Scattering from multiple PEC sphere using Translation Addition Theorems for Spherical Vector Wave Function

Scattering from multiple PEC sphere using Translation Addition Theorems for Spherical Vector Wave Function

Journal Pre-proof

Scattering from multiple PEC sphere using Translation Addition Theorems for Spherical Vector Wave Function Batool S., Frezza F., Mangini F., Yu-Lin Xu PII: DOI: Reference:

S0022-4073(19)30988-4 https://doi.org/10.1016/j.jqsrt.2020.106905 JQSRT 106905

To appear in:

Journal of Quantitative Spectroscopy & Radiative Transfer

Received date: Revised date: Accepted date:

23 December 2019 5 February 2020 11 February 2020

Please cite this article as: Batool S., Frezza F., Mangini F., Yu-Lin Xu, Scattering from multiple PEC sphere using Translation Addition Theorems for Spherical Vector Wave Function, Journal of Quantitative Spectroscopy & Radiative Transfer (2020), doi: https://doi.org/10.1016/j.jqsrt.2020.106905

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.

Scattering from multiple PEC sphere using Translation Addition Theorems for Spherical Vector Wave Function Batool S.*, 1 ,Frezza F.1 ,Mangini F.1,2 ,Yu-Lin Xu.3 (1)

1.

Department of Information Engineering, Electronics and Telecommunications, “La Sapienza” University of Rome, Via Eudossiana 18, 00184 Rome, Italy (2) Department of Information Engineering, University of Brescia, Italy (3) Jacobs, NASA Johnson Space Center, USA ∗ [email protected]

Abstract

In our manuscript, we are reporting the translation criteria of scattering from Perfect Electric Conductor (PEC) sphere along three-dimensional axes using semianalytical approach. The presented scattering model is based on a generalized Lorenz-Mie theory framework and ensemble with the vector translation Addition Theorem (AT) for the Vector Spherical Harmonics (VSH). Applying extended Mie theory on a sphere leads to a set of an unknown coefficients by the use of translation AT. In the literature, there are many authors reporting different sets of the vector translation coefficients, of which we mention those calculated by Stein, Cruzan and, Mackowski in particular. We have selected the Cruzan formulation of the vector translation coefficients for its structure based on the Wigner 3-j function. As an illustration, we want to present numerical examples than simulations. of total scattered field PEC sphere using vector translation AT. We used advanced computational tools and approaches for mathematical modeling of an observation. During our numerical test, we have deeply investigated generic truncation criteria in scattered electric field using translation AT. However, we have been obtained numerical validation by using computational approach. 2.

Introduction

Electromagnetic scattering from a sphere is a wide subject with a long and remarkable historical background. It is commonly referred to as the Mie series, and it has been studied extensively, as well as its wide range applications such as optics, nanoscience, astrophysics, climate modelling, and biomedical imaging are summarized in many textbooks [1–3]. The Rayleigh scattering approximation can be applied, when the spherical scatterer is much smaller than the wavelength of the incident wave, Mie approximation can be applied, when the size of the scatterer increase as compared to the wavelength of the incident wave, since

1

modern computers technology advancements have been applied to improve the stable and reliable algorithms of Mie series calculations [4]. Inspecting the light scattering from heterogeneous objects is a competent method to analyze some of the scatterer’s physical and electromagnetic properties, such as the shape, size, and the medium properties of the object. Moreover, light scattering simulations can be very informative and have many practical applications in cellular biology, which is associated with the corresponding complex structures, such as biological cells that comprise a nucleus and different tissues in human beings [5–7]. Analysis of the scattering by spherical objects conducting or dielectric is a classical topic in electromagnetic research. The first result, after the works by Mie on the electromagnetic scattering by a spherical object, can be considered the work on the scattering by two concentric spheres [8], and that on the scattering by a perfectly conducting sphere inclusion or core with a dielectric coating [9]. This problem, of great interest from both a theoretical and a practical point of view, has been deeply studied, also in the case of lossy and active media [10]. Moreover, the dynamic Mie-Lorentz based approach has been investigated [3, 11–14]. The approach followed in these works is based on Mie scattering by a sphere with the application of the addition theorem (AT) to consider spheres with different centers. The mutual interaction complicates light scattering by ensembles of particles. The problem therefore require the use of an AT to transform the relative basis function from a coordinate system centered on the scatterer to other reference systems centered outside the scatterer. For single particle of spherical geometry, Mie theory can be used to determine the exact analytic solutions for the electromagnetic fields [15]. In this study, the decomposition of the fields is obtained by using a representation in terms of translation AT for VSH. It affords the translation of the VSH between different spherical coordinate systems. Translation coefficients constitute the fundamental basis of the translation AT, that are essential for multiple scattering problems. In the literature, there are many different researcher investigated formalisms of the vector translation coefficients, among which we mention those calculated by Stein [16], Cruzan [17] and, Mackowski. Most practical implementation of translation AT are based on recursive relationship of Mackowski’s [18]. In the recent past, Xu reported that electromagnetic scattering by an aggregate sphere. He derived the solution of extended Mie theory that leads to interactive scattering coefficients [19–21]. In our manuscript, we have deeply investigated the generic truncation criteria in scattered electric field and it is outlined as a repeatable procedure to get an acceptable convergence. Antonio et.al. described the criteria of truncation in Mie series [22]. Sidra et.al. reported the effect of finite terms on the truncation error of AT for spherical vector wave function [23]. They showed that the, numerical results of spherical harmonics must be fluctuated due to the truncation effect. In the literature, Wiscombe’s approximated criteria can be used to determine how an infinite series convergences in Mie 2

Fig. 1. Representation of a plane wave effecting a PEC sphere

theory [24]. But it is insufficient for the translation AT in VSH. 3.

Scattering from PEC sphere in a free space

In this manuscript, we want to study the simplest case of electromagnetic radiation of a plane wave from a sphere in particular, we will first consider a sphere made up of PEC material, immersed in a homogeneous mediums such as air. Consider the space as a linear homogeneous, isotropic, and nondispersive medium, also refer to as a simple medium. According to that medium the permittivity  and permeability µ are constant. Fig. 1 shows as the plane wave impinging a PEC sphere. Consider a flat and elliptically polarized wave that propagates along a generic direction in a certain three dimension cartesian coordinate system can be written as

with

  0 0 0 0 0 Ei (r ) = epol exp(iki .r ) = Eθ θ + Eφ φ exp(iki .r )   0 0 0 0 0 0 0 0 0 k i = k1 sin θi cos φi x0 + sin θi sin φi y0 + cos θi z0   0 0 0 0 0 0 0 0 0 θ 0i = k1 cos θi cos φi x0 + cos θi sin φi y0 − sin θi z0 0

0

0

0

0

φ 0i = − sin φi x0 + cos φi y0

3

(1)

(2) (3) (4)

The incident plane wave in spherical harmonics may be written as [25] 0

Ei (r ) =

+∞ X ν h X ν=1 µ=−ν

i 0 (1) 0 aµν M(1) (r ) + b N (r ) µν µν µν

(1)

(5)

(1)

The vector spherical harmonics Mµν and Nµν are defined as h i 0 0 0 0 0 0 µ µ (1) Mµν = exp(iµφ )jν (kr ) iπν (cos θ )θ0 − τν (cos φ )φ0 0

0 jν (kr ) 0 0 N(1) ν(ν + 1)Pνµ (cos θ )r0 + 0 µν = exp(iµφ ) kr h i 1 ∂ 0 0 0 0 0 0 0 τνµ (cos θ )θ0 + iπνµ (cos φ )φ0 exp(iµφ ) 0 0 [r jν (kr )] kr ∂r

(6) (7)

By introducing modified spherical vector functions characterized by the dependence on angles 0 0 θ and φ . Using tesseral function for spherical vector function, the following is obtained [1] i h 0 0 0 0 0 (8) mµν = exp(iµφ ) iπνµ (cos θ )θ0 − τνµ (cos θ )φ0 h i 0 0 0 0 0 nµν = exp(iµφ ) τνµ (cos θ )θ0 + iπνµ (cos θ )φ0 (9) 0

0

0

pµν = exp(iµφ )ν(ν + 1)Pνµ (cos θ )r0

(10)

By simplifying Eq(6) and Eq(7), we get 0

(11)

M(1) µν = jν (kr )mµν N(1) µν =

0

jν (kr ) 1 ∂ 0 0 pµν + 0 0 [r jν (kr )]nµν 0 kr kr ∂r

(12)

with aµν bµν

i h + 1)(ν − µ)! 0 0 0 µ µ 0 0 exp (−iµφi ) − iEθ πν (cos θi ) − Eφ τν (cos θi ) =i i i ν(ν + 1)(ν + µ) h i (2ν + 1)(ν − µ)! 0 0 0 = iν−1 exp (−iµφi ) − iEθ0 τνµ (cos θi ) − Eφ0 πνµ (cos θi ) i i ν(ν + 1)(ν + µ) ν (2ν

(13) (14)

Now, the expression of the scattered field from the sphere through the development of the vector spherical harmonics, is a generic solution of the Helmholtz equation. Consider that the scattered field can be described as the superposition of the VSH, Mµν and Nµν represented by the third species Bessel functions, that is the first kind of Hankel functions, we obtain

0

Es (r ) =

+∞ X ν h i X 0 (3) 0 eµν M(3) (r ) + f N (r ) µν µν µν ν=1 µ=−ν

4

(15)

where eµν and fµν are the unknown coefficients that can be determined by the application of the boundary condition. The tangential component of the field must be continue on the spherical surface of the PEC scatterer. Therefore 0

0

(Ei + Es ) × r0 = 0

r =a

(16)

By using above Eq(5), Eq(16), the simplified expression may be written as [25] aµν jν (ka) + eµν h(1) ν (ka) = 0 0

0 (1)

bµν jν (ka) + eµν hν (ka) = 0

(17) (18)

By solving above equations, we can get the unknown coefficients to describe the scattered field. eµν = −aµν fµν = −bµν

jν (ka) (1)

hν (ka) 0 jν (ka) 0 (1)

hν (ka)

(19) (20)

The foregoing discussion can easily be extended to the case of lossy media [26]. 4.

Vector translation Addition Theorem (AT)

In this paragraph, we will represent a method that gives the scattered wave by an individual sphere in terms of the system of coordinates the origin of which differs from the center of the sphere. Fig. 2 shows the translation AT applied on PEC sphere between the different coordinate axes. For this purpose we need to describe the vector translation AT. It has many applications for the solution to various scientific problems including multiple scattering problems and asymmetry parameter of an arbitrary multi-particle configuration, etc. [19,21]. Vector translation coefficients have basically three forms of mathematical expression such as scalar translation coefficient [27], the vector translation coefficients in terms of Wigner 3j symbol and the gaunt coefficients [28], respectively. An adequate numerical technique of Cruzan formulas demands evaluation of the Gaunt coefficients in terms of Wigner 3-j symbol and has a key role in obtaining appropriate numerical results of vector translation coefficients in practical scattering calculations [29]. j1 j2 j3 m1 m2 m3

!

s

(j1 − m1 )!(j1 + m1 )!(j2 − m2 )!(j2 + m2 )!(j3 − m3 )!(j3 + m3 )! (j1 + j2 − j3 )!(j1 − j2 + j3 )!(−j1 + j2 + j3 )!(j1 + j2 + j3 + 1) ! ! ! kX max j + j − j j − j + j −j + j + j 1 2 3 1 2 3 1 2 3 × (−1)k k j1 − m1 − k j2 + m2 − k k=k

= (−1)j1 +j2 +m3

min

(21)

5

Fig. 2. Application of translation AT by a PEC sphere between different coordinates axis.

6

where

!

j k

represents the binomial coefficients, and (22)

kmin = max(0, j2 − j3 − m1 , j1 − j3 + m2 )

(23)

kmin = min(j1 + j2 − j3 , j1 − m1 , j2 + m2 )

The Wigner 3-j symbol must satisfy the condition m1 + m2 + m3 = 0, j3 approaches the triangular condition, j3 min ≤ j3 ≤ (j1 + j2 ), where j3 min = max(|j1 − j2 |, |m1 + m2 |). The Gaunt coefficient is closely related to Wigner 3-j. It can be defined as [21, 28] Z (2p + 1) (p − m − µ)! 1 m (24) P (x)Pµν (x)Ppm+µ (x)dx a(m, n, µ, ν, p) = 2 (p + m + µ)! −1 n where m,n,µ,ν are integers, |m| ≤ n, µ ≤ ν, and Pnm represents the associated Legendre function of the first kind. Cruzan formula associated with the Gaunt coefficient and the Wigner 3-j symbol is defined as [28] 

 21 (n + m)!(ν + µ)!(p − m − µ)! a(m, n, µ, ν, p) = (−1)m+µ (2p + 1) (n − m)!(ν − µ)!(p + m + µ)! ! ! n ν p n ν p × 0 0 0 m µ −m − µ 5.

(25)

Calculation of vector translation coefficient

Suppose that an electromagnetic field in terms of translation is represented by an infinite sum with respect to a coordinates system toward a different reference. So, Cruzan’s mathematical l,j expressions for Al,j mnµν and Bmnµν can be written as [17, 21] (2ν + 1)(n + m)!(ν − µ)! exp[i(µ − m)φlj ] 2n(n + 1)(n − m)!(ν + µ)! Qmax X × ip [n(n + 1) + ν(ν + 1) − p(p + 1)]aq

−m Al,j mnµν = (−1)

q=0

×jp(1) (kdlj )Ppµ−m (cos θlj )

l,j Bmnµν = (−1)−m+1 Qmax

×

X q=1

0

r > rlj

(26)

(2ν + 1)(n + m)!(ν − µ)! exp[i(µ − m)φlj ] 2n(n + 1)(n − m)!(ν + µ)! 1

ip+1 {[(p + 1)2 − (n − ν)2 ][(n + ν + 1)2 − (p + 1)2 ]} 2 (1)

µ−m ×b(−m, n, µ, ν, p + 1, p)jp+1 (kdlj )Pp+1 (cos θlj )

7

0

r > rlj

(27)

where k is propagation constant, aq = a(−m, n, µ, ν, p), q = 1, 2....., Qmax , p = n + ν − 2q, and   n + ν − |m + µ| (28) Qmax = min n, ν, 2 By using above Eq(26) Gaunt coefficient of b(m, n, µ, ν, p + 1, p) may be written as −m+µ

b(−m, n, µ, ν, p + 1) = (−1)



1 (n − m)!(ν + µ)!(p − m − µ + 1)! 2 (2p + 3) (n + m)!(ν − µ)!(p − m + µ + 1)! ! ! n ν p n ν p+1 × 0 0 0 −m µ m − µ

(29)

where Qmax 6.



n + ν + 1 − |m + µ| = min n, ν, 2



(30)

AT for spherical vector wave equation

From AT, the translation of the vector spherical harmonics depends on the relative separation in a spherical coordinate system with respect to different origins O and O0 . To express (i.e., to translate) the scattered field of lth body solved in the lth coordinate system in a translation coordinate system j or j0 , we have

M(3) µν (l)

=

N(3) µν (l) =

∞ X n X

(31)

(3) µν (3) Aµν mn (l, j0 )Mmn (j0 ) + Bmn (l, j0 )Nmn (j0 )

n=1 m=−n ∞ X n X

(32)

µν µν (3) Bmn (l, j0 )M(3) mn (j0 ) + Amn (l, j0 )Nmn (j0 )

n=1 m=−n

0

This is the case when r > rlj . Consequently the scattered field by using above Eq(31) and Eq(32) may be written as Es (l) =

ν h ∞ X X ν=1 µ=−ν

∞ X n n o X (3) µν (3) eµν Aµν mn (l, j0 )Mmn (j0 ) + Bmn (l, j0 )Nmn (j0 )

+fµν

n=1 m=−n ∞ n n XX

µν Bmn (l, j0 )M(3) mn (j0 )

n=1 m=−n

7.

+

(3) Aµν mn (l, j0 )Nmn (j0 )

(33)

oi

Numerical Results and Conclusion

Numerical results of the scattered electric field by a PEC sphere for near zone are described in this section. First, we used the basic test for accuracy measurement in the calculation 8

of translation coefficients. It is described as when the origin of the lth and jth coordinate system are overlapped (i.e., no translation is actually involved), when (m = µ, n = ν), µν µν µν Aµν mn ≡ 1, Bmn ≡ 0, and when (m 6= µ, n 6= ν), Amn ≡ 0, Bmn ≡ 0. These expressions (3) (3) (3) (3) stated as Mµν (l) ≡ Mµν (j), Nµν (l) ≡ Nµν (j). For this purpose equation (31)-(33) have been implemented using Matlab code. We have chosen the following parameters shown in Table 1. The position of two system of references are represented as a scattered electric field

Table 1.

# Symbols 1 a 1 f 2 θi 3 φi 4 Eθ 5 Eφ

Parametric values Description Numerical values for input Unit Radius of the sphere 0.25 m Frequency 0.3 GHz Incident angle 0 rad Incident angle π rad Elliptically polarized plane waves 0 V/m Elliptically polarized plane waves 1 V/m

is measured at a small line segment to twice the radius of sphere parallel to the x-axis, placed at a height equal to z = 6a; center of the system xc = yc = zc = 0; center of the sphere xp =0, yp = 0, zp = 0. Here actually, we want to validate the translation coefficients by using zero translation distance. For the validation, we draw a geometry on Comsol(Multi Layer Physics 5.2 version) for a single PEC sphere using above same parameters. Fig 3 show that the perfect comparison of Matlab and Comsol results. We have been used the small radius of sphere a=0.01m. It must satisfy the quasi-static approach a << λ. We want to translate sphere along z-axis by changing translation distance. For this purpose, we have been taken above parameters just move the center of sphere along z-axis zp = 2a by changing translation distance. Figs 4, 5 shows that the real case of the scattered field by a PEC sphere of Matlab results are overlapping with Comsol results. For the case of imaginary, we had found little shift in Comsol and Matlab Comparison test. The main cause of this shifting between these comparison test might be due to truncation of the sum to N. When we increased the number of terms N=30, both software results are perfectly overlapping shown in Fig. 4, 5. When we have been taken new parameters such as frequency f =3 GHz, 0.3 GHz, a = λ/2, center of the system xc = yc = zc = 0, center of sphere along axis is xp =0, yp = 0, zp = λ/4. Fig. 6 show the best comparison of Matlab and Comsol results. Figs 7, 8 shows the fluctuation in the plots of Matlab and Comsol (comparison test). This 9

0.02

10

10 -5

0.04 0.02

-0.02 -0.04

0

0

-5 -0.5

0.5

0.16

5

0.14

0

0.12

0

x[m]

0

-0.04 -0.5

0.5

x[m]

Imag[EY]

Imag[EX]

x[m]

0.1 -0.5

0 -0.02

0.5

10 -5

0.5

0.05

-5

-10 -0.5

0

x[m]

Imag[EZ]

-0.06 -0.5

5

Real[EZ]

Real[EY]

Real[EX]

0

0

x[m]

0.5

0

-0.05 -0.5

0

0.5

x[m]

Fig. 3. Real and imaginary parts of the scattered electric field by a PEC sphere. In particular, the solid red line refer to the Comsol result blue dot line refers to the Matlab result. Blue color dot line is perfectly overlapping with the red color solid line.

fluctuation is due to truncation. We have developed the criteria of truncating the inherent infinite series to achieve convergence with a finite version of the same, which leads a little bit of fluctuation in the comparisons test of the scattered electric field by a PEC sphere using Comsol and Matlab simulation software. When we increased the number of terms to N=30 by using above input, both software results are perfectly overlapping. During our numerical test, we have observed the trial and error criteria of truncation in the scattered electric field by using translation AT. We used Comsol and Matlab simulation software for investigation. We are sure that our improved approach will help researchers in developing efficient analysis of the problem of light scattering across general distributions of spheres. References 1. Bohren CF, Huffman DR. "Absorption and scattering of light by small particles". John Wiley and Sons, 26 (2008). 2. Hergert W, Wriedt T. "The Mie theory: basics and applications". Springer, 30 (2012).

10

Real[EX]

0.03 Comsol Matlab; N=10 Matlab; N=30

0.02 0.01 0 -0.02

Real[EY]

10

-0.015

-0.01

-0.005

5

0.005

0.01

0.015

0.02

0.01

0.015

0.02

Matlab N=10 Comsol Matlab N=30

0 -5 -10 -0.02

0

x[m]

-5

-0.015

-0.01

-0.005

0

0.005

Real[EZ]

x[m] 0.02

Matlab N=10 Comsol Matlab N=30

0 -0.02 -0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

x[m]

Fig. 4. Real part of the scattered electric field by a PEC sphere. In particular, the dot red line refer to the Comsol results, blue and green lines refer to the Matlab results at different value of the truncation sum N, when we were translated sphere along z-axis at z = 2a.

11

Imag[EX]

10 -3 Matlab; N=10 Comsol Matlab; N=30

6 4 2 -0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.01

0.015

0.02

x[m] Imag[EY]

10 -6 -5 Matlab N=10 Comsol Matlab N=30

-10 -0.02

-0.015

-0.01

-0.005

0

0.005

x[m] Imag[EZ]

10 -3 2

Matlab;N=10 Comsol Matlab; N=30

0 -2 -0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

x[m]

Fig. 5. Imaginary part of the scattered electric field by a PEC sphere. In particular, the dot red line refer to the Comsol results, blue and green lines refer to the Matlab results at different value of the truncation sum N, when we were translated sphere along z-axis at z = 2a.

12

0.25

10

10 -4 0.1

0.15 0.1

5

Real[EZ]

Real[EY]

Real[EX]

0.2

0

0.05 0 -0.1

0

0 -0.05 -0.1

-5 -0.1

0.1

0.05

0

x[m]

0.1

-0.1

x[m]

0

1

10

0

0.1

x[m]

-3

0.05

-0.2

Imag[EZ]

-0.1

Imag[EY]

Imag[EX]

0.5 0 -0.5

0

-1 -0.3 -0.1

0

x[m]

0.1

-0.05

-1.5 -0.1

0

x[m]

0.1

-0.1

0

0.1

x[m]

Fig. 6. Real and imaginary parts of the scattered electric field by a PEC sphere. In particular, the solid red line refers to the Comsol result and blue dot line refer to the Matlab result. When we have been taken frequency f =3 GHz, radius a = λ/2, center of sphere along axis is xp =0, yp = 0, zp = λ/4. As a consequence of these parameters, (blue color dot line of the Matlab) result, the scattered electric field is perfectly overlapping with the (red color solid line of the Comsol result) at a truncation sum N>5.

13

Real[EX]

Matlab, N=10 Comsol Matlab N=30

0.2 0.15 0.1 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x[m] Real[EY]

10

-3

10

Matlab N=10 Comsol Matlab N=30

5 0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Real[EZ]

x[m] 0.1

Matlab N=10 Comsol Matlab N=30

0 -0.1 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x[m]

Fig. 7. Real part of the scattered electric field by a PEC sphere. When we have been taken frequency f =3 GHz, radius a = λ/2, center of sphere along axis is xp =0, yp = 0, zp = λ/4.

14

Imag[EX]

0

Matlab N=10 Matlab N=30 Comsol

-0.1 -0.2 -0.3 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

Imag[EY]

x[m] 0

10

-3

Matlab N=10 Matlab N=30 Comsol

-5 -10 -15 -1

-0.8

-0.6

-0.4

-0.2

0

x[m] Imag[EZ]

0.1

Matlab N=10 Matlab N=30 Comsol

0 -0.1 -1

-0.8

-0.6

-0.4

-0.2

0

x[m]

Fig. 8. Imaginary parts of the scattered electric field by a PEC sphere. When we have been taken frequency f =3 GHz, radius a = λ/2, center of sphere along axis is xp =0, yp = 0, zp = λ/4.

15

3. Mishchenko MI, Travis LD. "Gustav Mie and the evolving discipline of electromagnetic scattering by particles". Bulletin of the American Meteorological Society, 89 (2008), 1853-1862. 4. Gogoi A, Choudhury A, Ahmed GA. "Mie scattering computation of spherical particles with very large size parameters using an improved program with variable speed and accuracy". Journal of Modern Optics, 57 (2010), 2192-202. 5. Chalut KJ, Ostrander JH, Giacomelli MG, Wax A. "Light scattering measurements of subcellular structure provide noninvasive early detection of chemotherapy-induced apoptosis". Cancer Research, 69 (2009), 1199-204. 6. Schmitt JM, Kumar G. "Optical scattering properties of soft tissue, a discrete particle model". Applied Optics, 37 (1998), 2788-2797. 7. Frezza F, Mangini F, Muzi M, Stoja E. "In silico validation procedure for cell volume fraction estimation through dielectric spectroscopy". Journal of Biological Physics, 41 (2015), 223-234. 8. Aden A, Kerker M. "Scattering of electromagnetic waves from two concentric spheres". Journal Applied Physics, 22 (1951), 1242-1246. 9. J. Rheinstein. "Scattering of electromagnetic waves from dielectric coated conducting spheres". IEEE Transaction Antennas Propagation, 12 (1964), 334-340. 10. Kim JS, Chang JK. "Light scattering by two concentric optically active spheres". Journal Korean Physics Society, 45 (2004), 352-365. 11. Fikioris JG, "Uzunoglu NK, Scattering from and eccentrically stratified dielectric sphere". Journal of the Optics Society of America, 69 (1979), 1359-1366. 12. Li J, Shanker B. "Time-dependent Lorentz-Mie-Debye formulation for electromagnetic scattering from dielectric spheres". Progress in Electromagnetics Research, 154 (2015), 195-208. 13. Parsons J, Burrows CP, Sambles JR, Barnes WL. "A comparison of techniques used to simulate the scattering of electromagnetic radiation by metallic nanostructures". Journal of Modern Optics, 10 (2010), 356-365. 14. Doicu A, Mishchenko MI. "Electromagnetic scattering by discrete random media. The coherent field". Journal of Quantitative Spectroscopy and Radiative Transfer, 230 (2019), 86-104. 15. Geng YL, He S. "Analytical solution for electromagnetic scattering from a sphere of uniaxial left-handed material". Journal of Zhejiang University-Science A, 7 (2006), 99104. 16. Stein S. "Addition theorems for spherical wave functions". Quarterly of Applied Mathematics, 19 (1961), 15-24. 17. Cruzan OR. "Translational addition theorems for spherical vector wave functions". Quar16

terly of Applied Mathematics, 20 (1962), 33-40. 18. Mackowski DW. "Analysis of radiative scattering for multiple sphere configurations". Proceedings of the Royal Society of London Series A: Mathematical and Physical Sciences, 433 (1991), 599-614. 19. Xu YL. "Electromagnetic scattering by an aggregate of spheres". Applied Optics, 34 (1995), 4573-4588. 20. Xu YL. "Calculation of the addition coefficients in electromagnetic multispherescattering theory." Journal of Computational Physics, 127 (1996), 285-298. 21. Xu YL. "Efficient evaluation of vector translation coefficients in multiparticle lightscattering theories". Journal of Computational Physics, 139 (1998), 137-165. 22. Neves AA, Pisignano D. "Effect of finite terms on the truncation error of Mie series". Optics Letters 37 (2012), 2418-2420. 23. Batool S, Benedetti A, Frezza F, Mangini F, Xu YL. "Effect of Finite Terms on the Truncation Error of Addition Theorems for Spherical Vector Wave Function". Proceeding of Progress in Electromagnetics Research Symposium PIERS in Rome, (2019). 24. Wiscombe WJ. "Improved Mie scattering algorithms". Applied Optics, 19 (1980), 15051509. 25. Frezza F, Mangini F, Tedeschi N. "Introduction to electromagnetic scattering: Tutorial". Journal of Optics Society Amercia, 35 (2018), 163-173. 26. Frezza F, Mangini F. "Vectorial spherical-harmonics representation of an inhomogeneous elliptically polarized plane wave". Journal of Optics Society Amercia, 32 (2015) 13791383. 27. Friedman B, Russek J. "Addition theorems for spherical waves". Quarterly of Applied Mathematics, 1 (1954), 13-23. 28. Xu YL. "Fast evaluation of the Gaunt coefficients". Mathematics of Computation of the American Mathematical Society, 65 (1996), 1601-1612. 29. Lai ST, Chiu YN. "Exact computation of the 3-j and 6-j symbols". Computer Physics Communications, 61 (1990), 350-360.

17

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: