Gustavsson, Magnus

Kristensson, Gerhard

Wellander, Niklas

English

Lund University Publications

LUND UNIVERSITYPO Box 117221 00 Lund+46 46-222 00 00The coherent electromagnetic field by a particulate media – numerical implementationin a planar geometryGustavsson, Magnus; Kristensson, Gerhard; Wellander, Niklas2021Link to publicationCitation for published version (APA):Gustavsson, M., Kristensson, G., & Wellander, N. (2021). The coherent electromagnetic field by a particulatemedia – numerical implementation in a planar geometry. Abstract from SIAM Conference on MathematicalAspects of Material Science.Total number of authors:3General rightsUnless other specific re-use rights are stated the following general rights apply:Copyright and moral rights for the publications made accessible in the public portal are retained by the authorsand/or other copyright owners and it is a condition of accessing publications that users recognise and abide by thelegal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private studyor research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portalRead more about Creative commons licenses: https://creativecommons.org/licenses/Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will removeaccess to the work immediately and investigate your claim.The coherent electromagnetic field by a particulate media – numerical implementation in a planar geometry Magnus Gustavsson1, Gerhard Kristensson2, and Niklas Wellander1 1Swedish Defence Research Agency, FOI, SE-581 11 Linköping, Sweden 2Dept. of Electrical and Information Technology, Lund University, SE-221 00 Lund, Sweden Scattering of electromagnetic waves by a particulate media is addressed. In general, the non-intersecting particles can be of arbitrary form, material and shape with number density n0 (number of scatterers per volume). The main aim of this paper is to calculate the coherent reflection and transmission characteristics for a collection of spheri-cal particles confined within two parallel planes. Typical applications of the results are found at a wide range of frequencies (radar up to optics), such as attenuation of electromagnetic propagation in rain, fog, and clouds etc. The integral representation constitutes the underlying framework of the solution of the deterministic problem, which then serves as the starting point for the solution of the stochastic problem. Conditional averaging and the employment of the Quasi-Crystalline Approximation lead to a system of integral equations in the unknown ex-pansion coefficients. The planar geometry and normal plane wave incidence imply a system of integral equations in the depth variable. The system of integral equations is solved by the Nyström’s method and Gauss-Legendre quadrature. Numerical examples are presented and the results are compared with the tenuous and low frequency approximations as well as the reflection and transmission of a homogenized slab. 

The coherent electromagnetic field by a particulate media – numerical implementation in a planar geometry

LUND UNIVERSITYPO Box 117221 00 Lund+46 46-222 00 00The coherent electromagnetic field by a particulate media - numerical implementationin a planar geometryGustavsson, Magnus; Kristensson, Gerhard; Wellander, Niklas2021Document Version:Publisher's PDF, also known as Version of recordLink to publicationCitation for published version (APA):Gustavsson, M., Kristensson, G., & Wellander, N. (2021). The coherent electromagnetic field by a particulatemedia - numerical implementation in a planar geometry. Paper presented atBremen Workshop on Light Scattering 2021, Bremen, Germany.Total number of authors:3General rightsUnless other specific re-use rights are stated the following general rights apply:Copyright and moral rights for the publications made accessible in the public portal are retained by the authorsand/or other copyright owners and it is a condition of accessing publications that users recognise and abide by thelegal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private studyor research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portalRead more about Creative commons licenses: https://creativecommons.org/licenses/Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will removeaccess to the work immediately and investigate your claim.The coherent electromagnetic eld by a particulatemedia  numerical implementation in a planargeometryMagnus Gustavsson(1), Gerhard Kristensson(2), and Niklas Wellander(1)(1) Swedish Defence Research Agency, FOI, SE-581 11 Linköping, Sweden.(2) Department of Electrical and Information Technology, Lund University,P.O. Box 118, SE-221 00 Lund, Sweden.1 IntroductionElectromagnetic scattering by randomly located particles is frequently encounteredin science in terrestrial and atmospheric research, biomedical and life sciences, as-trophysics, nanotechnology, just to mention a few topics. The theory is reviewedin [47]. The eective wave number approach is most commonly employed to solvethe problem, and the eective wave number is obtained by solving a determinantrelation. The new method presented in [1] solves the coherent transmitted and re-ected elds from a nite or an innite slab containing randomly located scatterersby the solution of a system of integral equations in the depth variable. In this paper,we present some numerical solutions with this method and also a comparison withthe eective wave number method.2 TheoryA plane wave impinges at normal incidence on the slab, z ∈ [0, d], containing spher-ical dielectric particles of radius a. The domain of possible locations of local origins,[z1, z2] = [a, d− a], is slightly smaller than the extent of the slab.The transmitted and reected coherent parts (ensemble average) of the totalelectric eld on either side of the slab areE±(r) =3f2(ka)3∑ni−l+τ−1An(±ẑ)k∫ z2z1e±ikz′fn(z′) dz′e±ikz,{z > dz < 0The summation over the multi-index n = {τ, σ,m, l} is over τ = 1, 2, σ = e, o,m = 0, 1, 2, . . . , l, and l = 1, 2, 3 . . ., and the vector-valued functions An(k̂i) arethe vector spherical harmonics, see [3] for more details. The volume fraction of thespheres is denoted f . The coecients fn(z) are the solution to a system of linear,one-dimensional integral equations in z, viz.,fn(z) = eikz∑n′Tnn′an′ + k∫ z2z1∑n′Knn′(z − z′)fn′(z′) dz′, z ∈ [z1, z2]The kernel entries, Knn′(z), can be computed analytically for the hole correction interms of a series of spherical waves [2]. The particles are completely characterized bythe transition matrix Tnn′ , which for a spherical particle is diagonal in its (pairwise)indices. The expansion coecients of the plane wave in terms of regular sphericalvector waves are denoted an, see [3].20 2 4 6 8 1010−1910−1610−1310−1010−710−410−1B-BkaT0.05 0.1 0.15 0.2 0.25 0.30.9980.9991kaTFigure 1: The transmissivity T (coherent part) as a function of the electrical sizeka for a slab thickness of d = 100a and constant volume fraction f = 0.1. Thedashed line is the result obtained by the Bouguer-Beer law (B-B) computed with aslab thickness of 98a. The insert shows the ne ripple that occurs at low frequencies.3 Numerical resultsIn Figure 1, we compare the transmissivity as a function of ka with the transmissivitycomputed with Bouguer-Beer law (B-B) for a slab with thickness 98a. The slabcontains non-magnetic dielectric spheres of radius a and εr = 1.332.There is a ne ripple in the transmissivity at low frequencies that is non-visibleon the scale of the gure and hidden in the line thickness. This is illustrated inthe insert in Figure 1. The eect diminishes at higher frequencies. The reason forthis ripple is interference eects between the front and trailing end discontinuitiesin particle densities at z = a and z = d− a. The period of the oscillation ∆(ka) is∆(ka) = 2πkRe keffa2Dwhere keff is the eective wave number computed from the transmission data. Theeective wave number is also compared with the existing technique of computationby the zeros of a determinant relation [5]. In Figure 2, the reection coecient in thecomplex plane is compared with the reection coecient for a homogeneous slab.Both the amplitude and the phase of the coherent reection coecient r(ka) andthe reection coecient of the homogenized slab agree perfectly at low frequencies.The homogenized slab has its left-hand sided located at z = a, and an additionalphase of 2ka is added to compensate for this oset. This is an additional numerical3−0.03−0.025−0.02−0.015−0.01−0.005−0.02−0.015−0.01−0.0050.0050.010.015ka = 0.1Re rIm rFigure 2: The black curve shows the complex-valued reection coecient, r(ka),in the complex plane as a function of the electrical size ka ∈ [0, 0.1] for a slab ofthickness d = 100a and volume fraction f = 0.1. The dashed curve shows thereection coecient for a homogenized slab with thickness 98a and left-hand sidelocation at z = a.verication that the correct location of the homogenized slab is [a, d − a] if theoriginal slab is located at [0, d].References[1] G. Kristensson. Coherent scattering by a collection of randomly located obstacles an alternative integral equation formulation. J. Quant. Spectrosc. Radiat.Transfer, 164, 97108, 2015.[2] G. Kristensson. Evaluation of some integrals relevant to multiple scattering byrandomly distributed obstacles. Journal of Mathematical Analysis and Applica-tions, 432(1), 324337, 2015.[3] G. Kristensson. Scattering of Electromagnetic Waves by Obstacles. Mario BoellaSeries on Electromagnetism in Information and Communication. SciTech Pub-lishing, Edison, NJ, USA, 2016.4[4] M. I. Mishchenko. Electromagnetic Scattering by Particles and Particle Groups.An Introduction. Cambridge University Press, New York, NY, 2014.[5] L. Tsang and J. A. Kong. Scattering of Electromagnetic Waves: Advanced Topics.John Wiley & Sons, New York, NY, 2001.[6] L. Tsang, J. A. Kong, and K.-H. Ding. Scattering of Electromagnetic Waves:Theories and Applications. John Wiley & Sons, New York, NY, 2000.[7] L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao. Scattering of ElectromagneticWaves: Numerical Simulations. John Wiley & Sons, New York, NY, 2001.

The coherent electromagnetic field by a particulate media - numerical implementation in a planar geometry

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The coherent electromagnetic field by a particulate media - numerical implementation in a planar geometry

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