in Fourier space. Since J→I is 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 [8, p. 27], we define the vectors(72)P→nmθ,ϕ=Pnmcos⁡θeimϕe^r,(73)B→nmθ,ϕ=ddθPnmcos⁡θe^θ+imPnmcos⁡θsin⁡θe→ϕeimϕ,(74)C→nmθ,ϕ=imPnmcos⁡θsin⁡θe^θ-ddθPnmcos⁡θe→ϕeimϕ.Then, the vector spherical harmonics can be written as(75)M→nmρ,θ,ϕ=jnρC→nmθ,ϕ,(76)N→nmρ,θ,ϕ=nn+1jnρρP→nmθ,ϕ+1ρddρρjnρB→nmθ,ϕ,(77)L→nmρ,θ,ϕ=ddρjnρP→nmθ,ϕ+jnρρB→nmθ,ϕ,and we recall that in our notation(78)M→nmρ,θ,ϕ=M→enmρ,θ,ϕ+iM→onmρ,θ,ϕ,(79)N→nmρ,θ,ϕ=N→enmρ,θ,ϕ+iN→onmρ,θ,ϕ,(80)L→nmρ,θ,ϕ=L→enmρ,θ,ϕ+iL→onmρ,θ,ϕ.Next, it can be shown [8, p. 28] that the functions (72), (73), and (74) satisfy the following integral relations:(81)L→nmkr,θ,ϕ=in-14π∫02π∫0πe-ikrr^·r^′P→nmθ′,ϕ′sin⁡θ′dθ′dϕ′,(82)M→nmkr,θ,ϕ=in4π∫02π∫0πe-ikrr^·r^′C→nmθ′,ϕ′sin⁡θ′dθ′dϕ′,(83)N→nmkr,θ,ϕ=in-14π∫02π∫0πe-ikrr^·r^′B→nmθ′,ϕ′sin⁡θ′dθ′dϕ′,where(84)r^·r^′=sin⁡θsin⁡θ′cos⁡ϕ-ϕ′+cos⁡θcos⁡θ′,and this integral relation will be used to evaluate the angular integration of the Fourier transform of (72), (73), and (74). Thus, by using Eq. (75) and taking its Fourier transform we obtain(85)∫VaM→nmksre-ip→′·r→d3r=∫02π∫0π∫0ajnksrC→nmθ,ϕe-ip′rp^′·r^sin⁡θr2drdθdϕ=∫0ajnksr∫02π∫0πC→nmθ,ϕe-ip′rp^′·r^sin⁡θdθdϕr2dr.Here we have written p→′·r→=p′r(p^′·r^) separating the directional dependence from the magnitude. Also here Va indicates that the integration volume is the volume of a sphere of radius a, and we have used the spherical basis {p′,θp′,ϕp′} for evaluating the integrals. In this basis, p→′=p′sin⁡θp′cos⁡ϕp′e^x+p′sin⁡θp′sin⁡ϕp′e^y+p′cos⁡θp′e^z, where {e^x,e^y,e^z} is the vector basis in which the spherical harmonics were defined, and e^z is chosen along the wave vector k→ of the incident plane wave.Now we use Eqs. (82) and (75) to get(86)∫VaM→nmksre-ip→′·r→d3r=∫0ajnksrM→nmp′r,θp′,ϕp′r2dr=4πin∫0ajnksrjnp′rr2drC→nmθp′,ϕp′.Thus, one can finally write(87)M→nmp′,θp′,ϕp′=4πa3inIn,nksa,p′a,2C→nmθp′,ϕp′,where the argument of the function indicates the corresponding Fourier space, p′,θp′,ϕp′, which are the spherical coordinates in the Fourier space. Also we have defined(88)In,n′x1,x2,N≡∫01jnx1ujn′x2uuNdu.Therefore, using this result, the Fourier transform of M→enm and M→onm can be written as(89)M→enmp′,θp′,ϕp′=4πa3inIn,nksa,p′a,2Re⁡C→nmθp′,ϕp′,(90)M→onmp′,θp′,ϕp′=4πa3inIn,nksa,p′a,2Im⁡C→nmθp′,ϕp′.Furthermore, it can also be shown [1] that the integral In,n(ksa,p′a,2) in Eq. (87) can be readily evaluated to yield(91)In,nx1,x2,2=1x12-x22x2jnx1jn-1x2-x1jn-1x1jnx2.One can also calculate the Fourier transform of N→nm using Eq. (76) and the same procedure as above. After some algebraic simplifications, one can write(92)N→enmp′,θp′,ϕp′=4πa3in-1nn+1J1n,ksa,p′aRe⁡P→nmθp′,ϕp′+J2n,ksa,p′aRe⁡B→nmθp′,ϕp′,(93)N→onmp′,θp′,ϕp′=4πa3in-1nn+1J1n,ksa,p′aIm⁡P→nmθp′,ϕp′+J2n,ksa,p′aIm⁡B→nmθp′,ϕp′,where we have defined(94)J1n,ksa,p′a=1a3∫0ajnksrksrddρjnρρ=p′r+1ρddρρjnρρ=ksrjnρρρ=p′rr2dr=jnaksaksjnap′ap′,and(95)J2n,ksa,p′a≡1a3∫0ann+1jnksrksrjnp′rp′r+1ρddρρjnρρ=ksr1ρddρρjnρρ=p′rr2dr.In fact it is possible to evaluate J2, defined in Eq. (95), in closed form, as it will be seen later on. It turns out to be given by(96)J2n,ksa,p′a=ip′aksa2-p′a2a~np′a,ksad~np′a,ksa,where a~n(p′a,ksa), d~n(p′a,ksa) are the coefficients in the result (54), (57) calculated without magnetic response, i.e., μs=μ0.Finally we need the Fourier transform ofeipz, which in our coordinate system, where the z axis coincides with the propagation direction p→, could be written as eip→·r→, and it is given by(97)F^eip→·r→=4πa3j1ap→-p→′ap→-p→′.Here F^ indicates the Fourier transform operator. As particular case, if p→=p→′=k0 and θ is the angle between them, this implies that p→-p→′=2k0sin⁡θ/2 and the former expression can be written as(98)j1ap→-p→′ap→-p→′=j12ak0sin⁡θ/22ak0sin⁡θ/2.


## 5.1. Fourier Transform of the Induced Current

Once we have an expression for the internal induced currentJ→I, we can take its Fourier transform and, with the aid of Eq. (16), we obtain an expression for the nonlocal conductivity tensor σ⃡NL(p→′,p→) in Fourier space. Since J→I is 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 [8, p. 27], we define the vectors(72)P→nmθ,ϕ=Pnmcos⁡θeimϕe^r,(73)B→nmθ,ϕ=ddθPnmcos⁡θe^θ+imPnmcos⁡θsin⁡θe→ϕeimϕ,(74)C→nmθ,ϕ=imPnmcos⁡θsin⁡θe^θ-ddθPnmcos⁡θe→ϕeimϕ.Then, the vector spherical harmonics can be written as(75)M→nmρ,θ,ϕ=jnρC→nmθ,ϕ,(76)N→nmρ,θ,ϕ=nn+1jnρρP→nmθ,ϕ+1ρddρρjnρB→nmθ,ϕ,(77)L→nmρ,θ,ϕ=ddρjnρP→nmθ,ϕ+jnρρB→nmθ,ϕ,and we recall that in our notation(78)M→nmρ,θ,ϕ=M→enmρ,θ,ϕ+iM→onmρ,θ,ϕ,(79)N→nmρ,θ,ϕ=N→enmρ,θ,ϕ+iN→onmρ,θ,ϕ,(80)L→nmρ,θ,ϕ=L→enmρ,θ,ϕ+iL→onmρ,θ,ϕ.Next, it can be shown [8, p. 28] that the functions (72), (73), and (74) satisfy the following integral relations:(81)L→nmkr,θ,ϕ=in-14π∫02π∫0πe-ikrr^·r^′P→nmθ′,ϕ′sin⁡θ′dθ′dϕ′,(82)M→nmkr,θ,ϕ=in4π∫02π∫0πe-ikrr^·r^′C→nmθ′,ϕ′sin⁡θ′dθ′dϕ′,(83)N→nmkr,θ,ϕ=in-14π∫02π∫0πe-ikrr^·r^′B→nmθ′,ϕ′sin⁡θ′dθ′dϕ′,where(84)r^·r^′=sin⁡θsin⁡θ′cos⁡ϕ-ϕ′+cos⁡θcos⁡θ′,and this integral relation will be used to evaluate the angular integration of the Fourier transform of (72), (73), and (74). Thus, by using Eq. (75) and taking its Fourier transform we obtain(85)∫VaM→nmksre-ip→′·r→d3r=∫02π∫0π∫0ajnksrC→nmθ,ϕe-ip′rp^′·r^sin⁡θr2drdθdϕ=∫0ajnksr∫02π∫0πC→nmθ,ϕe-ip′rp^′·r^sin⁡θdθdϕr2dr.Here we have written p→′·r→=p′r(p^′·r^) separating the directional dependence from the magnitude. Also here Va indicates that the integration volume is the volume of a sphere of radius a, and we have used the spherical basis {p′,θp′,ϕp′} for evaluating the integrals. In this basis, p→′=p′sin⁡θp′cos⁡ϕp′e^x+p′sin⁡θp′sin⁡ϕp′e^y+p′cos⁡θp′e^z, where {e^x,e^y,e^z} is the vector basis in which the spherical harmonics were defined, and e^z is chosen along the wave vector k→ of the incident plane wave.Now we use Eqs. (82) and (75) to get(86)∫VaM→nmksre-ip→′·r→d3r=∫0ajnksrM→nmp′r,θp′,ϕp′r2dr=4πin∫0ajnksrjnp′rr2drC→nmθp′,ϕp′.Thus, one can finally write(87)M→nmp′,θp′,ϕp′=4πa3inIn,nksa,p′a,2C→nmθp′,ϕp′,where the argument of the function indicates the corresponding Fourier space, p′,θp′,ϕp′, which are the spherical coordinates in the Fourier space. Also we have defined(88)In,n′x1,x2,N≡∫01jnx1ujn′x2uuNdu.Therefore, using this result, the Fourier transform of M→enm and M→onm can be written as(89)M→enmp′,θp′,ϕp′=4πa3inIn,nksa,p′a,2Re⁡C→nmθp′,ϕp′,(90)M→onmp′,θp′,ϕp′=4πa3inIn,nksa,p′a,2Im⁡C→nmθp′,ϕp′.Furthermore, it can also be shown [1] that the integral In,n(ksa,p′a,2) in Eq. (87) can be readily evaluated to yield(91)In,nx1,x2,2=1x12-x22x2jnx1jn-1x2-x1jn-1x1jnx2.One can also calculate the Fourier transform of N→nm using Eq. (76) and the same procedure as above. After some algebraic simplifications, one can write(92)N→enmp′,θp′,ϕp′=4πa3in-1nn+1J1n,ksa,p′aRe⁡P→nmθp′,ϕp′+J2n,ksa,p′aRe⁡B→nmθp′,ϕp′,(93)N→onmp′,θp′,ϕp′=4πa3in-1nn+1J1n,ksa,p′aIm⁡P→nmθp′,ϕp′+J2n,ksa,p′aIm⁡B→nmθp′,ϕp′,where we have defined(94)J1n,ksa,p′a=1a3∫0ajnksrksrddρjnρρ=p′r+1ρddρρjnρρ=ksrjnρρρ=p′rr2dr=jnaksaksjnap′ap′,and(95)J2n,ksa,p′a≡1a3∫0ann+1jnksrksrjnp′rp′r+1ρddρρjnρρ=ksr1ρddρρjnρρ=p′rr2dr.In fact it is possible to evaluate J2, defined in Eq. (95), in closed form, as it will be seen later on. It turns out to be given by(96)J2n,ksa,p′a=ip′aksa2-p′a2a~np′a,ksad~np′a,ksa,where a~n(p′a,ksa), d~n(p′a,ksa) are the coefficients in the result (54), (57) calculated without magnetic response, i.e., μs=μ0.Finally we need the Fourier transform ofeipz, which in our coordinate system, where the z axis coincides with the propagation direction p→, could be written as eip→·r→, and it is given by(97)F^eip→·r→=4πa3j1ap→-p→′ap→-p→′.Here F^ indicates the Fourier transform operator. As particular case, if p→=p→′=k0 and θ is the angle between them, this implies that p→-p→′=2k0sin⁡θ/2 and the former expression can be written as(98)j1ap→-p→′ap→-p→′=j12ak0sin⁡θ/22ak0sin⁡θ/2.


## 6. Results

With these expressions it is now possible to calculate the Fourier transform of each vector spherical harmonic and each plane wave that appear in the expressions for the internal induced current given in Eqs. (63), (64), and (65). We split the nonlocal conductivity tensor as(99)σ⃡NLp→′,p→=σ⃡Vp→′,p→+σ⃡Σp→′,p→,where σ⃡V corresponds to the conductivity associated to the induced currents in Eqs. (63), (64), and (65) located in the bulk of the sphere while σ⃡Σ corresponds to the conductivity associated to the induced currents located at the surface of the sphere, that is, the term with δ(r-a).


### 6.1. Nonlocal Conductivity Tensor

Finally, using Eq. (16) we arrive at the main results of this work.Bulk Contribution (100) σ ⃡ V p → ′ , p → · e ^ x = 4 π a 3 γ k ∑ n = 1 ∞ - 2 n + 1 n n + 1 c n p a I n , n k s a , p ′ a , 2 Im ⁡ C → n 1 θ p ′ , ϕ p ′ + d n p a n n + 1 J 1 n , k s a , p ′ a Re ⁡ P → n 1 θ p ′ , ϕ p ′ + J 2 n , k s a , p ′ a Re ⁡ B → n 1 θ p ′ , ϕ p ′ + 4 π a 3 γ p μ s μ 0 p 2 - k 0 2 p 2 - k s 2 j 1 a p → - p → ′ a p → - p → ′ e ^ x , (101) σ ⃡ V p → ′ , p → · e ^ y = 4 π a 3 γ k ∑ n = 1 ∞ 2 n + 1 n n + 1 c n p a I n , n k s a , p ′ a , 2 Re ⁡ C → n 1 θ p ′ , ϕ p ′ - d n p a n n + 1 J 1 n , k s a , p ′ a Im ⁡ P → n 1 θ p ′ , ϕ p ′ + J 2 n , k s a , p ′ a Im ⁡ B → n 1 θ p ′ , ϕ p ′ + 4 π a 3 γ p μ s μ 0 p 2 - k 0 2 p 2 - k s 2 j 1 a p → - p → ′ a p → - p → ′ e ^ y , (102) σ ⃡ V p → ′ , p → · e ^ z = 4 π a 3 γ k ∑ n = 1 ∞ 2 n + 1 d n L p a n n + 1 J 1 n , k s a , p ′ a Re ⁡ P → n 0 θ p ′ , ϕ p ′ (103) + J 2 n , k s a , p ′ a Re ⁡ B → n 0 θ p ′ , ϕ p ′ + 4 π a 3 γ p z ϵ 0 ϵ s j 1 a p → - p → ′ a p → - p → ′ e ^ z ,where γk,γp,γpz are given by Eqs. (66), (67), and (68). We must remember that the basis {e^x,e^y,e^z} is such that e^z coincide with the p→ direction. The angles θp′,ϕp′ are the spherical coordinates of p→′ with respect to p→; see Figure 1.Figure 1
Figure of the coordinate systems employed.Now we include the surface magnetization currents induced upon the sphere; for this we must include the contribution of the surface term (62) and take its Fourier transform.Because the surface term (62) includes a Dirac delta for r=a, then the Fourier transform with variable p→′ for the surface term is(104)J→Σp→′,p→=a21-μsμ0∫02π∫0πe^r×H→Ie-ip′ap^′·r^sin⁡θdθdϕ.The field H→I is given by (A.7)-(A.9), and by the fact that the vector spherical harmonics can be expressed in terms of the functions P→mn,B→mn,C→mn by the expression(105)M→nmρ,θ,ϕ=jnρC→nmθ,ϕ,(106)N→nmρ,θ,ϕ=nn+1jnρρP→nmθ,ϕ+1ρddρρjnρB→nmθ,ϕ.Also we have that(107)e^r×P→mnθ,ϕ=0,(108)e^r×B→mnθ,ϕ=-C→mnθ,ϕ,(109)e^r×C→mnθ,ϕ=B→mnθ,ϕ,and by (82), (83) we get that(110)∫02π∫0πe^r×B→nmeip′ap^′·r^sin⁡θdθdϕ=-4πinM→nmp′a,θp′,ϕp′,(111)∫02π∫0πe^r×C→nmeip′ap^′·r^sin⁡θdθdϕ=4πin-1N→nmp′a,θp′,ϕp′.With all this we have the next result.The surface term given by (62) associated to the surface current and by the application of the integration formulas (82)-(83) we find that its Fourier transform, when we consider that the external electric field has a unitary amplitude, gives the part of the nonlocal conductivity tensor associated to these surface currents; consider the following.Surface Contribution (112) σ ⃡ Σ p → ′ , p → · e ^ x = 4 π a 2 1 - μ s μ 0 ∑ n = 1 ∞ 2 n + 1 n n + 1 k s i ω μ s c n p a J n ∗ k s a + α ∗ p i ω μ 0 J n ∗ p a M → o n 1 p ′ a , θ p ′ , ϕ p ′ - k s i ω μ s d n p a j n k s a + α ∗ p i ω μ 0 j n p a N → e n 1 p ′ a , θ p ′ , ϕ p ′ (113) σ ⃡ Σ p → ′ , p → · e ^ y = - 4 π a 2 1 - μ s μ 0 ∑ n = 1 ∞ 2 n + 1 n n + 1 k s i ω μ s c n p a J n ∗ k s a + α ∗ p i ω μ 0 J n ∗ p a M → e n 1 p ′ a , θ p ′ , ϕ p ′ + k s i ω μ s d n p a j n k s a + α ∗ p i ω μ 0 j n p a N → o n 1 p ′ a , θ p ′ , ϕ p ′ (114) σ ⃡ Σ p → ′ , p → · e ^ z = 4 π a 2 μ 0 μ s - 1 k s i ω μ 0 ∑ n = 0 ∞ 2 n + 1 d n L p a j n k s a N → e n 0 p ′ a , θ p ′ , ϕ p ′where(115)α∗=p2-k02p2-ks2.Therefore, 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(116)σ⃡NLp→′,p→=σ⃡Vp→′,p→+σ⃡Σp→′,p→.We must remember that the base {e^x,e^y,e^z} is such that vector e^z coincides with the direction p→, and in this basis the nonlocal conductivity tensor only has five nonzero components; the rest are zero.


## 6.1. Nonlocal Conductivity Tensor

Finally, using Eq. (16) we arrive at the main results of this work.Bulk Contribution (100) σ ⃡ V p → ′ , p → · e ^ x = 4 π a 3 γ k ∑ n = 1 ∞ - 2 n + 1 n n + 1 c n p a I n , n k s a , p ′ a , 2 Im ⁡ C → n 1 θ p ′ , ϕ p ′ + d n p a n n + 1 J 1 n , k s a , p ′ a Re ⁡ P → n 1 θ p ′ , ϕ p ′ + J 2 n , k s a , p ′ a Re ⁡ B → n 1 θ p ′ , ϕ p ′ + 4 π a 3 γ p μ s μ 0 p 2 - k 0 2 p 2 - k s 2 j 1 a p → - p → ′ a p → - p → ′ e ^ x , (101) σ ⃡ V p → ′ , p → · e ^ y = 4 π a 3 γ k ∑ n = 1 ∞ 2 n + 1 n n + 1 c n p a I n , n k s a , p ′ a , 2 Re ⁡ C → n 1 θ p ′ , ϕ p ′ - d n p a n n + 1 J 1 n , k s a , p ′ a Im ⁡ P → n 1 θ p ′ , ϕ p ′ + J 2 n , k s a , p ′ a Im ⁡ B → n 1 θ p ′ , ϕ p ′ + 4 π a 3 γ p μ s μ 0 p 2 - k 0 2 p 2 - k s 2 j 1 a p → - p → ′ a p → - p → ′ e ^ y , (102) σ ⃡ V p → ′ , p → · e ^ z = 4 π a 3 γ k ∑ n = 1 ∞ 2 n + 1 d n L p a n n + 1 J 1 n , k s a , p ′ a Re ⁡ P → n 0 θ p ′ , ϕ p ′ (103) + J 2 n , k s a , p ′ a Re ⁡ B → n 0 θ p ′ , ϕ p ′ + 4 π a 3 γ p z ϵ 0 ϵ s j 1 a p → - p → ′ a p → - p → ′ e ^ z ,where γk,γp,γpz are given by Eqs. (66), (67), and (68). We must remember that the basis {e^x,e^y,e^z} is such that e^z coincide with the p→ direction. The angles θp′,ϕp′ are the spherical coordinates of p→′ with respect to p→; see Figure 1.Figure 1
Figure of the coordinate systems employed.Now we include the surface magnetization currents induced upon the sphere; for this we must include the contribution of the surface term (62) and take its Fourier transform.Because the surface term (62) includes a Dirac delta for r=a, then the Fourier transform with variable p→′ for the surface term is(104)J→Σp→′,p→=a21-μsμ0∫02π∫0πe^r×H→Ie-ip′ap^′·r^sin⁡θdθdϕ.The field H→I is given by (A.7)-(A.9), and by the fact that the vector spherical harmonics can be expressed in terms of the functions P→mn,B→mn,C→mn by the expression(105)M→nmρ,θ,ϕ=jnρC→nmθ,ϕ,(106)N→nmρ,θ,ϕ=nn+1jnρρP→nmθ,ϕ+1ρddρρjnρB→nmθ,ϕ.Also we have that(107)e^r×P→mnθ,ϕ=0,(108)e^r×B→mnθ,ϕ=-C→mnθ,ϕ,(109)e^r×C→mnθ,ϕ=B→mnθ,ϕ,and by (82), (83) we get that(110)∫02π∫0πe^r×B→nmeip′ap^′·r^sin⁡θdθdϕ=-4πinM→nmp′a,θp′,ϕp′,(111)∫02π∫0πe^r×C→nmeip′ap^′·r^sin⁡θdθdϕ=4πin-1N→nmp′a,θp′,ϕp′.With all this we have the next result.The surface term given by (62) associated to the surface current and by the application of the integration formulas (82)-(83) we find that its Fourier transform, when we consider that the external electric field has a unitary amplitude, gives the part of the nonlocal conductivity tensor associated to these surface currents; consider the following.Surface Contribution (112) σ ⃡ Σ p → ′ , p → · e ^ x = 4 π a 2 1 - μ s μ 0 ∑ n = 1 ∞ 2 n + 1 n n + 1 k s i ω μ s c n p a J n ∗ k s a + α ∗ p i ω μ 0 J n ∗ p a M → o n 1 p ′ a , θ p ′ , ϕ p ′ - k s i ω μ s d n p a j n k s a + α ∗ p i ω μ 0 j n p a N → e n 1 p ′ a , θ p ′ , ϕ p ′ (113) σ ⃡ Σ p → ′ , p → · e ^ y = - 4 π a 2 1 - μ s μ 0 ∑ n = 1 ∞ 2 n + 1 n n + 1 k s i ω μ s c n p a J n ∗ k s a + α ∗ p i ω μ 0 J n ∗ p a M → e n 1 p ′ a , θ p ′ , ϕ p ′ + k s i ω μ s d n p a j n k s a + α ∗ p i ω μ 0 j n p a N → o n 1 p ′ a , θ p ′ , ϕ p ′ (114) σ ⃡ Σ p → ′ , p → · e ^ z = 4 π a 2 μ 0 μ s - 1 k s i ω μ 0 ∑ n = 0 ∞ 2 n + 1 d n L p a j n k s a N → e n 0 p ′ a , θ p ′ , ϕ p ′where(115)α∗=p2-k02p2-ks2.Therefore, 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(116)σ⃡NLp→′,p→=σ⃡Vp→′,p→+σ⃡Σp→′,p→.We must remember that the base {e^x,e^y,e^z} is such that vector e^z coincides with the direction p→, and in this basis the nonlocal conductivity tensor only has five nonzero components; the rest are zero.


## 7. Some Special Cases

### 7.1. Nonlocal Conductivity Whenp→′=p→=k0 and Its Relation with the S Matrix

In this section we explore an approximation that we call the cone-light approximation and that corresponds to the case whenp→′=p→=k0; this means that the magnitudes of the wave vectors p→ and p→′ are equal to the wave number in vacuum. This is the most used form of the nonlocal conductivity tensor when related to the scattering S-matrix [11]. We will see that this substitution corresponds to the problem of calculating the scatter field in the far field region. To make the relationship with the S matrix when the magnitudes of p→ and p→′ are k0, we will denote p→ as k→i and p→′ as k→s, the wavevectors of the incident and scattered waves, respectively. However, we must remark that the treatment that follows is the most general, going beyond the so-called Mie theory since, even though ki=ks=k0, we still allow for incident and scattered waves with longitudinal components. As discussed previously, supporting longitudinal waves requires the existence of external currents.The electric field scatter by a sphere due to the presence of an incident plane wave is given, in the far field region as in [12, p. 63]:(117)E→sr→;e^i=eikr-ikrS⃡θ,ϕ·e^iE0,where S⃡(θ,ϕ) is the scattering matrix, θ is the angle between the incident wave vector and the observation position, E0 is the incident electric field amplitude, and e^i is its polarization. The scattered field by an isolated sphere, in terms of the induced current inside the sphere (1), is of the form(118)E→sr→;e^i=iωμ0∫R3d3r′G⃡r→,r→′·∫R3d3r′′σ⃡NLr→′,r→′′·e^ieik→i·r→′′,where the polarization of the incident wave is e^i and its wave vector k→i is such that k→i=k0, where k0 is the vacuum wave number.It is possible to write the free space Green’s function of the electromagnetic field as [8, pg. 55](119)G⃡r→,r→′=I⃡+1k02∇∇eik0r→-r→′4πr→-r→′.Substituting this expression in (118) and considering that r→>>0, then(120)k0r→-r→′=k0r1-2r^·r→′+r′2r21/2≈k0r-k→s·r→′,where k→s=k0r^, and r^ is the radial unitary vector. Using this far-field approximation, the electric field can be written as(121)E→sr→;e^i=iωμ0I⃡+1k02∇∇eik0r4πr∫R3d3r′∫R3d3r′′e-ik→s·r→′·σ⃡NLr→′,r→′′eik→i·r→′′·e^i.With these considerations and to order O(1/r) the electric field is written as(122)E→sr→;e^i=iωμ0eik0r4πrI⃡-k^sk^s·σ⃡NLk→s,k→i·e^i+O1r2.In addition, it is possible to get the expression for the electric field in the far-field region, when the incident wave is a plane wave propagating at vacuum from (49); if we make p=k0, thus the scatter electric field for each polarization e^x,e^y,e^z of the incident wave is(123)E→sr→;e^x=∑n=1∞2n+1nn+1inbnM→on13k0r-ianN→en13k0r(124)E→sr→;e^y=∑n=1∞-2n+1nn+1inbnM→en13k0r+ianN→on13k0r(125)E→sr→;e^z=∑n=1∞2n+1in-1anLN→en03k0r,where the second variable of E→s(r→;e^i) indicates the polarization of the incident wave, k0 is the vacuum wave number, the index (3) indicates that the radial part is given by the Hankel’s functions, and we have taken as unitary the amplitude of the incident wave. Here Eqs. (123) and (124) are exactly the Mie solution of the transverse scattered waves by a sphere due to an incident transverse wave, where the connection with the Mie theory is made evident [7]. With respect to the electric field in (125), this kind of scatter field requires the existence of an external current given by (27) with p→=k0. This response of the scattered field in the radial and transversal components of the sphere when excited by a longitudinal electromagnetic mode and the mixing with the transversal radiated field will be important in the formulation of effective field theories of many sphere scatterers, because the exciting field in a many-sphere system could have in effect a longitudinal component [1]. This opens the possibility of having longitudinal modes of electromagnetic radiation and needs the design of some experiment for its verification.The expressions for the Hankel functions and the vector spherical harmonic are given in AppendixB (B.1)-(B.8).Substituting these expressions in Eqs. (49) and (50) one finds that the electric scattered field to order O(1/r) is given by(126)E→sr→,e^x=eikr-ikrSθyθcos⁡ϕe^θ-Sϕxθsin⁡ϕe→ϕ,(127)E→sr→,e^y=eikr-ikrSθyθsin⁡ϕe^θ+Sϕxθcos⁡ϕe→ϕ,(128)E→sr→,e^z=eikr-ikrSθzθe^θ,where we have defined the scattering functions as(129)Sϕxθ=-∑n=1∞2n+1nn+1bnk0addθPn1cos⁡θ+ank0aPn1cos⁡θsin⁡θ,(130)Sθyθ=-∑n=1∞2n+1nn+1bnk0aPn1cos⁡θsin⁡θ+ank0addθPn1cos⁡θ,(131)Sθzθ=-∑n=1∞2n+1anLk0addθPncos⁡θ.(131) is a new term in the conventional S-matrix theory and comes from the fact that the sphere is excited by a longitudinal electric field. In addition, using the fact that limθ→0d/dθPn0(cos⁡θ)=0 we have that(132)Sθz0=0,and since that limθ→0Pn1cos⁡θ/sinθ=limθ→0d/dθPn1(cos⁡θ)=-(1/2)n(n+1), this implies(133)S0=Sθy0=Sϕx0=12∑n=1∞2n+1ank0a+bnk0a,where we compare these results with the expression for the electric field in the far-field region, Eq. (122), and choose k→s, such that its azimuthal coordinate is zero, which is equivalent to taking ϕ=π/2; we obtain [13](134)iωμ0σ⃡NLk→s,k→i·e^x=4π-ik0Sϕxθe^x,(135)iωμ0σ⃡NLk→s,k→i·e^y=4π-ik0Sθyθe^θ-4πa3k02ϵsϵ0-1Sryθe^r,(136)iωμ0σ⃡NLk→s,k→i·e^z=4π-ik0Sθzθe^θ+4πa3k02ϵsϵ0-1Srzθe^r,where Sθy, Sϕx, and Sθz are given in Eqs. (129)-(131), and(137)Sryθ=∑n=1∞2n+1dnk0ajnksaksajnk0ak0aPn1cos⁡θ(138)Srzθ=∑n=1∞2n+1dnLk0ann+1jnksaksajnk0ak0aPncos⁡θ,+ϵ0ϵsj12ak0sinθ/22ak0sinθ/2cos⁡θ.


### 7.2. Rayleigh Approximation

This approximation corresponds to the small-particle limit; to find out this limit, we truncate the Taylor’s series expansion for the radial dependence at the first term and we will suppose thatk0a,ksa,pa,p′a are small quantities ≪1; the details can be found in Appendix E, so that we have the following.In the same way, using the former approximations, we can find expressions in the small-particle limit for the nonlocal conductivity tensor for the sphere whenp→∥=p→∥′ and θ is the angle between p→ and p→′:(139)σ⃡p→′,p→·e^x=4πa3iωμ0μs-μ0μs+2μ0pp′cos⁡θ+k02ϵs-ϵ0ϵs+2ϵ0e^x+Oa3(140)σ⃡p→′,p→·e^y=4πa3iωμ0μs-μ0μs+2μ0pp′+k02ϵs-ϵ0ϵs+2ϵ0cos⁡θe^θ+k02ϵs-ϵ0ϵs+2ϵ0sin⁡θe^r+Oa3(141)σ⃡p→′,p→·e^z=4πa3iωμ0k02ϵs-ϵ0ϵs+2ϵ0-sin⁡θe^θ+cos⁡θe^r+Oa3,and recalling that(142)e^y=sin⁡θe^r+cos⁡θe^θ(143)e^z=cos⁡θe^r-sin⁡θe^θ,we get for a much smaller than the wavelength associated to the wave vectors p and p′, and μs=μ0:(144)σ⃡p→′,p→=4πa3iωμ0k02ϵs-ϵ0ϵs+2ϵ0I⃡+Oa3.Also we have that the small-particle limit for μs≠μ0 yields(145)σTp=4πa3iωμ0μs-μ0ϵs+2ϵ0p2+ϵs-ϵ0μs+2μ0k02ϵs+2ϵ0μs+2μ0+Oa3(146)σLp=4πa3iωμ0k02ϵs-ϵ0ϵs+2ϵ0+Oa3.


## 7.1. Nonlocal Conductivity Whenp→′=p→=k0 and Its Relation with the S Matrix

In this section we explore an approximation that we call the cone-light approximation and that corresponds to the case whenp→′=p→=k0; this means that the magnitudes of the wave vectors p→ and p→′ are equal to the wave number in vacuum. This is the most used form of the nonlocal conductivity tensor when related to the scattering S-matrix [11]. We will see that this substitution corresponds to the problem of calculating the scatter field in the far field region. To make the relationship with the S matrix when the magnitudes of p→ and p→′ are k0, we will denote p→ as k→i and p→′ as k→s, the wavevectors of the incident and scattered waves, respectively. However, we must remark that the treatment that follows is the most general, going beyond the so-called Mie theory since, even though ki=ks=k0, we still allow for incident and scattered waves with longitudinal components. As discussed previously, supporting longitudinal waves requires the existence of external currents.The electric field scatter by a sphere due to the presence of an incident plane wave is given, in the far field region as in [12, p. 63]:(117)E→sr→;e^i=eikr-ikrS⃡θ,ϕ·e^iE0,where S⃡(θ,ϕ) is the scattering matrix, θ is the angle between the incident wave vector and the observation position, E0 is the incident electric field amplitude, and e^i is its polarization. The scattered field by an isolated sphere, in terms of the induced current inside the sphere (1), is of the form(118)E→sr→;e^i=iωμ0∫R3d3r′G⃡r→,r→′·∫R3d3r′′σ⃡NLr→′,r→′′·e^ieik→i·r→′′,where the polarization of the incident wave is e^i and its wave vector k→i is such that k→i=k0, where k0 is the vacuum wave number.It is possible to write the free space Green’s function of the electromagnetic field as [8, pg. 55](119)G⃡r→,r→′=I⃡+1k02∇∇eik0r→-r→′4πr→-r→′.Substituting this expression in (118) and considering that r→>>0, then(120)k0r→-r→′=k0r1-2r^·r→′+r′2r21/2≈k0r-k→s·r→′,where k→s=k0r^, and r^ is the radial unitary vector. Using this far-field approximation, the electric field can be written as(121)E→sr→;e^i=iωμ0I⃡+1k02∇∇eik0r4πr∫R3d3r′∫R3d3r′′e-ik→s·r→′·σ⃡NLr→′,r→′′eik→i·r→′′·e^i.With these considerations and to order O(1/r) the electric field is written as(122)E→sr→;e^i=iωμ0eik0r4πrI⃡-k^sk^s·σ⃡NLk→s,k→i·e^i+O1r2.In addition, it is possible to get the expression for the electric field in the far-field region, when the incident wave is a plane wave propagating at vacuum from (49); if we make p=k0, thus the scatter electric field for each polarization e^x,e^y,e^z of the incident wave is(123)E→sr→;e^x=∑n=1∞2n+1nn+1inbnM→on13k0r-ianN→en13k0r(124)E→sr→;e^y=∑n=1∞-2n+1nn+1inbnM→en13k0r+ianN→on13k0r(125)E→sr→;e^z=∑n=1∞2n+1in-1anLN→en03k0r,where the second variable of E→s(r→;e^i) indicates the polarization of the incident wave, k0 is the vacuum wave number, the index (3) indicates that the radial part is given by the Hankel’s functions, and we have taken as unitary the amplitude of the incident wave. Here Eqs. (123) and (124) are exactly the Mie solution of the transverse scattered waves by a sphere due to an incident transverse wave, where the connection with the Mie theory is made evident [7]. With respect to the electric field in (125), this kind of scatter field requires the existence of an external current given by (27) with p→=k0. This response of the scattered field in the radial and transversal components of the sphere when excited by a longitudinal electromagnetic mode and the mixing with the transversal radiated field will be important in the formulation of effective field theories of many sphere scatterers, because the exciting field in a many-sphere system could have in effect a longitudinal component [1]. This opens the possibility of having longitudinal modes of electromagnetic radiation and needs the design of some experiment for its verification.The expressions for the Hankel functions and the vector spherical harmonic are given in AppendixB (B.1)-(B.8).Substituting these expressions in Eqs. (49) and (50) one finds that the electric scattered field to order O(1/r) is given by(126)E→sr→,e^x=eikr-ikrSθyθcos⁡ϕe^θ-Sϕxθsin⁡ϕe→ϕ,(127)E→sr→,e^y=eikr-ikrSθyθsin⁡ϕe^θ+Sϕxθcos⁡ϕe→ϕ,(128)E→sr→,e^z=eikr-ikrSθzθe^θ,where we have defined the scattering functions as(129)Sϕxθ=-∑n=1∞2n+1nn+1bnk0addθPn1cos⁡θ+ank0aPn1cos⁡θsin⁡θ,(130)Sθyθ=-∑n=1∞2n+1nn+1bnk0aPn1cos⁡θsin⁡θ+ank0addθPn1cos⁡θ,(131)Sθzθ=-∑n=1∞2n+1anLk0addθPncos⁡θ.(131) is a new term in the conventional S-matrix theory and comes from the fact that the sphere is excited by a longitudinal electric field. In addition, using the fact that limθ→0d/dθPn0(cos⁡θ)=0 we have that(132)Sθz0=0,and since that limθ→0Pn1cos⁡θ/sinθ=limθ→0d/dθPn1(cos⁡θ)=-(1/2)n(n+1), this implies(133)S0=Sθy0=Sϕx0=12∑n=1∞2n+1ank0a+bnk0a,where we compare these results with the expression for the electric field in the far-field region, Eq. (122), and choose k→s, such that its azimuthal coordinate is zero, which is equivalent to taking ϕ=π/2; we obtain [13](134)iωμ0σ⃡NLk→s,k→i·e^x=4π-ik0Sϕxθe^x,(135)iωμ0σ⃡NLk→s,k→i·e^y=4π-ik0Sθyθe^θ-4πa3k02ϵsϵ0-1Sryθe^r,(136)iωμ0σ⃡NLk→s,k→i·e^z=4π-ik0Sθzθe^θ+4πa3k02ϵsϵ0-1Srzθe^r,where Sθy, Sϕx, and Sθz are given in Eqs. (129)-(131), and(137)Sryθ=∑n=1∞2n+1dnk0ajnksaksajnk0ak0aPn1cos⁡θ(138)Srzθ=∑n=1∞2n+1dnLk0ann+1jnksaksajnk0ak0aPncos⁡θ,+ϵ0ϵsj12ak0sinθ/22ak0sinθ/2cos⁡θ.


## 7.2. Rayleigh Approximation

This approximation corresponds to the small-particle limit; to find out this limit, we truncate the Taylor’s series expansion for the radial dependence at the first term and we will suppose thatk0a,ksa,pa,p′a are small quantities ≪1; the details can be found in Appendix E, so that we have the following.In the same way, using the former approximations, we can find expressions in the small-particle limit for the nonlocal conductivity tensor for the sphere whenp→∥=p→∥′ and θ is the angle between p→ and p→′:(139)σ⃡p→′,p→·e^x=4πa3iωμ0μs-μ0μs+2μ0pp′cos⁡θ+k02ϵs-ϵ0ϵs+2ϵ0e^x+Oa3(140)σ⃡p→′,p→·e^y=4πa3iωμ0μs-μ0μs+2μ0pp′+k02ϵs-ϵ0ϵs+2ϵ0cos⁡θe^θ+k02ϵs-ϵ0ϵs+2ϵ0sin⁡θe^r+Oa3(141)σ⃡p→′,p→·e^z=4πa3iωμ0k02ϵs-ϵ0ϵs+2ϵ0-sin⁡θe^θ+cos⁡θe^r+Oa3,and recalling that(142)e^y=sin⁡θe^r+cos⁡θe^θ(143)e^z=cos⁡θe^r-sin⁡θe^θ,we get for a much smaller than the wavelength associated to the wave vectors p and p′, and μs=μ0:(144)σ⃡p→′,p→=4πa3iωμ0k02ϵs-ϵ0ϵs+2ϵ0I⃡+Oa3.Also we have that the small-particle limit for μs≠μ0 yields(145)σTp=4πa3iωμ0μs-μ0ϵs+2ϵ0p2+ϵs-ϵ0μs+2μ0k02ϵs+2ϵ0μs+2μ0+Oa3(146)σLp=4πa3iωμ0k02ϵs-ϵ0ϵs+2ϵ0+Oa3.


## 8. Some Numerical Results

In this section we present the polar plots for the scattering functions (129)-(131) and (137)-(138) which are the only nonzero components of the scattering matrix in the chosen basis.In Figure2 we show Sϕxθ/S0, Sθyθ/S0, and Sθzθ/S0 given by Eqs. (129), (130), and (131), respectively, as polar diagrams showing the scatter amplitude as function of the angle of the incident wave and the observation position. These represent the angular dependence of the scatter field when we are far away from the sphere. As we see these fields are related to the nonlocal conductivity in the special case that p→=p→′=k0. The scattering amplitudes (129), (130) coincide with the scattering amplitude of scattering matrix S1(θ) and S2(θ), but the scatter function (131) is the longitudinal component, due to the excitation of longitudinal wave, which is only possible with the artifice of introducing an external current, with wave vectors p→=p→′=k0. Figure 3 shows Sryθ/S0, Srzθ/S0. Eqs. (137) and (138) are the scatter field of the transversal field in the radial direction and the scatter field in the radial direction produced by a longitudinal unitary wave. These graphics serve the purpose of verification of the nonlocal conductivity, by yielding the scatter matrix elements, but additionally including longitudinal excitations and looking for the radial component of the electric scattered field.Figure 2
In this figure we show a polar diagram for scattering amplitudesSϕx, Sθy, and Sθz, normalized by the amplitude S(0)=3.2554-1.0662i, for a sphere with radius a=0.2μm, a wavelength λ=0.63μm, and electric permittivity ϵ/ϵ0=-18.0958+0.484224i corresponding to silver properties.Figure 3
In this figure we show a polar diagram for scattering amplitudesSry, Srz, normalized by the amplitude S(0)=3.2554-1.0662i, for a sphere with radius a=0.2μm, a wavelength λ=0.635μm, and electric permittivity ϵ/ϵ0=18.0958+0.484224i corresponding to the silver properties at that wavelength.Now we present the polar plots for the different scattering functions for a spherical particle for a sphere with radiusa=1/3λ, at a wavelength of λ=0.63μm. In Figure 4 we show Sϕxθ/S0, Sθyθ/S0, and Sθzθ/S0 given by Eqs. (129), (130), and (131), respectively, as polar diagrams showing the scatter amplitude as function of the angle of the incident wave and the observation position; these represent the angular dependence of the scatter field when we are far away from the sphere for several values of the magnetic permeability μ/μ0=1,10,100 and two values of the electric permittivity ϵ/ϵ0=2,6.Figure 4
Plots of polar scattering amplitudesSϕx(θ), Sθy(θ), and Sθz(θ) for several μ=1,10,100, normalized by the amplitude S0, for a sphere with radius a=1/3λ, a wavelength λ=0.63μm, and electric permittivity ϵ/ϵ0=2 (a, c, e) and ϵ/ϵ0=6 (b, d, f).
(a)
S ϕ x with ϵ/ϵ0=2 (b)
S ϕ x with ϵ/ϵ0=6 (c)
S θ y with ϵ/ϵ0=2 (d)
S θ y with ϵ/ϵ0=6 (e)
S θ z with ϵ/ϵ0=2 (f)
S θ z with ϵ/ϵ0=6As we see these fields are related to the nonlocal conductivity in the special case thatp→=p→′=k0. The scattering amplitudes Sϕx(θ) and Sθy(θ) coincide with the scattering amplitude of scattering matrix S1(θ) and S2(θ), but the scattering function Sθz(θ), Eq. (131), is the longitudinal component, due to the excitation of longitudinal wave, which is possible due to the external current, with wave vectors p→=p→′=k0.Figure5 shows both Sryθ/S0 and Srzθ/S0. Eqs. (137) and (138) for a magnetic particle μ/μ0=1,10,100 and ϵ/ϵ0=2,6 are the scatter fields of transversal field in the radial direction and the scatter field in the radial direction produced by a longitudinal unitary wave. These graphics serve the purpose of testing the nonlocal conductivity, by yielding the scatter matrix elements, but additionally including longitudinal excitations radiating radially and even transversally also.Figure 5
Plots of polar scattering amplitudesSry(θ) and Srz(θ) for several μ=1,10,100, normalized by the amplitude S0, for a sphere with radius a=1/3λ, a wavelength λ=0.63μm, and electric permittivity ϵ/ϵ0=2 (a, c) and ϵ/ϵ0=6 (b, d).
(a)
S r y with ϵ/ϵ0=2 (b)
S r y with ϵ/ϵ0=6 (c)
S r z with ϵ/ϵ0=2 (d)
S r z with ϵ/ϵ0=6We observe that the influence of the magnetic permeability and electric permittivity is that of changing the weights in the harmonic expansion of the scattered electric field; therefore we can have different scattering patterns by adjusting the magnetic permeability and electric permittivity.


## 9. Summary and Conclusions

In this paper we derived closed-form expressions for the nonlocal generalized conductivity tensor at frequencyω of a spherical particle with local permittivity and permeability different from those of its surroundings, all within the context of Maxwell’s equations. This tensor is proportional to the T-matrix operator at frequency ω used in multiple scattering theories by many authors. The derivation of our expressions is straightforward from the definition of the generalized conductivity tensor, and, thus, we believe it is a more physically transparent calculation. Additionally, our results go beyond those found in the literature in that we consider spheres with a magnetic response. Due to the magnetic response considered, the generalized conductivity tensor (and thus the T-matrix operator) acquires a surface contribution in addition to a bulk contribution. We gave closed-form expressions for both, the surface and bulk contributions. We also analyzed two special cases from the general expressions for σ⃡NL(p→,p→′): (i) the case when p→=p→′=k0, where k0 is the wavevector in vacuum, and we show its relation with the scattering matrix (S matrix), extending its application to cases when the incident field has a longitudinal component produced by external sources. (ii) The second is the Rayleigh limit or small-particle limit, and we show how to obtain from σ⃡NL(p→,p→′) the well-known expressions for the S matrix in this limit. Finally, we display some numerical results for the angular dependence of the scattering-matrix elements for spheres with and without an intrinsic magnetic response and also for the elements involving a longitudinal excitation.



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