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Multiple scattering by a collection of randomly located obstacles Part II: Numericalimplementation - coherent fields

Gustavsson, Magnus; Kristensson, Gerhard; Wellander, Niklas

2014

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Citation for published version (APA):Gustavsson, M., Kristensson, G., & Wellander, N. (2014). Multiple scattering by a collection of randomly locatedobstacles Part II: Numerical implementation - coherent fields. (Technical Report LUTEDX/(TEAT-7236)/1-15/(2014); Vol. TEAT-7236). The Department of Electrical and Information Technology.

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Electromagnetic TheoryDepartment of Electrical and Information TechnologyLund UniversitySweden

CODEN:LUTEDX/(TEAT-7236)/1-15/(2014)

Multiple scattering by a collectionof randomly located obstaclesPart II: Numerical implementation— coherent fields

Magnus Gustavsson, Gerhard Kristensson, and Niklas Wel-lander

Magnus GustavssonMagnus.Gustafsson@foi.se

Swedish Defence Research Agency, FOIP.O. Box 1165SE-581 11 LinköpingSweden

Gerhard KristenssonGerhard.Kristensson@eit.lth.se

Department of Electrical and Information TechnologyElectromagnetic TheoryLund UniversityP.O. Box 118SE-221 00 LundSweden

Niklas WellanderNiklas.Wellander@foi.se

Swedish Defence Research Agency, FOIP.O. Box 1165SE-581 11 LinköpingSweden

and

Department of Electrical and Information TechnologyElectromagnetic TheoryLund UniversityP.O. Box 118SE-221 00 LundSweden

Editor: Gerhard Kristenssonc© M. Gustavsson, G. Kristensson, and N. Wellander, Lund, December 11, 2014

1

Abstract

A numerical implementation of a rigorous theory to analyze scattering by

randomly located obstacles is presented. In general, the obstacles can be

of quite arbitrary shape, but, in this rst implementation, the obstacles are

dielectric spheres. The coherent part of the reected and transmitted intensity

at normal incidence is treated. Excellent agreement with numerical results

found in the literature of the eective wave number is obtained. Moreover,

comparisons with the results of the Bouguer-Beer law (B-B) are made. The

present theory also gives a small reected coherent eld, which is not predicted

by the Bouguer-Beer law, and these results are discussed in some detail.

1 Introduction

Electromagnetic scattering by randomly located objects are frequently encounteredin science. It is an important issue in terrestrial and atmospheric research, biomed-ical and life sciences, astrophysics, nanotechnology, just to mention a few. Theliterature is comprehensive, and we refer to the textbook literature and referencestherein, see e.g., [6, 7, 15, 16, 2124] for a survey of the eld.

The literature contains several methods of computing the eective wave numberkeff for a half space containing a collection of random spheres, see e.g., [1719, 25, 26]and [21, Chapter 6], and references therein. The eective wave number is obtainedby solving a determinant relation and there are in general many solutions. Thenew method presented in Part I, [11], does not suer from these deciencies andis able to compute the coherent transmitted and reected elds from an inniteslab containing random scatterers. In this paper results are presented for slabs withdierent thicknesses and spherical scatterers with the relative permittivity εr = 1.332

which corresponds to fresh water at optical frequencies. Both the electrical size ofthe spheres and the volume fraction have been varied.

2 Theory

The theory of electromagnetic scattering by an ensemble of nite scatterers is re-viewed in [6, 7, 13, 15, 16, 2123].

The underlying theoretical treatment of the problem handled in this paper ispresented in detail in Kristensson [11]. The purpose of this section is to review andhighlight some of the more important steps in the theory. For a more completereference we refer to Kristensson [11].

We simplify the theoretical results in [11] to a geometry of a slab (z ∈ [0, d])and to spherical scatterers of radius a (dielectric or perfectly conducting). Theseassumptions simplies the results considerably simpler, and make the numericalimplementation less demanding. The geometry is depicted in Figure 1. Notice thatthe domain of possible locations of local origins, [z0, zd], which denes the domainVs, is slightly smaller than the extent of the slab, i.e., the interval [z0, zd] = [a, d−a].Vectors are denoted in italic boldface, and matrices in roman boldface. A caret over

2

z

z = 0

z = d

z = z0

z = zd

ki

Vs

Figure 1: The geometry of the stratied scattering region. The yellow regiondenotes the region Vs, which is the domain of possible locations of local origins, i.e.,the interval [z0, zd].

a vector denotes a vector of unit length. In this paper, we adopt the multi-indexnotation n = τσml, where the integer indices τ = 1, 2, l = 1, 2, 3, . . ., m = 0, 1, . . . , l,and σ = e,o (even and odd in the azimuthal angle).

Assume the incident eld on the slab is

Ei(z) = E0eik0z

The coherent part of the total electric eld on either side of the slab is

<E> (z) = Ete±ik0z, z > d and <E> (z) = E0eik0z +Ere

−ik0z, z < 0

where the reected and transmitted amplitudes, Et and Er, respectively, are givenas

Et = E0 +2πn0

k20

∞∑l=1

i−l√

2l + 1

8π

(x

∫ zd

z0

e−ik0z′ (<f1o1l> (z′) + i<f2e1l> (z′)) dz′

− y∫ zd

z0

e−ik0z′ (<f1e1l> (z′)− i<f2o1l> (z′)) dz′

)(2.1)

3

and

Er =2πn0

k20

∞∑l=1

il√

2l + 1

8π

(x

∫ zd

z0

eik0z′ (<f1o1l> (z′)− i<f2e1l> (z′)) dz′

− y∫ zd

z0

eik0z′ (<f1e1l> (z′) + i<f2o1l> (z′)) dz′

)(2.2)

in terms of the number density n0 and the transmission matrix Tnn′ of the scatterers.The unknown coecients<fn> (z) are the solution to a system of linear, one-

dimensional integral equations in z, viz.,

<fn> (z) = eik0z∑n′

Tnn′an′ + k0

∫ zd

z0

∑n′

Knn′(z − z′)<fn′> (z′) dz′, z ∈ [z0, zd]

(2.3)where the kernel Knn′(z) can be expressed in terms of spherical waves [11, 12]. Theexplicit form of the kernel Knn′ is (ρ = xx+ yy)

Knn′(z) =n0

k0

∑n′′

Tnn′′

∫∫R2

g(|ρ− zz|)Pn′′n′(k0(ρ− zz)) dx dy, |z| < zd − z0

where g(r) is the pair distribution function [3, 14, 23, 28], and Pnn′(k0d) is the trans-lation matrix for the outgoing spherical vector waves [2]. The most simple pairdistribution function is the hole correction (HC), g(r) = H(r − 2a), where H(x)is the Heaviside function and a is the radius of the spheres. The double integralin the denition of the kernel can be solved analytically for the hole correction interms of a series of spherical waves [12]. More complex distributions functions, e.g.,the hypernetted-chain equation, the Percus-Yevick approximation (P-YA), the self-consistent approximation, and Monte Carlo calculations are not employed in thispaper [3, 14, 23, 28].

The spherical scatterers are completely characterized by the transition matrixTnn′ , which for a spherical scatterer is diagonal in all its indices. The coecients anare the expansion coecients of the incident plane wave in spherical vector waves. Ifthe incident direction is along the positive z-direction, i.e., ki = z, these are (σ = eis the upper line, and σ = o is the lower line)

a1σml = −ilδm1

√2π(2l + 1)

(z ×

xy

)·E0

a2σml = −il+1δm1

√2π(2l + 1)

xy

·E0

ki = z

where the vector E0 denotes the polarization state in the x-y plane.Equation (2.4) denes the complex valued transmission and reection coecients

that maps the incident eld to the transmitted and reected elds, i.e.,

Et = tE0, Er = rE0 (2.4)

4

respectively.The transmittivity T and the reectivity R of the slab is given by

T =|Et|2

|E0|2, R =

|Er|2

|E0|2(2.5)

3 Numerical implementation

To compute the reection and the transmission coecients of the slab, we needto solve (2.3) for given geometrical and material data. The equation is a linearsystem of Fredholm integral equations of the second kind [5]. The unknown quantity,<fn> (z), is evaluated at equally spaced points, z = z1, z2, . . . , zp, in the interval[z0, zd], and the integral in (2.3) is evaluated by the use of Simpson's quadrature atthe points of discretization. The spatially discretized vector<fn> is denoted F .Remembering that n is a multi-index of n = τσml, the entries of the vector arewritten as

F = (<f1e01> (z1) · · ·<f1e01> (zp) · · ·<f2ommaxlmax> (z1) · · ·<f2ommaxlmax> (zp))t

where mmax ≤ lmax. The discretized system has an overall linear dimension ofN = 4(mmax + 1)lmaxp, and the underlying integral equation in (2.3) discretized as

F = P + kB · F ⇔ (I− kB) · F = P (3.1)

where I is the identity matrix, and the elements of matrix B, Bnn′ , are matricesgiven by (3.2) which are the Simpson weighted discretized kernel in (2.3) for n andn′ respectively. The integration variable is discretized at the same points as the lefthand side and ordered the same way as the discrete vector F .

Bnn′ =

w1Knn′(0) w2Knn′(z1 − z2) · · · wpKnn′(z1 − zp)

w1Knn′(z2 − z1) w2Knn′(0) · · · wpKnn′(z2 − zp)...

.... . .

...w1Knn′(zp − z1) w2Knn′(zp − z2) · · · wpKnn′(0)

(3.2)

where wi, i = 1, 2, . . . , p, are the ordinary Simpson weights for numerical integration.The discretization of the single scattering contribution denes the vector P in thesame format as F with vector elements given by

Pn = eik0zi∑n′

Tnn′an′ i = 1, 2, . . . , p,

We solve for the unknown vector F by the solution of a linear system of equationsin MATLAB. The transmitted and reected elds are then found by using (2.1)and (2.2) respectively.

5

3.1 Computations of the eective wave number keff

The transmission coecient th for a normally incident plane wave onto a non-magnetic, i.e., µr = 1, homogeneous slab of thickness d and wave number k is [10]

th(k) =(1− Γ2

h)ei(k−k0)d

1− Γ2he

2ikd(3.3)

where Γh = (k0 − k)/(k0 + k) and k0 = ω/c0 is the wave number in vacuum.To compute the eective wave number keff of the slab containing randomly dis-

tributed non-magnetic scatterers we nd the zeros of the function G(k) = t− th(k)where t is computed by using (2.4) and th is given by (3.3). That is, the eectivewave number, keff , satises G(keff) = 0.

To nd the complex roots ki of G(k) in a given domain Ω in the complex planewe employ the method described in Theorem A.1 in Appendix A. The area Ωis subdivided into n suciently small rectangular domains, Ωq, with boundariesγq. For each γq, the expression (A.1) is calculated using the midpoint rule. Ifki ∈ Ωq a test is made with a smaller contour to ensure that ki is the single rootinside γq. The process is repeated for every γq ∈ Ω, and then for every frequency.Among the available roots, the root closest to the solution at the previous frequencyis chosen. For convenience, Ω was restricted to the region 0 ≤ Im Ω ≤

√0.1k0,√

0.99k0 ≤ Re Ω ≤√

2k0 in our application.

4 Results

The transmittivity using equation (2.5) is compared with the transmittivity com-puted by using Bouguer-Beer law, which is given below in (4.1).

The radiative transfer equation (RTE) is frequently used to infer the coherentand diuse intensities of scattering by random scatterers in a slab geometry [6].The coherent contribution in RTE is Bouguer-Beer law, which species the drop inthe coherent intensity Ic(z), due to scattering and absorption in the material. Theexplicit form of the law is [6]

Ic(z) = Ic(z0)e−n0σext(z−z0) (4.1)

where σext is the extinction cross section of the spheres [8].Slabs with dierent thicknesses and dierent electrical size of the scatterers,

which are assumed to be spheres with εr = 1.332, corresponding to rain drops atoptical frequencies, are studied. We also investigate how the volume fraction ofscatters, f , aects the results. The volume fraction is kept small so that the holecorrection is assumed to be an accurate model.

We also compare the method with the results of [21, Chapter 6]. This approachuses the same underlying theory translations of spherical vector waves toobtain a relation for the expansion coecients of the internal elds of the scatterer.A half-space geometry is employed and the expansion coecients are assumed tohave the form ane

ikeffz. This leads to an innite set of equations, and the eective

6

wave number, keff , is found by a determinant relation. To generate results from [21,Chapter 6] a MATLAB code was downloaded from [20]. For comparison reasons,the Percus-Yevick pair distribution function was modied to the hole distributionfunction, by changing Eq. (6.1.60) in [21, Chapter 6]. Even if the method in [21]and the one presented in this paper are based on the same underlying principles,the analysis diverges, and the agreement therefore is an accurate verication ofthe method in [21], which uses the assumption mentioned above. In addition, ourmethod predicts the reection properties of the slab as shown in Figures 9 and 10.

4.1 Computation parameters

The maximum number of terms included in the expansion is determined by theindex l which is denoted lmax. In all computations lmax = 12. This parametermay be set lower for smaller k0a. The Wiscombe criterion states that lmax = 20for k0a = 10 [29]. However, for our application a convergence study shows thatlmax = 9 is sucient. The spatial discretization is varied depending on the slabthickness and on k0a. The number of spatial discretization points, zp, is kept suchthat, ∆z = zi+1 − zi, is smaller than λ/3, where λ is the vacuum wavelength. Theindex m is xed and takes the value m = 1, due to the excitation and the propertiesof the transition matrix of a spherical object. This means that in practise the system(2.3) has 4lmaxp number of unknowns to be solved for.

4.2 Transmittivity as function of k0a and volume fraction f

We compare the transmittivity dened in (2.5) with the transmittivity computedwith Bouguer-Beer law (B-B) (4.1) for a slab with thickness d/a = 100 consistingof non-magnetic dielectric spheres of radii a and εr = 1.332 as a function of k0a. InB-B, the transmittivity is T = Ic(d)/Ic(0). Two dierent volume fractions are used,f = 0.01 and f = 0.1, see Figures 2 and 4, respectively.

In Figure 2 we notice that a very good agreement is achieved between dataobtained using the Bouguer-Beer law and the presented method for f = 0.01. Wealso get a very good agreement for f = 0.1 when k0a < 1 (see Figure 4). For largerk0a the agreement in a relative measure is less good, but in absolute values thedierence is negligible since both methods predict extremely low transmittivity. Inboth cases it is noted that the transmittivity has a global minimum, in the studiedfrequency interval, for k0a ≈ 6. The increase in transmittivity at larger k0a is dueto the fact that the extinction cross section σext decreases, see Figure 3. This meansthat the spheres scatter less, and, hence, the coherent transmittivity increases.

In Figure 5 the transmittivity T is plotted as a function of the volume fractionof the scatterers f at k0a = 10. We note that very good agreement between the twomethods is achieved for small f and then the curves start to deviate. A possibleexplanation to the discrepancies between the Bouguer-Beer law and the proposedmethod at higher volume fractions, f , is that fareld criterion is assumed betweenthe scatterers in the Bouguer-Beer law. At lower concentrations this assumptions ismore valid, hence the Bouguer-Beer law show better agreement with our method.

7

0 2 4 6 8 100.01

0.1

1

B-B

k0a

T

Figure 2: The transmittivity T (coherent part) in log scale as a function of theelectrical size k0a for a slab of thickness d/a = 100 and constant volume fractionf = 0.01 consisting of dielectric spheres of radii a. The material parameters of thespheres are εr = 1.332 and µr = 1. The dashed line is the result obtained by theBouguer-Beer law (B-B).

2 4 6 8 10

0.5

1

1.5

2

k0a

σext

Figure 3: The extinction cross section σext as a function of the electrical size k0afor a single dielectric sphere of radius a. The material parameters of the sphere areεr = 1.332 and µr = 1.

8

0 2 4 6 8 1010−19

10−16

10−13

10−10

10−7

10−4

10−1

B-B

k0a

T

Figure 4: The transmittivity T (coherent part) in log scale as a function of theelectrical size k0a for a slab of thickness d/a = 100 and constant volume fractionf = 0.1 consisting of dielectric spheres of radii a. The material parameters of thespheres are εr = 1.332 and µr = 1. The dashed line is the result obtained by theBouguer-Beer law (B-B).

0 0.1 0.210−19

10−16

10−13

10−10

10−7

10−4

10−1

B-B

f

T

Figure 5: The transmittivity T (coherent part) in log scale as a function of thevolume fraction f for a slab of thickness d/a = 100 and constant electrical sizek0a = 10 consisting of dielectric spheres of radii a. The material parameters of thespheres are εr = 1.332 and µr = 1. The dashed line is the result obtained by theBouguer-Beer law (B-B).

9

−1 −0.5 0.5 1

−0.5

0.5

1

k0a = 0.1

k0a = 0.9

k0a = 5

k0a = 9.6

k0a = 0.1

k0a = 0.9

k0a = 5

k0a = 9.9

Re t

Im t

Figure 6: The components of the complex-valued transmission coecient, t(k0a),in the complex plane as a function of the electrical size k0a for a slab of thicknessd/a = 100 and constant volume fraction f = 0.01 (black curve) and f = 0.1(green curve) consisting of dielectric spheres of radii a. The material parametersare εr = 1.332 and µr = 1.

Moreover, no boundary eects are present in the Bouguer-Beer law. In Figure 6the real and imaginary part of transmission coecient is plotted for f = 0.01 andf = 0.1 in the complex t plane with k0a as a parameter along the curves.

4.3 Computations of the eective wave number keff

The eective wave number keff is calculated using the transmission coecient forthe coherent eld, as described in Section 3.1, for three dierent slab-thicknessesd = 100a, d = 50a, and d = 10a, respectively. In Figures 7 and 8, we compareour results with the results of [21]. The wave number keff has been normalized withthe wave number of vacuum, k0. We notice that keff for d = 100a, d = 50a, andd = 10a, computed with the present method, and the results given in [21], agreewell. We also note that the imaginary part of keff is small for all k0a, and that thereal part of keff < 1 for k0a > 6.6, and that this holds for the coherent eld only.The propagation properties for the incoherent eld is not considered in this study.

10

2 4 6 8 10

1

1.001

1.002

1.003

k0a

Re keff/k0

Figure 7: The real component of the scaled eective wave number, keff/k0, as afunction of the electrical size k0a for a slab of thickness d/a = 100, 50, 10 (red, blueand green circles and curves, respectively) and constant volume fraction f = 0.01consisting of dielectric spheres of radii a. The material parameters are εr = 1.332

and µr = 1. The crosses and dashed curve are the result by [21].

2 4 6 8 10

0.001

0.002

k0a

Im keff/k0

Figure 8: The imaginary component of the scaled eective wave number, keff/k0, asa function of the electrical size k0a for a slab of thickness d/a = 100, 50, 10 (red, blueand green circles and curves, respectively) and constant volume fraction f = 0.01consisting of dielectric spheres of radii a. The material parameters are εr = 1.332

and µr = 1. The crosses and dashed curve are the result by [21].

11

0.1 1 10

10−10

10−9

10−8

10−7

10−6

10−5

k0a

R

Figure 9: Reectivity R (coherent part) v.s. reectivity of a homogeneous slab(dashed) with R in log scale as a function of the frequency for a slab of thicknessd/a = 10 and constant volume fraction f = 0.01 consisting of dielectric spheres ofradii a. The material parameters of the spheres are εr = 1.332 and µr = 1. Thedashed line is the result obtained by reection by a homogeneous slab.

0.1 1 10

10−10

10−9

10−8

10−7

10−6

10−5

k0a

R

Figure 10: Reectivity R (coherent part) v.s. reectivity of a homogeneous(dashed) with R in log scale as a function of the frequency for a slab of thick-ness d/a = 50 and constant volume fraction f = 0.01 consisting of dielectric spheresof radii a. The material parameters of the spheres are εr = 1.332 and µr = 1. Thedashed line is the result obtained by reection by a homogeneous slab.

12

4.4 Reectivity as function of k0a

The reection coecient for a homogeneous slab of thickness d and wave number kis given by [10]

rh(k) = Γh1− e2ikd

1− Γ2he

2ikd(4.2)

where Γh is given in (3.3).Figures 9 and 10 show the reectivity, R, see (2.5), for a volume fraction f = 0.01,

with slab thickness d/a = 10 and d/a = 50, respectively. The result is comparedwith the reectivity, computed with (4.2), of a homogeneous slab with wave numberkeff obtained in Section 4.3. We obtain good agreement between the two ways ofcomputing reection data for small k0a. This means that keff is a solution to bothG(k) = 0, see Section 3.1 for denition, and r − rh(k) = 0 when k0a is small. Forthese parameter values, when the wavelength of the electromagnetic eld is largecompared with the size of the scatterers, classical homogenization methods hold(e.g., see [1, 4, 9, 27]). However, at higher frequencies the deviation between the twocurves is signicant, both in amplitude and periodicity. The deviation is larger forthe thick slab, Figure 10, than for the thin slab in Figure 9. This last remark impliesthat the interference eects due to the thickness of the slab, at least for the thinslab, are the same in the solid slab and the slab of random scatterers. Note thatthe local extrema are somewhat shifted for the thicker slab (Figure 10) which weconjecture is due to increased damping of the wave in this case.

We believe that the deviations between the curves in Figure 9 and 10 representa measure of the accuracy of the homogenization procedure, and it provides anupper frequency limit of the validity of the use of homogenization. One possibleexplanation of the discrepancy between the responses is due to increased incoherentcontents in the reected eld at higher frequencies.

5 Discussion and conclusions

We have presented numerical results for the method described in [11] to modelthe coherent reected and transmitted eld for a slab of nite thickness containingspherical randomly distributed scatterers of equal size and relative permittivity, εr.The wave number for the transmitted eld agrees well with the eective wave numberobtained by the method given in [21, Chapter 6].

We have observed that the reection from this slab is not consistent with the re-ection from a homogeneous slab given the above mentioned eective wave number,except for low frequencies when homogenization methods apply.

We have used the hole correction which is applicable for gases. In the future an-other type of hole correction could be implemented. Extensions to oblique incidenceis also planned.

13

Appendix A Zeros and poles of an analytic function

The determination of the location of zeros of an analytic function is vital in thecomputations of the eective wave number in this paper. The following theorem isthen useful:

Theorem A.1. Let Ω be an open domain in the complex z-plane, and let f(z) be

an analytic function in Ω with a simple zero at z0. Then

z0 =

∮γ

z dz

f(z)∮γ

dz

f(z)

where γ is any contour that lies inside Ω, and that encircles the zero z0.

We give the proof of this theorem.

Proof. Since the zero is simple the function f(z) is

f(z) = g(z)(z − z0), z ∈ Ω

where g(z) has no zeroes in Ω. The residue theorem then gives∮γ

z dz

f(z)= 2πiRes

z

f(z)

∣∣∣∣z=z0

= 2πiz0

g(z0)

Similarly, ∮γ

dz

f(z)= 2πi

1

g(z0)

and the theorem is proved.

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